(c) cce-upes(c) cce-upes mbcq-721d. quantitative techniques for management applications course...

UNIT 20: Case Study

Quantitative Techniques for Management Applications

MBCQ-721D(c)

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Course Design

Advisory Council

Chairman Dr Parag Diwan

Members Dr Kamal Bansal Dean

Dr Anirban Sengupta Dean

Dr Ashish Bhardwaj CIO

Dr S R Das VP – Academic Affairs

Dr Sanjay Mittal Professor – IIT Kanpur

Prof V K Nangia IIT Roorkee

SLM Development Team Wg Cdr P K Gupta Dr Joji Rao Dr Neeraj Anand Dr K K Pandey

Print Production

Mr Kapil Mehra Mr A N Sinha Manager – Material Sr Manager – Printing

Author

S Jaisankar

All rights reserved. No part of this work may be reproduced in any form, by mimeograph or any other means, without permission in writing from MPower Applied Learning Enterprise.

Course Code: MBCQ-721D

Course Name: Quantitative Techniques for Management Applications

Version: July 2013

© MPower Applied Learning Enterprise (c) C

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UNIT 20: Case Study

Contents

Block-I

Unit 1 Decision Making ............................................................................................................. 3

Unit 2 Functions....................................................................................................................... 15

Unit 3 Equations ...................................................................................................................... 23

Unit 4 Applications of Linear Equations and Functions in Business ................................... 35

Unit 5 Case Studies.................................................................................................................. 53

Block-II

Unit 6 Matrices......................................................................................................................... 57

Unit 7 Determinants ................................................................................................................ 83

Unit 8 Probability................................................................................................................... 105

Unit 9 Random Variables and Probability Distributions..................................................... 123

Unit 10 Case Studies................................................................................................................ 153

Block-III

Unit 11 Decision Theory .......................................................................................................... 159

Unit 12 Linear Programming .................................................................................................. 167

Unit 13 Transportation Models ............................................................................................... 185

Unit 14 Assignment Problem .................................................................................................. 199


Block-IV

Unit 16 Game Theory............................................................................................................... 213

Unit 17 Markovian Model........................................................................................................ 223

Unit 18 Data Collection ........................................................................................................... 229

Unit 19 Presentation of Data................................................................................................... 255


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Block-V

Unit 21 Sampling ..................................................................................................................... 265

Unit 22 Basic Tools of Data Analysis–I .................................................................................. 287

Unit 23 Basic Tools of Data Analysis–II ................................................................................. 301

Unit 24 Forecasting.................................................................................................................. 309


Glossary ............................................................................................................................................ 375

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UNIT 1: Decision Making

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Detailed Contents UNIT 1: DECISION MAKING

Introduction

Managerial Decision Making System

Managerial Decision Making Environment

Quantitative Models

UNIT 2: FUNCTIONS

Introduction

Functions

Variables

Types of Functions

UNIT 3: EQUATIONS

Introduction

Linear Equations

UNIT 4: APPLICATIONS OF LINEAR EQUATIONS AND FUNCTIONS IN BUSINESS

Introduction

Supply and Demand Functions

Irregular, Unequal and Discontinuous Functions

Quadratic Equations

Fitting a Quadratic Cost Curve

UNIT 5: CASE STUDIES

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Unit 1

Decision Making

Objectives After completion of this unit, the students will be aware of the following topics:

Quantitative decisions

Importance of variables

Importance of eight decision making tools

Introduction

A decision always involves choice among several alternatives. In the most basic sense a decision always involves the answer to the question "to do or not to do?" Not to do (inaction) determines that decision. To do (action) usually involves different options. The mathematical model identifies the optimal way, but for a variety of reasons, other satisfying options may be selected and acted upon.

There are industry-wide and market-wide decisions that have to be made. Often these decisions must transcend domestic considerations to incorporate international aspects.

Managerial Decision Making System Every decision making task results in an output which is the evidence of the decision taken. In industry, it is ultimately some kind of product, that is, a good service or on idea. The reasoning takes place in the decision making rectangle which is sometimes referred to as, quite appropriately, the black box. Here a transformation of the inputs takes place that results in the output. The transformation process has both physical and mental properties. On the input side a large number of variables may be listed. These variables can be classified in terms of the traditional factors of production, i.e., land, labour and capital as well as the more recently emerged complex variables related to systems, technology and entrepreneurship. Underlying this input-output system is a feedback loop identified as managerial control system. Its function is to optimize the transformation of inputs into the desired output. Seen in a nutshell, in industry, optimization means

Activity Define the term decision.

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the minimization of costs and the maximization of profits subject to legal, social and ideological constraints.

The computer has forced the decision maker to very carefully delineate and quantify the variables that makeup the building blocks of the decision task. What is needed and how much is needed for decision optimization have become the important questions. In addition, the proper time sequencing of the decision variables within the decision process had to be understood. And all answers had to be unequivocally quantified. It soon became apparent to every decision maker that quantified variables had different properties and specific quantitative control mechanisms had to be designed. Not only was the decision maker confronted with variable-inherent properties, the decision tasks themselves have such peculiar quantitative properties.

A variable, the building block of the decision task, may be seen as a small piece of a complex behaviour. Buying a house, manufacturing a product, spending money on a show are examples of variables. Each variable represents a distinct dimension of the decision making task. So the decision space is always multidimensional, and it is a major task for the decision maker to find out which variables make up that space. If an important variable is overlooked, obviously the decision will be less than optimal. Furthermore, the quantitative impact of the variable must be ascertained. And here the special variable-inherent properties come into play. The following examples may show the differences among the three types of variables.

Deterministic variables can be measured with certainty. Thus, equal measures have equal cumulative impact, or, to use a simple example, a+a = 2a.

Stochastic variables are characterized by uncertainty. Thus, a+a=2a+X, where X is a value that comes about because of the uncertainty that is associated with the variable.

Heuristic variables are those that exist in highly complex, unstructured, perhaps unknown decision making situations. The impact of each variable may be explained contingent upon the existence of a certain environment. For example, a + a = 3a but only if certain conditions hold. Actual industrial decision making situations in each case may involve the number of gallons of aviation fuel obtained by cracking a barrel of crude oil (deterministic), projected product sales given amount spent on

Activity What is Managerial Decision Making System?

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advertising the product (stochastic) and the construction of a platform in outer space (heuristic).

Check Your Progress

Fill in the blanks:

1. Reasoning is ……………….. by nature, which can be rational or irrational.

2. All types of human decision-making are essentially ……………… processes.

3. The two most common flaws in decision making are inertia and……………….. .

Managerial Decision Making Environment The reason for the existence of a managerial hierarchy, that is, lower, middle and top management, finds itself in different parameters in which an organization operates. There are industry-wide and market-wide decisions that have to be made. Often these decisions must transcend domestic considerations to incorporate international aspects. Such decisions—usually made by top management—occur in a broad-based, complex, ill-defined and non-repetitive problem situation. Middle management usually addresses itself to company-wide problems. It sees to it that the objectives and policies of the organization are properly implemented and that operations are conducted in such a way that optimization may occur.

You may note that while most of the quantitative decision making tools —indeed virtually all of the deterministic tools—were developed to optimize the decision making process, actual managerial practice has sometimes moved away from that objective. The previously mentioned legal or social constraints often at times do not permit optimization and satisfying has been substituted for it. Satisfying refers to the attainment of certain minimum objectives. For example, a company may have the economic and technological power to smother the competition within its industry but refrains from doing so because of MRTP considerations. Big size per se may be considered in violation of the law or in the international arena, may result in the imposition of quotas.

Lower management is responsible for the conduct of operations—the firing line so to speak is this—in production, marketing,

Activity What is the reason for the existence of a managerial hierarchy?

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finance or any of the staff functions like personnel or research. This decision environment is usually well-defined and repetitive. Obviously, with reference to a given decision making situation, the distinction between top, middle and lower management may become blurred. In other words, in any ongoing business there is always a certain overlapping of the managerial decision making parameters.

The study and analysis of the existence and interaction of these parameters is of great importance to the management systems designer or communication expert. From the quantitative managerial decision making point of view, their importance lies in recognizing their peculiar constraints and then to build the appropriate decision models and to select the best suited quantitative decision tools. A brief discussion of each environment in this light may enhance the understanding of the tools that are discussed later on. The company’s approach to the domestic or international market is filtered through industry-wide considerations. What does the market want, what does the competition already supply? Where is our field of attack? Do we have the know-how, do we have the resources? What is the impact of our actions upon the market, our own industry and other industries? These are some of the questions that have to be asked, defined and answered. The problems are unstructured and complex. Thus, often a heuristic decision making process can be utilized to good advantage. Forecasting is of major importance and hence stochastic decision making is widely employed in this uncertain decision environment. But even a deterministic tool—usually intended for decision making situations that assume certainty—input-output analysis, can be effectively used in this environment.

Middle management decisions are primarily company-wide in nature. As mentioned before, these decisions steer the organization through its life cycle.

Major features of a firm’s life are objectives, planning, operation and the ultimate dissolution. The objectives are general and specific in nature. Obviously, the top management establishes the objectives, but middle management functions as their guardian. Indeed, every decision at this level must provide feedback control for each of the other components. (c) C

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Planning refers to both policy execution as well as policy development. Scale of production, pricing of the product, product mix, in short the orderly and efficient arrangement of the input factors is to be decided at this point. Making these factors into a product is the job of operations. Some operations have been traditionally called line (financing, production, and marketing) and others staff (personnel, research, etc.); yet, in the quantitative decision systems of the modern firm, such differences are difficult to trace in the decision patterns, because the same decision, making tools are employed. Since the decision environment at this level is somewhat more structured than at the top level but still highly uncertain, stochastic decision tools are frequently employed. In those finance, production and marketing situations that can be well-defined, may be repetitive, deterministic decision tools are found.

It may appear somewhat odd that the decision environment includes attention being paid to the dissolution of the firm. The life cycle concept has been mentioned, and it will be encountered again as one of the major underlying conceptual aids in forecasting. It is well known that business organizations are born, live and die like natural organisms.

Therefore, decision making should always be cognizant of the possibility of dissolution. That moment comes when, to use the vernacular, good money is thrown after bad. While market forces and the application of quantitative analysis normally show the approaching occurrence of that moment—even if the management involved shuts its eyes to the facts or is ignorant about them—at this point the decision is made or superimposed to opt for a turnaround or dissolution. Public agencies unfortunately are rarely subject to such stress producing alternatives.

The lower management decision making environment represents a specialized, narrowly defined area within a company’s total decision or operational field. Supervisory personnel of all types are operating in this environment. The decision tasks are normally well defined and repetitive. While the element of uncertainty never leaves the decision environment, here uncertainty can often be programmed into general or sub-routine, and stochastic decisions taken as if they were deterministic in nature. A good example is the pricing system of clothing discounters. Merchandise is put on the floor at price A on day one. On, say, day ten the price is (c)

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automatically reduced to price B and so on until the article is either sold or given to charity after thirty days. This is known as programmed decision making. It should be noted that while the nature of the decision environment remains intact, the decision maker’s tasks have been greatly reduced. The complex variables and unstructured decision environment of the merchandising task have been placed first into a model and then into decision making sequence (algorithm). This is the general idea behind model building and the development of algorithms.

It is highly important that every decision maker has a firm understanding of the philosophy upon which quantitative decision making is based. Under no circumstances is it sufficient to just know how to perform a certain quantitative analysis and to obtain a solution to be able to make a decision.

To turn to the specific aspects of the quantitative decision making process, it is possible to recognize three distinct phases in every decision situation. Given a carefully defined problem, a conceptual model is generated first. This is followed by the selection of the appropriate quantitative model that may lead to a solution. Lastly, a specific algorithm is selected. Algorithms are the orderly delineated sequences of mathematical operations that lead to a solution given the quantitative model that is to be used. The algorithms generate the decision which is subsequently implemented by managerial action programs.

Problem Definition

Problem definition is a cultural artifact which is especially visible in a society’s economic and industrial decision making process.

Obviously, if such cultural determinants are operative in the first phase of managerial decision making, their effect can be noticed at various stages in the process irrespective of the quantitative, thus hopefully objective, methods that are used in the design of the models and algorithms as well as the decision itself.

A brief digression into problem identification may be in order at this point. For purposes of this book and for quantitative management decision methodology in general, it is presupposed that a problem has been identified.

In the private sectors of free enterprise economies, however, a manager’s ability to recognize problems and even to anticipate problems that may emerge at some future time is vital to the

Activity What is Problem Definition?

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survival of the firm. Those managers that make effective decision concerning a known problem are good administrators; those that in addition can recognize and anticipate problems are creative. It is known that creativity is partially inborn and partially acquired. Thus, the quantitative decision maker will not only try to master the methodology but also attempt to sharpen his or her problem identification skills—his or her creativity.

Check Your Progress

Fill in the blanks

1. Inertia is often due to a fear of…………….

2. Problem definition is a cultural artefact which is especially visible in a society's economic and industrial………………

The Design of Conceptual Models

The conceptual model represents the logic that underlies a decision. Based on this logic the quantitative model and specific algorithms are constructed. The logic may be a priori or empirical in nature, e.g., when shooting craps in a casino, a gambler has pre-established a conceptual model concerning the odds of the game. On a priori ground—using only his or her intellect—in determining the odds of every roll of the dice, the concept dictates that the win of a seven or eleven on the first roll has likelihoods of 6/36 and 2/36 , respectively. (There are 6 possible combinations of spots showing on 2 dice that yield a seven and 2 combinations that yield an eleven with 36 combinations for all spots from two through twelve.) Given this conceptual model, quantitative models and algorithms can be designed that facilitate the betting decision.

Now suppose that our gambler stumbles across a floating craps game in some dark alley. After observing the action on the pavement for a while, he notices that seven’s and eleven’s do not occur on the first roll with the likelihood dictated by his conceptual model. Rather there seems to be a preponderance of two’s, three’s or twelve’s—which he knows are losses. Crooked dice, he may very quietly think to himself. For crooked dice, an a priori logic which is based on the ideal situation in which every spot on a dice has an equal probability of occurring (1/6) and any spot on two dice as well (1/6 × 1/6 = 1/36) according to the multiplication theorem) is unsuitable. Rather, he will now ascertain by observation (by

Activity What does conceptual model represents?

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experiment) the empirical probabilities which are determined by the weights that have been cleverly or crudely (it is a dark alley) concealed in or on the dice. Once this empirical conceptual model has been generated, our gambler may continue the betting decision process in terms of the amount of the bets at each roll, etc. He may also redefine the problem and leave.

Conceptual models may take many forms. In every case the general design intent is to understand and to depict the reality that relates to the problem. Most conceptual models show a functional relationship in graphic or matrix format. All models that are used in this book are of this type. But it is also possible, indeed necessary in some decision cases, to build a physical model. In the natural and engineering sciences, it is the usual form. If the decision involves mass production of some item, the physical model is known as a prototype.

In the design of the conceptual model, it is important to observe that the decision maker clearly delineates the interrelationships that make up the reality—or the systems—in which the problem occurs. But in the model building process it is virtually impossible to include all variables that have a bearing on the decision. The model includes only the major variables (endogenous variables) as seen from the decision maker’s vantage point. There will be always decision-related variables that exist outside of the decision space (exogenous variables) because of their unrecognized status or conscious exclusion due to time, cost or limited impact considerations. Such variables should be kept mentally ready because over a set decision horizon they may indeed become sufficiently important to be included into the system.

Once the conceptual model has been designed and its logic expressed in terms of some systems configuration such as the graph or matrix or perhaps network or flow diagram, the quantitative models are simply superimposed by quantifying the logic. Once that has been accomplished a relatively minor task remains in the selection of the algorithms and the computerization of the process. Many a decision process has been commenced needlessly and most of time injuriously, to some extent, because of faulty problem definition or poor conceptual model-building. Then there is no optimal or even satisfying outcome. To put it simply, number crunching and possible error correction is relatively easy, even though the reader may not immediately share this view as he (c) C

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or she does just that in the units that follow. Only the difficult tasks, that is, sound insight into the problem and its careful definition as well as proper logic employed in the conceptual model building process, will yield sound decisions and outcomes. Here errors are very difficult to correct.

Quantitative Models

Once the conceptual model has been properly designed, the quantitative model and its algorithms should almost “flow” out of it. The transition is natural, smooth, and almost automatic. The quantitative model is selected from the many such models that have been designed by mathematicians. So while the decision maker will always build a conceptual model, the quantitative model is typically selected from an available pool of such decision making tools. The selection is made on the basis of the predominantly stochastic, deterministic or heuristic nature of the variables. There are available quantitative models for each kind as discussed in the following units, and the decision makers task is to select the appropriate one for a given decision situation. “Know thy tools” should be inscribed on every decision maker’s desk. As it is possible to build a wall with a spade when the trowel would be the more appropriate tool, decision makers may sometimes misuse quantitative tools.

The Decision

A decision always involves choice among several alternatives. In the most basic sense a decision always involves the answer to the question “to do or not to do?” Not to do (inaction) determines that decision. To do (action) usually involves different options. The mathematical model identifies the optimal way, but for a variety of reasons, other satisfying options may be selected and acted upon. These other options are firmly rooted in an organization’s objectives and planning activities. As shown in greater detail later, a decision maker always has control over setting the objective and planning which interfaces with policies, strategies and tactics. But one has no control over the reaction to the decision within the market environment. Here various states, collectively known as the states of nature, emanating from customers, suppliers, competitors, public agencies, etc., render the final judgment about the soundness of the decision. The decision is the end product of a sequence of mental activities as illustrated in the preceding pages. To make a decision does not necessarily mean that it gets carried out. In order to accomplish that, numerous managerial action

Activity What is transition?

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programs are necessary. They represent the physical extension to the decision making process. This book stops at the point when the decision is rendered. The action programs, the physical component, cannot be discussed because they must be specifically designed for each situation. A good decision maker, however, will try to place the seeds for proper implementation into the decision.

Check Your Progress

Fill in the blanks:

1. A decision always involves choice among several ____________ .

2. The decision is the ____________ of a sequence of mental activities.

Summary Every decision making task results in an output which is the evidence of the decision taken. A variable, the building block of the decision task, may be seen as a small piece of a complex behaviour. Buying a house, manufacturing a product, spending money on a show are examples of variables.

The reason for the existence of a managerial hierarchy, that is, lower, middle and top management, finds itself in different parameters in which an organization operates. There are industry-wide and market-wide decisions that have to be made.

The conceptual model represents the logic that underlies a decision. Based on this logic the quantitative model and specific algorithms are constructed. Once the conceptual model has been properly designed, the quantitative model and its algorithms should almost "flow" out of it. The transition is natural, smooth, and almost automatic.

The mathematical model identifies the optimal way, but for a variety of reasons, other satisfying options may be selected and acted upon. These other options are firmly rooted in an organization's objectives and planning activities.

Lesson End Activity “A decision always involves choice among several alternatives”. Comment. (c) C

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Keywords

Deterministic Variables: Deterministic variables can be measured with certainty.

Stochastic Variables: Stochastic variables are characterized by uncertainty.

Heuristic Variables: Heuristic variables are those that exist in highly complex, unstructured, perhaps unknown decision making situations.

Questions for Discussion

1. Every decision making task results in an output which is the evidence of the decision taken.

2. Define variable. What are the different types of variables?

3. What are the key decisions taken at different managerial levels?

4. Write a note on problem definition.

5. Discuss the process of designing conceptual and quantitative models.

Further Readings

Books

R S Bhardwaj, Mathematics for Economics and Business, Excel Books, New Delhi, 2005

D C Sanchethi and V K Kapoor, Business Mathematics

Sivayya and Sathya Rao, An Introduction to Business Mathematics

Web Readings

www.managementstudyguide.com

www.textbooksonline.tn.nic.in

www.mathbusiness.com

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UNIT 2: Functions

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Unit 2

Functions


Functions

Variables

Types of Functions

Introduction

When we say that quantity demanded of a commodity is a function of its price, we mean that the quantity demanded depends upon the price and, therefore, for a given price, we can determine the quantity demanded. Here quantity demanded is a dependent variable while price is an independent variable. The behaviour of the dependent variable with respect to change in independent variable(s) is different in different situations. This fact gives rise to several types of functions. A function f from a set X (domain) to a set Y, a subset of X Y, is a rule that associates a unique element of set Y (target) to each element in X.

In this unit, we will study the functions and different types of functions.

Functions When we are talking about functions, we are not talking about marriage ceremonies or birthday parties but are talking about certain types of quantitative relationships between different variables mathematically. For example, sales revenue is a function of items sold and their price. We all know that. If we express it in function form it would look like this:

Sales Revenue = f (No. of items sold) (Function Form)

Sales Revenue = f (Price per item) (Function Form) (c) C

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Function form only tells us that there is a relationship between the variables. Here sales revenue, items sold and price per item are variables.

Check Your Progress

Fill in the blanks:

1. Function form only tells us that there is a ___________ between the variables.

2. Sales ___________ is a function of items sold and their price.

Variables

Variables are the terms used for mathematical quantities that can assume any values within a given set. The set of values of the variable is known as the domain of the variable, which could be limited (as is the case with water temperature which can vary between 0oC to 100oC) or can be unlimited (as distance).

Variables can be independent or dependent. Independent variables are those variables whose values are not governed by the values of another variable. In the case above, number of items sold and price per item are independent variables. Variables whose values are dependent on the values taken by another variable are called the dependent variables, e.g. sales revenue above.

In mathematics we say that whenever any variable Y is dependent on another variable X for its values then variable Y is a function of variable X. This means that whenever the value of variable X would change then there would be a corresponding change in the value of variable Y. Mathematically this is denoted as:

Y = f (X)

Now this relation between X and Y can take up any form, e.g., let us say Y is double of X. This means when X takes the value 2 than Y takes the value 2 × 2 = 4. So in the above function X is called the independent variable and Y is called the dependent variable.

Note that the function does not specify the exact relationship between X and Y, it only tells you that a relationship exists. The exact relationship is defined by equation, something that we will

Activity What are functions?

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UNIT 2: Functions

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see later. As stated above, the set of values of independent variable is called the domain. The set of values of dependent variable is called the range.

In the problems of business applications, generally the range and domain will be real numbers.

How do we apply these functions and equations in the business? The sales revenue example gives us a clue. Most of the activities in the business are dependent upon some other events happening. For example, if we want to know how the share prices move, we have to understand that the stock prices are dependent upon the economic situation, industry conditions and company performance.

So the share price would be a function of all these factors and the function would look like the following:

Share Price = f (economy, industry situation, company performance)

Therefore, we can use functions to define any relationships that exist between different processes and which can be measured mathematically. These models can be applied in making decisions when the business situations are complex. An important point to remember here is that, the functions do not imply a simple direct relationship.

The applications of these models are manifold and as diverse as the business situations we are in. The biggest benefit of using functions and equations for defining the relationships is that we understand how exactly the processes work and what are the constraints we are dealing with. For example, when we define the production as a function of number of machines and time and we know that we can produce a maximum of 500 units of a product per hour per machine. So this becomes our constraint and helps us in planning and scheduling. When we know that we need 5000 units of that product in 2 hours, we can either use five machines (so that each machine gives us 1000 pieces in 2 hours) or we get the work done from outside. Of course, this is a very simple situation and we will learn much more complex methods of mathematical treatments of solving complex situations.

Writing the relationship in function and equation form above, makes it easy for us to understand how the values of the dependent variables change because of a change in any of the independent variables on which it is based.

Activity What is share price?

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Most of the quantitative methods we will be dealing with in this book, would make use of mathematical models, some of which we will develop as we go along. We will only take the basic mathematical concepts as granted. Rest of the concepts would be dealt with as if we are talking to a novice.

Check Your Progress

Fill in the blanks

1. We can use functions to define any __________ that exist between different processes and which can be measured mathematically.

2. Variables can be __________ or_____________

Types of Functions

We said earlier that the mathematical models of the functions can take up different forms but we did not discuss the forms there. Now let us understand the different forms that these mathematical functions can take up.

1. Constant Function: Let A denote a fixed number. Consider that the function X has this value A for the value of X. So this constant function can be denoted by:

f (X) = A for all X

or more briefly by f (X) = A and we call it the constant function A

Therefore, there can be many constant functions.

2. Identity Function: The function that associates to each number X. The same number X is called the identity function. This is denoted by:

Y = f (X) = X for all X

3. Linear Function: Consider the expression f(Y) Y = 2X+ 10. For each number A, this associates the number 2A +10 to Y, got by substituting A for X. If X = 2 then the value of Y is 4+10 = 14. This function tells us about the relationship between the two variables X and Y.

4. Quadratic/Polynomial Function: Consider the expression f(Y)X2 + 2X + 10. For each number A, this associates the

Activity Identify different types of functions.

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UNIT 2: Functions

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number A2 + 2A + 10 to Y. This is a polynomial function and is used for solving complex situations.

5. Exponential Function: The function that associates the number ex to each real number x is called the exponential function. The properties of this function are given below:

ex+y = ex × ey

e-x = 1/ex

elog x = x

Here e= 1+12 +

12 + ..........

and ex = 1 +x1 +

x22

+ x33

......

ex is always positive.

6. Logarithmic Function: Log X is that number Y such that ey = x. Log x is defined only when x is positive. The function that associates log x to x is called the logarithmic function. Its domain is the set of positive real numbers. Log x=y implies loge x=y and real as ‘log x’ on base ‘e’ is equal to ‘y’ which implies ey = x but in frequent reading and writing we tend to take the base as understood. But if we are using two different bases then it must be mentioned. The two popular bases are ‘e’ and ‘10’. The properties of log function are given below:

Log 1 = 0

Log (xy)= Log x + Log y

7. Modulus Function: For each real number X let |X| denote the absolute value of X. It is also known as mod X.

|X|= X if X ≥ 0, and

|X|= -X if X < 0

For example |4| = 4 and |-4| = –(-4) = 4

These are only some of the examples of the different type of functions that can be there. These functions are translated into equations for use so that exact relationships are defined.

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Check Your Progress

Fill in the blanks:

1. The function that associates the number ex to each real number x is called the ____________ .

2. The function that associates log x to x is called the ____________ .

Summary

A function f from a set X (domain) to a set Y, a subset of X × Y, is a rule that associates a unique element of set Y (target) to each element in X. The unique element y in Y corresponding to an element x in X is denoted as f(x) and is called the image of x. In mathematics we say that whenever any variable Y is dependent on another variable X for its values then variable Y is a function of variable X. This means that whenever the value of variable X would change then there would be a corresponding change in the value of variable Y.

Lesson End Activity

When we say that quantity demanded of a commodity is a function of its price, what does it mean?

Keywords

Function: A function f from a set X (domain) to a set Y, a subset of X × Y, is a rule that associates a unique element of set Y (target) to each element in X.

Exponential Function: The function that associates the number ex to each real value of x is called exponential function.

Logarithmic Function: The function that associates log x to x is called the logarithmic function.


1. Define functions. What are the different types of functions?

2. Write a note on supply and demand functions. (c) C

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UNIT 2: Functions

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Further Readings

Books

R S Bhardwaj, Mathematics for Economics and Business, Excel Books, New Delhi, 2005 D C Sanchethi and V K Kapoor, Business Mathematics


Web Readings



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UNIT 3: Equations

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Unit 3

Equations


Equations

Linear equations

Graphical solution

Introduction

Theory of equations is frequently used in solving the problems of business. A statement of equality containing one or more variable is known as an equations. Two or more equations are said to be equivalent if they have the same solution.

While the functions tell us that a relationship exists, equations give us the exact relationship between the variables. These equations take many forms and have one or more variables. For example, we can say that Sales Revenue y = Number of items sold NX price per item P. This is an equation with three variables Y, N, P and defines an exact relationship amongst them. Here sales revenue Y is a dependent variable and other two are independent variables.

Here, we will discuss only linear and quadratic forms of equations.

Linear Equations

Linear Equation may be defined as an equation where the power of the variable(s) is one, and no cross or product terms are present.

The general expressions of these linear equations look like the following:

AX + B = 0

Here X = independent variable, A and B are numeric coefficients

This definition is a working definition.

Note that it is an accepted convention in mathematics that letters from the beginning of the alphabet are used to typify

Activity What does equation convey to us?

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known quantities and letters from the end of the alphabet are used to represent unknown quantities. In a linear equation, A and B are real numbers which can be either positive or negative and may involve fractions or decimals; it is also possible that B can be zero but A cannot be zero for then it is not an equation and B also has to be zero.

Example 3.1

A dealer sells a table and seven chairs for ` 6,000. The price of the table is known to be ` 2,500, what is the price of one chair?

Solution:

If X represents the price of each chair in units of ` 1 then we can say that 7 chairs × (Cost of each chair X) + Cost of table (` 2,500) = ` 6,000 ` 7´X + ` 2,500 = ` 6,000, or simply 7X + 2,500 = 6,000 7X - 6,000 + 2,500 = 0 7X - 3,500 = 0 (i.e., the form AX + B = 0) 7X = 3,500 X = 3,500 / 7 = 500

This means that each chair costs ` 500.

The number of variables in a linear equation can be one or more than one. For example, if

Y = AX + B

This equation can be rewritten as:

Y - AX - B = 0

That is the same form as we discussed earlier but it is an equation with two variables.

The purpose of this form of the linear equation, with two variables, is not to enable the problems to be solved but to state the relationship between Y and X. The value of X can vary from time to time, for some hypothetical future value of X may be under consideration; in each case, the corresponding value of Y is determined from the equation. (c) C

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Extending the example 3.1, let Y represent the value of the table, the equation can be rewritten as:

7X + Y = 6,000

If X takes up the values 200, 300, 400 and 500 we can easily see that Y will take the values 4,600, 3,900, 3,200, 2,500 respectively, which are calculated by substituting the value of X in the above equation.

Let us now plot this equation on the graph. On the graph the values of chair (X) are represented by rupees along the horizontal line and value of table (Y) along the vertical line. Any convenient scales may be chosen to represent the two variables; there is no reason why the same distance as used to represent an increase of ` 100 on horizontal axis cannot be used to represent ` 1,000 on the vertical axis.

Figure 3.1: Graph of 7x + y = 6,000

The point to be noted here is that both X and Y are dependent on each other. If the value of X changes the value of Y also changes and vice versa. This is because the net sum is a constant.

These kinds of equations are very easy to draw on a graph. You substitute zero for one variable to get the corresponding value of the other variable. In this example when we substitute zero for X we get Y = 6000 and when we substitute zero for Y we get X = 6000 / 7 = 857.14. When we want to plot this line on the graph we mark 6,000 on the vertical axis as this represents X = 0.

Similarly, we mark 857.14 on the horizontal axis as this represents Y = 0. Connecting these two points by a straight line gives us the line which can be used to find the value of one chair corresponding to value of one table, satisfying the above equation. (c)

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The word ‘linear’ is used to represent a function, which can be represented by a straight line (and not any function which can be represented by a line). Another point that must be remembered is that although we are using letter Y to represent the dependent variable, other letters can also be employed for denoting variables. Like S can represent Sales, D can represent Distance and so on.

Once we have the equation plotted on a graph we can very easily find out the value of one variable from the given value of another variable. In the example 3.1, we were given the value of Y, i.e., the value of one table as ` 2,500. So we draw a straight line from

Figure 3.2: Graph of 7x + y = 6,000

the vertical axis parallel to the horizontal axis at the value ` 2,500. The point where this line meets the equation line is used to draw a line parallel to the vertical line. Where this line meets the horizontal axis gives us the value of one chair. So use of graphs makes it very easy for us to solve these linear equations.

What happens when we are not given the value of both the variables? We cannot solve the equation if only a single equation is given but only give a range of values, which these two variables can take, all of which satisfy the equation in hand. But if we are given two simultaneous equations then it makes it easy for us to calculate the exact values of these two variables, which satisfy both the equations simultaneously.

In general it is usually possible to solve a set of equations if the number of variables is equal to number of equations, except in a case where the equations overlap.

Example 3.2

A manufacturer of printed fabrics has three machines, that prepare raw fabric and five machines that print on it. Two types of printed fabrics are produced; type A requires 3 minutes per meter (c) C

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UNIT 3: Equations

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to prepare and 6 minutes per meter to print, while type B requires 11 and 17 minutes per meter respectively. How much of each type of fabric should be produced per hour in order to keep all the machines fully occupied?

Solution:

The quantities to be produced per hour can be represented by X meters of type A and Y meters of type B. Then the situation above can be summarized in two simultaneous linear equations, one equation for each machine

3 X + 11 Y = 180 (1)

6 X + 17 Y = 300 (2)

The right-hand sides of these equations are obtained from the fact that there are 180 machine-minutes available per hour for preparing fabric (60 minutes x 3 machines) and 300 machine-minutes for printing (60 minutes × 5 machines).

There are three ways of solving any pair of simultaneous linear equations.

Elimination Method:

It will be observed that 6X is exactly twice 3X and so the first equation can be doubled to give:

6 X + 22 Y = 360

This is then subtracted from equation (2) to eliminate the terms involving X:

- 5 Y = -60

Y = 12

Substituting this value of Y in equation (1):

3 X + 132 = 180

X = 16

Substitution Method:

The second method of solution is by substitution. Equation (1) is rearranged so that one of the unknowns is expressed in terms of the other:

3 X = 180 - 11 Y

X =

180 - 11Y

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This formula for X is then substituted in equation (2):

6( 180 - 11Y

3 ) + 17 Y = 300

360 – 22Y + 17Y = 300 360 - 5 Y = 300 Y = 12

The value of X is then found using equation (1) and substituting them in equation (2) can check both values.

Cross Multiplication Method:

1 1 1 0a x b y c+ + = ...(1)

2 2 2 0a x b y c+ + = ....(2)

1 1 1 1 1

2 2 2 2 2

1x ya b c a ba b c a b

1 2 2 1 2 1 1 2 1 2 2 1

1yxb c b c a c a c a b a b

= =− − −

1 2 2 1 1 2 2 1

1xb c b c a b a b

∴ =− −

1 2 2 1

1 2 2 1

b c b cxa b a b

−∴ =

− which is the same as

2 1 1 2

2 1 1 2

−=

−b c b cxa b a b

and 2 1 1 2 1 2 2 1

1ya c a c a b a b

=− −

2 1 1 2

1 2 2 1

a c a cya b a b

−∴ =

−which is the same as

1 2 2 1

2 1 1 2

−=

−a c a cxa b a b

Note: While solving the system of linear equations, anyone of these methods can be used.

Example 3.3 + = −2 1x y

4x y− = Solution:

(A) By substitution method: + = −2 1x y ...(1)

x - y = 4 ...(2) (c) C

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(1) Substituting this in (2), we get ⇒ = − −2 1.y x

+ + = ⇒ =2 1 4 3 3x x x

Substituting this value of x in (1), we get

∴ = =3 i.e., 13

x x

2(1) 1y+ = − i.e. = −3y

(B) By elimination method:

+ = −2 1x y ...(1)

− = 4x y ...(2)

Adding (1) and (2), we get

=3 3x

∴ = =3 13

x

i.e. = 1x

Multiply (2) by 2

(1) ⇒ + = −2 1x y

× ⇒ − =2 (2) 2 2 8x y

× ⇒ − =2 (2) 2 2 8x y

Subtracting, we get

∴ = −3y

(C) By cross multiplication method: The equations are:

+ + =2 1 0x y

− − =4 0x y

− − −

12 1 1 2 11 1 4 1 1

x y

= =− + + − −

14 1 1 8 2 1

yx (c) C

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i.e. = =− −

13 9 3

yx

i.e. −∴ = =

− −3 9,3 3

x y

i.e. = = −1 , 3x y

The stepwise general procedure for solving these linear equations is given below for your reference:

Box 3.1: Stepwise Procedure for Solving 2×2 simultaneous Equations

1. Eliminate one of the variables using any or both of the properties specified below-

(a) Any linear equation can be multiplied or divided on both sides by any number without altering its truth or meaning.

(b) Any two linear equations can be added or subtracted (one from the other) to give a third, equally valid, equation.

2. Solve the resulting simple equation (to yield the value of the other variable).

3. Substitute this value back into one of the original equations, say equation 1 ( to yield the value of the first variable).

4. Check the solutions (by substituting both values into original equation 2).

Of course, graphical method can also be used to solve the 2x2 simultaneous equations. The first step is to draw the two lines represented by the two given equations on the same graph and the second step is to identify the X and Y values at the intersection of the lines. These X and Y values are the required solution for the pair of simultaneous equations.

Check Your Progress

Fill in the blanks:

1. Linear Equation may be defined as an equation where the power of the variable(s) is ________, and no cross or product terms are present.

2. The word ‘linear’ is used to represent a function, which can be represented by a ___________. (c)

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UNIT 3: Equations

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Graphical Solution

The situation above was summarized in two simultaneous linear equations:

3 X + 11 Y = 180 (1)

6 X + 17 Y = 300 (2)

Plotting the two equations simultaneously on the graph we find that they intersect at values Y=12 and X=16. This becomes the solution to the problem.

Similarly, 3 × 3 simultaneous equations can also be solved using an extension of the same technique that we have used above. There are other methods to solve these simultaneous equations which we would discuss in subsequent discussion on matrices and determinants and linear programming.

Figure 3.3: Graph of 3X +11Y=180 and 6X + 17Y = 300

The basic method for solving these 3 × 3 equations mathematically is given below:

Box 3.2: Procedure for Solving 3 × 3 Simultaneous Equations

1. Using any two of the given equations, eliminate one of the variables (using the equation-manipulating techniques previously described) to obtain an equation in two variables.

2. Using another pair of equations, eliminate the same variable as in (1), which will give a second equation in two variables.

3. Solve this 2 × 2 system of equations in the normal way.

4. Substitute into one of the three original equations to find the value of the third variable.

5. Check the solutions by substituting the values of these variables in the three equations. (c)

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Example 3.4

A furniture manufacturer sends Company A a bill for ` 10, 700 to cover 3 tables, 4 chairs and 3 stools. Company B is charged ` 14, 800 for 2 tables, 5 chairs and 7 stools. Company C is charged ` 15, 100 for 5 tables, 9 chairs and 2 stools. What are the respective prices for each of these items?

Solution

Representing the prices of one table, one chair and one stool by ` x, ` y and ` z respectively, the problem gives rise to three simultaneous linear equations:

3 x + 4y + 3z = 107 (1)

2x + 5y + 7z = 148 (2)

5x + 9y + 2z = 151 (3) These equations are still called ‘linear’ even though each could only be represented by a plane in a three-dimensional model and not by a straight line on a two-dimensional graph. The first step in their solution would be to multiply the first equation by 2 and the second equation by 3 in order to eliminate x and then subtracting first equation from the second one:

6 x + 8y + 6 = 214

6 x + 15y + 21z = 444

7y + 15z = 230 The second and third equations are then multiplied by 5 and 2 respectively in order to obtain a second equation in which x has been eliminated. The two equations involving only y and z are then solved as in example 3.2, to give y = 5, z = 13. Substituting these values in the first of the original equations gives x = 16. Substituting them in the other two original equations can check all three values.

Summary

While the functions tell us that a relationship exists, equations give us the exact relationship between the variables. A linear expression equated to zero is called a linear equation. Thus the general form of a linear equation is ‘ax + by +c = 0’, where a, b and c are real constants. An equation that can be written as ‘ax2 + bx + C = 0’, is called a quadratic equation. An equation of degree n has (c) C

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exactly n roots. Hence, a linear equation has exactly one root and a quadratic equation has exactly two roots.

Lesson End Activity

Write the Procedure for solving 3 x 3 simultaneous equations taking suitable example of the same.

Keywords

Equation: A statement that two expressions (connected by the sign=) are equal.

Linear Equation: A Linear expression equated to zero is called a linear equation.


1. Find two numbers whose difference is 8 and product is 20.

2. Find two consecutive numbers such that 5 times the smaller number is equal to 5 more than three times the greater number.

3. Find two numbers whose sum is 32 and their product is 252.

4. 2 years ago, elder brother's age was 3 times the square of his younger brother's age. After 3 years, elder brother's age will be 4 times his younger brother's age. Find their present ages.

5. Find two consecutive integers such that 10 times the smaller number is 5 times the bigger number.

6. When the solutions of quadratic equation are termed as:

(a) rational and equal,

(b) real and distinct and

(c) imaginary and distinct.

Further Readings

Books

R S Bhardwaj, Mathematics for Economics and Business, Excel Books, New Delhi, 2005 (c)

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Web Readings

www.textbooksonline.tn.nic.in www.mathbusiness.com

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UNIT 4: Applications of Linear Equations and Functions in Business

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Unit 4

Applications of Linear Equations and Functions in Business


Applications examples

Supply and Demand Functions

Quadratic equations

Introduction

In this unit, we have focused on various application problems related with the use of functions and equation. We have also focussed on supply and demand functions. We have tried to explain the concept by providing number of examples.

Example 4.1

The total production costs of a packaging machinery manufacturer are found to be an average of ` 60,000 per day. The cost accountant finds that the fixed costs are ` 32,000 per day and the direct costs average ` 7000 per machine. Calculate the average number of machines produced per day?

Solution:

This employs the accountants’ terms ‘fixed costs’ and ‘direct costs’ and uses the accountants model:

Total costs = fixed costs + (direct costs × quantity produced), i.e., T = F + Dx

Let x represent the no. of machines sold, the above model would look like:

` 60,000 = ` 32, 000 + (` 7,000 × x) (1)

Dividing the equation by ` 1,000 it reduces to

60 = 32 + 7x (2) (c) C

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In the above example, the model is very useful though approximate, since the direct costs per machine will probably vary quite widely. These types of models are used a great deal and are regarded as absolute truth by top management. But these models have their limitations as we will see.

Now if all the machines are sold at the same price, then the revenue is a linear function of the quantity produced. Putting R for revenue and p for price the function becomes

R = px

In the case of machine manufacturer, let us assume that the selling price is ` 18,000. By putting p = 18 (again assuming ` 1000 as the unit) a graph (Figure 4.1) can be drawn of the revenue function. However, it is much more informative to draw the line representing the cost function and revenue function on the same graph, as shown in the graph.

Extending the lines beyond their range in which their practical usefulness is proved is called extrapolation; it is a bad practice to extrapolate too far. The unreliable parts of the lines on the graph are shown by broken lines and the meaningless parts by dotted lines.

Figure 4.1: Machine Manufacturers' Cost and Revenue Functions (c) C

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The difference between revenue and total production costs can be described as gross profit G:

G=R – T

=px – (F+Dx) = 18x – (32 + 7x)

= 11x – 32

The break even point is when profit = 0, that is your revenue is equal to your costs. Putting this in the above equation we get:

11x – 32 = 0

11x = 32

x =

32

11= 2.91(approx. )

So your average production should be 2.91 machines for you to cover all your costs but make no profits.

This breakeven level can also be found from the graph where your revenue and costs curves cross each other. Alternatively, you can plot the equation (11x-32=0) and find the value of x where the line meets the x-axis as at that point the value of the function would be zero.

There are two possible ways in which you could have obtained the information related to fixed and direct costs as the cost accountant found. Either you take all the accounting records and classify each cost into the two headings, a tedious and time consuming process which is prone to error because of limited accounting knowledge and problems of classification.

Or a quicker and better method would be to record the actual total costs at two different levels of production and then find linear cost function which ‘fits’ these actual costs. For instance, the records might show that the average total cost per day was ` 49,500 when production averaged 2.5 machines per day and it rose to ` 63,500 when production rose to an average of 4.5 machines per day.

All that is necessary is to insert these two values of T and the corresponding values of x into the linear cost function, defined above.

T = F + Dx

This gives two equations, involving two unknowns F and D (c) C

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49.5 = F + 2.5D

63.5 = F + 4.5D

Solving these equations using the techniques already described, we get D = 7 and F = 32.

Substituting these values in the above function we get T = 32 + 7x, i.e., the same equation the cost accountant had with all his information.

Check Your Progress

Fill in the blanks:

1. If all the machines are sold at the same price, then the revenue is a ___________ function of the quantity produced.

2. Extending the linear beyond their range in which their practical usefulness is proved is called ___________.

Supply and Demand Functions Supply and Demand Functions are an important field of study for the economists. The amount of a particular product which a firm is willing to supply at a specified price will depend on the firms’ cost function and also on its marketing policy. The firm may be concerned with maximizing profit, increase market share, or just to keep the factory going in times of economic slow down. Once the firm decides what its policy is, the amount of products, which can be supplied to the market, is clearly a function of the price at which these products can be sold in the market. This forms the supply function of the firm. If the quantities of products that can be supplied by all the firms in this industry are totalled up for each price level of the product, this gives us the total supply function for the market as a whole.

As an example, let us assume that total supply of a particular type of phones in the market is 29,000 pieces per month when the price is ` 500 per piece. The same manufacturers are prepared to supply a total of 52,000 pieces per month if the price is raised to ` 600 per piece. A further rise in the price per piece would justify working overtime in the factory and also bring in foreign suppliers who were earlier not interested in selling at low prices in the market. It is found that a total of 75,000 pieces per month can be supplied when the price is ` 700 per piece.

Activity What are demand and supply functions?

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These three individual points can be plotted on a graph (Figure 4.2) letting ` P represent the price per phone (in ` hundred) and X the total quantity (in thousands of pieces per month) which would be supplied at that price. Although, here P is the independent

Figure 4.2: Demand & Supply Curves

variable and X is the dependent variable, it is customary for economist to plot prices on the vertical axis and quantities on the horizontal axis and this practice would be followed here. In the simplified example being considered, three points are found to lie on a straight line and so it can be assumed that the supply function for the market is approximately linear. The function is then found to be:

X = 23P – 86

For example, by substituting ` 500 as price (P = 5) we get X = 29 (i.e., 29,000 phones). This line represents the quantities which will be produced at different prices provided all the quantity produced can be sold. But to find out what can be sold in the market we need look at the demand function of the market. The demand function would indicate the total quantity that will be purchased at a particular price and therefore, represents the total individual demand functions of all the individual buyers.

Normally, large quantities would be bought when the price is lower and as the price goes up the quantities purchased come down. In this particular case it was found that only 24,000 telephone pieces can be sold at ` 700 per piece but that the sales would increase to 35,000 and 46,000 pieces per month at the prices of ` 600 and ` 500 respectively. These three points can be plotted on the same graph as supply curve so as to get the demand curve. The demand function is then found to be:

X = 101 – 11P (c) C

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It would be wrong to assume that we can extend these lines on either side for supply and demand functions. It would be absurd to assume that the demand is 2,000 pieces when the price is ` 900 and equally wrong to assume that demand is approximately 90,000 pieces when price is ` 100.

The reason for plotting supply and demand of the same graph is to found out the point of market equilibrium, which is the point of intersection of these two lines. It can also be find out using simple equation solving techniques mentioned earlier, by finding the value of P which makes the value of X same for both demand and supply functions. The prices and quantity at the point of market equilibrium are known as equilibrium price and the equilibrium quantity. Under the condition of free competition, the equilibrium quantity will be the quantity actually produced and the equilibrium price would be the price in the market.

Fitting demand and supply curves is much more tedious than solving other business situations. It is much more difficult to access how supply will respond to change in price than to access how the total production cost within a firm will vary with the quantities produced. Demand is also complicated because of the presence of substitute products in the market. These difficulties explain why mathematical economists need a lot of training and experience and why sometimes forecasted situations vastly differ from the actual situations in the market.

Check Your Progress

Fill in the blanks:

1. Fitting demand and supply curves is much ___________ than solving other business situations.

2. The reason for plotting supply and demand of the same graph is to found out the point of _______________.

Irregular, Unequal and Discontinuous Functions

Sometimes it is assumed that ‘y is a function of x’ means that there is a single formula connecting y with x. While it is easy to discuss functions which are described by a single formula, the only correct interpretation is that y is a function of x if the value of x determines the value of y, irrespective of the fact, whether it is in

Activity What are discontinuous functions?

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steps, multiple formulae are required or there are constraints attached. For example, since the price of a commodity determines the quantity supplied and the quantity demanded, these quantities are functions of the price, even in cases where the relationship is so irregular that it can be described only by a list of prices with the corresponding quantities.

Example 4.2

A certain car is so expensive that wealthy people buy it for prestige reasons because it is expensive. The demand falls from 30 per month when the price is ` 20,00,000 to 20 per month when the price is `16,00,000 and then rises to 100 per month when the price falls to ` 12,00,000.

Figure 4.3: Demand for a Luxury Car

Solution:

The methods discussed earlier can be used to find the equation for the straight line which passes through the points (30, 20 lacs) and (20, 16 lacs), and again for the straight line through the points (20, 16 lacs) and (100, 12 lacs). It is standard practice to denote points in the graph in the form (x, y). In order to state the range of values for which each of these equations is valid, it is necessary to use one of the family of symbols known as inequalities. The most important inequalities are

a > b means ‘a is greater than b’

a ≥ b means ‘a is greater than or equal to b’

a < b means ‘a is less than b’

a ≤ b means ‘a is less than or equal to b’ (c) C

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Only the last one is needed for the present. The formulae for the demand curve shown in the graph are:

x = (p/40000) – 20 when 16, 00,000 ≤ p ≤ 20, 00,000

x = 340 - (p/5000) when 12, 00,000 ≤ p ≤ 16, 00,000 The function is now defined for all values of p in the range 12 lacs < p < 20 lacs. It is perfectly mathematically sound to leave the function undefined outside this range if no information is available, though a keen sales manager would like to see that whether it would be highly profitable to fix the price higher than ` 20 lacs.

In this example, it is not correct to say that price p is a function of quantity x. The value of x does not determine the value of p, since p could be either of two values if the value of x is, for instance, 25. For normal commodities the demand curves slope downwards throughout their length and it is then correct to regard price and quantity each as a function of the other.

Some functions are represented by two or more lines which do not meet each other. A good example is a schedule of postage rates where the first slab is ` 2 up to 20 gms. and then Re 1 for every additional 10 gms.

Figure 4.4: Postal Rates Graph

Clearly, the postage rate is a function of the weight of the letter, since the latter determines the former. Any function which consists of two or more lines which do not meet is called a discontinuous function. There is said to be a discontinuity at each of the values of the independent variable at which there is a gap in the values of the dependent variable. (c) C

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It must be noted that the graph includes the points (20, 30), (30, 40)…… but does not include the points (20, 40), (40, 80)…… Sometimes appropriate marks are added to a graph to indicate whether or not these boundary points belong to the function.

Discontinuities and inequalities introduce awkward complications, and so mathematicians and scientists usually ignore them for as long as they can. Unfortunately, they arise far too frequently in managerial problems to be so lightly disregarded. Because of this reason, we use limitations and constraints whenever we develop problems and analyze them.

Check Your Progress

Fill in the blanks:

1. Any function which consists of two or more lines which do not meet is called a ___________ function.

2. There is said to be a discontinuity at each of the values of the ___________ variable at which there is a gap in the values of the ___________ variable.

Quadratic Equations

We saw that it is usually possible to sell larger quantities of a commodity where the price is lower. For a monopolist, the demand curve for the market is the price curve to be used in calculating the revenue of the firm. Where there is no monopoly, the amount a manufacturer can sell is still a function for the price at which he offers his goods, although in this case the price curve will not be the same as the demand curve for the market as a whole. Let us consider an example.

Example 4.3 (Extending example 4.1)

The same machine manufacturer finds that he could sell an average of four machines per day at a price of ` 18,000 per machine. Stepping up his production to an average of 4½ machines per day, he finds that he has to reduce the price to ` 17,500 per machine in order to sell all that he produces. Find the profit function.

Solution:

Now this problem abandons the unrealistic assumption of traditional cost accounting that the price is a constant and the

Activity When do we use quadratic equations?

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revenue function therefore linear. Putting the machines sold per day, x, as a linear function of the price (in units of ` 1,000), p:

x = ap + b

The method learned earlier makes it easy to find out a and b. Substituting their values we find that the equation reduces to:

x= 22 – p

The revenue R is the price multiplied by the number of machines sold:

Revenue R = Price p x Quantity x

R = px

In order to find the breakeven point, it is simplest to express p as a function of x; R then becomes a quadratic function of x:

p = 22 – x from the equation x = 22 - p above

Substituting this value of p in R = px we get

R = px = (22 – x) x = 22x – x2

Assuming that the linear cost function to be 32 + 7x, as found earlier, the gross profit G becomes a quadratic function of x:

G = R – T

= (22x – x2) – (32 + 7x) = -x2 + 15x – 32

This is the profit function for this manufacturer.

This quadratic function more closely approximates the real life situation. Now the question comes, how do we solve these quadratic functions/equations. There are three basic methods:

1. Factorization

2. Using Graphs

3. Using Formula

Solution of Quadratic Equations by Factorization

If the quadratic equation can be expressed as a product of two linear expressions (known as factors) it can be solved using factorization. This is possible only if the solution to the equation is an ordinary number or a fraction. For example:

2x2 – 17x + 21 = (x – 7) (2x – 3) = 0

Wm

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The identity sign (=) is used as a reminder that the two sides are equal for all values of x, which can be confirmed by multiplying out the right-hand side.

If the product of any two expressions is zero, then at least one of these expressions must be zero. So recognizing the factors immediately leads to the solution of the equation:

2x2 – 17x + 21 = 0

(x – 7) (2x – 3) = 0

Either x – 7 = 0 or 2x – 3 = 0

x = 7 or x = 32

The main difficulty in finding the solution of a quadratic equation by factorization lies in finding the factors. It is possible to do this by a routine procedure which will either find the factors systematically or prove that none exists.

Procedure for Finding Factors

Let the factors be (px + q) and (rx + s), where p, q, r and s are positive or negative integers and the product of the two factors is ax2 + bx + c. Multiplying the factors we get:

prx2 + psx + qrx + qs = ax2 + bx + c

The coefficients must be the same on both sides as this must be true for all values of x. This gives equations relating the unknown quantities to the coefficients in the expression to be factorized: pr = a; ps + qr = b; qs = c. It implies that the product of ps and qr is ac and so the first task is to find these two numbers whose sum and product are known.

Example 4.4

Find the factors of 10.8x2 + 93x + 140 Solution:

The first stage is to take out the fractional factor, resulting in the expression 0.2 (54x2 + 465x + 700). It is then necessary to look for two numbers whose sum is 465 and whose product is 54 × 700 = 37,800.

The simplest method is to start with any two numbers whose sum is 465, such as 80 and 385. If the product is too small, a suitable amount is added to the smaller one of the two and an equal (c)

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amount is subtracted from the larger one. A little exercise will show that the numbers are 105 × 360 = 37,800.

Putting ps = 105 and using the fact that pr = 54, the highest common factor 3 is then equated to p. Calculations would show that s = 35, r = 18 and q = 20 and the factors of the above equation are:

0.2 (3x + 20) (18x + 35)

Example 4.5

Find the factors of 11x2 + 56x + 21.

Solution:

Two numbers are to be found whose sum is 56 and the product is 11 × 21 = 231. Following the above procedure, it will be found that 4 × 52 is too small and 5 × 51 is too large. It can immediately be concluded that there are no integral factors.

Example 4.6

Find the factors of 12x2 + 39x – 105.

Solution:

Here two numbers have to be found whose sum is 39 and the product is 12 × (-105) = -1260.

It requires thinking to find that the one which is larger numerically (that is, disregarding any negative sign) must be positive and the other negative. Trying 70 × -31 = -2170 is too big; 60 × –21 proves correct. It is pointless to try any pair of numbers which does not include a multiple of 5 because the product is required to be a multiple of 5.

Depending on which number is chosen to represent ps, the factors are either (12x – 21) (x + 5) or (3x + 15) (4x – 7), both of which can be reduced to 3(4x – 7) (x + 5). It would have been better to take out the factor 3 right at the beginning if it had been recognized as a common factor of all the coefficients.

Solution of Quadratic Equations Using Graphs

One way of finding the solution of any equation which is in the form or can be rearranged in the form is:

f(x) = 0 (c) C

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is to draw the graph of the function:

Y = f(x)

If this graph cuts the x-axis at any point, the value of x at that point is the solution of the original equation, since it is the value of x at which y = 0. This was the method discussed earlier also for linear equations and can be applied to quadratic equation ax2 + bx + c as well. Here a, b and c can be positive or negative and may involve fractions or decimals. It is also possible for b or c to be zero, but if a was zero, the function would become a linear function.

Example 4.7 (example 4.3 extended)

Plot the function G = –x2 + 15x – 32 on a graph.

Solution:

To draw the graph of the function:

G = –x2 + 15x – 32

It is necessary to choose a range of values of x and calculate the corresponding values of G. In this example it is enough to consider values of x between 0 and 14:

x : 0 2 4 6 8 10 12 14 G : -32 –6 12 22 24 18 4 –18

It can be seen from the graph below that G is zero when x is about 2.6 or about 12.4. These two values are said to be the roots of the equation –x2 + 15x – 32 = 0.

This means that the manufacturer will make a profit between 2.6 to 12.4 machines and would make the maximum amount of profit when he makes 8 machines.

Every quadratic function in which the coefficient of x2 is negative gives a graph of shape shown above, which is termed a parabola. If the highest point is above the x-axis, the corresponding equation has two roots, but if the highest point of the function is below the x-axis there is no solution for the corresponding quadratic equation.

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Figure 4.5: Graph of -x2 + 15x -32=0

Solution of Quadratic Equation by Formula

It is necessary to have a method of solving an equation in which the quadratic expression is difficult to factorize such as examples 4.3 and 4.5.. The method used is equally applicable to equations where factorization is simple.

The derivation of the formula is of interest only to mathematicians the solution is given below for a quadratic equation ax2 + bx + c = 0.

X =-b b - 4ac

2a

2±

This formula can be applied to any quadratic equation irrespective of the fact whether the coefficients are positive or negative.

Example 4.7 (Cont....)

Taking the equation given in the example 4.7, here a = -1, b = 15 and c = -32. The solution by formula is:

x=

-15 (15) - 4(-1)

2(-1)

2 (-32)±

=

-15 225 - 128

-2=

-15 9.85

-2

± ±a f

= 12.425 or 2.575. (c) C

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This is the exact solution whereas from the graph we got an approximate solution. The sum of the roots is always equal to -b/a and their product is always equal to c/a. In this example the sum 15.05 and the product 32.62 can be checked with the original equation. This check makes it unnecessary for you to check the roots separately by substitution in the equation.

Check Your Progress

Fill in the blanks:

1. If the quadratic equation can be expressed as a product of two linear expressions then the two expression are called ______________.

2. Every quadratic equation in which are coefficient of x2 is negative gives a graph of _________ shape.

Fitting a Quadratic Cost Curve A linear function is never an ideal model of production costs. It is usually possible to obtain a much better model by fitting a quadratic curve as we did in the example 4.7 above.

T = ax2 + bx + c

The values of a, b and c will be positive. The term x2 implies that costs increase more steeply as the production goes up. If there was no such effect, it would never pay to enlarge a factory.

It must not be assumed that a quadratic curve will be a perfect model of production costs. There is nothing magical about the x2 term. A quadratic function will always be as good as a linear function and nearly always be a better deal.

Example 4.8 (Extension of example 4.1)

The machinery manufacturer, discussed earlier, found that the total production costs averaged ` 60,000 per day when an average of 4 machines per day are produced. An accurate assessment of costs when the average production is 3½ machines per day and again at 4½ machines per day gives figures of ` 56,600 and ` 63,600 respectively. Fit a quadratic cost curve.

Solution

Just as fitting a linear curve to two known points was shown earlier to give two simultaneous linear equations, so fitting a quadratic curve to three known points give three simultaneous

Activity Discuss fitting a quadratic cost curve.

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linear equations. From the given information, again using units of ` 1000, the equations are:

a (3½)2 + b(3½) + c = 56.6 (1)

a (4)2 + b(4) + c = 60.0 (2)

a (4½)2 + b(4½) + c = 63.6 (3)

This set of equations can be solved very easily by elimination, eliminating first c and then b to give a = 0.4, b = 3.8 and c = 38.4.

The cost function is therefore:

T = 0.4x2 + 3.8x + 38.4 (4)

When the quadratic revenue curve found in example 4.4 is applied, the gross profit function is found to be:

G = -1.4x2 +18.2x – 38.4 (5)

The breakeven point is then approx. 2.65, compared with 2.58 when a linear cost curve is assumed.

More advanced techniques based on more complicated models are available in managerial problems, but the practical benefits of increased accuracy will be negligible in most cases.

Summary There are two possible ways in which you could have obtained the information related to fixed and direct costs as the cost accountant found. Either you take all the accounting records and classify each cost into the two headings, a tedious and time consuming process which is prone to error because of limited accounting knowledge and problems of classification.

Lesson End Activity Solve the following equations by formula method and by factorisation method:

(a) − + =2 2 1 0x x

(b) − + =2 5 6 0x x

(c) − = −+ − − +1 1 1 1

1 1 2 2x x x x

Activity What is the way for fitting a quadratic cost curve?

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Keywords

Quadratic Equation: An equation of the form, where a 0, b, c are constant, is called a quadratic equation.

Root: A value of the variable which satisfies the given equation is called a solution or root.


1. Discuss any two methods of finding the solutions of a system of linear equations.

2. An investor wants to invest ` 15,000 in two types of bonds. He earns 12% in first type and 15% in the second. Find his investment in each of his total earning is ` 1950.

3. M/s Kalyani Forge pays its workers ` 70 for an 8-hour shift. In addition each worker is paid ` 10 for every one hour of overtime. However, overtime cannot exceed 4 hours per day.

(a) Cite the total wage paid to the worker as a function of overtime.

(b) Draw the graph of this function.

4. Ash Lubes sells X units of Supreme Lubes each day at the rate of ` 50 per unit of 100 gm. The cost of manufacturing and selling these units is ` 35 per unit plus a fixed daily overhead cost of ` 10,000. Determine the profit function. How would you interpret the situation if the company manufactures and sells 400 units of the lubes a day?

5. Parker India Ltd., manufacturers of quality stationary items, has introduced a new variety of pen in the market. The market supply function of the pen is represented by the function Q = 160 × 8P, where Q denotes the quantity supplies and P denotes the market price per unit. It costs ` 4 to produce a pen. If the total profit required is ` 500 what should be the market price per unit?

6. Ishaan Petrochemicals has introduced in the market its latest lube. The marketing manager has worked out that the demand function of this product, which can be expressed as:

Q = 30 – 4P,

Where, Q is the quantity and P is the per kilogram price.

(a) Write the total revenue as a function of price.

(b) Draw the graph of this function. (c) C

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7. The monthly supply of 2T Oil in Delhi is estimated to be 95,000 tons when the price is ` 13,000 per ton and 1,10,000 tons when the price is ` 16,000 per ton. The monthly demand is estimated to be 109,000 tons at ` 13,000 per ton and 99,000 tons at ` 16, 000 per ton. Assuming that the supply and demand functions are both linear find these functions and hence determine the equilibrium price and quantity.

8. A manufacturer of petrochemicals finds that his total production cost is ` 1,20,66,000 per week when he is producing 1240 tons per week. The fixed costs are ` 67, 34,000 per week, and the selling price is ` 11, 700 per ton. Find (a) the weekly revenue, (b) the weekly gross profit, and (c) the weekly production and total production cost at the break-even point.

9. Yarn is prepared from cotton by being passed successively through slubbing, roving and spinning frames. Each 20 kg of yarn A requires 6 minutes in a slubbing frame, 18 minutes in a roving frame, and 106 minutes in a spinning frame. For yarn B the times are 7, 27, and 150 minutes respectively, and for yarn C they are 8, 30 and 181 minutes respectively. If the plant consists of eight slubbing frames, 28 roving frames and 162 spinning frames, how much of each type of yarn should be produced per hour in order to keep all the machines fully occupied?

10. A rubber glove manufacturer finds that he can sell 1, 38,000 gloves per week pack sizes at ` 190 per pack. He increases the price to ` 200 per pack and finds that he can sell only 1, 28,000 packs `, per week. Assuming that price curve is linear, find (a) the price, (b) the weekly revenue of the number of packs sold per week, (c) find the prices and quantities for which the weekly revenue will be ` 60,990 per week.

Further Readings

Books

R S Bhardwaj, Mathematics for Economics and Business, Excel Books, New Delhi, 2005 D C Sanchethi and V K Kapoor, Business Mathematics Sivayya and Sathya Rao, An Introduction to Business Mathematics

Web Readings

www.textbooksonline.tn.nic.in www.mathbusiness.com (c) C

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UNIT 5: Case Study

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Unit 5

Case Studies

Objectives After analyzing these cases, the student will have an appreciation of the concept of topics studied in this Block.

Case Study 1: Gwalior Drums Ltd.

Gwalior Drums Ltd. is a medium scale company, engaged in the manufacture of drums of different qualities and sizes. It has a fixed cost of `10,000,000. The average cost of manufacturing a drum costs company ` 60 which the company sells at ` 100 assuming that every drum produced is sold off, find a formula for profit for the company. Question

Find the minimum number of drums that the company should produce and sell to meet exactly the cost.

Source: MBCQ-721D_Quantitative Techniques for Management Applications_CCE_UPES

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Case Study 2: Cost & Revenue Calculation

A switch manufacturer finds that his total monthly production costs are ` 10,600 when production is 16,000 units per month, ` 17,800 when it is 26,000 units and ` 27,000 when the production is 36,000 per month. He can sell 16,000 units per month at ` 104 each, but has to reduce the price to ` 94 each in order to sell 26,000 pieces. He can sell 36,000 pieces only at ` 80.

Question

Assuming that both cost curve and price curve are quadratic, find

(a) the monthly total cost,

(b) the price,

(c) the monthly revenue, and

(d) the monthly gross profit as functions of the quantity sold.

Find also

(e) the quantity sold,

(f) the price and

(g) the monthly revenue at the breakeven point and confirm that the monthly total cost is then equal to the monthly revenue.


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UNIT 6: Matrices

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BLOCK-II

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Detailed Contents UNIT 6: MATRICES

Introduction

Vectors

Multiplication of Vectors

Matrices

Properties of Matrix Multiplication and Addition

Use of Matrices for Production Planning

UNIT 7: DETERMINANTS

Introduction

Solving Linear Equation by the Use of Determinants

The Best Method of Solving Linear Equations

Higher Order Determinants

Main Properties of Determinants

Applications in Management

UNIT 8: PROBABILITY

Introduction

Concept of Probability

Objective and Subjective Probabilities

Basic Statement of Probability

Mutually Exclusive Events

Dependent and Independent Events

Decision Trees

Revision of Probabilities

Combinations

UNIT 9: RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Introduction

Random Variables

The Bernoulli Process

The Binomial Theorem

Probability Distributions

Poisson Distribution

The Normal Probability Distribution


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UNIT 6: Matrices

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Unit 6

Matrices


Vectors

Multiplication of vectors

Matrices

Types of matrices


Introduction

If the economic model is extended to include several commodities, it will have several equations in several variables. If all the equations of the model are linear, then matrix algebra provides an efficient method of their solution than the traditional method of elimination of variables. Matrix algebra also has the advantage of presenting a system of several equations in a compact form.

It may be pointed out here that a given system of equations may or may not be consistent. The method of testing the consistency of a system is also provided by the matrix algebra.

Since matrix algebra deals with a system of linear equations only, it is also referred to as Linear Algebra.

Matrices form one of the most powerful management tool of modern mathematics. They have innumerable applications in the analysis of material and machine requirements and the solution of problems in planning and organization. An understanding of matrices is also essential for most branches of advanced mathematics and statistics. As vectors lie at the base of matrices, let us start by understanding them first.

Vectors

The use of vectors can be illustrated by a very simple example. A small firm uses sheeting fabric to manufacture white sheets and pillowcases for hospitals and hotels, which are sold by the dozen.

Activity Define the term vector algebra.

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Orders received in the office are passed by telephone to the packing department, who is interested only in the quantity to be packed in each parcel.

Typical orders would be ‘4 dozen sheets and 2 dozen pillowcases’, ‘18 dozen sheets and 6 dozen pillowcases’, ‘12 dozen sheets’, ‘6 dozen pillowcases’ and so on. It would not be long before speaker and hearer agree to save a lot of time and breath by giving simply a pair of numbers for each order:

[4 2] [18 6] [12 0] [0 6]

Here the first number stands for dozens of sheets and the second number stands for pillowcases. The four brackets denotes four different orders. As long as the zero is inserted when necessary, there can be no confusion as to the meanings of these figures. As the orders are packed, the quantities can be added up. These pairs of numbers are examples of vectors. A vector is any row or column of figures in a specified sequence. The fact that [12 0] is an order for 12 dozen sheets while [0 12] would be an order for 12 dozen pillowcases indicates that the numbers acquire meaning from their positions in the sequence. A vector is normally printed between square or curved brackets or between a pair of double vertical lines.

The sum of the four orders is an example of vector addition. Two vectors are added together by adding the first number in the first vector to the first number in the second vector, the second number in the first vector to the second number in the second vector and so on. Each number is called an element of the vector. Vectors can have more than two elements, but two vectors can only be added together if both have the same number of elements. Clearly the sum of the above four orders is [34 14], i.e. 34 dozen sheets and 14 dozen pillowcases.

If the firm started to sell blankets also, a new convention would be needed by which [4 2 3] means 4 dozen sheets, 2 dozen pillowcases and 3 dozen blankets. The convention would have to be adopted completely for all orders, inserting 0 whenever an order did not include any blankets. The total quantities ordered would be given by the sum of these three-element vectors, which would itself be a three-element vector. Vectors are, thus, an ordered arrangement of numbers – it can be in a row or a column.

Activity Define the term vector

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UNIT 6: Matrices

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Multiplication of Vectors

If the customer, responsible for the order [4 2], asked for it to be doubled, this would be interpreted as [8 4]. If he asked for it to be tripled, it would become [12 6]. This is the rule for multiplying a vector by an ordinary number, which is called a scalar to distinguish it from a vector. Hence, the result of multiplying a vector [a b c] by a scalar k is the vector [ka kb kc].

A vector may also multiply a vector. But it would be meaningless to multiply together two vectors, both of which represent orders for goods. The definition of vector multiplication will be seen to make sense only when it is applied in a sensible situation.

When sheets and pillowcases have been ordered and packed, the next stage is to invoice them. If the prices are ` 1,800 for a dozen sheets and ` 700 for a dozen pillowcases, then the amount due for the order [4 2] will be:

(4 × 1,800) + (2×700) = ` 8,600

This suggests a use for the multiplication of vectors. The prices can be represented by a new vector, which, because it is a different kind of vector, will be written as a column:

1800700LNMOQP

Then multiplying an order vector by this price vector can be defined as multiplying the first element of the order vector by 1,800 and the second element of the order vector by 700 and adding the results together:

4 21 800700

8 600 18 61800700

36 600 12 01 800700

21600

0 61800700

4 200 34 141 800700

71000

,,

,,

,,

,,

,,

LNMOQP=

LNMOQP=

LNMOQP=

LNMOQP=

LNMOQP=

It is obvious that (34 x 1800) is the total value of all the sheets in the preceding four order vectors and (14 × 700) is the total value of all the pillowcases, so that the sum of these products, 71000, must be the sum of all the separate orders:

8,600 + 36,600 + 21,600 + 4,200 = ` 71,000

Activity Why it is meaningless to multiply two vectors?

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Two vectors can be multiplied together only if both have the same number of elements. Multiplication of a row vector by a column vector, which always results in a scalar, is called the scalar multiplication of vectors.

Note that the product has meaning in this case only because the first element in both the order vector and the price vector represents dozens of sheets and the second element in each type of vector represents dozens of pillowcases. Yet they are also distinctive; one vector gives the number of units ordered and the other vectors the price per unit, so that the product gives the value of the units ordered. It would be just as meaningless to add an order vector to a price vector as it would be to multiply an order vector by another order vector. Vectors have, in fact, been implicit in some of the earlier examples in this book even though they were not made explicit.

Check Your Progress

Fill in the blanks:

1. A vector __________ multiply a vector.

2. It would be __________to multiply together two vectors, both of which represent orders for goods.

Now let us turn our attention to matrices.

Matrices

A matrix is a rectangular or square array of numbers arranged into rows and columns, where the numbers acquire meaning from their position in the array. This means that vectors we discussed earlier are just simple example of matrices. Let us take up an example.

Example 6.1

Let us assume that the manufacturer of sheets and pillowcases discussed earlier has three types of machines. There is one machine for cutting the fabric, three machines for sewing and one for folding. The manufacturing times in minutes per dozen are:

Cutting Sewing Folding

Sheets 8 38 14

Pillowcases 6 32 4

Activity Define the term matrices.

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To form these facts into a matrix, it is only necessary to arrange the numbers between brackets or double vertical lines:

8 38 146 32 4LNM

OQP

Single vertical lines will not do, as these are used to represent a determinant. Even when it has the same number of rows as columns, a matrix is not at all the same thing as a determinant. There is no way of expanding or evaluating a matrix, since each element has its own distinctive meaning. The production time for an order [4 2] can be calculated by multiplying the order with the manufacturing time.

For this purpose, each column of the matrix will be treated as a column vector and the scalar multiplication of the order vector by these column vectors would give the three results.

(4 × 8) + (2 × 6) = 44 minutes cutting

(4 × 38) + (2 × 32) = 216 minutes sewing

(4 × 14) + (2 × 4) = 64 minutes folding

However, it is not necessary to separate out the column vectors; a convention of matrix multiplication is adopted which gives the same result:

[4 2]

8 38 146 32 4LNM

OQP = [44 216 64]

The production times now appear as the elements in a new row vector. Vectors are really simple examples of matrices. In general, a matrix has m rows and n columns. If m = 1, then the matrix is a row vector with n elements. If n = 1, then the matrix is a column vector with m elements. All general statements that we made or will make about matrices will also apply to vectors as well. Two matrices can be multiplied together only if the number of columns in the first matrix is equal to the number of rows in the second matrix. The first row of the first matrix is then multiplied by the first column of the second matrix, following the rules for the scalar multiplication of vectors and the result becomes the first element in the first row of the matrix forming the answer. In general, the product of the ith row of the first matrix and the jth column of the second matrix becomes the element in the ith row and the jth column of the matrix forming the answer. (c)

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Therefore, it follows that a matrix with m rows and n columns can be multiplied by a matrix with p rows and q columns only when n is equal to p. The product is then a matrix with m rows and q columns. In the above multiplication of a row vector by the manufacturing time matrix, m = 1, n = 2, p = 2, and q = 3. Let us now apply the rules of matrix multiplication.

The original four order vectors can be formed into a matrix in which each of the four rows represents a different order. The manufacturing time matrix can then multiply this and the answer is a matrix in which the columns represent the three types of machine and the rows represent the time taken to process each of the four orders.

4 218 612 00 6

8 38 146 32 4

44 216 64180 876 27696 456 16836 192 24

L

N

MMMM

O

Q

PPPPLNM

OQP=L

N

MMMM

O

Q

PPPP

Types of Matrices 1. Rectangular Matrix: A matrix consisting of m rows and n

columns, where m ≠ n, is called a rectangular matrix. For

example, 11 12 13

21 22 23

a a aA

a a a⎡ ⎤

= ⎢ ⎥⎣ ⎦

is a rectangular matrix of order

2 × 3. Similarly 11 12

21 22

31 32

b bB b b

b b

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

is a matrix of order 3 × 2.

2. Square Matrix: If the number of rows of a matrix is equal to its number of columns, the matrix is said to be a square

matrix. For example: 11 12 13

21 22 23

31 32 33

a a aA a a a

a a a

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

is a square matrix.

3. Row matrix: A matrix having only one row is called a row matrix (or row vector). For example, [ ]4 1 2 7 is a 1 4× row matrix.

4. Column Matrix: A matrix having only one column is called a

column matrix (or column vector). For example,214

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

is a 3 1×

column vector. (c) C

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5. Diagonal Matrix: A square matrix ij n nA a

×⎡ ⎤= ⎣ ⎦ is said to be

diagonal matrix if ija = 0 for i j≠ . For example, the matrix D given below is a 3 × 3 diagonal matrix.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

3

2

1

000000

dd

dD

We note that the elements of matrix A, for i = j, are called the diagonal elements and the line along which they lie is called principal diagonal.

6. Scalar Matrix: A diagonal matrix in which all the diagonal elements are equal, is called scalar matrix. The matrix

2 0 00 2 00 0 2

T⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

is a 3 × 3 scalar matrix.

7. Identity (or Unit) Matrix: A square matrix with each of its diagonal elements equal to unity and all non-diagonal elements equal to zero, is called an identity matrix. The

matrix 1 0 00 1 00 0 1

I⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

is a 3 × 3 identity matrix. We note that an

identity matrix is a particular case of a scalar matrix where each of its diagonal elements is equal to unity.

8. Null (or Zero) Matrix: A matrix (square or rectangular) having all its elements equal to zero, is called a null matrix

e.g. 0 0 00 0 0

O ⎡ ⎤= ⎢ ⎥⎣ ⎦

is a 2 × 3 null matrix.

9. Triangular Matrix: A triangular matrix can be: (a) Upper triangular or (b) Lower triangular matrix.

(a) A square matrix A = [ ija ] is said to be upper triangular if

ija = 0 for i j> . The matrix 11 12 13

22 23

33

00 0

a a aA a a

a

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

is upper

triangular.

Activity What is scalar matrix?

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(b) A square matrix A = [ ija ] is said to be lower triangular if

ija = 0 for i j< . The matrix 11

21 22

31 32 33

0 00

aA a a

a a a

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

is lower

triangle.

Remarks:

1. A diagonal matrix is both an upper and lower triangular.

2. If the diagonal elements of a triangular matrix are all zeros, it is said to be strictly triangular.

Check Your Progress

Fill in the blanks:

1. Two matrices can be multiplied together only if the number of columns in the first matrix is ________ to the number of rows in the second matrix.

2. A ____________ matrix is both an upper and lower triangular.

Properties of Matrix Multiplication and Addition

It is often useful to represent a matrix by a single symbol. A capital letter is usually printed in bold type to emphasize that it is a matrix. Putting the matrix of the four orders as A and the manufacturing time matrix as B, the product can be written as:

AB = C Therefore, C is a matrix giving the total production time on each type of machine for each of the four orders.

The equation would no longer be true if B were written before A. In fact, it is impossible to multiply B by A since B has three columns while A has four rows. If the fourth row of A was disregarded, there would then be two matrices which could be multiplied in either order, but the results would be different:

4 2 44 216 64 4 28 38 14 8 38 14 884 244

18 6 180 876 276 or 18 66 32 4 6 32 4 648 204

12 0 96 456 168 12 0

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

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The products are entirely different in the two cases. The second product is, in fact, completely meaningless, since it includes terms such as (8 × 2) where 8 is the cutting time for sheets and 2 is an order for pillowcases!

We showed earlier how a row vector could be multiplied by a column vector to obtain a scalar product. From the rules of matrix multiplication, we can now see that it is impossible to multiply a row vector by another row vector or to multiply a column by another column vector, except in the trivial case of vectors with only a single element. Multiplying a column vector by a row vector gives a matrix instead of a scalar product:

[ ]1,800 7,200 3,6004 2

700 2,800 1,400⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

In this matrix, the elements 3,600 and 2,800 are completely meaningless. For instance, 3600 is obtained by multiplying the price for sheets by the order for pillowcases.

Saying that the multiplication of ordinary numbers is commutative (a×b = b×a) while the multiplication of matrices is in general non-commutative (A×B ≠ B×A) summarizes the above result. This means that changing the order of multiplication of two matrices will generally change the answer. It will be seen later that there are a few cases where changing the order does not change the answer. A little thought will show that this can only be true for square matrices, that is, matrices with equal numbers of row and columns; but matrix multiplication is not in general commutative even for square matrices.

Since multiplying a vector by k means multiplying every element by k, irrespective of whether it is a row vector or a column vector, it is to be expected that the same rule would apply to matrices.

This is the case:

k

k kk kk

k

4 218 612 00 6

4 218 612 00 6

L

N

MMMM

O

Q

PPPP=

L

N

MMMM

O

Q

PPPP

If k = 0, one obtains a matrix in which all the elements are zero, termed a zero matrix. A zero matrix is also obtained if two matrices are multiplied together, one of which is a zero matrix. But it is also possible to obtain a zero matrix as the product of two matrices neither of which is a zero matrix: (c)

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2 1 1 4 0 06 3 2 8 0 0

− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Matrix addition has not been defined so far. It is possible to add together two matrices only when both have the same number of rows and the same number of columns. The sum is then obtained simply by adding together the corresponding elements:

3 8 5 5 30 9 3 5 8 30 5 92 7 2 4 25 2 2 4 7 25 2 2

+ + +⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Here, the first matrix could represent the machine loading and unloading times and the second matrix the machine running time for the sheets and pillowcases in above example, so that the sum would be the manufacturing time matrix already employed.

It will be seen that the rule for vector addition conforms to this rule for matrix addition. Unlike matrix multiplication, matrix addition is commutative; changing the order of the matrices, which are added together, does not change the result.

A few words must be added on the equality, subtraction and division of matrices. Two matrices are said to be equal only if they are identical; they must have the same number of rows and the same number of columns and every element in the second matrix must be equal to the corresponding element in the first matrix.

A matrix can be subtracted from another matrix only when both have the same number of rows and the same number of columns. Subtraction is then simply the reverse of addition:

8 38 14 5 30 9 3 8 56 32 4 4 25 2 2 7 2

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Matrix subtraction is non-commutative, but this is to be expected since the subtraction of ordinary numbers is also non-commutative.

Matrix division is quickly dealt with, as it is impossible to divide a matrix by another matrix directly. There is a round about method which we will learn later in the unit.

Check Your Progress

Fill in the blanks:

1. Matrix subtraction is ___________________.

2. Unlike matrix multiplication, matrix addition is _______

Activity What is the use of matrices in production planning?

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Before the advent of the computer, use of matrix methods in production planning were of theoretical interest rather than real practical value. The position is now completely reversed. Any firm, which has access to a computer and does not use matrix methods, must be regarded as backward and inefficient. By these standards most of the Indian firms would be!

If a firm makes m products using n different types of machines, matrix A can represent the machine time requirements with m rows and n columns. One such matrix was shown earlier.

The total costs per minute for running each type of machine, including both capital and labour costs, form the n elements in a column vector. Small letters in bold type can represent such vectors; let c be the machine-cost vector. Then Ac will be a column vector with m elements, giving the total machine cost per unit for each product.

If the manufacturer of sheets and pillowcases in the example above finds that the machine costs are Re 0.2 per minute for cutting, Re 0.1 for sewing and Re 0.3 for folding, then the total machine costs per product are given by:

0.28 38 14 9.6

0.16 32 4 5.6

0.3

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

For most manufacturers there would, of course, be much larger numbers of machines and products. In this section only a very simple example can be followed through as it makes it easier to understand the process. Computers can handle matrices with dozens of rows and columns, each element having three or four digits and they still use the same process.

Matrix B with m rows and q columns may represent the material contents of the different products. With four ingredients such as sheeting fabric, thread, labels and packing material, there would be a four-column ingredients matrix for sheets and pillowcases, such as:

46 7 12 2816 3 12 13

⎡ ⎤⎢ ⎥⎣ ⎦

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The units of measurement may be different for each column, being chosen to suit the nature of the ingredient. The same unit will be used in each case when preparing the ingredient-cost vector. This will be a column vector with q elements and may be represented by d. Then Bd will be a column vector with m elements, giving the total ingredient cost per unit for each product:

2.546 7 12 28 0.3 123.916 3 12 13 0.1 44.7

0.2

⎡ ⎤⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

Any labour or other costs not already included in the machine-cost and ingredient-cost vectors will be computed for each product to form an additional cost vector e with m elements. The total cost per unit for each product is then obtained by adding together the three vectors each with m elements:

Ac + Bd + e = total cost vector

9.6 123.9 2.0 135.55.6 44.7 1.5 51.8

⎡ ⎤⎢ ⎥⎣ ⎦

All the information about machine time requirements, material contents and costs is collected by work study and costing staff and the matrices A and B and vectors c, d and e are stored in the computer. When new techniques or changed prices or wage rates make it necessary, the matrices and vectors are brought up to date. The products Ac and Bd and the total cost vector Ac + Bd + e are also computed and kept up to date. Depending on the pricing policy of the firm, there may also be a selling price vector p, which is computed from the total cost vector by adding a suitable percentage for fixed costs and profit.

When an enquiry is received, the prices and delivery dates are quoted by reference to the computer. If this results in an order, it is recorded as a row vector with m elements. A row vector is distinguished from a column vector by a distinctive mark:

x′= [4 2]

The order vector x′ may be multiplied by A to obtain a row vector giving the production times for the order, as calculated earlier. At the same time, x′ may be multiplied by Ac to give the total machine cost for the order: (c) C

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[ ] 9.64 2 49.6

5.6⎡ ⎤

=⎢ ⎥⎣ ⎦

One would expect to obtain the same result if the production-time vector x′ A is multiplied by the machine-cost vector c and this is in fact the case:

[ ]0.2

44 216 64 0.1 49.60.3

⎡ ⎤⎢ ⎥ =⎢ ⎥⎢ ⎥⎣ ⎦

The fact that x′A multiplied by c always gives the same result as x′ multiplied by Ac is called the associative property of matrix multiplication. The product can be written simply as x′Ac.

The computer stores a vector of running totals of machine-time commitments, to which the vector x′A is added. As each order is completed, its production times are deducted from the running totals. The commitment for each type of machine may then be divided by the number of machines of that type, obtaining the number of minutes and hence the number of weeks it will take to produce all outstanding orders. This information forms the basis for quoting delivery dates for new order and perhaps also for planning overtime work or the purchase of additional machines.

The vector x′B is computed and similarly incorporated in the running totals of ingredient requirements. The scalar quantity x′Bd gives the total ingredient cost of the order and may also be added to a continuous running total if it is necessary to keep a check on the amount of capital needed to finance work in progress. When the order is delivered, the amount of money due is given by x′p. This serves as a check on the invoice total.

This presentation is far from complete, but it is enough to show how computers using matrices can keep a check on production commitments and stocks of materials. The most important extension necessary in most firms will take into account production dates. The final delivery date of each order will determine the dates by which various stages of manufacture must be completed. Running totals of commitments will be kept by dates so that no type of machine can be overcommitted at any stage.

Ideally, the computer will be used to print out the production orders for each department at the appropriate times. Because t makes all calculations extremely rapidly and keeps a complete

Activity Give and example of row operation.

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check on all machine and material requirements, the computer will print each production order only when it is time to commence production. This makes it possible to accept any rush order, which does not conflict with existing commitments; such orders are usually very profitable if they can be successfully handled.

The computer will similarly record all ingredient requirements by dates, so that supplies are not obtained unnecessarily early. It will be programmed to give a warning if supplies do not arrive when they are due.

Properly used, the computer is the manager’s most efficient assistant, reminding him of any supplies that need to be chased and warning him of any future production bottlenecks. It is a complete waste of the computer’s powers to employ it in churning out masses of detail which the manager has to pore through in order to find where things may go wrong. This laborious and inefficient management by detail must give way to management by exception.

Let us now turn our attention to other applications of matrices.

Solving Linear Equations

There are three methods that can be used to solve linear equations: (1) Row operations (2) Matrices and (3) Determinants.

First two of these will be discuss here and third will be discuss in next unit.

Row Operations

A few hints have already been given that matrices can be used in the solution of sets of simultaneous linear equations. But before considering the role of matrices, it is useful to consider a technique known as row operations. Example 6.2

Solving linear equations by row operations is in principle the same as solving them by elimination. The difference is that row operations, in turn, aim systematically at a coefficient of 1 for each unknown. For instance, in given example

3x + 11y = 180 (1)

6x + 17y = 300 (2)

The first stage in row operations is to divide the first equation by 3 in order to make the coefficient of x into 1. This equation is then (c) C

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multiplied by 6 and the result subtracted from the second equation to eliminate x from that equation. So the equations become:

x + 3 23

y = 60 (3)

0x + 5y = – 60 (4)

The next stage is to divide the second equation by – 5, in order to make the coefficient of y equal to 1. This equation is then multiplied by 32/3 and the result subtracted from the first equation in order to eliminate y from that equation:

x + 0y = 16 (5)

0x + y = 12 (6)

This gives the solution, x = 16, y = 12. If one is able to multiply and subtract mentally, the whole procedure is very rapid. The other special feature of row operations is that it is unnecessary to keep writing the letters and the addition signs. All that is needed is to write down the figures, keeping the zeros as in the above equations and to insert a vertical line to separate the two sides of each equation:

3 11 1806 17 300⎡ ⎤⎢ ⎥⎣ ⎦

2/31 603

0 60–5⎡ ⎤⎢ ⎥−⎣ ⎦

1 0 160 1 12⎡ ⎤⎢ ⎥⎣ ⎦

A set of three equations in three unknowns requires three stages. It is necessary at each stage to write first the row, which has to be divided through to give the coefficient 1. This is the first row in the first stage, the second row in the second stage and so on. This row is then used to eliminate the corresponding coefficients in all the other rows, both above and below. Hence, after the first stage the first column of figures reads 1 0 0; after the second stage the second column of figures reads 0 1 0, while the first columns remains as 1 0 0 and so on.

The solution of example 3.4 by row operations reads:

3 4 3 1072 5 7 1485 9 2 151

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1/3 2/3

1/3 2/3

1/3 1/3

1 1 1 350 2 5 760 2 3 27

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥− −⎣ ⎦

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6 /7 1/7

1/7 6 /7

1 0 1 80 1 2 320 0 8 104

⎡ ⎤− −⎢ ⎥⎢ ⎥⎢ ⎥− −⎣ ⎦

1 0 0 160 1 0 50 0 1 13

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

The obvious disadvantage of the method is that it introduces fractions even when the final solution does not include any fractions. Its advantage is that it follows a strict routine, which is always necessary if computers are to be used.

Since each row represents an equation, it is permissible to rearrange the rows in order to get out of a difficulty. For instance, the equations:

2x - 4y + 3z = 14 (1)

3x - 6y + 2z = 11 (2)

6x - 3y + z = 2 (3)

After the first stage of row operations give:

1/2

1/2

1 2 1 70 0 2 100 1 ½ 5

⎡ ⎤−⎢ ⎥

− −⎢ ⎥⎢ ⎥− − −⎣ ⎦

To obtain 1 in the second position of the second row, the simplest procedure is first to interchange the second and third rows. It would alternatively be permissible to add the third row to the second row. The reader should follow through both methods, obtaining the solution x = 7, y = 3, z = 4.

Row operations can deal with a set of equations that are not all independent.

In example 6.2 after collecting all the terms involving unknowns on the left, the initial set of rows is:

11 11 032 44

1 1 13 06 6 841 1 2 1

06 4 3 41 51 1

04 86 6

−

−

−

−

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After three stages of row operations they become:

1 0 0 1.575 00 1 0 0.75 00 0 1 1.05 01 0 0 0 0

−−−

It is now clear that the fourth row contributes no information other than that contained in the first three rows; in other words, the equations were not all independent. There has been no attempt to discard an equation arbitrarily, as done when solving this example here. If the solution of a set of equations using determinants gives zero divided by zero for each of the unknowns, row operations will always lead to discarding the correct row since it will eventually produce a row of zeros. If the equations had been contradictory, row operations would have produced a row of zeros to the left of the vertical line with a non-zero value on the right.

Since example 6.2 leads to three independent equations in four unknowns, it is only possible to express each unknown in terms of one of the others. The first row in the final set of rows can be interpreted as b = 1.575w and the other rows give the corresponding solutions for f and h.

Using Matrices to Solve Linear Equations

Before we understand this method it is necessary for us to understand the concept of 'inverse' of a matrix.

Inverse of a Square Matrix

Example 6.3

The set of simultaneous equations used in examples 3.2 and 3.4 can be written as:

3 11 x 1806 17 y 300⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

3 4 3 x 1072 5 7 y 1485 9 2 z 151

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

It is only necessary to follow out the rules for matrix multiplication to see that these matrix products are identical to the sets of equations. If the symbol A is used to represent the matrix of (c)

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coefficients, x to represent the vector of unknowns and B to represent the vector on the right-hand side, any set of simultaneous linear equations can be expressed in the form:

Ax = B

Both A and B are known, but x is unknown. If there was a simple rule for dividing B by A, the result x would be obtained. However, it is impossible to divide a matrix directly by another matrix.

The operations of division are best understood as the reverse of multiplication. ‘Divide B by A’ is a way of saying, 'find the matrix or vector which when pre multiplied by A will give B’. Since matrix multiplication is in general non-commutative, it is necessary to say ‘when pre-multiplied by A’ to indicate that A precedes x in the above equation.

If the required matrix or vector exists, it can be found. But the procedure is not very straightforward. It involves the new concepts termed a unit matrix and the inverse of a matrix.

The unit matrix of order n, written In, is the square matrix with n rows and columns which has the figure 1 for each element in the principal diagonal, the diagonal from the top left-hand corner to the bottom right-hand corner and 0 for all other elements. Thus:

2 2

1 0 01 0

I I 0 1 00 1

0 0 1

⎡ ⎤⎡ ⎤ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦

In a set of n simultaneous linear equations in n unknowns, A will always be a square matrix. Let I be the unit matrix with the same number of rows and columns as A, which is also the number of elements in x, it is easy to see that:

A I = A

I A =A

I x = x

These equations can be confirmed by writing any elements one chooses in A and x and then following the rules of matrix multiplication. Since the product of I with any matrix or vector is identical to the matrix or vector itself, I is sometimes called the identity matrix.

For a given matrix A, it is usually possible to find a matrix which when multiplied by A gives the answer I. This matrix is a sort of (c) C

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reciprocal of A. The matrix which has this property is represented by A –1 and is called the reciprocal or more commonly the inverse of A.

The inverse is accordingly defined by the equation:

A-1 A = I

It can be shown that the multiplication of a square matrix and its inverse always commutes, that is:

A A-1 = I

In this discussion, it has been assumed that a matrix can have only one inverse. Although academic proofs are not the main purpose of this book, it is of interest to see how this can be proved. Let B be any matrix, which satisfies the equation:

B A = I Then:

B = BI = BAA-1 = IA-1 = A-1

Each step in this proof uses one of the results previously obtained, including the associative property of matrix multiplication mentioned earlier. Since two matrices are only equal if they are identical in all respects, this proves that B is identical to A-1 and therefore, that there is only one inverse of A.

This proof gives a glimpse at the foothills of what mathematicians’ term matrix algebra. The usefulness of matrices lies in the fact that one can employ them in all kinds of proofs and manipulations, simply representing each matrix by a bold letter without specifying what its elements are or even how many rows and columns it has, provided that one always conforms to the rules of matrix multiplication and addition.

If the inverse of A can be found, it is easy to solve the equation:

Ax = B

It follows that :

A-1 Ax = A-1 B

Ix = A-1 B which implies x = A-1 B

So the method of solving the equations is to find the inverse of A and then multiply this by B. This is not the same thing as dividing B by A, but it is the nearest one can get to division of matrices.

The simplest method of finding the inverse of a square matrix is to use row operations. Using the matrix interpretation of a set of (c)

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equations, row operations involved converting the matrix A on the left of the vertical line into the matrix I. The effect of these operations was to convert the vector b on the right of the vertical line into the solution vector, which is now seen to be equal to A–1 B. It is reasonable to deduce that the same operations will convert the matrix I into A–1I, which is the same thing as A-1.

For example 3.2, the matrix of coefficients is inverted as follows:

3 11 1 06 17 0 1

2/3 1 01 3

30 5 2 1− −

17 111 0 15 150 1 2 1

5 5

−

−

As a check, the final matrix obtained on the right of the vertical line may be multiplied by the original matrix on the left to show that it produces the unit matrix. It is then multiplied by the vector B to obtain the solution vector:

17 11180 1615 15

2 1 300 125 5

⎡ ⎤−⎢ ⎥ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦⎢ ⎥−⎢ ⎥⎣ ⎦

In using the inverse, it is often convenient to keep fractions outside the brackets:

17 1116 315

−⎡ ⎤⎢ ⎥−⎣ ⎦

To multiply this by the vector with elements 180 and 300 it is then obviously easiest to start by dividing the latter numbers by 15.

For example 3.4 it is necessary to invert the matrix: 3 4 32 5 75 9 2

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

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Using row operations the inverse proves to be: 53 19 13

1 31 9 1556

7 7 7

− −⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥−⎣ ⎦

Rather large numbers become involved when the vector multiplies this matrix formed by the right-hand sides of the original equations, but eventually the number 56 cancels out:

53 19 13 107 161 31 9 15 148 556

7 7 7 151 13

− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

It is unnecessary to check that the inverse was correct provided that the solution vector is now checked in the original equations:

3 4 3 16 1072 5 7 5 1485 9 2 13 151

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Following this example though involves a lot of arithmetic, it must be remembered that all modern mathematical methods assume that computers are available, so these large amounts of routine calculations are no drawback.

If row operations are used to try to find the inverse of the matrix from example 6.2, it will be found that the third stage gives a fourth row consisting entirely of zeros on the left of the vertical line. This matrix has no inverse and is known as a singular matrix. Whenever a set of equations has no unique solution, it will be found that the coefficients form a singular matrix.

Check Your Progress

Fill in the blanks

1. Properly used, the ______________is the manager’s most efficient assistant, reminding him of any supplies that need to be chased and warning him of any future production bottlenecks.

2. It is a complete ___________ of the computer’s powers to employ it in churning out masses of detail which the manager has to pore through in order to find where things may go wrong. (c)

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Summary

Matrices form one of the most powerful tools management and of modern mathematics. They have innumerable applications in the analysis of material and machine requirements and the solution of problems in planning and organization. An understanding of matrices is also essential for most branches of advanced mathematics and statistics. As vectors lie at the base of matrices, let us start by understanding them first.

Lesson End Activity

A manufacturer produces three products A, B and C, which are sold in Delhi and Calcutta. The annual sales of these products are given below:

ProductA B C

Delhi 5000 7500 15000Calcutta 9000 12000 8700

If the sale price of the products A, B and C per unit be ` 2, 3 and 4 respectively, calculate total revenue from each centre by using matrices.

Keywords

Matrix: An array of numbers arranged in certain numbers of rows and columns.

Rectangular Matrix: A matrix consisting of m rows and n columns, where.

Square Matrix: If the number of rows of a matrix is equal to its number of columns, the matrix is said to be a square matrix.

Row Matrix: A matrix having only one row.

Column Matrix: A matrix having only one column.

Singular Matrix: A matrix whose determinant is zero.


1. Discuss with examples, different types of matrices.

2. Illustrate the use of vectors.

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3. Given the following column vectors

1 2 2

A 2 , B -1 , C -23 3 1

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Find (i) A+B, (ii) A+B+C, (iii) 2A-B+C

4. If A = 2 3 1 2 3A and B

3 7 3 2 4⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Then find AB.

5. For square matrices A and B, expand (A+B) (A-B) and (A-B) (A+B). When will these products be equal?

6. Nahar Chemical Mills produces three varieties of base oil, Super fine Grade (A grade), fine grade (B grade) and coarse grade (C grade). The total annual sales in lacs of rupees of these products for the year 1999 and 2000 in the four cities is given below, find the total sales of three varieties of base oil for two years. For the year 1999

City Product Calcutta Mumbai Chennai Delhi Superfine base oil (A grade) 30 16 12 24 Fine base oil (B grade) 10 48 14 16 Coarse base oil (C grade) 16 8 62 12

For the year 2000

City Product Calcutta Mumbai Chennai Delhi Superfine base oil (A grade) 34 20 10 14 Fine base oil (B grade) 10 44 22 8 Coarse base oil (C grade) 26 12 78 10

7. A 2T oil manufacturer produces three products A, B, C which he sells in the market. Annual sale volumes are indicated as follows:

Market Products A B C I 8,000 10,000 15,000 II 10,000 2,000 20,000

If the unit sale price of A, B and C are ` 2.25, 1.50 and ` 1.25 respectively, find the total revenue in each market with the help of matrices. (ii) If the unit costs of above three products are ` 1.60, ` 1.20 and ` 0.90 respectively, find the gross profit with the help of matrices. (c)

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8. In a development plan of a cracker complex, a contractor has taken a contract to construct certain buildings for which he needs building materials like stone, sand, etc. There are three firms A, B, C that can supply him these materials. At one time these firms A, B, C supplied him 40, 35 and 25 truck loads of stones and 10, 5 and 8 truck loads of sand respectively. If the cost of one truck load stone and sand are ` 1,200 and ` 500 respectively then find the total amount paid by the contractor to each of these firms A, B and C separately.

9. HEG Ltd. maintains the records of the daily cost C of operating the “stress relieving furnaces division", which is a linear function of the number of incoming electrodes 1 and outgoing electrodes P, plus a fixed cost a, i.e.

C = a + bp = d1

Given the following data for 3 days find the values of a, b, and d by setting up a linear system of equations and using the matrix inverse.

Day Cost (in `)

No. of Incoming Electrodes to stress relieving furnaces

division

No. of Outgoing electrodes from stress Relieving

furnaces division 1 6,950 40 10 2. 6,725 35 9 3. 7,100 40 12

10. Mr. Bhattacharya has retired from service in 1999 from IOC. He received ` 14 lacs as the provident fund and retirement benefits from the company. He decides to invest a sum of ` 40,000 in three different stocks that yield 10%, 12 % and 15% respectively. The income from the third stock (which yields 15%) is twice the income from the first stock (which yields 10%). After one year, Mr. Bhattacharya earned an income of ` 5000 from his investments. What is the amount that he has invested in each type of stock.

11. Robin Singh & Company Ltd. stocks lubes of Castrol brand and Mak brand. The matrix of transition probabilities of the lubes is shown below:

Castrol Mak Castrol 0.9 0.1

Mak 0.3 0.7

Determine the market share of each of the brand in equilibrium position. (c)

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Further Readings

Books




Web Readings


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Unit 7

Determinants


Determinants

Solution of Linear Equations using Determinants.

Properties of Determinants

Introduction

In this unit, we will discuss the concepts of determinants and their importance in solving real world problems of business. While a matrix is an array of numbers arranged into certain number of rows and columns, a determinant is a scalar associated with a square matrix. Unlike scalars, the basic operations such as addition, subtraction and multiplication can be performed only if certain conditions are satisfied by the participating matrices. Like scalars division of one matrix by another is not defined.

Solving linear Equation by the use of Determinants Any pair of simultaneous equations in two unknowns can be written in the form:

a1x + b1y = h1 (1)

a2x + b2y = h2 (2)

The symbols, other than x and y, represent known quantities and the solution by elimination can be carried out in the same way as done earlier. The first step is to multiply the first equation throughout by a2 and the second equation throughout by a1.

a1 a2x + a2 b1y = a2 h1 (3)

a1 a2x + a1 b2 y = a1 h2 (4)

Then, subtract the first equation from the second in order to eliminate x:

a1b2y - a2 b1y = a1 h2 - a2 h1

Activity Differentiate matrices and determinants

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1 2 2 1

1 2 2 1

a h a hy =a b a b

−−

(5)

Substituting the value of y in the first equation and rearranging we find that

1 2 2 1

1 2 2 1

h b h bxa b a b

−=

− (6)

These two formulae are called the general formulae for the solution of any pair of simultaneous linear equations.

The next important step is to introduce a particular way of representing this solution so that it is easy to remember. The method adopted is to write:

1 11 2 2 1

2 2

a ba b a b

a b= −

The left-hand side of this equation is known as a determinant and the symbols between the two vertical lines are termed the elements of the determinant. Any set of four symbols or numbers arranged in this way between vertical lines is always interpreted as the difference of two products in the order shown. It should also be noted that this is a special kind of equation called an identity and is true for any values or symbols. Therefore, it is a general mathematical truth or definition rather than a statement about particular values or symbols. An identity is often represented by a triple hyphen in place of the double hyphen for an ordinary equation.

Two simple properties of determinants are immediately obvious. The value of the determinant is unchanged if the first row becomes the first column and the second row becomes the second column; but the value is multiplied by – 1 if the first and second rows are interchanged.

These two properties can be represented by the identities:

1 1 1 2

2 2 1 2

a b a aa b b b

≡

1 1 2 2

2 2 1 1

a b a b1

a b a b≡ −

These identities can be proved by multiplying both sides in accordance with the original definition of a determinant. This is (c) C

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known as expanding the determinants. There is a third simple property which can be similarly proved; if any multiple of one row or column is added to the other row or to the other column respectively, the value of the determinant is unchanged:

1 1 1 2 2 2 1 1 1

2 2 2 2 2 2 2

a b (a pa ) (b pb ) (a qb )ba b a b (a qb )b

+ + +≡ ≡

+

whatever the values of p or q.

So determinants have properties which can be proved and utilized in solving linear equations. So far the determinants considered have two rows and two columns and called second order determinants, used in solving a pair of simultaneous linear equations in two unknowns. Once the method of solution is understood, it can be extended to larger systems of equations where the practical merits of the method are much more apparent.

Given the two equations at the beginning of this section, the first step is to take the four coefficients on the left-hand sides and write them as a determinant. This forms the denominator of the formula for x. The numerator is the same determinant with the coefficients of x replaced by the column of values from the right-hand sides of the equations. So the formula is:

1 1

2 2

1 1

2 2

h bh b

xa ba b

=

By expanding these determinants, it can be seen that this is the same solution as that obtained by elimination. Since the setting-up of the determinants is a routine procedure and their evaluation is also routine, the original objective has been achieved. The equations can now be solved using a computer, which is able to carry out routine calculations but unable to think for itself.

We still need to point out that the value of y is represented in the same way as the value of x. The denominator is the same and the numerator is obtained by replacing the coefficients of y by the column of values on the right-hand sides of the equations:

1 1

2 2

1 1

2 2

a ha h

ya ba b

= (c) C

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Check Your Progress

Fill in the blanks

1. The left-hand side of any equation is known as a __________.

2. The symbols between the two vertical lines are termed the ____________ of the determinant.

The Best Method of Solving Linear Equations

Three different systematic techniques have been presented for the solution of sets of simultaneous linear equations: determinants, row operations and matrices. It is reasonable to ask which is the best method.

The method of determinants is usually the quickest if one requires the value of a few unknowns. This often applies when the equations arise from problems in probability. The determinant of the coefficients on the left-hand side of the equations should always be evaluated if there is a doubt whether a set of equations has a unique solution.

The quickest way to obtain a complete solution for all the unknowns is usually by row operations. If the method of determinants gives zero divided by zero for each of the unknowns, only row operations will provide the solution, which expresses the unknown in terms of one another, there being no unique solution. Row operations are obviously quicker than matrix methods, since they obtain the whole solution with no more work than is involved in inverting the matrix.

Matrix methods are extremely valuable if there are a large number of different sets of equations, all of which have the same matrix of coefficients. For instance, it might be desired in example 6.2 to consider all the possible product mixes obtainable by adding or subtracting one or two machines of each of the four kinds. There would then be several hundred possible combinations of machines, each giving rise to a set of equations with the same matrix of coefficients. The inverse of this matrix can be found and then each set of equations is solved simply by multiplying this inverse matrix by the appropriate vector.

Activity What is the best method to solve linear equations?

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All three methods can be applied using computer, but in practice computer programs tend to prefer matrix methods because of the wider applications of matrices.

It will be observed that a set of equations which has no unique solution always gives a determinant of value zero and always provides a singular matrix. It is customary to speak of ‘the determinant of a matrix’ provided that it is a square matrix. A singular matrix may be defined either as a matrix which has no inverse or as a matrix whose determinant is zero.

Check Your Progress

Fill in the blanks:

1. It will be observed that a set of equations which has no unique solution always gives a determinant of value _______ and always provides a ___________matrix.

2. A set of equations which has no unique solution always gives a __________ matrix

Higher Order Determinants

The practical importance of determinants increases when it is necessary to solve large numbers of equations with correspondingly large number of unknowns. It would be very convenient if the solution of a large set of linear equations could be written out in determinant form by the same routine as has been described for a pair of equations. If we followed this procedure, the solution of the equations: a1x + b1y + c1z = h1 (1)

a2x + b2y + c2z = h2 (2)

a3x + b3y+ c3z = h3 (3)

ought to be:

1 1 1

2 2 2

3 3 3

1 1 1

2 2 2

3 3 3

h b ch b ch b c

xa b ca b ca b c

=

But is it that simple?

Activity What are higher order matrices?

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Until now, a determinant with three rows and three columns, termed a Third Order determinant, has not been defined; there are, therefore, no rules for evaluating it. Since the purpose of determinants is to enable the solution of a set of equations to be determined, the rules for evaluating determinants of higher order are designed to fulfil this purpose.

The value of x in the solution of the given set of equations is found by elimination to be:

1 2 3 3 2 1 2 3 3 2 1 2 3 3 2

1 2 3 3 2 1 2 3 3 2 1 2 3 3 2

h (b c – b c ) – b (h c – h c ) + c (h b – h b )a (b c – b c ) – b (a c – a c ) + c (a b – a b )

The terms have here been arranged so that all the expression in brackets can be expressed as second order determinants. It can now be seen that this will be the same formula as given previously. So the rule for evaluating a third order determinant is:

1 1 1 2 2 2 2 2 2

2 2 2 1 1 1

3 3 3 3 3 3 3 3 3

a b c b c a c a ba b c a b ca b c b c a c a b

≡ − +

The values of y and z are obtained by expressing the information in the original equations in the form of determinants by the same routine as for a pair of equations; that is, the denominator is the same as in the formula for x, and the column of values from the right hand sides of the equations replaces the coefficients of y or z in the determinants forming the respective numerators.

2

1 1 1 1 1 1

2 2 2 2 2

3 3 3 3 3 3

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

a h c a b ha h c a b h a h c a b h

y za b c a b ca b c a b ca b c a b c

= =

The same principles can be applied to solve larger sets of equations, the determinants being set up by exactly the same routine. A determinant always has the same number of rows as columns, corresponding to the number of equations and unknowns that have to be solved.

The rule for evaluating a determinant of n rows and columns is to take the n elements in the first row and multiply each by the smaller determinant obtained by missing out the (c) C

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first row and the column which contains the elements in question. Thus, the first element is multiplied by the original determinant reduced by the first row and the first column; the second element is multiplied by the original determinant reduced by the first row and the second column and so on. Finally, the products are collected together by adding all the odd ones and subtracting all the even ones. So a fourth order determinant is expanded as:

1 1 1 12 2 2 2 3 2

2 2 2 21 3 3 3 1 3 3 3

3 3 3 34 4 4 4 4 4

4 4 4 4

a b c db c d a c d

a b c da b c d b a c d

a b c db c d a c d

a b c d

= −

2 2 2 2 2 2

1 3 3 3 1 3 3 3

4 4 4 4 4 4

a b d a b cc a b d d a b c

a b d a b c+ −

Third order determinants, which occur in this expansion are termed as minors of the original determinant. If the minor is prefaced by the appropriate sign, it is termed as the co-factor of the element by which it is multiplied. Thus, the cofactor of b1 is:

2 2 2

3 3 3

4 4 4

a c da c da c d

−

Check Your Progress

Fill in the blanks:

1. The rule for evaluating a determinant of n rows and columns is to take the n elements in the first row and multiply each by the smaller determinant obtained by __________out the first row and the column which contains the elements in question.

2. The practical importance of determinants increases when it is required to solve _________ no of equations.

Main Properties of Determinants

Unless a computer is being used, a larger determinant is hardly ever multiplied out as it stands. It is much easier to start off by simplifying it, using the principal properties of determinants to

Activity Discuss the main properties of determinants.

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reduce large numbers to small numbers or to zero wherever possible.

Three of the principal properties have already been mentioned when discussing second order determinants and it can be proved that they remain true for determinants of higher order. The main properties of the determinants can be summarized as under: 1. The value of the determinant is unchanged if the rows and

columns are interchanged with each other. 2. The value of the determinant is multiplied by –1 if any row is

interchanged with any other row. It follows from these first two properties that the value of the determinant is multiplied by – 1 if any two columns are interchanged.

3. If any multiple of any row or column is added (or subtracted) to any other row or any other column respectively, the value of the determinant is unchanged. However, one cannot add a multiple of a row to a column, or vice versa.

4. If any row or column has a factor common to all its elements, then this factor may be divided out. For instance, it can be seen by expanding both sides that:

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

a b c a b cpa pb pc p a b ca b c a b c

≡

5. It follows from (4) that if any determinant has a row of zeros or a column of zeros, the value of the determinant is zero because then you can take out 0 common and multiply the whole determinant with it resulting in a zero.

6. It follows from (3) and (5) that if any row is identical to any other row or a multiple of any other row (or if any column is identical to any other column or a multiple of any other column) then the value of the determinant is zero. We would like to illustrate the use of these six properties before we introduce the final three properties.

Example 1 01 0

The value is obviously zero, using the properties (5) or (6) discussed above. (c)

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Example

3 62 4

is zero, using (6)

The first row is 1½ times the second row.

Example

4 7 4 7is equal to using (3)

11 13 3 1−

By subtracting twice the first row from the second row, the numbers are made smaller and so easier to manipulate.

In evaluating a large determinant, property (4) is first applied wherever possible. Then property (3) is used where possible to make the numbers small and in particular to obtain as many zeros as possible.

Example

12 17 36 19 12 17 36 1911 13 28 14 11 13 28 14

39 15 12 6 3 5 4 2

13 5 8 11 13 5 8 11

− −=

− − − −

=−

− −

=

−− −

− −

12

12 17 9 1911 13 7 143 5 1 2

13 5 2 11

12

12 1 9 111 1 7 03 3 1 013 1 2 15

=

−− −

− −

=

−− −

− − −

12

13 1 9 110 1 7 00 3 1 0

14 1 2 15

12

12 1 9 110 1 7 00 3 1 01 1 2 15

The numbers have now been reduced quite a lot and there is a row which includes two zeros. The next step is to make a third zero in this row. The third row is then interchanged with the second row and again interchanged with the first row. The fourth order determinant then reduces to a single third order determinant.

12

12 28 9 110 22 7 00 0 1 01 5 2 15

12

12 28 9 10 0 1 010 22 7 01 5 2 15

−− −

− −

= −

−

− −− − (c) C

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=−

− −− −

=−

− −−

12

0 0 1 012 28 9 110 22 7 01 5 2 15

1212 28 110 22 01 5 15

The same procedure can now be applied to this smaller determinant. A second zero is obtained in the final column, this column is moved to the first column by applying property (2) twice and property (1) is then applied before the final evaluation.

12 28 1 12 28 1 12 1 2824 5 11 0 24 5 11 0 24 5 0 11

1 5 15 181 425 0 181 0 425

− − −− − = − − = − − −− − −

=−

− −−

= − −− −

241 12 280 5 110 181 425

241 0 0

12 5 18128 11 425

=

− −−

245 18111 425

= 24 [(- 5 × 425) – (- 11 × – 181)]

= 24 (-2125 – 1991) = 24 × – 4,116

= - 98,784

This example is a long one, since the original determinant was quite a large one. It should also be remembered that it is much easier to apply the rules for simplifying and evaluating determinants oneself than to follow the steps which someone else has chosen. Often there is more than one route to the answer and the route adopted may be a matter of choice.

The last three of the principal properties of determinants combine together some of the earlier properties, enabling one to amalgamate two or three steps into one.

7. If any row is moved up or down an even number of rows, the value of the determinant remains unchanged. If it is moved an odd number of rows, the value of the determinant is multiplied by –1. Similarly, if any column is moved an even number of columns, the value is unchanged; if it is moved an odd number of columns, the value is multiplied by – 1. (c)

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12 28 9 1 0 0 1 010 22 7 0 12 28 9 1

10 0 1 0 10 22 7 01 5 2 15 1 5 2 15

−− − −

= −− −

− − − −

12 28 1 1 12 28

5 11 0 0 5 11181 425 0 0 181 425

− −− − = − −

− −

8. A determinant can be expanded using the elements in the first column instead of the elements in the first row. Each element is multiplied by its minor, the smaller determinant obtained by deleting the first column and the row which contains the element. The products are then alternately added and subtracted in the same way as when expanding by the first row.

( )12 28 1

11 10 28 1 28 15 11 0 12 ( 5) 1

5 15 5 15 11 01 5 15

−− − −

− − = − − + −−

−

9. If there is any row or column in which all the elements, except one, are zero, the determinant is equal to the product of that one non-zero element and its cofactor. For an element in the jth row and the kth column, the cofactor is obtained by deleting the jth row and kth column to obtain the minor and then multiplying by –1 if (j + k) is odd but leaving it unchanged if (j + k) is even.

12 28 9 112 28 1

10 22 7 010 22 0

0 0 1 01 5 15

1 5 2 15

−−

− −= − −

−− −

(Positive since j = 3 and k=3)

12 28 15 11

5 11 0181 425

181 425 0

−− −

− − =−

−(Positive since j = 1 and k = 3)

2 7 0 52 7 5

1 3 0 47 1 3 4

4 2 0 24 2 2

5 4 7 1

−= − −

−−

−

(Negative since j=4 and k=3) (c) C

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All these properties of determinants can be proved, but the proofs are of interest only to mathematicians. They are all intuitively reasonable and the manager who needs to use determinants should be content to accept that they are valid for determinants of any order.

Check Your Progress

Fill in the blanks:

1. A determinant can be expanded using the elements in the first column instead of the elements in the__________.

2. If any row is moved up or down an even no. of rows, the value of determinant remains ______________ .

Applications in Management

Managerial Problems Involving Determinants

The systematic nature of determinants makes it possible to use them in solving sets of simultaneous linear equations with the aid of a computer and so take the drudgery out of mathematics. It is still necessary to understand the nature of determinants and their principal properties in order to know what the computer is doing and why it may sometimes fail to produce a solution. It is also an advantage to be able to solve fairly simple sets of equations without the complications of computers, either by calculation on papers or by using a calculator.

An obvious managerial application of determinants is in the type of problem illustrated below, where it is required to find the correct product mix in order to make full use of machine capacities.

Example 7.1

In a petroleum engineering workshop there are seven machines for drilling, two for turning, three for milling and one for grinding. Four types of brackets are made. Type A is found by work study to require 7 minutes drilling, 3 minutes turning, 21/2 minutes milling, and 11/2 minutes grinding, and the corresponding times in minutes for the other types are: B: 5, 0, 1 1/2, 1/2; C: 14, 6, 9, 3 1/2; D: 26, 9, 11, 11/2. How many of each type of brackets should be produced per hour in order to keep all the machines fully occupied?

Activity Discuss the applications of determinants in management.

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Solution

The four equations could be set up in the same way as done earlier, each equation representing the total minutes of work per hour on a particular type of machine. The numbers on the right-hand sides would therefore be 420, 120, 180, and 60 respectively. The coefficients on the left-hand sides would form the determinant constituting the denominator in the solution. This can be written out directly from the information given in the question and then evaluated:

7 5 14 267 5 14 26 0 5 0 53 0 6 91 0 2 3 1 0 2 33 31 12 1 9 11 5 3 18 22 0 3 8 74 42 2

1 1 1 1 3 1 7 3 0 1 1 61 3 12 2 2 2

= =

−

The operations has been as follows. From second row 3 and from 3rd and 4th row ½ has been taken out as common. Then, 1 in the second determinant the second row multiplied by 7, 5 and 3 respectively is subtracted from 1st, 3rd and 4th row respectively. In the fifth determinant first column is subtracted from third column.

5 0 5 1 0 13 153 8 7 3 8 74 4

1 1 6 1 1 6

−= − =− −

1 0 015 3 8 44

1 1 7

−=−

8 4 2 115 151 7 1 74

−= = −− −

= –15 (–14 –1) = 225

The numerator for type A appears more formidable at first, but can be quickly simplified:

420 5 14 26 7 5 14 26 7 5 14 26120 0 6 9 2 0 6 9 2 0 6 9

60 50180 3 9 11 3 3 9 11 6 3 18 2260 1 3 1 1 1 3 1 2 1 7 3

= = (c) C

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3 0 21 113 21 11

2 0 6 915 15 2 6 9

0 0 3 130 3 13

2 1 7 3

− −− −

= =−

−

3 7 1145 2 2 9

0 1 13

− −=

−

3 7 16

3 8045 2 2 35 45

2 350 1 9

− −− −

= =−

3 16

2252 7− −

=

= 225(–21 + 32) = 225 × 11 Since the denominator is 225, the number of brackets per hour of type A in the solution is 11. There are no new problems involved in finding the solution for the other three types of bracket and so this task is left for you. You should note that it has not been necessary either to introduce symbols for the four unknowns or to write out the four equations.

Most managerial applications of determinants are as straightforward as this example.

Determinants are useful when you have to solve a number of equations, but are difficult to use when you have to look at computerization of the firm’s operations. We then turn to matrices, to see how these can be used to solve managerial problems.

Brand Switching and Markov Chains

Let us take up a managerial application of matrices, which is of theoretical as well as practical interest. This is mostly used by advertising agencies and big companies in brand management.

Example 7.2

Three brands of detergent share the market, 40% of customers buying brand A, 50% brand B, and 10% brand C. Each week there are changes in the customers’ choices. Of those who bought brand A previous week, 50% buy it again, but 15% change to brand B and 35% to brand C. Of those who bought brand B, 60% buy it again,

Activity What is brand switching?

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10% buy brand A and 30% buy brand C. Of those who bought brand C, 85% buy it again, 5% buy brand A and 10% buy brand B. What proportion of the market will each of the three brands eventually hold?

Solution:

It is simplest to express the brand switching percentage as decimals, keeping percentage figures for the market shares. The change in market shares in the first week can be obtained as the product of a matrix representing the brand switching and a vector representing the initial market shares:

0.50 0.10 0.05 40 25.50.15 0.60 0.10 50 37.00.35 0.30 0.85 10 37.5

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

It will easily be seen that the terms involved in this product, (0.50 × 40), (0.10 × 50), etc., are the correct calculations from the information given in the example in order to obtain the new market shares. In this type of model, each of the matrices adds up to 1 and the elements in each vector total 100.

For the following week, the new market-share vector must multiply the same brand switching matrix:

0.50 0.10 0.05 25.5 18.3250.15 0.60 0.10 37.0 29.7750.35 0.30 0.85 37.5 51.900

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

As could have been guessed from the original information, brand C is getting a steadily larger share of the market. It cannot, however, obtain a monopoly. A little arithmetic will show that if brand C has 80% of the market one-week it cannot have more than 75% the next week. Clearly its eventual share will be somewhere between 51.9% and 75%, but obviously a direct method of finding the eventual share is desirable.

Let the brand switching matrix be M and the successive market-share vector a, b, c… so that the above two equations can be written as:

Ma = b

Mb = c (c) C

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A vector x will represent the final market shares such that pre-multiplying it by the brand switching matrix produces an unchanged result. That is:

Mx = x

Using I for the unit matrix and 0 for a zero matrix or vector, this can be rearranged:

Mx = Ix

Mx – Ix = 0

(M – I) x = 0

The fact that the last equation is equivalent to the preceding one depends on the property of matrix multiplication termed distributive. The full summary of the properties of matrix multiplication is that it is associative and distributive but not, in general, commutative. The latter property makes it essential to place M and I before x in both equations to be sure that they are equivalent.

Since M must be a square matrix and I is chosen to have the same number of rows and columns as M, there is no difficulty in finding the matrix (M –I):

0.50 0.10 0.05 1 0 0 0.50 0.10 0.050.15 0.60 0.10 0 1 0 0.15 0.60 0.100.35 0.30 0.85 0 0 1 0.35 0.30 0.15

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− = −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

One cannot find x by obtaining the inverse of (M-I), because it is a singular matrix. Since each column of M totals 1, each column of (M-1) must total zero. The determinant of this matrix must therefore be equal to zero, applying properties (3) and (5) of determinants section in adding the second and subsequent rows to the first row. It follows that (M-I) will always be a singular matrix in this type of problem.

The equations represented by the matrix equation are clearly not all independent. When it has been stated what proportions of previous customers are retained or gained by all brands except the last, the equations giving the market share of the last brand is made up of what is left.

There is an additional fact not included in the matrix equation. Expressing the market shares as percentages, all the elements in (c) C

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the vector x must total 100. This gives an additional row to permit a unique solution to be found by row operations:

0.50 0.10 0.05 00.15 0.40 0.10 00.35 0.30 0.15 0

1 1 1 1000

−−

−

In the process of solution, one of the rows will become a row of zeros and the remaining rows will give a unique solution. It is left to you to confirm that the final shares are respectively 12/109, 23/109 and 74/109, or approximately 11.0%, 21.1% and 67.9%.

Forming it into a vector and pre-multiplying this vector by the brand switching matrix can check the solution. It will be noted that the information about the initial market shares is not used in finding the solution. The final market shares will be exactly the same whatever shares the brands started with.

As a practical application, the brand switching example has two drawbacks. First, repeated matrix multiplications will eventually involve fractions of customers, which is impossible. Second, it is highly improbable that there will be a persistent pattern of brand switching; either customer will become less inclined to switch brands or the pattern of switching will be disrupted by special sales campaigns or dynamic market forces.

However, the model has many other applications more realistic than brand switching, particularly when the discussion is transferred from proportions to probabilities. Where an operator attends several machines which are subject to random stoppages at differing average frequencies, so that two or more may be stopped at the same time causing machine interference, the technique here discussed enables the average productivity of each machine to be accurately calculated. It can also be used to calculate average stock levels and the probability of running out of stock when demand and supply are random and so assist in finding the optimum stockholding policy. The model has the impressive title ergodic Markov Chains.

You would have noticed that one or two theoretical difficulties have been ignored in presenting the technique. How can it be proved that the market shares will settle down to a stable vector? Is it always permissible to use an equation, which assumes the existence of the stable vector? (c)

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The techniques are, in fact, valid for the types of problems, which have been discussed. But it would be very unwise to apply it unthinkingly in quite different situations, where not all the elements in the matrix are positive. Pure mathematicians go deeply into all questions of validity and it is necessary to seek their guidance whenever the appropriateness of certain techniques is in doubt.

Several examples of problems involving sets of simultaneous linear equations were discussed in unit 3, which could be solved either by elimination or by substitution. However, all these methods require certain amount of ingenuity and a great deal of calculation. Attempting to solve ten or twelve simultaneous equations by elimination would be a task not meant for the fainthearted and usually managers do not have the time to do it. Yet in many important applications, such as finding the product mix that will keep a number of different types of machines fully occupied, it is not unusual to have large numbers of equations and unknowns. If this amount of detail cannot be avoided, it is reasonable to look for a routine method of solving equations which is so automatic that a computer can be employed to find the solution. Such routine methods, involving determinants and matrices, are discussed here.

Keeping all the machines fully occupied is not necessarily the manager’s main objective. There may be other product mixes which leave some machines partly unoccupied but yield a greater total profit. The manager’s usual objective is to find the product mix which will maximize the total profit without exceeding the capacities of the machines.

Check Your Progress

Fill in the blanks

1. The manager’s usual objective is to find the product mix which will __________ the total profit without exceeding the capacities of the machines.

2. The equations represented by the matrix equation are clearly ________all independent.

Summary

The method of determinants is usually the quickest if one requires the value of a few unknowns. This often applies when the (c) C

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equations arise from problems in probability. The determinant of the coefficients on the left-hand side of the equations should always be evaluated if there is a doubt whether a set of equations has a unique solution

Lesson End Activity

A manufacturer produces three products A, B and C, which he sells in two markets. Annual sales volumes are indicated as follows:

8000200006000II18000200010000I

CroductPBroductPAroductPMarket

If each unit of A, B and C is sold at ` 2.50, 1.25 and 1.50 respectively, find revenue from each market.

Keywords

Determinant: A numeric value that indicate singularity or non-singularity of a square matrix.

Identity: An identity is often represented by a triple hyphen.

Minor: A minor is the det. of the square matrix formed by deleting one row and one column from some larger square matrix.

Cofactor: Cofactor is the multiplication of (–1)i + j with minor.


1. Find the value of the following determinant:

1 3 23 9 51 3 2

2. What are the main properties of determinants? Illustrate with suitable examples.

3. By expanding the determinants, prove the identities:

(a) 1 1 1 2

2 2 1 2

a b a aa b b b

≡ (b) 1 1 2 1

2 2 1 1

a b a ba b a b

≡ −

(c) 1 1 1 2 1 2 1 1 1

2 2 2 2 2 2 2

a b a pa b pb a qb ba b a b a qb b

+ + +≡ ≡

+ (c)

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4. Evaluate the determinants:

(a) 1 00 1

(b) 1 01 1

(c) 1 21 4

(d) 1 23 4

(e) 3 12 4

(f) 1 23 4−−

(g) 3 62 4

(h) 4 117 13

5. Evaluate the following determinants, first using property (9) and then the other properties as appropriate:

(a) 3 0 27 0 54 1 2−

(b) 2 3 2

0 5 03 2 21

−

−

(c) 9 21 110 0 35 15 7

−−

(d)

3 0 6 22 0 4 51 2 7 3

5 0 10 17

−−− −

−

(e)

2 10 5 34 8 0 70 0 0 53 6 1 9

−−

−

6. Evaluate the following determinants, first using property (4) three times, then property (3) once or twice to make the number smaller and finally property (9), or other properties as appropriate.

(a) 2 3 14 6 1

12 24 2 (b)

3 44 302 5 187 35 63

− −−−

(c) 32 35 328 10 434 40 3

(d)

1 3 1 26 12 4 121 6 3 84 9 2 2

(e)

2 3 6 43 1 3 85 3 3 4

12 6 9 12

−−−

(c) C

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7. Four boys order in a fish-and-chips restaurant. A orders fish, chips and coke. B orders two fish with chips. C orders fish and coke. D orders chips and coke. The prices are ` 50 for fish, ` 18 for chips, and ` 15 for coke.

(a) Express each boy’s order as a row vector.

(b) Add together these four vectors to obtain a fifth row vector representing the total quantities ordered.

(c) Express the prices as a column vector.

(d) Multiply each of the five row vectors by the price vector, to obtain the amount owed by each boy and the total amount owed.

(e) Check that the fifth result in (d) is equal to the sum of the other four results.

Further Readings

Books




Web Readings



(c) C

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UNIT 8: Probability

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Unit 8

Probability


Probability

Concept of probability

Objective and subjective probability

Introduction

If all business decisions could be made under conditions of certainty, the only valid justification for a poor decision would be failure to consider all the pertinent facts. With certainty, one can make a perfect forecast of the future. Unfortunately, however, the manager rarely, if ever, operates in a world of certainty. Usually, the manager is forced to make decisions when there is uncertainty as to what will happen after the decisions are made. In this latter situation, the mathematical theory of probability furnishes a tool that can be of great help to the decision maker.

Concept of Probability

The idea of probability is normally associated and remembered in connection with games or gambling. If a bookmaker offers odds of 2 to 1 against a team winning the world cup, it means that he considers that the probability, that the team will win, is not more than 1/3. When he accepts a bet of ` 1, he believes he has a probability of at least 2/3 that he will gain the money and a probability of not more than 1/3 that he will have to pay out ` 2 as well as return the original ` 1. So he expects to make a profit, not necessarily on this particular bet but on all the bets he takes when applying the same general principles.

Herein lies the basis for a definition of probability as the ‘degree of belief’ that an event will occur. Most statisticians would not accept this as a satisfactory definition, but there is an important minority, usually known as Bayesians, who do in fact take ‘degree of belief’

Activity What does probability mean?

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as their definition of probability. The fact that there are differences of opinion even over the definition of probability will serve as a warning to the reader of the difficulties of the subject, for there were no such disagreements over the subjects covered in the previous units.

It may be added that it is only the early quotations of betting odds that are based on the bookmaker’s degree of belief. When a reasonable number of bets have been taken, the bookmaker will adjust the odds in such a way that he makes a profit whichever team wins. It is the balance of opinion among punters, which really decide the odds.

An alternative definition of probability is based on counting numbers of equally likely events. In a situation in which there are ‘n’ possible outcomes, all equally likely, the probability of each of these outcomes is 1 divided by ‘n’. The outcomes must be defined in such a way that exactly one will occur; that is, it must be impossible for none of them or for more than one of them to occur. They are termed as simple events.

If an ordinary six-sided die is thrown, the possible outcomes are the numbers 1, 2, 3, 4, 5 and 6. One and only one of these will in fact occur. If the die is unbiased, which is a way of saying that all the outcomes are equally likely, then the probability of each simple event is 1/6.

It is now possible to discuss compound events, such as ‘less than 3’, ‘3 or more’, or ‘an even number’ when throwing an unbiased die. If a compound even is defined to include m equally likely simple events, then the probability of the compound event is m/n. The first example given consists of the simple events 1, 2; the second consists of 3, 4, 5, 6; and the third consists of 2, 4, 6. So their respective probabilities are 1/3, 2/3, and 1/2.

This definition of probability is called the classical or a priories definition of probability. It is ‘a priories’ because the probability can be determined before the die has even been thrown. But it has two great weaknesses. Its theoretical weakness is that the phrase ‘equally likely’ is synonymous with ‘equally probable’, and so the definition is circular. Its practical weakness is that it cannot cope with situations where the simple events are not all equally likely; it would be an unusual horse-race in which one could say with confidence that all the horses are equally likely to win. (c) C

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UNIT 8: Probability

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Later in this unit a third definition of probability will be introduced. However, the classical definition is perfectly adequate for discussing permutations and combinations, and nothing said later will detract from the validity of the conclusions reached using the classical definition.

Check Your Progress

Fill in the blanks:

1. An alternative definition of probability is based on __________ numbers of equally likely events.

2. An ordinary die has _________ faces.

Objective and Subjective Probabilities

Most of us are familiar with the laws of chance regarding coin flipping. If someone asks about the probability of a head on one toss of a coin, the answer will be one-half, or 0.50. This answer is based on common experience with coins, and assumes that the coin is a fair coin and that it is “fairly” tossed. This is an example of objectivity probability. There are two interpretations of objective probability. The first relies on the symmetry of outcomes and implies that outcomes that are identical in essential aspects should have the same probability. A fair coin is defined to be one that is evenly balanced and has two sides that are identical (except for minor differences in the image). Hence, each side should have equal probability of one-half (ignoring the possibility of the coin landing on its edge). If the coin was bent, weighted, or two-tailed, the answer would be different. As another example, suppose we have a box containing three red and seven black balls (that are of the same size, have the same feel, and are otherwise identical except for colour), and the balls are thoroughly mixed. The symmetry of outcome interpretation would assign a 0.10 probability to each ball, and hence a 0.30 chance of drawing a red ball.

The relative frequency interpretation of objective probability relies on historical experience in identical situations. Thus, if a coin has been flipped 10,000 times with 4,998 heads, we would conclude that the probability was 0.50 (i.e. 4,998/10,000 rounded) for a head the next time the coin was flipped in the same manner as before.

Activity What are objective and subjective probabilities?

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A subjective interpretation of probabilities is often useful for business decision making. In the case of objective probability, definitive historical information, common experience (objective evidence) or rigorous analysis lies behind the probability assignment. In the case of subjective interpretation, quantitative historical information may not be available; and instead of objective evidence, personal experience becomes the basis of the probability assignment. For managerial decision-making purposes, the subjective interpretation is frequently required, since reliable objective evidence may not be available.

Assume that a manager is trying to decide whether or not to build a new factory, and the success of the factory depends largely on whether or not there is a recession in the next five years. A probability assigned to the occurrence of a recession would be a subjective weight, which would be assigned after. There would certainly be less agreement on this probability than there would be on the probabilities of drawing a red ball, or of a fair coin coming up heads. Since we are primarily concerned in this book with management decisions, we shall often assign subjective probabilities to events that have a critical bearing on the management decision. This procedure aims to ensure consistency between a decision-maker’s judgment about the likelihood of the possible states of nature and the decision that is made.

One important objective of the suggested decision process is to allow the decision maker to think in terms of the possible events that may occur after a decision, the consequences of these events and the probabilities of these events and consequences, rather than having the manager jump immediately to the question of whether or not the decision is desirable.

Check Your Progress

Fill in the blanks

1. A _________ interpretation of probabilities is often useful for business decision making.

2. In the case of objective probability, definitive historical information, common experience (objective evidence), or rigorous analysis lie behind the _________.

Activity What is the basic statement of probability?

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UNIT 8: Probability

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Basic Statement of Probability

Two fundamental statements about probabilities are:

1. Probabilities of all the various possible outcomes of a trial must sum to one.

2. Probabilities are always greater than or equal to zero (i.e., probabilities are never negative) and are less than or equal to one. The smaller the probability, the less likely the chance of the event happening.

The first statement indicates that if A and B are the only candidates for an office, the probability that A will win plus the probability that B will win must sum to one (assuming a tie is not possible).

The second statement results in the following interpretations. If an event has a positive probability, it may possibly occur; the event may be impossible, in which case it has a zero probability; or the event may be certain to occur, in which case the probability is equal to one. Regardless of whether probabilities are interpreted as objective probabilities or as subjective weights, it is useful to think in terms of a weighting scale running from zero to one. If someone tosses a coin of unknown characteristics 500 times to obtain an estimate of objective probabilities and the results are 225 heads and 275 tails, the range of possible results may be converted to a zero-to-one scale by dividing by 500. The actual results are 225/500 = 0.45 heads and 275/500 = 0.55 tails. Hence, if we wish to derive probabilities, we shall manipulate the data so as to adhere to the zero-to-one scale. The 0.45 and the 0.55 may be used as the estimators of the true probabilities of heads and tails (the true probabilities are unknown).

Check Your Progress

Fill in the blanks:

1. Probabilities of all the various possible outcomes of a trial must sum to _________.

2. Probabilities are never _________ .

Mutually Exclusive Events

Two or more events are mutually exclusive if only one of the events can occur on any one trial. The probabilities of mutually exclusive

Activity What are mutually exclusive events?

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events can be added to obtain the probability that one of a given collection of the events will occur.

Example 8.1

The probabilities shown in Table 8.1 reflect the subjective estimate of a newspaper editor regarding the relative chances of four candidates for a public office (assume a tie is not possible).

Table 8.1: Election Probabilities

Event: Elect Probability

Candidate A 0.18

Candidate B 0.42

Candidate C 0.26

Candidate D 0.14

1.00

These events are mutually exclusive, since in one election (or in one trial) only one event may occur; therefore, the probabilities are additive. The probability of a Democratic victory is 0.60; of a Republican victory, 0.40; or of either B or C winning, 0.68. The probability of both B and C winning is zero, since only one of the mutually exclusive events can occur on any one trial.

Check Your Progress

Fill in the blanks:

1. Two or more events are mutually exclusive if __________ of the events can occur on any one trial.

2. The probability of mutually exclusive events can be added to obtain to probability that one of a given _______ of the events will occur.

Dependent and Independent Events

Events may be either independent or dependent. If two events are (statistically) independent, the occurrence of one event will not affect the probability of the occurrence of the second event.

When two (or more) events are independent, the probability of both events (or more than two events) occurring is equal to the product of the probabilities of the individual events. That is:

P (A and B) = P(A) × P(B) if A, B independent (c) C

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Where

P (A and B) = Probability of events A and B both occurring

P (A) = Probability of event A occurring

P (B) = Probability of event B occurring

The above equation indicates that the probability of A and B both occurring is equal to the probability of A multiplied by the probability of B, if A and B are independent. If A is the probability of a head on the first toss of the coin and B is the probability of a head on the second toss of the coin, then:

P (A) = ½

P (B) = ½

P (A and B) = ½ x ½ = ¼

The probability of A and B occurring (two heads) is one-fourth. P (A and B) is the joint probability of events A and B. Where appropriate, the word and can be omitted to simplify the notation and the joint probability can be written simply as P (AB).

To define independence mathematically, we need the symbol P (B½A). The symbol P (B½A) is read “the probability of event B, given that event A has occurred.” P (B½A) is the conditional probability of event B, given that event A has taken place. Note that PB½A) does not mean the probability of event B divided by A – the vertical line followed by A means “given that event A has occurred.”

With independent events:

P (B½A) = P (B) if A, B independent

That is, the probability of event B, given that event A has occurred, is equal to the probability of event B if the two events are independent. With two independent events, the occurrence of the one event does not affect the probability of the occurrence of the second [in like manner, P (A½B) =P (A)].

Two events are dependent if the occurrence of one of the events affects the probability of the occurrence of the second event.

Let’s take an example. (c) C

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Flip a fair coin and determine whether the result is heads or tails. If heads, flip the same coin again. If tails, flip an unfair coin that has a three-fourths probability of heads and a one-fourth probability of tails. Is the probability of heads on the second toss in any way affected by the results of the first toss? The answer here is yes, since the result of the first toss affects which coin (fair or unfair) is to be tossed the second time.

Another example of dependent events involves mutually exclusive events. If events A and B are mutually exclusive, they are dependent. Given that event A has occurred, the conditional probability of B occurring must be zero, since the two events are mutually exclusive. Let’s take up two examples where we clear the difference between joint probabilities, conditional probabilities and unconditional (or marginal) probabilities.

Example 8.2

Assume we have three boxes, which contain red and black balls as follows:

Box 1 : 3 red and 7 black



Suppose we draw from a ball from box 1; if it is red, we draw a ball from box 2. If the ball drawn from box 1 is black, we draw a ball from box 3. Consider the following probability questions about this game:

1. What is the probability of drawing a red ball from box 1? This probability is an unconditional or marginal probability; it is 0.30. (The marginal probability of getting a black is 0.70).

2. Suppose we draw a ball from box 1, and it is red; what is the probability of another red ball when we draw from box 2 on the second draw? The answer is 0.60. This is an example of a conditional probability. That is, the probability of a red ball on the second draw if the draw from box 1 is red is a conditional probability. (c)

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3. Suppose our first draw from box 1 was black; then the conditional probability is 0.80. The draw from box 1 (the conditioning event) is very important in determining the probabilities of red (or black) on the second draw.

4. Suppose, before we draw any balls, we ask the question: What is the probability of drawing two red balls? This would be a joint probability; the event would be a red ball on both draws. The computation of this joint probability is a little more complicated than the above questions, and some analysis will be of value. Computations are as follows:

Figure 8.1

Table 8.2 shows the joint probability of two red balls as 0.18 [i.e. P(R and R) or more simply P(RR), the top branch of the tree]. The joint probabilities may be summarized as follows:

Two red balls P(RR) = 0.18 A red ball on first draw and a black ball on second draw P(RB) = 0.12 A black ball on first draw and a red ball on second draw P(BR) = 0.56 Two black balls P(BB) = 0.14 1.00

Table 8.2: Probabilities Calculations

Marginal Conditional = Joint Event P(A) P(B½A) = P(A and B) RR P(R) = 0.30 P(R|R) = 0.60 P(RR) = 0.18 RB P(R) = 0.30 P(B|R) = 0.40 P(RB) = 0.12 BR P(B) = 0.70 P(R|B) = 0.80 P(BR) = 0.56 BB P(B) = 0.70 P(B|B) = 0.20 P(BB) = 0.14

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Check Your Progress

Fill in the blanks:

1. With two independent events, the occurrence of the one event ___________ affect the probability of the occurrence of the second.

2. The events are dependent if the occurrence of one of the events ___________ two probability of the second event occurrence.

Decision Trees

This is a very useful device for illustrating uncertain situations. The first fork shows that either a red or a black may be drawn, and the probabilities of these events are given. If a red is drawn, we go to box 2, where again a red or black may be drawn, but with probabilities determined by the fact that the draw will take place in box 2. For the second forks, we have conditional probabilities (the probabilities depend on whether a red or a black ball was chosen on the first draw). At the end of each path are the joint probabilities of following that path. The joint probabilities are obtained by multiplying the marginal (unconditional) probabilities of the first branch by the conditional probabilities of the second branch.

Table 8.3 presents these results in a joint probability table; the intersection of the rows and columns are joint probabilities. The column on the right gives the unconditional probabilities (marginals) of the outcome of the first draw; the bottom row gives the unconditional or marginal probabilities of the outcomes of the second draw. Table 8.3 effectively summarizes the tree diagram. Now, let us compute some additional probabilities: 1. Probability of one red and one black ball, regardless of order: = 0.56 + 0.12 = 0.68

2. Probability of a black ball on draw 2:

Explanatory calculation:

Probability of red-black = 0.12

Probability of black-black = 0.14

Probability of black on draw 2 = 0.26

3. Probability of second draw being red if first draw is red:

= 0.60

Activity What is the importance of decision tree ?

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If first draw is red, we are in the R row of

Table 8.3, which totals 0.30. The question is

what proportion is 0.18 of 0.30?

The answer is 0.60; or in terms of the appropriate formula:

2 12 1

1

P(R and R ) 0.18P(R |R ) = = = 0.60P(R ) 0.30

Table 8.3: Joint Probability Table

Example 8.3 We give a further example of the basic probability definitions. A survey is taken of 100 families; information is obtained about family income and about whether the family purchases a speciality food product. The results are shown in Table 8.4.

Table 8.4: Survey of 100 Families, Classified by Income and Buying Behaviour

Low Income (family income below ` 30,000)

High Income (family income of ` 30,000 or more)

Total number of Families

Family is: Buyer of speciality 18 20 38 food products Non-buyer 42 20 62

Total number of families 60 40 100

Suppose a family is to be selected at random from this group.

1. What is the probability that the family selected will be a buyer? Since 38 of the 100 families overall are buyers, the probability is 0.38. Note that this is a marginal or unconditional probability.

2. What is the probability that the selected family is both a buyer and with high income? Note that this is a joint probability. P(Buyer and High income) = 20/100 = 0.20. (c)

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3. Suppose that a family is selected at random and you are informed that it has high income. What is the probability that this family is a buyer? Note that this asks for the conditional probability P (Buyer|High income). Of the 40 families with High income, 20 are Buyers. Hence, the probability is 20/40 = 0.50.

4. Are the events Buyer and High income independent for this group of families? Note from question 1 that P(Buyer) = 0.38; from question 3, P(Buyer|High income) = 0.50. These are not the same. Hence, the two events are dependent. Knowing the family has high income affects the probability that it is a buyer. Another way of expressing this dependence is to say that the percentage of buyer is not the same for the high – and low-income families.

Revision of Probabilities

Having discussed joint and conditional probabilities, let us investigate how probabilities are revised to take account of new information.

Example 8.4 Suppose we do not know whether a particular coin is fair or unfair. If the coin is fair, the probability of a tail is 0.50; but if the coin is unfair, the probability of a tail is 0.10. Assume we assign a prior probability to the coin being fair of 0.80 and a probability of 0.20 to the coin being unfair. The event “fair coin” will be designated A1, and the event “unfair coin” will be designated A2. We toss the coin once; say, a tail is the result. What is the probability that the coin is fair?

The conditional probability of a tail, given that the coin is fair, is 0.50; that is P(tail|A1) = 0.50. If the coin is unfair, the probability of a tail is 0.10; P(tail|A2) = 0.10

Let us compute the joint probability P (tail and A1). There is an initial 0.80 probability that A1 is the true state; and if A1 is the true state, there is a 0.50 conditional probability that a tail will result. The joint probability of state A1 being true and obtaining a tail is (0.80 x 0.50) = 0.40. Thus:

P(tail and A1) = P(A1) × P(tail|A1) = 0.80 x 0.50 = 0.40

The joint probability of a tail and A2 is equal to:

P(tail and A2) = P(A2) × P(tail|A2) = 0.20 × 0.10 = 0.02

Activity What is combination means?

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A tail can occur in combination with the state “fair coin” or in combination with the state “unfair coin”. The probability of the former combination is 0.40; of the latter, 0.02. The sum of the probabilities gives the unconditional probability of a tail on the first toss; that is, P(tail) = 0.40 + 0.02 = 0.42:

P(tail and A2) = 0.02 P(tail and A1) = 0.40

P(tail) = 0.42

If a tail occurs, and if we do not know the true state, the conditional probability of state A1 being the true state is:

11

P(tail and A ) 0.40P(A |tail) = 0.95P(tail) 0.42

= =

Thus, 0.95 is the revised or posterior probability of A1, given that a tail has occurred on the first toss. Similarly:

21

P(tail and A ) 0.02P(A |tail) = 0.05P(tail) 0.42

= =

In more general symbols:

ii

P(A and B)P(A |B) =P(B)

Conditional probability expressed in this form is known as Bayes theorem. It has many important applications in evaluating the worth of additional information in decision problems.

In this example, the revised probabilities for the coin are 0.95 that it is fair and 0.05 that it is unfair (the probabilities were initially 0.80 and 0.20). These revised probabilities exist after one toss when the toss results in a tail. It is reasonable that the probability that the coin is unfair has decreased, since a tail appeared on the first toss, and the unfair coin has only a 0.10 probability of a tail.

Check Your Progress

Fill in the blanks:

1. Decision trees is a very useful device for illustrating _____________ situations.

2. The joint probabilities are obtained by ________ the marginal probabilities of the first branch by conditional probabilities of the second branch. (c)

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Combinations

It is sometimes necessary to know the number of different ways in which ‘r’ objects can be selected from n objects without regard to sequence. For instance, the number of different permutations of given cards from a pack of 52 cards is 52P5. The same five cards dealt in different sequences are different permutations. In practice, the interest is usually in the number of different possible hands irrespective of the sequence in which they were dealt. To obtain this, one must divide by the number of ways in which five cards can be arranged among themselves, which is 5P5.

Each different way of selecting r objects from n without regard to the sequence in which they were selected is termed a combination, and each such combination consists of rPr permutations. The number of combinations is represented by nCr, and so nCr divided by

rPr, since rPr is r! :

nr

n!C(n r)!r !

=−

It will be seen that nCr works out the same as nCn-r. This is reasonable, since the number of ways of selecting five cards from a pack of 52 is obviously the same as the number of ways of selecting 47 cards and leaving a hand of five cards behind.

Example 8.5

In how many different ways three bolts can be selected from a box containing eight bolts?

Solution:

The answer is 8C3. It is convenient to adopt the practice of using dots in place of multiplication signs when several numbers are all multiplied together:

83

8! 8.7.6C 565!3! 1.2.3

= = =

There is no need to write out the factorials in full. The number of integers to be multiplied in both numerator and denominator is the smaller of r and (n-r). The denominator always cancels completely, since nCr must be an integer.

Activity What is the importance of decision tree ?

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Example 8.6 If three bolts are selected at random from a box containing six sound and two faulty bolts, what is the probability of obtaining (i) three sound, (ii) two sound and one faulty, (iii) one sound and two faulty bolts?

Solution: The word ‘random’ indicates that all the 56 different ways of selecting three bolts from a total of eight bolts are equally likely, and so the classical definition of probability can be applied. The number of ways of obtaining three sound bolts is 6C3, which is 20, and dividing this by 56 gives 0.357.

The number of ways of obtaining two sound bolts is 6C2, which is 15. Each of these combinations can be associated with either of the 2C1 ways of obtaining one faulty bolt, and so 15 is multiplied by two and divided by 56 to give 0.536.

Following the same principles, the probability of obtaining one sound and two faulty bolts is:

6

8

21 2

3

C C 6 1 3 0.10756 28C×= = =

A check confirms that the three probabilities calculated of possible combinations of sound and faulty bolts total 1, which must be so since one of the three results must occur.

Probabilities may be similarly calculated in any situation involving permutations or combinations provided that, all the outcomes are equally likely. Using example 8.4, the probability of choosing the winning list from ten models in a fashion competition is 1 divided by 1,51,200 if six models have to be listed, provided that no skill is involved.

Check Your Progress

Fill in the blanks

1. There is _________need to write out the factorials in full.

2. Probabilities may be calculated in any situation involving permutations or combinations provided that, all the outcomes are __________likely (c)

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Summary

The probabilities assigned to various events on the basis of the conditions of the experiment or by actual experimentation or past experience or on the basis of personal judgement are called prior probabilities.

Lesson End Activity

A and B stand in a ring with 10 other persons. If arrangement of 12 persons is at random, find the chance that there are exactly three persons between A and B.

Keywords

A Sample Event: The basic possible outcome of an experiment, it cannot be broken down into simpler outcomes.

Event: Any set of all possible outcomes or simple events of an experiment.

Probability: A numerical measures of the likelihood of occurrence of an uncertain event.

Conditional Probability: The probability of an event occurring, given that another event has occurred.


1. Define the term 'probability' by (a) The Classical Approach, (b) The Statistical Approach. What are the main limitations of these approaches?

2. Discuss the axiomatic approach to probability. In what way it is an improvement over classical and statistical approaches?

3. What do you mean by objective and subjective probability?

4. Discuss the basic statements of probability.

5. Distinguish between objective probability and subjective probability. Give one example of each concept.

6. State and prove theorem of addition of probabilities for two events when (a) they are not independent, (b) they are independent. (c)

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Further Readings

Books




Web Readings




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Unit 9

Random Variables and Probability Distributions


Random Variables

Binomial distribution

Poisson distribution

Introduction

A probability function is a rule that assigns probabilities to each element of a set of events that may occur. In this unit, we will discuss random variables and probablity distribution.

Random Variables

We can assign a specific numerical value to each element of the set of events, a function that assigns these numerical values is termed a random variable. The value of a random variable is the general outcome of a random (or probability) experiment. It is useful to distinguish between the random variable itself and the values that it can take on. The value of a random variable is unknown until the event occurs (i.e., until the random experiment has been performed). However, the probability that the random variable will be any specific value is known in advance.

The probability of each value of the random variable is equal to the sum of the probabilities of the events assigned to that value of the random variable.

For example, suppose we define the random variable Z to be the number of heads in two tosses of a fair coin. Then the possible values of Z, and the corresponding probabilities, are:

Activity What is probability distribution?

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Possible Values of Z Probability of Each Value

0 ¼

1 ½

2 ¼

Random variables can be grouped into probability distribution, which can be either discrete or continuous. Discrete probability distributions are those in which the random variable can take on only specific values. The table above is an example of such a distribution since the random variable Z can be only 0, 1, or 2. A continuous probability distribution is one in which the value of the random variable can be any number within some given range of values – say, between zero and infinity. For example, if the random variable was the height of members of a population, a person could be 5.3 feet, 5.324 feet, 5.32431 feet, and so on, depending on the ability of instruments to measure. Some additional examples of random variables are shown in the Table 9.1.

A discrete probability distribution is sometimes called a probability mass function (p.m.f.) and a continuous one is called a probability density function (p.d.f.) Graphs of the two types of distributions are shown in Figure 9.1 and 9.2.

For a discrete distribution, the height of each line represents the probability for that value of the random variable. For example, 0.30 is the probability that tomorrow’s demand will be 0.2 tons in figure. For a continuous random variable, the height of the probability density function is not the probability for an event. Rather, the area under the curve over any interval on the horizontal axis represents the probability of taking on a value in that interval.

Table 9.1: Examples of Random Variables

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Figure 9.1: Discrete Probability Distribution

(Probability Mass Function or p.m.f.)

Now let us take a look at the first discrete probability distribution, the binomial probability distribution and how it is derived. The binomial probability distribution is the base on which several other probability distributions are based upon.

Figure 9.2: Continuous Probability Distribution

Check Your Progress

Fill in the blanks:

1. The binomial probability distribution is the base on which several other probability distributions are _______________.

2. ___________ variables can be grouped into probability distribution, which can be either discrete or continuous.

Activity What is probability mass function? What is continuous probability distribution?

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The Bernoulli Process

Understanding of the Bernoulli process is necessary before we understand the binomial distribution. The characteristics of a Bernoulli process are described below:

1. The outcomes or results of each trial in the process are characterized as one of two types of possible outcomes, such as:

a. Success, failure.

b. Yes, no.

c. Heads, tails.

d. Zero, one.

2. The probability of the outcome of any trial is “stable” and does not change throughout the process. For example, the probability of heads, given a fair coin, is 0.50 and does not change, regardless of the number of times the coin is tossed.

3. The outcome of any trial is independent of the outcome of any previous trial. In other words, the past history of the process would not change the probability assigned to the next trial. In our coin example, we would assign a probability of 0.50 to the next toss coming up heads; even if we had recorded heads on the last 10 trials (we assume the coin is fair).

4. The number of trials is discrete and can be represented by an integer such as 1, 2, 3, and so on.

Given a process, we may know that it is Bernoulli, but we may or may not know the stable probability characteristic of the process. With a fair coin, we may know the process is Bernoulli, with probability 0.50 of a success (say heads) and probability 0.50 of a failure (tails). However, if we are given a coin and told it is not fair, the process (flipping the coin) may still be Bernoulli, but we do not know the probability characteristic. Hence, we may have a Bernoulli process with a known or unknown probability characteristic.

Many business processes can be characterized as Bernoulli for analytical purposes, even though they are not true Bernoulli in every respect. If the “fit” is close enough, we may assume that the Bernoulli process is a reasonable characterization. Let us discuss some examples.

Activity What is Bernoulli?

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Suppose we are concerned with a production process where a certain part (or product) is produced on a machine, we may be interested in classifying the parts as “good” or “defective”, in which case the process may be Bernoulli. If the machine is not subject to fast wear and tear – that is, if a setting will last for a long run of parts 0 the probability of good parts may be sufficiently stable for the process to qualify as Bernoulli. If, on the other hand, more defects occur as the end of the run approaches, the process is not Bernoulli. In many such processes, the occurrence of good and defective parts is sufficiently stable (no pattern over time is observable) to call the process Bernoulli. The probability of good and defective parts may remain stable through a production run, but it may vary from run to run (because of machine setting, for example). Here, the process could still be considered Bernoulli, but the probability of a success (or failure) will change from run to run.

A different example of a Bernoulli process is a survey to determine whether or not consumers prefer liquid to powdered soaps. The outcome of a survey interview could be characterized a “yes” (success) or “no” (failure) answers to the question. If the sample of consumers was sufficiently randomized (no pattern to the way in which the yes or no answers occur), Bernoulli (with an unknown probability) may be a useful description of the process.

Note that if the probability of a success in a Bernoulli process is 0.50, the probability of a failure is also 0.50 (since the probabilities of the event happening and the event not happening add to one). If the probability of a success is p, the probability of a failure is (1 – p).

Check Your Progress

Fill in the blanks:

1. A different example of a Bernoulli process is a __________ to determine whether or not consumers prefer liquid to powdered soaps.

2. Many business processes can be characterized as Bernoulli for _______ purposes.

The Binomial Theorem

Let us start by using an example.

Example 9.1

The probability that a salesman makes a sale on a visit to a prospect is 0.2. (c)

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What is the probability, in 2 visits, of:

making no sales

making one sale

making two sales

Solution:

Here p = probability of sale = 0.2

And q = probability of no sale = 1 – 0.2 = 0.8

The various outcome possibilities are

Visit 1 Visit 2 Probabilities

Sale Sale i.e. p x p = p2 = 0.22 = 0.04

Sale No sale i.e. p x q = 0.02 x 0.8 = 0.16

No sale Sale i.e. q x p = 0.8 x 0.2 = 0.16

No sale No sale i.e. q x q = q2 = 0.82 = 0.64

1.00

Thus P (no sales) = 0.64

P (one sale) = 0.32

P (two sales) = 0.04

In this simple example, it is easy to show the whole process but this becomes lengthy and cumbersome where the number of trials (visits, in the above example) becomes larger.

Fortunately there is a simpler approach which is by the expansion of the binomial expression. The general form of the binomial expression is

(p + q)n

Where p = probability of an event occurring

q = probability of an event not occurring

and n = number of trials

In the above example p = probability of a sale, i.e., 0.2, q = probability of no sale, i.e., 0.8 and n = number of visits i.e. 2.

Use the binomial expansion to confirm the probabilities.

∴ (p + q)2 = p2 + 2pq + q2

= 0.22 + 2(0.2 x 0.8) + 0.82

= 0.04+0.32 + 0.64, i.e., the values previously obtained. (c) C

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Where n becomes larger it is useful to be able to calculate the coefficients of each part of the expansion in a direct manner rather than writing out the whole expansion.

It is easy to show by direct multiplication that:

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

These are called expansions of the expressions on the left, which are called binomial expressions, because they each contain two terms within the bracket. The binomial theorem is concerned with the general form of the expansion of (a+b).

It is not difficult to see from the above examples that the first term in the expansion of (a + b)n is an, the last term is bn, and there are (n + 1) terms in all. In general, the jth term consists of an+1–j bj-1 preceded by an appropriate coefficient, where j takes all integral values from 1 to (n + 1).

To find the appropriate coefficient, one can investigate why the coefficient of a2 b2 in the expansion of (a + b)4 works out as 6. Writing out the four terms, which have to be multiplied together:

(a + b) (a + b) (a + b) (a + b)

It is easy to see that multiplying a in the first bracket by a in the second, b in the third, and b in the fourth will give the term a2b2. But the same result is obtained by taking the sequence a, b, a, b, from the four brackets, or by taking any of the other possible sequences which include ‘a’ from any two of the brackets and ‘b’ from the remaining two brackets.

The number of different ways of selecting two brackets from four is

4C2 is six. The brackets selected are the ones which contribute the symbol a, whilst the remaining brackets contribute the symbol b.

We can now consider the general term in the expansion of (a + b)n. It is usual to refer to the jth term as the (r + 1)th term, so that the (r + 1)th term has the symbol an-r br and r takes all integer values from 0 to n. The coefficient in the (r + 1)th term is then the number of ways of selecting (n-r) brackets from n, which is nCn –r.

It would be equally correct to select the r brackets which are to provide the symbol b, letting the remaining (n - r) brackets provide the symbol a. The coefficient is then nCr, but it has already been (c)

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seen earlier that nCr is always equal to nCn–r and so there is no contradiction. It follows that the series of coefficients is always symmetrical; the coefficient of a n–r br is always the same as the coefficient of ar bn-r.

Hence the binomial theorem states that for any positive integer n:

(a + b)n = an + nC1an-1b + nC2an-2b2 +....+ nCran–rbr +....+nCn-1abn-1 + bn

Sometimes nC1 and nCn-1 are written simply as n. On the other hand, one can complete the uniformity of the expansion by writing the coefficients in the first and last terms as nC0 , nCn respectively, these both being equal to 1.

Example 9.2

For components of a certain type, the probability that a component is faulty when it leaves the production line is 0.05. If 10 components are selected at random, what is the probability of obtaining two faulty components?

Solution: This is the same problem as in example 9.2 except that the size of the batch must now be regarded as infinite. It is not difficult to see that the previous approximate formula now becomes exact, so that the answer is 0.0746.

The exact probability distribution for the number of faulty components in example 9.2 is called the hyper geometric distribution. It is not very important in practice. If a sample is drawn from a large batch, the binomial distribution is usually a good enough approximation. If only a small batch is involved, it can be examined in its entirety and then no probability problem arises.

In some type of problems the binomial distribution can be derived directly from a consideration of probabilities without reference to batch sizes.

Example 9.3

The probability of meeting the buyer at a random visit to a certain firm is 0.05. If a salesman makes 10 random visits, what is the probability that he will meet the buyer on two occasions? (c) C

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Solution: The probability that the salesman will meet the buyer on both the first two visits is 0.05 multiplied by 0.05. The probability law of multiplication states that the probability that both of two events will occur is the product of their separate probabilities, provided that the events are independent. Two events are said to be independent if the fact that one has occurred makes no difference to the probability that the second will occur. We can assume that this will apply to random visits by the salesman.

Further visits are again independent. The probability of not meeting the buyer is 0.95. Multiplying all the probabilities, it follows that the probability of meeting the buyer on the first two occasions and not on any of the following eight occasions is:

(0.05)2 (0.95)8

The probability of meeting the buyer on none of the first eight visits but on both of the last two visits is similarly found to be:

(095)8 (0.05)2

This is the same as the previous probability, and in fact any arrangement of two successes among the 10 visits will have the same probability.

The probability law of addition states that, the probability that either of the two events will occur is the sum of their separate probabilities, provided that they are mutually exclusive; that is, provided that it is impossible for both to occur. This applies to any two different arrangements of two successes among 10 visits, and so adding the probabilities for the 10C2 different arrangements gives the total probability of two successes in 10 visits as:

10C2(0.95)8 (0.05)2

which is again 0.0746

In this example the concept of probability was not derived from counting equally likely events. The figure 0.05 indicates that on an average one will meet the buyer five times in every 100 visits. This does not mean that there will be exactly five successes in each 100 visits, but that the relative frequency of successes will tend to the figure 0.05 in an unlimited number of visits. So probability is here defined as the limiting relative frequency of a success; this is the definition of probability most widely applied by statisticians. (c)

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Check Your Progress

Fill in the blanks:

1. The concept of probability was not derived from counting ___________ events.

2. Probability is also defined as the limiting relative frequency of a ___________.

Probability Distributions

A probability distribution is a rule that assigns a probability to every possible outcome of an experiment. An event whose numerical value is determined by the outcome of an experiment is called a variate or often a random variable.

There are two kinds of probability distributions, discrete and continuous. Discrete probability distributions are those where only a finite number of outcomes are possible. For example, the throw of a dice has a discrete probability distribution as only six outcomes are possible and are known beforehand. Continuous probability distributions are those which represent continuously variable random variables.

The Binomial Probability Distribution

The Binomial Probability Function If the assumptions of the Bernoulli process are satisfied and if the probability of a success on one trial is p, then the probability distribution of the number of successes, r, in n trials, is a binomial distribution.

Binomial Probabilities on Spreadsheets

The Excel and Quattro spreadsheet programs have a function that can be used for evaluating binomial probabilities, both individual terms and cumulative probabilities. The form of the function is:

= BINOMDIST(r, n, p, 0 or 1)

Where r is the number of successes, n is the number of trials, and p is the probability of success on each trial. The last term is either a zero or one; if a zero is entered, the individual binomial term is given; if a one is used, the cumulative value (of the £ type) is given. Suppose, for example, we want the probability of exactly three (c) C

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successes in five trials, with a probability of success of p = 0.4 - P(r = 3½n = 5, p = 0.4). This is:

= BINOMDIST(3,5,0.4,0), which equals 0.2304

For the probability of 3 or fewer successes in five trials, with p = 0.4 - P(r £ 3½n = 5, p = 0.4)

= BINOMDIST (3, 5, 0.4, 1), which equals 0.9130

Example 9.4 Suppose we plan to toss a fair coin three times and would like to compute the following probabilities:

a. The probability of three heads in three tosses.

b. The probability of two or more heads in three tosses.

c. The probability of fewer than two heads in three tosses.

Possible Outcomes

HHH

HHT

HTH

THH

TTH

THT

HTT

TTT

In this example, the Bernoulli process p is 0.50, and a head constitutes a success. The number of trials (n) is three.

The first probability is the probability of three heads (successes) in three tosses (three trials), given that the probability of a head on any one toss is 0.50. This probability can be written as follows:

P(r = 3½p = 0.50, n = 3) =?

Where P = Probability, r = Number of successes, n = Number of trials, and p = Probability of success on any one trial. The left side of the equation should be read “the probability of three successes, given a process probability of 0.50 and three trials.”

In answering the probability questions, let us first list all the possible outcomes of the three trials and compute the probabilities

The above probabilities can also be calculated using the equation. You can try it yourself. (c)

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Example 9.5 (Unsolved) A very large lot of manufactured goods are to be sampled as a check on its quality. Suppose it is assumed that 10 percent of the items in the lot are defective and that a sample of 20 items is drawn from the lot. What are the following probabilities:

1. Probability of exactly zero defectives in the sample?

2. Probability of more than one defective in the sample?

3. Probability of fewer than two defectives in the sample?

Mean and Standard Deviation of the Binomial Distribution

The mean of a binomial distribution is found by multiplying the probability of the event in which we are interested by n, the number of trials:

Mean = np

This value is the same as the expected value as previously discussed.

The variance of a binomial distribution is calculated as:

Variance = npq so that the standard deviation is npε

Characteristics of the Binomial Distribution

1. It is a discrete distribution of the occurrences of an event with two outcomes - success or failure, good or bad.

2. The trials must be independent of one another. This assumption implies sampling from an infinite population. Sampling with replacement fulfils this requirement, but where sampling without replacement is used, the binomial distribution is still useful provided that the sample size is less than 20.

3. As the number of trials grows and if p = 0.5 then the binomial distributions approaches the normal distribution. For the normal distribution to be an appropriate approximation, np should be > 5.

Example 9.6 (unsolved) Components are placed into bins containing 100. After inspection of a large number of bins the average number of defective parts was found to be 10 with a standard deviation of 3. (c) C

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Assuming that the same production conditions continue, except that bins containing 300 were used:

1. What would be the average number of defective components per larger bin?

2. What would be the standard deviation of the number of defectives per larger bin?

3. How many components must each bin hold so that the standard deviation of the number of defective components is equal to 1% of the total number of components in the bin?

Check Your Progress

Fill in the blanks:

1. The Excel and Quattro spreadsheet programs have a function that can be used for evaluating binomial probabilities, both __________ terms and __________ probabilities

2. Mean = ______________

Poisson Distribution

The probability of getting specified number of successes from a repeated number of trials can be obtained with the help of binomial distribution provided the probability of success or failure is known. However, the probability of success or failure is finite in this case. As the number of trials approaches infinity, Poisson distribution is the limit of the binomial distribution.

Here c is the number of defects per sample, a is the expected number of defects per sample, and e = 2.71828, the base of the natural logarithm.

To explain this, we may think about the defects as being distributed over the area of a surface, any unit being defective if it contains one or more defects. If the surface area is divided into very small units, so that no unit of the area has more than one defect, the distinction between defect c and defective d disappears. As the number n of units increases, nP remaining constant, probability P must get smaller and smaller, and the binomial distribution gradually approaches the Poisson.

Activity What is probability distribution need for?

(c) C

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Table 9.2: Calculation of a Poisson Distribution for c = 0 through 5 and a = 0.5.

The Poisson distribution is useful in ways other than as an approximation to the binomial; for example, the analysis of queues, or arrival and waiting line patterns at bridges and airports, tool distribution points in factories, etc.

Characteristics of Poisson Distribution

(a) It is a discrete distribution and is a limiting form of the binomial distribution when n is large and p or q is small.

(b) Mean and variance are equal.

(c) It is usually, definitely positively skewed but cannot be negatively skewed

(d) As n becomes very large the Poisson distribution approximates to the normal distribution

(e) The mean = np

Applications of Poisson Distribution

The Poisson distribution is similar to the binomial but is used when n, the number of items or events, is large or unknown and p, the probability of an occurrence, is very small relative to q, the probability of non-occurrence. A rule of thumb is that the Poisson distribution may be used when n is greater than 50 and the mean np is less than 5. Some examples follow but it is important to realize that the Poisson distribution only applies when the events occur randomly, i.e., they are independent of one another. (c) C

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Example 9.7 Customers arrive randomly at a service point at an average rate of 30 per hour. Assuming a Poisson distribution, calculate the probability that:

(a) no customer arrives in any particular minute.

(b) exactly one customer arrives in any particular minute.

(c) two or more customers arrive in any particular minute.

(d) Three or fewer customers arrive in any particular minute.

Solution:

The time interval to be used is one minute with a mean of 30/60 = 0.5

1. P (no customer) = 0.6065 from Table VI(a)

2. P (1 customer) = 0.3033 from Table VI(a)

3. P (2 or more) = 1 - 0.9098

= 0.0902

The value of 0.9098 is the cumulative probability of 1 or fewer customers arriving in a particular minute. As the sum of the probability of every possible number of arrivals equals 1, the probability of 2 or more = 1 - P(1 or fewer).

4. P(3 or fewer) = 0.9982

Example 9.8

A firm buys springs in very large quantities and from past records, it is known that 0.2% is defective. The inspection department sample the springs in batches of 500. It is required to set a standard for the inspectors so that if more than the standard number of defectives is found in a batch the consignment can be rejected with at least 90% confidence that the supply is truly defective.

How many defectives per batch should be set as the standard?

Solution:

With 0.2% defective and a sample size of 500 m= 500 × 0.2% = 1. (c) C

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To find the probability of 0, 1, 2, 3, etc. or more defectives the respective probabilities are deducted from 1.

P(0 or more defectives) = certainty = 1

P(1 or more defectives) = 1 - 0.3679 = 0.6321




These probabilities mean, for example, that there is a 26.42% chance that 2 or more defectives will occur at random in a batch of 500 with a 0.2% defect rate. If batches with 2 or more were rejected then there can be 73.58% (1 - 0.2642) confidence that the supply is defective.

As the firm wishes to be at least 90% confident, the standard should be set at 3 or more defectives per batch. This level could only occur at random in 8.03% of occasions so that the firm can be 91.97% confident that the supply is truly defective.

Check Your Progress

Fill in the blanks:

1. In Poisson distribution Mean and variance are _________.

2. The Poisson distribution is similar to the binomial but is used when___________

The Normal Probability Distribution

Till recently we have been making curves which illustrated some of the forms that a frequency distribution may assume. These curves were based upon data of a few tens or hundreds of cases; each was a sample drawn from a much larger, possibly infinite, universe. Being a sample, a given curve would not necessarily have exactly the same shape as the curve for the universe, but if the sample is properly selected, the curve for the sample will tend to be of the same general shape as the curve for the universe.

The normal curve represents a distribution of values that may occur, under certain conditions, when chance is given full play. In every case the necessary conditions include the existence of a large (c) C

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number of causes, each operating independently in a random manner.

Graphically, the normal distribution looks like a bell shaped curve:

There are certain properties of the normal distribution that you can notice from the above curve:

1. It is symmetrical on both sides of the mean, i.e., mean, median and mode coincide at the central value.

2. The curve never touches the x-axis and extends to -infinity on the left hand side to +infinity on the right hand side.

The normal distribution is an extremely important distribution. It is easier to manipulate mathematically than many other distributions and is a good approximation for several of the others. In many cases, the normal distribution is a reasonable approximation for binomial probability distribution for business decision purposes; and in the following units, we shall use the normal distribution in many of the applications. Despite its general application, it should not be assumed that every process can be described as having a normal distribution.

The normal distribution is a function of z, the standard normal variate, and is defined as:

Here the value of z is given by:

The normal distribution is completely determined by its expected value or mean (denoted by m) and standard deviation (s); that is, once we know the mean and standard deviation, the shape and location of the distribution is set. The curve reaches a maximum at the mean of the distribution. One half of the area lies on either side of the mean. The greater the value of s, the standard deviation, the more spread out the curve?

With any normal distribution, approximately 0.50 of the area lies within ±0.67 standard deviation from the mean; about 0.68 of the area lies within ±1.0 standard deviations; and 0.95 of the area lies with ±1.96 standard deviation.

Normal Probabilities on Spreadsheets

The Excel and Quattro spreadsheet function NORMSDIST provides exactly the same values as in the Appendix 3. That is, for any value of the standardized value of Z, it provides the left tail cumulative normal probability. For example, for P(Z ≤ 0.67), use: (c)

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NORMSDIST (0.67) = 0.7486

The next logical question might be: Given a normal distribution with mean 4 and standard deviation s, what percentage of the total area lies to the right or the left of a given X value? Alternatively, we may ask: What is the probability of obtaining a value of X that is as large as or larger than one specified? Since there are infinitely many normal curves, each depending upon a particular combination of mean and SD, the answer would vary from normal curve to normal curve. The problem would be much less difficult if we could cause all the normal distributions to have the same mean and SD.

This is exactly what we do when we standardize distributions; i.e., we cause them to have a mean of zero and a variance of one. Thus, if we transform the given X value into a z value, where z is defined as given above, and call Q(z) the probability of obtaining a value of z that is equal to or larger than the one specified, tables are already calculated to allow us to find Q(z), given z. Such a table is given in Appendix 3.

Example 9.9 Assume that your working hours X are distributed normally with m = 5 and s = 2. What is the probability of your working 9 hours or more than 9 hours?

Solution: First we standardize the specified X value: z=

Looking in the Appendix 3, we find that Q(z), the probability of obtaining a value of z as large as or larger than specified, is

Q(z) = Q(2) = 0.02275

Alternatively, we may say that 2.275 per cent of the area in the distribution is to the right of z = 2. This means that there is 2.275 per cent probability that you would be working for 9 or more than 9 hours in the day.

This could be looked at in another way. What is the probability that you would be working less than 9 or 9 hours in a day? The solution of this question can be found by subtracting the above value from 1, so that we get the value which lies on the left of the 9 hour line and not on the right in the normal distribution. The (c) C

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answer is 0.97725 or 97.725 per cent probability that you would be working for 9 or less than 9 hours on any given day.

Example 9.10 An assembly line contains 2,000 components each one of which has a limited life. Records show that the life of the components is normally distributed with a mean of 900 hours and a standard deviation of 80 hours.

1. What proportion of components will fail before 1,000 hours? 2. What proportion will fail before 750 hours? 3. What proportion of components fails between 850 and 880

hours? 4. Given that the standard deviation will remain at 80 hours

what would the average life have to be to ensure that not more than 10% of components fail before 900 hours?

Solution: 1. Here the value being investigated, 1000 hrs, is 1.25 standard

deviation away from the mean of 900 hours. If Appendix 3 is examined it will be seen that the value for a z

score of 1.25 is 0.3944. As one half of the distribution is less than 900, the proportion which fails before 1,000 hours is 0.5 + 0.3944 = 89.44%.

If required this could be expressed as the number of components which are expected to fail, thus

2,000 × .8944 = 1788.8 which be rounded to 1789 2. From the tables in appendix 3 we obtain the value 0.4696. In

this case as we require the proportion that will fail before 750 hours, the table value is deducted from 0.5.

∴ Proportion expected to fail before 750 hours. = 0.5 = 0.4696 = 0.0304, i.e., 3.04% 3. When it is required to find the proportion between two values

(neither of which is the mean) it is necessary to use the tables to find the proportion between the mean and one value and the proportion between the mean and the other value. Then find the difference between the two proportions.

Which gives a proportion of 0.2340? Which gives a proportion of 0.0987? ∴ Proportion between 850 and 880 is 0.2340 - 0.0987 = 0.1353, i.e., 13.53% (c)

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Note: This part of the example illustrates the proportion between two values on the same side of the means. If the two values are on opposite sides of the mean, the calculated proportions would be added.

This problem is the reversal of the earlier questions based on the same principles. The earlier problems started with the mean and standard deviation, found the z score and then the proportion from the tables. We now start with the proportion and work back, through the tables, to find a new mean value. If not more than 10% should be under 900 it follows that 90% of the area of the curve must be greater than 900.

Bearing in mind that the tables only show values for half the distribution (because both halves are identical) we have to look in the tables for a value close to 0.4 (i.e. 0.9 - 0.5).

It will be seen that three is a value in the Table in Appendix 3 of 0.3997 i.e. virtually 0.4. This value has a z score of 1.28.

Thus

∴ 102.4 = mean - 900

∴ mean = 1002.4 hours.

Thus if the mean life of the components is 1002.4 hours with a standard deviation of 80 hours, less than 10% of the components will fail before 900 hours.

Hints on what Type of Distribution Applies

It is not always easy to recognize when a Normal, Binomial or Poisson distribution should be used. The following hints might help.

Binomial Distribution

(a) Outcomes have discrete values and do not have continuous ranges of possible values. For example, in dealing with people, there can only be discrete values; 1, 2, 3 etc., 2.35 people is not possible.

(b) There are only two possible conditions; good/bad, black/white, male/female, acceptable/not acceptable and so on.

(c) The probability of an item having one of the two possible conditions is p, thus the probability of the item having the (c)

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other condition is (1 - p). (1 - p) is usually referred to as q, so that (p + q) = 1.

(d) When the number of items, n, is large and p is not close to 0 or 1 so that the distribution is approximately symmetric, the binomial probabilities can be approximated using a Normal Distribution with the same mean (m = np) and standard deviation (s = Önpq).

Poisson distribution

Similar to a binomial distribution, but used for rare events:

(a) The number of items, n, is large; say greater than 50.

(b) p is small in relation to q so that np (the mean of a Poisson distribution) is less than, say, 5.

Normal Distribution

The most commonly applied probability distribution.

(a) It applies to variables with a continuous range of possible values. Examples include: time, weights, distances, sizes, growth rates, etc.

(b) Where the quantities are large, the Normal Distribution can also be used for discrete variables (see note above binomial distribution).

Check Your Progress

Fill in the blanks:

1. Poisson distribution is ______ to a binomial distribution, but used for rare events.

2. Normal Distribution is applies to variables with a ________ range of possible values.

Summary

A probability distribution is a rule that assigns a probability to every possible outcome of an experiment. An event whose numerical value is determined by the outcome of an experiment is called a variate or often a random variable. There are two kinds of probability distributions, discrete and continuous. Discrete probability distributions are those where only a finite number of

Activity What is the difference between poisson distribution and normal distribution?

(c) C

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outcomes are possible. Continuous probability distributions are those which represent continuously variable random variables.

Lesson End Activity

Two different digits are chosen at random from the set 1, 2, 3, 4, 5, 6, 7, 8. Find the probability that sum of two digits exceeds 13.

Keywords

Probability Distribution: A probability distribution is a rule that assigns a probability to every possible outcome of an experiment.

Random Variable: An event whose numerical value is determined by the outcome of an experiment is called a variate or often a random variable.

Discrete Probability Distributions: Discrete probability distributions are those where only a finite number of outcomes are possible.

Continuous Probability Distributions: Continuous probability distributions are those which represent continuously variable random variables


1. Explain the meaning of conditional probability. State and prove the multiplication rule of probability of two events when (a) they are not independent, (b) they are independent.

2. What is the probability of getting a sum of 2 or 8 or 12 in single throw of two unbiased dice?

3. Two cards are drawn at random from a pack of 52 cards. What is the probability that the first is a king and second is a queen?

4. What is the probability of successive drawing of an ace, a king, a queen and a jack from a pack of 52 well shuffled cards? The drawn cards are not replaced.

5. 5 unbiased coins with faces marked as 2 and 3 are tossed. Find the probability of getting a sum of 12.

6. If 15 chocolates are distributed at random among 5 children, what is the probability that a particular child receives 8 chocolates? (c)

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7. A batch of 5,000 electric lamps has a mean life of 1,000 hours and a standard deviation of 75 hours. Assume a Normal Distribution.

(a) How many lamps will fail before 900 hours?

(b) How many lamps will fail between 950 and 1,000 hours?

(c) What proportion of lamps will fail before 925 hours?

(d) Given the same mean life, what would the standard deviation have to be to ensure that not more than 20% of lamps fail before 916 hours?

8. A mail-order company is analyzing a random sample of its computer records of customers. Among the results are the following distributions:

Size of order Number of customers No. of lubes April September Less than 1 8 4 1 and less than 5 19 18 5 and less than 10 38 39 10 and less than 15 40 69 15 and less than 20 22 41 20 and less than 30 13 20 30 and over 4 5 Total 144 196

Required:

(a) Calculate the arithmetic mean and standard deviation order size for the April sample;

(b) Find 95% confidence limits for the overall mean order size for the April customers and explain their meaning;

(c) Compare the two distributions, given that the arithmetic mean and standard deviation for the September sample were 13.28 and 7.05 orders respectively.

9. The chief accountant of the Hotels Group is analyzing the profitability of the group’s smallest hotel, the Unity. The Unity has 120 bedrooms, each of which can be let as either a single or a double room. The price of a single room is ` 45 per night whereas for a double room it is ` 35 per person per night. The hotel’s other main source of revenue is the restaurant where the average price of an evening meal is ` 9. (c)

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A proportion of residents can be expected to have an evening meal in the restaurant, and experience has shown that for the Unity this proportion is usually about 60%.

Required:

(a) Develop an expression for the average daily revenue (R) from residents at the hotel, including both the letting of rooms and evening meals in the restaurant.

Assume that:

i. the overall proportion of rooms occupied is p, and

ii. a proportion q of the rooms that are occupied are let as single rooms.

(b) The proportion (q) of rooms which are let as single rooms is usually between 10% and 30%, and it can be assumed that this is independent of the overall proportion of rooms occupied (p).

Determine the average daily revenue (R) in terms of the parameter p for values of q of 10%, 20% and 30% and plot these three functions on a graph. What is the lowest percentage occupancy (p) which will yield an average daily revenue of ` 8,000, and for which of the given values of q does this occur?

(c) What is the overall relationship between R and p if the percentage of rooms let as single rooms is given by the following probability distribution?

q Probability

10% 0.3

20% 0.5

30% 0.2

(d) Alternatively, if q has a normal distribution with a mean of 19% and a standard deviation of 7%, determine a 95% confidence interval for R in terms of the parameter p. (For the purposes of this calculation, you may ignore any variability in the proportion of residents who take an evening meal in the restaurant.)

10. Castrol Company Limited is planning to introduce a new motor lube. The company’s marketing department estimates that the prior distribution for likely sales is normal with a (c)

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mean of 10,000 tonnes. In addition it has determined that there is a probability of one half that the likely sales will lie between 8,000 and 12,000 tonnes.

The lube will sell for ` 10 per 100 gm but the publishing company pays the parent company 10% of revenue in royalties and the fixed costs of printing and marketing the lube are calculated to be ` 25,000. Using current facilities, the variable production costs are ` 4 per 100 gm. However, the Castrol Company has the option of hiring a special machine for ` 14,000 which will reduce the variable production costs to ` 2.50 per 100 gm.

Required:

(a) Show that the standard deviation of likely sales is approximately s = 3,000.

(b) Using s = 3,000 determine the probability that the company will at least break even if

i) existing facilities are used,

ii) the special machine is hired.

(c) By comparing expected profits, decide whether or not the company should hire the special machine.

(d) By using the normal distribution it can be shown that the following probability distribution may be applied to lube sales:

Sales (‘000) 0-5 5-8 8-10 10-12 12-15 15-20 Probability 0.05 0.20 0.25 0.25 0.20 0.05

By assuming that the actual sales can only take the midpoints of these classes, determine the expected value of perfect information and interpret its value.

11. Assume that the probability of a salesperson making a sale at a randomly selected petrol station is 0.1. If a salesperson makes 20 calls a day, determine the following:

(a) The probability of no sales.

(b) The probability of one sale.

(c) The probability of four or more sales. (c) C

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(d) The probability of more than four sales.

(e) The probability of four sales.

12. Daily sales of a certain product are known to have a normal distribution of 20 per day, with a standard deviation of 6 per day.

(a) What is the probability of selling fewer than 16 on a given day?

(b) What is the probability of selling between 15 and 25 units on a given day?

(c) How many units would have to be in hand at the start of a day in order to have less than a 10 percent chance of running out?

13. On a midterm exam, the scores were distributed normally with mean of 72 and standard deviation of 10. Student Wright scored in the top 10 percent of the class on the midterm.

(a) Wright’s midterm score was at least how much?

(b) The final exam also had a normal distribution, but with mean of 150 and standard deviation of 15. At least what score should Wright get in order to keep the same ranking (i.e., top 10 percent)?

14. An investor wishes to invest in one of two projects. The returns for both projects are uncertain, and the probability distribution for returns can be expressed by a normal distribution in each case. Project A has a mean return of ` 240,000 with a standard deviation of ` 20,000. Project B has a mean return of ` 250,000 and a standard deviation of ` 40,000.

(a) Consider a return of ` 280,000. Which project has a higher chance of returning this much or more?

(b) Consider a return of ` 2,20,000. Which project has a higher chance of returning this much or more?

15. A survey was conducted among the readers of a certain magazine. The results showed that 60 percent of the readers were homeowners and had incomes in excess of ` 25,000 per month; 20 percent were homeowners but had incomes of less than ` 25,000; 10 percent had incomes in excess of ` 25,000 (c)

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but were not homeowners; and the remaining 10 percent were neither homeowners nor had incomes in excess of ` 25,000.

(a) Suppose a reader of this magazine is selected at random and you are told that the person is a homeowner. What is the probability that the person has income in excess of ` 25,000?

(b) Are home ownership and income (measured only as above or below ` 25,000) independent factors for this group?

16. A survey was conducted of families in an urban and the surrounding suburban area. The families were classified according to whether or not they customarily watch two TV programs. The data are shown in the table in percentages of the total.

(a) If a family is selected from this group at random, what is the probability that it views both programs?

(b) If the family selected views program A, what is the probability that it also views program B?

(c) Are the events (views program A) and (views program B) independent events?

(d) Is the event (views program B) independent of the event (urban)?

(e) Consider the event (view either program A or B or both). Is this event independent of the event (urban)?

Watch Watch Program A

Program B Yes No

Urban Suburban Urban Suburban Total Yes 10% 14% 5% 1% 30% No 15 21 20 14 70 Total 25% 35% 25% 15% 100%

17. Newspaper articles frequently cite the fact that in any one year, a small percentage (say 10 percent) of all drivers is responsible for all automobile accidents. The conclusion is often reached that if only we would single out these accident-prone drivers and either retrain them or remove them from the roads, we could drastically reduce auto accidents. You are told that of 100,000 drivers who were involved in one or more accidents in one year, 11,000 of them were involved in one or more accidents in the next year. (c)

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(a) Given the above information, complete the entries in the joint probability table below.

(b) Do you think searching for accident-prone drivers is an effective way to reduce auto accidents? Why?

Table of accidents

Second Year First Year Accident No Marginal Probability of Accident Event in First Year

Accident 0.10

No Accident 0.90

Marginal Probability of

Event in Second Year 0.10 0.90 1.00

18. Gati India Ltd., maintains kilometre records on all of its rolling equipment. Here are weekly kilometre records of its trucks.

810 450 756 789 210 657 589 488 876 689

1450 560 469 890 987 559 788 943 447 775

(a) Calculate the median kilometre a truck travelled.

(b) Calculate the mean for 20 trucks.

(c) Compare part (a) and (b) and explain which one is better measure of central tendency of the data.

19. Premier Automobiles Ltd. does statistical analysis for an automobile racing team. Here are the fuel consumption figures in kilometre per litre for the team’s cars in the recent races.

4.77 6.11 6.11 5.05 5.99 4.91 5.27 6.01

5.75 4.89 6.05 5.22 6.02 5.24 6.11 5.02 (a) Calculate the median fuel consumption.

(b) Calculate the mean fuel consumption.

(c) Group the given data into equally sized classes. What is the fuel consumption value of the modal classes.

(d) Which of the three measures of central tendency is best for Premier to use when she orders fuel? Explain.

20. A machine is assumed to depreciate 40% in value in first year, 25% in second year and 10% per annum for the next three (c)

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years. Each percentage is calculated on the diminishing value. What is the average percentage depreciation for the five years?

21. Philips India Ltd., manufactures the famous Philips tube lights of 40 watts. The company has developed a new variety of Fluorescent 24 watt tube lights for specific applications in control equipments used in defence components. Before it is commercially launched the manager R and D desires to ensure its reliability and quality. The test results conducted on 400 such tube lights are shown below. Compute the coefficient of skewness.

Life Time (hours) No. of Tubes 300-400 14 400-500 46 500-600 58 600-700 76 700-800 68 800-900 62

900-1000 48 1000-1100 22 1100-1200 6

22. From past experience it is known that a machine is set up correctly on 90% of occasions. If the machine is set up correctly then 95% of good parts are expected but if the machine is not set up correctly then the probability of a good part is only 30%.

On a particular day the machine is set up and the first component produced and found to be good. What is the probability that the machine is set up correctly?

23. If the probability of obtaining heads when tossing a certain coin is ½, what is the probability of obtaining heads four times in nine tosses?

24. If the probability of obtaining a 6 when throwing a certain dice is ½, what is the probability of obtaining a 6 four times in nine throws?

25. In how many different ways the first three places can be filled in a race in which there are 11 horses?

26. From our statisticians and seven engineers, a committee is to be formed which must consist of two statisticians and two engineers. In how many different ways, the committee can be formed if (i) There is no other restriction on the membership (c)

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of the committee. (ii) A particular statistician must be included, (iii) Two of the engineers are unable to serve on the committee?

27. Five bolts are selected at random from a box containing six sound and three faulty bolts. What is the probability of obtaining (i) five sound, (ii) four sound and one faulty, (iii) three sound and two faulty, (iv) two sound and three faulty bolts?

Further Readings

Books




Web Readings




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UNIT 10: Case Studies

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Unit 10

Case Studies


Case Study 1: Selecting Distribution Type

In each of the following three situations, use the binomial, Poisson or normal distribution according to which it is most appropriate. In each case, explain why you selected the distribution and draw attention to any feature which supports or casts doubt on the choice of distribution.

Situation 1

The lifetimes of a certain type of electrical component are distributed with a mean of 800 hours and a standard deviation of 160 hours.

Questions

1. If the manufacturer replaces all components that fall before the guaranteed minimum lifetime of 600 hours, what percentage of the components has to be replaced?

2. If the manufacturer wishes to replace only the 1% of components that have the shortest life, what value should be used as the guaranteed lifetime?

3. What is the probability that the mean lifetime of a sample of 25 of these electrical components exceeds 850 hours?

Situation 2

A green grocer buys peaches in large consignments directly from a wholesaler. In view of the perishable nature of the commodity, the green grocer accepts that 15% of the supplied peaches will usually be unsaleable. As he cannot check all the peaches individually, he selects a single batch of 10 peaches on which to base his decision of whether to purchase a large consignment or not. If no more than two of these peaches are unsatisfactory, the greengrocer purchases the consignment.

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Question

Determine the probability that, under normal supply conditions, the consignment is purchased.

Situation 3

Vehicles pass a certain point on a busy single-carriageway road at an average rate of two per ten-second interval.

Question

Determine the probability that more than three cars pass this point during a twenty-second interval.


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Case Study 2: Functional Probabilities

Thirty chief executive officers in an oil and gas industry are classified by age and by their previous functional position as shown in the table below:

Previous Functional Age

position Under 55 55 and older Total

Finance 4 14 18

Marketing 1 5 6

Other 4 2 6

Total 9 21 30

Suppose an executive is selected at random from this group.

Questions

1. What is the probability that the executive chosen is under 55? What type (marginal, conditional, joint) of probability is this?

2. What is the probability that an executive chosen at random is 55 or older and with Marketing as the previous functional position? What type of probability is this?

3. Suppose an executive is selected, and you are told that the previous position was in Finance. What is the probability that the executive is under 55? What kind of probability is this?

4. Are age and previous functional position independent factors for this group of executives?


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UNIT 11: Decision Theory

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BLOCK-III

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Detailed Contents UNIT 11: DECISION THEORY

Introduction

Key Terms

Steps in Decision Theory

UNIT 12: LINEAR PROGRAMMING

Introduction

Linear Programming Problem

Graphical Method

Simplex Method

Duality

UNIT 13: TRANSPORTATION MODELS

Introduction

Demand, Transport Demand and Market Models

Types of Transportation Problem

UNIT 14: ASSIGNMENT PROBLEM

Introduction

Hungarian Method for Solving Assignment Problem


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Unit 11

Decision Theory


Key Terms


Decision Tree

Introduction

It assists the managers/executives in making the decisions. It deals with the methods helpful to decision makers to select the best course of action from amongst the alternative plans of action. It provides a method of rational decision making when consequences are not fully known. It provides a framework for better understanding of decision situations and for evaluating alternatives.

Key Terms

Decision Maker: An individual/group responsible for making a choice of appropriate course of action.

Objectives: Which decision maker wants to achieve.

Preference/value System: Criteria that decision maker uses in decision making a choice of best course of action.

Courses of Action: Decision alternatives under control available to decision maker.

States of Nature: Decision maker prepares the list of possible future events before applying theory. These future events are called states of nature, are mutually exclusive and collectively exhaustive. It can have/not have a numerical description (Low/high demand of items, lockout or strike etc.)

Payoff: It is the effectiveness of particular combination of a course of action and state of nature. These are also called as conditional values/profits.

Activity What is payoff means?

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Opportunity Loss: It is incurred due to failure of not adopting most favourable course of action or strategy.

Decisions: Broadly there are three types of decisions which are as follows:

Strategic decisions are concerned with external environment of the organization. (Selection of location, product, market, technology, etc.)

Administrative Decisions are concerned with structuring and acquisition of the organization’s resources so as to optimize the performance of the organization. (Layout, distribution channel, purchase of assets, etc.)

Operating Decisions are primarily concerned with day to day operations of the organization. (Production scheduling, inventory, packing and dispatching).

Check Your Progress

Fill in the blanks

1. Strategic decisions are concerned with ________ environment of the organization.

2. Administrative Decisions are concerned with structuring and acquisition of the organization’s ___________


(i) Decision making environment

(ii) Objective of a decision maker

(iii) Alternative plans of actions or strategies

(iv) Decision Payoff

(i) Decision making environment Deterministic situation: Where the information is

completely known and the outcome of a specified decision can be predetermined with certainty. Decision maker has the complete information of impact of each course of action. The techniques used under such situations are: (i) Linear Programming, (ii) Input-output analysis, (iii) Transportation and assignment models (iv) CPM.

Activity What is scholastic situation?

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Stochastic situation: In this situation, the decision maker knows the likelihood of the occurrence of each of state of nature. The probability of occurrence of each state is known. The techniques used are Decision Theory, PERT, Simulation, Markov Chain, and Bayesian Theorem.

Situation of uncertainty: Wherein the probabilities associated with the states of nature are unknown. For example, the success of new product launched in the market or the success of branch office opened abroad. Game Theory is used to analyse such situations.

(ii) Objective of a decision maker: The objective should be defined explicitly. Whether he wants to continue the existing state or switch over to other state. The problem relates to single goal or multiple goals. Example: Maximizing return or profit, minimizing loss or wastage.

(iii) Alternative plans of actions or strategies: There must be more than one course of action; otherwise there is no need of any decision. Exhaustive list of alternatives is prepared and then feasible alternatives should be considered for the analysis.

(iv) Decision Payoff: It represents the effectiveness of the strategies. Generally, it is measured in monetary terms. It can be fixed in advance or can be random variable.

Payoff Table: List of states of nature (events) which are mutually exclusive and collectively exhaustive and a set of given courses of action (strategies) and for each combination of these, payoff is calculated.

Table 11.1: Payoff Table

State of nature Courses of action S1 S2 S3 ……… Sn O1 a11 a12 a13 a1n O2 a21 a22 a23 …… a2n O3 a31 a32 a33 ……. a3n . .. .. .. .. Om am1 am2 am3 ……. amn

EMV (Expected Monetary Value): For a given course of action is the weighted average payoff, which is the sum of the product of payoff for the several combinations of strategies and states of (c)

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nature, multiplied by the probability of occurrence of each outcome.

Steps for Calculating EMV

(i) Define systematically the states of nature (Oi) and course of action (Sj).

(ii) List the payoff associated with each combination of state of nature and course of action along with probability of each state of nature.

(iii) Calculate EMV for each course of action by multiplying the conditional payoffs by associated probabilities and sum up these weighted values for each course of action.

(iv) On the basis of specified decision objective, determine the courses of action corresponding to optimal EMV.

EPPI (Expected Profit with Perfect Information): It is the maximum obtainable expected monetary value based on perfect information as to which state of nature will occur.

EVPI (Expected Value of Perfect Information): It is the maximum amount one would be willing to pay to obtain perfect information.

EVPI = EPPI – EMV

EOL (Expected Opportunities Loss): It occurs due to lack of perfect information. It shows the expected differences between the payoff of right decision (maximum) and payoff of actual decision.

EOL = Summation of COL (Oi, Sj). P(Oi)

Where COL is the conditional opportunity loss

Decision Tree

This approach is used for complicated situations. A decision tree is a decision flow diagram that includes branches leading to alternatives one can select among the usual branches leading to events that depend on probabilities. Expectation principle is used i.e. to choose the alternative that maximizes the expected profit or minimizes the expected cost.

Advantages

(i) It provides a graphical presentation of sequential actions or decisions.

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(ii) It makes the analysis simple as the computed values can be written on the tree diagram.

(iii) It clearly depicts when decisions are expected to be made along with their possible consequences and results.

(iv) The complex problems can be solved with tree diagram.

Limitations

(i) The probability estimates may not be accurate thus, may not give true results.

(ii) Certain extraneous variables may be out of control of the decision maker and thus, objective may not be achieved completely.

(iii) When number of states of nature is large, the problem becomes complicated.

Check Your Progress

Fill in the blanks

1. The probability estimates ______ be accurate

2. A decision tree is a _______________ that includes branches leading to alternatives one can select among the usual branches leading to events that depend on probabilities

Summary

Decision theory provides a framework for better understanding of decision situations and for evaluating alternatives. Broadly there are three types of decisions including Strategic decisions, Administrative Decisions and Operating Decisions.

A decision tree is a decision flow diagram that includes branches leading to alternatives one can select among the usual branches leading to events that depend on probabilities.

Lesson End Activity

The marketing department of a certain petrochemicals manufacturing company has worked out the payoff (given in the following table) in terms of profit expressed in million dollars,

Activity What is decision tree meant for?

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concerning a technical proposal depending upon the rate of technological changes.

Technological Course of Action

Changes Accept Reject

Fast 2 3

Slow 5 2

None –1 4

Whether the technical proposal should be accepted or not?

Keywords

Decision Maker: An individual/group responsible for making a choice of appropriate course of action.

Payoff: It is the effectiveness of particular combination of a course of action and state of nature. These are also called as conditional values/profits.

Decision Tree: A decision tree is a decision flow diagram that includes branches leading to alternatives one can select among the usual branches leading to events that depend on probabilities.




1. What are the key characteristics of decision theory?

2. What are the different types of strategic decisions?

3. What are the key steps in decision theory?

4. Illustrate the calculation of EMV.

5. Define the approach of decision tree. What are the key advantages and disadvantages of decision tree approach?

6. A distributor Doon Enterprises buys tankers of diesel for ` 10,00,000 and sells them at ` 12,00,000 each. All the tankers left at the end of the day are worthless. Over the last 100 days (c)

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he recorded the daily sale and the distribution of sale is as follows:

Tankers Sold No. of Days 20 5 21 20 22 30 23 35 24 10

Determine the number of tankers the distributor should buy so as to maximize his profit.

7 An auto parts retailer sells headlights for ` 35 each which he buys at ` 30 each. He cannot return the unsold headlights. The daily demand has the following distribution: No. of Headlights 23 24 25 26 27 28 29 30 31 Probability 0.01 0.03 0.06 0.1 0.2 0.25 0.1 0.050 0.05

If each day’s demand is independent of previous day’s demand, how many headlights should he order each day?

8. An Oil exploration company wants to make a decision regarding buying the site on lease or not, the cost has been calculated for these two states for getting oil or not getting oil. Draw a decision tree for the data given in the cost matrix:

State of Nature Prob.

Getting Oil 0.01

Not getting Oil 0.99

Go for Lease ` 1,00,000 ` 1,00,000 Not go for Lease ` 80,00,000 ` 0

Also advise the company which decision it should take.

(Ans. ` 80,000)

9. A company owns a lease on a certain property in a European country. It may sell the lease for US $ 18,000 or it may drill this property for exploring gas. Various possible drilling results were obtained after a research conducted by its engineers. The data is given in the following table:

Possible Results Probability

Outcomes (US$)

Dry well 0.10 - 1,00,000 Oil well 0.20 45,000 Gas well only 0.40 98,000 Oil and gas combination 0.30 1,99,000 (c)

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Construct a decision tree diagram for the above problem and determine EMV for the company. What will you suggest the company to sell or drill?

10. A medical practitioner purchases a specific vaccine on Sunday evening each week. The vaccine has to be used in the following week otherwise it becomes worthless. It costs ` 20 per dose and he charges ` 22 only per dose. The practitioner administered the vaccine in following quantities in last 50 weeks:

Number of weeks 20 25 40 60 Number of doses/ week 5 15 25 5

Calculate the number of doses the practitioner must buy every week.

Further Readings

Books




Web Readings


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UNIT 12: Linear Programming

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Unit 12

Linear Programming


The significance and advantages of Linear Programming

The conversion of the LP problem into mathematical model

How to solve the LP problems of minimization and maximization using graphical and simplex methods

How to obtain the dual of primal

Introduction

It was introduced by George Dantzig in 1947. It is used as a scientific approach to decision making and an optimization technique used in operations research. It is applied to minimize the cost or maximize the profit.

Linear Programming problem

A Linear Programming problem is a special case of a Mathematical Programming problem. From an analytical perspective, a mathematical program tries to identify an extreme (i.e., minimum or maximum) point of a function, which furthermore satisfies a set of constraints, e.g., Linear programming is the specialization of mathematical programming to the case where both, function Z - to be called the objective function - and the problem constraints are linear.

It allows the rationalization of many managerial and/or technological decisions required by contemporary techno-socio-economic applications. An important factor for the applicability of the mathematical programming methodology in various application contexts is the computational tractability of the resulting analytical models. Under the advent of modern computing technology, this tractability requirement translates to the existence of effective and efficient algorithmic procedures able to provide a systematic and fast solution to these models.

Activity What is L.P?

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Let us discuss the meaning of terms Linear and Program used in LP:

Linear: The relationship between the variables is directly proportional. For example, if a wooden table requires 30 cubic feet of wood; then 10 tables would require 300 cubic feet wood.

Program: A program is a set of instructions arranged in a logical sequence.

Optimal: It means if a programme maximizes or minimizes some measure or criterion of effectiveness. Ex. Maximization of profit / sales or minimization of cost or distance etc.

Limited: Availability of resources during planning horizon.

The related problem of Integer Programming requires some or all of the variables to take integer (whole number) values. Integer programs (IPs) often have the advantage of being more realistic than LPs, but the disadvantage of being much harder to solve. The most widely used general-purpose techniques for solving IPs use the solutions to a series of LPs to manage the search for integer solutions and to prove optimality. Thus, most IP software is built upon LP software.

Linear programming has proved valuable for modelling many and diverse types of problems in planning, routing, scheduling, assignment, and design. Industries that make use of LP and its extensions include transportation, energy, telecommunications, and manufacturing of many kinds.

Check Your Progress

Fill in the blanks

1. Linear programming is the specialization of mathematical programming to the case where both function Z - to be called the ___________function and the problem constraints are linear.

2. ____________means maximisation of profit or minimisation of cost of a business subject to constraints or hurdles.

Assumptions

Certainty- in Linear Programming, it is assumed that all model parameters such as available resources, coefficients of objective (c) C

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function and coefficients of the constraints are known with certainty.

Proportionality- all relationships in objective functions and constraints are linear. In economic terminology, this means that there are constant returns to scale i.e., if one unit of a product contributes `10 as profit, then five units will contribute ` 50.

Additivity- the total of all the activities is given by sum total of each activity performed separately. For example, the total profit in the objective function will be sum of the profits contributed by each of products separately.

Continuity- means that the decision variables are continuous. Accordingly, the solutions of decision variables and resources are assumed to have whole numbers or fractions. When only integers are required such as the number of tables/books/workers/machines, Integer Programming is more suitable.

Finite Choices- implies that finite numbers of choices are available to a decision maker and decision variables do not assume negative values.

Advantages

It is used to determine the product mix.

It helps in attaining the optimum use of production factors.

It gives possible and practical solutions, optimal solution for the decision maker.

It improves the quality of decisions as these are objective in nature.

It is also helpful in re-evaluation of a basic plan for changing conditions.

Limitations

Linearity: It treats all relationships among decision variables and objective functions as linear but in real world situation it may not be so.

Integer Value: The solution of LPP may not result into integer values, the values are in fractions, rounding off to the nearest integers does not give the optimal solution. Integer Programming is the modified LPP which overcomes this limitation. (c)

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Multiple Goals: It deals with only single objective (Minimizing the cost or maximizing the profit) but practically a manager has to make decision considering various goals at the same time i.e. sales maximization, minimizing idle labour time and maximizing capacity utilization etc. Goal Programming is the technique which is suitable for such situations.

Constant Parameters: All the parameters are assumed to be constant in LPP but it is not so in real life situations.

Number of Variables: It can be solved manually when there are two or three variables, in case of large number of variables computing can be done through software only.

Deterministic Nature: It is assumed that all the coefficients and resources are known with certainty and it is not affected by time and uncertainty.

Problem Formulation (Mathematical Model): When a decision has to be taken regarding the number of units of two or more than two products to be manufactured under certain constraints like manpower, material or space etc., the mathematical or analytical model of linear programming has to be used. There are three components of mathematical model:

(i) Objective Function (Z): The linear function which is to be optimized is called Objective Function. The objective functions can be in terms of minimization or maximization. In case of minimization, we can have cost, time, distance etc. whereas in case of maximization, these can be profit, revenue or sales etc. For example:

Max. Z = 100 X1 + 200X2

Where X1 and X2 are the quantities (number of units) of two products P1 and P2

`100 and `200 are the profit values for these two products P1 and P2.

(ii) Constraint (Inequality) Equations: These represent the linear relationship of constraints in a given situation. These can be of types such as: material, manpower, space, machine or budget etc. The equations will have the signs less than equal to (<=), greater than equal to (>=) or equality (=).

2X1 + 5X2 <= 2000 (c) C

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2 and 5 are the quantities of material consumed (cubic feet for wooden material) for P1 and P2.

2000 is the total quantity of material available.

3X1 + 4X2 <= 4200

3 and 5 are the man-hours for P1 and P2.

4,000 is the total man-hours available.

(iii) Non-negative Equations: The decision variables have the values zero or positive, not negative. Thus, these are also called as variable sign restrictions.

X1 and X2 >= 0

Check Your Progress

Fill in the blanks:

1. All the parameters are assumed to be ________ in LPP but it is not so in real life situations.

2. The solution of LPP may not result into __________ values.

Graphical Method

Linear programming problems with two variables can be represented and solved graphically with ease. Though in real-life, the two variable problems are practiced very little, the interpretation of this method will help to understand the simplex method. The solution method of solving the problem through graphical method is discussed with an example given below.

Example 12.1: A company manufactures two types of boxes, corrugated and ordinary cartons. The boxes undergo two major processes: cutting and pinning operations. The profits per unit are ` 6 and ` 4 respectively. Each corrugated box requires 2 minutes for cutting and 3 minutes for pinning operation, whereas each carton box requires 2 minutes for cutting and 1 minute for pinning. The available operating time is 120 minutes and 60 minutes for cutting and pinning machines. Determine the optimum quantities of the two boxes to maximize the profits. (c)

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Solution:

Key Decision

To determine how many (number of) corrugated and carton boxes are to be manufactured.

Decision Variables

Let xl be the number of corrugated boxes to be manufactured.

Let x2 be the number of carton boxes to be manufactured

Objective Function

The objective is to maximize the profits. Given profits on corrugated box and carton box are ` 6 and ` 4 respectively.

The objective function is,

Zmax = 6x1 + 4x2

Constraints

The available machine-hours for each machine and the time consumed by each product are given.

Therefore, the constraints are,

2x1 + 3x2 ≤ 120 (i)

2x1+ x2 ≤ 60 (ii)

where x1, x2 ≥ 0

Graphical Solution

As a first step, the inequality constraints are removed by replacing equal to sign to give the following equations:

2x1 + 3x2 = 120 (1)

2x1 + x2 = 60 (2)

Find the coordinates of the lines by substituting x1 = 0 and x2 = 0 in each equation. In equation 1, put x1 = 0 to get x2 and vice versa

2x1 + 3x2 = 120

2(0) + 3x2 = 120, x2 = 40

Similarly put x2 = 0,

2x1 + 3x2 = 120

2x1 + 3(0) = 120, x1 = 60 (c) C

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The line 2x1 + 3x2 = 120 passes through coordinates (0, 40) (60, 0).

The line 2x1 + x2 = 60 passes through coordinates (0, 60) (30, 0).

The lines are drawn on a graph with horizontal and vertical axis representing boxes x1 and x2 respectively. Figure 12.1 shows the first line plotted.

Figure 12.1: Graph Considering First Constraint

The inequality constraint of the first line is (less than or equal to) ≤ type which means the feasible solution zone lies towards the origin. The no shaded portion can be seen is the feasible area shown in Figure 12.2 (Note: If the constraint type is ≥ then the solution zone area lies away from the origin in the opposite direction). Now the second constraints line is drawn.

Figure 12.2: Graph Showing Feasible Area (c) C

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When the second constraint is drawn, you may notice that a portion of feasible area is cut. This indicates that while considering both the constraints, the feasible region gets reduced further. Now any point in the shaded portion will satisfy the constraint equations. For example, let the solution point be (15, 20) which lies in the feasible region.

If the points are substituted in all the equations, it should satisfy the conditions. 2x1 + 3x2 ≤ 120 = 30 + 60 ≤ 120 = 90 ≤ 120

2x1 + x2 ≤ 60 = 30 + 20 ≤ 60 = 50 ≤ 60

Now, the objective is to maximize the profit. The point that lies at the furthermost point of the feasible area will give the maximum profit. To locate the point, we need to plot the objective function (profit) line.

Equate the objective function for any specific profit value Z,

Consider a Z-value of 60, i.e. 6x1 + 4x2 = 60

Substituting x1 = 0, we get x2 = 15 and

if x2 = 0, then x1 = 10

Therefore, the co-ordinates for the objective function line are (0, 15), (10, 0) as indicated by dotted line L1 in figure. The objective function line contains all possible combinations of values of xl and x2.

The line L1 does not give the maximum profit because the furthermost point of the feasible area lies above the line L1. Move the line (parallel to line L1) away from the origin to locate the furthermost point. The point P is the furthermost point, since no area is seen further. Take the corresponding values of x1 and x2 from point P, which is 15 and 30 respectively, and are the optimum feasible values of x1 and x2.

Therefore, we conclude that to maximize profit, 15 numbers of corrugated boxes and 30 numbers of carton boxes should be produced to get a maximum profit. Substituting x1 = 15 and x2 = 30 in objective function, we get Zmax = 6x1 + 4x2

= 6(15) + 4(30)

Maximum profit: ` 210.00 (c) C

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Check Your Progress

Fill in the blanks:

1. Linear programming problems with two variables can be represented and solved ___________ with ease.

2. Linear programming methods can be applied to various fields such as production, marketing, sales, banking and farm management, to solve the problems arising due to limited resources and ___________ of resources.

Simplex Method

Two families of solution techniques are in wide use today. Both visit a progressively improving series of trial solutions, until a solution is reached that satisfies the conditions for an optimum. Simplex methods visit “basic” solutions computed by fixing enough of the variables at their bounds to reduce the constraints to a square system, which can be solved for unique values of the remaining variables. Basic solutions represent extreme boundary points of the feasible region. The simplex method can be viewed as moving from one such point to another along the edges of the boundary.

Barrier or interior-point methods, by contrast, visit points within the interior of the feasible region. These methods derive from techniques for nonlinear programming that were developed and popularized in the 1960s by Fiacco and McCormick, but their application to linear programming dates back only to Karmarkar’s innovative analysis in 1984.

Box 12.1: Simplex Method for Standard Maximization Problem

Step 1: Convert to a system of equations by introducing slack variables to turn the constraints into equations, and rewriting the objective function in standard form.

Step 2: Write down the initial tableau.

Step 3: Calculate Zj and then Cj and then calculate Cj–Zj. Select the pivot column: Choose the positive number with the largest magnitude in the index row. Its column is the pivot column. (If there are two candidates, choose either one.) If all the numbers in the bottom row are zero or negative (excluding the rightmost entry- Minimum Ratio), then the solution obtained is the feasible solution.

Activity When do we use simplex method?

Contd….(c) C

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Step 4: Select the pivot in the pivot column: The pivot must always be a positive number. For each positive entry b in the pivot column, compute the ratio a/b, where a is the number in the Answer column in that row. Of these test ratios, choose the smallest one. The corresponding number b is the pivot.

Step 5: Divide the pivot row by pivot element to make it unity. Construct the new tableau by writing the previous pivot row first at the same position (as it was having previously).

Step 6: Write the values of other rows so as the corresponding element in the pivot column becomes zero. Again calculate Zj and Cj–Zj and iterate step 3 onwards to reach the feasible solution.

Check Your Progress

Fill in the blanks

1. Two families of _________________ techniques are in wide use today.

2. Interior-point methods, by contrast, visit points within the interior of the ________________ region.

3. The concept of ________________ was introduced by W.W. Leontief, who was of the opinion that the variables are in a linear relationship with each other and therefore, a problem can be stated in mathematical terms.

Duality

For every given linear programming problem, there exists an intimately related L.P. Problem referred to as its Dual. The given (original) problem is known as Primal. The duality theorem states that “for every maximization (minimization) problem there is a unique similar problem of minimization (maximization) involving the same data which describes the original problem”. The ‘DUAL’ of a ‘DUAL’ is PRIMAL.

The characteristics of dual problem: If the objective of the primal is maximization, the objective of

the dual is minimization.

The primal has m-constraints while its dual has m-unknowns and vice-versa.

Activity What is duality?

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The coefficients of the objective function of primal become the constraints of its dual and vice-versa.

The variables of the primal are replaced by the new variables of its dual.

The sign of the inequalities in the set of restrictions of the primal (<=) is reversed in the set of restrictions in its dual (>=).

For finding the dual of the given maximization problem, all the constraint inequalities should be of (<=) type and for minimization, these should be of (>=) type.

Example 12.2: For obtaining the dual of following primal problem: Max. Z = 3X1 + X2 + 2X3 – X4

St: 2X1 – X2 + 3X3 + X4 = 10

X1 + X2 – X3 + X4 = 11

X1, X2 >= 0, X3, X4 unrestricted in sign

For coefficients of objective function, the matrix is: [3 1 2 -1] Another matrix for coefficients and resources are: [2 -1 3 1] [X1] <= [10]

[1 1 -1 1] [X2] [11]

The variables X3 and X4 in Primal are restricted in sign; therefore, the third and fourth constraints in the dual will have equality sign. As both the constraints in primal are of equality sign, corresponding dual variables will be unrestricted in sign. Let W1, W2 be the corresponding dual variables. The dual is as follows:

[2 1] [-1 1] [W1 W2] <= [10]

[3 –1] [11]

[1 1] 2W1 + W2 >= 3

–W1 + W2 >= 1

3W1 – W2 = 2

W1 + W2 = -1

W1, W2 are unrestricted in sign. (c) C

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Check Your Progress

Fill in the blanks:

1. For every given __________ programming problem, there exists an intimately related L.P.

2. The coefficients of the __________ function of primal become the constraints of its dual and vice-versa.

3. __________ introduced the concept of linear programming, which is one of the most widely used methods of operations research.

4. In linear programming, the problems can be solved by using linear programming methods such as ___________and graphical, which aim at either maximising the profit or minimising the cost.

Summary

A Linear Programming problem is a special case of a Mathematical Programming problem. It allows the rationalization of many managerial and/or technological decisions required by contemporary techno-socio-economic applications. Linear programming has proved valuable for modelling many and diverse types of problems in planning, routing, scheduling, assignment, and design.

Two families of solution techniques are in wide use today. Both visits a progressively improving series of trial solutions, until a solution is reached that satisfies the conditions for an optimum. Simplex methods visit "basic" solutions computed by fixing enough of the variables at their bounds to reduce the constraints to a square system, which can be solved for unique values of the remaining variables. Barrier or interior-point methods, by contrast, visit points within the interior of the feasible region.

Lesson End Activity

Which two families of solution techniques are in wide use today?

Keywords

Linear: The relationship between the variables is directly proportional. For example, if a wooden table requires 30 cubic feet of wood then 10 tables would require 300 cubic feet wood. (c) C

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Program: A program is a set of instructions arranged in a logical sequence.


Objective Function (Z): The linear function which is to be optimized is called Objective Function.


1. Define a linear programming problem. What are the key assumptions of a linear programming problem?

2. What are the key advantages and limitations of LP?

3. What are the key components of mathematical model?

4. State the difference between the simplex methods and barrier or interior point methods.

5. Write a note on duality theorem.

6. A home decorator Glamour Enterprises manufactures two types of lamps which go under two first technicians, a cutter, second a finisher. Lamp A requires 2 hours of cutter’s time and 1 hour of finisher’s time, Lamp B requires 1 hour of cutter’s time and 2 hour of finisher’s time. The cutter has 104 hours and finisher 76 hours of time available each month. Profit on one lamp A is ` 6 and on one lamp B is ` 11. Assuming that he can sell all the lamps he produces, formulate the problem.

7. A firm manufactures three products A, B and C. The profits are ` 30, ` 20 and ` 40 respectively. The firm has two machines M1 and M2 and required processing time in minutes on each product in given below:

Product Machine A B C M1 4 3 5 M2 2 2 4

Machines M1 and M2 have 2,000 and 2,500 machine minutes respectively. The firm manufactures 100 A’s, 200 B’s and 50 C’s but not more than 150 A’s. Formulate the problem as a LPP. (c)

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8. The objective of a diet problem is to ascertain the quantities of foods that should be eaten to meet certain nutritional requirement at a minimum cost. The consideration is limited to milk, beef and eggs, and to vitamins A, B and C. The number of grams of each of these vitamins contained in a unit of each food is given below: Vitamin Gallon of

Milk Pound of

Beef Dozens of

Eggs Minimum daily

requirement A 1 1 10 1 B 100 10 10 50 C 10 100 10 10

Cost (Rs.)

40 90 20

Formulate the mathematical model.

9 A textile unit has two grades of inspectors, I and II, who are to be assigned for the quality control inspection. It is required that 2,000 pieces be inspected per 8 hours a day. Grade I inspectors can check pieces at the rate of 50 per hour with an accuracy of 97%, and grade II inspectors can check pieces at the rate of 40 per hour with an accuracy of 95%. The wage rate of grade I inspectors is ` 4.50 per hour and that of grade II is ` 2.50 per hour. Each time an error is made by an inspector, the cost to the company is one rupee. The company has available I 10 grade 1 and 5 grade II inspectors for the inspection job. Formulate the problem to minimize the total cost of inspection.

10. (a) Solve the following problems graphically: Max Z = 20 X1 + 30 X2

Sub. to: X1 + X2 <= 1

3X1 + X2 <= 4

X1, X2 >= 0 (Ans. X1=0, X2=1, Z=30)

(b) Max Z = 30 X1 + 50 X2

Subject to: X1 + 2X2 <= 2000

X1 + X2 <= 1500

X2 <= 600

X1, X2 >= 0 (Ans. X1=1000, X2=500, Z=55,000) (c) C

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(c) Max. Z = 4X1 + 5X2

Subject to: X1 + X2 >= 1

-2X1 + X2 <= 1

4X1–2X2 <= 1

X1, X2 >= 0 (Ans. Unbounded Solution)

(d) Min. Z = -X1 + 2X2

Subject to: -X1 + 3X2 <= 10

X1 + X2 <= 6

X1 – X2 <= 2 (Ans. X1=2, X2=0, Z= - 2)

(e) Min. Z = 20 X1 + 10 X2

Subject to: X1 + 2X2 <= 40

3X1 + X2 >= 30

4X1 + 3X2 >= 60

X1, X2 >= 0 (Ans. X1=6, X2=12, Z=240)

(f) Min. Z = 30 X1 + 15 X2

subject to: 5X1 + X2 >= 10

X1 + X2 >= 6

X1 + 4X2 >= 12

X1, X2 >=0 (Ans. X1=1, X2= 5, Min. Z = 105)

(g) Min. Z = 12 X1 + 15 X2

subject to: X1 <= 5

X2 >= 3

X1 + X2 = 6

X1, X2 >= 0 (Ans. X1 =3, X2 = 3, Z = 81) (c) C

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11. Solve the following problems by simplex method: Max. Z = 2X1 + 4X2

Subject to: 2X1 + 3X2 <= 48

X1 + 3X2 <= 42

X1 + X2 <= 21

X1, X2 >= 0 (Ans. X1=6, X2=12, Z=60)

12 Max. Z = 4X1 + 10X2

Subject to: 2X1 + X2 <= 50

2X1 + 5X2 <= 100

2X1 + 3X2 <= 90

X1 X2 >= 0 (Ans. X1=0, X2=20, Z=200)

13. Min. Z = 3X1 + 2X2+ X3

Subject to: -3X1 + 2X2 + 2X3 = 8

-3X1 + 4X2 + X3 = 7

X1, X2, X3 >=0

(Ans. Unbounded Solution, Hint: all the elements in key column have negative sign and Cj-Zj >0)

(Ans. 4 insertions in TV & 0 in FM radio)

14. Obtain the dual of the following: Max. Z = -3X1 – 2X2

subject to: X1 + X2 >= 1

X1 +2X2 >= 10

X1 + X2 <= 7

X2 <= 3

and X1, X2 >=0 (c) C

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15. Find out the dual of the primal given below: Min. Z = 10 W1 + 8 W2

subject to: W1 + W2 >= 1

W1 + 3W2 >= 4

2W1 - W2 <= 12

and W1, W2 >=0

16. Write the dual of the following primal problem: Max. Z = 5W1 + 6 W2

subject to: W1 + 2W2 = 5

-W1 + 5W2 >= 3

4W1 + 7W2 <= 8

W1 unrestricted in sign and W2 >=0

Further Readings

Books



Sivayya and Sathya Rao, an Introduction to Business Mathematics

Web Readings

http://www.math.ucla.edu/~tom/LP.pdf




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UNIT 13: Transportation Models

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Unit 13

Transportation Models


The significance of transportation models

The types of transportation models

The applications of transportation models

How to solve the transportation problems using IFS and optimality test

Introduction

The TRANSPORTATION model of linear programming is a flow optimization technique. It is a special case that is somewhat easier to solve than the general L.P. model. It is used to produce optimal assignments of origin quantities to destinations. A condition is that the sum of the origin quantities must equal the destination demands. F.L. Hitchcock contributed significantly in developing transportation models. Since transportation is an economic activity all the various accounting, financial, economic, & econometric models are used. Much of the transport activity that uses ‘civil engineering’ is involved with ‘public works’. Economic and accounting models for private and public enterprises are generally used to assist major decisions & policy formulation.

The transportation problem is a special case of Linear Programming which is concerned with the distribution of a certain product/commodity from a number of sources (origins) to the number of destinations. It can be tabulated in a matrix called transportation matrix. In this matrix, each row denotes sources (factories), each column denotes the destination (warehouses) and each cell shows the cost of transportation per unit (Cij) from ith factory to jth warehouse. Ai and Bj denote the supply and demand respectively.

Factory/ Warehouse

D1 D2 D3 D4 Supply

O1 C11 C12 C13 C14 A1 Contd…

Activity What is transportation models?

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O2 C21 C22 C23 C24 A2 O3 C31 C32 C33 C34 A3 O4 C41 C42 C43 C44 A4 Demand B1 B2 B3 B4 Total

Objective: The objective of transportation problem is to minimize the cost of transportation under the given supply and demand constraints.

Check Your Progress

Fill in the blanks:

1. The ________________ model of linear programming is a flow optimization technique.

2. The objective of transportation problem is to minimize the cost of transportation under the given supply and ________________.

Demand, Transport Demand and Market Models

“At any one price, there is some quantity of a product which an individual consumer is willing and able to purchase over a given period of time. If the price changes, the quantity purchased will also change. Economists call the relationship between the price of a commodity and the quantity purchased during some specified period of time the demand for that commodity.

When a consumer is willing and able to purchase some quantity of a commodity at the existing market price, he is said to have an effective demand for that good. This means that the buyer has the:

1. desire to make a purchase

2. willingness to pay the price

3. ability to pay the price

Transport demand is usually categorized as:

Commodity or Goods movements, i.e. quantified by mass, weight, volume or number of items.

Person Travelling, Passengers, Passenger Trips, People Movements, etc. (c)

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Demand or Market Models tend to be disaggregated by trip purpose, or commodity. These models are either:

Cross Section, i.e. they quantify the movements for a short time period for a number of Origins and Destinations.

Time Series, i.e. they represent one movement over a number of time periods.

General, i.e. they attempt to combine cross section and time series aspects much like space-time representations.

The models may focus on a single market segment or several. Some further categorization:

Aggregate, i.e. the producing activity is treated as single strata

Disaggregate which subdivides the activity into a number of strata

Multimodal where a number of competing and complimentary modes are used to satisfy a market

Abstract Mode, an econometric approach that is useful in multimodal models

Applications

The common uses are for planning, design, operations and management. Each of the functions requires that the models reflect behaviour of the chosen system and scenario. Each of the comments below applies equally to the four activities mentioned above. However the costs and difficulties of complicated modelling should be matched with the expected payoff from the exercise.

1. Identify technically efficient solutions to transport resource allocation: The models can be run as:

(a) Transport Costing Models (also net revenue)

(b) Transport Production Models

(c) Resource Costing Models (also net revenue)

(d) Resource Production Models

2. They can be used to OPTIMIZE the system for a given level of production for a given amount of service. This can lead to development of cost effective relationships with different (c)

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levels of production with fixed levels of plant, labour or investment.

3. Give the ability to model as described above the domain can be increased to simulate performance with some of the inputs as random variables, and for different scenarios.

Check Your Progress

Fill in the blanks:

1. Cross Section quantifies the movements for a _______ time period for a number of Origins and Destinations.

2. Time Series represent one movement over a ____________ time periods.

Types of Transportation Problem

(i) Minimization (Cost or Distance)

(ii) Maximization (Return or Revenue)

(iii) Balanced (Supply = Demand)

(iv) Unbalanced (Supply not equal to Demand)

(v) Restricted

(vi) Transshipment

Balanced Transportation Problem

When the total supplies of all the sources are equal to the total demand of all destinations, the problem is a balanced transportation problem.

Total supply = Total demand

1 1

m n

i ji j

a b= =

=∑ ∑

Example 13.1

Consider the following transportation problem and develop a linear programming (LP) model.

Activity What are the different types of transportation problems?

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Table 13.1: Transportation Problem

Destination Source 1 2 3 Supply

1 15 20 30 350 2 10 9 15 200 3 14 12 18 400

Demand 250 400 300

Solution: Let xij be the number of units to be transported from the source i to the destination j, where i = 1, 2, 3,…m and j = 1, 2, 3,…n.

The linear programming model is

Minimize Z = 15x11+20x12+30x13+10x21+9x22+15x23+14x31+12x32+18x33

Subject to constraints,

x11+x12+x13 ≤ 350 (i)

x21+x22+x23 ≤ 200 (ii)

x31+x32+x33 ≤ 400 (iii)

x11+ x12+x31 = 250 (iv)

x12+x22+x32 = 400 (v)

x13+x23+x33 = 300 (vi)

xij ≥ 0 for all i and j.

In the above LP problem, there are m × n = 3 × 3 = 9 decision variables and m + n = 3 + 3 = 6 constraints.

Unbalanced Transportation Problem

When the total supply of all the sources is not equal to the total demand of all destinations, the problem is an unbalanced transportation problem.

Total supply ≠ Total demand

1

m

ii

a=∑ ≠

1

n

jj

b=∑

Demand Less than Supply In real-life, supply and demand requirements will rarely be equal. This is because of variation in production from the supplier end, and variations in forecast from the customer end. Supply (c)

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variations may be because of shortage of raw materials, labour problems, improper planning and scheduling. Demand variations may be because of change in customer preference, change in prices and introduction of new products by competitors. These unbalanced problems can be easily solved by introducing dummy sources and dummy destinations. If the total supply is greater than the total demand, a dummy destination (dummy column) with demand equal to the supply surplus is added. If the total demand is greater than the total supply, a dummy source (dummy row) with supply equal to the demand surplus is added. The unit transportation cost for the dummy column and dummy row are assigned zero values, because no shipment is actually made in case of a dummy source and dummy destination.

Example 13.2: Check whether the given transportation problem shown in Table is a balanced one. If not, convert the unbalanced problem into a balanced transportation problem.

Table 13.2: Transportation Model with Supply Exceeding Demand

Destination Source 1 2 3

Supply

1 25 45 10 200 2 30 65 15 100 3 15 40 55 400

Demand 200 100 300

Solution: For the given problem, the total supply is not equal to the total demand.

3

1i

ia

=∑ ≠

3

1j

jb

=∑

since,3 3

1 1

700and 600i ji j

a b= =

= =∑ ∑

The given problem is an unbalanced transportation problem. To convert the unbalanced transportation problem into a balanced problem, add a dummy destination (dummy column).

i.e., the demand of the dummy destination is equal to,

3 3

1 1

i ji j

a b= =

−∑ ∑ (c) C

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Thus, a dummy destination is added to the table, with a demand of 100 units. The modified table is shown in Table which has been converted into a balanced transportation table. The unit costs of transportation of dummy destinations are assigned as zero.

Table 13.3: Dummy Destination Added

Destination Source

1 2 3 4

Supply

1 25 45 10 0 200

2 30 65 15 0 100

3 15 40 55 0 400

Demand 200 100 300 100 700/700

Similarly,

If 1 1

n m

j ij i

b a= =

>∑ ∑ then include a dummy source to supply the excess

demand.

Demand Greater than Supply

Example 13.3: Convert the transportation problem shown in table into a balanced problem.

Table 13.4: Demand Exceeding Supply

Destination Source

1 2 3 4

Supply

1 10 16 9 12 200

2 12 12 13 5 300

3 14 8 13 4 300

Demand 100 200 450 250 1000/800

Solution: The given problem is,

4 3

1 1j j

j ib a

= =

>∑ ∑

3

1800i

ia

=

=∑ and 4

11000j

jb

=

=∑

The given problem is an unbalanced one. To convert it into a balanced transportation problem, include a dummy source (dummy row) as shown in the table given below. (c)

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Table 13.5: Balanced TP Model

Destination Source 1 2 3 4

Supply

1 10 16 9 12 200 2 12 12 13 5 300 3 14 8 13 4 300

Demand 100 200 450 250 1000/800

Rim Condition: This is the necessary and sufficient condition for determining the optimal solution any transportation problem. The condition is:

Total Supply = Total Demand ai = bj

Check Your Progress

Fill in the blanks

1. ________ is the necessary and sufficient condition for determining the optimal solution any transportation problem.

2. In real-life, supply and demand requirements will ________ be equal.

Degeneracy in the Transportation Problem

The degeneracy occurs in the transportation problem when we find IFS and observe that if the number of occupied cells is less than the total number of rows plus columns minus one). Thus the formula becomes: Co < m + n – 1

Where Co = No. of occupied cells m = No. of rows n = No. of columns

Methods of Transportation Problem

The steps for solving the transportation problem:

1. Formulate the problem and set up in the matrix form.

2. Obtain IFS by applying any of the methods.

3. Apply the optimality test (MODI or STEPPING STONE METHOD) for obtaining the optimal solution. (c)

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The transportation problem can be solved in two phases. In the first phase, IFS (Initial Feasible Solution) is calculated and in the second phase, optimal solution is calculated.

For obtaining the IFS, the following methods are used:

(i) NWCM (North West Corner Method)

(ii) LCM/MMM (Least Cost Method/Matrix Minima Method)

(iii) VAM (Vogel’s Approximation Method)

For feasible solution, the methods used are:

(i) MODI (Modified Distribution Method)

(ii) SSM (Stepping Stone Method)

Before applying IFS, check the rim condition (ai = bj), which must be satisfied.

NWCM (North West Corner Method):

1. Choose the cell in north west corner.

2. Find out minimum of supply and demand i.e. Min. (ai, bj).

3. Allocate min.(ai, bj) in the north west cell and exhaust the row (column) if supply (demand) is satisfied and adjust the balance.

4. Repeat the steps 1 onwards till all the supply and demand are satisfied.

LCM/MMM (Least Cost Method/Matrix Minima Method)

1. Choose the cell with minimum cost.

2. Find out minimum of supply and demand i.e. Min. (ai, bj).

3. Allocate min.(ai, bj) in the north west cell and exhaust the row (column) if supply (demand) is satisfied and adjust the balance.

4. Repeat the steps 1 onwards till all the supply and demand are satisfied.

VAM (Vogel’s Approximation Method)

1. Calculate the penalty for each row or column by subtracting the smallest element from the next smaller element.

2. Choose the highest penalty out of all the penalties and choose the cell with minimum cost in the corresponding row (column). (c)

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In case of tie between two highest penalties, choose arbitrarily.

3. Find out minimum of supply and demand i.e. Min. (ai, bj) corresponding to the chosen cell.

4. Allocate min.(ai, bj) in the cell with minimum cost and exhaust the row (column) if supply (demand) is satisfied and adjust the balance.

5. Repeat the steps I onwards till all the supply and demand are satisfied.

Optimality Test

MODI Method:

Prior to applying this method, the following condition must be satisfied:

Co = m+n-1

If this condition is not satisfied, degeneracy occurs. Degeneracy can be removed by putting Delta in the unoccupied cell having minimum cost.

1. Construct a transportation table with given cost and allocations as per IFS through any of these methods.

2. For occupied cells, calculate index numbers Ui and Vj for rows and columns respectively. Values of these index numbers are calculated by:

Cij = Ui + Vj

3. Opportunity cost is computed for all the unoccupied cells by using the following equation:

Dij = Cij – (Ui + Vj)

4. Examine unoccupied cells evaluation for opportunity cost (Dij);

(a) If Dij > 0, Cost of transportation will increase, the solution is optimal.

(b) If Dij < 0, Cost of transportation will decrease, the solution is not optimal.

Activity What is optimality test?

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(c) If Dij = 0, Cost of transportation will not change, alternate solution exists.

In case of Dij < 0, loop is constructed for which following steps are required:

5. Select an unoccupied cell with largest negative opportunity cost.

6. Constructed a closed path for the unoccupied cell determined in previous step and assign plus (+) and minus (-) sign alternatively beginning with plus sign for the selected unoccupied cell.

7. Assign as many units as possible to the unoccupied cell satisfying the rim condition. The smallest allocation in a cell with –ve sign on the closed path indicates number of units that can be shipped to the unoccupied cells. This quantity is added to all the occupied cells on the path marked with +ve sign.

8. Go to step 2 and iterate all the steps until all Dij become positive to reach the optimal solution. Then calculate the transportation cost.

* The method discussed here is applicable for case of minimization. In case of maximization, the problem is converted into minimization by subtracting all the elements from the highest element.

Check Your Progress

Fill in the blanks:

1. The __________ occurs in the transportation problem when we find IFS and observe that if the number of occupied cells are less than the total number of rows plus columns minus one.

2. In MODI optimality test C0 = m+ ________________.

Summary

The transportation problem is a special case of Linear Programming which is concerned with the distribution of a certain product/commodity from a number of sources (origins) to the number of destinations. It can be tabulated in a matrix called transportation matrix. (c)

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The degeneracy occurs in the transportation problem when we find IFS and observe that if the numbers of occupied cells are less than the total number of rows plus columns minus one).

The transportation problem can be solved in two phases. In the first phase, IFS (Initial Feasible Solution) is calculated and in the second phase, optimal solution is calculated.

Lesson End Activity

Illustrate the preparation of transportation matrix with a suitable example.

Keywords

Transportation Model: The transportation model of linear programming is a flow optimization technique.

Effective Demand: When a consumer is willing and able to purchase some quantity of a commodity at the existing market price, he is said to have an Effective Demand for that good.


1. Define transportation problem. 2. What are the different types of transportation problems? 3. What are the key methods of obtaining IFS? 4. An oil company has three refineries located in the country.

The daily oil production (in million tonnes) is as follows: Refineries I II III Oil produced 6 1 10

Each day, the refineries must satisfy the needs for their distribution centres. Minimum requirement at each centre is as under:

Distribution Centre I II III IV Oil supply 7 5 3 2

The cost in thousands of rupees of shipping one million tonnes from each refinery to each distribution centre is as per the table given below:

Dist. Centres/Refineries D1 D2 D3 D4 R1 2 3 11 7 R2 1 0 6 1 R3 5 8 15 9 (c)

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Determine initial feasible solution by a) NWCM b) LCM c) VAM method.

(Ans. `11,600, `11,200, `10,200) 5. A car hire company has one car at each of five depots A,B,C,D

and E. A customer requires a car in each town P,Q,R,S,T. Distances (given in kms) between depots and towns are given in the following table:

Depot/Town A B C D E P 160 130 175 190 200 Q 135 120 130 160 175 R 140 110 155 150 185 S 150 50 80 80 110 T 55 35 70 80 105

(Ans. 570 kms)

Further Readings

Books




Web Readings

http://www.math.ucla.edu/~tom/LP.pdf




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UNIT 14: Assignment Problem

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Unit 14

Assignment Problem


The objective of assignment models

The applications of assignment models

The Hungarian method for solving the assignment problems

Introduction

An assignment problem is a special type of transportation problem. The method to solve assignment problems was introduced by D. Konig, a Hungarian mathematician. In such models, only one unit can be supplied to each destination from each source, for example, to assign one job to each facility in order to achieve the minimum possible cost.

Objective: The objective of assignment model is to assign a number of resources to an equal number of activities so as to minimize the cost or maximize the profit by optimal allocation.

Applications

(i) Assignment of machines to jobs

(ii) Assignment of workers to various tasks

(iii) Assignment of sales representatives to sales territories

(iv) Assignment of contracts to bidders

(v) Assignment of buses/airlines/trains to various routes

(vi) Assignment of officers to various offices

Types of assignment problems:

(i) Minimization (Cost, time or Distance)

(ii) Maximization (Profit or Revenue)

(iii) Balanced (No. of rows = No. of columns)

(iv) Unbalanced (No. of rows /= No. of columns)

Activity What is assignment problem?

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(v) Restricted

(vi) Crew assignment

Check Your Progress

Fill in the blanks:

1. An assignment problem is a special type of ___________ problem.

2. The objective of assignment model is to assign a number of resources to an equal number of activities so as to ___________ the cost or maximize the profit by optimal allocation

Hungarian Method for Solving Assignment Problem

Step 1: In a given problem, if the number of rows is not equal to the number of columns and vice versa, then add a dummy row or a dummy column. The assignment costs for dummy cells are always assigned as zero.

Step 2: Reduce the matrix by selecting the smallest element in each row and subtract with other elements in that row.

Step 3: Reduce the new matrix column-wise using the same method as given in step 2.

Step 4: Draw minimum number of lines to cover all zeros.

Step 5: If Number of lines drawn = order of matrix, then optimality is reached, so proceed to step 7. If optimality is not reached, then go to step 6.

Step 6: Select the smallest element of the whole matrix, which is NOT COVERED by lines. Subtract this smallest element with all other remaining elements that are NOT COVERED by lines and add the element at the intersection of lines. Leave the elements covered by single line as it is. Now go to step 4.

Step 7: Take any row or column which has a single zero and assign by squaring it. Strike off the remaining zeros, if any, in that row and column (X). Repeat the process until all the assignments have been made.

Step 8: Write down the assignment results and find the minimum cost/time.

Activity What is Hungarian Method for Solving Assignment Problem?

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Note: While assigning, if there is no single zero exists in the row or column, choose any one zero and assign it. Strike off the remaining zeros in that column or row, and repeat the same for other assignments also. If there is no single zero allocation, it means multiple numbers of solutions exist. But the cost will remain the same for different sets of allocations.

Example: Assign the four tasks to four operators. The assigning costs are given in Table 14.1

Table 14.1: Assignment Problem

Operators 1 2 3 4

A 20 28 19 13

Tasks B 15 30 31 28

C 40 21 20 17

D 21 28 26 12

Solution:

Step 1: The given matrix is a square matrix and it is not necessary to add a dummy row/column

Step 2: Reduce the matrix by selecting the smallest value in each row and subtracting from other values in that corresponding row. In row A, the smallest value is 13, row B is 15, row C is 17 and row D is 12. The row wise reduced matrix is shown in Table 14.2:

Table 14.2: Row-wise Reduction Operators

1 2 3 4

A 7 15 6 0

Tasks B 0 15 16 13

C 23 4 3 0

D 9 16 14 0

Step 3: Reduce the new matrix given in table by selecting the smallest value in each column and subtract from other values in that corresponding column. In column 1, the (c)

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smallest value is 0, column 2 is 4, column 3 is 3 and column 4 is 0. The column-wise reduction matrix is shown in Table 14.3.

Table 14.3: Column-wise Reduction Matrix Operators

1 2 3 4

A 7 11 3 6

Tasks B 0 11 13 13

C 23 0 0 0

D 9 12 11 0

Step 4: Draw minimum number of lines possible to cover all the zeros in the matrix given in Table 14.4

Table 14.4: Matrix with all Zeros Covered Operators

1 2 3 4

A 7 11 3 0

Tasks B 0 11 13 13

C 23 0 0 0

D 9 12 11 0

The first line is drawn crossing row C covering three zeros, second line is drawn crossing column 4 covering two zeros and third line is drawn crossing column 1 (or row B) covering a single zero.

Step 5: Check whether number of lines drawn is equal to the order of the matrix, i.e., 3 ≠ 4. Therefore optimality is not reached. Go to step 6.

Step 6: Take the smallest element of the matrix that is not covered by single line, which is 3. Subtract 3 from all other values that are not covered and add 3 at the intersection of lines. Leave the values which are covered by single line. Table 14.5 shows the details.

No. of lines drawn ≠ order of matrix

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Table 14.5: Subtracted or Added to Uncovered Values and Intersection Lines Respectively

Operators

1 2 3 4

A 7 9 0 0

Tasks B 0 9 10 13

C 26 0 0 3

D 9 9 8 0

Step 7: Now, draw minimum number of lines to cover all the zeros and check for optimality. Here, in Table 14.6, minimum number of lines drawn is 4, which is equal to the order of matrix. Hence optimality is reached.

Table 14.6: Optimality Matrix Operators

1 2 3 4

A 7 9 0 0

Tasks B 0 9 10 13

C 26 0 0 3

D 9 9 8 0

Step 8: Assign the tasks to the operators. Select a row that has a single zero and assign by squaring it. Strike off remaining zeros if any in that row or column. Repeat the assignment for other tasks. The final assignment is shown in Table 14.7.

Table 14.7: Final Assignment Operators

1 2 3 4

A 7 9 0

Tasks B 9 10 13

C 26 0 3

D 9 9 8

No. of lines drawn = order of matrix

0

0

0

0

×

×

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Therefore, optimal assignment is:

Task Operator Cost

A 3 19

B 1 15

C 2 21

D 4 12

Total Cost = ` 67.00

Check Your Progress

Fill in the blanks

1. While assigning, if there is no single zero exists in the row or column, choose _______zero and assign it.

2. If there is no single zero allocation, it means _____________ of solutions exist

Summary

An assignment problem is a special type of transportation problem. The objective of assignment model is to assign a number of resources to an equal number of activities so as to minimize the cost or maximize the profit by optimal allocation.

Lesson End Activity

Study the application areas where Hungarian method can be beneficial.

Keywords

Assignment Problem: An assignment problem is a special type of transportation problem. Objective of Assignment Model: Objective of assignment model is to assign a number of resources to an equal number of activities


1. Define assignment model.

2. What is the objective and application of assignment model? (c) C

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3. What are the different types of assignment problems?

4. Write a note on Hungarian method.

5. Due to absence of a workman, an officer has to assign four out of five different jobs to four workers with the performance (cost) matrix given below, determine an optimal solution.

Worker/Jobs A B C D 1 3 6 5 3 2 4 9 3 2 3 11 2 4 6 4 10 4 6 5 5 11 12 14 10

(Ans. ` 13,000)

6. A company is faced with the problem of assigning five different machines to five different jobs with a view to minimize total cost. The costs (in thousands) are estimated and shown in the following table:

Jobs Machines 1 2 3 4 5 I 2.5 5 1 6 2 II 2 5 1.5 7 3 III 3 6.5 2 8 3 IV 3.5 7 3 9 4.5 V 4 7 3 9 6

(Ans. ` 20,000)

Further Readings

Books R S Bhardwaj, Mathematics for Economics and Business, Excel Books, New Delhi, 2005 D C Sanchethi and V K Kapoor, Business Mathematics Sivayya and Sathya Rao, An Introduction to Business Mathematics

Web Readings http://www.math.ucla.edu/~tom/LP.pdf www.managementstudyguide.com www.textbooksonline.tn.nic.in www.mathbusiness.com (c)

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Unit 15

Case Studies


Case Study 1: Constructing Payoff Table

A company wants to increase its production beyond its existing capacity. It has finally arrived at two approaches to increase the capacity (1) Expansion, at a cost of ` 80 Lakh or (2) Modernization at a cost of ` 50 Lakh. Both approaches would require 8 months for implementation. The Board of Directors feels that during implementation and thereafter the demand will either be very high or moderate. The probability for very high demand is estimated as 0.35 and for moderate it is 0.65. If demand is very high, expansion would result additional profit of 120 Lakh, but on the other hand modernization would bring additional 60 Lakh only. It is estimated that when demand is moderate, the comparable profit would be 70 Lakh and 50 Lakh for modernization.

Questions

(a) Construct the Payoff (Profit) Table.

(b) What is the optimum strategy for the company?

(c) Calculate EMV and EVPI.

(d) Calculate EOL. Source: MBCQ-721D_Quantitative Techniques for Management Applications_CCE_UPES

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Case Study 2: Media Mix Determination

An advertising agency wants to finalise its campaign and it plans to target two types of audiences: customers with cell phone (Type A) and customers not having cell phones (Type B). The total ad budget is ` 2 lakhs. One insertion of TV ad on movie channel costs ` 50,000 and one insertion on FM radio costs ` 20,000. As per agreement, at least three insertions must be there for movie channel and it can not exceed five in number. As per the findings of research agency, a single TV ad reaches 350,000 customers in target audience A and 150,000 in target audience B. One FM radio ad reaches 10,000 in target audience A and 90,000 in target audience B.

Question

Determine the media mix to maximize the total reach. (Formulate the problem and use simplex method)


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Case Study 3: Determining Optimal Solution

A company has plants P1, P2, P3 which supply to distribution centres D1, D2 and D3. Monthly factory capacities are 200, 160 and 90 units respectively whereas requirement of distribution centres are 180, 120 and 150 respectively. Unit transportation cost is given in the following table:

Distribution Centres\Plants

D1 D2 D3

P1 16 20 12

P2 14 8 18

P3 26 24 16

Question

Determine the optimal solution for the company. Source: MBCQ-721D_Quantitative Techniques for Management Applications_CCE_UPES

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Case Study 4: Determining Optimal Assignment

A trip from Delhi to Ajmer takes six hours by bus. The manager has designed the time table of bus service both ways which is as under:

Departure from Delhi

Route No.

Arrival at

Ajmer

Arrival at Delhi

Route No.

Departure from Ajmer

06.00 I 12.00 11.30 a 05.30

07.30 II 13.30 15.00 b 09.00

11.30 III 17.30 21.00 c 15.00

19.00 IV 01.00 00.30 d 18.30

00.30 V 06.30 06.30 e 00.00

The cost of providing service by the travel agency depends upon the time spent by the bus crew away from their places in addition to service time. There is a constraint that every crew should be provided with at least 4 hours of rest before the return trip and at the most he can stay for 24 hours before he goes for the return trip. The crews are residing at rest house hired by travel agency in Delhi as well as Ajmer. The manager wants to minimize the waiting time of crews.

Question

Determine the optimal assignment for the crews for various routes. (There are five crews and five routes)


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UNIT 16: Game Theory

Notes

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BLOCK-IV

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Detailed Contents UNIT 16: GAME THEORY

Introduction

Game

Assumptions

Limitations

Two-person Zero-sum Game

Rules (Principles) of Dominance

UNIT 17: MARKOVIAN MODEL

Introduction

Properties of Markov Models

Applications

Types

UNIT 18: DATA COLLECTION

Introduction

Data Characteristics

Requirements of Data in the Organization

Problems and Limitations of Data Collection

Basic Classification of Data

Primary Data

Methods of Primary Data Collection

Secondary Data

UNIT 19: PRESENTATION OF DATA

Introduction

Categorical data and numeric data


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Unit 16

Game Theory


The game theory and its assumptions

The types, limitations and rules of game theory

The value of game using various methods

Introduction

Professor John Von Neumann and Oscar Morgenstern published their book entitled “The Theory of Games and Economic Behaviour” wherein they provided a new approach to many problems involving conflicting situations. The theory of games attempts to provide the rational decision in the confronting situations.

Game

The term “game” represents a conflict between two or more individuals or groups or organizations. It is the science of conflict. It is applicable to those competitive situations which are technically known as “competitive games”. The objective of game theory is to determine the rules of rational behaviour in situations wherein the outcome resulting from a decision made by one individual depends not only on that individual’s choice but also on the course of action taken by other interested individuals.

Assumptions

1. The players act rationally and intelligently.

2. The players attempt to maximise gains and minimise losses.

3. All relevant information is known to each player.

4. Each player has available to him a finite set of possible courses of action.

Activity What is theory of game?

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5. The players make individual decisions without direct communication.

6. The players simultaneously select their respective course of action.

7. The payoff is fixed in advance.

Types of Games

1. Two-person games and n-person games

2. Zero sum and non-zero sum games

3. Games of perfect information and imperfect information

4. Games with finite number of moves

5. Cooperative and non-cooperative games

6. 2 × 2 two-person games, 2 × m and m × 2 games

7. 3 × 3 and larger games

8. Constant sum games

Limitations

Businessmen do not have the knowledge of game theory and all the alternative strategies available to them or their competitors.

The business environment is turbulent and there is a lot of uncertainty. Thus game theory may not be giving accurate results in such cases as outcome of a strategy may not be known with certainty.

The game theory may not be suitable for oligopoly situations where there are number of companies/firms involved.

Larger size games are very complicated and cannot be solved manually.

The assumption in game theory that one player tries to maximise the gains and the other tries to minimise the losses may not be true in case of today’s dynamic businessman.

Strategy: The strategy for a player is the list of all possible courses of actions that he will take for every pay-off that might arise. (c) C

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Pure Strategy: It is the decision rule which is always followed by the player to select the particular course of action.

Mixed Strategy: When the player has alternative courses of action and he has to select combination of these with some fixed probabilities.

Check Your Progress

Fill in the blanks:

1. The ____________ for a player is the list of all possible courses of actions that he will take for every pay-off that might arise.

2. ____________ theory may not be giving accurate results in many cases.

Two-person Zero-sum Game

In a game with two players, if the gain of one player is equal to the loss of another player, then the game is a two person zero-sum game.

A game in a competitive situation possesses the following properties:

1. The number of players is finite.

2. Each player has finite list of courses of action or strategy.

3. A game is played when each player chooses a course of action (strategy) out of the available strategies. No player is aware of his opponent's choice until he decides his own.

4. The outcome of the play depends on every combination of courses of action. Each outcome determines the gain or loss of each player.

Pure Strategies: Game with Saddle Point The aim of the game is to determine how the players must select their respective strategies such that the pay-off is optimized. This decision-making is referred to as the minimax-maximin principle to obtain the best possible selection of a strategy for the players.

In a pay-off matrix, the minimum value in each row represents the minimum gain for player A. Player A will select the strategy that gives him the maximum gain among the row minimum values. The

Activity How will you account that game is a competitive situation?

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selection of strategy by player A is based on maximin principle. Similarly, the same pay-off is a loss for player B. The maximum value in each column represents the maximum loss for Player B. Player B will select the strategy that gives him the minimum loss among the column maximum values. The selection of strategy by player B is based on minimax principle. If the maximin value is equal to minimax value, the game has a saddle point (i.e., equilibrium point). Thus the strategy selected by player A and player B are optimal.

Example 16.1: Consider the example to solve the game whose pay-off matrix is given in table as follows:

Table 16.1: Game Problem Player B

1 2

1 1 3 Player A 2 -1 6

The game is worked out using minimax procedure. Find the smallest value in each row and select the largest value of these values. Next, find the largest value in each column and select the smallest of these numbers. The procedure is shown in table below.

Table 16.2: Minimax Procedure Player B

1 2 Row Min

1 1 3 Player A 2 -1 6 -1 Col Max 6

If Maximum value in row is equal to the minimum value in column, then saddle point exists.

Max Min = Min Max

1 = 1

Therefore, there is a saddle point.

The strategies are,

Player A plays Strategy A1, (A → A1).

Player B plays Strategy B1, (B → B1).

Value of game = 1.

1

1

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Mixed Strategies: Games without Saddle Point

For any given pay off matrix without saddle point the optimum mixed strategies are shown in Table.

Table 16.3: Mixed Strategies

Player B

B1 B2

A1 a11 a12

Player A A2 a21 a22

Let p1 and p2 be the probability for Player A.

Let q1 and q2 be the probability for Player B.

Let the optimal strategy be SA for player A and SB for player B.

Then the optimal strategies are given in tables given below.

Table 16.4: Optimum Strategies

A1 A2 B1 B2

SA = and SB =

p1 p2 q1 q2

p1 and p2 are determined by using the formulae,

p1 = ( ) ( )

22 21

11 22 12 21

a –aa a – a a+ +

and p2 = 1 – p1

q1 = ( ) ( )

22 12

11 22 12 21

a –aa a – a a+ +

and q2 = 1 – q1

and the value of the game w.r.t player A is given by,

a11 a22 – a12a21

Value of the game, v = (a11+a22) – (a12+a21)

Check Your Progress

Fill in the blanks

1. The game is worked out using _______ procedure.

2. In a _________ matrix, the minimum value in each row represents the minimum gain for a player.

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Rules (Principles) of Dominance

Row Dominance: When each element in a row are less than or equal to the corresponding element in another row, this row is dominated and hence can be deleted from the payoff matrix.

Column Dominance: When each element in a column are less than or equal to the corresponding element in another column, this column is dominated and hence can be deleted from the payoff matrix.

Average Dominance: A strategy can also be dominated when it is inferior to an average of two or more pure strategies.

Value of a game: It is the average payoff per play of game over an extended of time.

Saddle Point: In a payoff matrix saddle point is one which is the smallest value in its row and the largest value in the column. In other words, it is the point where maximin is equal to minimax.

Example 16.2: Determine the value of the following payoff matrix: Player B

Player A [ 2 3] [ 5 4]

Solution: Player B Player A [ 2 3] Row min. 2 Maximin = 4 [ 5 4] Row min. 4 Col. Max. 5 4 Minimax = 4

Thus in this game, saddle point is 4. This is value of the game. Such game is known as pure strategy game.

Activity What are the principles of dominance?

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Flow Chart of Game Theory Approach

Formulate the pay – off matrix

Apply maximin or manimax principle

Is there a saddle

point?

Identify the value of game and write the optimal strategy

of the players

Solve by using algebraic or matrix method

for mixed strategic games

Is it a 2 x 2 pay-off

matrix game?

Use dominance rule to reduce the size of the pay-off matrix to either 2 x 2.

2 x n or m x 2 size (order)

Is pay-off matrix

reduced to a 2 x 2

size?

Is pay-off matrixreduced to a

2 x n or m x 2size?

Formulate and solve as an LP problem

Yes

Yes

Yes

No

Yes

No

No

No

No

Use graphical method to solve the problem

Figure 16.1: Game Theory Approach

Methods for Solving Games

(i) Short cut method

(ii) Graphical method

(iii) Algebraic method

(iv) Linear Programming method

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Check Your Progress

Fill in the blanks:

1. When each element in a column are ________ or ________ the corresponding element in another column.

2. In a payoff matrix ____________ is one which is the smallest value in its row and the largest value in the column.

Summary

The term "game" represents a conflict between two or more individuals or groups or organizations. It is the science of conflict. It is applicable to those competitive situations which are technically known as "competitive games". The objective of game theory is to determine the rules of rational behaviour in situations wherein the outcome resulting from a decision made by one individual depends not only on that individual's choice but also on the course of action taken by other interested individuals.

The key methods of solving games are:

(i) Short cut method

(ii) Graphical method

(iii) Algebraic method

(iv) Linear Programming method

Lesson End Activity

Determine the value of the game and optimum strategies for the following matrix:

Player B3 4

Player A 6 27 12 9

− −− −− −− −

Keywords

Strategy: The strategy for a player is the list of all possible courses of actions that he will take for every pay-off that might arise. (c) C

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Pure Strategy: It is the decision rule which is always followed by the player to select the particular course of action.


Saddle Point: In a payoff matrix saddle point is one which is the smallest value in its row and the largest value in the column.


1. Define game and its types.

2. Define the following:

(a) Saddle Point

(b) Pure and Mixed Strategies

(c) Zero-sum game

3. What are the assumptions made in the theory of games?

4. Describe rules of dominance with examples.

5. What are the various methods for solving a game theory and their suitability?

6. Find out the value of the game, the payoff (in rupees) is given in the following matrix:

Player A / Player B Strategy I Strategy II Strategy I 10 14 Strategy II 7 12

7. The conditional gains to the workers association (in thousands) against management strategies are given in the following payoff table:

Association Management Strategies Strategies M1 M2 M3 M4 A1 20 15 12 35 A2 25 14 8 10 A3 40 2 10 5 A4 -5 4 11 0

8. Two petroleum companies X and Y are competing for business. The matrix shows the gains to the company X (Assume the game is zero sum). (c)

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Company X’s Gain X/Y Sales Promotion Advertising ExhibitionSales Promotion 60 50 40 Advertising 70 70 50 Exhibition 80 60 75

Determine the optimal strategies and value of the game. 9. Determine the value of the following game:

Player Y I II III IV I 3 2 4 0 II 3 4 2 4 Player X III 4 2 4 0 IV 0 4 0 8

10. The payoff matrix is given below. Determine the optimum strategies and value of the game. (Solve graphically)

Player B 3 6 -1 0 -3 Player A 2 3 -4 2 -1

Further Readings

Books




Web Readings

http://www.math.ucla.edu



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UNIT 17: Markovian Model

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Unit 17

Markovian Model


The properties and applications of Markovian model

The types of Markov process and various states involved

How to determine the future market share and market share in steady state condition

Introduction

Markov models are used to analyze states of stochastic system to describe its position at any instant of time.

Properties of Markov Models

(i) It is a stochastic (probabilistic) process.

(ii) Markov Process is a sequence of experiments in which each experiment has certain possible outcomes.

(iii) There is a finite set of states.

(iv) The process can be only in one state at a given time.

(v) The probability of moving from one state to another or remaining in the same state in a single time period is called transition probability (Pij). It always remains constant and 0 <= Pij <= 1

(vi) The probability of transition from a given state to future state is dependent on the present state.

Check Your Progress

Fill in the blanks

1. Markov models are used to analyze states of _________ system to describe its position at any instant of time.

2. Markov Process is a sequence of experiments in which each experiment has certain possible ___________.

Activity What is markovian model?

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Applications

(i) It is a technique applied to solve various management problems. For example it can be used to determine the future market share of various products/brands. It can be used in inventory management also to decide the order size.

(ii) It is very useful for studying the buying pattern of consumers or organizations particularly in terms of brand loyalty and switching patterns.

Check Your Progress

Fill in the blanks:

1. It is very useful for studying the _________ pattern of consumers.

2. Markov technique is used to solve various ________ problems.

Types

(i) First order Markov process

(ii) Second order Markov process

(iii) Higher order Markov process

Steady State Condition: When state probabilities may become constant. The system is in steady state condition if following conditions are satisfied:

(i) The transition matrix elements remain positive from one period to the next. This property is known as the regular property of Markov chain.

(ii) It is possible to move from one state to another state in a finite number of steps, irrespective of the present state.

Absorbing State: A state is said to be absorbing (trapping) state if it does not leave the state. It occurs when if any transition probability in the retention diagonal from upper left to lower right is equal to one (1).

Transient State: A state is said to be transient if it is not possible to move to that state from any other state except itself.

Cycling Process: A cycling (periodic) process is one in which transition matrix contains all zero elements in the retention cells (diagonal elements) and all other elements are either 0 or 1.

Activity Discuss the applications of markovian technique.

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Methods for solving Markov Chains

Transition Probability Matrix

Decision Tree Diagram

Check Your Progress

Fill in the blanks:

1. The system is in steady state when state probabilities may become ______________.

2. A _________ (periodic) process is one in which transition matrix contains all zero elements in the retention cells.

Summary

Markov models are used to analyze states of stochastic system to describe its position at any instant of time. It is a technique applied to solve various management problems. It is very useful for studying the buying pattern of consumers or organizations particularly in terms of brand loyalty and switching patterns.

The system is in steady state condition if following conditions are satisfied:

1. The transition matrix elements remain positive from one period to the next. This property is known as the regular property of Markov chain.

2. It is possible to move from one state to another state in a finite number of steps, irrespective of the present state.

Lesson End Activity

Three brands of toothbrush are available in a provision store. It has been observed that 50% of customers buy brand Oral-B, 30% brand Pepsodent and 20% brand Ajanta. The owner of provision store Sajjan Khatri found that each quarter the customers change their preference. Of those who bought Oral-B last quarter, 50% buy it again, but 15% change to brand Pepsodent and 35% to brand Ajanta. Of those who bought brand Pepsodent, 70% buy it again, 10% switched to Oral-B and 20% to Ajanta. Of those who bought brand Ajanta 80% buy it again, 5% switched to Oral-B and 15% to Pepsodent. Construct the transition matrix and determine their share in this situation.

Activity What is transient state?

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Keywords

Absorbing State: A state is said to be absorbing (trapping) state if it does not leave the state.




1. What are the key properties of Markov models?

2. What are the key uses of Markov models?

3. What are the different types of Markov models?

4. What are steady state conditions?

5. Define absorbing state.

6. There are two brands of oil engine A and B. Both have exactly equal market share in the town presently. The market size is also fixed. The transition matrix is given below:

To A B A [0.8 0.2] From B [0.5 0.5]

Determine their future market share for the next year and market share in the steady state.

7. (a) M/s. Manoj Kumar Kamal Lal stocks three brands of lubes at its various petrol pumps. Calculate the equilibrium market share for three brands of lubricants; the transition matrix is as follows:

To

Castrol Elf Mak Castrol [0.8 0.1 0.1] From Elf [0.05 0.85 0.10] Mak [0.10 0.06 0.84]

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(b) If the present market share of these three brands is 40%, 30% and 30% respectively, determine their market share for the year 2008.

Further Readings

Books




Web Readings

www.textbooksonline.tn.nic.in www.mathbusiness.com (c)

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UNIT 18: Data Collection

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Unit 18

Data Collection


Data characteristics




Methods of Data Collection

Introduction

To solve any managerial problem that you face in the organization you need relevant information. This relevant information has to meet the tests of sufficiency and accuracy to be useful to solve the problem in hand. This information, which is the processed form of data, refers to collection of numbers, letters, or symbols, maintained or produced for the management when required.

Data Characteristics

In order, the numbers that you have collected, may be called data, the following characteristics must be present:

(i) It should be an aggregate of facts; for example, single unconnected figures cannot be called data, as they cannot be used to study characteristics of any event or operation of any industry or organization.

(ii) There should be a reasonable standard of accuracy as is required for the problem in hand, for example, in the measurement of length one may measure correctly up to 0.01 cm or 1cm or 1m as required, the quality of the product could be reasonably estimated by certain tests of small samples drawn from big lots of products.

(iii) It should be collected in a systematic manner for a predetermined objective.

Activity What is data?

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(iv) The elements of the data must be related to one another. The base used in the data should be the same for the data of different times or firms to arrive at any meaningful decision. For example, you cannot compare two different companies figures if one company reports in rupees and another reports in dollars without making the base currency same.

(v) It must be numerically expressed in measurable units.

Data, when processed and presented in proper context, becomes information which controls the activity of the organization. Data is one of the major resources of the organization, developed over a period of time and therefore needs to be properly managed and safeguarded. It can be treated as inventory because it may be procured, stored and supplied when needed. Also just like any other physical stock it suffers from deterioration and obsolescence. Data may have different interpretations if not properly defined so the proper definition is very important. Data also has time dimension as its use and value will change with time and obsolete data is not very relevant for information needs. So it is important to understand the methods of collection of data so that the most relevant data can be collected and used as soon as possible for the effective management of the organization.

Check Your Progress

Fill in the blanks:

1. To solve any managerial problem that you face in the organization you need __________ information.

2. Data, when processed and presented in proper context, becomes information which __________ the activity of the organization


All the managerial processes require information in some form or the other and therefore, accurate and relevant data is required to accomplish almost all the tasks a manager has to perform. The following tasks are only indicative:

To set the objectives of the company, organization, industry, Government or any other business entity.

Activity What are data characteristics?

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To formulate major strategies and policies to meet specific objectives.

To report the result of operations of the business to the share holders.

To inform others of selected policies of the company.

To keep abreast of current operations of the business.

To inform employees on various matters.

To prepare long range plans.

To explore new opportunities.

To allocate capital resources.

To exercise necessary control over day-to-day operations.

To determine the costs underlying various activities of the firm.

To provide for proper co-ordination and control of business activities.

All organizations, whether social, political, religious or economic, are designed to achieve certain objectives. Notwithstanding the differences in the nature of their activities, the underlying management processes are common. The management must plan for and control the usage of various organizational resources namely, manpower, materials, production facilities and capital in the most effective manner to achieve the organization’s objectives. This involves decision making, which is dependent on the data and its quality. Data management is, therefore, being increasingly recognized as fifth organizational resource, which needs to be managed just like other four traditional resources of man, machine, material and money.

Check Your Progress

Fill in the blanks:

1. Accurate and relevant data is required to accomplish almost all the tasks a _________ has to perform.

2. The management must plan for and control the usage of various ________________ resources. (c)

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The major problems and limitations of data collection can be classified under the following broad headings:

(a) Lack of identification of data needs: The tasks performed by various levels of management are different. Therefore, data needs of the manager vary with the level at which they are operating and the function within which they are working. Very few organizations have made a conscious and deliberate effort to identify specific data needs of various managerial positions in the organization.

(b) Response time: This is one of the major data processing problems. The data is not collected and processed fast enough to allow enough time for mangers to react quickly and in time. There is a gap between supply of data and requirement of the data. i.e., Data is supplied with a lead time. In many cases data is not of any value when it is made available to the managers.

(c) Inaccessibility of data: Useful and necessary data is available but is often in a form or location that makes it uneconomical and infeasible to retrieve.

(d) Differing and conflicting data: Due to different sources used for collection, data about the some item may differ and may conflict with each other, for example, two market research agencies give you a different size of the prospective market for your products.

(e) Duplication of efforts: Identical data is maintained and similar reports are generated at several points in the organization, thereby wasting both time and manpower resources.

(f) Lack of training: The lack of scientific training in methodology of data collection is a great handicap in most of the organizations.

(g) Absence of code of conduct: There does not exist a code of conduct for use of data and managers often mould the data in the way they want to suit their needs without caring for the accuracy of the same. (c)

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(h) Inaccurate and unreliable data: The sheer volume of data and human intervention makes it humanly impossible to be consistently accurate and reliable.

Check Your Progress

Fill in the blanks:

1. Due to different sources used for collection, data about the some item may differ and may _________with each other.

2. There __________ exist a code of conduct for use of data and managers often mould the data in the way they want to suit their needs without caring for the accuracy of the same.


Data may be classified as:

(i) Primary

(ii) Secondary

Primary Data

Primary data represents those items that are collected for the first time and first hand. The data is recorded as observed or encountered. Essentially, this data is the raw material and may be combined, or structured in any form. The point to be noted here is that the data has not been statistically processed. For example, data obtained by counting the number of bad pieces and good pieces in the production is the primary unprocessed data. After this the data can be statistically processed to yield the required information.

The main advantages of collecting primary data are the following:

(i) They are accurate and reliable as they are collected from the original source.

(ii) They provide detailed information according to requirements of the users.

(iii) It is more reliable and less prone to error.

(iv) Definitions and meaning of terms used in data are explained to make it understandable and the process transparent.

Activity Give an example of primary data.

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(v) Method of collection, its limitations and other aspects are generally highlighted.

Where there are roses there will also be thorns. Following are the main limitations of the primary data:

(i) Cost: It is expensive to collect primary data.

(ii) Time: It is time consuming method of data collection.

(iii) Training: It requires experts/trained personnel to collect data.

Secondary Data

This is also known as published data. Data which is not originally collected but rather obtained from published sources and is normally statistically processed is known as secondary data. For example, data published by Reserve Bank of India, Ministry of Economic Affairs, Commerce Ministry as well as international bodies such as World Bank, Asian Development Bank, etc.

As is the case with primary data there are advantages and disadvantages associated with secondary data also. The advantages are:

(i) Cost: It is more economical than primary data, since data is already available.

(ii) Time: It is faster to collect and process as time has already been spent to collect the data.

(iii) Information insight: It provides a base on which further information can be collected to update it and finally use it. It provides valuable insights and contextual familiarity with the subject matter.

The limitations of secondary data are as follows:

(i) It may not be too relevant for the problem in hand as it was originally collected for some other context.

(ii) It could be outdated and hence not of much use in a dynamically changing environment.

(iii) The accuracy of secondary data as well as its reliability would depend on its source as the assumptions made during the data collection are not specified.

Activity Give suitable example of secondary data.

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(iv) Locating appropriate source and finally getting access to the data could be time consuming.

(v) The data available might be too extensive and a lot of time and money may be spent going through it.

Table 18.1: Distinction between Primary Data and Secondary Data

Parameter Primary Data Secondary Data

Source of Data Original source Secondary source

Method of Data Collection Observation method, Published data from Questionnaire method, etc. various sources

Statistical Processing Not Processed Usually processed

Originality of Data Original Not original. Data collected by First time collected by user some other agency

Use of Data Data is compiled for specific There may not be a specific purpose purpose

Terms and Definitions of Incorporated May not be incorporated Units

Copy of the Schedule Included Excluded

Method of Data Collection Given May not be given

Description of Sample Given May not be given Selection

Time Required More Less

Cost to the Organization Expensive Comparatively cheaper

Efforts Spent More Less

Accuracy of Data More accurate Less accurate

Training Experts/ trained people Less trained personnel required required

On closer investigation, it will be noticed that the distinction between primary and secondary data in many cases is of degree only. Data, which would be secondary in the hands of one, could be primary for others. For example, to a bank the details of the customer are primary data, but to a reader of the report of the bank these details are secondary.

Primary Data

Considerations in Selection of Primary Data Study

While selecting the subject for primary data collection, the following considerations should be kept in mind:

(i) Economic Considerations:

(a) Data collection efforts cost money. The value of the anticipated results must commensurate with the efforts put in. (c)

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(b) Short-term data collection studies that can yield appreciable dividends quickly should be preferred to long term studies whose benefits may be difficult to foresee.

(ii) Technical Considerations:

(a) It should be made sure that adequate technical knowledge is available to carry out the right process of data collection.

(b) Where a large problem throws up a number of subjects which are independent of each other, it is better to have small individual data collected on each subject.

(c) Where a problem brings to light two or more subjects, which are interrelated, independent studies on each might be carried out in the preliminary stages, but they should later be continuously integrated by co-ordinating the recording of the different teams working on each subject. The critical examination has to be the completeness of the data and it has to be carried out by the team as a whole.

(d) The scope and magnitude of the problem would determine the data required.

(iii) Human Considerations Where resistance to change or reaction is likely to be there the

data collection should not be proceeded with until acceptance has been gained.

(iv) Other Limitations and Constraints

(a) Time Limit: Data collection must be completed within time frame specified so as to be of maximum utilization.

(b) Cost Considerations: Data must be collected within the cost framework .

(c) Accuracy: Reasonable accuracy, as is required for the problem, should be ensured.

Methods of Primary Data Collection

The following four methods of primary data collection are most widely used:

1. Observation method

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3. Questionnaire method

4. Case study method

Let us look at each method, one by one:

Observation Method

This is the most commonly used method of data collection, especially in studies relating to production management and behavioural sciences. Accurate watching and noting down of phenomenon, as they occur in nature or at shop floor with regard to cause and effect, is called the observation method of data collection.

Differentiating characteristics of observation method are as follows:

(i) Direct Method: Direct contacts of sensory organs particularly eyes and ears are involved to gather and record the data.

(ii) Observe and Record: The observer first observes the phenomenon carefully and then records data.

(iii) Selective and Purposeful Collection: The observations are made with a definite purpose in mind and only relevant data is collected.

(iv) Cause and Effect Relationship: Observation method leads to development cause and effect relationship.

Box 18.1: Merits and limitations of Observation Method

Merits

This method of observation is common to all the discipline of research is simple to use.

It is realistic as it is based on actual and first hand experience.

The conclusions are more accurate reliable and dependable.

This method is used for formulation of hypothesis.

This method is successfully used for verification of hypothesis.

It is useful when in-depth study is required.

Limitations

Some events cannot be observed without biases. For example, it is not possible to observe emotions and sentimental factors, like and dislikes without bias about the degree of emotions.

Activity What is observation method?

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It sometimes results in illusory observations.

Being a long drawn process, the techniques of observation are expensive and time consuming.

Sometimes the atmosphere tends to become artificial and this leads to a sense of self-consciousness among the individuals who are being observed. This defeats the purpose of observation.

The slowness of observation methods leads to disheartening and disinterest among both the observer and observed.

The final results of observation depend upon the interpretation and understanding of the observer, the defects of the subjectivity in the explanation creep in the description of the observed and deductions from it.

As the purpose of the observation is known to observers, therefore, it is their own wish to record or view a particular thing.

The control can be of two forms. The observer could be a participant or a silent observer. In group discussions he is normally a silent observer but in interview techniques he becomes a part of the interview and hence his lack of objectivity may hamper the quality of his observations.

Controlled and uncontrolled observation methods are the two sub-methods used to watch and understand the observation.

(i) Controlled Observation: This is a systematic observation based on logic and reasoning. This is done on a preconceived plan and deliberate effort is made to control the phenomenon.

(ii) Uncontrolled Observation: In this method observations are made in a natural surrounding. There is no planning, no control and no use of any deliberate effort to change the working of the phenomenon.

Table 18.2: Distinction between Controlled and Uncontrolled Observation

Parameter Controlled Uncontrolled Observation Observation

Control Dimensions Control over the pheno- No control. Observations menon, conditions of light, under natural conditions temperature, humidity, etc. Control over the observer or observed

Techniques of Control Planning of observations No need to use control Used situations, Use of mechanical techniques appliances such as recorders,

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watch, etc. Maps and sociometric scales Hypothesis Detailed notes Group discussions

Degree of Bias Subjective study and bias This is an objective study comes in during study and keeps the observations bias free

Cause and Effect Well established Difficult to establish Relationship

Degree of Reliability High Low of Data

The process of observation method is used most effectively in the field observations where the presence of the observer does not make a difference to the observed. For example, if you want to know how many people enter the New Delhi railway station from the Paharganj side, you just have to stand at the gate and count. Your presence there or not being there does not matter to people who are being observed.

Steps in Organization of Field Observation

Following are the main steps generally followed in the organization of field observations:

(i) Determination of nature and limits of observation: Depending upon the nature of research and hypothesis, an outline of the research is prepared. This helps the observer to guide him on what should be observed and on what should be left out.

(ii) Determination of time, place and subject of study: A project can be of short or long duration, it may be studied under laboratory conditions or in the open. It should also be decided that whether we shall observe the behaviour of phenomenon as a whole or of the individual items in relation to the total.

(iii) Determination of the investigators: Depending on the nature, work content and objectives, the needs for individual investigator or of a team are to be identified.

(iv) Provision of mechanical appliances needed: The mechanical appliances for recording such as tape recorder, movie camera, etc., required should be identified in the beginning and used when needed.

When you take care of these basic steps, your data would be useful and relevant to the problem in hand.

Activity Discuss the main points in personnel interview.

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Personal Interview Method

Under this method of collecting data there is a face to face contact with the persons from whom the information is to be obtained (known as informants). The interviewer asks them questions pertaining to the survey and collects the desired information. For example, if a person wants to collect data about the working conditions of the workers of Hindustan Lever Ltd., Mumbai, he would go to the HLL factory at Mumbai, contact the workers and obtain the required information. The information obtained is direct and original. This is the most suitable method of data collection for business and economic problems.

Table 18.3: Merits and Demerits of Personal Interview Method

Merits Demerits

In this method, direct contact between researcher and informants is established and effective communication is built, which helps in getting direct information about paradigms, inner feelings, emotions and sentiments.

There are certain matters, which can be written in privacy but about which one does not wish to speak before others. If these matters are the subjects of interview, the likelihood is that only a disguised version of these will be presented.

Fine tuning of the responses can be done so as to get out the best possible by rephrasing the questions and probing deeper wherever required

If an interviewee is of low level intelligence he is usually unfit to give correct information. Same goes for interviewer also as interviewing is an art rather than science and the art has to be mastered

An interview gives us knowledge of facts, which are inaccessible to observation. The emotional attitude, secret motivation and incentives governing human life come to surface in an interview though these are unobservable. Therefore, interview has a quality which may be called supra-observational

If the interviewer is unable to suppress his prejudices, his understanding and interpretation of data given in the interview will be defective

Through this method it is possible to verify the information that has been collected from other sources.

In the interview, certain aspects of the human behaviour get overemphasized at the expense of others. There is a tendency to give too much importance to personal factors and minimize the role of environmental factors. This has to be guarded against.

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Types of Interviews

Interviews can be classified according to their basic characteristics.

(i) According to Formalness:

(a) Formal Interview: In formal interviews, the interviewer presents a set of well defined questions and notes down the answers of informants in accordance with the prescribed rules. Here, emphasis is given on the order and on sequence of question.

(b) Informal Interview: Here the interviewer has the freedom of alterations in questions to suit a particular situation in formal interview. He may revise, reorder or rephrase the questions to suit the needs of the respondents. The emphasis is on situation and questioning generally depends on the situation and individual.

(ii) According to Number:

(a) Personal Interview: In personal interview only a single person is interviewed at one time. Detailed knowledge about intimate and personal aspects of individual can be obtained as it is face-to-face talk.

(b) Group Interview: In this method two or more persons are interviewed at the same time. The group interview is, therefore, more suited for gathering routine information rather than personal information.

(iii) According to Purpose:

(a) Diagnostic Interview: In this type of interview, interviewer tries to understand the cause or causes because of which a particular fact or incident happened. For example, diagnostic interviews are held with the operators with a purpose to grasp the cause and nature of failure of machines and not to ascertain whether failure has occurred.

(b) Research Interview: These interviews are held to gather information pertaining to certain problems but may not be as specific as diagnostic interviews. The questions to be asked to gather the desired information are predetermined. In as much as this data is gathered for the purpose of research into a problem, this is called research interview.

Activity What are the different types of interview?

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(iv) On the basis of Function and Methodology:

(a) Non–directed Interview (Non Directional Interview): This is also known as free or unstructured interview. This is a type of interview in which the interviewer exercises no control, provides no direction and has no brief or predetermined set of questions to ask. The interviewer merely engages the interviewee in talks and encourages him to tell about his experiences and feelings. This type of interview is suitable when the researcher wishes to assess the amount of awareness a person has about certain problems and the manner in which he views them.

(b) Focussed Interview: This method is employed for studying the socio-psychological effects of mass media like radio, television, cinema, etc. The specialty of the focused interview is that by its means the personal reactions, emotions and intellectual orientation of the persons to be interviewed towards specific issues can be studied.

(v) Classification according to Subject Matter:

(a) Qualitative Interview: The qualitative interviews are about complex and non-quantifiable subject matter. For example, interviews held for case studies for specific problem study are qualitative, because the interviewer has to cover past, present and future to know a case. In this a qualitative analysis associated with a situation is performed. The subjective opinion of the interviewer is sought.

(b) Quantitative Interview: The quantitative interviews are those in which certain set facts are gathered about large number of cases. The census interviews are an example of this type.

Many combinations of these types can be made to suit a particular situation.

Getting Correct Response in an Interview

The main concern of the researcher employing the method of interview is to get correct and to the point answers to the topic of research. A research can be less expensive and economical only if deviations from the main line of approach are kept under control. Normally, the accuracy of the responses depends upon the skill and tactful approach of the interviewer and no rules can be framed in (c) C

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this connection. Still the following points can be kept at the back of your mind:

Prior to start the business with the interviewee, interviewer must develop rapport with the interviewer, so that he feels comfortable with him.

For allowing maximum opportunity of self-expression to the interviewee, he should be allowed to narrate his experience in the story form.

The interviewee and interviewer should be free and frank. The interviewee should be allowed to describe whatever he thinks worthwhile. Even if some irrelevant facts are being described the interviewee need not be checked. He should not be discouraged. Though maximum freedom of self-expression is desirable, this can only be within the scope of the problem being discussed. This requires alertness and direction at the suitable occasion. Good humour is the essence of successful direction.

The interviewer must hear the interviewee with full interest. Nobody should be able to guess from his expression that he is bored or his mind is elsewhere.

If an interviewer can convince the interviewee that he appreciates his cooperation and greatly values the information given by him, this word of encouragement has a salutary effect on the interviewee, who then gives more focussed responses.

The information given by the interviewee, if suspected, can be tested through cross-examination of the interviewee. Moreover, the emotional expression accompanying the responses give a clue to the interviewer about the veracity or otherwise of the answer being given.

Precautions to be taken while Interviewing

Following are the main causes, which render an interview unusable. These should be taken care of when interviewing:

(i) Often interviewees, under emotional spells, exaggerate the facts in order to satisfy their vanities and create impression. It should be taken with a pinch of salt.

(ii) Sometimes there is communication gap between the interviewer and interviewee with the result that interviewees (c)

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say one thing and the interviewer understands something else.

(iii) Some interviewees deliberately try to mislead the interviewer and make a fool of him, the interviewer must be mature and experienced enough to tell off and rebuff such fake interviewees.

(iv) Sometimes an inexperienced interviewer is offended by the behaviour of interviewees and in a revengeful mood distorts the facts in his report.

(v) Interviewer should critically examine those aspects of the interview in which the relationship of cause and effect seems to hold. This helps to determine whether the causes are always present or not when certain effects appear.

Questionnaire Method

Under this method, a formal list of questions pertaining to the survey (known as questionnaire) is prepared and sent to the various informants. Questionnaire contains the questions and provides the space for answers. A request is made to the respondents through a covering letter to fill up the questionnaire and send it back within a specified time.

The questionnaires could be structured or unstructured. Structured questionnaires are those that pose definite, concrete and preordained questions with fixed response categories. In unstructured questionnaires, questions are not necessarily presented to the respondents in the same wording and do not have fixed responses. Respondents are free to answer the questions the way they like in their own wording and style. Questionnaires could be a mix of the two types also leaving the field wide open to the designer of the questionnaire.

Types of Questions used in Devising a Questionnaire

Dichotomous Questions: When reply to a question is in the form of one out of two alternatives given, one answer being given in negative and other positive, it is called a dichotomous question. Both the negative and positive answers combined together form the whole range of answers given. For example: “Whether respondent is educated………………..Yes/No.”

Multiple Choice Questions: In these questions normally three to five alternative answers are given. These alternatives (c)

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are quite comprehensive and the respondent has to select one of them. In framing these types of questions, the framer has to be cautious enough that all the possible alternatives are included in it and they are mutually exclusive.

Ranking Item Questions: A variation on multiple choice questions, these questions are so designed as to record the preferences of the respondent. In ranking item questions there may be several preferences arranged item wise.

Open-ended Questions: Questions, which are of descriptive type and allow the respondent to cite his experiences, are known as open-ended questions.

Leading Questions: These are suggestive questions. In these types of questions the reply is suggested in a particular direction. They should be avoided as far as possible.

Ambiguous Questions: The questions that lack clarity and are so worded that the meaning is not clear are known as ambiguous questions. Such questions normally should not be included in the questionnaire as they are likely to confuse the respondent. The meaning of such questions is not uniformly convulsed to all the respondents.

Table 18.4: Merits and Demerits of Questionnaire Method

Merits Demerits In comparison to other methods,

the questionnaire method is both cheaper and quicker

Lack of interest on the part of respondents lowers the number of responses, making the study unreliable

It requires less skill to administer than other methods

Incomplete and illegible responses renders the whole response bad

If the informants or the respondents are scattered in large geographical areas, this is the most suitable method

If a problem requires deep and long study, it cannot be studied through this method

Besides saving money, time is also saved as simultaneously hundreds of persons can be approached

This method is very rigid since no alteration and rephrasing of questions can be used

It is more reliable in special cases although in most cases the reliability is suspected

Prejudices and bias of the researcher influences the framing of the questions

The respondent is free from external influences, such as researcher and therefore provides reliable, valid and meaningful information

Sometimes the questionnaire is itself incomplete and leaves out certain critical questions which are unearthed later rendering the whole exercise fruitless

Contd…

Activity What are Dichotomous Questions?

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Chances of errors are low because respondent supplies information himself

There is no provision in this method for coming face to face with the respondent. This may result in manipulation of replies by the respondents.

The informants are directly involved in the supply of information, so the method is more original

The impersonal nature of questionnaire ensures uniformity from one measurement situation to another.

Considerations in Questionnaire Design

Questionnaire is always framed with the help of certain background material and the problem statement. The first requirement always is the design of the problem statement and this is the area where most of the questionnaires go wrong. If your problem statement is faulty, your questions are not going to point to the required direction and you are bound to get wrong inferences. One should spend the maximum time on it, since it will be well spent.

After the problem statement comes, the issue of the respondents as their intellectual level has to be kept in mind while designing questions. If the questions, language and wordings are not in accordance with the intellectual level of respondents then it would not be possible for them to furnish correct replies. In such a situation the purpose of the research would not be fulfilled. The outcome of past experiences enables the researcher to know the shortcomings beforehand, enabling him to remove these deficiencies so as to improve the response rate.

Other factors to be taken into account in the construction of a questionnaire:

Appeal: Each questionnaire should be attached with an appeal in which the aim and purpose of the questionnaire is set forth and the sincere co-operation of the respondents is requested. The appeal may be made more effective by giving appropriate incentives in the form of money, books, and with a promise to give a copy of the report to the respondents.

Instructions for filling up the questionnaire: The questionnaire must carry a list of instructions for filling it up and dispatching it. The respondent must not have to pay for return postage, unless you are promising a prize for (c)

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responses. If the questionnaire is time bound, the last date of receiving completed response and the address should be clearly written.

Clarity of questions: For desired response it is of utmost importance to formulate questions that are direct, clear and precise.

Order of questions: The questions should be broken up into sections and each section should have a number of questions, which are mutually interrelated. Question about personal detail should be avoided or should be asked in the end.

Protesting of Response in Questionnaire

The basic thing that has to be kept in mind is that ambiguity should be avoided in collecting data through questionnaire method. For this, it is necessary that the questionnaire should be tested before it is actually used in a business research study. Pre-testing is nothing but testing of questionnaire before it is actually used. If testing is to be done the right way the following steps are required:

Testing the validity in a representative sample: The questionnaire should be tested in every respect, before it is actually mailed to the target segment. This testing can be done on a limited number of people through sampling method but while testing it within the sample, it should not be forgotten that the sample should be perfectly representative of the target segment.

Protesting to check whether the results are in tune with objectives: The questionnaire should meet the objective of research study. It means that it should help in getting maximum possible relevant responses. It is, therefore, necessary that it should be made suitable to objectives of study even if it requires testing more than once.

Poor response requires modification of the questionnaire: The questionnaire is mailed to the informants who are required to fill it and send it back. If the response of the informant is poor and very few questionnaires are returned, it means that there is something wrong with the form and style of the questionnaire and it requires modifications/change and reframing. Furthermore, if the questionnaires returned are incomplete or the replies are not

Activity What are the problems of response?

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satisfactory and up to the mark, it should be presumed that the questionnaire is defective and it requires modification. After modification the questionnaire should again be subjected to pre-testing.

Problem of Response: Difficult Situations for the Researcher

When the questionnaire is not leading to any response, one of the following factors is usually responsible for it:

Importance of the problem to the respondents: It is generally seen that those who are concerned with the problem give better response than those who are not.

Characteristics of the respondents and prestige of the sponsoring body: It is seen that educated people with social consciousness are more responsive as compared to people belonging to lower economic group. If the research study is sponsored by a well-known organization it is likely to have better response.

Form and nature of questionnaire and arrangements of the questions: Questionnaire also plays its part in the matter of response. If the questionnaire is short and has been printed in attractive manner, its layout is neat and attractive, the arrangements of questions are scientifically planned, and it is likely to invite a better response.

To get better response, inducement is needed. Inducement may be classified under two heads: monetary and non-monetary. Monetary inducement is given generally to people who are economically weak or likely to be influenced by money. This money is given in advance or after receiving the filled questionnaire. Non-monetary inducement may be in the form of a reward. It may be a letter of appreciation or mentioning of the name in the report of study and so on. The suitability of the inducement to the study and the respondent’s expectations has to be kept in mind when deciding upon which inducement to use.

To take care of the poor response situations companies normally get students to help to find respondents and to filling up the questionnaires. Excellent way for you to make money while studying! (c) C

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Schedule

Schedule is a variation of the questionnaire and can be defined as a proforma that contains a set of questions which are asked and filled by an interviewer himself in a face-to-face situation with interviewee. Unlike a questionnaire, the schedule acts as a guideline to the interviewer trying to get the required response from the interviewee. Schedule is a standardized device or a tool of observation to collect data in an objective manner. Same guidelines as mentioned in the questionnaire are to be kept in mind while making these schedules.

Case Study Method

Case study method may be defined as small, inclusive and intensive study of a situation in which investigator uses all his skills and methods for systematic gathering of enough information about a situation to understand the problem and its solution. The case study is a form of qualitative cum quantitative analysis involving the very careful and complete observation of a person, situation or institution.

Table 18.5: Merits and Demerits of Case Study Method

Merits Demerits Intensive and deep study of

the problem is possible Several unrealistic assumptions may

be made when structuring the case, making it difficult to relax them later on

Study of subjective aspects of the problem is possible and more elaborative than other methods

It is expensive in terms of money, time and energy

Comparison of possible problem statements is easier

If there is improper understanding between the developer and the respondents, the data and hence the inferences could be false and misleading

Valid hypothesis can be formulated and tested while the case is in development

Prejudices and biases come in more easily as the study is more subjective

Is very useful when you have to study processes and not isolated incidents

It is not possible to apply sampling methods and generalization often leads to false conclusions.

Very useful in situations where more of qualitative rather than quantitative decision making is involved.

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Check Your Progress

Fill in the blanks:

1. _____________ is a variation of the questionnaire and can be defined as a proforma that contains a set of questions which are asked and filled by an interviewer himself in a face-to-face situation with interviewee.

2. _____________method may be defined as small, inclusive and intensive study of a situation in which investigator uses all his skills and methods for systematic gathering of enough information about a situation to understand the problem and its solution.

Secondary Data

Although primary data is required for most of the internal business situations, many of the strategic decisions depend upon the information that is external to the organization. The criticality of the decision and the time factor involved would decide whether secondary data is to be used or the situation calls for primary data.

If the situation calls for secondary data, this data would normally be either published or unpublished. Unpublished records, although dealing with the matters of public interest, are not available to people in published form. It means that everybody cannot have access to these records. Proceedings of the meetings, noting on the files, private research, etc., form the category of unpublished records. Normally these records are very reliable since there is no fear of their being made public; the writers give out their views clearly.

Published records are available to people for investigation, perusal and for further use, survey reports, magazine articles, published studies, etc., fall under this category. The data contained in these documents can be considered reliable or unreliable depending upon the agency that is collecting the data and the sources it had used for collecting this data. Most of the information that is now available to people and researchers in regard to business environment are to be found in the form of reports. The reports published by governments are considered more dependable on one hand and on the other hand some people think that the reports (c) C

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that are published by certain private individuals and agencies are more dependable and reliable.

There are so many sources of published data that it is impossible to name them all here. In spite of so many sources, the published data usually suffers from the following drawbacks:

(a) Data about all the aspects of business and economic activity are not collected.

(b) Even the Government of India does not have an up-to-date and latest data about many socio-economic aspects as well as the business environment, although it is now working towards it.

(c) Data lacks in homogeneity and continuity.

(d) The data collected by the Government agencies is not beyond doubt. This is due to the approach of the administration and also because of the method of data collection. The resources that are put at the disposal of the machinery that is entrusted for the task of collection of data are very meagre.

(e) Data collected by private agencies run the risk of their biases coming into picture, as also their own aims and objectives could make them present the data in an improper way rendering it unuseful for you.

Therefore, before using the secondary data, it is essential that the investigator should satisfy himself that the data is: (a) Reliable, (b) Suitable, (c) Adequate and (d) Timely. Reliability of data can be established by asking yourself the following questions: Who collected the data and from which sources? Are the methods used in collecting are standard methods and reliable? Whether both the compiler and source are dependable? The purpose for which the data were originally collected is in tune with the purpose that you are going to use the data for; the secondary data should be suitable for the purpose of enquiry. Even if the data is reliable it should not be used if the same is found to be unsuitable for the enquiry. For checking the suitability of data one should see: What was the object of the enquiry? The definitions of various items and units of collection must be carefully scrutinized. What was the accuracy (c)

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aimed at? What is the time of collection of data required? Can it be regarded a normal time? Is the data homogeneous?

The secondary data may be reliable and suitable but the same may be inadequate for the purpose of investigation. The data collected earlier may refer to a problem area which could be narrower or wider than the area required for the present enquiry and if it is such, the data should be carefully scrutinized to test whether it meets the requirements or not. If it does not meet the requirements of the scope or the time frame of study, do not use the data just because it is the only data that is there. Although knowledge of the matter under consideration and proper use of the statistical methods is presupposed, great care is necessary in dealing with published statistics because of the limitations or inaccuracies that may be present.

Check Your Progress

Fill in the blanks:

1. If the situation calls for _____________ data, this data would normally be either published or unpublished.

2. ___________ records, although dealing with the matters of public interest, are not available to people in published form.

3. The cost of collection of ................... is much higher than the collection of secondary data.

4. Under ................... method, the investigator collects data from a third party or witness or head of an institution, etc.

5. The collected data are arranged into ................... groups.

6. A classification is said to be ................... if it brings out essential features of the collected data.

Summary

Any set of numerical figures cannot be regarded as statistics or data. A set of numerical figures collected for the investigation of a given problem can be regarded as data only if these are comparable and affected by a multiplicity of factors. Data from a primary source are original and correspond to the objective of investigation. (c) C

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However, the secondary data are often available in published form, collected originally by some other agency with a similar or different objective. Generally, the primary data are more reliable than the secondary data which, however, are more economical.

Lesson End Activity

How will you design a questionnaire?

Keywords

Data: The facts and figures that is collected, analyzed and interpreted.

Observation: The set of measurements obtained for a single element in the data set.

Qualitative Data: Data that are labels or names used to identify an attribute of each element. Qualitative data use the nominal or ordinal scale of measurement and may be non-numeric or numeric.

Statistics: The art and science of collecting, analyzing, presenting and interpreting data.

Quantitative Data: Data that indicate that how much or how many of an element. Quantitative data use the interval or ratio scale of measurement and are always numeric.


1. Define data. What are the key characteristics of data?

2. Why data is required in the organisations?

3. What are the key problems in data collections?

4. What are the different types of data?

5. Make a distinction between primary and secondary data.

6. What are the key methods of collecting primary data?

7. What are the different types of interviews?

8. What are the key drawbacks of collecting data from secondary sources? (c)

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Further Readings

Books




Web Readings




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UNIT 19: Presentation of Data

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Unit 19

Presentation of Data


Categorical data and numeric data

Utility and Advantages of Diagrammatic Presentation

Numeric data

Categorical data

Introduction

An important function of statistics is the presentation of complex mass of data in a simple way so that it becomes easier to understand. Classification and tabulation are the techniques that help in presenting the data in an intelligible form. But with increase in volume of data, it becomes more and more inconvenient to understand even after its classification and tabulation. Thus, to understand various trends of the data at a glance and to facilitate the comparison of various situations, the data are presented in the form of diagrams and graphs.

Categorical Data and Numeric Data

Both categorical data and numeric data can be used to create statistics. 1. Numerical Data: Numeric data are data that exist in

numeric form, such as height, the number of children in a household and annual income. Numeric data is most definitely quantitative, and can be either "continuous," such as time, or income, or "discrete," such as number of children.

2. Categorical Data: Categorical data are data that take a finite set of values that can be either numeric or categorical. For example, eye colour, ethnicity, and country of residence are all possible types of categorical data that are not numeric.

Activity What are numeric data?

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Statisticians often refer to categorical data as "qualitative data," while social scientists might consider categorical data to be "quantitative."

Check Your Progress

Fill in the blanks:

1. Numeric data are data that exist in ______ form, such as height, the number of children in a household and annual income.

2. Statisticians often refer to categorical data as ___________ data

Utility and Advantages of Diagrammatic Presentation

Data presented in the form of diagrams are useful as well as advantageous in many ways, as is obvious from the following: 1. Diagrams are attractive and impressive: Data presented

in the form of diagrams are able to attract the attention of even a common man. It may be difficult for a common man to understand and remember the data presented in the form of figures but diagrams create a lasting impression upon his mind. Due to their attractive and impressive character, the diagrams are very frequently used by various newspapers and magazines for the explanation of certain phenomena. Diagrams are also useful in modern advertising campaign.

2. Diagrams simplify data: Diagrams are used to represent a huge mass of complex data in simplified and intelligible form which is easy to understand.

3. Diagrams give more information: In addition to the depiction of the characteristics of data, the diagrams may bring out other hidden facts and relations which are not possible to know from the classified and tabulated data.

4. Diagrams save time and labour: A lot of time is required to study the trend and significance of voluminous data. The same data, when presented in the form of diagrams, can be understood in practically no time.

5. Diagrams are useful in making comparisons: Many a times the objective of the investigation is to compare two or more situations either with respect to time or places. The task of comparison can be very conventionally done by the use of diagrams. (c)

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6. Diagrams have universal applicability: Diagrams are used in almost in every field of study like economics, business, administration, social institutions and other fields.

Check Your Progress

Fill in the blanks:

1. Data presented in the form of diagrams are able to attract the ___________of even a common man.

2. Diagrams are used in almost in _____________ of study like economics, business, administration, social institutions and other fields.

3. _____________ is a systematic presentation of numerical data in rows and columns.

Summary

Presentation of data is provided through tables and charts. A frequency distribution is the principal tabular summary of either discrete or continuous data.

Lesson End Activity

Collect data on IOC, HPCL, BPCL & ONGC company’s financial performance from different sources like internet, newspapers, magazines, etc. Are there any differences between the same? What inferences do you draw on the objectives of that particular type of media when they are presenting data?

Keywords

Data Presentation: The presentation of complex mass of data in a simple way so that it becomes easier to understand.

Frequency Distribution: A frequency distribution is the principal tabular summary of either discrete or continuous data.

Numerical Data: Numeric data are data that exist in numeric form, such as height, the number of children in a household and annual income.

Categorical Data: Categorical data are data that take a finite set of values that can be either numeric or categorical. (c)

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1. What is data presentation means?

2. What is frequency distribution meant for?

3. What is numerical data?

4. What is Categorical data?

5. What are the different techniques of data presentation?

6. Try and collect as much data as possible from different sources about the health levels of the people residing in your area. What problems come in while collecting this data?

7. If you have to use the sampling method in question 6 what method would you use and how would you reduce sampling and non-sampling errors in your sample?

8. Ansal Builders is engaged in the construction of a multi-storey building for setting up a lube factory. It has recently conducted a cost audit. The manager (cost accounting) has collected the figures of total cost and its major constituents. The information collected as percentage of expenditure is shown below. Represent the data with the help of a suitable diagram.

Item Expenditure % Wages 25 Bricks 15 Cement 20 Steel 15 Wood 10 Supervision and Misc. 15

9. Chand Contractors supplies contract labour to various industrial units for carrying out their various production activities in and around Bhilai. Mr. R.B. Tripathi is the chief consultant and is responsible to manage the continuous supply of contract labour on weekly basis. The daily wages of contract labour varies from ` 25 to 95 per day depending on the skill, experience and the nature of work in the industry utilizing the services of contract labourers. The daily wages and number of workers data have been compiled by Shri Tripathi for estimating the number of workers demanded and their average wages.

Draw a suitable diagram of the data to enable the chief executive of Chand Contractors to understand the relations between wages and number of workers. (c)

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Find out the number of workers getting wages lower than 57 and more than 77 using Ogive graphs.

Daily Wages (`)

No. of Workers

Daily Wages (`)

No. of workers

20-25 21 60-65 36 25-30 29 65-70 45 30-35 19 70-75 27 35-40 39 75-80 48 40-45 43 80-85 21 45-50 94 85-90 12 50-55 73 90-95 5 55-60 68

10. IBM Computers (I) Ltd. has been entrusted with the responsibility of developing a relationship between number of employees and salary structure in Arian Pharmaceuticals Ltd. The statistics manager, Mr. Ayyar has collected the following data. Draw the frequency distribution and superimpose frequency polygon and frequency curve on it.

Salary No. of Employees Salary No. of Employees 300-400 20 700-800 115 400-500 30 800-900 100 500-600 60 900-1000 60 600-700 75 1000-1200 40

Further Readings

Books




Web Readings



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Unit 20

Case Studies


Case Study 1: ASSOCHAM

Associated Chamber of Commerce and Industry (ASSOCHAM) is very much concerned about the employment of youths and their pay rolls in small oil an industries, with special reference to ancillary parts manufacturing, transport for hire, taxis, dealers of new and old vehicles, petrol stations and automobile repair garages. The chamber has employed you to collect the data regarding employment and payroll as on 31st March, 2000 and present it suitably through diagram so that it can be include in the final memorandum to be submitted to the Minister for Industries.

The data that you have collected is as follows:

Industry Employment on 31-3-2000

Avg. Earnings per employee per year (`)

1. Parts manufacturers 4,34,856 56,540

2. Transport for hire 15,26,897 26,348

3. Taxis 11,32,560 42,685

4. Dealers of new and used vehicles

1,09,805 13,684

5. Retail filling stations 22,25,960 15,008

6. Automobile repair garages

12,35,200 12,048

Question

Present the data using a suitable diagram(s) so as to bring out the finer points.


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Case Study 2: Mount Shivalik

Mount Shivalik Distilleries is a progressive manufacturer of ‘Wasp’ brand export quality rum. It follows the modern practices of presentation of data in various board meetings. The data collected by its Finance Director over a period of 3 years pertaining to its operations is shown below.

Particulars 1997-1998

1998-1999

1999-2000

(a) Raw Material Cost per bottle of rum 9 15 21

(b) Other costs 6 10 14

(c) Packing and Distribution 3 5 7

Sales proceeds per bottle (excluding excise) 20 30 40

Profit / (Loss) 2 0 (2)

Question

The Finance Director desires that the data should be presented diagrammatically. Would you please help him in presenting the data?


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Detailed Contents UNIT 21: SAMPLING

Introduction

Sampling and Sampling Design

Theory of Sampling

Methods of Sampling

Precautions in Using Sampling Methods

Sampling Reliability

Sampling and Non-Sampling Errors

UNIT 22: BASIC TOOLS OF DATA ANALYSIS–I

Introduction

Frequency Distributions

Interpretation of Frequency Distributions

Understanding and Using Averages

Kinds of Averages

UNIT 23: BASIC TOOLS OF DATA ANALYSIS–II

Introduction

Commercial Averages

Comparison of 3 M's of Statistics

Measuring Dispersion

Coefficient of Variance

UNIT 24: FORECASTING

Introduction

The Conceptual Model

The Mathematical Model

Simple Forecasting System

Algorithms and Applications

The Regression Algorithm

Simple Linear Regression Analysis

Correlation Analysis (The Parametric Case)

Correlation Analysis (The Non-parametric Case)

Time Series Analysis

Linear Analysis

Non-linear Analysis

Multiple Regression Analysis

The Smoothing Algorithm


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Unit 21

Sampling


Sampling and Sampling Design

Law of Statistical Regularity

Theory of Sampling


Introduction

If we want to make a study, which involves the total population, there are two methods of doing it. One, we talk to each and every member of the population and another which takes a representation of the whole population and does the study on it. It is obvious that the second method is less costly, faster and easier for us to use. The only problem that can crop up is that the representation is not the true representation of the whole population or is not a representation at all. For this there are certain special statistical techniques used which help in checking that the representation used actually and truly represents the population. These techniques are called sampling techniques and the representation is called the sample.

Sampling and Sampling Design Sampling can be defined as the selection of some part of an aggregate or totality on the basis of which a judgment or inference about the aggregate or totality is made. Thus, only after studying a part of the whole population, inference is drawn on the whole population. The whole population (or the desired group that we want to study from which a sample is drawn) is called the universe. Here onwards we will only use the term population.

Population could be finite or infinite depending upon the number of elements in it. For example, the number of books that one publisher sells is finite and can be known but the same thing cannot be said about the number of people which have gone

Activity What is sampling means?

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through these books so that becomes an infinite population. Population can also be divided into real and hypothetical. Real population refers to hard facts which in the above case are the number of books published. It can be hypothetical or imaginary, for example, the number and types of emotions that you displayed in last one hour! These can only be projected or imagined but you can never be sure.

Characteristics and elementary units are the other two terms that you need to know about here. Characteristics refer to the attributes (non-quantified qualities) which are the objects of the study. Elementary units refer to those units which possesses the characteristics of the population. The total of such elementary units is called the population.

A sampling design is a definite plan for obtaining a sample from the sampling frame. It refers to the technique or procedure of selecting some sampling units from which inferences about the population are drawn.

Sampling Errors: Since in sample survey, only a small part of the universe is studied, as such there is every possibility that its result would differ to some extent from that of the universe. Even if two or three samples of the same universe are taken, the result would differ from each other. These differences constitute the errors due to sampling and are known as sampling errors. Errors due to calculations or improper convention of observation are called non-sampling errors.

Sampling Distribution: If we take certain number of samples and for each sample and compute various statistical measures such as mean, standard deviation, etc., then we can find that each sample may give its own value for statistics under consideration. All such values of a particular statistic, say mean, together with their relative frequencies will constitute the sampling distribution.

The confidence level or reliability of the sample is found from the sampling distribution. It is the expected percentage of times that the actual value will fall within the stated precision limits. Thus, if we take confidence level of 95%, we mean that there are 95 chances in 100 that sample results represent the true condition of the universe. The significance level has the opposite interpretation of that of confidence level and indicates the likelihood that the answer will fall outside the range. You should always remember that if the confidence level is 95% then the significance level will be

Activity What is sampling design?

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5%, which essentially means that there are five per cent chances that the sample will not represent the true condition of the universe. The sample you have selected must follow the following law.

Law of Statistical Regularity

The law states that, if a moderately large number of items are selected at random from a given universe, the characteristics of those items will reflect, to a fairly accurate degree, the characteristics of the entire universe. For example, if 500 leaves are picked from a tree at random and the average length is found out, the result will be nearly the same if all the leaves of the tree are picked up and measured.

The reliability in the Law of Statistical Regularity depends on two factors:

(i) The larger the sample, the more reliable are its indications for the population. The reliability of a sample is proportional to the square root of the number of items it contains and larger the sample the more representative and stable it will be.

(ii) The sample must be chosen at random.

With the use of law we can say that a part of the population can represent the population. When census is not possible due to paucity of time, money and/or labour, then with the help of this law and random sampling, investigations can be made about the properties of the population. This is possible because the selection is made at random and by this law all types of units, whether good, bad or average, have equal chance of being selected.

However, there are certain precautions which we should keep in mind. The selection of units should be unbiased, i.e., random. The two characteristics of randomization – One can not fit any relationship between occurred values. Secondly, the probability of occurrence of every item should be the same. The inferences drawn from this are applicable on an all units of the population so the sample should be identical to universe. By collecting information from smaller units, we cannot apply the results drawn from it to the whole universe.

The Law of Inertia of large numbers is a corollary of the Law of Statistical Regularity. It lays down that in large masses of data,

Activity Define law of statistical regularity.

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abnormalities will occur, but in all probability, exceptional items will offset each other, leaving the average unchanged subject to where the elements of the time enters in the general trend of data. The law of Inertia of large numbers asserts that large aggregates are relatively more stable than small ones. The movements of an aggregate are the result of the movements of its separate parts and it is improbable that the later will be moving in the same direction at the same time. Consequently, their movements will tend to compensate one another and the larger the numbers involved, the more complete will this compensation is. Thus, the law states that the larger the number of items we take out of a given universe, the greater is the probability of accuracy.

Check Your Progress

Fill in the blanks:

1. The Law of Inertia of large numbers is a corollary of the Law of ___________.

2. The reliability of a sample is proportional to the _______ of the number of items it contains

Theory of Sampling

The theory of sampling is a study of relationships existing between a universe and sample drawn from the universe. It is applicable only to random sampling.

The theory of sampling is concerned with estimating the properties of universe from those of the sample and also with gauging the precision of estimation. Sampling theory deals with the following aspects:

(a) Statistical Estimation: Sampling theory helps in estimating unknown population parameters from the knowledge of statistical measures based on sample studies. The estimation can be a point estimate or it may be an interval estimate. Point estimate is a single estimate expressed in the form of a single figure and interval estimate has two limits viz., the upper limit and lower limit within which the parameter value may lie. For example, we can say that the number of defective parts in 100 pieces is 10 based on the sample of 10 in which 1 defective part was found or we can say that the defective parts could be from 8-12 based on the many samples.

Activity What is theory of sampling?

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(b) Testing of Hypothesis: The second objective of the sampling theory is to accept or reject the hypothesis. It helps in determining whether observed differences are actually due to chance or whether they are really significant.

(c) Statistical Inferences: Sampling theory helps in making generalization about the universe from the studies based on sample drawn from it. It also helps in determining, the accuracy of such generalization.

In the quantified research, the sampling technique is made maximum use of and in no field of research can its importance and value be belittled. In researches in the educational, economic, commercial and scientific domains, the sampling technique is used and considered most apt for research. Sampling technique also has a very high value in day to day activities. In making our daily purchases of foodstuff, vegetables, fruits, etc., it is not considered necessary to examine each and every piece of the commodity. Only a handful of goods are examined and the idea about the whole lot is formed and this usually proves justified. For example, the physicians make inference about a patient’s blood through examination of a single drop.

Sampling technique has the following features, which highlight its importance:

(a) Economy: The sampling technique is in expensive and less time consuming than the census technique.

(b) Reliability: If the choice of sample units is made with due care and the matter under survey is not heterogeneous, the conclusion of the sample survey can have almost the same reliability as those of census survey.

(c) Detailed study: Since the number of sample units is fairly small, it can be studied intensively and elaborately. These can be examined from multiple points of view.

(d) Scientific base: This is a scientific technique because the conclusions are verifiable from other units. By taking random samples we can determine the amount of deviation from the norm.

(e) Greater suitability in most situations: Most of the surveys are made by the technique of sample survey, because if matter (c)

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is homogeneous, the examinations of few units suffice. This is the case in the majority of situations.

The question that arises is whether we can use sampling techniques in any situation. No, sampling is useful only when:

(i) Data is vast: When the number of units is very large, sampling technique must be used as it economizes money, time and effort.

(ii) When utmost accuracy is not required: The sampling technique is very suitable in those situations where cent per cent accuracy is not required otherwise census technique is unavoidable, because 100% accuracy is achievable only by this means.

(iii) Where census is not feasible: If we want to know the mineral wealth in the country we cannot dig all the mines to discover and count, we have to use the sampling technique.

(iv) Homogeneity: If all the units of a domain are alike, sampling technique is easier to use and is much more accurate.

The point to remember is that if due care is not taken in the selection of samples or if they are arbitrarily selected, the conclusions derived from them about the universe will be misleading, if not totally wrong. For example, in assessing the monthly expenditure of university students, if we select for our sample only the students who come in cars, our results will be highly erroneous if extended to all students.

You should remember that the sampling technique can be successful only if a competent and able investigator makes the selection. If the sampling is done by an average investigator, the selection may be prone to error.

Check Your Progress

Fill in the blanks:

1. The ___________ is in expensive and less time consuming than the census technique.

2. Most of the surveys are made by the technique of ___________ .

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Methods of Sampling

Sampling methods can be grouped under two broad categories:

(i) Probability sampling methods or random sampling methods.

(ii) Non-probability sampling methods or non-random sampling methods.

Probability Sampling Methods

Probability sampling methods are those in which every item in the universe has a known chance or probability of being included in the sample. This implies that the selection of item for the sample is independent of the person making the study and the items will be chosen strictly at random.

Probability sampling can be divided into four types. We will take a look at each one of them, one by one.

(i) Simple random sampling.

(ii) Stratified sampling.

(iii) Systematic sampling.

(iv) Cluster sampling.

(i) Simple Random Sampling: A procedure of sampling will be called simple random sampling where individual items (units) constituting the samples are selected at random. Random sampling is the form applied when the method of selection assures each individual element or unit in universe can have an equal chance of being chosen. In other words, if in a sample size of ‘n’ all the possible combinations of ‘n’ element items have the same probability of being included; it is called simple random sampling. It can be performed with replacement of the taken out element or without replacement of the taken out element.

Selecting a Random Sample

A random sample can generally be selected in following four methods:

(a) Lottery method.

(b) Tippet’s numbers method.

Activity What is lottery method?

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(c) Selection from sequential list.

(d) Grid system.

A brief description of the above methods is given below:

(a) Lottery method: In this method, a unit is drawn by writing the numbers or the names of various units and putting them in a container. They are thoroughly mixed and certain numbers are picked up from the container, and those picked up are taken up for sampling.

(b) Tippet’s numbers method: It is called Tippet’s numbers method because it was evolved by L.H.C. Tippet who constructed a list of 10,4000 four digit numbers written at random on every page. From those numbers it is not very difficult to draw samples at random. For example, if 50 persons are to be selected for study out of the total number of 500, then we can open any page of Tippet’s numbers and select first 50 that are below 500 and take them up for study. On the basis of the experiments carried out through this technique, it has been found that the results that are drawn on the basis of this method of random sampling are quite reliable.

(c) Selection from sequential list: In this method, the names are arranged serially according to a particular order. The order may be alphabetical, geographical or only serial. Then out of the list any number may be taken up. Beginning of selection may be made from anywhere.

For example, if we want to select 10 persons, we can start right from the 10th, and select 10, 20, 30, 40 and so on.

(d) Grid System: This method is generally used for selecting the sample of an area and so in this method, a map of entire area is drawn. After that a screen with the squares is placed upon the map and some of the squares are selected at random. Then screen is placed upon the map and the areas falling within the selected squares are taken as samples.

(ii) Stratified Sampling Method: In this method, the entire population is divided into a number of groups called strata. Then a number of items are taken from each group at random. (c)

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This means that a stratified sample is equivalent to a set of random samples on a number of sub-populations. It can be performed with replacement of the taken out element or without replacement of the taken out element.

As you would have guessed by now, in this method much depends on the process of stratification. For taking the right strata the following precautions need to be taken:

(a) Each stratum in the population should be large enough in size so that selection of items may be done on random basis.

(b) There should be a perfect homogeneity in different units of any one stratum.

(c) Stratification should be well defined and clear cut. By this we mean that each unit or stratum should be free from influence of the other.

Types of Stratified Sampling

Stratified samplings are of three kinds:

(i) Proportionate stratified sampling: In this method the number of units drawn from each stratum are in exact proportion to proportion of strata to the population.

(ii) Disproportionate stratified sampling: In this type of stratified sampling an equal number of cases are taken from each stratum without any consideration to the size of strata in proportion to population. It is also called “controlled sampling” or a quota is decided for each state.

(iii) Stratified weight sampling: In this method, an equal number of units are selected from each stratum and averages are drawn from each stratum, but in doing so they are given weight in proportion to the size of stratum in relation to the whole population.

(iii) Systematic Sampling: Systematic sampling is a variation of simple random sampling. It requires that the universe or a list of its units may be ordered in such a way that each element of the universe can be uniquely identified by its order. A voters list, a telephone directory, a card index system would all generally satisfy this condition. Suppose, there are 5000 (c)

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cases (and hence 5000 units of the population) and we want a sample of 50. We can select a number of between (and including) 1 and 10 at random, say 8. Then we can select the units whose case is in the following position 18, 28, 38,........., 1008,.........., 4998. This would be a systematic random sample or commonly known as systematic sampling.

Systematic selection implies that the sample units are picked out in a definite sequence, at equal intervals from one another. Reduction or increase in the variability of estimates yielded by systematic sampling depends on the way population is arranged. If the population is thoroughly mixed with respect to the characteristics under study, the variability of the estimates will be affected.

In practice, it is essential to use systematic sampling only when we are sufficiently acquainted with the data to be able to demonstrate that periodicities do not exist, or that the interval between the elements of the sample is not multiple or submultiple of the period.

(iv) Cluster Sampling: Cluster sampling is also called multistage sampling or sub-sampling as it uses various stages to reach or make samples. This method is generally used in selecting a sample from a very large population. The original units into which the population is divided are known as primary units. Each primary unit that falls into the sample is subdivided into secondary units in preparation for the second stage of sampling. In three stage sampling; there will be primary, secondary and tertiary units. Sometimes four stages are also used.

Let us take an example to understand the procedure as it is slightly tedious. Let us say that we want to take sample from the universe of professors/lecturers associated with Delhi University. The list consists of 100 pages with approximately 25 names per page. These pages are numbered and constitute the sampling units. All names are numbered and arranged alphabetically to constitute the ultimate sampling units. Let us suppose we want a sample of 100 professors/lecturers. The sample may be selected as follows: we may decide to select 5 professors/lecturers each from 20 pages. Select a number from 1 to 5 at random say 3. Select pages 3, 8, 13, 28, ... and so on to 98. Then by the use of random numbers (c) C

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select 5 names from each of 20 pages. This is a combination of systematic and simple random sampling.

The point to remember is that you should use a form of random sampling in each of the sampling stages where there are two or more than two stages.

The variability of estimates yielded by multistage sampling may be greater than that of estimates yielded by simple random sampling for equal size. The variability of estimates in multistage sampling depends on the composition of primary units. There are three reasons for hesitating to recommend this method.

(i) The cost of the travel would be too high.

(ii) Control of the non-sampling errors would be difficult; and

(iii) A probability sample of small units drawn at one stage requires a form which lists all the small units and such a plan becomes costly by comparison.

Non-probability Sampling Methods

Sampling methods which do not provide every element in the universe a known chance of being included in the sample are collectively known as non-probability sampling methods. Here the selection process is partially subjective and does not use randomization. In other words we use judgements based on convenience and other considerations rather than probability considerations.

Non-probability sampling methods can be divided into basically three groups:

(1) Judgement or purposive sampling,

(2) Convenience sampling, and

(3) Quota sampling.

Judgement or Purposive Sampling

In this technique the investigator has complete freedom in choosing his sample according to his wishes and desires. Although he will try his best to get the sample which is representative of the population, his judgment plays a major part in determining which the best sample is and no other considerations are used for the same. (c)

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When only a small number of sampling units are in the population, simple random selection may miss the more important elements, where judgement selection would certainly include them in a sample. For example, when we want to know the effectiveness of HR policies of the company and we randomly choose the sample of 5 from a company of 50 people, it is possible that we only get marketing people and not a representation of other functions in the sample.

This is bound to give us improper ideas about the effectiveness of HR policies if the marketing people do not like the HR people in the organization for whatever reasons. This personal selection can become a disadvantage also when not used properly. Another disadvantage that is associated with this method is that there is no objective way of evaluating the reliability of sample results.

Still when we want to study some unknown traits of a population, some of whose characteristics are known; we may then stratify the population according to these known properties and select sampling units from each stratum on the basis of judgement. This method will then result in a more representative sample.

Convenience Sampling

As the name suggests, in this method the sample of the population being investigated is selected neither by probability nor by judgement but by convenience of reach. A sample obtained from readily available lists such as automobile registrations, telephone directories, etc., is a convenience sample and not a random sample even if the sample is drawn at random from these lists. For example, if you do a survey on the internet about the social issues, your sample is a convenience sample and not representative of all the people to whom these issues concern. Therefore, the results obtained by convenience sampling methods are generally biased and unsatisfactory.

So convenience sampling is normally suitable for doing pilot studies and in cases where the population is not well defined or sample units cannot be clearly defined or when the complete data about the population is not available. (c) C

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Quota Sampling

Quota sampling is a special form of stratified sampling. In this method, first the classification of the population into various strata is done in terms of properties known or assumed to be pertinent to the characteristics being studied. Then proportion of the population falling into each stratum on the basis of known or estimated composition of the population is defined. After that the quotas for each interviewer or investigator are determined so that the total sample interviewed contains a proportion of each stratum so that all investigators study all the stratums thereby doing a complete study of the population in a mini form.

The advantage of using this method is that items, which are close to each other, are clubbed together, thereby saving costs and introducing some stratification effect. The disadvantage is that the bias of investigator is introduced in classification of subjects and in random selection within various strata. Another disadvantage is that since random sampling is not involved at any stage, the errors of the method can not be estimated by statistical procedures.

This method is mostly used in marketing surveys and election polls and is pretty successful in that.

Multiphase Sampling

Sometimes it is economical and organizationally convenient also, to collect certain items of information from the all the units of a sample and other items of information from some of these units only, these latter units being selected so as to form a sub sample of the original sample. This may be termed as two phase sampling or double sampling. If necessary another phase may be added.

Multiphase or sequential sampling is of great use when the desired accuracy of different items is widely different, either owing to the fact that the variability of the associated variants is different or because the desired accuracy is different.

The major advantage is that it is possible to select one sample from a universe, analyze it and use the inferences in designing a second sample from the same universe. But this method can only be used where a small sample can represent the universe very well and where the number of observations can be increased easily at any stage of enquiry. (c)

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Check Your Progress

Fill in the blanks:

1. Quota sampling is a special form of _________ sampling.

2. In ______________ method, the classification of the population into various strata is done in terms of properties known or assumed to be pertinent to the characteristics being studied.

3. _________ or sequential sampling is of great use when the desired accuracy of different items is widely different.

Precautions in Using Sampling Methods

In order to be useful, the study has to be representative in character. If it does not possess all the characteristics of the universe it shall not be representative enough and thus, it shall not be able to fulfil the objective of the study. In order to enable the investigator to keep himself away from the biased samples, he has to take the following precautions:

If the universe is subject to change, an enquiry carried out on a single occasion, howsoever accurate, cannot by itself give any information on the nature of the rate of such change. In such cases, provision must be made for studying the samples at successive intervals if up to date information is desired. This successive study also gives some idea about the nature and question of change.

The size of the sample should not be too small as compared to the universe. The size of the sample should be large enough so that its representative character may not be lost and selection on random basis may be possible. Apart from it, the sampling should not be done purposely. In such an event, the sample generally gets biased.

If the sampling is done through stratified method; it should not be governed by the principle of perfect stratification. Elements of unsuitability, overlapping or lack of proportion have no place in sampling. When these elements are there, the samples become biased.

Activity What is multiphase sampling?

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Lack of source list or incomplete source list makes the sample biased.

If the cases, which were originally selected for the study, are lost or not available for enquiry, they are replaced by new ones. In such a situation, there is a danger of the bias influencing the selection of samples.

When field workers are given the liberty to select samples according to their wishes and no specific guidelines are given to them, they are likely to select samples according to their convenience. In that event prejudices and bias are likely to influence the sample.

If the method of drawing samples is inadequate or not suitable to the project, the samples drawn may be biased. Sometimes the nature of the phenomenon makes the selection of representative samples extremely difficult. It generally happens in case of complex, heterogeneous and widespread cases.

The investigator has to safeguard against the bias and try to find out perfectly representative samples. In this, the investigator and his skills play a vital role.

If the investigator is well equipped with the knowledge of the universe, know the importance and the nature of the study and makes efforts to collect the representative sample, he would be successful in selecting representative samples and take precautions for removing the bias. Pre-testing is very helpful in determining whether a particular sample is truly representative or not.

Check Your Progress

Fill in the blanks:

1. ________ is very helpful in determining whether a particular sample is truly representative or not.

2. Lack of source list or incomplete source list makes the sample __________. (c)

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Sampling Reliability

That the sample should be reliable and free from biases goes without saying, but how that needs to be tested. The size of the sample, its relevance and suitability to the problem, its representation of the universe, etc., are some of the factors that determine the reliability of the sample. Reliability may be tested on the following parameters:

Size of the sample: The size of the sample for study very much determines, not only its representative ness but also its utility for study. The investigator must test that the size is adequate for scientific and convenient study of the problem.

By testing the representative ness of the sample: The representativeness of the sample should also be tested. It means that the sample selection should be representative and possess the characteristics of other units.

By drawing a parallel sample: It means that apart from the samples that have been drawn, another sample may be drawn from the same universe for testing. On the basis of these tests, the reliability of the sample, primarily selected, may be tested. The comparison of two sample values gives you a better understanding of sample.

By testing the homogeneity of the samples: Samples should be homogeneous. They should possess all the characteristics that are present in the population.

Through comparison of the measurement of the sample with those of the population: Sometimes, different measurements about the universe are also known. In order to test the reliability of the samples, the investigator may apply his knowledge and thereby test the reliability of the sample.

Unbiased selection: The selection of sample should be done through a method that is free from bias and prejudices.

By taking a sub-sample from a main sample: This is a method of sampling within sampling. Out of the universe, we draw a sample, but in order to test the reliability of the sample, we draw a sub-sample, from main sample and study it intensively and compare the findings of the study of the sub-sample with the findings of the study of the main sample. This (c)

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helps the investigator to detect any error that might have crept in.

Check Your Progress

Fill in the blanks:

1. The size of the sample for study very much determines not only its representativeness but also its _________ for study.

2. The _______________must test that the size is adequate for scientific and convenient study of the problem.


Errors in any statistical investigation, i.e., in collection, processing and analysis of the data, may be broadly classified as: (i) Sampling errors and (ii) Non-sampling errors.

Sampling Errors

In a sample survey, only a small part of the universe is studied. As such there is every possibility that its results would differ to some extent from that of the universe. Even if two or more samples of the same universe are taken the results will differ to some extent from that of the universe, as well as from each other. The difference would be always present even if the sample is drawn at random. These differences constitute the errors due to sampling and are known as sampling errors. This is the error which is the result of sample or sampling procedural and it always exists in same quantity.

Sampling errors creep in because of the following reasons:

(i) Faulty selection of the sample: Purposive selection of sample would result in biases, as in this case, the investigator deliberately selects the representative sample. If, however, the selection of sample is haphazard, the chances of bias errors are great.

(ii) Incomplete investigation (or non-response): If all the items to be included in the sample are not covered there will be bias. This occurs frequently in case of sample collected through questionnaire method. All the questions in the questionnaires are not responded properly. Again if the

Activity What are sampling errors?

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selected person is not interviewed at any time to collect information, bias may arise.

(iii) Faulty collection of data: During the process of collection of data certain errors and mistakes may creep in due to the following reasons:

(a) Negligence or prejudice of enumerator in putting questions or recording answers.

(b) Lack of knowledge on the part of person furnishing information.

(c) Poorly designed questionnaire.

(d) Unorganized method of collection of data.

(iv) Substitution: Due to non-availability of the selected person (or item) the investigator may interview another person from the same sample. The second person may not have the same characteristics as the original one. This will introduce the substitution bias in the sample and as such deviate the result.

(v) Faulty Analysis: Faulty method of analysis of data may also introduce the sampling error.

Enough has been written about the biases of the investigator or the respondents. There could also be unbiased errors that creep in due to accident or by natural course of events without any bias by enumerator or informant. They occur due to chance factor that of the member of universe being excluded or included in the sample selection. Further, this type of error occurs when only a partial observation of universe is made and is equal to the difference between sample statistic and parameter of universe.

If the overestimated and underestimated values of observations are nearly equal, then errors in one direction will compensate the errors of other direction. Therefore, the unbiased errors are usually known as compensatory errors as they tend to offset each other and leave little effect on the general results.

Non-sampling Errors

The above discussion about the sampling errors seems to imply that studies of the entire population are free from any errors. (c) C

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Nothing could be farther from the truth. Errors may occur at any stage of enquiry, i.e., planning, collection, processing and analysis of data. Apart from sampling errors, errors arise due to following reasons:

Faulty planning resulting in improper definition of the problem statement.

Vague or incomplete definition of universe.

Imperfect questionnaire which might result in incomplete or wrong information.

Defective data collection.

Acceptance of exaggerated or irrelevant or wrong answers to the questions that satisfy the pride or self-interests of the respondents.

Personal bias of the user of the report.

Improper understanding and definition of the variables.

Improper use of averages to replace the actual figures.

Application of wrong methods.

Defect in measuring instruments.

And this list is by no means exhaustive! The magnitude of the errors that can creep in indicate that sampling is a technique which must be used selectively and objectively. Reducing sampling and non-sampling errors can be achieved by keeping in mind the above mentioned points and not loosing the sight of the objectives of the study at any stage. Still some errors would invariably creep in.

Check Your Progress

Fill in the blanks:

1. There could also be ___________ errors that creep in due to accident or by natural course of events without any bias by enumerator or informant.

2. Selection of the ___________ problem, as has already been stated, should be in line with the researcher's interest, chain of thinking and existing research in the same area and should have some direct utility.

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3. What is most important in selecting a research problem is that the research topic should be within ___________ .

4. Grid system is applied for selection of sample from a ___________ .

5. Purposive selection of samples, as the name goes, depends more on the researcher's ___________ .

6. Stratified sampling combines the characteristics of random sampling and ___________ .

Summary

Sampling can be defined as the selection of some part of an aggregate or totality on the basis of which a judgment or inference about the aggregate or totality is made. Thus, only after studying a part of the whole population, inference is drawn on the whole population. Population could be finite or infinite depending upon the number of elements in it.

A sampling design is a definite plan for obtaining a sample from the sampling frame. It refers to the technique or procedure of selecting some sampling units from which inferences about the population are drawn. The law states that, if a moderately large number of items are selected at random from a given universe, the characteristics of those items will reflect, to a fairly accurate degree, the characteristics of the entire universe. The theory of sampling is concerned with estimating the properties of universe from those of the sample and also with gauging the precision of estimation.

Sampling methods can be grouped under two broad categories: (i) Probability sampling methods or random sampling methods and (ii) Non-probability sampling methods or non-random sampling methods.

Errors in any statistical investigation, i.e., in collection, processing and analysis of the data, may be broadly classified as: (i) Sampling errors and (ii) Non-sampling errors.

Lesson End Activity

Write a note on non-probability sampling methods. (c) C

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Keywords

Sampling: Sampling can be defined as the selection of some part of an aggregate or totality on the basis of which a judgment or inference about the aggregate or totality is made.

Sampling Design: A sampling design is a definite plan for obtaining a sample from the sampling frame.

Simple Random Sampling: A procedure of sampling will be called simple random sampling where individual items (units) constituting the samples are selected at random.

Non-sampling Errors: Errors due to calculations or improper convention of observation are called non-sampling errors.


1. Define sampling.

2. What is the law of statistical regulatory?

3. Write a note on sampling theory.

4. What are the key features of sampling techniques?

5. What are the key methods of sampling?

6. Make a difference between simple random sampling and stratified sampling.

7. Differentiate between the sampling and non-sampling errors.

Further Readings

Books




Web Readings



www.mathbusiness.com (c)

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UNIT 22: Basic Tools of Data Analysis–I

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Unit 22

Basic Tools of Data Analysis–I




Kinds of Averages

Introduction

Collecting and collating data is one serious matter as we saw in the last unit, getting a meaningful analysis out of it is another. In the next few units we will focus on understanding the tools that we require to analyze this data.

The data collected could be in terms of qualitative variables or in terms of quantitative variables. Examples of qualitative variables include items termed as defective or effective, persons classified as rich or poor, etc. Care must be taken before we quantify these qualitative variables for this is one of the major sources of errors. Quantitative variables may be discrete, continuous or a combination of the two. Discrete variables take on only whole number values, for example, number of defective parts in a sample, number of married people in a city, etc. Continuous variables can be measured to any arbitrary degree of accuracy, for example, the weight of a person can be measured to the nearest kilograms, grams, milligrams, etc. The result may or may not be a whole number. The accuracy desired should be such that the relevancy of the data is not lost and it is not too difficult to get the desired data.


The data that you have collected till now, either through sampling or otherwise, are called raw data. Now this data can be arranged in an array. For example, if you collected data on electricity consumption for one day of 1000 households, you would get an array with 1000 rows and two columns, 100 rows for the houses and the two columns for house numbers and electricity

Activity What is the difference between collecting and collating data?

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consumption respectively. As it is very difficult to draw inference from this raw data, we can process this data so as to show the number of houses, which are using electricity within a particular range, together. This table of electricity consumption ranges and number of houses is shown below:

Table 22.1 Frequency Distribution of Electricity Consumption

Electricity Consumption (kilowatts) Number of houses 0-9 1

10-19 3 20-29 5 30-39 10 40-49 20 50-59 35 60-69 50 70-79 70 80-89 100 90-99 130

100-109 130 110-119 100 120-129 70 130-139 50 140-149 35 150-159 20 160-169 10 170-179 5 180-189 3 190-199 1

Note that, we are not using the house numbers as the information is irrelevant for the purpose of determining how many houses use how much electricity per day. Also note that, this is only one day’s utilization and is not indicative of the average utilization of electricity by that household. If it is Sunday, the overall average would be lower that what you can draw an inference from this data. This kind of over-generalization is very common and is a frequent source of errors in real life situations. It is important that a frequency distribution should have a suitable number of class intervals. Class intervals mean the ranges for which we classified the number of items. In the above case 10 is the class interval used for electricity consumption. If too few classes are used, the original data would be so compressed that little information will be available. If too many classes are used, there will be too few items (c) C

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in the classes, and the frequency polygon would be irregular in appearance.

There are basically three precautions that must be kept in mind when determining the class intervals. First we must select the class interval so that the mid-values of the classes will coincide, as far as possible, with the concentration of items that may be present. Second, we should avoid open-ended classes. Third, the class intervals should usually be uniform.

Check Your Progress

Fill in the blanks:

1. The data that you have collected till now, either through sampling or otherwise, is called ________ data

2. There are basically three precautions that must be kept in mind when determining the class intervals. First ___________ Second, ______________ Third, ___________.

Interpretation of Frequency Distributions

Frequency distributions may differ in average value, dispersion, shape or any combination of the three. Mathematicians have named almost all kinds of shapes which combine the properties of these three variations. They do not add too much value to your learning and therefore, are not mentioned here.

We have discussed averages, dispersion and skewness graphically and you must be thinking that there must be a way to measure these quantities mathematically. There is and that is what we are going to discuss next before going on to probability distributions.


In a series of statistical data that parameter which reflects a central value of the series is called the central tendency. Central tendency refers to the middle point of a statistical distribution and is also known as an average.

Average as a Measure of Central Tendency

An average can be defined as a central value around which other values of series tend to cluster. An average is computed to give a concise picture of a large group: By the use of average complex

Activity Define the term average

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groups, large numbers are presented in a few significant words or figures. Averages help in obtaining a picture of the universe with the help of sample. Although sample and the universe differ in size, still their average may be very much identical.

Averages give a mathematical concept to the relationship between different groups, for example, the trees in one forest are taller than in another forest but in order to find any definite ratio of heights it is essential to resort to averages.

But is an average a representative? Yes, essentially because of three reasons:

(i) Ordinarily most of the values of a series cluster in the middle,

(ii) At the extreme ends the number of items is usually very little, and

(iii) Ordinarily items with values less than the average cancel out the items whose values are greater than the average – The average of 4, 5, 6 is 5. The average 5 is less in value and is more in value by one towards both the extremes. Thus, the two deviations -1 and +1 cancel each other.

An average should be affected as little as possible by sampling fluctuations, i.e., for different sample of same population the variation in the average is very little. An average should be capable of algebraic treatment so that it can be used for further mathematical manipulation.

Check Your Progress

Fill in the blanks:

1. An average should be affected as little as possible by sampling ____________.

2. For different sample of same population the variation in the average is ________________

Kinds of Averages

Averages may be classified into three broad types:

1. Mathematical Averages:

(a) Arithmetic mean (c) C

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(b) Geometric mean

(c) Harmonic average

2. Positional Averages:

(a) Mode

(b) Median

3. Commercial Averages:

(a) Moving average

(b) Progressive average

(c) Quadratic average Mathematical averages are those which utilize mathematical formula for the calculation of their values. Positional averages do not use mathematical calculations but give you an indication about the positional characteristics of certain items. Commercial averages are the applications of averages in commercial situations.

If so many varieties of averages are there, the question that arises is which one to use. As we go ahead we would see that each type has a specific application and should be used only in that case.

Mathematical Averages

Arithmetic Mean

Most of the time when we refer to the average we are talking about arithmetic mean. This is true in cases like average winter temperature in Delhi; average life of a flash light battery, average working hours of an executive, etc. The arithmetic mean (or simply mean) is the quantity obtained by dividing the sum of the values of items (∑X) in a variable by their number (n), i.e., number of items.

XXnΣ=

The mean of 3, 4, 5, 6, 7 is. X 3 4 5 6 7X 5n 5Σ + + + += = =

Looking at the formula above we can say that the algebraic sum of the deviations of the individual items from the arithmetic mean is zero. If the sum of the deviations of individual items from the

Activity Establish the relation between mean, median and mode.

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mean is zero, then the sum of squares of the deviations is minimum when taken from the arithmetic mean than taken from any other item.

This means that if any one or more items in the group are replaced by new items, the new arithmetic mean would be changed by the net change divided by number of items. For example, if the values 3 and 4 in the above example changes to 8 and 9 (total change of 10) then the mean can be calculated in either of the two ways mentioned below:

X 3 4 5 6 7X 7n 5Σ + + + += = =

OR

Change in Value 10New X Old X 5 7n 5

= + = + =

Although in this example it would have been faster to do it the original way, the alternate method assumes more and more significance as the number of items go up.

When a frequency distribution is given, as in the table 22.1, the mean is calculated using a variation on the above formula.

fxXnΣ=

Here f stands for frequency of the class, x stands for mid-value of the class and n stands for total of all frequencies in all classes.

Revised Table 22.1 is reproduced below as Table 22.2.

Table 22.2: Frequency distribution of Electricity Consumption

Electricity Consumption (kilowatts)

Mid-value (x) Number of houses (f)

fx

0-9 5 1 5 10-19 15 3 45 20-29 25 5 125 30-39 35 10 350 40-49 45 20 900 50-59 55 35 1925 60-69 65 50 3250 70-79 75 70 5250 80-89 85 100 8500

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90-99 95 130 12350 100-109 105 130 13650 110-119 115 100 11500 120-129 125 70 8750 130-139 135 50 6750 140-149 145 35 5075 150-159 155 20 3100 160-169 165 10 1650 170-179 175 5 875 180-189 185 3 555 190-199 195 1 195

Total Value 848 84800

Applying this formula to the table we get:

fx 84800X 100n 848Σ= = =

In the case where cumulative percentage distribution is given, grouped frequency distribution is derived from the cumulative percentage distribution and then the usual procedure is applied for computing the mean.

If two or more groups contain respectively N1, N2, … observations with means X

1, X2, … respectively, then the combined mean (X) of

the composite group is given by the relation:

1 2 3N

1 2 3

NX NX NX ...XN N N ...

+ + +=

+ + +

Here N stands for the sum of the denominator (N1+N2+N3+…) Table22.3: Critical Evaluation of Arithmetic Mean

Merits Demerits It is rigidly defined and is

definite When distribution is highly skewed on

either side, arithmetic mean looses its representativeness

Its calculation is easy and generally understood

Its calculation requires information about all units, either individually or collectively. Therefore, it can not be safely used in open end tables

The data needs very little preparation, e.g., it need not be arrayed

It is not suitable for non-homogeneous series

It utilises all the data in the groups

Can't be applied for extremely large values on either side

It is suitable for arithmetic and algebraic manipulation

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Weighted Average

In calculating simple arithmetic mean it is assumed that all items were equal in importance. It may not be the case always. When items vary in importance they should be assigned weights in order of their relative importance. For calculating the weighted arithmetic mean the value of each item is multiplied by its weight, product summated and divided by the total of weights and not by the number of items. The result is the weighted arithmetic average. Symbolically:

1 1 2 2 3 3w

1 2 3

x w x w x w ...Xw w w ...+ + +

=+ + +

Here w1, w2, w3, … stands for the respective weights of each of the items.

Weighted averages have important applications in trend analysis and forecasting. But it should be used when any of the following conditions holds true:

1. When the importance of all the items in a series is not equal.

2. When the items falling into different grades of the classes of the same group show considerable variation and it is desired to obtain an average which would be representative of the whole group, weighted average is the only proper average to be used. In other words, when the classes of the same group contain widely varying frequencies.

3. When the percentages, rates or ratios are being averaged.

4. When there is a change either in the proportion of frequencies of items or in the proportion of their values.

It would not be improper to remind you that simple mean and weighted mean are two of the mostly used means.

Geometric Mean

The most important application of geometric mean is in the construction of index numbers, i.e., averaging rates of change. For example, if you are investing in the stock markets, and your money grows from ` 1,00,000 to ` 2,50,000 in three years and you want to know what is the average percentage gain you are making over the three years, you can use this mean. (c) C

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This could simply be written as:

33

0

XGX

=

This leads us to the general formula for geometric mean

nn

0

XGX

=

Mathematically speaking, the geometric mean is the nth root of the product of n items of a series.

The only problem with using this formula is that, you cannot do it on a simple calculator and this is the biggest drawback of it.

Geometric mean is also useful in skewed distributions and averaging ratios.

Table 22.4: Critical Evaluation of G.M.

Merits Demerits It gives comparatively little weight

to extreme values. It cannot be used when any of the

quantities is zero or negative

It is suitable for arithmetic and algebraic manipulation

It is difficult to compute and requires more time in computation

It is reversible both ways and therefore, suitable for ratios and percentages

It is difficult to understand

It gives less weight to large items which sometimes may be a limitation, viz., computing average cost per unit.

Harmonic Mean

It is defined as the reciprocal of the arithmetic mean of the reciprocals. Thus, for a simple harmonic mean:

1 2 3 4 n

nH 1 1 1 1 1...x x x x x

=+ + + + +

For a weighted harmonic mean, the above equation is rewritten as:

w

1 2 3 4 n1 2 3 4 n

wH1 1 1 1 1w w w w ....wx x x x x

Σ=⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

+ + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

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Although harmonic mean is of limited use, it is less affected by extremely large observations than any other average. It is properly used to average rates where the weights are the numerators of the fractions used to compute the rates.

Positional Averages

Mode

Mode is that value which has the maximum frequency (i.e. occurs most often) in a given set of values. Thus the mode of a set of data is simply the value that is repeated most often. It is the most typical value and, therefore, the clearest example of a measure of central tendency.

For example, if you leave for your office everyday in the morning and you recorded the following times for two weeks:

8.30 8.25 8.35 8.29 8.31 8.30 8.32 8.31 8.31 8.31

One thing is obvious, you are quite punctual! Anyway if you arrange the data in increasing order

8.25 8.29 8.30 8.30 8.31 8.31 8.31 8.31 8.32 8.35

Here 8.31 occurs most frequently and is therefore the mode of the given range.

You must be thinking that there usually be two items of exactly the same size for a continuous variable, (if measurements are made with sufficient precision), it is apparent that our definition of the mode is somewhat vague. For this we group the data and then use this simple equation:

Mode = l1 + 1

1 2

dd d+

× C, where

l1 = lower boundary of the class containing the largest frequency

d1 = difference of the largest frequency and the frequency of the last class d2 = difference of the largest frequency and the frequency of the next class

C = class interval

The main advantage of mode is that the value of mode is not affected by the extreme values of the series. Plotting also is not difficult, if there are more than one mode in the series, then it is not difficult to determine it and it can be located graphically also, very easily. (c) C

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Table 22.5: Critical Evaluation of Mode

Merits Demerits It is very easy to locate. In many cases

it can be obtained by inspection. It is frequently ill defined

It is not influenced by the presence of a small number of extreme items.

It sometimes is indeterminable without modifying the data

It may be ascertained even when the details of extreme items are not available.

It cannot be calculated by simple arithmetic process

It is easily understandable. It is unsuitable for arithmetic and algebraic manipulation

It may be determined with considerable accuracy from a well selected sample data.

Median

Median is the value of that item in a series which divides the series into two equal parts, one part consisting of all values less and the other all value greater than it. Defined in another way median is that value of the central tendency, which divides the total frequency into two halves.

Table 22.6: Critical Evaluation of Median

Merits Demerits It is easy to understand and calculate

It cannot be calculated by mathematical methods and therefore is not suitable for algebraic treatment

It eliminates the effect of extreme items

Median is usually affected by fluctuations of sampling. Data must be arranged before calculation of median

In many cases it may be obtained by inspection

It is unsuitable when greater importance to large or small values is to be given

It is easy to locate, subject to the actual number of items being known

It may be difficult to locate and when located it may not be representative in case the items in a series are not suited closely together

Its position is more definite than that of mode

A correct total cannot be obtained by multiplying the median by the number of items.

It is clearly and rigidly defined.

Calculation of median from simple series is very simple. If the data set contains an odd number of items, the middle item of the array after arrangement is the median. If there is an even number of items the median is the average of the two middle items after arrangement.

Activity What are grouped and ungrouped distribution?

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Calculation of median from simple frequency distribution (Ungrouped) is also easy. The cumulative frequency (less than type) corresponding to each distinct value of the variable is calculated. If the total frequency is N, the value of the variable corresponding to cumulative frequency gives the median.

Calculation of median from simple frequency distribution (Grouped) is slightly more complex. The following formula is used:

m

N F2Median = L+ CF

−×

Where,

L = Lower boundary of the median class

N = Total frequency

F = Cumulative frequency below the class immediately preceding the median class

Fm = Frequency of the median class

C = Class interval or width of the median class.

Check Your Progress

Fill in the blanks:

1. Median is the value of that item in a series which divides the series into two ________ parts.

2. Mode is that value which has the __________ frequency (i.e. occurs most often) in a given set of values

Summary

The data collected could be in terms of qualitative variables or in terms of quantitative variables. The data that you have collected till now, either through sampling or otherwise, are called raw data. As it is very difficult to draw inference from this raw data, we can process this data so as to show the number of houses, which are using electricity within a particular range, together.

In a series of statistical data that parameter which reflects a central value of the series is called the central tendency. Central tendency refers to the middle point of a statistical distribution and is also known as an average. (c) C

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An average can be defined as a central value around which other values of series tend to cluster. Averages may be classified into three broad types: Mathematical Average, Positional Averages and Commercial Average.

Lesson End Activity

What are the key situations in which weighted average is more suitable than other methods of computing averages?

Keywords

Average: An average can be defined as a central value around which other values of series tend to cluster.

Mode: Mode is that value which has the maximum frequency (i.e. occurs most often) in a given set of values.

Median: Median is the value of that item in a series which divides the series into two equal parts, one part consisting of all values less and the other all value greater than it.

Mathematical Averages: Mathematical averages are those which utilize mathematical formula for the calculation of their values.

Positional Averages: Positional averages do not use mathematical calculations but give you an indication about the positional characteristics of certain items.

Moving Average: The moving average is an arithmetic average of data over a period and is updated regularly by replacing the first item in the average by the new item as it comes in.


1. Illustrate the preparation of frequency distribution with a suitable example.

2. Define average. What are the different types of averages and their implication?

3. What are the different types of positional average? (c) C

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Further Readings

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Web Readings




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UNIT 23: Basic Tools of Data Analysis–II

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Unit 23

Basic Tools of Data Analysis–II


Commercial Averages



Introduction

In this unit, we will discuss about commercial averages which will include moving average, progressive average and quadratic average. We will further study comparison of the mean, median and mode. We will also focus on measuring dispersion.

Commercial Averages

These are classified as follows:

Moving Average

The moving average is an arithmetic average of data over a period and is updated regularly by replacing the first item in the average by the new item as it comes in. It is useful in eliminating the irregularity of time series and is generally computed to study the trend.

Suppose the prices for 12 months are given and a three monthly average is to be computed. Then the first item in the 3-month moving average would be the average [(a1+a2+a3)/3], the second item would be the average of the next three months [(a2+a3+a4)/3] and so on. The last item would be the average [(a10+a11+a12)/3]. As the next month would come in a10 would be dropped and a13 would be added in [(a11+a12+a13)/3] and so on.

Progressive Average

Progressive average is also calculated with the help of simple arithmetic mean. It is a cumulative average. In computation of

Activity What is meant by commercial average?

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progressive average, figures of all previous years are added and divided by the number of items. As the number of items go up and reach a desired number, we switch to moving average.

Quadratic Average

The quadratic mean or average is estimated by taking the square root of the average squares of the items of a series.

Symbolically,

2 2 2 2

ma b c ... nQ

N+ + + +=

Where Qm = Quadratic Mean

a2, b2, c2 .....n2 = squares of the different values

Quadratic average is useful when some items have negative values and other positive values because in such cases the mean is not very representative. It is also used in averaging deviations, rather than original values, when the standard deviation is computed.

Check Your Progress

Fill in the blanks:

1. The moving average is an _________ average of data over a period.

2. Quadratic average is useful when some items have ______________ values and other positive values because in such cases the mean is not very representative.


A comparison of the mean, median and mode is necessary for you to understand the positive and negative characteristics of them.

Table 23.1: Comparison of 3 M's of Statistics

Parameter of Comparison Average Mean is a calculated average. Median and Mode are

averages of positions. If all the items in a variable are the same, then only AM=GM=HM otherwise AM > GM > HM

Calculation Mean is the sum of the values of the items divided by the number of items in the series. Median is the middle value which divides a series into two equal parts. Mode is the value around which the items of a series tend to concentrate in density, i.e., most occurred frequency.

Activity Give an example of progressive average.

Contd….(c) C

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Treatment Mean is capable of mathematical treatment. Median and Mode are not capable of such treatment.

Items All the items in a series are taken into account in the calculation of Mean. Median and Mode calculations do not consider all the items in a series.

Extreme Values Mean is affected by extreme values of the items in a series but it is not so in case of Median and Mode.

Calculation in Mean calculations of frequency distribution with open-ended

Open-ended distributions Reliability

class intervals at both ends is not possible. Median and mode of such distribution can he easily calculated. Mean is considered to be more reliable measures of central tendency than Median and Mode.

Result In a series of distribution of data, there is only one value of mean or median. But there could be more than one mode or no mode at all. Mean is simple to understand and to calculate.

Use Mean is widely used. Median and Mode have limited use.

Check Your Progress

Fill in the blanks:

1. _____ is a calculated average.

2. Median and Mode are averages of __________


It is only because of variability that we compute averages. But if there is too much variability among the data, an average is so unreliable that it is almost useless. Usually, a high degree of uniformity (i.e. a small amount of dispersion) is a desirable quality. Mass production would usually be uneconomical if there was a large amount of variability in materials or manufactured parts, for standardization and interchangeability of parts it is essential to have low variability.

Therefore, we will consider several measures of dispersion, but there are only two that are used much; the range and the standard deviation.

Range

The range is the difference between the largest value and the smallest value.

R = xn – x1

Activity What are 3M’s of statistics?

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For the example on leaving for office times, the range was 10 minutes (8.35 - 8.25).

The problem with using range is that it considers the extreme values only and does not use all the data in the sample. It is therefore less reliable than some other measures of dispersion and varies too much from sample to sample to be of effective use. It is also very sensitive to the size of the sample, it usually increases with the increase in the sample size, although not proportionately.

In spite of these shortcomings, there are special situations where the range is useful. When the sample is from a 'normal' universe with a small sample size, the quantity is nearly as reliable as the more laboriously calculated standard deviation. There are also certain types of data and certain purposes for which the use of range is appropriate, e.g., the range of temperature in Delhi.

Mean Deviation

Also known as average deviation, mean deviation is the mean of the absolute amounts by which the individual items deviate from the mean. The following procedure is usually applied:

1. Calculate the absolute deviation from the mean, removing any negative signs.

2. Sum all the deviations.

3. Divide the sum of the deviations by the total number of items.

Symbolically, these steps may be summarized as follows:

E xMD

n=

Item number

X Deviation from mean Absolute deviation from mean

1 10 10 10 2 20 20 20 3 30 30 30 4 -10 -10 10 5 -20 -20 20 6 -30 -30 30 7 -25 -25 25 8 25 25 25

Total 0 0 170

Activity What is range?

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E x 170MD 21.25n 8

= = =

Mean deviation is simple and easy to understand and unlike R, it is affected by the value of each item. But it is unreliable because it varies from sample to sample taken from the same universe. Also it is a biased estimator of the population. Therefore standard deviation, discussed below, is the most often used measure of population dispersion.

Standard Deviation

The standard deviation of a sample SD is similar to the mean deviation in that it considers the deviation of each X value from the mean. However, instead of using the absolute values of the deviations, it uses the squares of the deviations. These are summed, divided by n, and the square root extracted.

The formula for standard deviation (SD or s as it is usually represented)

2XSD( )n

Σσ =

Variance is the square of SD and is represented by: 2XVariance ( ) =

nΣσ

The detailed calculation is shown below:

Item number X Deviation from mean

Square of Deviation

1 10 10 100 2 20 20 400 3 30 30 900 4 -10 -10 100 5 -20 -20 400 6 -30 -30 900 7 -25 -25 625 8 25 25 625

Total 0 0 4050

2X 4050SD ( ) 22.5n 8

σ = = =

The concept of using sums of squares of deviations about the arithmetic mean of a distribution is very important and we would use it extensively in the units that follow.

Activity Make a difference between M.D and S. D.

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Coefficient of Variance

To get an indication of the variation that is related to the mean, we divide the standard deviation by the mean to get the coefficient of variance. This enables us to compare two groups, which have different standard deviations and means more easily.

Coefficient of Variance = σμ

× 100

Skewness

Skewness may be defined as the lack of symmetry or degree of distortion from symmetry exhibited by a probability distribution. Any measure of skewness indicates the difference between the manner in which the items are distributed in a particular distribution compared to a normal distribution.

The most useful measure of skewness is the Karl Pearson's coefficient of skewness.

Karl Pearson’s Coefficient of Skewness

Mean–ModeCoefficient of Skewness = Stan ndard Deviation

When the mode is not clear or where there are two or three modes, the following formula is used:

3(Mean–Mode)Coefficient of Skewness =Stan dard Deviation

Check Your Progress

Fill in the blanks:

1. ………………mean the ranges for which we classified the number of items.

2. An …………..can be defined as a central value around which other values of series tend to cluster.

3. ……………is defined as the reciprocal of the arithmetic mean of the reciprocals.

4. ……………is that value which has the maximum frequency (i.e. occurs most often) in a given set of values.

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Notes

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5. ____________ is the value of that item in a series which divides the series into two equal parts, one part consisting of all values less and the other all value greater than it.

6. Skewness may be defined as the lack of ____________ or degree of __________ from symmetry exhibited by a probability distribution.

Summary

In this unit, we have studied commercial averages which include the study of moving average, progressive average, quadratic average. Further, we have focussed on 3M’s of statistics i.e. mean, median and mode. Finally, we have considered mean, deviation and standard deviation.

Lesson End Activity

Do a comparative study amongst 3M’s of Statistics.

Keywords

Commercial Averages: Commercial averages are the applications of averages in commercial situations.

Standard Deviation: The standard deviation of a sample SD is similar to the mean deviation in that it considers the deviation of each X value from the mean.

Skewness: Skewness may be defined as the lack of symmetry or degree of distortion from symmetry exhibited by a probability distribution.


1. Make a comparison between mean, mode and median.

2. What are the key methods of measuring dispersion?

3. Write a note on the following:

(a) Standard deviation

(b) Coefficient of variation

(c) Skewness (c) C

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Further Readings

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UNIT 24: Forecasting

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Unit 24

Forecasting


The mathematical model of forecasting

Correlation analysis

How regression analysis works?

Multiple regression analysis and their applications in business

Introduction

The future has always held a great fascination for mankind. Perhaps this is biologically determined. Man and the higher apes seem to have brains that are equipped to engage in actions for which a future reward is anticipated. In extreme situation reward is anticipated not in this life but in the next life.

There are two methodologies to anticipate future. They are called qualitative and quantitative. But both start with the same premise, that an understanding of the future is predicted on an understanding of the past and present environment. In this unit, we will mainly deal with quantitative methods. We will also distinguish between forecast and prediction. We use the word forecast when some logical method is used.

The quantitative decision maker always considers himself or herself accountable for a forecast—within reason. Let us look at the conceptual model first and then the mathematical model and algorithms in turn which are used for making forecast.

The Conceptual Model

The qualitative school has generated many philosophical, religious or political conceptual models according to which the ideology and dogma is structured and forecasts prepared. Quantitative decision making, defined here as anything that is not based on underlying belief, offers three conceptual models. They are quite quantitative (c)

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to highly technical. They are guesstimate, fundamental and technical models.

In the guesstimate conceptual model the forecast is based on expert opinion. It is almost like qualitative decision making except that the bias of many is pooled. This method of forecast basically revolves around Delphi Method.

This conceptual model for forecasts should not be used when ample data bases are available. It is also known as option methodology. The Delphi method consists of a panel of experts and a series of rounds during which forecasts are made via questionnaire. Whether expertise or ignorance is pooled in each round, the result is the same: a forecast is born. But in the absence of sufficient data, it may be preferable to develop heuristics first rather than to rely initially on guesstimates.

The second conceptual model stresses the fundamentals that impinge upon the environment at any given time. In this case the forecaster tries to ascertain the functional relationships among variables defining the environment. In addition, attention is paid to changes in the magnitude of the variables that make up the environment. This conceptual model is superior because it is based on logical considerations and not merely on expert opinion.

The reason why not all forecasters wholeheartedly embrace the fundamental conceptual model is that it takes a pretty good mind to understand the variables and their interrelationships that represent the environment. It takes constant study, constant learning, constant testing and then the intellectual ability to synthesize it all. To cite an example, it does not take much to come up with a fundamental conceptual model to forecast a nation’s economic activity. We know that Gross National Product (GNP) is a function of consumption (C), investment (I), government spending (G) and net exports (E). In equation form it appears as GNP=C+I+G+E. Now each variable, i.e., consumption, investment, etc., can be carefully quantified. For example, do we consume more goods or services? More hard or soft goods? and so on. Beautiful econometric models have been generated on the basis of this conceptual model. Beautiful forecasts have also been presented.

Activity What is Quantitative decision making?

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Notes

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The third conceptual model was called technical. It is used by forecasters who call themselves technocrats. Whenever, a predetermined parameter that the technocrats follow reaches a certain magnitude, they forecast a change in the environment irrespective of the behaviour of other variables. Sometimes, this model gives accurate results and some times not.

Check Your Progress

Fill in the blanks:

1. In the guesstimate conceptual model the forecast is based on ___________.

2. The qualitative school has generated many philosophical, religious or __________ conceptual models

The Mathematical Model

The mathematical models play a very important role in forecasting. The quantitative analysis that underlies a forecast is based on the type of conceptual model that has been chosen. The guesstimate and technical models typically result in mathematical models that are less rigorous than the fundamental model, although the decision tools of the former may be applied in the case of the latter as well. Again in this unit only those decision tools are discussed that are considered efficient and wisely used. The mathematical model that is considered here should be used in conjunction with the fundamental conceptual model.

The logic used in mathematical model is twofold. First, it is based on the idea that the past and present environments may be used to extrapolate the future, the forecasted environment. Second, the environment itself comes about because of a functional relationship that exists between the variable whose value is to be forecasted and one or more other variables that determine the forecasted variable’s magnitude. The environment can be an economy, the market for a product, the productivity of a shift of assembly line workers, the track performance of 1500 meter runners and so on. It is clear that in forecasting there are always two types of variables: the one that is being forecasted and one or more from which the forecast is made. The first one is known as the dependent variable,

Activity What is the role being played by mathematical models?

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the latter as the independent variable(s). The functional relationship between the two can be visualized within a system of coordinates where the dependent variable is shown on the y and independent variable(s) on the x-axis. Since both types of variables have usually positive values, the entire environment (past, present and future) is shown in the first quadrant. Those variables that may affect the dependent variable’s magnitude but which are not considered in the decision space of the forecast either because of oversight or their effect is deemed negligible, are known as intervening variables.

Check Your Progress

Fill in the blanks:

1. The quantitative analysis that underlies a forecast is based on the type of ___________ model that has been chosen.

2. The functional relationship can be visualized within a system of coordinates where the dependent variable is shown on the _____ and independent variable(s) on the ________ axis.

Simple Forecasting System

Such a system may be called a simple forecasting system. If there is more than one independent variable, a multiple forecasting results. Visualize each independent variable as representing a dimension in the decision space. For the case y=f (x1, x2), the space may still be shown on a plane (a two-dimensional piece of paper). If there are more than two independent variables then imagine a different dimension for each variable—all starting at the origin (O).

Now the question arises as to how the dependent and independent variables may be calculated mathematically so that a forecast can be made. Let us stay with the simple system (only one independent variable x) for example purposes and argue that we can forecast gross national product on the basis of consumption alone, that is, GNP=f (Consumption). In developed economy we would not be too far off the mark. Past and present GNP and corresponding consumption data may then be obtained and plotted. The resulting cluster of coordinates is known as a plot or scattergram.

Activity What is Simple Forecasting System?

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Now suppose that someone asked you to forecast GNP, given a certain consumption value. Such a value may be obtained by polling a sample of consumers about the amounts that they are planning to spend. Then calculate the sample mean, construct a confidence interval, and then an average line or curve may be drawn through the plot and the y-value obtained by extrapolation. The immediately apparent problem is the mathematically proper selection of the line or curve, because there are many possible ways of drawing such an average, as shown and hence many possible forecasted GNP values. There are two basic methods to solve this problem. They are known as regression and smoothing. Each one has spawned a number of offspring. In this book only those are discussed that minimize the possibility of injecting the decision maker’s bias into the forecast. Furthermore, the decision tools that are illustrated represent the strongest mathematical link in the chain.

Both methods use the arithmetic mean which represents the forecasted value. Fitting an average line by freehand process, points A and C fall above while points B and D fall below this average line. The distances between them and the line are shown as a, b, g, d and the points a, b, c, d determine the line with its familiar equation yc =a + dx1 where yc is any estimated (forecasted) y-value given a certain x-value, a is the y-intercept and b the slope of the line. Obviously the best and tightest fit is obtained when the sum of the distances equal 0, since the positive distances (above the line) and negative distances (below the line) would be offset against each other. Mathematically the tightest fit is obtained by the method of least squares,

a2 + b2 + g2 + d2 = |Minimum|

This conditions holds when the normal equations are used for calculating the a and b values, or, for the straight line where all values were previously defined. The method is known as regression analysis and was developed by Francis Galton. You may read a book on statistics for deeper understanding of regression and correlation analysis.

It should be noted that the known environment extends between Points A and D. Given this environment and the line that has been fitted to it, a forecast yc may be made given x as shown in Figure 24.1. However, we are now extrapolating into an unknown (c)

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environment, that is, its real x and y values are not known. Therefore, we must make a crucial assumption, namely, for our forecasted yc value to be correct, no material alternations in the functional relationship y=f(x) must have taken place. If, for example, a much larger or smaller real value corresponds to each real x value in this unknown environment or if a new important variable has entered the decision space, the forecasted yc value is probably false.

Figure 24.1: Fitting a line yc = a+bx, to four points A, B, C, D

As we will see shortly, fitting lines and curves to given data sets is simple. In fact the computer usually does it for us. Technocrats do this very well. But the good forecaster knows his environment. The skill lies in understanding the interrelationships in a multi-variable decision space and to know when a change can be expected in the functional relationships or when new variables (henceforth intervening variables) must be considered an integral part of that decision space. It is in this aspect of forecasting that a solid fundamental conceptual model pays its dividends. A good forecaster never loses sight of the fundamentals. The master forecaster already knows the fundamentals of the unknown environment. This is a skill that cannot be taught. It can only “happen” to the individual who is willing to completely immerse himself in a veritable flood of data and information blocks that (c) C

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may in any conceivable way have some bearing on the forecast. If he ever comes up again, it has happened: one of the few master forecasters has joined the ranks.

A distinction is made between linear and curvilinear regression analysis. Linear analysis fit straight lines to data sets. Curvilinear or nonlinear analysis does the same with curves. Furthermore there is a simple regression analysis with only one independent variable, i.e.; y=f (x), and multiple regression with more than one independent variables or y=f (x1, x2,..., xn). A special type of regression analysis uses time as the independent variable. Sales forecasts are examples where Sales =f (Time). This type of regression analysis is known as time series analysis. Finally, before turning to the algorithms, a word may be added about smoothing, a major forecasting method. As the name implies observed values are smoothed and a weighted average is obtained which represents the behaviour of the variable under consideration. It is to be noted that the smoothing method is mathematically much less rigorous than regression analysis. Also, the method is limited in its scope of application.

Check Your Progress

Fill in the blanks:

1. If there are more than one independent variable, a _________forecasting results.

2. ___________ each independent variable as representing a dimension in the decision space.

Algorithms and Applications

Each of the two forecasting methods—regression and smoothing—have their distinct algorithms. They will be discussed in turn. In order to set the mental stage for this discussion, it is helpful to rethink the approach taken in the particular type of forecasting analysis that is examined here. It is not a forecast out of the blue. Rather it is based on functional relationships among variables for which there is a stipulated logic. In other words there is a fundamental conceptual model upon which the quantitative analysis is based. Furthermore it is assumed that the past and present environments, that is, the joint behaviour of the dependent and independent variables, are indicators of the future environment. Mathematically speaking, the forecast is nothing but (c)

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an extrapolation of the past and present. As previously pointed out, if no fundamental changes take place, with respect to the magnitude of the functional relationships among the variables and no new variables enter the decision space, this logic has been found to work well when applied. It has stood the test of time. But the above assumption also means that the direction of the forecast, that is, an increase or decrease in the magnitude of the forecasted behaviour of the variable, must be clearly visible and mathematically substantiated in the past and present environments.

The Regression Algorithm It may be recalled that, depending upon the forecasting problem, regression analysis may take several forms. There is linear and non-linear regression. There is simple and multiple analyses. And there is time series analysis. For each, however, the same algorithm or solution methodology may be used. This algorithm has following steps.

(1) Prepare a plot.

(2) Fit a line or curve to the plot and define either mathematically by the method of test square.

(3) Test the significance of the slope. Sometimes people skip this test.

(4) Construct a confidence interval for the forecasted yc value.

(5) Estimate the quantitative effect of the independent variable(s) x on the behaviour of the dependent variable y. This is known as correlation analysis.

(6) Test the significance of the correlation. This, too, is sometimes ignored.

After you have done this, what have you got? You got yourself a forecast and you may be 95% confident (remember that in forecasting it is always 95%) that your forecast will be on the mark when, that mark ultimately becomes known assuming that the logic of your conceptual model is sound. This is a big assumption, because anybody can come up with a functional relationship y=f(x); which may not necessarily be sound.

Let us proceed in our examination of the decision tools from the simple to the more complex ones and explain each step of the algorithm in detail as we go along. We begin with simple linear (c) C

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regression and correlation analysis, yc=f (x), then switch to linear and non-linear time series analysis, yc=f (x, time), which is only a special case as may be recalled, and finally look at multiple regression and correlation analysis, yc=f(x1,x2,...,xk).

Check Your Progress

Fill in the blanks:

1. Depending upon the forecasting problem, regression analysis may take _________ forms.

2. There is _______ and __________ regression

Simple Linear Regression Analysis

A time honoured functional relationship exists between the amounts spent on advertising and sales generated by these amounts. It is a popular belief that this is a positive relationship in the sense that each rupee spent on advertising generates so many additional rupees (say, about 10 for consumers’ goods) in sales revenue. Indeed, the relationship appears to be linear within meaningful ranges of the advertising budget. Thus, very little money spent on advertising may have very little effect on sales or none at all. Similarly, extremely large sums that are spent on advertising will not generate that much more in sales revenue. But, a meaningful advertising budget (meaningful in terms of market constraints, demand factors, customer income, etc.), usually shows a linear relationship to sales revenue. In order to perform regression analysis, data sets of between n=15 and n=25 should be used as a minimum sample size. After all, the past and present environment is described and too small a sample of observations will not do. For example purposes we will use only n=5 so that the manual calculations do not detract from an understanding of the procedures. On the job, regression analysis should be performed on the computer exclusively once the method has been understood. For easier referencing all calculations for each step of the regression algorithm are shown in one work sheet with the column number identified for each step. Let us get to work on an assignment that has been just received from the controller of our company. Given advertising expenditures of ` 8 million for the next fiscal year, how much (c)

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sales revenue can be expected? Is the question that the company wants an answer for?

Let us try to answer this question. First, the conceptual model: what, if any, relationship exists between sales and advertising? This has been taken care of. We know that sales =f (Advertising). Secondly, how does this relationship look in the case of our particular firm? OK. We need data. We get the data from our accounting department. They tell us that in the past, five advertising expenditures (budgets) resulted in certain corresponding sales figures. Remember that we use n=5 in order to keep the calculations simple. Notice that they gave us a random sample of observations that is deemed representative of the environment. The data are recorded in ascending order (not a requirement) in the work sheet below. Sales are the dependent variable and advertising the independent variable. Already at this point begin to think about the intervening variables that are likely to be operative in this decision situation. We will have to come back to them later on. And now let us activate the algorithm.

Plotting the Data

The data, as they appear in columns 1 and 2 of the worksheet, are placed into a system of co-ordinates. This can be done manually or by computer. When making a forecast about a certain phase of your operations, you usually have a pretty good idea of how the result “ought” to look. This mental picture is the result of your collective experience with the operations. The plot allows you to verify the mental picture with reality. You can see, for example, if there is a positive, negative or zero relationship between the variables. You can see whether the relationship is consistent (small variance) when the data cluster is close or inconsistent (large variance) when they are all over the quadrant. These observations help you to shape the conceptual model.

Also, you can see if by freehand method you may want to fit a line or curve to the data as previously mentioned. Furthermore you can see whether linearity or non-linearity governs the entire data set or if there are combinations. These observations help you to decide on the Mathematical model for your forecast. Obviously your choice of a linear versus non-linear analysis has great impact on the forecasted value and the implications of this choice will be discussed in detail when performing the non-linear analysis later on. (c) C

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Table 24.1: Worksheet for a Simple Linear Regression Analysis

A good forecaster will think about the plot and inspect it again and again for a considerable period of time before fixing in his mind the (c)

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conceptual and mathematical models. Once the models are determined, the rest of the quantitative analysis is routine. The work can be turned over to the computer. But relating a mental picture to reality and vice versa goes a long way in making a valid forecast. The plot helps in this endeavour.

Our plot is shown in Figure 24.2. Disregard for the moment everything except the connected original data points shown as circles. These data are taken from the worksheet. You can readily see that if you had to fit an average to this data set by freehand method, you would use a straight line. The line that represents the best fit is calculated next in Step 2 of the algorithm.

Figure 24.2: Plot of Original Values and Line of Best Fit

Fitting the Straight Line

The mathematically best fit of the line yc=a+bx to a data set is obtained by the method of least squares as previously discussed. The calculations for the normal equations

Σ Σ

Σ Σ Σ

y =na+b x

xy= a x+b x2

are shown in the worksheet resulting in

245 = 5a + 25b

1327 = 25a + 136.98b

which now may be solved simultaneously by solving for b and substituting in the first equation in order to find a. Thus, (c) C

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[245 = 5a + 25b ] × 5

⇒ 1225 = 25a + 125b

1327 = 25a + 136.98b

102 = 11.98b

b = 8.5142

Therefore, the best fitting line has been defined as yc=6.429 + 8.5142x.

Rather than to solve a and b algebraically, the normal equations may be solved for a and b. After simplification

b =n xy - x y

n x - ( x)2 2

Σ Σ ΣΣ Σ

is obtained. Using these direct formulae which are more efficient

b =

(5)(1327) - (25)(245)

(5)(136.98) - (25)2

=510

59.9

= 8.5142

and

a =245

5-

25

5(8.5142) = 6.429

The line yc=6.429 + 8.5142x is called the forecasting equation. It is the decision tool that allows us to answer the controller’s question. You remember that she stipulated an advertising budget of ` 8 million or x=8.

We can now estimate sales revenue by

yc = 6.429+(8.5142) (8)

= ` 74.5426 million

You know that the straight line is defined by two points. Given the forecasting equation, pick any two (simple) x-values. Let us say (c)

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that we use x=0 because this is the y-intercept and x =5.5. The estimated y-values are then found by

yc = 6.429+ (8.5142) (0)

= 6.429

and

yc = 6.429 + (8.5142) (5.5)

= 53.2571 These values are shown in Figure 24.2 as squares and the regression line has been constructed. As you read up to the line at x=8; you find the forecasted yc –value. Note the managerial meaning of the y-intercept a and the slope b. At zero advertising expenditures, sales amount to a= ` 6.429 million. This tells you that there is not exactly a perfect relationship between sales and advertising. If there were, you would see zero sales. You are still thinking about the intervening variables? Always do. Now, the slope b= ` 8.5142 million tells you that as the controller authorizes the advertising budget, each Re. 1 million will result in ` 8.5142 million in sales. You realize, of course from the preceding units, that since only sample data are used, this value must be seen as the midpoint of a confidence interval. So don’t call the controller and say ` 1 million advertising results exactly in ` 8.5142 million in sales. This is stochastic decision making after all. Find the standard error of the slope and look up the proper t - value and put everything into the 95% confidence interval for the regression coefficient b (note the new term) which takes the form

b ± t 0.025 σb

While you are at it, you suddenly remember (from the preceding unit) that the forecasted sales value yc= ` 74.5426 million should be communicated to the controller in confidence interval form as well. We are still talking sample statistics and always will be. We never know the population regression line Y=a+bX. All we know is the sample regression line yc=a+bx. So, better hold that call and let us figure out σb first. Needless to say that you could also calculate σa—if the urge ever struck you. Should it strike you? Look at it this way. At the y-intercept (a), x=0. But you postulated y=f (x)! If you want to set x=0, you don’t have much of a relationship. Do you? You want to study the effect x has on y, and therefore, you are (c) C

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interested in the slope b. Leave σa to the statisticians. They have the most peculiar urges at times.

The Significance Test of the Regression Coefficient b

So you are interested in the slope b. The question is at what angle to the x-axis a slope becomes significant. Only a significant slope means that y=f(x) is real in the stochastic decision making sense. If the slope is significant case there is a forecasting tool. In the insignificant case there is nothing except a waste paper basket for the study or smoothing.

Regression analysis is based on the assumption that the y-variable is normally distributed. Figure 24.3 shows a pictorial presentation. Then, it will be recalled from the preceding unit, the proper significance test for b is the one-sample t-test.

The algorithm for that test is as follows:

Step 1: Ho : b = 0

Ho : b > 0

or

B≠ 0

It is left to the discretion of the forecaster whether to use a one or two-sided alternative hypothesis. In this particular case the plot shows clearly a positive slope. Therefore to test b < 0 does not make much sense. This test, however, is included in b ≠ 0. So why use it? The answer is that it is always easy to reject Ho when using a one-tailed test. Remember the mathematically expected values (MEV) at P .05 for the one and two-tailed tests? They are 1.64 and 1.96, respectively. You can see that a smaller experimentally obtained value (EOV) leads to the rejection of Ho. Take, for example, EOV=1. 75. Then with a one-tailed test you reject. But not with the two-tailed test. Rejecting Ho in the regression problem means that the regression coefficient b is significant which means in turn that the regression equation may be used as a valid forecasting tool. A conservative forecaster may argue that the selection of such a tool should be as severely constrained as possible. Hence, the two-sided alternative hypothesis.

Activity Discuss significance test of the regression coefficient.

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Step 2: P.05

The 95% confidence level is selected as standard operating procedure in forecasting studies involving regression analysis for reasons previously stated.

Step 3: t =

b - 0

bσ

The one-sample mean problem formula for the t-test may be recalled.

It reads

t =x -

x

μ

Remember that each point on the regression line with slope b represents a mean. Thus, we are talking, in the regression case, simply about a “many-sample” mean problem where each data consists of two values, x and y. So you can see the similarities in the two t configurations: b takes the place of x, o the place of because we want to test whether the population regression coefficient equals zero (this is the null hypothesis in all regression problems) and finally the standard error of the regression coefficient b, σb is substituted for the standard error of the mean, σ You recall that σ is unknown and must be estimated by

σ x =s

n Similarly, σb must be estimated, but this estimation is somewhat more complex because of the broader scope of the problem. It is

σb

yx2

2=S

Z(x-x)

where σ2yx is the standard error of the estimated regression

equation of the y values on x. This standard error σyx is defined by

S =

(y - y )

n - myx2 c

2Σ

Where n-m are the degrees of freedom with m the number of regression coefficients. In the case of the straight line m=2 because there are two quantities a and b. The calculations for Step 3 of the algorithm may now be performed as shown in Table 24.1 columns 5 (c) C

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through 7. Note the distinction between the observed y value (column 1) and the estimated yc value (column 5). The latter is obtained by solving the regression equation yc=6.429+8.5142x for every observed x value (2.8, 4.3, 5.0, 5.5, 7.5) as shown in column 2. In other words every estimated yc value is obtained for every observed x value. Then, sum the squared difference between y and yc as shown in Table 24.1. The SS of the x values (column 7) you calculated in the preceding unit. Note that as shown in column 2, the t test may now be performed as follows,

S =

51.5524

5 - 2yx

2

= 17.1841

and

17.1841

11.98 =1.1977 If still interested, you may now construct the confidence interval for the slope b. Finally

t =

8.5142 - 0

1.1977

= 7.11σ

IV EOV>MEV or 7.11 > 2.353 (one-tailed)

or 7.11 > 3.182 (two-tailed)

∴ The slope is not zero.

∴ The regression equation may be used as a valid forecasting tool.

The Confidence Interval for yc

We are now ready to calculate a proper stochastic answer to the forecasting problem at hand. Remember that since sample data are used, the answer always takes interval form. The 95% regression interval is

yc ± Sd (t.0250)

yc ± Sd (t.0250) (c) C

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Where Sd is the standard error for a certain yc value given its corresponding x value.

This standard error is defined by

S =1

n+

(x - x)

(x - x)S

d

g

yx

22

22[ ]

Σ All values are known. Note the xg value in the numerator which is the given x value corresponding to the yc value. In other words for each yc, a new interval must be constructed. As shown by Figure 24.3 each point on the regression line represents the mean of a normally distributed sample of y values. The upper and lower confidence limits of these samples are not parallel to the regression line but are curvilinear as shown in somewhat exaggerated form.

Figure 24.3: Normal Distribution of y-values

Now, our controller wants to know the forecasted sales revenue (yc) given an advertising budget of ` 8 million (x=8). We calculated yc = ` 74.5426. Calculating first the standard error

S = [1

5+

(8 - 5)

11 99] 17.18412

22

. =16.3464

the interval is 74.542 ( 16.3464) (3.182)±

74.542 ± 12.8650

or

(61.677, 87.407) (c) C

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Remember that t is distributed with (n-m) or 3 degrees of freedom and that, since an interval is involved with upper and lower limits, the two-tailed MEV must be employed.

And now our final answer to the company: we tell the controller that we are 95% certain that with an advertising budget of ` 8 million and assuming no major changes in the market environment, between ` 61.677 and ` 87.407 million in sales revenue may be expected.

But the sales revenue may be the result of some other factor than advertising.

Check Your Progress

Fill in the blanks:

1. It is a popular belief that this is a ___________ relationship in the sense that each rupee spent on advertising generates so many additional rupees (say, about 10 for consumers’ goods) in sales revenue.

2. Each point on the regression line with slope b represents a _________.

Correlation Analysis (The Parametric Case)

A correlation analysis which measures the closeness of fit of the regressions line while assuming that both x and y are normally distributed (bivariate normal distribution) is necessary to set at rest the above doubt. Hence a parametric analysis is performed. Correlation analysis may also be seen as a measure of mutuality of x and y. Indeed, it is the more prevalent approach and computer programs are usually based on it. Of course, the results are the same. Let us start with closeness of fit because we have most of the calculations completed and the reasoning process of fitting a line is still fresh in our minds. But first a general word about this new decision tool.

The degree of correlation between normally distributed dependent and independent variables is signified by the correlation coefficient r. The symbol r was used in order to honour Francis Galton’s work. You recall that he gave us regression analysis and coined the term regression. Perhaps it would have made more sense to call the regression coefficient rather than the correlation coefficient r. But

Activity Discuss the parametric case of correlation analysis.

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the slope of the line was already labelled (b is generally known as the regression coefficient although there is a as well), so the correlation coefficient r it had to be. The coefficient r is a pure number. It is constrained by ±1 and defined by

r = 1 -S

S

xy2

2y

Where 2yS is the variance of the y values. But since these y-values

are part of a regression problem, n – m degrees of freedom apply. Therefore, in the case of the straight line, m=2 previously discussed and as shown in Table 24.1 columns 8 and 9. Then

S =(y - y)

n - m

=(y - y)

n - 2

=920

3

= 306.6667

yz i

2

i2

Σ

Σ

As pointed out, the correlation coefficient r is a pure number. Its sign is positive in this case because the slope of the regression line is positive. If b were negative, r would be negative as well. This fact is mentioned because it is not readily apparent since the radical has both the positive and negative sign.

Various types of correlation and their respective plots are shown in figure 24.4

Figure 24.4 (c)

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Since perfect positive correlation is r=+1.00, but r is a pure number, we cannot specify an exact quantitative meaning of the value that we obtained, r = + 0 .9716. We can only say in general terms that our findings show a “high” positive correlation between sales and advertising. In order to establish a specific quantitative relationship, we must calculate r2=0.9440. This is the coefficient of determination which indicates the percentage of the variation in the y variable that is caused by the x variable.

For purposes let us say rather loosely—the specifics are discussed in the section about the significance of r —that 94.40% of the increases or decreases in the sales revenue are caused by advertising expenditures and only 5.6% are caused by other intervening variables. These findings should pacify our controller, because our product sales seem to be very sensitive to advertising expenditures. More is to be said about the significance of correlation, let us hold this judgement for a while.

First, though, let us take a look at the other method, mutuality of x and y of correlation analysis. It was mentioned previously that parametric correlation analysis underlies the assumption of bivariate normal distribution. If this is the case, the distinction between independent and dependent variable may be dropped. We may perform a regression of y on x values—as we did in the forecasting study—or regress x on y values. In the first case we obtain our familiar equation yc=a+bx. In the second case we interchange the variable titles for the given data sets in columns 1 and 2 and obtain xc=a'+b' y. You may do this right now and plot the two regression lines. Obviously, with perfect correlation the two lines would be superimposed. Less than perfect correlation results in two interesting lines at ever wider angles as the correlation decreases. Obviously, the position of the lines are caused by the slopes b and b'. And by now you probably sense already something interesting, namely, that there is a direct relationship between the correlation coefficient r and the two regression coefficient b and b'. Indeed, the correlation coefficient is the geometric mean of the

regression coefficients, or 1r bb=

Normally, this book does not discuss the derivation of the decision tools. But in this case you can get a little flavour of that too because a knowledge of higher mathematics is not necessary. You recall that we solved the normal equations for b yielding (c)

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b =n xy - ( x)( y)

n x - ( x)2 2

Σ Σ ΣΣ Σ

This was simplified and in a form that uses totals rather than deviations. The matter was discussed in the preceding unit, formula for the variance. Remember? Now let us take one step back and show the formulae for b and b' in deviation form.

b =x - x) (y - y)

x - x)2

ΣΣ(

( and

b =(y - y) (x - x)

- y)1

y 2

ΣΣ(

then

bb =(x - x) (y - y)

x - x) y - y)2 2

[Σ ]2

Σ( Σ( and

bb =(x - x) (y - y)

x - x)2 y - y)2

Σ

Σ( Σ( or in simplified “totals” form

r =n xy - ( x) ( y)

[n x - ( x) ][n y - ( y) ]2 2 2 2

Σ Σ Σ

Σ Σ Σ Σ

This formula is known as the Product-Moment Formula or Spearman- Brown Formula. It is identified as such in the computer libraries. Solving for our sales and advertising example, using the calculations in Table 24.1. We obtain

r =(5)(1327) - (25)(245)

[(5)(136.98) -(25) ] [(5)(12925) -(245) ]

=510

524.919

=0.9716

2 2

which is the same value as obtained previously.

A concluding comment about parametric correlation analysis may sharpen further our understanding of the subject. Remember that, the idea behind correlation analysis is to specify the variation in (c) C

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the dependent variable that is caused by the independent variable or variables in the multiple correlation case as discussed later. We may show this idea in basic equation form like this -

Total Variation = Unexplained + Explained

Variation Variation

Where the explained variation is the portion that has been explained by the regression of y on x, or, 94.40% in our example and the unexplained variation, equal to 5.6%, which is due to the influence of intervening variables and statistical error. In symbolic form this equation may be written as

Σ Σ Σ(y - y) = (y - y) + (y - y)2 2 2c

The coefficient of determination is the ratio

r2 =

ExplainedVariation

TotalVariation

and perfect correlation would exist, r2=1.00, if the unexplained variation equalled zero. Rewriting the symbolic form of the basic equation with the degrees of freedom and previously introduced symbols we obtain

y

2

yx

2

y(c)

2S =S +S

Where 2( )y cS is the variance of the yc values and has not been

calculated, but it is known by substraction if 2yS (total variance)

and 2yxS (unexplained variance or standard error) are known. This

is indeed the case and r2 may be solved in the closeness of fit

approach.

r =

(y - y)

n - m(y - y)

n - m

=S

S

=S -S

S

2

c2

2

y(c)2

y2

y2

yx2

y2

Σ

Σ

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and, as shown before,

r = 1 -

S

S2 yx

2

y2

Check Your Progress

Fill in the blanks:

1. Correlation analysis may also be seen as a measure of ___________.

2. Symbol r was in correlation analysis to honour ___________ ___________ move.

Correlation Analysis (The Non-parametric Case)

When the bivariate normal distribution cannot be assumed or the observed data represent rankings, a non-parametric correlation analysis must be performed. The reasoning that underlies this technique is the same as discussed in the previous unit. The appropriate decision tool is known as the Spearman Rank Order correlation coefficient.

r =1-6 d

n(n - 1)s

2

2

Σ

Where rs stands for the Spearman r and d represents the differences between ranks as already encountered in the previous unit. In order to illustrate the application of this tool, let us use the problem on supermarket stools. Ten cashiers were tested for productivity increase after a stool had been installed at the supermarket checkout. The data for “with stool” and “without stool” conditions are ranked, the squared rank differences obtained and summed.

The worksheet in Table 24.2 shows the computations.

Table 24.2: Worksheet

Cashier With Stool

WithoutStool

Rank “With”

Rank “Without” d d2

1 21 21 2.0 3.0 -1.0 1.00 2 24 23 7.0 0.0 0.5 0.25 3 28 27 10.0 10.0 0.0 0.00

Activity Discuss the non parametric case of correlation analysis.

Contd…(c) C

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4 22 24 4.5 8.0 -3.5 12.25 5 22 21 4.5 3.0 1.5 2.25 6 25 25 8.5 9.0 -0.5 0.25 7 21 20 2.0 1.0 1.0 1.00 8 21 23 2.0 6.5 -4.5 20.25 9 23 21 6.0 3.0 3.0 9.00

10 25 22 8.5 5.0 3.5 12.25 Total 58.5

Therefore,

r = 1-6(58.5)

10(10 - 1)

= 1 -351990

= 0.6455

22

It may be recalled that when this problem was phrased as a non-parametric dependent two-sample mean comparison test, the Wilcoxon’s Test proved to be insignificant. What do you think is the significance or rs in this case? When the introduction of a stool did not make a difference in productivity, would it not make sense to forecast the “with stool” condition by the “without” condition? With Stool Productivity =f (without stool productivity)? Of course it would. Now, what does this mean concerning the significance of rs? You’ll find out in the next section. Meanwhile ponder the problem and note that =0.4167 which means that 41.67% of the variation in with-stool-productivity is “caused” by without-stool-productivity.

Significance Test of the Correlation Coefficients r and rs

Correlation analysis examines the functional relationship between dependent and independent variables. The question arises after an r or rs has been calculated whether the relationship that is expressed by it happens to be meaningful or not. The question is: what is meaningful? Meaningfulness may be defined in a statistical sense and an operational sense. And so it is in any correlation analysis. The decision maker has to look at significance (meaningfulness) from two perspectives. Let us look at the easier one—statistical significance first.

As in the case of the regression coefficient b, the regression coefficient r must be tested for significance; for, if it turns out to be insignificant, only a change relationship exists between the dependent and independent variables. In those correlation analysis (c)

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that assume a bivariate normal distribution (parametric correlation), the t test may be used as before in the case of b. For the Spearman Rank Order Correlation (non-parametric case), significance values developed by E. Olds must be used. Let us run the test for the parametric example first. The steps in the algorithm are as follows:

Step 1: HO : ρ = 0 (Rho, population correlation coefficient)

HA: ρ¹ 0

or

ρ > O

Step 2: P.05

Step 3:

t =r

1 - rn - 2

=0.9716

1 - 0.94405 - 2

=0.97160.1366

= 7.11

2

σ

Note that this t-value is exactly equal to the t-value that was obtained before when testing the significance of b. Minor rounding errors may be expected.

EOV > MEV or 7.11> 2.353 (one-tailed)

or 7.11> 3.182 (two-tailed)

∴ The correlation coefficient is not zero.

∴ There is a statistically significant correlation between advertising expenditures and sales revenues.

The direct relationship between r and b was noted. Therefore the equality of the t value for both r and b in Step 3 of the algorithm does not come as a complete surprise. Obviously if b is known to be significant, r is also significant and vice versa. You notice that the significance test for r is much easier to perform manually than the (c) C

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test for b. It may pay therefore to test the significance of r if you want to find out the significance of b. Similarly, as seen in the multiple regression case many computer programs provide only the significance for b. If you happen to be interested only in a correlation analysis, run the regression equation to obtain b, test its significance and you know whether the r in which you are interested is significant or insignificant.

Let us now turn to the nonparametric correlation example and the application of the Olds’ tables. Our experimentally obtained value was rs =0.6455. Remember that we have an answer outstanding from you. Using the two-sided alternative hypothesis that was used in the corresponding means comparison test, we find in the table, MEV= 0.564 for n = 10. Since EOV>MEV we have obtained a statistically significant correlation. Was that your outstanding answer a few pages back? If so, you are getting the hang of it. And, by the way, it makes good sense. If the means difference had been significant, all we could expect was a chance relationship for With-Stool-Productivity= f (Without-Stool-Productivity).

Now let us turn to the much more difficult answer concerning the operational significance of a correlation. To give a comprehensive answer, we consider three types of correlations, all of which may be statistically significant but which have vastly different operational implications for the decision. The first one is known as causal correlation. Here one variable causes the behaviour of the other. Of course this is hoped for status in any regression study, e.g., advertising causes sales, or, sales = f (advertising). Could the casualty be reversed? Nothing says that it can’t give the assumption of a bivariate normal distribution. Indeed, we saw that in the Product Moment formula that it did not make any difference which variable was dependent or independent. Then sales cause advertising. Why not? It is the old chicken and egg story. But note that you and the controller would approach the decision itself (there is ` 8 million involved, after all) from a completely different perspective.

The second type of correlation may come about because of co-dependence. In this case the behaviour of the dependent and independent variables is caused by an intervening variable or variables. In the sales advertising study, sales = f (advertising) may be statistically and operationally significant but only because our customers have sufficient disposable income to buy. You can (c)

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see that it takes much more for a good forecaster than to throw around a few formulae. It takes a sound conceptual model.

The third type of correlation is due to coincidence. There is no logical tie between the variables, but there may be a beautiful statistical significance nevertheless. For example, the correlation between the growth of a child and a plant such, correlations need not be considered in decision making.

Check Your Progress

Fill in the blanks:

1. The first type of correlation is __________ correlation

2. The second type of correlation is __________ correlation

3. The third type of correlation is __________ correlation

Time Series Analysis

Time has strange, fascinating and little understood properties. Virtually every process on earth is determined by a time variable. One of the most frequently encountered managerial decision situations involving forecasting is to measure the effect that time has on the sales of a product, the market price of a security, the output of individuals, work shifts, companies, industries, societies and so on. A fundamental conceptual model in all of these situations is the product life cycle concept which goes through four stages – introduction, growth, maturity and decline. Let us look at this concept in greater detail before we apply it.

Figure 24.5: Product Life-Cycle

Activity What is time series analysis?

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The sales performance of this product goes through the four stages—introduction, growth, maturity and decline. Data have been plotted and regression lines fitted to each of the four environments. Thus, when a sales forecast is made and the target horizon falls within the same stage, the linear fit yields valid results. If, however, the target horizon falls into a future stage, a linear forecast may be erroneous. In this case a curve should be fitted as shown. It is usually lightly speculative to select a forecasting horizon that spans more than two stages.

Another point of interest is the behaviour of the sales variable over the short run. It fluctuates between a succession of peaks and troughs. How do these come about? In order to answer this question, the time series, must be decomposed. Then four independent motors for this behaviour become visible. First, there is a long-term or secular trend (T) which is primarily noticeable within each stage of the cycle and over the entire cycle. Secondly, cyclical variations (C) which are caused by an economy’s business cycles affect product sales. Such cycles, whose origins are little understood, exist for all economies. Thirdly, the product’s sales may be influenced by the seasonality (S) of the item, and finally, there may be the irregular (I) effects of inclement such as weather, strikes and so forth. In equation form the decomposed time series appears as TS = T + C + S + I.

This creates a complex situation in time series analysis. Each factor must be quantified and its effect ascertained upon product sales. Let us see how this is done. The long-term trend effect T is reflected in the slope b of the regression equation. We already know how b is calculated even though minor modifications of the decision formulae will be encountered soon. The quantification of the cyclical component C is beyond the scope of this book. However, since business cycles always proceed from peak to trough to new peak and so on, their positive and negative effects upon a product’s sales cancel out in the long-run. Hence in managerial, as opposed to economic, decision making, the sum effect of the business cycles may be set equal to zero. This eliminates the C factor from the equation. Seasonality, if present, is something that must be taken into consideration because it is a product-inherent variable and (c)

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therefore, it is under the immediate control of the decision maker. We will quantify the S component and keep it in the equation.

Finally, there are the irregular variations. Do we know in July whether the weather will be sunny and mild during the four weeks before Diwali? We don’t, but we know that if this happens, Diwali sales will be severely impacted. Can we forecast such horrible weather conditions? Not really. We cannot forecast them because they cannot be quantified—a rather unpleasant characteristic they share with all other type of irregular variations like strikes, earthquakes, power failure, etc. Yet, something strange usually happens after such an irregular variation from “normal” has occurred. Whatever people did not do because of it, like not buying a product, they attempt to catch up with quickly. Therefore, the I factor effect may also be assumed to cancel out over time and it may be dropped from the equation which then appears to the manager as TS = T + S.

Check Your Progress

Fill in the blanks:

1. When a sales forecast is made and the target horizon falls within the same stage, the ________ fit yields valid results.

2. If, the target horizon falls into a future stage, a linear forecast may be _____________.

Linear Analysis We will construct again the best fitting regression line by the method of least squares. In order to illustrate the procedure, let us use a data set from understand Case 24.1 at the end of the unit. It involves the dividend payments per share of the Smart, a well-known discount store chain, for the years 1990 through 1999. Suppose that a potential investor would like to know the dividend payment for 2001. The data are recorded in the work sheet (Figure 24.6) that appears below. First, however, turn your attention to Figure 24.6 which shows the plot for this problem.

Activity What is Target Horizons?

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Figure 24.6: Plot of Dividend Values

Think for a moment about the qualitative nature of the time variable. It is expressed in years in this case but could be quarters, months, days, hours, minutes or any other time measurement unit. How does it differ from advertising expenditures, the independent variable that we examine in the preceding section? Is there a difference in the effect that a unit of each has on the dependent variable, or, ` 1 million in one case and 1 year in the other? Time, as you can readily see is constant. One year has the same effects as any other. This is not true for advertising expenditures, especially when you leave the linear environment and enter the non-linear environments. Then there may be qualitative difference in the sales impact as advertising expenditures are increased or decreased by unit.

Since time is constant in its effect, we may code the variable rather than to use the actual years or other time units x values. This code assigns a 1 to the first time period in the series and continues in unit distances to the nth period. Do not start with a zero as this may cause some computer programs to reject the input. The code is based on the fact that the unit periods are constant, and therefore, their sum may be set equal to zero. See what effect this has on the normal equations for the straight line.

∑ y = na + b ∑ x

∑ xy = a ∑ x + b ∑ x2

If ∑ x = 0, the equations reduce to

∑ y = na

∑ x y = b∑ x2 (c) C

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which allow the direct solution for a and b as follows

a =y

n

b =xy

x2

Σ

Σ

Σ This form simplifies the calculations substantially compared to the previous formulae. The code, however, that allows to set å x = 0 must incorporate the integrity of a unit distance series. Thus, if the series is odd-numbered, the midpoint is set equal to zero and the code completed by negative and positive unit distances of x=1, where each x unit stands for one year or other time period. If the series is even-numbered, let us say it ran from 2003 to 2012, the two midpoints (2007/2008) are set equal to-1 and +1, respectively. Since there is now a distance of x = 2 between + 1 (-1, 0, +1), the code continues by negative and positive units distance of x = 2 where each x unit stands for one-half year or other time period.

The worksheet is in Table 24.3 and calculations are as follows:


YEAR Code for an Even Series

X

YEAR Code for an Odd Series

X

Dividend payments

in ` Y

XY X2

2003 -9 2004 -7 2004 -4 2.2 -8.8 162005 -5 2005 -3 2.4 -7.2 92006 -3 2006 -2 3.0 -6.0 42007 -1 2007 -1 5.0 -5.0 12008 1 2008 0 6.8 0 02009 3 2009 1 8.1 8.1 12010 5 2010 2 9.0 18.0 42011 7 2011 3 9.5 28.5 92012 9 2012 4 9.9 39.6 16Total 0 0 55.9 67.2 60

Then

a =5.59

9= 6.211

b

67 2

601 12=

.= .

and Yc = 6.211 + 1.12 x

origin 1995 x in 1 year units (c) C

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Note that in the case of time series analysis, the origin of the code and the x units must be defined as part of the regression equation. In our problem the investor would like to obtain a dividend forecast for 2014. Since the origin is 2008 (x = 0) and x = 1 year units, the code for 2014 is x = 6. Therefore the forecast is yc = 6.211+1.12 (6) = ` 12.9. If the time series had been even numbered, let us say that dividend payments for 2003 had been included in the forecasting study, the definition under the regression equation would have read

origin 2007/08

x in 6 month units.

Thus, we know that for 2008, x=1; and since we must use x=2 units for each year, the code value for 2014 would be x=13. Once the yc value has been obtained, b is tested for significance and the 95% confidence interval constructed as previously shown.

Time series analysis is a long-term forecasting tool. Hence, it addresses itself to the trend component T in our time series equation TS = T + S. In the dividend forecast, b=1.120 was calculated which means that in the environment that is reflected in the set, Smart increased the dividend payments on an average by ` 1.12 per year. Let us now turn out attention to the seasonal variation component, that may be present in a time series. A product’s seasonality is shown by the regularly recurring increases or decreases in sales or production that is caused by seasonal influences. In the case of some products, their seasonality is quite apparent. As an obvious example virtually all non-animal agricultural commodities may be cited. Seasonality of other products may be more difficult to detect. Take hogs in order to stay on the farm. Are they seasonal? They are lusty breeders and could not care less about seasonal influences. Yet, there is an induced season by the corn harvest. If corn is plentiful and cheap, farmers raise more hogs. This is known as the corn-hog cycle. Or take automobiles, Indian manufacturers are used to introduce major design or technological changes once every generation. This “season” has now been shortened somewhat. How about computers? There the season even has a special name. It is called a generation and prior to increased competitive pressures within the industry it used to be about seven years long. Our stock market investor knows that stock trades on the Stock Exchanges are (c)

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seasonal. The daily season is V-shaped starting the trading with a relatively high volume which tapers off toward the lunch hour to pick up again in the afternoon. And so it goes with many other products, not ordinarily thought of as being seasonal.

Let us quantify this seasonality and illustrate how it may be used in a decision situation. There are, as is often the case, a number of decision tools that may be applied. The reader may be familiar with the term ratio-to-moving-average. It is a widely used method for constructing a seasonal index and programs are available in larger computer libraries. Usually, the method assumes a 12 - period season like the twelve months of the year. There is a more efficient method which yields good statistical results. It is especially helpful in manual calculations of the seasonal index and when the number of seasonal periods is small like the four quarters of a year, the six hours of a stock exchange trading day or the five days of a work week. This method is known as simple average and will be used for example purposes.

To stay with the investment environment of this unit section, let us calculate a seasonal index for shares traded on the Stock Exchange from July 2 through July 7, 2012. This period includes the July 4 weekend. Volume of shares (data) for each trading day (season) is given in thousands of shares per hour. The Individual steps of the analysis (operations) are discussed in detail for each column of the worksheet below.


Column (2)

(3) (4) (5) (6) (7) (8) (9) (10)

Hour Total Trend Seasonal Seasonal 7/2 7/3 7/6 7/7 Avg. Variation Variation Variation Index (TS) (T) TS-T

10-11 0.965 0 0.965 110.6 12.00 12.25 15.44 16.72 14.10 11-12 0.245 0.159 0.086 103.7 10.40 11.75 15.04 16.32 13.38 12-13 -0.885 0.318 -1.117 94.2 10.55 10.06 12.95 15.44 12.25 13-14 -1.555 0.477 -2.032 87.1 9.55 9.46 12.05 15.24 11.58 14-15 0.395 0.636 -0.241 101.1 11.02 11.55 14.82 16.73 13.53 15-16 0.835 0.795 0.040 103.3 11.58 12.25 15.38 16.69 13.97 Average -0.383 600 10.85 11.22 14.28 16.19 13.135

As you inspect the data columns, you notice the V-shaped season for each trading day. You also notice in the total daily volume that there is an increase in shares traded. Hence, you can expect a positive slope of the regression line. The hourly mean number of shares is also indicated. This is the more important value because (c) C

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we are interested in quantifying a season by the hour for each trading day. Now turn to the operations. In last column, the hourly trading activity for the four days has been summed. In this total all time series factors are assumed to be incorporated. You will recall that the positive or negative cyclical and irregular component effect is assumed to cancel out over time. Hence averaging the trading volume over a long term data set eliminates both components, yielding TS=T+S. You may ask, are four days a sufficiently long time span? The answer is NO. In a real study you would probably use 15 to 25 yearly averages for each trading hour. In an on-the-job application of this tool, you will have to know the specific time horizon in order to effectively eliminate cyclical and irregular variations. But by and large, what is a long or short time span depends upon the situation.

In order to isolate the trend component (T) so that it may be subtracted from column (2) in the Table 24.4, yielding seasonal variation, the slope (b) of the regression line must be calculated. (Remember: b is T.) The necessary calculations are performed below using the mean hourly trading volume for each day. But since we are interested in an index by the hour, the calculated daily b value must be apportioned to each hour. This is accomplished by a further division by six—the number of trading hours. The result is entered in column (3). Note that the origin of a time series is always zero. The origin of the time series is always the first period of the season. In our case this is the 10-11 trading hour. Therefore the first entry in column (3) is always zero to be followed by the equal (since this is a linear analysis) summed increment of the apportioned b-value.

Table 24.5

Average Hourly Day Code Trading Volume

Per Day x x1 y x y x2

7/1 -3 10.85 -32.55 9 7/5 -1 11.22 -11.22 1 7/6 1 14.28 14.28 1 7/7 3 16.19 48.57 9

Total 52.54 19.08 20 (c) C

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b =xy

x

=19.08

20

2

Σ

Σ

= 0.954

and the apportioned b-value is

0.954

6= 0.159

It is not necessary to calculate the y-intercept (a) in this analysis unless of course, you wish to combine it with a long-term forecast of daily trading volume. Then, just to review the calculations, you would find:

a =y

n

=52.54

4

Σ

= 13.135

and

yc = 13.135 + 0.954 x

origin 7/5 and 7/6

x in half trading day units.

In column (4) TS - T = S is performed. Column (4) is already a measure of seasonal variation. But in order to standardize the answer so that it may be compared with other stock exchange, for example, it is customary to convert the values in column (4) to a seasonal index. Every index has a base of 100 and the values above or below the base indicate percentages of above or below “normal” activity, hence the season. Since the base of column (5) is 100, the mean of the column should be 100 and the total 600 since there are 6 trading hours. In order to convert the obtained values of column (4) to index numbers, each of its entries is added to the total mean and then is divided by the column mean added to total mean and multiplied by 100 yielding the corresponding entry in column (5). It is customary to show index numbers with one significant digit.

Column (6) shows the seasonal effect of this decision variable—share trading on the Stock Exchange. Regardless of heavy or light

Activity What are the limitation of linear analysis?

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daily volume, the first hour volume is the heaviest by far. It is 7.4% above what may be considered average trading volume for any given day. Keep in mind that a very limited data set was used in this analysis and while the season, reaching its low point between 1 and 2 p.m., is generally correctly depicted, individual index members may be exaggerated. What managerial action programs would result from analysis such as this? Would traders go out for tea and samosas between 10-11? How about lunch between 1-2? When would brokers call clients with hot or lukewarm tips? Assuming that a decrease in volume means a decrease in prices in general during the trading day, when would a savvy trader buy? When would he sell? Think of some other intervening variables and you have yourself a nice little bull session in one of Dalal Street’s watering holes. If, in addition, you make money for yourself or firm, then, you have got it.

Check Your Progress

Fill in the blanks:

1. Time series analysis is ________ term forecasting tool.

2. Product seasonablity is shown by regular occuring______ or _______ is sale

Non-linear Analysis

Any number of different curves may be fitted to a data set. The most widely used program in computer libraries, known as CURFIT, offers a minimum of 5 curves plus the straight line. The curves may differ from program to program. So, which ones are the “best” ones? There is no answer. Every forecaster has to decide individually about his pet forecasting tools. We will discuss and apply three curves in this section. They appear to be promising decision tools especially in problem situations that in some way incorporate the life cycle concept and the range of such problems is vast, indeed.

If you take a look again at Figure 24.5, you see that three curves have been plotted. As we know from many empirical studies, achievement is usually normally distributed. Growth, on the other hand, seems to be exponentially distributed. The same holds true for decline. As the life cycle moves from growth to maturity, a parabolic trend may often be used as the forecasting tool. These are two of the curves that will be considered. The third one is (c)

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related to the exponential curve. As you look at the growth stage and mentally extrapolate the trend, your eyes will run off the page. Now, we know—again from all sorts of empirical evidence—that trees don’t grow into the high heavens. Even the most spectacular growth must come to an end. Therefore, when using the exponential forecast, care must be taken that the eventual ceiling or floor (in the case of a decline) is not overlooked. The modified exponential trend has the ceiling or floor build in. It is the third curve to be discussed.

One final piece of advice before we start fitting curves. If you can do it by straight line, do it. For obvious reasons, just look at Figure 24.5, any possible error—and there is always a built-in five percent chance—is worse when a curve is fitted. By extending the planning and forecasting horizon over a reasonable shorter period rather than spectacular but dangerous longer period, the straight line can serve as useful prediction tool.

The Parabola Fit

The parabola is defined by

yc = a + bx + cx2

Where a, b and c are constants a and b have been dealt. c can be treated as acceleration. The normal equations are (method of least square).

Σy = na + bΣx +cΣx2

Σxy = aΣx + bΣx2 + cΣx3

Σx2y = aΣx2 + bΣx3 + cΣx4

Setting Σx= 0 as previously explained, Σx3 will also be zero.

Σ Σ

Σ Σ Σ

ΣΣ

y = na + c x

x y = a x + c x

b =xy

x

2

2 2 4

2

There are direct formulae for a and c as well, but because of the possible compounding of arithmetic error in manual calculations, it is safer to solve a and c algebraically in this case.

To illustrate the parabolic trend let us forecast earnings per share in dollars for Storage Technology Corporation for the years 2014

Activity How does parabola fit is applied in managerial applications?

(c) C

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and 2015. Storage Technology manufactures computer data storage equipment, printers, DVD-ROMS and telecommunication products. The company was founded in 1969 and after going through a period of explosive growth seems to be moving into the maturity stage. Data, code and calculations are shown below in the usual worksheet format.

Year Code

(x) Earnings Per Share

(y) xy x2 x2y x4

2007 -3 0.39 -1.17 9 3.51 81

2008 -2 0.54 -1.08 4 2.16 16

2009 -1 1.13 -1.13 1 1.13 1

2010 0 1.58 0 0 0 1

2011 1 1.72 1.72 1 1.72 1

2012 2 2.50 5.00 4 10.00 16

2013 3 1.84 5.52 9 16.56 81

Total 0 9.70 8.86 28 35.08 196

Then 8.8628

=b

= 0.3164

and solving simultaneously

9.70 = 7a + 28c × 4

⇒ 38.8 = 28a + 112c

35.08= 28a + 196c

3.72 = 84c

c = 0.0443

a = 1.2085

Therefore,

yc = 1.2085 + 0.3164 x- 0.0443x2

origin 1996

x in 1 year units

and specifically,

y2014 = 1.2085 + 0.3164(5) - 0.0443(5)2

= ` 1.68, (c) C

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y2015 = 1.2085 + 0.3164(6) -0.0443(6)2

= ` 1.51

Remember that the data set is small. Quarterly earnings per share figures for the period may have been better because of the larger sample size. The significance test and construction of the confidence interval is performed as previously shown. Furthermore, as soon as new earnings per share figures become available, the regression line should be re-calculated, because there is always the chance that there may be a change in the environment.

The Exponential Fit

This illustrative forecasting study is performed for the Acme Company Ltd. that manufactures toy rubber ducks to be used in bathtubs. Over the past few quarters, the company has become a major defence contractor. The Navy is buying an ever increasing number of the ducks as part of its rearmament program. Shipment figures are kept secret to confuse the enemy—and the media. Therefore, the data in the accompanying table are hypothetical. We may fit an exponential trend which takes the form

yc = abx

As previously mentioned, exponential trends are difficult to plot, because you run very quickly off the top of the page. However, when using semi-log paper (the y-axis is in logarithmic scale), the trend appears as a straight line. This phenomenon may be used to good advantage when calculating a and b.

Thus using the logarithmic form of the exponential trend

log yc = log a + x log b

The straight line equations may be used, or,

log a =

log y

n

Σ

and

log b =

log y

x2

Σ

Σ

Activity What is the scope of exponential fit?

(c) C

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when Σx = 0. The data set and calculations appear in the worksheet below. Logarithms are obtained from a pocket calculator or any standard table.

Quarter Since Initial Navy

Contract

Code (x)

Shipments in Thousands

of Units (y)

log y x log y x2

1 -5 2 0.30103 -1.50515 25 2 -3 4 0.60206 -1.80618 9 3 -1 9 0.954243 -0.954243 1 4 1 20 1.30103 1.30103 1 5 3 55 1.740363 5.221089 9 6 5 110 2.041393 10.206965 25

Total 0 7.241149 12.463511 70

Then

log a =7.241149

6

= 1.2069

and

log b =12.463511

70

= 0.1781

when expressed in logarithmic form. The regression equation is

log yc = 1.2069 + 0.1781 x

origin third and fourth quarters

x in half quarter units.

The equation may be used in this form for forecasting purposes. Suppose that Acme would like to have a forecast for the next two quarters.

The forecasts are

log y7 = 1.2069 + 0.1781(7)

= 2.4536 and finding the anti-log

y7 = 284 thousand more rubber ducks

and

log y8= 1.2069 + 0.1781 (9)

= 2.8098 and finding the anti-log

y8 = 645 thousand still more rubber ducks (c) C

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If you transform the logarithmic form of the regression equation, there is something interesting to be seen if you remember the compound interest formula. It works like this-

log yc= 1.2069 + 0.1781x and finding the anti-log

yc= (16.1) (1.51)x

which may be rewritten

yc= (16.1) (1+0.51)x,

And you recognize that it takes the form of the compound interest formula where the rate is 0.51 or 51%. This is Acme’s average quarterly increase in its defence business.

The Modified Exponential Fit

In any case other than military procurement, except in those countries that have bled themselves dry because of it and now have neither the money for military extravaganza nor civilian necessities/amenities, trees don’t grow into the high heavens. Given this profound observation, there must be a decision tool that places a cap or ceiling on overly exorbitant growth forecasts. But they may want to consider the other option: exponential economic declines do not always result in a merciful state of down and out but gradually approach a floor which may be called subsistence, making do, squeezing by or other nice and flowery allegories. At any rate the asymptote of the modified exponential curve is,

yc = k + abx,

Where k is the asymptote, provides us with such a tool. There are four cases.

Figure 24.7 (c) C

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A least squares fit is not efficient in this case. Rather a solution method is discussed that is based on the theorem that the ratio of successive first differences between points on the exponential curve is constant and equal to the slope b.

The decision tool is known as the method of semi-averages. It is based on the calculation of three sums of successive points of the time series. Therein lies the limitation of this technique, because the number of data points must be divisible by three. Thus, a minimum of six points is necessary and if the time series consists of n=20 data points, the two earliest one (to preserve the most relevant environment) must be eliminated. The formulae for a, b and k are generated as follows from six general y-values starting with the origin of the series.

y0 = k + a

y1 = k + ab ⎫⎬⎭

Σ1 = y0 + y1

y2 = k + ab2

y3 = k + ab3 ⎫⎬⎭

Σ2 = y2 + y3

y4 = k + ab4

y5 = k + ab5 ⎫⎬⎭

Σ3 = y4 + y5

then

Σ1 = 2k + a(b+1)

Σ2 = 2k + ab2 (b+1)

Σ3 = 2k + ab4 (b+1)

and

b =-

-

2k =-

b

a = ( -b

(b b

2 3 2

2 1

12 1

2

2 1 2 2

Σ Σ

Σ Σ

ΣΣ Σ

Σ Σ

--

-

- -

11

1 1)

)( )

or, in the general case involving a time series of n data points and n is divisible by three. (c)

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b =-

-

nk =-

b

a = ( -b

(b b

n 3 2

2 1

12 1

n

2 1 n n

Σ Σ

Σ Σ

ΣΣ Σ

Σ Σ

--

-

- -

11

1 1)

)( )

Suppose a set consists of the following data points.

Year Code Sales Units

2007 0 100

2008 1 160

⎫⎬⎭

Σ1 = 260

2009 2 200

2010 3 230

⎫⎬⎭

Σ2 = 430

2011 4 245

2012 5 250 ⎫⎬⎭

Σ3 = 495

Then

b =495 - 430

430 - 260

= 0.38

b = 0.62

2k = 260 -430 - 2600.38 - 1

= 260 - (-274.19)

= 534.19

k = 276.10

a = (430 - 260)0.62 - 1

(6.38 - 1)(0.38 - 1)

=(170) (- 0.99)

= -168.3

2

and

yc= 267.10 + (-168.3) (0.62)x

which makes it a Case 4 curve with a sales ceiling of 267.10. The forecast is made in the usual manner. For 2013 it is (c) C

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y2013 = 267.10 +(-168.3) (0.62)6

= 267.10 +(-9.56)

= 257.54 units.

Check Your Progress

Fill in the blanks

1. The decision tool is known as method of _________ .

2. Shipment figures are kept secret to confuse the __________ and the ___________ .

Multiple Regression Analysis

At this point we have a fairly good understanding about how to approach, a forecasting problem with one independent variable. We know how to fit a straight line or three widely applicable curves to a data set. But even as we generated the functional relationship of our first problem—Sales= f (Advertising) — the thought must have occurred to us if advertising were the only predictor variable in this case. And as you can sense, the decision situation will be considerably broadened thereby. It is not only a matter of incorporating more variables into the decision space, or, mathematically speaking the move from y=f(x) to y = f (x1, x2,...,xn) and in the form of the regression equation from y=a+bx to yc = a + b1 x1 + b2 x2 + ...... + bn xn, but rather by testing additional variables effect and then by substituting another one, we are able to simulate various environments. In this sense multiple regression analysis may not only be a more sophisticated forecasting tool, but it can nicely serve to sharpen the decision maker’s understanding of the forecasting environment and thus, serve as a managerial training device.

To get us started in this new light of somewhat more sophistication, a few background review comments may be in order. The relationships between independent and dependent variables in the multiple regression problems are assumed to be linear. The methodology for a non-linear multiple regressions have not really advanced beyond the research stage. This does not mean that inherently non-linear variables cannot be accommodated within the analysis system, but they must be transformed. The growth situation of the exponential trend problem comes to mind. Remember that we then introduced linearity back into the picture by using the logarithm of the variable’s data—y in that case. We

Activity What is the limitation of modified exponential fit?

(c) C

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transformed the data to make them appear linear. Other techniques than logarithmic transformation are available.

The fitting technique for multiple regressions is again the least squares method which is based on the Gauss-Markov Theorem. It holds that when the population variables are fixed and the Y-variable random, the variances of the sub- population ys for corresponding xs are equal, and the probability of the system being in a certain state (environment) initially is the same probability throughout the system, the best estimation of the population regression y = a + b x by the sample regression yc = a+bx is the method of least squares. In Figure 24.1 we noted the deviation y-yc, which is known as stochastic disturbance or simply error (e). You remember that the method of least squares minimizes the sum of the squared error. Further, it is assumed that the effort is normally distributed, hence y is normally distributed and therefore, we can test a and b by t-test after estimating s a and s b. We did only the latter, but soon you will see computer printouts that include the former as well.

Finally a few comments may be made about operational aspects. The normal equations are as follows for the multiple regression problem yc= a + b1 x1 + b2 x2 + b3 x3

ΣY = na + b1 Σx1 + b2 Σx2 + b3 Σx3

Σx1y = aΣx1 + b1 + b2 Σx1 x2 + b3 Σx1 x3

Σx2y = aΣx2 + b1 Σx2 x1 + b2 + b3 Σx2 x3

Σx3y = aΣx3 + b1 Σx3 x1 + b2 Σx3 x2 + b3 Σx3

This is the general system of equations and we will consider an actual problem shortly. By comparing the normal equations for the straight line with the ones above, you can readily see how they were developed and you can easily generate the equations for 10 or 15 variable problems as frequently encountered in econometrics. What is not so easy, indeed it may be impossible, is to solve this many equations simultaneously. Here matrix algebra helps. But even then manual calculations are forbidding. Therefore, multiple regression analysis must be performed by computer and we will not even bother with a simplified manual example as before. However, so that we may have an inkling about the computational procedures, let us take a look at the multiple correlation formula for a problem with two independent variables

R =r r r r r

r123

122

132

232

+ +-

21

12 13 23

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Where R and r are the multiple and simple correlation coefficient, respectively, and y=1, x1 = 2 and x2 = 3. As you can see the procedures rest on performing iterative simple correlation analysis. This, of course, is what computers love to do. Managers don’t.

Having solved this operational problem, let us look at the second and trickier, one. It has to do with a relaxation of the underlying assumptions of regression analysis. Such relaxation may be necessary because of aspects that exist in the environment. First, it is assumed that the error (e) terms a, b, g, d are independent, or re = 0. In trend analysis, however, a long term increase or decrease virtually assures dependency. This is known as serial or auto correlation. Tests and significance tables have been prepared. The Durbin-Watson statistic, soon to be encountered in our computer analysis, allows the identification of serial correlation. Secondly, it is assumed that the independent variables x1, x2, ..., xn are mutually independent. If this assumption is relaxed, sb tends to become large resulting in an insignificant b. Small potatoes, as you recall. This problem is known as multi-collinearity, and techniques are available to test for its existence. Remember, however, that the logic of a well-designed conceptual model should identify the problem already at the stage of forecasting. If it exists, there are only two ways out: get additional data, or more typical, get other variables.

There are three more, though minor, problems that may arise because of the underlying assumptions. The first one was mentioned already and is only repeated here. The assumption is made of linearity or constant variance s2. It has the name of homoscedasticity. We already know that the behaviour of some variables tends to result in non-linearity and that some “linear” variables — as shown in Figure 24.5— may display non-linearity at certain magnitudes. It was suggested to use transformations in those cases. This is sound advice as long as the forecaster knows which way s2 varies. If unknown, the transformation cannot be performed. The second problem involves lagged variables such as in

yc = a + b1 x b + b2 xt–1

Where xt is the x value of time period t and xt-1 the x value of the previous time period. We will encounter lagged variables again in (c)

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the later discussion about smoothing. Indeed, since lagged variables may contain both multicollinear and serial correlation, perhaps the initial forecasting method should rely on smoothing. There are at present no meaningful methods to cope with this problem in the regression framework.

Finally, there is the general assumption governing the whole body of quantitative analysis, namely, the variables must be quantifiable. We said before rather non-chalantly that this would not pose a problem. May be not, but, as you may have noticed, our discussion is moving away from the conceptually simple decision problems into the more complex ones. In the case of the first type, once the problem has been defined and a solution method algorithm explained, the situation is clear. The second type calls for extensive model building abilities and training. Here the problem and algorithm are clear but not necessarily how to quantify complex variables and to show their interrelationships.

There are three qualitatively different types of variables in any decision situation. They may be called concrete, abstract and nonsense. Concrete variables can be measured easily. Take heat, we obtain a physical measurement in centigrade. Take advertising expenditure, we obtain a dollar or rupee measurement. Now take beauty which, as anyone knows, is in the eye of the beholder. This is an abstract variable. But it is easily quantifiable by rankings as any beauty contest demonstrates. The same holds true for a nonsense variable concept as found in many product names. But now suppose that one of the variables in a sales forecasting study is gender. Let us say that it is known that women buy more of the product than men. In this case, data collection procedures assign a 1 when women respond and a 0 for the response of men. Such variables are known as dummy variables and, as of now, this term has been removed from your catalog of insults. Make it part of the model building knowledge. Here is how it works.

Take

yc = a + b1 x1 + b2 x2

Where yc is sales revenue, b1 advertising expenditures and b2 gender. In this equation yc and x1 are quantitative and x2 qualitative in nature. Now setting

x2 = 1, when a women responded

x2 = 0, for responses from all others, (c) C

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we obtain the following equations

yc = a + b1 x1 for “all other response”

yc = (a + b2) + b1 x1 for “woman response”.

You notice that the slope of the regression lines is b1, but that a the y - intercept differs. In the case of “Woman response”, the line shifts upward, parallel to the all other responses line. Similarly regression equations may be solved when all independent or the dependent variables are qualitative in nature. The disadvantage of the dummy variable, however, is the fact that it can take only the 0 or 1 value. There are methods that allow qualitative rankings with more than 2 response choices like agree, don’t know, disagree. Such a variable is known as polychotomous and just from the term you know that it is beyond the scope of this book. But if you ever meet such a variable in some dark decision situation, brush up on multivariate analysis. That allows you to handle this stranger.

Now let us look at an old friend in new clothes and demonstrate how multiple problems are actually solved. From now on it is strictly computer work and every computer library has one or more MULREG programs, as they are often called. (To use the most used problems we will use ‘Regression Analysis’ in ‘Tools’ section of ‘Microsoft Excel 97’). The problem is the same that we solved for our controller via simple linear regression except that we change it from Sales = f (Advertising) to Sales = f (Advertising, Percent spendable income allocated to product type by our customers as shown by a market research study, Time). Perhaps we thought that an analysis like this would put us on the promotional fast-track of the company. We shall see. First the data set.

Sales Advertising Per cent Time

` lacs ` lacs Allocated Year Code

y x2 x3 x4

30 2.8 13.8 1995 1

40 4.2 12.2 1996 2

55 5.0 15.0 1997 3

50 5.5 16.0 1998 4

70 7.5 18.0 1999 5

Let us mention the caveat again: this data set lands us in the dog house rather than on the fast-track because of its paucity. But we (c)

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are interested only in an example of the procedures. Note that the dependent and three independent variables are called y and x respectively and x variable are numbered consecutively. This is for data input purposes. Thus the multiple regression equation is

y = a + b2 x2 + b3 x3 + b4 x4.

Each computer system calls for slightly different instructions and data input for the Mulreg program. By looking carefully at the 'Microsoft Excel 97' procedures below, you can readily infer how your system works.

Before we look at them let us explain the reasons for this analysis. Remember that multiple regression analysis is as much as forecasting tool as it is a managerial training device. Obviously we emphasis the training aspect here. First we want to perform the simple regression analysis by using 'Regression Analysis' in 'Microsoft Excel 97' and you will see how quickly we can verify the results to the manual calculations that we suffered through a number of pages ago. So the first regression that we perform is simple and reads

y = f (x2)

Next we want to do the whole Schemer. You get

y= f (x2,x3,x4)

and you see how nice and easy life is for the quantitative decision maker.

The third regression is a time series analysis, or,

y = f (x2, x3)

and

y = f (x3, x4)

As you realize, by now, in this little training exercise we have incorporated all the regression options that have been discussed so far except the dummy variable. Its inclusion, let us say that the product was seasonal and we had differentiated between “high season” and “all others”, would pose no problem, except that then we would have had to change the sample size and could not have used the original problem. It works like this. In this multiple regression analysis

df = n - m ≥ 1 (c) C

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or, to rewrite this assumption in terms of the bs

n > k + 1

where k is the number of bs or variables in the equation. Otherwise a degenerate model results and that’s why the dummy variable was not included. You may remember this point when you run your own multiple regression problem with 15 variables.

In order to perform these five regressions, the instructions about the 'Regression' dialog box are shown in Exhibit 24.1.

Exhibit 24.1: About the Regression Dialog Box

Input Y range

Enter the reference for the range of dependent data. The range must consist of a single column of data. Here the sales data is the Y range.

Input X Range

Enter the reference for the range of independent data. Microsoft Excel orders independent variables from this range in ascending order from left to right. The maximum number of independent variables is 16. In this question the inputs have to be specified for each of the regression calculation separately.

Labels

Select if the first row or column of your input range or ranges contains labels. Clear if your input has no labels; Microsoft Excel generates appropriate data labels for the output table.

Confidence Level

Select to include an additional level in the summary output table. In the box, enter the confidence level you want applied in addition to the default 95 per cent level.

Constant is Zero

Select to force the regression line to pass through the origin.

Output Range

Enter the reference for the upper-left cell of the output table. Allow at least seven columns for the summary output table, which includes an ANOVA table, coefficients, standard error of y estimate, r2 values, number of observations and standard error of coefficients.

Contd….(c) C

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New Worksheet Ply Click to insert a new worksheet in the current workbook and past the result starting at cell A1 os the new worksheet. To name of new worksheet, type a name in the box. New Workbook Click to create a new workbook and past the results on a new worksheet in the new workbook. Residuals Select to include standardized residuals in the residuals output table. Standardized Residuals Select to include standardized residuals in the residuals output table. Residual lots Select to generate a chart for each independent variables versus the residual. Line Fit plots Select to generate a chart for predicted values versus the observed values. Normal Probability Plots Select to generate a chart that plots normal probability.

After we have discussed each regression, it is suggested that you run this very problem with your computer program unless you are already familiar with it.

Regression 1 (y = a + b2 x2).

The printout for this simple regression problem is shown in Exhibit 24.2. It was previously solved manually.

Exhibit 24.2: Summary Output of Regression of Sales (y1) and Advertising (x2)

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Let us proceed step-by-step through the printout. The correlation coefficient value shows a minor rounding error between the manual and computer solutions (r= 0.9716 and r = 0.9715784, respectively). The Anova matrix is of no concern to us here. This matrix is followed by the values of the regression equation where coefficient indicates the y - intercept (a) value and the other X Variable 1 value the b value that corresponds to the numbers of the independent variables. Thus, yc = 6.429 + 8.514x, as previously obtained. Note the Standard. Error and T-STAT. The latter is rounded t = 7.11 for b and as previously calculated.

Let us now discuss the main regression statistics, the R-squared and R-values. These are the adjusted values for the multiple coefficients of determination and correlation, respectively. These coefficients carry a positive bias in unadjusted form, that is, an unadjusted R2 obtained by the formula results in an overestimate of the true R2. In order to find the true R2

1 the following adjustment formula is used

ADJ

2 2R = 1 - (1 - R )(

n - 1

n - m)

Since in this case of a simple correlation problem R2 = r2 = (0.9715782)2. Then

ADJ

2R = 1 - (1 - 0.9439641)(

5 - 1

5 - 2)

= 0. 9252855

while the printout shows 0.925286105. It should be noted that the relatively large difference comes between the adjusted and unadjusted R2 values, it is due to the small sample size. With a meaningful sample size, the difference between R2 and R2

ADJ is usually negligible.

Regression 2 (y = a + b2 x2 + b3 x3 + b4 x4)

Now we are entering the realm of the multiple.

Notice the regression equation

y = yc = -8.3958 - 3.9583 x4 +0.9375x3 +11.0416 x2

and the T-STAT that is insignificant in every case. R2 ADJ has dropped from 92.5% of the variation explained (Regression 1) to 80.4%. Obviously, the additional variables did not give us a better (c)

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forecasting tool. With an analysis like this, we won’t make the fast-track.

Regression 3 (y =a + b4 x4).

The b is significant again and R2 ADJ has improved. Overall it does not look as good as Regression 1 as a forecasting tool. Unless, after elimination of the serial correlation, its R2 ADJ represents an improvement over this value in Regression 1, we might as well stay with Regression 1.

Regression 4 (y = a + b3 x3 + b2 x2)

Well, there is a flicker of hope in Table 24.6.

R2 ADJ has improved to 89% from 84% in Regression 3. But look at the T-STAT of b3 which indicates an insignificant value and b2 just makes it by one-tailed test with df = 2, or MEV = 2.92 according to the t table. Conclusion: no improvement over Regression 1.

It shows you very nicely the interface between the conceptual and mathematical models as you can test any linear variable or combination of variables keeping in mind the time and cost constraints that apply to data collection. It also teaches you to differentiate between statistical and operational significance which helps you to sharpen the conceptual model building skills—the most important asset of the forecaster.

Table 24.6: Summary Output of Regression of Sales (y) with Advertising (x2) and Percent Allocated (x3)

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Check Your Progress

Fill in the blanks:

1. There are _________ qualitatively different types of variables in any decision situation.

2. Variables must be _____________

The Smoothing Algorithm

If you have understood the basic aspects of forecasting up to this point and are reasonably certain that you can use the regression algorithm, you may not want to take this final section of the unit too seriously. Read it, nevertheless, to know how the simple minded folk of forecasters spend its days. Indeed, many authors feel compelled to point out when introducing the subject of smoothing that it is used when (a) “real” forecasting would be prohibitively expensive (b) “real” forecasting talent has not found its way into an organization’s staff (c) “real” forecasting talent is not necessary because in the case of the operations or firms involved, it wouldn’t amount to much anyway and so forth. You remember the comment about how simple minded decision making folk knows how to phrase its utterances to make them sound “scientific”? Well, this is it. Final introductory advice: Beware of smooth smothers.

What do such types do anyhow? For some psychological reason they do not like the charming 1 and revealing little wiggles and waggles (scientific: peaks and troughs) of time series and smooth them out. They throw away the good stuff and eat chaff. How do they do it? They average them out. But that is all. No further sophistication is visible. Remember, they are simple minded folk. Here is how it works. Suppose that you have the stock closing prices of Joy Manufacturing Co. for the month of April, 1999. During this month there were 21 trading days with their inevitable wiggles and, unfortunately, waggles. By calculating a moving average, they may disappear. The moving averages are known as the smoothed values and are obtained by

s =x + x + x +...+x + x

nt

t-m/2 t-m/2+1 t tx-m/2+1 tx-m/2

Where St is the smoothed value for time period t, m a movement specification that is to be determined, x the data, and n the number of adjacent data points that are to be averaged. (c)

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Furthermore m is defined by m = (n-1)/2. Nobody will argue that this does not look very scientific, indeed.

Let us do it for a five-day moving average. The m = (5-1)/2=4 which means that four data points are lost. That is, two daily stock prices at the beginning of the series and two at the end. Hence

s =x + x + x + x + x

5=

26.25 + 27.25 + 28 + 28.125 + 28.25

5= 27.575

s =x + x + x + x + x

5=

27.25 + 28 + 28.125 + 28.25 + 28.375

5= 28

s =x + x + x + x + x

5=

28 + 28.125 + 28.25 + 28.375 + 28.25

5= 28.2

3

1 2 3 4 5

4

2 3 4 5 6

5

3 4 5 6 7

s =x + x + x +...+x + x

nt

t-m/2 t-m/2+1 t tx-m/2+1 tx-m/2

Where St is the smoothed value for time period t, m a movement specification that is to be determined, x the data, and n the number of adjacent data points that are to be averaged. Furthermore m is defined by m = (n-1)/2. Nobody will argue that this does not look very scientific, indeed.

Let us do it for a five-day moving average. The m = (5-1)/2=4 which means that four data points are lost. That is, two daily stock prices at the beginning of the series and two at the end. Hence

s =x + x + x + x + x

5=

26.25 + 27.25 + 28 + 28.125 + 28.25

5= 27.575

s =x + x + x + x + x

5=

27.25 + 28 + 28.125 + 28.25 + 28.375

5= 28

s =x + x + x + x + x

5=

28 + 28.125 + 28.25 + 28.375 + 28.25

5= 28.2

3

1 2 3 4 5

4

2 3 4 5 6

5

3 4 5 6 7

and so on. The lagging is necessary because of St-1 which obviously does not exist at X t-1. As you notice by doing a few more St values manually, exponential smoothing uses the weighted average of past time series values in order to compute the smoothed values and it assigns greater weight to the most recent time series values. The method assumes no significant long-term trend or seasonal variation in the data. So what kind of a forecasting tool is this? As pointed out before, if smoothing is to be used at all, it should be used over the very short term only. Perhaps a trader, as opposed to investor, may base stock purchases or sales on exponential (c) C

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smoothing. But check the effectiveness of this decision tool yourself in the cases of the given two common stocks. Disregarding intra-day trading highs and lows, how often would our trader have been successful in buying or selling the stocks of the companies if he had bid the estimated price for each of the trading days? Then calculate his profits or losses. The proof is in the pudding.

Check Your Progress

Fill in the blanks:

1. __________is management’s most important task, but not many managers in the public or private sectors are good forecasters.

2. It is final introductory advice: __________ of smooth smothers.

Case 24.1: There are 2117 Smart stores at petrol stations in the India (the chain is building up). At present Smart has reached an “upgrading” phase like so many discounters before.

Given the data below, perform the indicated analysis.

YEAR 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990

EARNINGS

PER SHARE 19.0 17.5 20.7 28.4 27.4 23.9 21.1 16.1 8.5 11.1

DIVIDENDS 9.9 9.5 9.0 8.1 6.8 5.0 3.0 2.4 2.2 1.9

PER SHARE

PRE-TAX 2.1 2.0 3.1 4.9 5.4 5.7 5.8 5.8 3.3 5.3

MARGIN

1. To what extent does the Board of directors regard dividend payments as a function of earnings? Test whether there is a significant relationship between the variables. Use a parametric analysis.

2. Find the linear forecasting equation that would allow you to predict dividend payments based on earnings and test the significance of the slope.

3. Is there a significant difference in pre-tax margin when comparing the periods 1995-1999 and 1990-1994. Perform a non-parametric analysis. Explain the managerial implications of your findings. (c)

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Summary There are two methodologies to anticipate future. They are called qualitative and quantitative. The quantitative decision maker always considers himself or herself accountable for a forecast-within reason. The qualitative school has generated many philosophical, religious or political conceptual models according to which the ideology and dogma is structured and forecasts prepared. In the guesstimate conceptual model the forecast is based on expert opinion. The Delphi method consists of a panel of experts and a series of rounds during which forecasts are made via questionnaire. The second conceptual model stresses the fundamentals that impinge upon the environment at any given time. In this case the forecaster tries to ascertain the functional relationships among variables defining the environment. The mathematical models play a very important role in forecasting. The quantitative analysis that underlies a forecast is based on the type of conceptual model that has been chosen. Each of the two forecasting methods – regression and smoothing – has their distinct algorithms. The forecast is nothing but an extrapolation of the past and present. As previously pointed out, if no fundamental changes take place, with respect to the magnitude of the functional relationships among the variables and no new variables enter the decision space, this logic has been found to work well when applied. When the bivariate normal distribution cannot be assumed or the observed data represent rankings, a non-parametric correlation analysis must be performed. Correlation analysis examines the functional relationship between dependent and independent variables.

Lesson End Activity

Glass company is headquartered in Indore, M.P. It manufactures glass and plastic storage jars for petrochemical industry.

Given the data below, perform the indicated analysis. YEAR 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 EARNINGS PER SHARE 1.25 2.89 2.81 3.09 3.52 2.97 2.71 2.13 1.59 1.77

DIVIDENDS 1.36 1.36 1.28 1.20 1.07 1.00 0.90 0.80 0.72 0.72 PER SHARE

PRE-TAX 1.1 4.6 5.2 5.5 8.9 7.2 10.1 7.9 7.3 9.7 MARGIN% (c) C

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(1) To what extent does the Board of directors regard dividend payments as a function of earnings? Test whether there is a significant relationship between the variables. Use a paramedic analysis.

(2) Find the linear forecasting equation that would allow you to predict dividend payments based on earnings and test the significance of the slope.

(3) Is there a significant difference in pre-tax margin when comparing the periods 1996-2000 and 1991-1995? Perform a parametric analysis.

Keywords Intervening Variables: Those variables that may affect the dependent variable's magnitude but which are not considered in the decision space of the forecast either because of oversight or their effect is deemed negligible, are known as intervening variables.

Correlation Coefficient: The degree of correlation between normally distributed dependent and independent variables is signified by the correlation coefficient.


1. Define forecasting. What are the key methodologies of forecasting?

2. State the difference between the conceptual and mathematical model of forecasting.

3. Discuss the concept of algorithms and their applications.

4. How will you conduct a simple linear regression analysis?

5. Write a note on time series analysis.

6. Illustrate the concept of multiple regression analysis.

7. Employment figures in thousands for Neo-Classical City and suburbs are given below. Perform the required analysis.

(a) Using linear forecasts, predict the year in which employment will be the same for the two locations.

(b) Construct the NCC confidence interval for that year. (c) C

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(c) Correlate the employment figures for the two areas using both parametric and non-parametric methods and test the significance of the correlation coefficients.

(d) Fit a modified exponential trend to SUB data and discuss the results in terms of your findings in (1) above.

(d) Are NCC employment figures uniformly distributed over the period 1994 through 2000?

YEAR 1994 1995 1996 1997 1998 1999 2000

NYC 64.1 60.2 59.2 59.0 57.6 54.4 50.9

SUB 20.7 21.4 22.1 23.8 24.5 26.3 26.5

8. Shown below are data sets that have been compiled by the Reserve Bank of India and The Department of Commerce. All amounts are in billions of rupees. Perform the following analysis:

(a) Fit a modified exponential trend to Other Checkable Deposits such as NOW accounts and predict the 2002 value. Compare this value to the actual one.

(b) Is there a significant difference in the Percent Cash Purchases when comparing the first half of the series against the second half?

(c) Predict Personal Consumption on the basis of Consumer Credit in the amount of ` 500 billion and test the significance of the slope b.

(d) Predict Demand Deposits for 2002 by linear trend. Year Demand

Deposits(` bn)

Other Checkable Extensions

(` bn)

Consumer Credit

Expenditures (` bn)

Personal Consumption

Purchases (` bn)

Percent Cash (` bn)

1990 53.6 0.4 187.1 634.1 70.50%

1991 58.6 0.5 215.8 692.6 68.8

1992 65.4 0.6 240.8 767 68.6

1993 70.1 0.8 269 834.3 67.8

1994 73.3 0.9 269.4 914.1 70.5

1995 78 1.6 280.7 1016.9 72.4

1996 82.6 3.2 318.2 1127.9 71.8

1997 91 4.8 373.5 1254.5 70.2

1998 97.4 7.8 424.2 1416.6 70.1

Contd… (c) C

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1999 99.2 17.7 465.8 1582.3 70.6

2000 102.4 27.4 449.3 1751 74.3

2001 86.6 74.4 477.2 1909.5 74.1

9 Graph is petrol additives manufacturer. Perform the analysis specified below and briefly discuss your findings in terms of the managerial if not national implications.

(a) Quarterly sales figures in thousands for car petrol additives are given below

i. Construct a seasonal index.

ii. Predict sales for the first and second quarters of 2001.

iii. Construct a confidence interval for the 2001 Q2 forecast.

Q1 Q2 Q3 Q4

1998 2 4 3 5

1999 4 7 5 10

2000 9 10 10 12

(b) While car petrol additives have had difficulties getting off the ground, Graph successfully introduced car diesel additives. For a modified exponential trend and predict 2001 Q3 sales.

1998 Q1 Q2 Q3 Q4 1999 Q1 Q2 Q3 Q4 2000 Q1 Q2 Q3 Q4

2 2 7 20 40 70 150 200 250 400 750 50

(b) Given data referring to sales of three wheeler full additives and two wheeler fuel additives, calculate the correlation coefficient by (a) parametric and (b) non-parametric methods and test the significance.

3-wheeler 20 40 60 30 20 70

2-wheeler 2 3 4 4 2 6

Further Readings

Books

R S Bhardwaj, Mathematics for Economics and Business, Excel Books, New Delhi, 2005 D C Sanchethi and V K Kapoor, Business Mathematics Sivayya and Sathya Rao, An Introduction to Business Mathematics (c)

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Web Readings




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Unit 25

Case Studies


Case Study 1: Exxon Oil Marketing Ltd.

After three months on the job at Exxon Oil Marketing Ltd.; Nidhi Batra—a recent MBA graduate— was called into the office of Mr Sunil, Vice President and General Manager. Mr Sunil said that Nidhi’s initial progress at the firm had been quite satisfactory and that, because she had acquired certain quantitative and computer skills at the University and on the job, she was to be put in charge of a sales and productions study. The study itself was rather complex, but the technical problems boiled down to the following.

Sales Portion of the Study

In order to get a better feel for a certain target market, average annual family income in that market had to be estimated. Before a sample could be drawn and income figures obtained, Nidhi had to determine the proper sample size. She set the following criteria: (a) maximum sampling error not more than 10,000 above or below the true population mean, (b) confidence interval should be 99%, and (c) standard deviation of the population, based on a previous study, was known to be 2000. What was the sample size?

Another aspect of the sales portion of the study called for the determination of a seasonal index for trend. Nidhi decided to use the method of simple averages and obtained the following input data for the firm’s major sales outlet in the target market.

Quarterly Sales in lacs

Year Q1 Q2 Q3 Q4

1981 10 8 14 16

1982 18 16 22 24

1983 20 30 27 35

Question:

What was the seasonal index adjusted for trend for each quarter? Contd….(c)

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Production Portion of the Study

Here two technical problems were encountered. The first consisted of a production forecast for 2001 based on the known estimating equating log yc =1.082 + 0.013x with origin at July 1, 1994, x in one year units and y in thousand-ton units. What was the forecasted tonnage?

Finally, in order to study Astra’s training program effectiveness; Nidhi obtained pertinent data for five workers and calculated a coefficient of determination. What was r2?

Worker A B C D E

Hours of Training 1 2 3 4 5

Units Produced 4 3 6 5 9

Mindful that Mr Sunil would want to have some idea about the implications, that is, validity or managerial explanation, of the four value sets calculated, Nidhi wrote a brief statement concerning each.

Question

What would you have advised her to cover in each statement for the four value sets?


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Case Study 2: Parametric Correlation and Non-parametric Correlation

We do not know precisely why formerly great nations like the Italian city states, Spain, France and England in this order declined in economic and political importance during the modern era. Exact data is not available. It is safe to assume, however, that fiscal mismanagement was a major contributing factor in each case. Today we do have data like the ones for India. Get this from any good source. Perhaps one day historians trained in statistics will perform appropriate analysis and know precisely the reasons for this nation’s decline.

Given your own understanding of our economic and political situation today, briefly explain your findings after solving the problems below.

Questions

1. Calculate the parametric correlation between Receipts and Outlays for the 1970s and test the significance.

2. Calculate the non-parametric correlation between Receipts and Outlays for the 1990s and test the significance.

3. Compare your answers for 1 and 2. What does remain in each case and is it operationally meaningful


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Glossary

A Sample Event: The basic possible outcome of an experiment, it cannot be broken down into simpler outcomes.

Absorbing State: A state is said to be absorbing (trapping) state if it does not leave the state.

Assignment Problem: An assignment problem is a special type of transportation problem. Average: An average can be defined as a central value around which other values of series tend to cluster.

Categorical Data: Categorical data are data that take a finite set of values that can be either numeric or categorical.

Column Matrix: A matrix having only one column. Commercial Averages: Commercial averages are the applications of averages in commercial situations.

Conditional Probability: The probability of an event occurring, given that another event has occurred.

Continuous Probability Distributions: Continuous probability distributions are those which represent continuously variable random variables

Correlation Coefficient: The degree of correlation between normally distributed dependent and independent variables is signified by the correlation coefficient.


Data Presentation: The presentation of complex mass of data in a simple way so that it becomes easier to understand.

Data: The facts and figures that is collected, analyzed and interpreted.

Decision Maker: An individual/group responsible for making a choice of appropriate course of action. (c)

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Decision Tree: A decision tree is a decision flow diagram that includes branches leading to alternatives one can select among the usual branches leading to events that depend on probabilities.

Determinant: A numeric value that indicate singularity or non-singularity of a square matrix.

Deterministic Variables: Deterministic variables can be measured with certainty. Discrete Probability Distributions: Discrete probability distributions are those where only a finite number of outcomes are possible.

Effective Demand: When a consumer is willing and able to purchase some quantity of a commodity at the existing market price, he is said to have an Effective Demand for that good.


Equation: A statement that two expressions (connected by the sign=) are equal. Event: Any set of all possible outcomes or simple events of an experiment.


Exponential Function: The function that associates the number ex to each real value of x is called exponential function. Frequency Distribution: A frequency distribution is the principal tabular summary of either discrete or continuous data.

Function: A function f from a set X (domain) to a set Y, a subset of X Y, is a rule that associates a unique element of set Y (target) to each element in X.

Heuristic Variables: Heuristic variables are those that exist in highly complex, unstructured, perhaps unknown decision making situations. Intervening Variables: Those variables that may affect the dependent variable's magnitude but which are not considered in the decision space of the forecast either because of oversight or (c) C

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their effect is deemed negligible, are known as intervening variables.

Linear Equation: A Linear expression equated to zero is called a linear equation.

Linear: The relationship between the variables is directly proportional. For example, if a wooden table requires 30 cubic feet of wood then 10 tables would require 300 cubic feet wood.

Logarithmic Function: The function that associates login to x is called the logarithmic function. Mathematical Averages: Mathematical averages are those which utilize mathematical formula for the calculation of their values.

Matrix: An array of numbers arranged in certain numbers of rows and columns. Median: Median is the value of that item in a series which divides the series into two equal parts, one part consisting of all values less and the other all value greater than it.


Mode: Mode is that value which has the maximum frequency (i.e. occurs most often) in a given set of values.

Moving Average: The moving average is an arithmetic average of data over a period and is updated regularly by replacing the first item in the average by the new item as it comes in.

Non-sampling Errors: Errors due to calculations or improper convention of observation are called non-sampling errors.

Numerical Data: Numeric data are data that exist in numeric form, such as height, the number of children in a household and annual income.

Objective Function (Z): The linear function which is to be optimized is called Objective Function. Objective of Assignment Model: Objective of assignment model is to assign a number of resources to an equal number of activities

Observation: The set of measurements obtained for a single element in the data set. (c)

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Payoff: It is the effectiveness of particular combination of a course of action and state of nature. These are also called as conditional values/profits. Positional Averages: Positional averages do not use mathematical calculations but give you an indication about the positional characteristics of certain items.

Probability Distribution: A probability distribution is a rule that assigns a probability to every possible outcome of an experiment.

Probability: A numerical measures of the likelihood of occurrence of an uncertain event. Program: A program is a set of instructions arranged in a logical sequence. Pure Strategy: It is the decision rule which is always followed by the player to select the particular course of action.

Quadratic Equation: An equation of the form, where a 0, b, c are constant, is called a quadratic equation. Qualitative Data: Data that are labels or names used to identify an attribute of each element. Qualitative data use the nominal or ordinal scale of measurement and may be non-numeric or numeric.

Quantitative Data: Data that indicate that how much or how many of an element. Quantitative data use the interval or ratio scale of measurement and are always numeric.

Random Variable: An event whose numerical value is determined by the outcome of an experiment is called a variate or often a random variable.

Rectangular Matrix: A matrix consisting of m rows and n columns, where.

Root: A value of the variable which satisfies the given equation is called a solution or root.

Row Matrix: A matrix having only one row. Saddle Point: In a payoff matrix saddle point is one which is the smallest value in its row and the largest value in the column. (c) C

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Sampling Design: A sampling design is a definite plan for obtaining a sample from the sampling frame.

Sampling: Sampling can be defined as the selection of some part of an aggregate or totality on the basis of which a judgment or inference about the aggregate or totality is made.

Simple Random Sampling: A procedure of sampling will be called simple random sampling where individual items (units) constituting the samples are selected at random.

Singular Matrix: A matrix whose determinant is zero. Skewness: Skewness may be defined as the lack of symmetry or degree of distortion from symmetry exhibited by a probability distribution.

Square Matrix: If the number of rows of a matrix is equal to its number of columns, the matrix is said to be a square matrix. Standard Deviation: The standard deviation of a sample SD is similar to the mean deviation in that it considers the deviation of each X value from the mean.

Statistics: The art and science of collecting, analyzing, presenting and interpreting data.

Stochastic Variables: Stochastic variables are characterized by uncertainty. Strategy: The strategy for a player is the list of all possible courses of actions that he will take for every pay-off that might arise.


Transportation Model: The transportation model of linear programming is a flow optimization technique.

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