introduction(new)

© 2008 Prentice-Hall, Inc.

Chapter 1

Introduction to Quantitative Analysis

© 2009 Prentice-Hall, Inc. 1 – 2

Learning Objectives

1. Describe the quantitative analysis approach2. Understand the application of quantitative

analysis in a real situation3. Describe the use of modeling in quantitative

analysis4. Discuss possible problems in using

quantitative analysis5. Perform a break-even analysis

After completing this chapter, students will be able to:After completing this chapter, students will be able to:


Chapter Outline

1.1 Introduction1.2 What Is Quantitative Analysis?1.3 The Quantitative Analysis Approach1.4 How to Develop a Quantitative Analysis

Model1.5 The Role of Computers and Spreadsheet

Models in the Quantitative Analysis Approach

1.6 Possible Problems in the Quantitative Analysis Approach

1.7 Implementation — Not Just the Final Step


Introduction

Mathematical tools have been used for thousands of years

Quantitative analysis can be applied to a wide variety of problems

It’s not enough to just know the mathematics of a technique

One must understand the specific applicability of the technique, its limitations, and its assumptions


Examples of Quantitative Analysis

Taco Bell saved over $150 million using forecasting and scheduling quantitative analysis models

NBC television increased revenues by over $200 million by using quantitative analysis to develop better sales plans

Continental Airlines saved over $40 million using quantitative analysis models to quickly recover from weather delays and other disruption


MeaningfulInformation

QuantitativeAnalysis

Quantitative analysisQuantitative analysis is a scientific approach to managerial decision making whereby raw data are processed and manipulated resulting in meaningful information

Raw Data

What is Quantitative Analysis?


Management Science TechniquesManagement Science Techniques

In Operations Research, we will cover the following topics:

(i) Linear Programming (ii) Transportation Problems (iii) Assignment Problems (iv) Waiting Line Models (v) Simulation (vi) Decision Analysis (vii) Forecasting


Linear Programming

Linear Programming is a problem-solving approach developed for situations involving maximizing or minimizing a linear function subject to linear constraints that limit the degree to which the objective can be pursued.


Applications of Linear Programming

1. Marketing Applications (i) Media Selection (ii) Marketing Research (iii) Product selection and

competitive strategies (iv) Utilization of salesman, their

time and territory control (v) Marketing advertising decision



(2) Financial Applications (i) Financial Planning (ii) Portfolio Selection (iii) cash flow analysis (iv) Investment policy for

maximum return (v) Profit planning (vi) Risk analysis



(3) Production Management Applications

(i) A Make or Buy Decision (ii) Production Scheduling (iii) Workforce Assignment (iv) Maintenance Policies (v) Crew sizing and Replacement

system (vi) Quality Control decision and

material handling facilities



(4) Blending Problems (5) Revenue Management


Transportation Model

The transportation problem is applied to distribution type problems in which supplies of goods that are held at various locations are to be distributed to other receiving locations.


Applications of transportation Problem

(1) Minimize the total cost of transportation

(2) Taking loans from bank so that the total interest payable be the least

(3) Minimizing Adv. Cost for Exposure

(4)Optimal sales distribution


Assignment Models

Assignment Problems are characterized by a need to pair items in one group with items in another group in a one-for-one matching.


Applications of Assignment Model

(1) Assign the given jobs to some workers on a one-to-one basis

(2) The salesmen of a company be assigned to different sales zones so that the total expected sales are maximized

(3) Given the order of preferences that different managers have for the different rooms in a hotel on one of its floor

(4) Schedule the flights or bus


simulation

Simulation is a technique used to model the operation of a system.


Applications of Simulation modeling

1. Lead time for inventory2. Times between machine breakdowns3. Times between arrivals4. Service times5. Inventory demand6. Times to complete project activities7. Number of employees absent


Decision Analysis

Decision Analysis can be used to determine optimal strategies in situations involving several decision alternatives and an uncertain or risk-filled pattern of events


Forecasting

Forecasting methods are techniques that can be used to predict future aspects of a business operation.


Applications of Forecasting

1. trend equations used for making forecasts

2. Sales be predicted


Quantitative factorsQuantitative factors might be different investment alternatives, interest rates, inventory levels, demand, or labor costQualitative factorsQualitative factors such as the weather, state and federal legislation, and technology breakthroughs should also be considered

Information may be difficult to quantify but can affect the decision-making process

What is Quantitative Analysis?


Alternative Starting Salary

Potential for Advancement

JobLocation

1. Rochester $ 38,500 Average Average

2. Dallas $ 36,000 Excellent Good

3. Greensboro $ 36,000 Good Excellent

4. Pittsburgh $ 37,000 Average Good

Job evaluation decision-making problem


Implementing the Results

Analyzing the Results

Testing the Solution

Developing a Solution

Acquiring Input Data

Developing a Model

The Quantitative Analysis Approach

Defining the Problem

Figure 1.1


Defining the Problem

Need to develop a clear and concise statement that gives direction and meaning to the following steps

This may be the most important and difficult step

It is essential to go beyond symptoms and identify true causes

May be necessary to concentrate on only a few of the problems – selecting the right problems is very important

Specific and measurable objectives may have to be developed


Developing a Model

Quantitative analysis models are realistic, solvable, and understandable mathematical representations of a situation

There are different types of models

$ Advertising

$ S

ales Y = b0 + b1X

Schematic models

Scale models


Developing a Model

Models generally contain variables (controllable and uncontrollable) and parameters

Controllable variables are generally the decision variables and are generally unknown

Parameters are known quantities that are a part of the problem


Acquiring Input Data

Input data must be accurate – GIGO rule

Data may come from a variety of sources such as company reports, company documents, interviews, on-site direct measurement, or statistical sampling

Garbage In

Process

Garbage Out


Developing a Solution

The best (optimal) solution to a problem is found by manipulating the model variables until a solution is found that is practical and can be implemented

Common techniques are SolvingSolving equations Trial and errorTrial and error – trying various approaches

and picking the best result Complete enumerationComplete enumeration – trying all possible

values Using an algorithmalgorithm – a series of repeating

steps to reach a solution


Testing the Solution

Both input data and the model should be tested for accuracy before analysis and implementation

New data can be collected to test the model Results should be logical, consistent, and

represent the real situation


Analyzing the Results

Determine the implications of the solution Implementing results often requires change

in an organization The impact of actions or changes needs to

be studied and understood before implementation

Sensitivity analysisSensitivity analysis determines how much the results of the analysis will change if the model or input data changes

Sensitive models should be very thoroughly tested


Implementing the Results

Implementation incorporates the solution into the company

Implementation can be very difficult People can resist changes Many quantitative analysis efforts have failed

because a good, workable solution was not properly implemented

Changes occur over time, so even successful implementations must be monitored to determine if modifications are necessary


Modeling in the Real World

Quantitative analysis models are used extensively by real organizations to solve real problems

In the real world, quantitative analysis models can be complex, expensive, and difficult to sell

Following the steps in the process is an important component of success


How To Develop a Quantitative Analysis Model

An important part of the quantitative analysis approach

Let’s look at a simple mathematical model of profit

Profit = Revenue – Expenses



Expenses can be represented as the sum of fixed and variable costs and variable costs are the product of unit costs times the number of units

Profit = Revenue – (Fixed cost + Variable cost)

Profit = (Selling price per unit)(number of units sold) – [Fixed cost + (Variable costs per unit)(Number of units sold)]

Profit = sX – [f + vX]

Profit = sX – f – vX

wheres = selling price per unit v = variable cost per unitf = fixed cost X = number of units sold



Expenses can be represented as the sum of fixed and variable costs and variable costs are the product of unit costs times the number of units

Profit = Revenue – (Fixed cost + Variable cost)

Profit = (Selling price per unit)(number of units sold) – [Fixed cost + (Variable costs per unit)(Number of units sold)]

Profit = sX – [f + vX]

Profit = sX – f – vX

wheres = selling price per unit v = variable cost per unitf = fixed cost X = number of units sold

The parameters of this model are f, v, and s as these are the inputs inherent in the model

The decision variable of interest is X


Pritchett’s Precious Time Pieces

Profits = sX – f – vX

The company buys, sells, and repairs old clocks. Rebuilt springs sell for $10 per unit. Fixed cost of equipment to build springs is $1,000. Variable cost for spring material is $5 per unit.

s = 10 f = 1,000 v = 5Number of spring sets sold = X

If sales = 0, profits = ––$1,000$1,000

If sales = 1,000, profits = [(10)(1,000) – 1,000 – (5)(1,000)]

= $4,000



0 = sX – f – vX, or 0 = (s – v)X – f

Companies are often interested in their break-even break-even pointpoint (BEP). The BEP is the number of units sold that will result in $0 profit.

Solving for X, we havef = (s – v)X

X = f

s – v

BEP = Fixed cost

(Selling price per unit) – (Variable cost per unit)



0 = sX – f – vX, or 0 = (s – v)X – f

Companies are often interested in their break-even break-even pointpoint (BEP). The BEP is the number of units sold that will result in $0 profit.

Solving for X, we havef = (s – v)X

X = f

s – v

BEP = Fixed cost

(Selling price per unit) – (Variable cost per unit)

BEP for Pritchett’s Precious Time Pieces

BEP = $1,000/($10 – $5) = 200 units

Sales of less than 200 units of rebuilt springs will result in a loss

Sales of over 200 units of rebuilt springs will result in a profit


Advantages of Mathematical Modeling

1. Models can accurately represent reality2. Models can help a decision maker

formulate problems3. Models can give us insight and information4. Models can save time and money in

decision making and problem solving5. A model may be the only way to solve large

or complex problems in a timely fashion6. A model can be used to communicate

problems and solutions to others


Mathematical Models

Objective Function – a mathematical expression that describes the problem’s objective, such as maximizing profit or minimizing cost

Constraints – a set of restrictions or limitations, such as production capacities

Uncontrollable Inputs – environmental factors that are not under the control of the decision maker

Decision Variables – controllable inputs; decision alternatives specified by the decision maker, such as the number of units of Product X to produce


Example : O’Neill Shoe Manufacturing company

The O’Neill Shoe Manufacturing company will produce a special-style shoe if the order size is large enough to provide a reasonable profit. For each special-style order, 5 hours are required to manufacture and only 40 hours of manufacturing time are available. The profit is $ 10 per pair. Let x denotes the number of pairs of shoes produced. Develop a mathematics model.


Mathematical ModelsMathematical Models

Deterministic Model – if all uncontrollable inputs to the model Deterministic Model – if all uncontrollable inputs to the model are known and cannot varyare known and cannot vary

Stochastic (or Probabilistic) Model – if any uncontrollable are Stochastic (or Probabilistic) Model – if any uncontrollable are uncertain and subject to variationuncertain and subject to variation

Stochastic models are often more difficult to analyze.Stochastic models are often more difficult to analyze.


Model Solution

The analyst attempts to identify the alternative

(the set of decision variable values) that provides the “best” output for the model.

The “best” output is the optimal solution. If the alternative does not satisfy all of the model

constraints, it is rejected as being infeasible, regardless of the objective function value.

If the alternative satisfies all of the model constraints, it is feasible and a candidate for the “best” solution


Model Solution

One solution approach is trial-and-error. Might not provide the best solution Inefficient (numerous calculations required)

Special solution procedures have been developed for specific mathematical models. Some small models/problems can be solved by

hand calculations Most practical applications require using a

computer

45


Trial and Error Solution For The O’Neill Shoe Manufacturing

Company

46


Example: Iron Works, Inc.

Iron Works, Inc. manufactures two products

made from steel and just received this month's allocation of b pounds of steel. It takes a1 pounds of steel to make a unit of product 1 and a2 pounds of steel to make a unit of product 2. Let x1 and x2 denote this month's production level of product 1 and product 2, respectively. Denote by p1 and p2 the unit profits for products 1 and 2, respectively. Iron Works has a contract calling for at least m units of product 1 this month. The firm's facilities are such that at most u units of product 2 may be produced monthly.

47


Example: Iron Works, Inc

Mathematical Model The total monthly profit =

(profit per unit of product 1) x (monthly production of product 1)

+ (profit per unit of product 2) x (monthly production of product 2)

= p1x1 + p2x2 We want to maximize total monthly profit:

Max p1x1 + p2x248


Example: Iron Works, Inc

Mathematical Model (continued) The total amount of steel used during

monthly production equals: (steel required per unit of product 1)

x (monthly production of product 1) + (steel required per unit of product 2)

x (monthly production of product 2)

= a1x1 + a2x2 This quantity must be less than or equal to the allocated b pounds of steel:

a1x1 + a2x2 < b 49



Mathematical Model (continued) The monthly production level of product 1 must be

greater than or equal to m :

x1 > m The monthly production level of product 2 must be

less than or equal to u :

x2 < u However, the production level for product 2 cannot

be negative:

x2 > 0 50



Mathematical Model Summary

Max p1x1 + p2x2

s.t a1x1 + a2x2 < b

x1 > m

x2 < u

x2 > 0

51

ObjectiveObjectiveFunctionFunction

ConstraintsConstraints

““Subject Subject to”to”



Question:

Suppose b = 2000, a1 = 2, a2 = 3, m = 60, u = 720, p1 = 100, p2 = 200. Rewrite the model with these specific values for the uncontrollable inputs.



Answer:Substituting, the model is:

Max 100x1 + 200x2

s.t. 2x1 + 3x2 < 2000

x1 > 60

x2 < 720

x2 > 0


Models Categorized by Risk

Mathematical models that do not involve risk are called deterministic models We know all the values used in the model

with complete certainty Mathematical models that involve risk,

chance, or uncertainty are called probabilistic models Values used in the model are estimates

based on probabilities


Possible Problems in the Quantitative Analysis Approach

Defining the problem Problems are not easily identified Conflicting viewpoints Impact on other departments Beginning assumptions Solution outdated

Developing a model Fitting the textbook models Understanding the model


Possible Problems in the Quantitative Analysis Approach

Acquiring input data Using accounting data Validity of data

Developing a solution Hard-to-understand mathematics Only one answer is limiting

Testing the solutionAnalyzing the results


Implementation – Not Just the Final Step

Lack of commitment and resistance to change

Management may fear the use of formal analysis processes will reduce their decision-making power

Action-oriented managers may want “quick and dirty” techniques

Management support and user involvement are important


Implementation – Not Just the Final Step

Lack of commitment by quantitative analysts

An analysts should be involved with the problem and care about the solution

Analysts should work with users and take their feelings into account


Summary

Quantitative analysis is a scientific approach to decision making

The approach includes Defining the problem Acquiring input data Developing a solution Testing the solution Analyzing the results Implementing the results


Summary

Potential problems include Conflicting viewpoints The impact on other departments Beginning assumptions Outdated solutions Fitting textbook models Understanding the model Acquiring good input data Hard-to-understand mathematics Obtaining only one answer Testing the solution Analyzing the results


Summary

Implementation is not the final step Problems can occur because of

Lack of commitment to the approach Resistance to change

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