practice questions.pdf

24
Quantitative Techniques - II Topic wise unsolved questions from Previous Years Exam papers EXERCISE 1 EXERCISE 1 EXERCISE 1 EXERCISE 1 INTRODUCTION TO INTRODUCTION TO INTRODUCTION TO INTRODUCTION TO DECISION MODELING APPROACH: DECISION MODELING APPROACH: DECISION MODELING APPROACH: DECISION MODELING APPROACH: Role of Quantitative Analysis in Decision Making and Operations Research Techniques their Nature and Characteristics feature. Different types of OR problems. Qt 1: “Modern day managerial decision making uses various Quantitative Technique tools.” Comment on this statement mentioning the utility of at least five such tools (with examples). Qt: 2 Answer the following questions. Write down only the correct option in your answer sheet. (a) ................................ is the first step in Quantitative Analysis. Qt 3: Write a detailed note on the decision modeling approach.

Upload: superking-super

Post on 30-Oct-2014

1.057 views

Category:

Documents


0 download

DESCRIPTION

Uploaded from Google Docs

TRANSCRIPT

Page 1: Practice Questions.pdf

Quantitative Techniques - II Topic wise unsolved questions from Previous Years Exam papers

EXERCISE 1EXERCISE 1EXERCISE 1EXERCISE 1

INTRODUCTION TO INTRODUCTION TO INTRODUCTION TO INTRODUCTION TO DECISION MODELING APPROACH:DECISION MODELING APPROACH:DECISION MODELING APPROACH:DECISION MODELING APPROACH: Role of Quantitative Analysis in Decision Making and Operations Research Techniques their Nature and Characteristics feature. Different types of OR problems. Qt 1: “Modern day managerial decision making uses various Quantitative Technique tools.” Comment on this statement mentioning the utility of at least five such tools (with examples).

Qt: 2 Answer the following questions. Write down only the correct option in your answer sheet.

(a) ................................ is the first step in Quantitative Analysis.

Qt 3: Write a detailed note on the decision modeling approach.

Page 2: Practice Questions.pdf

Quantitative Techniques - II Topic wise unsolved questions from Previous Years Exam papers

EXERCISE 2EXERCISE 2EXERCISE 2EXERCISE 2

TTTTHEORY OF GAMES:HEORY OF GAMES:HEORY OF GAMES:HEORY OF GAMES: Game Models, Two-person Zero Sum Game, Solution of 2 x n and n x 2 games, Games of Pure and Mixed Strategy, Principle of Dominance.

Qt 1: Two competing firms R and S, must open their next branch at one of the three cities A, B and C, whose distance profile has been given below. If both the companies open their branches in the same city, they will split the business evenly. If, however, they build the branches in different cities, the company that is closer to a given city will get all that business. If all the cities have the same amount of business, where should the firm open the branch? From A to B : 25 Km From A to C : 18 Km From B to C : 22 Km

(10 Marks) (Suggested Time : 30 Minutes) Qt: 2 Answer the following questions. Write down only the correct option in your answer sheet. (a) The solution to the mixed strategy 2 x 2 game can be found by: (i) equating a player's expected winning for one of the opponent's strategies with opponent's other strategy (ii) equating the value of the game for player A with the value of the game for the player B. (iii) setting 'q' equal to 'p' and solving (iv) setting the maximin and minimax as equality and solving the game. (v) Taking out the saddle point and getting the value of the game. (b) Which of the following methods are used to reduce the size of the game of mixed strategy: (i) Inversion and Reduction (ii) Rotation and Reduction (iii) Co-factorization and Reduction (iv) Transposition and Reduction (v) None of the above (c) Index of Optimism is :

(i) Discretionary (ii) Confusion (iii) Subjective Probability (iv) Opportunity Cost / Penalty (v)Expectation Principle

Qt 3: There are two companies in a competitive market. Both these companies compete to increase their market share. Assuming that, at present both the company's market share is at equilibrium and it will move in favour of Company A by 8 %. The payoff matrix from company A's perspective has been given below which was used for arriving at the movement of market share.

Page 3: Practice Questions.pdf

3 4 5

9 8 Q

4 p 5 Find the possible value(s) for p and q. Mention further assumption (if any) made for solving the game. Qt 4: There are two companies A and P, who are competing for the same product in the same market. Each of these companies attempts to raise its share of the market. An advertising campaign by A can increase its share of the market by 7 percent provided P does nothing. Company P contemplates invoking a price cut if it is worthwhile. The price cut shall cause a 5 percent gain to the company P, but the gain is subject to the condition that A takes no action. Price cut by P, accompanied by the advertising campaign of A shall lead to a 2 percent rise in P's share of market. No action on part of both the companies shall leave the market share for them undisturbed at 40:60. Write the information in form of a pay-off table. Find out the optimal strategies and the value of the game. Comment on the results. Qt 5: Game theory is the study of interactions among players whose payoffs depend on one another's choices and who recognize the interdependence in trying to maximize their respective payoffs. Consider, therefore, the pharmaceutical case study. One of the most profitable products sold by the incumbent (hereinafter, I) dominated its category but faced the introduction of a substitute product by a major competitor. As the late mover, the entrant (hereinafter, E) was expected to launch its own product at a very large discount. It was uncertain exactly how E's launch price would be, and whether I should reduce its own price in anticipation or reaction. The cash flows involved were very large enough, however, to compel careful analysis of I's options. Since there were only two players, the information on their strategies and payoffs could conveniently be captured in a payoff matrix, as depicted in table 1 with rows representing I's strategic options, and the columns E's. The first entry in each cell corresponds to the estimated NPV, in million of rupees, for I, and the second entry to the estimated NPV for E; the three crossed cells were judged to be arithmetically infeasible. Within this set-up, firms select strategies – rows or columns – and not individual outcomes, as in particular cells. It is in this sense that payoff matrices capture the dependence of each firm's payoff on its rival's strategies as well as its own.

Table 1 : The pharmaceutical Payoff Matrix (million of rupees for I,E) Incumbent (I's) Price Entrant's (E's) Price

Very Low Low Moderate High

No Price Change 358,190 507, 168 585,129 624,116

Large Price Advantage for E 418,163 507,168 X X

Small Price Advantage for E 454,155 511,138 636,126 X

Neutralization of E's Price Advantage

428,50 504,124 585,129 669,128

Read the given literature carefully and observe the table keenly as well. Answer the following questions:

Page 4: Practice Questions.pdf

(a) Briefly describe (Not more than five sentences) the contents of the case in simpler words with some quantitative inputs. (b) Recommend the optimal strategies for both the players. (c) What we understand by first mover advantage. Does this concept has a fitment in the given case? (d) Do you think Nash Equilibrium holds true for the given case. Give quantitative reasoning for your answer. (Nash Equilibrium can be defined as - A strategy pair, one for each player, from which neither player had an incentive to deviate unilaterally.)

(2½ x 4 = 10 Marks) Qt 6: Dilemma of two athletes winning GOLD medal in Commonwealth Games but caught in doping test

Beedi, the retired sportsman is currently serving the nation as a coach of the athletics team. These athletes are competing in the Common Wealth Games. In the athletics arena, two of the team members have won a GOLD medal in a team event. Subsequent to the awarding of the medal one of the athletes tested positive in the dope test. Beedi as their coach objected vehemently to the results of the dope test. As a result of the protest lodged by Beedi, a second sample was drawn. To the dismay of all the organizers, this time, the second athlete tested positive. Since both these two athletes were part of the team event, the organizers were in a perplexed state, what to do with the awarded medal. Should the athletes be stripped of the medal or should they be allowed to keep it. The problem is that, as per rules, for stripping them from the medal both the samples should test positive. To come out of this interesting but complicated situation, the organizers decided to adopt an entirely new strategy. Both the athletes will have to appear individually before a jury. The jury will interrogate them and then will arrive at conclusion. Accordingly, the first athlete appeared before the jury. The jury promised no punishment to him if he confesses and a stringent verdict of life time ban for the second athlete. Similarly, if the second athlete will confess, the first athlete will be banned for life. The jury had internally decided to award No Punishment, if none of them confessed. The jury had further decided that if both of them confesses, they would be awarded a two years ban. Construct a game theory matrix arising out of the above facts. Discuss the various strategies that are possible in the given situation.

(8 marks)

Page 5: Practice Questions.pdf

Quantitative Techniques - II Topic wise unsolved questions from Previous Years Exam papers

EXERCISE 3EXERCISE 3EXERCISE 3EXERCISE 3

DECISION THEORY:DECISION THEORY:DECISION THEORY:DECISION THEORY: Payoff table, Opportunity loss or regret table, decision rules, decision making under the conditions of certainty, uncertainty, risk and competition or conflict.

Qt 1: Dr Jhamjham is considering the possibility of opening a small dress shop at Greenbanks Circle, a few blocks from the university. She has located a good mall that attracts students. Her options are to open a small shop, a medium size shop or no shop at all. The market for a dress shop be good, average or bad. The probabilities for these three possibilities are 0.2 for a good market, 0.5 for an average market and 0.3 for a bad market. The profit or loss for the medium sized and small sized shops for the various market conditions are given below. Building no shop at all yields no loss and no gain. What do you recommend.

(figures in Rupees) Alternative Good Market Average Market Bad Market

Small Shop 75000 25000 - 40000

Medium Shop 100000 35000 - 60000

No Shop 0 0 0

(10 Marks) (Suggested Time : 25 Minutes) Qt 2: Answer the following questions. Questions (a) to (e) carry 1 mark each. Write down only the correct option in your answer sheet. (a). The summated value of the product of the probabilities of mutually exclusive events & the collectively exhaustive states of nature with the conditional payoffs of those same states of nature is called as : (i) EPPI (ii) EVPI (iii) EVP (iv) Hurwicz Criterion (v) None of the above (b) Select the odd one out. (i) Decision can be made under the conditions of Risk. (ii) Decision can be made under the conditions of Uncertainty. (iii) Decision can be made under the conditions of Competition. (iv) Decision can be made under the conditions of Complacency (v) Decision can be made under the conditions of Certainty. (c) A risk taker in Decision Theory assigns:

(i) A higher utility value to losses than to gains (ii) A higher utility value to gains than to losses (iii) Equal utility value to gains ans losses (iv) All the above (v) None of the above

Page 6: Practice Questions.pdf

(d) Identify the option which is most similar to the Savage Rule

(i) Maximization Assignment Problems (ii) Maximization Maxima – Minima Problems (iii) Maximization Sensitivity Analysis (iv) Maximization Vogel's Approximation Method (v) None of the above

Qt 3: Rampack and Rimzhim have known each other since high school. Two years ago they entered the same university and today they are taking a PGDM course in ITM Business School. Both hope to graduate in RMM. In an attempt to make extra money and to use some of the knowledge gained from business course Rampack and Rimzhim decided to look into the possibility of starting a word company which would provide word processing services to students who needed model test papers and other reports prepared in a professional manner. Using the systems approach, Rampack and Rimzhim have identified three strategies. Strategy 1 is to invest in a fairly expensive microcomputer system with a high quality laser printer. In a favourable market they should be able to obtain a net profit of Rs 10000 over the next two years. If the market is unfavourable, they can lose Rs 8000. Strategy 2 is to purchase a less expensive system and with a favourable market they could get a return of Rs 8000 during the next two years but with an unfavrouable market, they can get a loss of Rs 4000. Their strategy 3 is to do nothing. Rampack is baiscally a risk taker while Rimzhim tries to avoid risk. (a) What type of decision procedure should Rampack use? What would Rampack's decision be in terms of profit or loss ? (b) What type of decision maker is Rimzhim? What decision will Rimzhim make? (c) If Rampack and Rimzhim were indifferent to risk, what type of decision approach should they use? (d) Assuming that you are an expert in the area of business plan implementation and you provide suggestions to the budding entrepreneurs. Give the process which you will follow and the assumptions made if any. Show relevant calculations made.

(4 x 5 = 20 Marks) (Suggested Time : 25 Minutes) Qt 4: A Television (TV) dealer finds that the cost of holding a TV in stock for a week is Rs 50. Customers who cannot obtain new TV sets immediately tend to go to other dealers and he estimates that for every customer who cannot get immediate delivery he loses an amount of Rs 200. For one particular model of TV, the probabilities of demand of 0, 1, 2, 3, 4 and 5 TVs in a week are 0.05, 0.10, 0.20, 0.30, 0.20, and 0.15 respectively. (a) How many TV sets per week the dealer should order? Assume that there is no time lag between ordering and delivering.

(b) The dealer is thinking of spending some amount on a small market survey to obtain additional information regarding the demand levels. How much he should be willing to spend on such survey?

Solve the above problem first taking the maximization approach and then taking the minimization approach.

Page 7: Practice Questions.pdf

Qt 5: Rampack wishes to invest in one of the three companies A, B, and C for taking the benefits of the economic prosperity of the nation. Although all three companies are making good profit and are operating in the same industry but it has been observed from past experience that the distribution pattern of dividends are not identical for these three companies. The following table shows the dividend paid by 3 companies (in percentage) at various levels of stock market index. Also the chance factor for the particular level of stock market index is given in the additional table. Table 1 : Payoff table for dividend payment (in%) at Various Levels of stock market index

Company Level of stock market (Thousands)

5 10 15 20 25 30

A 50 60 80 90 110 120

B 40 50 70 110 130 140

C 40 60 80 90 100 110

Table 2 : Level of stock market and Associated Chance factor

Level of Stock Market Index (Thousands) 5 10 15 20 25 30

Chance factor 10% 15% 40% 20% 10% 5%

(a) Assuming that Rampack is a risk taker, which company will he be investing? (b) Rampack with his conventional wisdom thinks that the stock market index will be between 15000 to 20000. Further, he has a belief that stock market will be precisely at 18000. Now, he is in a state of confusion, which company between A and C is better. Both these companies are giving the same return at stock market index of 15000 and 20000. Can you help him out in selecting the company to invest upon. (c) Calculate the EVPI and comment on the results. (d) Since Rampack is currently in TERM IV of the MBA programme, he is well aware of the Simulation technique. He considers the following random numbers for ascertaining the stock market index but is not able to recollect the concept that he studied in TERM II. Can you help him out in selecting a company to invest in.

Company Random Numbers

A 87 4 8 78 56 90 87 52 68 26

B 56 76 20 53 35 93 25 9 77 89

C 76 90 73 34 23 5 99 9 63 87

Suggestions : Rampack is a student of MBA programme, accordingly he is well aware about the following concepts

Page 8: Practice Questions.pdf

-> Decision making under the conditions of Uncertainty -> Regression Analysis -> Decision making under the conditions of Risk -> Simulation technique.

Suggested Time : 40 Minutes

(4 x 5 = 20 Marks) Qt 6: Beedi the retired footballer is running a SME. Beedi has come with entirely new product line. His representatives have been offered the new product line. If they accept the new line, they can handle it either with the existing sales-force or set up a separate sales-force exclusively for this line. The setting up of a separate sales-force would involve an initial investment of Rs 100000 on office, equipment and hiring and training of sales persons. If the new line is handled by the existing sales-force using the existing facilities, the initial investment would only be Rs 30000 primarily for training the present sales staff. The new product sells for Rs 2500. The representative normally receives 20 percent of the sales price on each unit sold. They, in turn pay 10 percent to the sales person as his commission. In order to encourage the representatives to set up a separate sales-force to handle the new product, Beedi offers them to pay 30 percent of the sales price of each unit sold to the representative, if they set up a separate sales-force. Otherwise the normal 20 percent will be paid. The commission to the sales person will remain fixed at 10 percent in both cases. Based on the size of the territory and experience with other products, the representative estimates the following probabilities for annual sales of the new product.

Sales in units Probability

1000 0.10 2000 0.15 3000 0.40 4000 0.30 5000 0.05

Set up a Regret Table and ascertain the best course of action. Comment on the Results.

(8 Marks) Qt 7: Beedi, the owner of a Tennis shop must decide how many tennis shirts to order for the summer season. For a particular type of shirt, he must order in batches of 100. If he orders 100 shirts, his cost is Rs 100 per shirt; if he orders 200, his cost is Rs 90 per shirt; and if he orders 300 or more shirts, his cost is Rs 85 per shirt. The selling price is Rs 120, but if any shirts are left unsold at the end of the summer, they will sold for half price. For simplicity, the owner believes that demand for this shirt will be either 100, 150 or 200. Of course, he cannot sell more shirts that he stocks. If, however, he under stocks, there is a goodwill loss of Rs 5 for each shirt a person wants to buy but cannot because it is out of stock. Furthermore, the owner must place the order now for the forthcoming summer season; he cannot wait to see how the demand is running for this shirt before he orders, nor can he place several orders.

Page 9: Practice Questions.pdf

For the above situation, define: (a) The decision maker (b) The alternative courses of action (c) The events (d) The consequences

(1½ x 4 = 6 Marks) Qt 8: Jhamjham Inc : Entering in Baby Food Industry After watching a movie about a young women who quit a successful corporate career to start her own baby food company, Dr Jhamjham, a Dentist by profession decided that she wanted to do the same. In the movie, the baby food company was very successful. Dr Jhamjham knew, however that it is much easier to make a movie about a successful women starting her own company than to actually do it. The product had to be of the highest quality, and Dr Jhamjham had to get the best people involved to launch the new company. Dr Jhamjham closed her polyclinic and launched new company – Jhamjham Inc. Dr Jhamjham decided to target the upper end of the baby food market by producing baby food that contained no preservatives but had a great taste. Although the price would be slightly higher than for existing baby food, Dr Jhamjham believed that parents would be willing to pay more for a high quality baby food. Instead of putting baby food in jars, which would require preservatives to stabilize the food, Dr Jhamjham decided to try a new approach. The baby food would be frozen. This would allow for natural ingredients, no preservatives, and outstanding nutrition. Getting good people to work for the new company was also important, Dr Jhamjham decided to find people with experience in finance, marketing, and production to get involved with Jhamjham Inc. With her enthusiasm and charisma, Dr Jhamjham was able to find such a group. Their first step was to develop prototypes of the new frozen baby food and to perform a small pilot test of the new product. The pilot test received rave reviews. The final key to getting the young company off to a good start was to raise funds. Three options were considered corporate bonds, preferred stock, and common stock. Dr Jhamjham decided that each investment should be in blocks of INR 30000. Furthermore, each investor should have an annual income of at least INR 40000 and a net worth of INR 100000 to be eligible to invest in Jhamjham Inc. Corporate bonds would return 13% per year for the next five years. Dr Jhamjham furthermore guaranteed that investors in the corporate bonds would get at least INR 20000 back at the end of five years. Investors in preferred stock should see their initial investment increase by a factor of 4 with a good market or see the investment worth only half of the initial investment with an unfavourable market. The common stock had the greatest potential. The initial investment was expected to increase by a factor of 8 with a good market, but investors would lose everything if the market was unfavourable. During the next five years, it was expected that inflation would increase by a factor of 4.5% each year. (a) Rimzhim is considering an investment. She is very conservative and do not want to repent later for taking wrong decision. What do you recommend? (b) Prof. Tamtam is also considering an investment, although he believes that there is only 11% chance of success. What do you recommend. (c) Beedi, a footballer believes that there is an equally likely chance for success. What is your recommendation.

Page 10: Practice Questions.pdf

(d) Rampack is extremely optimistic about the market for the new baby food. What is your advice to Rampack. Qt 9:

Dr Jhamjham's tryst for the ascertainment of inventory of laptops Dr Jhamjham is running a small retail electronics shop. Going by the past experience, she has estimated that the cost of holding one laptop in the shop comes out to be Rs 100. She keeps certain amount of laptops in her shop. Customers keep on visiting the shop and they buy the laptops from her shop. If, she runs out of stock and a customer returns without purchasing a laptop, she losses an amount of Rs 400. The probabilities worked out for the different levels of demand is given in the following table:

Demand 0 1 2 3 4 5

Probability 0.05 0.1 0.2 0.3 0.2 0.15 (a) Assuming the above situation as Minimization of Costs, give an advice to Dr Jhamjham as to how many laptops should be kept in her retail store? Use the expectation principle. Calculate EVPI. (b) Assuming the above situation as Maximization of Profits , give an advice to Dr Jhamjham as to how many laptops should be kept in her retail store? Use the expectation principle. Calculate EVPI.

(2 x 6 = 12 marks)

Page 11: Practice Questions.pdf

Quantitative Techniques - II Topic wise unsolved questions from Previous Years Exam papers

EXERCISE 4EXERCISE 4EXERCISE 4EXERCISE 4

LINEAR PROGRAMMINGLINEAR PROGRAMMINGLINEAR PROGRAMMINGLINEAR PROGRAMMING:::: Formulation of LPP, Graphical Solutions, Simples Methods, Post Optimality Analysis, Duality

Qt: 1 Answer the following questions. Write down only the correct option in your answer sheet. (a) A graphical method consists of:

(i) Objective Function : Multiple Linear Equations : Single Corner Point (ii) Objective Equation : Multiple Linear Functions : Multiple Corner Point (iii) Multiple Objective Functions : Multiple Linear Equations : Multiple Corner Points (iv) Objective Function : Multiple Linear Equations : Multiple Corner Points (v) Multiple Functions : Single Linear Equation : Multiple Corner Points

(b) A feasible region of a LPP problem:

(i) needs to satisfy all of the problem's constraints simultaneously (ii) needs to satisfy all of the problem's constraints not simultaneously (iii) It can be any corner point (iv) All corner points of the graph (v) must always gives maximum profit.

(c) A manufacturing company's production problem has been represented through a LPP. In this LPP, if two corner points are giving same cost then:

(i) The solution is infeasible (ii) Two corner points giving the same cost is not possible (iii) One corner point essentially will be out of feasible region. (iv) The company can run efficiently at two different production levels (v) The company is running at a loss.

(d) A graphical method consists of : (i) Objective Function : Multiple Linear Equations : Single Corner Point (ii) Objective Equation : Multiple Linear Functions : Multiple Corner Point (iii) Multiple Objective Functions : Multiple Linear Equations : Multiple Corner Points (iv) Objective Function : Multiple Linear Equations : Multiple Corner Points (v) Multiple Functions : Single Linear Equation : Multiple Corner Points (e) A manufacturing company's production problem has been represented through a LPP. In this LPP, if two corner points are giving same cost then:

(i) The solution is infeasible (ii) Two corner points giving the same cost is not possible (iii) One corner point essentially will be out of feasible region. (iv) The company can run efficiently at two different production levels (v) The company is running at a loss.

Page 12: Practice Questions.pdf

Qt 2: Rampack and Rimzhim are successfully running an Electronics Company. As a part of quality improvement initiative, Rampack and Rimzhim wants to give more and more training to their employees. Their employees complete a three day training program on teaming and a two day training program on problem solving. The HR manager has suggested that, for the improvement of quality at least 8 training programs on teaming and at least 10 training programs on problem solving should be offered during the next six months. Rampack and Rimzhim have accepted this proposal and in addition to this have also decided that at least 25 training programs must be offered during this period. Rampack and Rimzhim engages a Corporate trainer to conduct the training programs. During the next six months, the corporate trainer has 84 days of training time available. Each training program on teaming costs Rs 10000 and each training program on problem solving costs Rs 8000. (a) Formulate a LPP that can be used to determine the number of training programs on teaming and the number of training programs on problem solving that should minimize the total cost. (b) Solve Graphically and comment on the results.

(2 X 5 = 10 Marks) (Suggested Time : 40 Minutes)

Qt 3: Prof. Tamtam (investor) has money making activities A1, A2, A3 and A4. He has only Rs 1 lakhs to invest. In order to avoid excessive investment, no more than 50% of the total investment can be placed in activity A2 and activity A3 put together. Activity A1 is very conservative, while activity A4 is speculative. To avoid excessive speculation at least Rupee 1 must be invested in activity A1 for every Rs 3 invested in activity A4. The data on the return on investment are as follows:

Activity Anticipated return On investment (%)

A1 10 A2 12 A3 14 A4 16

Prof Tamtam wishes to know how much to invest in each activity to maximize the total return on the investment. Formulate the above problem as LPP and ascertain whether the equations can be solved directly (system of linear equations) or not?

(6 Marks) Qt 3:

Dr Jhamjham’s Innovative Packaging system for Christmas Season Dr Jhamjham is running a manufacturing unit producing items which are of daily consumption like biscuits, cookies, chocolates etc. Currently she is having a standardized system of packaging and distributing the products that are manufactured in her company on stand-alone basis. For the forthcoming Christmas festival, she has received a suggestion from Prof. Tamtam to go for revamping of the packaging system and introduction of gift packs. These gift packs would contain a mixture of these products. Accordingly, she has considered four types of gift packs containing three types of products: Orange Juice (OJ), Cocoa Chocolates (CC) and Potato Wafers (PW). Market research conducted to assess the preferences of the consumer shows the following types of assortments to be in good demand:

Page 13: Practice Questions.pdf

Assortment Contents Selling price / kg (Rs) A Not less than 40% of OJ 400

Not more than 20% of CC B Not less than 20% of OJ 500

Not more than 40% of CC C Not less than 50% of OJ 450

Not more than 10% of CC D No restrictions 240

For the biscuits the manufacturing capacity and costs are given below:

Variety Plant Capacity (Kg / day) Manufacturing cost (Rs / Kg) OJ 200 150 CC 200 200 PW 150 140

Dr Jhamjham wants to formulate the above data as a LP model to find the production schedule which maximizes the profit assuming that there are not market restrictions. Since she is unable to come up with the proper mathematical model, please help her out.

(6 marks)

Qt 4:

Optimal Product mix of cricketing balls – White Fire and Red Fury

Beedi, the retired footballer has decided to manufacture two types of world class cricketing balls. A white ball to be used for ODI matches. The name given to this product was ‘White Fire’ . The second variety of ball which will be used for Test Matches was named ‘Red Fury’. Each unit of Red Fury takes twice as long to produce as one unit of White Fire and if the company were to produce only White Fire it would have time to produce 2000 units per day. The availability of raw material is sufficient to produce 1500 units per day of both White Fire and Red Fury combined. Product Red Fury contains a special type small iron ball at the core so that the ball becomes heavier. This small iron ball is imported from Korea and thus has a limited availability accordingly only 600 units can be made per day. Product White Fire fetches a profit of Rs 2 per unit while Product Red Fury gets a profit of Rs 4 per unit. Find the product mix by the LPP method by using the graphical method so that Beedi can make optimal profit.

(8 marks) Qt 5: (a) What are the various assumptions of Linear Programming Models? (b) Write an account of the areas in which Linear Programming Models are used.

Page 14: Practice Questions.pdf

Quantitative Techniques - II Topic wise unsolved questions from Previous Years Exam papers

EXERCISE 5EXERCISE 5EXERCISE 5EXERCISE 5

TRANSPORTATION MODELS TRANSPORTATION MODELS TRANSPORTATION MODELS TRANSPORTATION MODELS – Methods of Feasible Solution: NW Corner Method, LCM, VAM, Row / Column Minima Methods, Optimal Solutions, Balanced and Unbalanced Types, Maximization and Minimization Types.

Qt: 1 Answer the following questions. Write down only the correct option in your answer sheet.

(a) Find the combination which is least cost sensitive:

(i) NW Corner Method and LCM (ii) LCM and Row Minima Method (iii) VAM and Column Minima Method (iv) VAM and NW Corner Method (v) LCM and VAM

(b) Transportation model deals with: (i) Multiple factories : Multiple Markets : Multiple Products (ii) Multiple factories : Single Market : Independent of products (iii) Single factory : Multiple markets : Multiple products (iv) Multiple factories : Multiple Market : Independent of products (v) Single factory : Multiple Market : Independent of products

(c) Find the combination which is least cost sensitive: (i) NW Corner Method and LCM (ii) LCM and Row Minima Method (iii) VAM and Column Minima Method (iv) VAM and NW Corner Method (v) LCM and VAM (d) If certain routes are not available (prohibited) for transportation:

(i) We assign a zero cost to those routes (ii) We assign a very high cost to those routes (iii) The lowest value in the matrix is subtracted from all the other elements of the matrix (iv) We add dummy rows / columns in the matrix (v) Both (i) and (iv)

(e) Transportation model deals with:

(i) Multiple factories : Multiple Markets : Multiple Products (ii) Multiple factories : Single Market : Independent of products (iii) Single factory : Multiple markets : Multiple products (iv) Multiple factories : Multiple Market : Independent of products (v) Single factory : Multiple Market : Independent of products

Page 15: Practice Questions.pdf

Qt 2: Build the mathematical model for the following transportation problem. Warehouses

Plant W1 W2 W3 Supply

P1 C11 C12 C13 S1

P2 C21 C22 C23 S2

P3 C31 C32 C33 S3

Demand D1 D2 D3 (5 Marks) (Suggested Time : 10 Minutes)

Qt 3: A dairy firm has three plants located in a state. The daily milk production at each plant is as follows:

Plant 1 : 6 million litres Plant 2 : 1 million litres Plant 3 : 10 million litres

Each day, the firm must fulfill the needs of its four distribution centres. Minimum requirement at each centre is as follows:

Distribution centre 1 : 7 million litres Distribution centre 2 : 5 million litres Distribution centre 3 : 3 million litres Distribution centre 4 : 2 million litres

Cost in hundreds of rupees of shipping one million litre from each plant to each distribution centre is given in the following table:

D1 D2 D3 D4

P1 2 3 11 7

P2 1 0 6 1

P3 5 8 15 9

Find the Initial Basic Feasible Solution by all the FIVE methods. Comment on the results Suggestions: If a transporter has agreed to carry the goods free of cost then the cost of transportation as recorded in the books of the company will be zero rupees. In short there is no cost of transportation. Suggested Time: 30 Minutes

Qt 4: A leading firm has three auditors and each auditor can work up to 160 hours during the next month, during which time three projects must be completed. Project I will take 130 hours project II will take 140 hours, and project III will take 160 hours. The amount per hour that can be billed for assigning each auditor each project is given below:

Page 16: Practice Questions.pdf

Projects

Auditor I II III

1 Rs 1200 Rs 1500 Rs 1900

2 Rs 1400 Rs 1300 Rs 1200

3 Rs 1600 Rs 1400 Rs 1500 (i) Formulate this as a transportation problem and find the Initial basic feasible solution by VAM and LCM methods (ii) Also find out the total billing during the next month. Comment on the results

Suggestions : -> Demand and Supply gap may happen. -> Amount may be taken in hundreds of rupees Suggested Time : 25 minutes

(2 x 5 = 10 Marks) Qt 5: Rampack is running a company whereby, he is having manufacturing units at four locations and from these four centers, he transports the goods to three consumption locations. A special type of product is manufactured in factories A, B, C and D; and is sold at centers 1,2 and 3. The cost in Rupees of product per unit and capacity in Kg per unit time of each plant is given below:

Factory Cost (Rs) per unit Capacity (Kg) per unit A 12 100 B 15 20 C 11 60 D 13 80

The sales price in Rupees per unit and the demand in Kg per unit time are as follows:

Sales Center Sales Price (Rs) per unit Demand (Kg) per unit 1 15 120 2 14 140 3 16 60

Give an advice to Rampack s so that it becomes easy for him to ascertain the method through which the sales distribution is involving the lowest possible cost. Comment on the results. Qt 6: After passing out their MBA, Rampack and Rimzhim wants to become entrepreneurs. They are planning to start a factory to manufacture Fruits & Nuts Bread. They are going to set-up two plants for the same. The capacity and the cost of production for these plants are given below:

Plant Production Capacity Production Cost (Rs per bread)

A 2500 2.30

B 2100 2.50

Page 17: Practice Questions.pdf

Rampack and Rimzhim have already entered into a tie-up with four restaurants to buy their product. These restaurants are ready to buy the products in different quantities and are also offering different purchase price for the Fruits & Nuts bread. Their demand and the prices they are willing to pay are given in the following table:

Chain Demand Price offered (Rs / bread)

1 1800 3.90

2 2300 3.70

3 550 4.00

4 1750 3.60

The cost (in Rupees) of transportation of one bread from plant to restaurant chain is given below:

Plant Chain 1 Chain 2 Chain 3 Chain 4

A 0.06 0.08 0.11 0.09

B 0.12 0.06 0.08 0.05

Using the NW Corner Method and the Vogel's Approximation Method(VAM), determine the delivery schedule that will maximize the total profit for Rampack and Rimzhim. Comment on the results obtained from these two methods.

Suggestions � Calculate per unit profit.

� This is a Profit matrix while Transportation problems works on minimization problems

� multiply with 100 to avoid decimals

Suggested Time: 30 minutes (10 Marks)

Qt 7: Rampack is running a company whereby, he is having manufacturing units at four locations and from these four centers, he transports the goods to three consumption locations. A special type of product is manufactured in factories A, B, C and D; and is sold at centers 1,2 and 3. The cost in Rupees of product per unit and capacity in Kg per unit time of each plant is given below:

Factory Cost (Rs) per unit Capacity (Kg) per unit A 12 100 B 15 20 C 11 60 D 13 80

The sales price in Rupees per unit and the demand in Kg per unit time are as follows:

Sales Center Sales Price (Rs) per unit Demand (Kg) per unit 1 15 120 2 14 140 3 16 60

Page 18: Practice Questions.pdf

Give an advice to Rampack s so that it becomes easy for him to ascertain the method through which the sales distribution is involving the lowest possible cost. Comment on the results.

(8 Marks) Qt 8: ABC Limited has three production shops supplying a product to five warehouses. The cost of production varies from shop to shop and cost of transportation from one shop to a warehouse also varies. Each shop has a specific production capacity and each warehouse has certain amount of requirement. The costs of transportation are given as follows:

Shop Warehouse Supply I II III IV V

A 6 4 4 7 5 100 B 5 6 7 4 8 125 C 3 4 6 3 4 175

Demand 60 80 85 105 70 The cost of manufacturing the product at different production shops is:

Shop A B C Variable cost (Rs) 14 16 15

Find the optimum distribution pattern from all the five methods so as to minimize the cost. Comment on the results.

(10 marks)

Page 19: Practice Questions.pdf

Quantitative Techniques - II Topic wise unsolved questions from Previous Years Exam papers

EXERCISE 6EXERCISE 6EXERCISE 6EXERCISE 6

ASSIGNMENT MODELSASSIGNMENT MODELSASSIGNMENT MODELSASSIGNMENT MODELS – Maximization and Minimization Types, HAM, Traveling Salesman Problems

Qt: 1 Answer the following questions. Write down only the correct option in your answer sheet. (a) At the time of allocating jobs, the zeros which are covered by two lines: (i) always gets job allocated. (ii) never gets job allocated. (iii) get preference at the time of job allocation (iv) All the above (v) None of the above

Qt 2: The Captain of a Test Match Cricket team has to allot five middle batting positions to five batsman. The average runs scored by each batsman at these positions are as follows:

Batting Position→→→→

Batsman ↓↓↓↓ 1 2 3 4 5

P 40 40 35 25 50

Q 42 30 16 25 27

R 50 48 40 60 50

S 20 19 20 18 25

T 58 60 59 55 53 (a) Find the assignment of batsman to positions which would give the maximum number of runs (b) If another batsman 'U' is added to the team who has the following average runs in these batting positions :

Batting Position→→→→

Batsman ↓↓↓↓ 1 2 3 4 5

U 45 52 38 50 49 Should the new batsman included to play in the team. If yes, then whom to be replaced. Suggestions : The captain will always be interested in scoring as many runs as possible. Suggested time : 35 minutes

( 7 + 3 = 10 Marks)

Page 20: Practice Questions.pdf

Qt 3: A company has four salesmen who are to be assigned to four different territories. The monthly increases estimated for each salesman in different territories (in lakh of rupees), are shown in the table below:

Territory/Salesman 1 2 3 4 5 A 75 80 85 70 90 B 91 71 82 75 85 C 78 90 85 80 80 D 65 75 88 85 90

(a) Suggest optimal assignment for the salesman. Which sales territory will remain unassigned? (b) If for certain reasons salesman D cannot be assigned to territory 3, will the optimal assignment schedule be different? If so, show the new assignment schedule / schedules

Qt 4: Dr Jhamjham is the owner of a factory which produces electrical goods. She is in the process of allotting five jobs to five different persons with different competency level and varied experience. The following two tables shows the details of qualification & experience and the job description. Table 1 : Qualifications and experience of the employees

Manpower Qualification Experience

M1 Diploma in Electrical Engineering 15 years

M2 High Schooling 5 years

M3 Electrical Engineer (BE and ME), PhD

10 years (mostly in production and R&D)

M4 Electrical Engineer 6 years

M5 Electrical Engineer, MBA, 10 years (mostly on the shop floor)

Table 2 : The job Description

Jobs Job Title Job Description

1 Manager Overseeing the production process, leave records, feedback to the management about improvements to be carried out, optimization of resources

2 Senior Manager General Management of Operations, Salary disbursements, Recruitments, Dissemination of information, production planning etc

3 Worker Starting of the machines, Supply of all raw materials and other inputs

4 Floor Manager General Management of Operations, Salary disbursements, Recruitments, Dissemination of information, production planning etc

5 Supervisor Maintenance of the machines and job allocation of workers

Page 21: Practice Questions.pdf

The hierarchy of the company is given below from left to right (Extreme left is the lowest rank).

Worker --> Supervisor --> Manager -- > Senior Manager & Floor Manager Using the following matrix, you are required to give suggestions to Dr Jhamjham related to job allocation. Give sufficient reasons for your style of job allocation.

Jobs

Manpower 1 2 3 4 5

M 1 85 75 65 125 75

M 2 90 78 66 132 78

M 3 75 66 57 114 69

M4 80 72 60 120 72

M 5 76 64 56 112 68

Suggestions : Assume the above matrix as minimization matrix Suggested time : 30 minutes

(10 Marks) Qt 5: A company has four salesman who are to be assigned to four different territories. The monthly increases estimated for each salesman in different territories (in lakh of rupees), are shown in the table below: Territory/Salesman 1 2 3 4 5 A 75 80 85 70 90 B 91 71 82 75 85 C 78 90 85 80 80 D 65 75 88 85 90 (a) Suggest Optimal assignment for the salesman. Which sales territory will remain unassigned? (b) If for certain reasons salesman D cannot be assigned to territory 3, will the optimal assignment schedule be different? If so, show the new assignment schedule / schedules

Page 22: Practice Questions.pdf

Quantitative Techniques - II Topic wise unsolved questions from Previous Years Exam papers

EXERCISE 7EXERCISE 7EXERCISE 7EXERCISE 7

SIMULATION MODELSSIMULATION MODELSSIMULATION MODELSSIMULATION MODELS –Process of Simulation, Monte Carlo Simulation techniques and Simulation queuing models, Inventory models, planning etc.

Qt 1: Dr Jhamjham is running a small hospital in a rural area. She receives a delivery of fresh blood plasma once each week from a city blood bank. The supply varies according to the demand from other clinics and hospitals in the catchment region of this blood bank. But ranges between four to nine litres of the most widely used blood type (Group O ; Rh +ve). The number of patients per week requiring this blood varies from zero to four and each patient may need from one to four litres of blood. The delivery quantities, patients distribution and demand per patient are given in the following tables: Table 1 : Delivery Quantity

Litres / Week 4 5 6 7 8 9

Probability 0.15 0.20 0.25 0.15 0.15 0.10 Table 2 : Patient Description

Patient / Week (Requiring Blood) 0 1 2 3 4

Probability 0.25 0.25 0.30 0.15 0.05 Table 3 : Demand per Patient

Litres 1 2 3 4

Probability 0.40 0.30 0.20 0.10 Assuming that the blood is storable and the initial stock is 3 litres and Using the following random numbers answer the questions :

Delivery Quantity Patient Description Demand per Patient

10 85 21 06 61

31 28 96

02 72 12 67

53 44 23

16 16 -

40 83 65 34 82

Page 23: Practice Questions.pdf

(a) Construct Random number interval table(s) for the above problem. (b) Determine the number litres of blood in excess or short for a six week period using the simulation technique. (c) Comment on the results obtained. Suggestions : Use the random numbers in the order in which they are appearing in your answer sheet. Suggested Time : 40 Minutes

(6 + 10 + 4 = 20 Marks)

Qt 2: The casualty room of a hospital receives between zero to six emergency calls each night according to the following probability distribution

Calls 0 1 2 3 4 5 6

Probability 0.05 0.12 0.15 0.25 0.22 0.15 0.06 The medical team at the casualty room classifies each emergency call into one of the three categories : minor; medium and major emergency. The probability that a particular call may be anyone out of these categories is given below:

Emergency Type Minor Medium Major

Probability 0.30 0.56 0.14 The type of emergency call determines the size of medical team required to treat the emergency. A Minor emergency requires two person team, a medium emergency call requires three person team and a major emergency requires a team of five persons. Simulate the emergency call received by the casualty room for 10 nights, compute the average number of each type of emergency call each night and determine the number of medical team members that might be needed on any given night. Random numbers for the number of calls : 65 48 08 04 89 06 62 16 77 68 The following random numbers can be used for getting the final results. Use the random number in order it is printed. 71 18 12 17 89 18 83 90 - 18 08 26 47 94 72 47 68 60 88 36 43 28 31 06 39 71 22 76

Page 24: Practice Questions.pdf

Qt 3: Using Simulation technique design a flow chart for a company which is facing a dilemma in the decision making with respect to manufacture of Cakes on daily basis. The company would take a decision to produce either 42 cakes or 46 cakes based upon the results of the quantitative analysis. Keeping in mind all possible costs including opportunity cost and wastages. Qt 4: The occurrence of rain in a city on a particular day is dependent upon whether or not it rained on the previous day? If it rained on the previous day, the rain distribution is given by:

Event No rain 1 cm rain 2 cm rain 3 cm rain 4 cm rain 5 cm rain Probability 0.50 0.25 0.15 0.05 0.03 0.02

If it did not rain the previous day, the rain distribution is given by:

Event No rain 1 cm rain 2 cm rain 3 cm rain Probability 0.75 0.15 0.06 0.04

Simulate the city’s weather for 10 days and determine the total days without rain as well as the total rainfall during the period. Use the following random numbers:

67 63 39 55 29 78 70 06 78 76 (6 marks)