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About the Book O This book 'Operations Research' promises to be the first Indian book that shows applications of simple topics in Mathematics, ranging from linear functions, quadratic functions, concave and convex functions, sequences, to combinatorial analysis in P OPERATIONS helping an Indian practising manager in stating, defining, structuring, and resolving complex managerial problems. E This book is descriptive only to the extent of describing a practical problem that the authors want the readers to formulate and provide reasoned recommendations. It is only conditionally prescriptive in the sense that it says that if a person wants a logical R approach in helping him in deciding a course of action, then the person would find this book useful. The book includes comprehensive material from Management Functions to Pricing of Natural Resources. RESEARCH A Extensive use of MS Excel is made in helping the readers in carrying out complicated computations without any errors. Similarly, T Excel embedded solver programme for solving large Linear Programming Problems, Integer Programming Problems (including problems where the solution is in binary numbers) is frequently used. I Understanding probability is a necessary prerequisite in analyzing any problem of Operations Research dealing with O uncertainty; hence the concepts of probability are covered in four chapters. Bayesian approach linking prior probability with posterior probability in any situation involving physical probability is quite important; hence our chapter on Decision Theory has N these ideas. Further, an exclusive chapter on Special Loss Functions provides insight into the resolution of modern problems that deal with uncertainty. The book provides the ideas of finance, modeling of functions, understanding of permutations and S combinations, grasp of sequences. All these will give a thorough grounding in handling Operations Research under certainty. The appendices at the end of chapters provide mathematical formulations which can be of interest for readers who wish to get into the details. R Many real-life cases to sharpen the readers' skills in applying the ideas of Mathematics for analysis and resolution of E unstructured problems are included. S E The book addresses 10 important problems: A 1. Economic Order Quantity (Wilson's Economic Order Quantity) R 2. Scrap Allowance Problem 3. A Grain Dealer's Buying and Selling Strategy C 4. Selection of Cargo for Loading in a Ship H 5. Formulating a Budget in WAC Ltd 6. Revising our Judgements after seeing the Results of an Experiment 7. Testing a Statistical Hypothesis 8. Compiling Data to Test a Person's Claim M 9. Computing the Probability that among a Group of r Persons there is at least One Common Birthday O 10. Problem of making a Choice Under Uncertainty T E | M A Wiley India Pvt. Ltd. ISBN 978-81-265-5638-0 D 4435-36/7, Ansari Road, Daryaganj H New Delhi-110 002 A VASANT LAKSHMAN MOTE | T. MADHAVAN Customer Care +91 11 43630000 V Fax +91 11 23275895 A [email protected] N www.wileyindia.com 9 788126 556380 follow us on OPERATIONS RESEARCH OPERATIONS RESEARCH Vasant Lakshman Mote (Retired) Professor, Indian Institute of Management, Ahmedabad Planning Officer, Calico Mills President Business Planning and Trustee, Strategic Help and Relief to Distressed Areas (Arvind Mills) T. Madhavan (Retired) Professor, Indian Institute of Management, Ahmedabad OPERATIONS RESEARCH Copyright © 2016 by Wiley India Pvt. Ltd., 4435-36/7, Ansari Road, Daryaganj, New Delhi-110002. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or scanning without the written permission of the publisher. 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Disclaimer: The contents of this book have been checked for accuracy. Since deviations cannot be precluded entirely, Wiley or its author cannot guarantee full agreement. As the book is intended for educational purpose, Wiley or its author shall not be responsible for any errors, omissions or damages arising out of the use of the information contained in the book. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. Trademarks: All brand names and product names used in this book are trademarks, registered trademarks, or trade names of their respective holders. Wiley is not associated with any product or vendor mentioned in this book. Other Wiley Editorial Offices: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030, USA Wiley-VCH Verlag GmbH, Pappellaee 3, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 1 Fusionopolis Walk #07-01 Solaris, South Tower Singapore 138628 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada, M9W 1L1 First Edition: 2016 ISBN: 978-81-265-5638-0 ISBN: 978-81-265-8232-7 (ebk) www.wileyindia.com Preface In the recent years we have seen a rapid expansion of Institutes of Management, Business schools, and Integrated 5-Year Programmes in Management set up in reputed universities. The Institutes of Management, include 19 IIMs; XLRI (Xavier Institute of Institute of Management), Jamshedpur; Faculty of Management Studies, University of Delhi; S.P. Jain Institute of Management and Research, Mumbai; Management Development Institute (MDI), Gurgaon; Narsee Monjee Institute of Management (NMIMS), Mumbai; Jamnalal Bajaj Institute of Management Studies; B. K. School of Management in Ahmedabad and the like. The rapid expansion in Management Education shows the growing demand for Management Education in India. Both Madhavan and I were lucky that we had an opportunity to teach in the Indian Institute of Management, Ahmedabad. This experience gave us the insights into what to teach and how to teach in a Management Programme that had about 60% of students with no background in Mathematics. (Now approximately 10% of students have no background in Mathematics.) Our book includes comprehensive material from Management Functions to Pricing of Natural Resources. Salient Features of the Book This book promises to be the first Indian book that shows applications of simple topics in Mathematics, ranging from linear functions, quadratic functions, concave and convex functions, sequences, combinatorial analysis in helping an Indian practising manager in stating, defining, structuring, and resolving complex managerial problems. This book is descriptive only to the extent of describing a practical problem that the authors want the readers to formulate and provide reasoned recommendations. It is only conditionally prescriptive in the sense that it says that if a person wants a logical approach in helping him in deciding a course of action, then the person would find this book useful. The case Ranchi Refractories is a good example to illustrate the idea of a prescriptive approach. We make extensive use of MS Excel in helping the readers in carrying out complicated computations without any errors. Similarly, we frequently use the Excel embedded solver programme for solving large Linear Programming Problems, Integer Programming Problems (including problems where the solution is in binary numbers). Finally, we include real-life cases to sharpen the readers’ skills in applying the ideas of Mathematics for analysis and resolution of unstructured problems. Undoubtedly, many of the cases in the book are old but they are not outdated. Target Readers Our teaching experience in the first year of the IIMA’s Two-Year Post Graduate Programme has convinced us that even the students with background in Mathematics need to understand the use of Mathematics in defining, structuring, and resolving complex unstructured managerial problems. And, convince through discussions, other classmates about the effectiveness of the solution that they are proposing. Therefore, all the students studying in the first year of the 19 Institutes of Managements’ Two-Year Post Graduate Programme and the students studying in the first year of the other Institutes listed in the beginning of this Preface are also the target readers of the book. In addition there are many colleges that have a Five-Year Integrated Programme in Management. These colleges are also an important member of our target readers. Structure of the Book The book is organised in 22 chapters. We present the summary below. OR_Preface.indd 5 4/6/2016 2:08:11 PM vi Preface Chapter 1 opens by listing the functions of management. It states that, in a simplified sense, the management of any organisation has to carry out five major functions. The chapter lists these functions. Chapter 2 highlights the commonality in the analysis and resolution of the problems that the management of any organisation faces. The chapter lists steps involved in resolving the managerial problems that the management of any organisation faces. The central foci of Chapter 3 are a discussion of quadratic, exponential, logarithmic, concave and convex functions. This chapter introduces the quadratic functions through a discussion of quantity discounts. The chapter also includes the definition of sequences and presents the ideas of Arithmetic and Geometric Progressions, which are sequences having special structures. Exponential and logarithmic functions, which have many applications, also find a place in Chapter 3. We discuss convex sets, concave and convex functions which have applications in Applied Statistical Decision theory. Chapter 4 introduces the basics of combinatorial analysis. These basics comprise ideas of permutations and combinations. It introduces these ideas with everyday examples. For instance, it presents the idea of permutations by posing two questions. Why does Lord Vishnu have 24 names? And, how many configurations are there for transferring n persons to m available positions? These concepts have wide applications in theory of Probability and Statistics. Chapter 5 presents the idea underlying the Linear Programming Problem, gives a mathematical formulation of the problem and explains the nature of the Linear Programming (LP) Problems. The cases included, ‘Yusuf Bhai’s problem’ and ‘Ranchi Refractories’, show how effectively LP has been applied to solve practical problems. There are other important problems which are a special class of Linear Programming problem. These include the Transportation problem and Trans- shipment problem, which we discuss in Chapters 6 and 7, respectively. Chapter 8 unfolds the ideas in Critical Path Method (CPM) and Programme Evaluation and Review Technique (PERT). The CPM is a procedure for scheduling a set of activities that a project manager must carry out for completing a project on time and within the budget. It is an essential means for managing projects effectively and efficiently. Now it is widely applied in a variety of large projects comprising construction, defence and aerospace projects, software development, research projects, projects such as developing newer products, plant maintenance, etc. In short, CPM has a potential application for any project that involves inter-dependent activities. In CPM, the basic assumption is that the duration of every activity is completely known. In other words, duration of activities is deterministic. But in Program Evaluation and Review Technique (PERT), the duration of activities is governed by uncertainty. PERT is event-oriented while CPM is activity-oriented. PERT focuses on time only while CPM considers both time and cost. Chapter 9 introduces the Integer Programming Problem. The linear programming problems that we have discussed in Chapter 5 do not guarantee an integer solution. In many applications, the non-integral solution does not pose a problem. However, in many practical applications we need an integer solution, since a fractional solution is meaningless to the problem at hand. We begin Chapter 9 with the discussion of an integer programming problem in which the solution is in binary numbers (0, 1) and show its application to set covering and portfolio selection problems. Chapter 10 opens by introducing the idea and formal definition of a “sequence” which, Mathematicians define as a function defined on the natural numbers. The chapter illustrates the formal definition of sequences with some practical examples. Further, Chapter 10 presents sequences with special structures comprising sequences in arithmetic and geometric progression. We also show the application of the geometric progression in answering the question “Do banks create money?”. After we answer this question, we show the major weapons the Reserve Bank of India uses to control inflation. Chapter 11 introduces the idea of a sample space, which comprises all simple events. Further, in Chapter 11 we consider only a sample space, which is discrete. We present three examples in which the sample space contains a finite number of points N. In all these examples, we have attached the probability of 1/N to each of the sample point in the sample space. We must note that in every finite sample space it is not necessary that all the sample points will have equal probability. A new born baby is either a boy or a girl. But experience shows that the two possibilities are not equally likely. Chapter 12 develops the readers’ insights to the ideas and concepts of conditional probability, an important topic in the probability theory. There are no new ideas in this chapter but the chapter introduces new notation to express the idea of conditional probability. We observe some event, let us call it as event H, which has occurred. The question that now concerns us is, how does this knowledge about the event H change our judgment about the probability of another event A’s occurrence? Consider a family having exactly two children. If we are told that one of them is a boy and then asked “What is the probability that the other is also a boy?”. The sample space with new information comprises only three points. This formulation paves the way for proving Bayes’ theorem. OR_Preface.indd 6 4/6/2016 2:08:11 PM Preface vii Chapter 13 we discuss Random Variables. A random variable x(cid:31) is a function defined on a given sample space, that is, an assignment of a real number to each sample point. The random variable is always associated with a mass function and distribution function. Let x(cid:31) be a random variable taking the values x ,x ,x ,…,x . Then the aggregate of all sample points on which 1 2 3 n x(cid:31) assumes the value x forms the event x(cid:31)=x , and we denote its probability by j j P(x(cid:31)=x )= f(x ) j j We will call the function f that we have defined above as the mass function of the random variable x(cid:31) when the sample space is discrete. Associated with the random variable are some summary measures comprising expected value or the mean and fractile. These are called summary measures because they summarise the entire data by one number. Similarly associated with a random variable are measures of dispersion, which are called variance and standard deviation. When we have two random variables, we have joint distributions, marginal distributions and conditional distributions. They have marginal expectation and conditional expectation and joint expectation. In the case of two random variables, covariance is also an important summary measure. Chapter 14 presents two important mass functions, the Binomial and the Poisson. This chapter also introduces the den- sity function, called the ‘Normal distribution’, a continuous distribution widely used in applications from diverse fields such as agriculture and engineering. The chapter also provides the mean and variance for all the distributions presented here. The chapter provides numerous examples to illustrate the application of all these distributions. We introduce Excel function and R programming function to calculate various probabilities. Pikko Company is a case involving the application of Binomial distribution. U. S. Public Health Service is also a nice application of Binomial distribution. Chapter 15 deals with the topics like judging the number of customers (service seekers) that would arrive per unit of time, formation of waiting lines, conditions under which the waiting lines continue to get longer and longer. Similarly, we have answered the question whether a service provider’s starting more counters (place where the customers get the service they are seeking) helps in reducing the time a person spends in a waiting line? In what way would the impatient customers affect the economics of a service provider? Are there methods available to a textile mill’s operations manager in deciding the number of machines the manufacturing department allocates to a repair-person? We have illustrated with examples, the ideas underlying each of these topics and presented the mathematical approach for answering the question posed in the topics under discussion. Chapter 16 introduces the idea of decision tree by structuring Mr Kannan’s decision to drill a well on his farm. In making this decision, Mr Kanan has to make many other decisions before he can decide whether or not he should drill a well on his farm. The first decision he has to make is whether he should conduct an imperfect test to test the presence of water and then decide whether he should drill or should he drill without making the test. Further, if he does decide to drill then he must make other decision to what depth should he dig to find water before abandoning the drilling? These decisions are important because the cost of drilling in Mr Kannan’s village was Rs 50 per foot. In spite of all the care that Mr Kannan takes before drilling the well, the well can fail. Since Mr Kannan has borrowed funds from the unorganised sector (euphemism for money lenders − loan sharks), a failed well can cause severe financial hardship to Mr Kannan, may drive him to commit suicide. There are several cases of farmers’ suicides in Andhra Pradesh and Vidarbha in Maharashtra. Realising the woes of the farmers, the Finance Minister, in his budget for the year 2016–17 has taken several steps to alleviate the farmers’ problems when the crop fails. The decision tree presents an analytical tool for making decisions under uncertainty. The decision tree maps actions we take, the uncertain events that affect the results of the actions, probabilities of the events and consequences. We considered several criteria to decide on the alternative to choose. We proposed that expected monetary value (EMV) as a possible crite- rion which takes into account the probabilities and consequences. We also indicated situations where EMV is not applicable. We also provide preference functions in such cases. Expected value of perfect information is a good tool for seeking additional information for resolving uncertainty. It serves as a useful upper bound on how much we should spend to seek additional information. Chapter 17 begins with the discussion of loss functions with a problem known as the “Newsboy problem” in the statistical decision theory. A newsboy must decide every morning the number of copies of a newspaper that he sells. He buys the newspaper for Rs. k per copy and sells the newspaper at a price that gives him a contribution of Rs. k per paper. Obviously, k >0 0 u u and k >0. At the end of the day, the unsold newspapers have no value and the newsboy disposes off the unsold copies as scrap. 0 How many copies of the newspaper should the newsboy stock? Many problems like reservation problem, garland problem, etc. have similar structure. We provide a formal method for solving Newsboy Problem and similar problems and obtaining the optimal act. Besides linear loss function, we also consider Quadratic Loss Function. We provide an insight into reservation OR_Preface.indd 7 4/6/2016 2:08:13 PM viii Preface problem and scrap allowance problem. In addition, we assumed that we know the mass function f of the random variable that we encountered in the problems. In all the problems that we discussed, we chose the optimal act by minimising the expected loss. In addition, we interpreted the expected loss of the best act as the Expected Value of Perfect Information (EVPI). Chapter 18 deals with Statistical Inference concerning Testing Hypothesis and Estimation. In this chapter, there is no decision to make. The title of the chapter “Statistical Inference” correctly describes the problem at hand. Our problem is to understand “what does the data say?” (draw inference). We start with an illustrative example of terrorist attacks. All the six attacks on India, between 2008 and 2011, were on either 13th or 26th of a month. Does this data support the hypothesis that the terrorist have a system? The first question is concerned about the inference that we can draw from the data concerning the terrorist attacks in India. What does the data say? This is the central question in this problem. And the second question is concerned about the estimation of the probability p that the terrorist attack is either on 13th or 26th of a month. In the literature in statistics, this estimation problem is called the point-estimation problem. In these problems there is no decision to be made. The problem is to answer the question “what does the data say?” The test of hypothesis involves specifying the null hypothesis H and the alternative hypothesis H . The act of rejecting a 0 1 null hypothesis when it is true is called Type I error and the act of accepting a null hypothesis when it is false is known as Type II error. It is not possible to minimise the probability of committing Type I error and probability of committing Type II error simultaneously. We keep the probability of committing Type I error at a level a and minimise the probability of committing Type II error. If we want to ensure the probability of committing Type I error and probability of committing Type II error at specified level, then we have to decide the sample size n to cater to these conditions. We provide the approach for testing simple H against simpleH . We provide a formal mechanism to find the critical 0 1 region. Our next problem is to find the value of n to achieve the given value of b for a given value of a, where a is the probability of committing Type I error and b is the probability of committing Type II error. In all but one example discussed, we did not have any specific decision problem that we wanted to resolve. Our problem is only on understanding “what does the data say?”. Only in one example we had to decide the number of bulbs that an industrial establishment must stock to ensure that it does not run the risk of more than 1 in 1000 of running short of bulbs in 2 years. We use Maximum Likelihood Method to arrive at Point Estimates. For example, the likelihood function in the case of Binomial Distribution is ænö f (p|n,r)=ç ÷pr(1- p)n-r b èrø To decide the maximum likelihood estimate of p, we maximise this function with respect to p. We supplement this approach by “the Bayesian approach” in determination of sample size. In the Bayesian approach, we formally use the decision maker’s judgement and whether the sample would improve the decision based on the decision maker’s prior judgement without sampling. Chapter 19 highlights three motives for holding cash (inventory), referred to by Keynes as the transaction, precautionary and speculative motives. Roughly speaking, the speculative motive is the possibility of earning profit through changes in prices and interest rates or other demand and supply conditions. The precautionary motive stems from the need for protection against uncertainty and the transaction motive is the result of managing the costs of transactions. In transaction motive, we obtain the Economic Order Quantity (EOQ) considering the carrying cost and setup cost. We modify the EOQ when a Customer Gets a Quantity Discount and under the case when backorders are allowed. We move on to precautionary motive for holding inventory, static inventory management when demand is uncertain. We use Newsboy problem to illustrate this approach. We then decide the reorder level when the lead-time is uncertain. We then discuss the speculative motive for holding inventories. We describe the problem concerning managing in-process inventory. The ideal before every manufacturing organisation is to hold in-process inventory just equal to the quantities that are on machines undergoing some processing. However, a manufacturing company hardly achieves this ideal. There are many reasons that prevent a manufacturing company from achieving this ideal. We will remain content by citing just three reasons. 1. Unforeseen breakdown of some key machines. 2. The “line” is not balanced if the manufacturing is organised in “lines”. 3. Employee absenteeism. In India, the wedding season is a nightmare for the supervisory staff in a manufacturing organisation since in this season employee absenteeism is at its zenith. OR_Preface.indd 8 4/6/2016 2:08:13 PM Preface ix We then provide a note on Material Requirement Planning (MRP). The recently developed idea and tools of MRP make it a forward-looking method. It does not rely on historical data to determine reorder levels. In those cases where the product-mix (customer demand) is not static, use of MRP can have a telling effect on inventory carrying costs. The MRP cycle (e.g., weekly, monthly or quarterly) has to be decided by the management depending upon flexibilities in manufacturing (setup time), com- plexity in line balancing (machine scheduling and capacity planning − how much to use in-house or how much to outsource). The essential data that the user must provide for carrying out the MRP program for a planning cycle comprises: 1. Item demand matrix (the dimensions are item name, item quantity, date on which required, plant where required). 2. Stock on hand (of finished goods, intermediate goods at each conversion centre, raw material and packing materials). 3. Pending orders with expected deliveries in the planning cycle (item name, quantity, expected receipt date). 4. BOM (Bill of Material – for each item to be manufactured, what are the input items and respective quantities). 5. Item master data (reorder quantity, lead time). However, all the inventory models have lost their sheen after the Japanese introduced the idea of “Just in time” deliveries. For example, a garment factory will inform the textile mill that supplies fabrics to them precisely how the mill must supply the fabric. For instance, the garment factory will specify the exact order in which the mill must supply the shades of the dyed fabric and in quantities the garment factory wants the fabric. The garment factory will not inspect the fabric but will send it directly to the cutting table for cutting the fabric to its requirements. Chapter 20 describes the mathematical theory of games of strategy dealing with situations involving two or more participants with conflicting interests. The outcome of such games is usually controlled partly by one side and partly by the opposing side or sides; it depends to some extent on chance, but primarily on the intelligence and skill employed by the participants. We shall first consider only two-person zero-sum games, that is, games with only two participants (competing persons, teams firms, nations) in which one participant wins what the other participant loses. A fundamental concept in game theory is that of a strategy. A strategy for Player I is a complete enumeration of all actions Player I will take for every contingency that might arise, whether the contingency be one of chance or one created by move of the opposing player. A second fundamental concept in game theory is that of the pay-off. The pay-off is the connecting link between the set of strategies open to Player I and Player II. Specifically, it is a rule that tells how much Player I may be expected to win from Player II if Player I chooses a strategy from his set of strategies and Player II chooses a particular strategy from his set. In zero-sum games, the gain of one is the loss of the other. There can be non–zero-sum games, wherein, the sum of the pay offs of the two players is not zero. It is possible that both of them can win. A player uses a pure strategy if he or she uses the same strategy at each round of the game. A saddle point is a payoff which is simultaneously a row minimum and a column maximum. When there is no saddle point, we can obtain the mixed strategy through Linear Programming. We can also get the pure strategy with Linear Programming as we have demonstrated. Chapter 21 has the central theme “the pricing of water, a scarce natural resource”. We have chosen pricing water, a subset of scarce natural resources, as the central theme for this chapter because of our strong belief that pricing of natural resources like water will gain importance in India over time, since water is among five other scarce natural resources including oil and natural gas, which are likely to run out within a foreseeable future. Chapter 22 highlights the duality theory. The purpose of the LP’s dual is to value the productive resources (assets) the organisation has acquired. The dual therefore allows the organisation’s management to see whether the cost it incurred for acquiring the assets, is less than the value of the production the assets generate. We first formulate a problem, called the primal problem, to maximise a given objective function subject to the given set of constraints including the non-negativity constraints. The second problem we formulate is the problem to minimise the linear function formed by attaching non-negative variables y to 1 constraint number 1, y to constraint 2, y to constraint 3 and multiplying y by the right-hand side of constraint 1, y by the 2 3 1 2 right-hand side of constraint 2 and y by the right-hand side of constraint 3 and adding the three products. The variables y ,y 3 1 2 and y are called the dual variables. These variables measure the economic value of the machine capacities that the organisation 3 uses to get the optimum contribution and thus provides the value of the organisation’s productive assets. Important Problems Considered in the Book 1. Economic Order Quantity (Wilson’s Economic Order Quantity): The annual requirement for a product is 40,000 units. The ordering cost is Rs. 20 per order. The cost of the item is Rs. 4. Carrying cost is 10% of the unit price. What is the economic order quantity? OR_Preface.indd 9 4/6/2016 2:08:13 PM