Overview of Applications by Discipline

ECONOMICS
Estimating sensitivity of demand to price
352–356
Pricing problems 352–366, 422–427
Estimating cost of power 363–366

47–56,

Assessing a utility function 554–556
Estimating demand for products 632–638,
649–650, 764–771, 965
Subway token hoarding 792

FINANCE AND ACCOUNTING
Collecting on delinquent credit accounts 14–16
Cost projections 29–33
Finding a breakeven point 33–41
Calculating NPV 57–62
Calculating NPV for production capacity decision 58–62
Portfolio management 173–178, 345–346,
387–394, 442–444, 689–691
Pension fund management 178–182
Financial planning 210–214, 676–681,
734–735
Arbitrage opportunities in oil pricing 215–219
Currency trading 220
Capital budgeting 290–295
Estimating stock betas 396–401
Hedging risk with put options 407–408

Stock hedging 407–408
Asset management 409–410
New product development 503–504, 574, 673–676,
715–722
Bidding for a government contract 513–518,
523–533, 653–657
Investing with risk aversion 557–560
Land purchasing decision 575
Risk analysis 582–583
Liquidity risk management 651–653
Estimating warranty costs 657–661
Retirement planning 681–685
Modeling stock prices 685–686
Pricing options 686–689, 691–693
Investing for college 732
Bond investment 733

HUMAN RESOURCES AND HEALTH CARE
Fighting HIV/AIDS 23–24
DEA in the hospital industry 184–189
Salesforce allocation problems 454–456
Assigning MBA students to teams 462

Selecting a job 484–492
Selecting a health care plan 519–521
Drug testing for athletes 535–538, 539–542

MARKETING
Determining an advertising schedule 133–141,
373–376, 465–471, 480–483
Estimating an advertising response function 369–373
Retail pricing 422–427
Estimating a sales response function 437–441
Cluster analysis of large cities 445–449

Classifying subscribers of the WSJ 450–453
New product marketing 543–552
Valuing a customer 695–699
Reducing churn 699–703
Estimating market share 703–706
Estimating sales from promotions 703–706

MISCELLANEOUS
Investment in U.S. Space Systems 285–286
Prioritizing projects at NASA 463–464
Biotechnical engineering 576–577

OPERATIONS MANAGEMENT
Queueing problems 4–7, 796–850
Ordering problems (newsboy) 25–28, 604–613,
617–631, 632–639, 649, 760–763
Ordering with quantity discounts 42–46, 747–748
Manufacturing operations 71–72
Choosing an optimal diet 75–92
Product mix problems 98–107, 127–130, 168–172,
298–307
Production scheduling 108–117, 150–159,
432–437, 650
Production, inventory management 131–132,
499–501, 661–666
Scheduling workers 142–148
Aggregate planning 150–159
Gasoline, oil blending 161–166, 207–209
Logistics problems 221–222, 223–234, 241–249
Assigning workers to jobs 235–236
Assigning school buses to routes 237–240
Finding a shortest route 250–254
Equipment replacement 254–258
Airline crew scheduling 260–265
Airline flight scheduling 265–271
Aircraft maintenance 272
Global manufacturing and distribution 280–281
Motor carrier selection 282–284

SPORTS AND GAMES
Rating NFL teams 382–386
Playing craps 708–710
NCAA basketball tournament

710–713

Airline hub location 309–314
Locating plants and warehouses 314–325,
378–381
Cutting stock problems 327–330
Plant expansion and retooling 341–342
Telephone call processing 343, 857–858
Railway planning 411–412
Loading a gas truck 429–432
Traveling salesperson problem 454–457
Determining trade–off between profit and pollution 477–479
Airline boarding strategies 579–580
Deming’s funnel experiment 667–671
Global supply chain decisions 737–738
Economic order quantity models 743–758
Ordering decisions with demand uncertainty
764–771, 773–778
Production planning in fashion industry 779–784
Reducing work in progress 793–794
Operations at banks 859–860
Scheduling multiple projects 861–862, 885–889
Project scheduling with CPM 865–885, 890–895
Forecasting problems 903–904, 910–919, 921–929,
938–941, 944–956, 964–965

This page intentionally left blank

REVISED

3RD
EDITION

Practical Management Science

Wayne L. Winston
Kelley School of Business, Indiana University

S. Christian Albright
Kelley School of Business, Indiana University

With Cases by

Mark Broadie
Graduate School of Business, Columbia University

Lawrence L. Lapin
San Jose State University

William D. Whisler
California State University, Hayward

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

Practical Management Science,
Revised Third Edition
Wayne L. Winston,
S. Christian Albright
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Jack W. Calhoun
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Alex von Rosenberg
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To Mary, my wonderful wife, best friend, and constant companion
And to our Welsh Corgi, Bryn, who still just wants to play ball

To my wonderful family
Vivian, Jennifer, and Gregory

W.L.W.

S.C.A.

About the Authors
S. Christian Albright got his B.S. degree in Mathematics from
Stanford in 1968 and his Ph.D. degree in Operations Research from Stanford in 1972. Since then he has been teaching in the
Operations & Decision Technologies Department in the Kelley
School of Business at Indiana University. He has taught courses in management science, computer simulation, and statistics to all levels of business students: undergraduates, MBAs, and doctoral students. He has also taught courses on database analysis to the
U.S. Army. He has published over 20 articles in leading operations research journals in the area of applied probability, and he has authored several books, including Practical Management
Science, Data Analysis and Decision Making, Data Analysis for Managers, Spreadsheet
Modeling and Applications, and VBA for Modelers. He jointly developed StatTools, a statistical add-in for Excel, with the Palisade Corporation. His current interests are in spreadsheet modeling and the development of VBA applications in Excel, as well as Web programming with Microsoft’s .NET technology.
On the personal side, Chris has been married to his wonderful wife Mary for
37 years. They have one son, Sam, who is currently finishing a law degree at Penn Law
School. Chris has many interests outside the academic area. They include activities with his family (especially traveling with Mary), going to cultural events at Indiana University, playing golf and tennis, running and power walking, and reading. And although he earns his livelihood from statistics and management science, his real passion is for playing classical music on the piano.

Wayne L. Winston is Professor of Operations & Decision
Technologies in the Kelley School of Business at Indiana
University, where he has taught since 1975. Wayne received his
B.S. degree in Mathematics from MIT and his Ph.D. degree in
Operations Research from Yale. He has written the successful textbooks Operations Research: Applications and Algorithms,
Mathematical Programming: Applications and Algorithms,
Simulation Modeling Using @RISK, Data Analysis and
Decision Making, and Financial Models Using Simulation and
Optimization. Wayne has published over 20 articles in leading journals and has won many teaching awards, including the schoolwide MBA award four times. He has taught classes at Microsoft, GM, Ford, Eli Lilly, Bristol-Myers Squibb,
Arthur Andersen, Roche, PriceWaterhouseCoopers, and NCR. His current interest is in showing how spreadsheet models can be used to solve business problems in all disciplines, particularly in finance and marketing.
Wayne enjoys swimming and basketball, and his passion for trivia won him an appearance several years ago on the television game show Jeopardy, where he won two games. He is married to the lovely and talented Vivian. They have two children, Gregory and Jennifer.

Brief Contents
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Preface xi
Introduction to Modeling 1
Introduction to Spreadsheet Modeling 23
Introduction to Optimization Modeling 71
Linear Programming Models 131
Network Models 221
Optimization Models with Integer Variables 285
Nonlinear Optimization Models 345
Evolutionary Solver: An Alternative Optimization Procedure 411
Multiobjective Decision Making 463
Decision Making Under Uncertainty 503
Introduction to Simulation Modeling 579
Simulation Models 651
Inventory Models 737
Queueing Models 793
Project Management 861
Regression and Forecasting Models 903
References 966
Index 969

v

This page intentionally left blank

Contents
Preface xi
CHAPTER 1 Introduction to Modeling 1
1.1
1.2
1.3
1.4
1.5

Introduction 3
A Waiting-Line Example 4
Modeling Versus Models 7
The Seven-Step Modeling Process 8
A Successful Management Science
Application 14
1.6 Why Study Management Science? 17
1.7 Software Included in This Book 18
1.8 Conclusion 20

CHAPTER 2 Introduction to Spreadsheet
Modeling 23
2.1 Introduction 24
2.2 Basic Spreadsheet Modeling: Concepts and Best
Practices 24
2.3 Cost Projections 29
2.4 Breakeven Analysis 33
2.5 Ordering with Quantity Discounts and Demand
Uncertainty 42
2.6 Estimating the Relationship Between Price and
Demand 47
2.7 Decisions Involving the Time Value of Money 57
2.8 Conclusion 63
Appendix: Tips for Editing and Documenting
Spreadsheets 67

CHAPTER 3 Introduction to Optimization
Modeling 71
3.1
3.2
3.3
3.4
3.5
3.6

Introduction 72
Introduction to Optimization 73
A Two-Variable Model 74
Sensitivity Analysis 85
Properties of Linear Models 93
Infeasibility and Unboundedness 96

3.7 A Product Mix Model 98
3.8 A Multiperiod Production Model 108
3.9 A Comparison of Algebraic and Spreadsheet
Models 118
3.10 A Decision Support System 118
3.11 Conclusion 120
Appendix: Information on Solvers 126
CASE 3.1 Shelby Shelving 127
CASE 3.2 Sonoma Valley Wines 129

CHAPTER 4 Linear Programming Models 131
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

Introduction 132
Advertising Models 133
Workforce Scheduling Models 142
Aggregate Planning Models 150
Blending Models 160
Production Process Models 167
Financial Models 173
Data Envelopment Analysis (DEA) 184
Conclusion 191
CASE 4.1 Amarco, Inc. 207
CASE 4.2 American Office Systems, Inc. 210
CASE 4.3 Lakefield Corporation’s Oil
Trading Desk 215
CASE 4.4 Foreign Currency Trading 220

CHAPTER 5 Network Models 221
5.1
5.2
5.3
5.4
5.5
5.6
5.7

Introduction 222
Transportation Models 223
Assignment Models 235
Minimum Cost Network Flow Models 241
Shortest Path Models 250
Other Network Models 260
Conclusion 273
CASE 5.1 International Textile
Company, Ltd. 280 vii CASE 5.2 Optimized Motor Carrier Selection

at Westvaco 282

CHAPTER 6 Optimization Models with Integer
Variables 285
6.1 Introduction 286
6.2 Overview of Optimization with
Integer Variables 287
6.3 Capital Budgeting Models 290
6.4 Fixed-Cost Models 297
6.5 Set-Covering and Location-Assignment
Models 309
6.6 Cutting Stock Models 327
6.7 Conclusion 331
CASE 6.1 Giant Motor Company 341
CASE 6.2 Selecting Telecommunication Carriers to
Obtain Volume Discounts 343

CHAPTER 7 Nonlinear Optimization Models 345
7.1
7.2
7.3
7.4

Introduction 346
Basic Ideas of Nonlinear Optimization 347
Pricing Models 351
Advertising Response and Selection
Models 368
7.5 Facility Location Models 378
7.6 Models for Rating Sports Teams 382
7.7 Portfolio Optimization Models 387
7.8 Estimating the Beta of a Stock 396
7.9 Conclusion 401
CASE 7.1 GMS Stock Hedging 407
CASE 7.2 Durham Asset Management 409

CHAPTER 8 Evolutionary Solver: An Alternative
Optimization Procedure 411
8.1
8.2
8.3
8.4
8.5
8.6
8.7

viii

Introduction 412
Introduction to Genetic Algorithms 415
Introduction to Evolutionary Solver 416
Nonlinear Pricing Models 422
Combinatorial Models 428
Fitting an S-Shaped Curve 437
Portfolio Optimization 442

Contents

8.8
8.9
8.10
8.11

Cluster Analysis 444
Discriminant Analysis 450
The Traveling Salesperson Problem 454
Conclusion 458
CASE 8.1 Assigning MBA Students to Teams 462

CHAPTER 9 Multiobjective Decision Making 463
9.1
9.2
9.3
9.4
9.5

Introduction 464
Goal Programming 465
Pareto Optimality and Trade-Off Curves 475
The Analytic Hierarchy Process (AHP) 484
Conclusion 494
CASE 9.1 Play Time Toy Company 499

CHAPTER 10 Decision Making Under
Uncertainty 503
10.1
10.2
10.3
10.4
10.5
10.6
10.7

Introduction 505
Elements of a Decision Analysis 506
The PrecisionTree Add-In 522
Bayes’ Rule 534
Multistage Decision Problems 538
Incorporating Attitudes Toward Risk 554
Conclusion 561
CASE 10.1 Jogger Shoe Company 574
CASE 10.2 Westhouser Paper Company 575
CASE 10.3 Biotechnical Engineering 576

CHAPTER 11 Introduction to Simulation
Modeling 579
11.1 Introduction 581
11.2 Real Applications of Simulation 582
11.3 Probability Distributions for
Input Variables 583
11.4 Simulation with Built-In Excel Tools 603
11.5 Introduction to @RISK 615
11.6 The Effects of Input Distributions on Results 632
11.7 Conclusion 640
Appendix: Creating Histograms with Excel Tools 646
CASE 11.1 Ski Jacket Production 649
CASE 11.2 Ebony Bath Soap 650

CHAPTER 12 Simulation Models

651

12.1
12.2
12.3
12.4
12.5
12.6

Introduction 653
Operations Models 653
Financial Models 672
Marketing Models 695
Simulating Games of Chance 708
Using TopRank with @RISK for Powerful
Modeling 714
12.7 Conclusion 722
CASE 12.1 College Fund Investment 732
CASE 12.2 Bond Investment Strategy 733
CASE 12.3 Financials at Carco 734

CHAPTER 13 Inventory Models 737
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8

Introduction 739
Categories of Inventory Models 740
Types of Costs in Inventory Models 741
Economic Order Quantity (EOQ) Models 743
Probabilistic Inventory Models 759
Ordering Simulation Models 773
Supply Chain Models 778
Conclusion 784
CASE 13.1 Subway Token Hoarding 792

CHAPTER 14 Queueing Models
14.1
14.2
14.3
14.4
14.5
14.6

793

Introduction 795
Elements of Queueing Models 796
The Exponential Distribution 799
Important Queueing Relationships 804
Analytical Steady-State Queueing Models 807
Approximating Short-Run Behavior
Analytically 830

14.7 Queueing Simulation Models 835
14.8 Conclusion 852
CASE 14.1 Catalog Company Phone Orders 857
CASE 14.2 Pacific National Bank 859

CHAPTER 15 Project Management 861
15.1
15.2
15.3
15.4
15.5
15.6

Introduction 863
The Basic CPM Model 865
Modeling Allocation of Resources 874
Models with Uncertain Activity Times 890
A Brief Look at Microsoft Project 896
Conclusion 898

CHAPTER 16 Regression and Forecasting
Models 903
16.1
16.2
16.3
16.4
16.5
16.6
16.7
16.8

Introduction 904
Overview of Regression Models 905
Simple Regression Models 909
Multiple Regression Models 921
Overview of Time Series Models 932
Moving Averages Models 937
Exponential Smoothing Models 942
Conclusion 958
CASE 16.1 Demand for French Bread at Howie’s 964
CASE 16.2 Forecasting Overhead at Wagner
Printers 964
CASE 16.3 Arrivals at the Credit Union 965

References 966
Index 969

Contents

ix

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Preface
Practical Management Science provides a spreadsheet-based, example-driven approach to management science. Our initial objective in writing the book was to reverse negative attitudes about the course by making the subject relevant to students. We intended to do this by imparting valuable modeling skills that students can appreciate and take with them into their careers. We are very gratified by the success of the first two editions. The book has exceeded our initial objectives. We are especially pleased to hear about the success of the book at many other colleges and universities around the world. The acceptance and excitement that has been generated has motivated us to revise the book and make the third edition even better.
By incorporating our own teaching experience and the many user comments and suggestions, we believe this third edition is a significant improvement over the first two. We hope you will agree.
When we wrote the first edition, management science courses were regarded as irrelevant or uninteresting to many business students, and the use of spreadsheets in management science was in its early stages of development. Much has changed since the first edition was published in 1996, and we believe that these changes are for the better. We have learned a lot about the best practices of spreadsheet modeling for clarity and communication. We have also developed better ways of teaching the materials, and we understand more about where students tend to have difficulty with the concepts. Finally, we have had the opportunity to teach this material at several Fortune 500 companies
(including Eli Lilly, Price Waterhouse Coopers,
General Motors, Tomkins, Microsoft, and Intel).
These companies, through their enthusiastic support, have further enhanced the realism of the models included in this book.
The success of the book outside of the classroom justifies our approach in the third edition. Although we have retained many of the features that have made this book a bestseller, we have enhanced the coverage to make it more relevant and more accessible to students of varying backgrounds. Throughout the book, you will find many new models that are based upon real problems, and you will find a much clearer presentation of the modeling, solution, and interpretation of the examples. We have found that professionals, like students, have differing backgrounds in terms of their

command of mathematics and Excel, but they both desire skills and knowledge they can immediately apply to the challenges they face in their professions.
Those of you who are sympathetic to this approach will find the third edition is better suited to meet these needs. Our objective in writing the first edition was very simple—we wanted to make management science relevant and practical to students and professionals.
This book continues to distinguish itself in the market in four fundamental ways:
■

■

■

Teach by Example. The best way to learn modeling concepts is by working through examples and solving an abundance of problems.
This active learning approach is not new, but our text has more fully developed this approach than any book in the field. The feedback we have received from many of you has confirmed the success of this pedagogical approach for management science.
Integrate Modeling with Finance, Marketing, and Operations Management. We integrate modeling into all functional areas of business.
This is an important feature because the majority of business students major in finance and marketing. Almost all competing textbooks emphasize operations management-related examples. Although these examples are important, and many are included in the book, the application of modeling to problems in finance and marketing are too important to ignore. Throughout the book, we use real examples from all functional areas of business to illustrate the power of spreadsheet modeling to all of these areas. At Indiana University, this has led to the development of two advanced MBA electives in finance and marketing that build upon the content in this book. The inside front cover of the book illustrates the integrative applications contained in the book.
Teach Modeling, Not Just Models. Poor attitudes among students in past management science courses can be attributed to the way in which they were taught: emphasis on algebraic formulations and memorization of models.
Students gain more insight into the power of

xi

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management science by developing skills in modeling. Throughout the book, we stress the logic associated with model development and formulation, and we discuss solutions in this context. Because real problems and real models often include limitations or alternatives, we include many “Modeling Issues” sections to discuss these important matters. Finally, we include “Modeling Problems” in most chapters to help develop these skills.
Provide Numerous Problems and Cases.
Whereas all textbooks contain problem sets for students to practice, we have carefully and judiciously crafted the problems and cases contained in this book. The third edition contains many new problems and cases. Each chapter contains four types of problems: Skill-Building
Problems, Skill-Extending Problems, Modeling
Problems, and Cases. A new feature to this edition is that almost all of the problems following sections of chapters ask students to extend the examples in the preceding section.
The end-of-chapter problems then ask students to explore new models. Selected solutions are available to students who purchase the Student
Solution Files online and are denoted by the second color numbering of the problem.
Solutions for all of the problems and cases are provided to adopting instructors. In addition, shell files (templates) are available for most of the problems for adopting instructors. The shell files contain the basic structure of the problem with the relevant formulas omitted. By adding or omitting hints in individual solutions, instructors can tailor these shell files to best meet the individual/specific needs of their students.

■

Enhancements to the Third Edition
Our extensive teaching experience has provided more insights about the instruction of a spreadsheet-based course in management science, and we have incorporated many suggestions from users of the first two editions to improve the book. In addition, there have been several advances in spreadsheet-based technology in recent years. The software accompanying new copies of the third edition includes the most extensive and valuable suite of tools ever available in a management science textbook. Important changes to the third edition include the following:
■

xii

Improved Spreadsheet Readability and
Documentation. Many professionals we have
Preface

■

taught instinctively document their spreadsheet models for the purpose of sharing them with colleagues or communicating them in presentations and reports. This is an important element of good spreadsheet modeling, and the third edition does even more to emphasize good practices. Furthermore, grading homework assignments and exams can be a very timeconsuming chore if students are permitted to construct their models in any form. Therefore, we place early and consistent emphasis on good spreadsheet habits. This should benefit both students and instructors. Although we try not to force any one approach on everyone, we do suggest some good habits that lead to better spreadsheet models.
To achieve this goal of better readability and documentation, we have reworked many examples in the chapters, and we have incorporated our new habits in the many new examples. This is especially important because this edition’s most important feature continues to be the set of examples that illustrates the concepts in each chapter. For users of the first two editions, the changes will sometimes appear subtle, but they make a significant difference pedagogically. Pedagogical Enhancements. We have borrowed the pedagogical enhancements from our
Spreadsheet Modeling and Applications text.
These include (1) “Objectives” sections following most examples that briefly describe the objectives of the example; (2) “Where Do the
Numbers Come From” sections following most examples that discuss how the model inputs might be obtained in real business situations;
(3) tables of key variables and constraints following most examples in the optimization chapters that form a bridge from the statement of the problem to the spreadsheet implementation;
(4) “Fundamental Insight” sections that highlight important concepts; and (5) summaries of key terms at the ends of chapters. All of these enhancements make it easier for students to study the material and prepare for exams.
Interpretation of Results and Sensitivity
Analysis. Some users mentioned that we often show solutions to models and then move too quickly to other models. In this edition, we spend more time discussing the solutions to models for the insights they contain. We often try to

■

■

■

■

■

understand why a solution comes out as it does, and we spend plenty of time doing sensitivity analyses on the solutions. As we have heard from practitioners, this is often where the real benefits of management science are found. In this sense, a developed model is not an end; it is a beginning for understanding the business problem. Range Names. In the second edition, we perhaps overemphasized the use of range names. We still believe strongly that range names are an important way to document spreadsheet formulas, so we continue to use them but to a somewhat lesser extent. We also emphasize how easy it is to create range names, in many cases, by using adjacent labels.
New Problems. We have added more than
100 new problems, many of which follow sections of the chapters. These problems ask students to extend the examples covered in the preceding sections in a variety of directions.
These “model extension” problems are arguably as important as any other enhancement to this edition. We have always emphasized model development, and we continue to do so. But it is also extremely important to understand how completed models can be extended to answer various business questions. Students now have plenty of opportunity to extend existing models. New Chapter on Project Scheduling. The material on project scheduling included in the optimization and simulation chapters of previous editions has now been assembled into its own chapter, Chapter 15. This allows many instructors who feel strongly about this topic to cover it in a more integrated manner.
New Examples in Queueing. Given the support for the Erlang loss model from the queueing chapter, we have reinstated it in the third edition.
We have also added a very insightful queueing spreadsheet model that approximates transient behavior of queues. This model is not very difficult to implement in Excel, and we’re surprised that it hasn’t been taught more often in management science courses. This model illustrates interesting short-run behavior of queues that does not appear in the traditional steady-state analysis.
Combination of Regression and Forecasting
Chapters. These two topics are of obvious

■

importance to management scientists, but we decided to combine them into a single chapter, focusing on the most essential concepts and methods. Our rationale is twofold. First, we doubt that many instructors have enough time to cover regression and forecasting in as much depth as we included in the second edition.
Second, we suspect that many students have already been exposed to this material in a statistics class, so a quick refresher chapter should suffice.
New Chapter Opening Vignettes and Other
Real Applications. It is important for students to realize the important role management science plays in today’s business world. The Interfaces journal chronicles the many successful applications that have saved companies millions or even billions of dollars. The third edition contains descriptions of many new Interfaces applications, both in the chapter opening vignettes and throughout the chapters.

Reason for this Revised Edition
In 2007, Microsoft released its newest version of Office,
Office 2007 (also called Office 12). This was not just another version with a few changes at the edges. It was a completely revamped package. Suddenly, many of the screenshots and instructions in our books were no longer correct because of the extensive user interface changes in Excel 2007. To add to the confusion, thirdparty developers of add-ins for Excel, particularly
Palisade for our books, had to scramble to update their software for Excel 2007. We also had to scramble as the
Fall semester of 2007 approached. By the time we obtained the updated software, even in beta version, it was too late to produce updates to our books by Fall
2007. Therefore, we put “changes” files on our Web site
(http://www.kelley.iu.edu/albrightbooks) to help many of you to make the transition.
Of course, a more permanent solution was needed—hence, this revised edition. It is entirely geared to Excel 2007 and the updated add-ins for
Excel 2007. If you have moved to Excel 2007, you should use this revised edition. If you are still using an earlier version of Excel, you should continue to use the original third edition. Almost all of the content in the two versions are identical. We changed as little as possible, mostly just the screenshots and accompanying explanations.
Two other comments are in order. First, Palisade was in the midst of an extensive rewrite of its
Preface

xiii

DecisionTools™ suite when Office 2007 came out. To get through the 2007–2008 academic year, Palisade developed a slightly revised version of its old
DecisionTools suite (for example, version 4.5.7 of
@RISK) that was compatible with Excel 2007, and we made that version available to users for the 2007–2008 year. Fortunately, their new version of the
DecisionTools suite, version 5.0, is now out.
Therefore, the educational version of it accompanies this book, and the screenshots and explanations are geared to it. We believe you will like its much friendlier user interface.
Second, the example files on the CD-ROM inside the revised edition have been updated slightly. They still have the same content, but they are now in Excel
2007 format (.xlsx or .xlsm). Also, although we still provide templates and finished versions of all these example files (in different folders), we have appended
“Finished” to the names of all the finished files. This should avoid the confusion of having the same name for two different files that some of you have mentioned.
Finally, we have included “Annotated” versions of these finished example files, plus a few “Extra” application files, to instructors. The annotated versions include our insights and suggestions for the examples.
We hope these help you to teach the material even more effectively. Accompanying Student Resources
We continue to be very excited about offering the most comprehensive suite of software ever available with a management science textbook. The commercial value of the enclosed software exceeds $1000 if purchased directly. This software is packaged free with NEW copies of the third edition. It is for students only and, in contrast to the previous edition, it requires no online registration. The following software is included on the accompanying Palisade CD-ROM:
■

xiv

Palisade’s DecisionTools™ Suite, including the award-winning @RISK, PrecisionTree,
TopRank, and RISKOptimizer, and Palisade’s
StatTools add-in for statistics are included in the box that is packaged with new copies of the book. This software is not available with any competing textbook and comes in an educational version that is only slightly scaled down from the expensive commercial version. (StatTools replaces Albright’s StatPro add-in that came with the second edition. If you are interested, StatPro is still freely available from
Preface

■

■

http://www.kelley.iu.edu/albrightbooks, although it will not be updated for Excel 2007.) For more information about the Palisade Corporation, the
DecisionTools Suite, and StatTools, visit
Palisade’s Web site at http://www.palisade.com.
Also available with new copies of the revised third edition is a Student CD-ROM. This resource provides the Excel files for all of the examples in the book, as well as the data files required for a number of problems and cases. As in the second edition, there are two versions of the example files: a completed version and a template to get students started. To make sensitivity analysis useful and intuitive, we continue to provide Albright’s SolverTable add-in (which is also freely available from http://www.kelley.iu.edu/albrightbooks). SolverTable provides data table-like output that is easy to interpret. Finally, the newest version of
Frontline Systems’ Premium Solver™ for
Education is included for use in Chapter 8. Its
Evolutionary Solver uses genetic algorithms to solve nonlinear optimization problems. For more information on Premium Solver or
Frontline Systems, visit Frontline’s Web site at http://www.frontsys.com. (Note that the versions of SolverTable and Premium Solver included are updates compatible with Excel 2007.)
Additionally, a 60-day trial version of Microsoft
Project 2007 is packaged with new copies of the revised third edition.

Companion VBA Book
Soon after the first edition appeared, we began using
Visual Basic for Applications (VBA), the programming language for Excel, in some of our management science courses. VBA allows us to develop decision support systems around the spreadsheet models. (An example appears at the end of Chapter 3.) This use of
VBA has been popular with our students, and many instructors have expressed interest in learning how to do it. For additional support on this topic, a companion book, VBA for Modelers, 2e (ISBN 0-495-10683-6) is available. It assumes no prior experience in computer programming, but it progresses rather quickly to the development of interesting and nontrivial applications.
The revised third edition of Practical Management
Science depends in no way on this companion VBA book, but we expect that many instructors will want to incorporate some VBA into their management science courses. Ancillary Materials
Besides the Student CD-ROM that accompanies new copies of the third edition, the following materials are available: ■

■

An Instructor’s Resource CD (ISBN-10: 0-32459557-3; ISBN-13: 978-0-324-59557-4) that includes PowerPoint slides, Instructor’s
Solutions Files, and Test Bank files is available to adopters. This useful tool reduces class preparation time, aids in assessment, and contributes to students’ comprehension and retention. Student Solution Files (ISBN-10: 0-495-01643-8;
ISBN-13: 978-0-495-01643-4) are available for purchase through the Student Resources
Web page at the book support Web site. Visit academic.cengage.com/decisionsciences/ winston for more information on purchasing this valuable student resource.

If you are interested in requesting any of these supplements or VBA for Modelers, contact your local
South-Western Sales Rep or the Academic Resource
Center at 800-423-0563.

Acknowledgments
This book has gone through several stages of reviews, and it is a much better product because of them. The majority of the reviewers’ suggestions were very good ones, and we have attempted to incorporate them. We would like to extend our appreciation to:
Timothy N. Burcham, University of Tennessee at
Martin
Spyros Camateros, California State University,
East Bay
Michael F. Gorman, University of Dayton
Cagri Haksoz, City University of London
Levon Hayrapetyan, Houston Baptist University
Bharat K. Kaku, Georgetown University
Bingguang Li, Albany State University

Jerrold May, University of Pittsburgh
Paul T. Nkansah, Florida A&M University
Ferdinand C. Nwafor, Florida A&M University
Lynne Pastor, Carnegie Mellon University
Ronald K. Satterfield, University of South Florida
Phoebe D. Sharkey, Loyola College in Maryland
John Wang, Montclair State University
Kari Wood, Bemidji State University
Roger Woods, Michigan Tech
Zhe George Zhang, Western Washington University
We would also like to thank two special people.
First, we want to thank our previous editor Curt
Hinrichs. Although Curt has moved from this position and is no longer our editor, his vision for the past decade has been largely responsible for the success of the first and second editions of Practical Management
Science. We want to wish Curt the best of luck in his new position at SAS. Second, we were lucky to move from one great editor to another in Charles
McCormick Jr. Charles is a consummate professional, he is both patient and thorough, and his experience in the publishing business ensures that the tradition Curt started will be carried on.
In addition, we would like to thank Marketing
Manager, Bryant Chrzan; Senior Developmental
Editor, Laura Bofinger; Content Project Manager,
Emily Nesheim; Art Director, Stacy Shirley; Editorial
Assistant, Bryn Lathrop; and Project Manager at ICC
Macmillan Inc., Leo Kelly.
We would also enjoy hearing from you—we can be reached by e-mail. And please visit either of the following Web sites for more information and occasional updates:
■
■

http://www.kelley.iu.edu/albrightbooks academic.cengage.com/decisionsciences/winston Wayne L. Winston (winston@indiana.edu)
S. Christian Albright (albright@indiana.edu)
Bloomington, Indiana
January 2008

Preface

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CHAPTER

Introduction to Modeling

© Getty Images/PhotoDisc

1

COMPLEX ALGORITHMS AND THE “SOFT OR”
APPROACH SOLVE REAL-WORLD PROBLEMS s you embark on your study of management science, you might question the usefulness of quantitative methods to the “real world.” A front-page article in the December 31, 1997 edition of USA Today entitled “Higher Math
Delivers Formula for Success” provides some convincing evidence of the applicability of the methods you will be learning. The subheading of the article,
“Businesses turn to algorithms to solve complex problems,” says it all. Today’s business problems tend to be very complex. In the past, many managers and executives used a “seat of the pants” approach to solve problems—that is, they used their business experience, their intuition, and some thoughtful guesswork to solve problems. But common sense and intuition go only so far in the solution of the complex problems businesses now face. This is where management science models—and the algorithms mentioned in the title of the article—are so useful.When the methods in this book are implemented in user-friendly computer software packages that are applied to complex problems, the results can be amazing. Robert Cross, whose company, DFI
Aeronomics, sells algorithm-based systems to airlines, states it succinctly: “It’s like taking raw information and spinning money out of it.”
The methods in this book are powerful because they apply to so many problems and environments. The article mentions the following “success stories” in which management science has been applied; others will be discussed throughout this book.

A

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United Airlines installed one of DFI’s systems, which cost between $10 million and $20 million. United expected the system to add $50 million to $100 million annually to its revenues.
The Gap clothing chain uses management science to determine exactly how many employees should staff each store during the Christmas rush.
Management science has helped medical researchers test potentially dangerous drugs on fewer people with better results.
IBM obtained a $93 million contract to build a computer system for the
U.S. Department of Energy that would do a once-impossible task: make exact real-time models of atomic blasts. It won the contract—and convinced the DOE that its system was cost-effective—only by developing management science models that would cut the processing time by half.
Hotels, airlines, and television broadcasters all use management science to implement a method called yield management. In this method, different prices are charged to different customers, depending on their willingness to pay. The effect is that more customers are attracted and revenues increase. Although most of this book describes how quantitative methods can be used to solve business problems, solutions do not always need to be quantitatively based. In a recent article, Kimbrough and Murphy (2005), two academics located in Philadelphia, describe how they were commissioned by the city to study the knowledge economy of the region and make recommendations on ways to improve its rate of growth. Unlike most of the success stories chronicled in the Interfaces journal (which is described on page 16), the authors state right away that they used no quantitative methods or mathematical models to develop recommendations for the city. Instead, they used a soft OR approach.1 By this, they imply that they used a systematic approach to formulate and solve their client’s problem, even though the approach does not employ quantitative methods.
Specifically, Kimbrough and Murphy used two interrelated approaches in their study. First, using ideas in the management science and economics literature, they developed a comprehensive framework for thinking about regional economic development. This allowed them to identify the many factors that influence a region’s economic vitality or lack thereof. Second, they interviewed a wide range of people from the region, including researchers in science and technology, business people, government officials, and academics.
Instead of asking these people what ought to be done, they asked them to propose ideas or policy initiatives that might improve the region’s economy.
As they state, “The results were gratifying. The framework we developed focuses people’s thinking on problems, bottlenecks, and leverage points in the knowledge economy. Asking for specific ideas produced a rich and constructive list of more than 50 promising, realistic, and detailed policy initiatives.”
However, the study went beyond brainstorming. After conducting the interviews and analyzing the responses, the authors made specific recommendations to their client on initiatives that might be implemented to improve the knowledge economy. Based on these recommendations, the board
1

OR is an abbreviation for operations research, another term for management science. Over the years, the two terms have become practically synonymous, although some people in the field still prefer to be called management scientists, whereas others prefer to be called operations researchers. 2

Chapter 1 Introduction to Modeling

of directors of Greater Philadelphia First adopted Six for Success, a strategy that commits leaders to (1) attract more research dollars and expertise;
(2) implement strategies to accelerate science and technology; (3) promote an entrepreneurial climate; (4) launch a business marketing plan; (5) leverage quality-of-life infrastructure and amenities; and (6) streamline and rationalize business-oriented nonprofits. Granted, these ideas are not necessarily ground breaking, but they made sense to leaders in Philadelphia. The important point is that they were developed through a systematic approach to solving a problem—even if it wasn’t the quantitative approach discussed through most of this book. ■

1.1 INTRODUCTION
The purpose of this book is to expose you to a variety of problems that have been solved successfully with management science methods and to give you experience in modeling these problems in the Excel spreadsheet package. The subject of management science has evolved for more than 50 years and is now a mature field within the broad category of applied mathematics. This book will emphasize both the applied and mathematical aspects of management science. Beginning in this chapter and continuing throughout the rest of the book, we discuss many successful management science applications, where teams of highly trained people have implemented solutions to the problems faced by major companies and have saved these companies millions of dollars. Many airlines and oil companies, for example, could hardly operate as they do today without the support of management science. In this book, we will lead you through the solution procedure of many interesting and realistic problems, and you will experience firsthand what is required to solve these problems successfully. Because we recognize that most of you are not highly trained in mathematics, we use Excel spreadsheets to solve problems, which makes the quantitative analysis much more understandable and intuitive.
The key to virtually every management science application is a mathematical model.
In simple terms, a mathematical model is a quantitative representation, or idealization, of a real problem. This representation might be phrased in terms of mathematical expressions
(equations and inequalities) or as a series of interrelated cells in a spreadsheet. We prefer the latter, especially for teaching purposes, and we concentrate primarily on spreadsheet models in this book. However, in either case, the purpose of a mathematical model is to represent the essence of a problem in a concise form. This has several advantages. First, it enables a manager to understand the problem better. In particular, the model helps to define the scope of the problem, the possible solutions, and the data requirements. Second, it allows analysts to employ a variety of the mathematical solution procedures that have been developed over the past half-century. These solution procedures are often computer intensive, but with today’s cheap and abundant computing power, they are usually feasible.
Finally, the modeling process itself, if done correctly, often helps to “sell” the solution to the people who must work with the system that is eventually implemented.
This chapter begins with a relatively simple example of a mathematical model. You’ll then examine the difference between modeling and a collection of models and learn about a seven-step model-building process that you should follow—in essence if not in strict conformance—in all management science applications. Next, you’ll learn about a successful application of management science and how the seven-step model-building process was followed. Finally, you’ll see why the study of management science is valuable, not only to large corporations, but also to students like you who are about to enter the business world.

1.1 Introduction

3

1.2 A WAITING-LINE EXAMPLE
As indicated earlier, a mathematical model is a set of mathematical relationships that represent, or approximate, a real situation. Models that simply describe a situation are called descriptive models. Other models that suggest a desirable course of action are called optimization models. To get started, consider the following simple example of a mathematical model. It begins as a descriptive model, but then expands to an optimization model.
Consider a convenience store with a single cash register. The manager of the store suspects that customers are waiting too long in line at the checkout register and that these excessive waiting times are hurting business. Customers who have to wait a long time might not come back, and potential customers who see a long line might not enter the store at all.
Therefore, the manager builds a mathematical model to help understand the problem. The manager wants the model to reflect the current situation at the store, but it should also suggest improvements to the current situation.

A Descriptive Model
This example is a typical waiting line, or queueing, problem. (Such problems are studied in detail in Chapter 14.) The manager first wants to build a model that reflects the current situation at the store. Later, he will alter the model to predict what might make the situation better. To describe the current situation, the manager realizes that there are two important inputs to the problem: (1) the arrival rate of potential customers to the store and (2) the rate at which customers can be served by the single cashier. Clearly, as the arrival rate increases and/or the service rate decreases, the waiting line will tend to increase and each customer will tend to wait longer in line. In addition, more potential customers likely will decide not to enter at all. These latter quantities (length of waiting line, time in line per customer, fraction of customers who don’t enter) are commonly referred to as outputs. The manager believes he has some understanding of the relationship between the inputs and the outputs, but he is not at all sure of the exact relationship between them.
This is where a mathematical model is useful. By making several simplifying assumptions about the nature of the arrival and service process at the store (as discussed in
Chapter 14), you can relate the inputs to the outputs. In some cases, when the model is sufficiently simple, you can write an equation for an output in terms of the inputs. For example, in one of the simplest queueing models, if A is the arrival rate of customers per minute, S is the service rate of customers per minute, and W is the average time a typical customer waits in line (assuming that all potential customers enter the store), then the following relationship can be derived mathematically:
A
W ϭ ᎏᎏ
S(S Ϫ A)

(1.1)

This relationship is intuitive in one sense. It correctly predicts that as the service rate
S increases, the average waiting time W decreases; as the arrival rate A increases, the average waiting time W increases. Also, if the arrival rate is just barely less than the service rate—that is, the difference S Ϫ A is positive but very small—the average waiting time becomes quite large. [This model requires that the arrival rate be less than the service rate; otherwise, equation (1.1) makes no sense.]
In many other models, there is no such closed-form relationship between inputs and outputs (or if there is, it is too complex for the level of this book). Nevertheless, there may still be a mathematical procedure for calculating outputs from inputs, and it may be possible to implement this procedure in Excel. This is the case for the convenience store problem. Again, by making certain simplifying assumptions, including the assumption

4

Chapter 1 Introduction to Modeling

Figure 1.1
Descriptive
Queueing Model for
Convenience Store

1
2
3
4
5
6
7
8
9
10
11

A
DescripƟve queueing model for convenience store

B

Inputs
Arrival rate (customers per minute)
Service rate (customers per minute)
Maximum customers (before others go elsewhere)
Outputs
Average number in line
Average Ɵme (minutes) spent in line
Percentage of potenƟal arrivals who don't enter

0.5
0.4
5

2.22
6.09
27.1%

that potential customers will not enter if the waiting line is sufficiently long, you can develop a spreadsheet model of the situation at the store.
Before developing the spreadsheet model, however, we must determine how the manager can obtain the inputs he needs. There are actually three inputs: (1) the arrival rate A,
(2) the service rate S, and (3) the number in the store, labeled N, that will induce future customers not to enter. The first two of these can be measured with a stopwatch. For example, the manager can instruct an employee to measure the times between customer arrivals.
Let’s say the employee does this for several hours, and the average time between arrivals is observed to be 2 minutes. Then the arrival rate can be estimated as A ϭ 1͞2 ϭ 0.5
(1 customer every 2 minutes). Similarly, the employee can record the times it takes the cashier to serve successive customers. If the average of these times (taken over many customers) is, say, 2.5 minutes, then the service rate can be estimated as S ϭ 1͞2.5 ϭ 0.4
(1 customer every 2.5 minutes). Finally, if the manager notices that potential customers tend to take their business elsewhere when 5 customers are in line, he can let N ϭ 5.
These input estimates can now be entered in the spreadsheet model shown in Figure 1.1.
Don’t worry about the details of this spreadsheet—they are discussed in Chapter 14. The formulas built into this spreadsheet reflect an adequate approximation of the convenience store’s situation. For now, the important thing is that this model allows the manager to enter any values for the inputs in cells B4 through B6 and observe the resulting outputs in cells B9 through B11. The input values in Figure 1.1 represent the store’s current input values. These values indicate that slightly more than 2 customers are waiting in line on average, an average customer waits slightly more than 6 minutes in line, and about 27% of all potential customers do not enter the store at all (due to the perception that waiting times will be long).
The information in Figure 1.1 is probably not all that useful to the manager. After all, he probably already has a sense of how long waiting times are and how many customers are being lost. The power of the model is that it allows the manager to ask many what-if questions. For example, what if he could somehow speed up the cashier, say, from 2.5 minutes per customer to 1.8 minutes per customer? He might guess that because the average service time has decreased by 28%, all the outputs should also decrease by 28%. Is this the case? Evidently not, as shown in Figure 1.2. The average line length decreases to 1.41, a
36% decrease; the average waiting time decreases to 3.22, a 47% decrease; and the percentage of customers who do not enter decreases to 12.6%, a 54% decrease. To illustrate an
Figure 1.2
Queueing Model with a Faster Service
Rate

1
2
3
4
5
6
7
8
9
10
11

A
DescripƟve queueing model for convenience store

B

Inputs
Arrival rate (customers per minute)
Service rate (customers per minute)
Maximum customers (before others go elsewhere)

0.5
0.556
5

Outputs
Average number in line
Average Ɵme (minutes) spent in line
Percentage of potenƟal arrivals who don't enter

1.41
3.22
12.6%

1.2 A Waiting-Line Example

5

Figure 1.3

1
2
3
4
5
6
7
8
9
10
11

Queueing Model with an Even Faster
Service Rate

A
DescripƟve queueing model for convenience store
Inputs
Arrival rate (customers per minute)
Service rate (customers per minute)
Maximum customers (before others go elsewhere)
Outputs
Average number in line
Average Ɵme (minutes) spent in line
Percentage of potenƟal arrivals who don't enter

B

0.5
0.8
5

0.69
1.42
3.8%

even more extreme change, suppose the manager could cut the service time in half, from
2.5 minutes to 1.25 minutes. The spreadsheet in Figure 1.3 shows that the average number in line decreases to 0.69, a 69% decrease from the original value; the average waiting time decreases to 1.42, a 77% decrease; and the percentage of customers who do not enter decreases to 3.8%, a whopping 86% decrease. The important lesson to be learned from the spreadsheet model is that as the manager increases the service rate, the output measures improve more than he might have expected.
In reality, the manager would attempt to validate the spreadsheet model before trusting its answers to these what-if questions. At the very least, the manager should examine the reasonableness of the assumptions. For example, one assumption is that the arrival rate remains constant for the time period under discussion. If the manager intends to use this model— with the same input parameters—during periods of time when the arrival rate varies a lot
(such as peak lunchtime traffic followed by slack times in the early afternoon), then he is almost certainly asking for trouble. Besides determining whether the assumptions are reasonable, the manager can also check the outputs predicted by the model when the current inputs are used. For example, Figure 1.1 predicts that the average time a customer waits in line is approximately 6 minutes. At this point, the manager could ask his employee to use a stopwatch again to time customers’ waiting times. If they average close to 6 minutes, then the manager can have more confidence in the model. However, if they average much more or much less than 6 minutes, the manager probably needs to search for a new model.

An Optimization Model
So far, the model fails to reflect any economic information, such as the cost of speeding up service, the cost of making customers wait in line, or the cost of losing customers. Given the spreadsheet model developed previously, however, incorporating economic information and then making rational choices is relatively straightforward. To make this example simple, assume that the manager can do one of three things: (1) leave the system as it is, (2) hire a second person to help the first cashier process customers more quickly, or (3) lease a new model of cash register that will speed up the service process significantly. The effect of
(2) is to decrease the average service time from 2.5 to 1.8 minutes. The effect of (3) is to decrease the service time from 2.5 to 1.25 minutes. What should the manager do?
He needs to examine three types of costs. The first is the cost of hiring the extra person or leasing the new cash register. We assume that these costs are known. For example, the hourly wage for the extra person is $8, and the cost to lease a new cash register (converted to a per-hour rate) is $11 per hour. The second type of cost is the “cost” of making a person wait in line. Although this is not an out-of-pocket cost to the store, it does represent the cost of potential future business—a customer who has to wait a long time might not return. This cost is difficult to estimate on a per-minute or per-hour basis, but we assume it’s approximately $13 per customer per hour in line.2 Finally, there is the opportunity cost for
2

Here we are charging only for time in the queue. An alternative model is to charge for time in the queue and for time in service.

6

Chapter 1 Introduction to Modeling

customers who decide not to enter the store. The store loses not only their current revenue but also potential future revenue if they decide never to return. Again, this is a difficult cost to measure, but we assume it’s approximately $25 per lost customer.
The next step in the modeling process is to combine these costs for each possible decision. Let’s find the total cost per hour for decision (3), where the new cash register is leased. The lease cost is $11 per hour. From Figure 1.3, you see there is, on average, 0.69 customer in line at any time. Therefore, the average waiting cost per hour is 0.69($13) ϭ
$8.91. (This is because 0.69 customer-hour is spent in line each hour on average.) Finally, from Figure 1.3 you see that the average number of potential arrivals per hour is 60(1͞2) ϭ
30, and 3.8% of them do not enter. Therefore, the average cost per hour from lost customers is 0.038(30)($25) ϭ $28.52. The combined cost for decision (3) is $11 ϩ $8.91 ϩ $28.52 ϭ
$48.43 per hour.
The spreadsheet model in Figure 1.4 incorporates these calculations and similar calculations for the other two decisions. As you see from row 24, the option to lease the new cash register is the clear winner from a cost standpoint. However, if the manager wants to see how sensitive these cost figures are to the rather uncertain input costs assessed for waiting time and lost customers, it’s simple to enter new values in rows 10 and 11 and see how the “bottom lines” in row 24 change. This flexibility represents the power of spreadsheet models. They not only allow you to build realistic and complex models, but they also allow you to answer many what-if questions simply by changing input values.
Figure 1.4
Queueing Model with Alternative
Decisions

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

A
Decision queueing model for convenience store
Inputs
Arrival rate (customers per minute)
Service rate (customers per minute)
Maximum customers (before others go elsewhere)

B

C

D

Decision 1
0.5
0.4
5

Decision 2
0.5
0.556
5

Decision 3
0.5
0.8
5

$0
$0
$13
$25

$8
$0
$13
$25

$0
$11
$13
$25

2.22
6.09
27.1%

1.41
3.22
12.6%

0.69
1.42
3.8%

Cost informaƟon
Cost of extra person per hour
Cost of leasing new cash register per hour
Cost per hour of waiƟng Ɵme
Cost per hour of lost customers

$0
$0
$28.87
$203.29

$8
$0
$18.31
$94.52

$0
$11
$8.91
$28.52

Total cost per hour

$232.16

$120.82

$48.43

Cost of extra person per hour
Cost of leasing new cash register per hour
Cost per customer per hour waiƟng in line
Cost per customer who doesn't enter the store
Outputs
Average number in line
Average Ɵme (minutes) spent in line
Percentage of potenƟal arrivals who don't enter

1.3 MODELING VERSUS MODELS
Management science, at least as it has been taught in many traditional courses, has evolved as a collection of mathematical models. These include various linear programming models
(the transportation model, the diet model, the shortest route model, and others), inventory models, queueing models, and so on. Much time has been devoted to teaching the intricacies of these particular models. Management science practitioners, on the other hand, have justifiably criticized this emphasis on specific models. They argue that the majority of real-world management science problems cannot be neatly categorized as one of the handful of models typically included in a management science textbook. That is, often no “off-the-shelf” model can be used without modification to solve a company’s real problem. Unfortunately,

1.3 Modeling Versus Models

7

management science students have gotten the impression that all problems must be “shoehorned” into one of the textbook models.
The good news is that this emphasis on specific models has been changing in the past decade, and our goal in this book is to continue that change. Specifically, this book stresses modeling, not models. The distinction between modeling and models will become clear as you proceed through the book. Learning specific models is essentially a memorization process—memorizing the details of a particular model, such as the transportation model, and possibly learning how to “trick” other problems into looking like a transportation model. Modeling, on the other hand, is a process, where you abstract the essence of a real problem into a model, spreadsheet or otherwise. Although the problems fall naturally into several categories, in modeling, you don’t try to shoe-horn each problem into one of a small number of well-studied models. Instead, you treat each problem on its own merits and model it appropriately, using whatever logical, analytical, or spreadsheet skills you have at your disposal—and, of course, drawing analogies from previous models you have developed whenever relevant. This way, if you come across a problem that does not look exactly like anything you studied in your management science course, you still have the skills and flexibility to model it successfully.
This doesn’t mean you won’t learn some “classical” models from management science in this book; in fact, we’ll discuss the transportation model in linear programming, the
M/M/1 model in queueing, the EOQ model in inventory, and others. These are important models that should not be ignored; however, we certainly do not emphasize memorizing these specific models. They are simply a few of the many models you will learn how to develop. The real emphasis throughout is on the modeling process—how a real-world problem is abstracted into a spreadsheet model of that problem. We discuss this modeling process in more detail in the following section.

1.4 THE SEVEN-STEP MODELING PROCESS
The discussion of the queueing model in Section 1.2 presented some of the basic principles of management science modeling. This section further expands on these ideas by characterizing the modeling process as the following seven-step procedure.

Step 1: Problem Definition
The analyst first defines the organization’s problem. Defining the problem includes specifying the organization’s objectives and the parts of the organization that must be studied before the problem can be solved. In the simple queueing model, the organization’s problem is how to minimize the expected net cost associated with the operation of the store’s cash register.

Step 2: Data Collection
After defining the problem, the analyst collects data to estimate the value of parameters that affect the organization’s problem. These estimates are used to develop a mathematical model (step 3) of the organization’s problem and predict solutions (step 4). In the convenience store queueing example, the manager needs to observe the arrivals and the checkout process to estimate the arrival rate A and the service rate S.

Step 3: Model Development
In the third step, the analyst develops a model of the problem. In this book, we describe many methods that can be used to model systems.3 Models such as the equation for W,
3
All these models can generically be called mathematical models. However, because we implement them in spreadsheets, we generally refer to them as spreadsheet models.

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Chapter 1 Introduction to Modeling

where you use an equation to relate inputs such as A and S to outputs such as W, are called analytical models. Most realistic applications are so complex, however, that an analytical model does not exist or is too complex to work with. For example, if the convenience store had more than one register and customers were allowed to join any line or jump from one line to another, there would be no tractable analytical model—no equation or system of equations—that could be used to determine W from knowledge of A, S, and the number of lines. When no tractable analytical model exists, you can often rely instead on a simulation model, which enables you to approximate the behavior of the actual system. Simulation models are covered in Chapters 11 and 12.

Step 4: Model Verification
The analyst now tries to determine whether the model developed in the previous step is an accurate representation of reality. A first step in determining how well the model fits reality is to check whether the model is valid for the current situation. As discussed previously, to validate the equation for the waiting time W, the manager might observe actual customer waiting times for several hours. As you’ve already seen, the equation for W predicts that when A ϭ 0.5 and S ϭ 0.4, the average customer spends 6.09 minutes in line. Now suppose the manager observes that 120 customers spend a total of 750 minutes in line. This indicates an average of 750͞120 ϭ 6.25 minutes in line per customer. Because 6.25 is reasonably close to 6.09, the manager’s observations lend credibility to the model. In contrast, if the 120 customers had spent 1,200 minutes total in line, for an average of 10 minutes per customer, this would not agree very well with the model’s prediction of 6.09 minutes, and it would cast doubt on the validity of the model.

Step 5: Optimization and Decision Making
Given a model and a set of possible decisions, the analyst must now choose the decision or strategy that best meets the organization’s objectives. (We briefly discussed an optimization model for the convenience store example, and we discuss many others throughout the book.) Step 6: Model Communication to Management
The analyst presents the model and the recommendations from the previous step to the organization. In some situations, the analyst might present several alternatives and let the organization choose the best one.

Step 7: Model Implementation
If the organization has accepted the validity and usefulness of the study, the analyst then helps to implement its recommendations. The implemented system must be monitored constantly (and updated dynamically as the environment changes) to ensure that the model enables the organization to meet its objectives.

Flowchart of Procedure and Discussion of Steps
Figure 1.5 illustrates this seven-step process. As the arrows pointing down and to the left indicate, there is certainly room for feedback in the process. For example, at various steps, the analyst might realize that the current model is not capturing some key aspects of the real problem. In this case, the analyst should revise the problem definition or develop a new model.

1.4 The Seven-Step Modeling Process

9

Figure 1.5 Flowchart for the Seven-Step Process

Problem definition Data collection Model development Model verification Optimization and decision making Model communication to management Model implementation Possible feedback loops

Now that you’ve seen the basic outline of the seven steps and the flowchart, let’s delve into the modeling proces in more detail.

Step 1: Problem Definition

It’s important to solve the correct problem, and defining that problem is not always easy. Typically, a management science model is initiated when an organization believes it has a problem. Perhaps the company is losing money, perhaps its market share is declining, perhaps its customers are waiting too long for service—any number of problems might be evident. The organization (which we refer to as the client) calls in a management scientist (the analyst) to help solve this problem.4 In such cases, the problem has probably already been defined by the client, and the client hires the analyst to solve this particular problem.
As Miser (1993) and Volkema (1995) point out, however, the analyst should do some investigating before accepting the client’s claim that the problem has been properly defined. Failure to do so could mean solving the wrong problem and wasting valuable time and energy.
For example, Miser cites the experience of an analyst who was hired by the military to investigate overly long turnaround times between fighter planes landing and taking off again to rejoin the battle. The military (the client) was convinced that the problem was caused by inefficient ground crews—if they worked faster, turnaround times would presumably decrease. The analyst nearly accepted this statement of the problem and was about to do classical time-and-motion studies on the ground crew to pinpoint the sources of their inefficiency. However, by snooping around, he found that the problem lay elsewhere. It seems that the trucks that refueled the planes were frequently late, which in turn was due to the inefficient way they were refilled from storage tanks at another location. After this latter problem was solved—and its solution was embarrassingly simple—the turnaround times decreased to an acceptable level without any changes on the part of the ground crews.
If the analyst had accepted the client’s statement of the problem, the real problem would never have been located or solved.
The moral of this story is clear: If an analyst defines a problem incorrectly or too narrowly, the best solution to the real problem might never emerge. In his article,
Volkema (1995) advocates spending as much time thinking about the problem and defining it properly as modeling and solving it. This is undoubtedly good advice, especially in real-world applications where problem boundaries are often difficult to define.

Step 2: Data Collection
The data collection step often takes the most time.

This crucial step in the modeling process is often the most tedious. All organizations keep track of various data on their operations, but the data is often not in the form the analyst requires. In addition, data is often stored in different places throughout the organization and in all kinds of formats. Therefore, one of the analyst’s first jobs is to gather exactly the right data and put the data into an appropriate and consistent format for use in the model. This
4

Most organizations hire outside consultants, sometimes academics, to help solve problems. However, a number of large organizations employ a staff of management scientists who function as inside consultants.

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typically requires asking questions of key people (such as the accountants) throughout the organization, studying existing organizational databases, and performing time-consuming observational studies of the organization’s processes. In short, it typically entails a lot of legwork. In this book, as in most management science textbooks, we shield you from this data-collection process by supplying the appropriate data to develop and solve a model.
Although this makes the overall modeling process seem easier than it really is, in most class settings, having you go to companies and gather data is just not practical. (In many cases, it would not even be allowed for proprietary reasons.) Nevertheless, we provide some insights with “Where Do the Numbers Come From?” sections. If nothing else, these sections remind you that in real applications, someone has to gather the necessary inputs. Step 3: Model Development
Steps 3 and 5, developing and optimizing models, are the steps emphasized most heavily in this book.

This step, along with step 5, is where the analyst brings his or her special skills into play.
After defining the client’s problem and gathering the necessary data, the analyst must develop a model of the problem. Several properties are desirable for a good model. First, it should represent the client’s real problem accurately. If it uses a linear (straight-line) function for costs when the real cost function is highly nonlinear (curved), the recommendations of the model could be very misleading. Similarly, if the model ignores an important constraint, such as an upper bound on capacity, its recommendations might not be possible to implement.
On the other hand, the model should be as simple as possible. Most good models
(where “good” really means useful) capture the essence of the problem without getting bogged down in less important details. They should be approximations of the real world, not mirror images in every last detail. Overly complex models are often of little practical use. First, overly complex models are sometimes too difficult to solve with the solution algorithms available. Second, complex models tend to be incomprehensible to clients. If a client cannot understand a model, the chances are that the model’s recommendations will never be implemented. Therefore, a good model should achieve the right balance between being too simple and too complex.

Step 4: Model Verification
This step is particularly important in real management science applications. A client is much more likely to accept an analyst’s model if the analyst can provide some type of verification. This verification can take several forms. For example, the analyst can use the model with the company’s current values of the input parameters. If the model’s outputs are then in line with the outputs currently observed by the client, the analyst has at least shown that the model can duplicate the current situation.
A second way to verify a model is to enter a number of sets of input parameters (even if they are not the company’s current inputs) and see whether the outputs from the model are reasonable. One common approach is to use extreme values of the inputs to determine whether the outputs behave as they should. For example, for the convenience store queueing model, you could enter an extremely large service rate or a service rate just barely above the arrival rate in the equation for W. In the first case, you would expect the average waiting time to approach 0, whereas in the latter case, you would expect it to become very large. You can use equation (1.1) for W to verify that this is exactly what happens. This provides another piece of evidence that the model is reasonable.
If you enter certain inputs in the model, and the model’s outputs are not as expected, there are two possible causes. First, the model could simply be a poor approximation of the actual situation. In this case, the analyst must refine the model until it lines up more accurately with reality. Second, the model might be fine, but the analyst’s intuition is not very

1.4 The Seven-Step Modeling Process

11

good. That is, when asked what he or she thinks would happen if certain input values were used, the analyst might provide totally wrong predictions. In this case, the fault lies with the analyst, not the model. Sometimes, good models prove that people’s ability to predict outcomes in complex environments is lacking. In such cases, the verification step becomes harder because of “political” reasons (office politics).

Step 5: Optimization and Decision Making

A heuristic is a relatively simple solution method that often provides “good” but not necessarily optimal solutions. After the problem has been defined, the data has been collected, and the model has been developed and verified, it’s time to use the model to recommend decisions or strategies. In the majority of management science models, this requires the optimization of an objective, such as maximizing profit or minimizing cost.
The optimization phase is typically the most difficult phase from a mathematical standpoint. Indeed, much of the management science literature (mostly from academics) has focused on complex solution algorithms for various classes of models. Fortunately, this research has led to a number of solution algorithms—and computer packages that implement these algorithms—that can be used to solve real problems. The most famous of these is the simplex algorithm. This algorithm, which has been implemented by many commercial software packages (including Excel’s Solver), is used on a daily basis to solve linear optimization models for many companies. (We take advantage of the simplex method in
Chapters 3 through 5.)
Not all solution procedures find the optimal solution to a problem. Many models are either too large or too complex to be solved exactly. Therefore, many complex problems use heuristic methods to locate “good” solutions. A heuristic is a solution method that is guided by common sense, intuition, and trial and error to achieve a good, but probably not optimal, solution. Some heuristics are “quick and dirty,” whereas others are sophisticated.
As models become larger and more complex, good heuristics are sometimes the best that can be achieved—and frequently they are perfectly adequate.

Step 6: Model Communication to Management
Sooner or later, an analyst must communicate a model and its recommendations to the client. To appreciate this step, you must understand the large gap that typically exists between the analyst and the managers of organizations. Managers know their business, but they often don’t understand much about mathematics and mathematical models—even spreadsheet implementations of these models. The burden is therefore on the analyst to present the model in terms that nonmathematical people can understand; otherwise, a perfectly good model might never see the light of day.
The best strategy for successful presentation is to involve key people in the organization, including top executives, in the project from the beginning. If these people have been working with the analyst, helping to supply appropriate data and helping the analyst to understand the way the organization really works, they are much more likely to accept the eventual model. Step 6, therefore, should really occur throughout the modeling process, not just toward the end.
The analyst should also try to make the model as intuitive and user-friendly as possible. Clients appreciate menu-driven systems with plenty of graphics. They also appreciate the ability to ask what-if questions and get answers quickly in a form that is easy to understand. This is one reason for developing spreadsheet models. Although not all models can be developed in spreadsheets due to size and/or complexity, the spreadsheet approach in this book is an excellent choice whenever possible because most business people are comfortable with spreadsheets. Spreadsheet packages support the use of graphics, customized menus and toolbars, data tables and other tools for what-if analyses, and even macros (that can be made transparent to users) for running complex programs.

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Step 7: Model Implementation
A real management science application is not complete until it has been implemented. A successful implementation can occur only when step 6 has been accomplished. That is, the analyst must demonstrate the model to the client, and the client must be convinced that the model adds real value and can be used by the people who need to use it. For this reason, the analyst’s job is not really complete until the system is up and running on a daily basis. To achieve a successful implementation, it isn’t enough for management to accept the model; the people who will run it every day must also be thoroughly trained to use it.
At the very least, they should understand how to enter appropriate inputs, run what-if analyses, and interpret the model’s outputs correctly. If they conclude that the model is more trouble than it’s worth, they might simply refuse to use it, and the whole exercise has been a waste of time. An interesting trend (as evidenced in many of the Interfaces articles discussed shortly) is for analysts to build a user-friendly Excel “front end” for their clients, even if the actual number crunching is performed behind the scenes in some non-Excel package. Because many employees understand at least the basics of Excel, such a userfriendly front end makes the system much more attractive for daily use.
Many successful management science applications take on a life of their own after the initial implementation. After an organization sees the benefits of a useful model—and of management science in general—it is likely to expand the model or create new models for uses beyond those originally intended. Knowing that this is often the case, the best analysts design models that can be expanded. They try to anticipate problems the organization might face besides the current problem. They also stay in contact with the organization after the initial implementation, just in case the organization needs guidance in expanding the scope of the model or in developing new models.
This discussion of the seven-step modeling process has taken an optimistic point of view by assuming that a successful study employs these seven steps, in approximately this chronological order, and that everything goes smoothly. It does not always work out this way. Numerous potential applications are never implemented even though the technical aspects of the models are perfectly correct. The most frequent cause is a failure to communicate. The analyst builds a complex mathematical model, but the people in the organization don’t understand how it works and hence are reluctant to use it. Also, company politics can be a model’s downfall, especially if the model recommends a course of action that top management simply does not want to follow—for whatever reasons.
Even for applications that are eventually implemented, the analyst doesn’t always proceed through the seven steps exactly as described in this section. He or she might backtrack considerably throughout the process. For example, based on a tentative definition of the problem, a model is built and demonstrated to management. Management says that the model is impressive, but it doesn’t really solve the company’s problem. Therefore, the analyst goes back to step 1, redefines the problem, and builds a new model (or modifies the original model). In this way, the analyst generates several iterations of some or all the seven steps before the project is considered complete.

The Model as a Beginning, Not an End
This book heavily emphasizes developing spreadsheet models, which is step 3 of the seven-step modeling process. We lead you, step-by-step, through the model development process for many examples, and we ask you to do this on your own in numerous problems.
Given this emphasis, it’s easy to think of the completed model as the end of the process— you complete the model and then go on to the next model. However, a completed model is really a starting point. After you have a working model of the problem, you can—and should—use it as a tool for gaining insights. For most models, many what-if questions can

1.4 The Seven-Step Modeling Process

13

be asked. If the model has been developed correctly, it should be capable of answering such what-if questions fairly easily. In other words, it should be relatively easy to perform sensitivity analysis on the model. This is, in fact, how management science models are used in the business world. They are typically developed to solve a particular problem, but they are then used as a tool to analyze a number of variations of the basic problem.
For most of the examples in the book, we not only develop a model to obtain an
“answer,” but we often include a section called “Discussion of the Solution” (or a similar title) and a section called “Sensitivity Analysis.” The first of these gets you to step back and look at the solution. Does it make sense? Does it provide any insights, especially surprising ones? The second section expands the model in one or more natural ways. What happens if there is more or less of some scarce resource? What happens if a new constraint is added? The point is that before moving to the next model, you should spend some time taking a close look at the model you just developed. This is not just for pedagogical purposes; it is exactly the way real management science projects proceed.

1.5 A SUCCESSFUL MANAGEMENT SCIENCE APPLICATION
This section discusses a successful management science application. Besides a detailed
(but nonquantitative) description of the application, you’ll see how it ties into the sevenstep model-building process discussed in the previous section.

GE Capital
GE Capital, a subsidiary of the General Electric Company’s financial services business, provides credit card service to 50 million accounts. The average total outstanding balance exceeds $12 billion. GE Capital, led by Makuch et al. (1992), developed the PAYMENT system to reduce delinquent accounts and the cost of collecting from delinquent accounts.
The following describes how the seven-step model-building process played out for GE
Capital:
1. At any time, GE Capital has more than $1 billion in delinquent accounts. The company spends $100 million annually processing these accounts. Each day employees contact more than 200,000 delinquent credit card holders with letters, taped phone messages, or live phone calls. However, there was no real scientific basis for the methods used to collect on various types of accounts. For example, GE Capital had no idea whether a two-month due account should receive a taped phone message, a live phone call, some combination of these, or no contact at all. The company’s goal was to reduce delinquent accounts and the cost of processing these accounts, but it was not sure how to accomplish this goal. Therefore, GE Capital’s retail financial services component, together with management scientists and statisticians from GE’s corporate research and development group, analyzed the problem and eventually developed a model called PAYMENT. The purpose of this model was to assign the most cost-effective collection methods to delinquent accounts.
2. The key data requirements for modeling delinquent accounts are delinquency movement matrices (DMMs). A DMM shows how the probability of the payment on a delinquent account depends on the collection action taken (no action, live phone call, taped message, or letter), the size of the unpaid balance, and the account’s performance score. (The higher the performance score associated with a delinquent account, the more likely the account is to be collected.) For example, if a $250 account is two months delinquent, has a high performance score, and is contacted with a phone message, then certain events might occur with certain probabilities. The

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events and the probabilities listed in Table 1.1 illustrate one possibility. The key is to estimate these probabilities for each possible collection action and each account type.

Table 1.1

Sample DMM Entries

Event
Account completely paid off
One month is paid off
Nothing is paid off

Probability
0.30
0.40
0.30

Fortunately, because GE Capital had millions of delinquent accounts, plenty of data was available to estimate the DMMs accurately. To illustrate, suppose there are
1000 two-month delinquent accounts, each with balances under $300 and a high performance score. Also, suppose that each of these is contacted with a phone message. If 300 of these accounts are completely paid off by the next month, then an estimate of the probability of an account being completely paid off by next month is
0.30 (ϭ 300͞1000). By collecting the necessary data to estimate similar probabilities for all account types and collection actions, GE Capital finally had the basis for determining which collection strategies were most cost-effective.
3. After collecting the required data and expressing it in the form of DMMs, the company had to discover which collections worked best in which situations. Specifically, they wanted to maximize the expected delinquent accounts collected during the following six months. However, GE Capital realized that this is a dynamic decision problem. For example, one strategy is called creaming. In this strategy, most collection resources are concentrated on live phone calls to the delinquent accounts classified as most likely to pay up—the best customers. This creaming strategy is attractive because it’s likely to generate short-term cash flows from these customers. However, it has two negative aspects. First, it’s likely to cause a loss of goodwill among the best customers. Second, it gains nothing in the long run from the customers who are most likely to default on their payments. Therefore, the analysts developed the PAYMENT model to find the best decision strategy, a contingency plan for each type of customer that specifies which collection strategy to use at each stage of the account’s delinquency. The constraints in the PAYMENT model ensure that available resources are not overused.
4. A key aspect of GE Capital’s problem is uncertainty. When the PAYMENT model specifies the collection method to use for a certain type of account, it implies that the probability of collecting on this account with this collection method is relatively high. However, there is still a chance that the collection method will fail. With this high degree of uncertainty, it’s difficult to convince skeptics that the model will work as advertised until it has been demonstrated in an actual environment. This is exactly what GE Capital did. It piloted the PAYMENT model on a $62 million portfolio for a single department store chain. To see the real effect of PAYMENT’s recommended strategies, the pilot study used manager-recommended strategies for some accounts and PAYMENT-recommended strategies for others. (They referred to this as the
“champion” versus the “challenger.”) The challenger (PAYMENT) strategies were the clear winners, with an average monthly improvement of $185,000 over the champion strategies during a five-month period. In addition, because the PAYMENT strategies included more “no contact” actions (don’t bother the customer this month), they led to lower collection costs and greater customer goodwill. This demonstration was very convincing. In no time, other account managers wanted to take advantage of PAYMENT.

1.5 A Successful Management Science Application

15

5. As described in step 3, the output from the PAYMENT model is a contingency plan.
The model uses a very complex optimization scheme, along with the DMMs from step 2, to decide what collection strategy to use for each type of delinquent account at each stage (month) of its delinquency. At the end of each month, after the appropriate collection methods have been used and the results (actual payments) have been observed, the model then uses the status of each account to recommend the collection method for the next month. In this way, the model is used dynamically through time. 6. In general, the analyst demonstrates the model to the client in step 6. In this application, however, the management science team members were GE’s own people—they came from GE Capital and the GE corporate research and development group.
Throughout the model-building process, the team of analysts strived to understand the requirements of the collection managers and staff—the end users—and tried to involve them in shaping the final system. This early and continual involvement, plus the impressive performance of PAYMENT in the pilot study, made it easy to “sell” the model to the people who had to use it.
7. After the pilot study, PAYMENT was applied to the $4.6 billion Montgomery Ward department store portfolio with 18 million accounts. Compared to the collection results from a year earlier, PAYMENT increased collections by $1.6 million per month, or more than $19 million per year. (This is actually a conservative estimate of the benefit obtained from PAYMENT, because PAYMENT was first applied to the Montgomery
Ward portfolio during the depths of a recession, when it’s much more difficult to collect delinquent accounts.) Since then, PAYMENT has been applied to virtually all of
GE Capital’s accounts, with similar success. Overall, GE Capital estimates that
PAYMENT has increased collections by $37 million per year and uses less resources than previous strategies. The model has since been expanded in several directions. For example, the original model assumed that collection resources (such as the amount available for live phone calls) were fixed. The expanded model treats these resource levels as decision variables in a more encompassing optimization model.

A Great Source for Management Science Applications: Interfaces
The GE Capital application is reported in the Interfaces journal, which is a highly respected bimonthly journal that chronicles real applications of management science that have generated proven benefits, often in the millions or even hundreds of millions of dollars. The applications are in a wide range of industries, are global, and employ a variety of management science techniques.
Of special interest are the January-February and (since 1999) the September-October issues. Each January-February issue contains the winner and finalists for that year’s Franz
Edelman Award for Achievement in Operations Research and the Management Sciences.
This is the profession’s most honored prize for the practice of management science. The prize is awarded for “implemented work that has had significant, verifiable, and preferably quantifiable impact.” Similarly, each September-October issue contains the winner and runners-up for that year’s Daniel H. Wagner Prize for Excellence in Operations Research
Practice. Each prize is named after a pioneer in the field of operations research and management science, and the winning papers honor them by documenting the practice of management science at its best. Many of the chapter openers and citations in this book are based on these winning articles, as well as on other Interfaces articles.
The journal is probably available from your school’s library. Alternatively, you can browse the abstracts of the articles online at http://pubsonline.informs.org. You must pay a membership fee to read the entire articles, but you can log in as a guest to read the abstracts.

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1.6 WHY STUDY MANAGEMENT SCIENCE?
After reading the previous section, you should be convinced that management science is an important area and that highly trained analysts are needed to solve the large and complex problems faced by the business world. However, unless you are one of the relatively few students who intends to become a professional management scientist, you are probably wondering why you need to study management science. This is a legitimate concern. For many years, those in the field of management science education received criticism from students and educators that management science courses were irrelevant for the majority of students who were required to take them. Looking back, it’s difficult to argue with these critics. Typical management courses were centered primarily around a collection of very specific models and, worse, a collection of mind-numbing mathematical solution techniques—techniques that students were often required to implement by hand! (Some courses are probably still taught this way, but we hope the number is decreasing rapidly.)
Two forces have helped to change this tendency toward irrelevance. First, the many vocal critics motivated many of us to examine our course materials and teaching methods.
Certain topics have been eliminated and replaced by material that is more relevant and interesting to students. The second force is the emergence of powerful computers and the accompanying easy-to-use software, especially spreadsheet software. With the availability of computers to do the number crunching, there is no need—except in advanced courses— to delve into the mathematical details of the solution techniques. We can delegate this task to machines that are far better at it than we are. We can now use the time formerly spent on such details to develop modeling skills that are valuable to a wide audience.
The intent in this book is not just to cover specific models and specific approaches to these models, but to teach a more general approach to the model-building process. The spreadsheet approach is the best way to do this because it appeals to the largest audience.
We have been teaching our own courses with this spreadsheet-modeling approach for more than a decade—to a wide range of business students—and have received very few complaints about irrelevance. In fact, many students have stated that this is the most valuable business course they have taken. The following are some of the reasons for this new-found relevance: ■

■

■

The modeling approach emphasized throughout this book is an important way to think about problems in general, not just the specific problems we discuss. This approach forces you to think logically. You must discover how given data can be used
(or which data are necessary), you must determine the elements of the problem that you can control (the decision variables), and you must determine how the elements of the problem are logically related. Students realize that these logical thinking skills are valuable for their careers, regardless of the specific fields they enter.
Management science is admittedly built around quantitative skills—it deals primarily with numbers and relationships between numbers. Some critics object that not everything in the real world can be reduced to numbers, but as one of our reviewers correctly points out, “a great deal that is of importance can.” As you work through the many models in this book, your quantitative skills will be sharpened immensely.
In a business world driven increasingly by numbers, quantitative skills are an obvious asset.
No matter what your spreadsheet abilities are when you enter this course, by the time you’re finished, you’ll be a proficient spreadsheet user. We deliberately chose the spreadsheet package Excel, which is arguably the most widely used package (other than word-processing packages) in the business world today. Many of our students state that the facility they gained in Excel was worth the price of the course. That

1.6 Why Study Management Science?

17

■

doesn’t mean this is a course in spreadsheet fundamentals and neat tricks, although you will undoubtedly pick up a few useful tricks along the way. A great spreadsheet package—and we strongly believe that Excel is the greatest spreadsheet package written to date—gives you complete control over your model. You can apply spreadsheets to an endless variety of problems. Spreadsheets give you the flexibility to work in a way that suits your style best, and spreadsheets present results (and often catch errors) almost immediately. As you succeed with relatively easy problems, your confidence will build, and before long, you’ll be able to tackle more difficult problems successfully. In short, spreadsheets enable everyone, not just technical people, to develop and use their quantitative skills.
Management science modeling helps you develop your intuition, and it also indicates where intuition alone sometimes fails. When you confront a problem, you often make an educated (or maybe not so educated) guess at the solution. If the problem is sufficiently complex, as many of the problems in this book are, this guess will be frequently wide of the mark. In this sense, the study of management science can be a humbling experience—you find that your unaided intuition is often not very good. But by studying many models and examining their solutions, you can sharpen your intuition considerably. This is sometimes called the “Aha!” effect. All of a sudden, you see why a certain solution is best. The chances are that when you originally thought about the problem, you forgot to consider an important constraint or a key relationship, and this caused your poor initial guess. Presumably, the more problems you analyze, the better you become at recognizing the critical elements of new problems. Experienced management scientists tend to have excellent intuition, the ability to see through to the essence of a problem almost immediately. However, they are not born with this talent; it comes through the kind of analysis you’ll be performing as you work through this book.

1.7 SOFTWARE INCLUDED IN THIS BOOK
Very few business problems are small enough to be solved with pencil and paper. They require powerful software. The software included in this book, together with Microsoft®
Excel, provides you with a powerful software combination that you will use for this course and beyond. This software is being used—and will continue to be used—by leading companies all over the world to solve large, complex problems. The experience you obtain with this software, through working the examples and problems in this book, will give you a key competitive advantage in the marketplace.
It all begins with Excel. All the quantitative methods that we discuss are implemented in Excel. Specifically, in this edition, we use Excel 2007. Although it’s impossible to forecast the state of computer software into the long-term or even medium-term future, as we are writing this book, Excel is the most heavily used spreadsheet package on the market, and there is every reason to believe that this state will persist for quite awhile. Most companies use Excel, most employees and most students have been trained in Excel, and Excel is a very powerful, flexible, and easy-to-use package.
Although Excel has a huge set of tools for performing numerical analysis, we have included several add-ins with this book (available on the CD-ROM inside this book and the package bundled with the book) that make Excel even more powerful. We discuss these briefly here and in much more depth in the specific chapters where they apply. Throughout the text, you will see icons that denote where each of these add-ins is used.
Together with Excel and the add-ins included in this book, you have a wealth of software at your disposal. The examples and step-by-step instructions throughout the book will help you to become a power user of this software. This takes plenty of practice and a

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Chapter 1 Introduction to Modeling

willingness to experiment, but it’s certainly within your grasp. When you are finished, don’t be surprised if you rate improved software skills as one of the most valuable things you have learned from the book.

PREMIUM
SOLVER

PREMIUM SOLVER
Excel contains a built-in add-in called Solver that is developed by Frontline Systems. This add-in is used extensively throughout the book to find optimal solutions to spreadsheet models. The version of Solver that ships with Excel is powerful and suffices for the majority of the optimization models we discuss. However, Frontline Systems has developed several other versions of Solver for the commercial market, and one of these, called Premium
Solver for Education (for Excel 2007), is included in the book. The primary advantage of
Premium Solver is that it enables you to solve some problems with a kind of algorithm
(called a genetic algorithm) that is different from the algorithms used by the built-in Solver.
Only a small percentage of all optimization models require a genetic algorithm, but in
Chapter 8 (and in a few examples in later chapters), we illustrate several very interesting models that take advantage of genetic algorithms and hence Premium Solver. These models could either not be solved with the built-in Solver, or they would require nonobvious modeling tricks to make them amenable to the built-in Solver.

Palisade Software
The Palisade Corporation has developed several powerful add-ins for Excel that we have included in this book. These are educational versions of commercial software packages used widely in the business world.

Decision Tools® Suite
Decision Tools is a collection of add-ins that Palisade sells separately or as a suite. All of the items in this suite are Excel add-ins—so the learning curve is not very steep. The four separate add-ins in this suite are @RISK, PrecisionTree, TopRank, and RISKOptimizer.5
The first two are the most important for our purposes, but all are useful for certain tasks.
@ RISK

@RISK The @RISK add-in is extremely useful for the development and analysis of spreadsheet simulation models. First, it provides a number of probability functions that enable you to build uncertainty explicitly into Excel models. Then when you run a simulation,
@RISK automatically keeps track of any outputs you select, displays the results in a number of tabular and graphical forms, and enables you to perform sensitivity analyses, so that you can see which inputs have the most effect on the outputs.

PRECISION
TREE

PrecisionTree® The PrecisionTree add-in is used in Chapter 10 to analyze decision prob-

TOP
RANK

TopRank® Although we do not use the other Palisade add-ins as extensively as @RISK

lems with uncertainty. The primary tool for performing this type of analysis is a decision tree. Decision trees are inherently graphical, and they have always been difficult to implement in spreadsheets, which are based on rows and columns. However, PrecisionTree does this in a very clever and intuitive way. Equally important, after the basic decision tree has been built, PrecisionTree makes it easy to perform sensitivity analysis on the model inputs.

and PrecisionTree, they are all worth investigating. TopRank is a “what-if” add-in used for sensitivity analysis. It starts with any spreadsheet model, where several inputs are used,
5

The Palisade suite has traditionally included two stand-alone programs, BestFit and RISKview. The functionality of both of these is now included in @RISK, so they are no longer included in the suite. As this book is going to press, we have been told by Palisade that two new add-ins, NeuralTools and Evolver, will be added to the suite, but we haven’t seen them yet, and they won’t be discussed in the book.

1.7 Software Included in This Book

19

together with spreadsheet formulas, to produce one or more outputs. TopRank then performs sensitivity analysis to see which inputs have the largest effects on the outputs. For example, it might tell you which affects after-tax profit the most: the tax rate, the riskfree rate for investing, the inflation rate, or the price charged by a competitor. Unlike @RISK,
TopRank is used when uncertainty is not explicitly built into a spreadsheet model. However, TopRank considers uncertainty implicitly by performing sensitivity analysis on the important model inputs.

RISKOptimizer RISKOptimizer combines optimization with simulation. Often, you want to use simulation to model some business problem, but you also want to optimize a summary measure, such as the mean, of an output distribution. This optimization can be performed in a trial-and-error fashion, where you try a few values of the decision variable(s) and see which provides the best solution. However, RISKOptimizer provides a more automatic (and time-intensive) optimization procedure.

STAT StatTools™
TOOLS
Palisade has also developed a statistics add-in called StatTools, which enhances the statistical capabilities of Excel. Excel’s built-in statistical tools are rather limited. It has several functions, such as AVERAGE and STDEV for summarizing data, and it includes the
Analysis ToolPak, an add-in that was developed by a third party. However, these tools are not sufficiently powerful or flexible for the heavy-duty statistical analysis that is sometimes required. StatTools provides a collection of tools that help fill this gap. Admittedly, this is not a statistics book, but StatTools will come in particularly handy in Chapter 16 when you study regression analysis and forecasting.

1.8 CONCLUSION
In this chapter, we have introduced the field of management science and the process of mathematical modeling. To provide a more concrete understanding of these concepts, we reviewed a simple queueing model and a successful management science application. We also explored a seven-step model-building process that begins with problem definition and proceeds through final implementation. Finally, we discussed why the study of management science is a valuable experience, even if you do not intend to pursue a professional career in this field.
Don’t worry if you don’t understand some of the terms, such as linear programming, that were used in this chapter. Although the seven-step process is not too difficult to comprehend, especially when discussed in the context of real applications, it typically entails some rather complex logical relationships and mathematical concepts. These ideas are presented in much greater detail in the rest of this book. Specifically, you’ll learn how to build spreadsheet models in Excel, how to use them to answer what-if questions, and how to find optimal solutions with the help of a spreadsheet Solver. For practical reasons, most of your work will take place in the classroom or in front of your own PC as you work through the examples and problems. The primary emphasis of this book, therefore, is on steps 3 through 6, that is, developing the model, testing the model with different inputs, optimizing the model, and presenting (and interpreting) the results to a client—probably your instructor. Keep in mind, however, that with real problems you must take crucial steps before and after the procedures you’ll be practicing in this book. Because real problems don’t come as nicely packaged as those we discuss and because the necessary data are seldom given to you on a platter, you’ll have to wrestle with the problem’s scope and precise data

20

Chapter 1 Introduction to Modeling

requirements when you solve problems in a real setting. (We have included “modeling problems” at the ends of most chapters. These problems are not as well structured as the
“skill” problems, so the burden is on you to determine an appropriate structure and decide the necessary data.) Also, because a mathematically accurate model doesn’t necessarily result in a successful implementation, your work is not finished just because the numbers check out. To gain acceptance for a model, an analyst must have the right combination of technical skills and people skills. Try to keep this in mind as you write up your solutions to the problems in this book. Don’t just hand in a mass of numbers with little or no explanation. Sell your solution!

1.8 Conclusion

21

This page intentionally left blank

CHAPTER

Introduction to Spreadsheet Modeling

© Dan Burn-Forti/Photographer’s Choice/Getty Images

2

ANALYSIS OF HIV/AIDS any of management science’s most successful applications are traditional functional areas of business, including operations management, logistics, finance, and marketing. Indeed, many such applications are analyzed in this book. However, another area where management science has had a strong influence over the past decade has been the analysis of the worldwide HIV/AIDS epidemic. Not only have theoretical models been developed, but even more important, they have also been applied to help understand the epidemic and reduce its spread. To highlight the importance of management science modeling in this area, an entire special issue (May–June 1998) of Interfaces, the journal that reports successful management science applications, was devoted to
HIV/AIDS models. Some of the highlights are discussed here to give you an idea of what management science has to offer in this important area.
Kahn et al. (1998) provides an overview of the problem. They discuss how governments, public-health agencies, and health-care providers must determine how best to allocate scarce resources for HIV treatment and prevention among different programs and populations. They discuss in some depth how management science models have influenced, and will continue to influence,
AIDS policy decisions. Other articles in the issue discuss more specific problems. Caulkins et al. (1998) analyze whether the distribution of difficult-toreuse syringes would reduce the spread of HIV among injection drug users.
Based on their model, they conclude that the extra expense of these types of syringes would not be worth the marginal benefit they might provide.

M

23

Paltiel and Freedberg (1998) investigate the costs and benefits of developing and administering treatments for cytomegalovirus (CMV), an infection to which HIV carriers are increasingly exposed. (Retinitis, CMV’s most common manifestation, is associated with blindness and sometimes death.) Their model suggests that the costs compare unfavorably with alternative uses of scarce resources. Owens et al. (1998) analyze the effect of women’s relapse to high-risk sexual and needle-sharing behavior on the costs and benefits of a voluntary program to screen women of childbearing age for HIV. They find, for example, that the effect of relapse to high-risk behaviors on screening program costs and benefits can be substantial, suggesting that behavioral interventions that produce sustained reductions in risk behavior, even if expensive, could be cost-saving.
The important point is that these articles (and others not mentioned here) base their results on rigorous management science models of the
HIV/AIDS phenomenon. In addition, they are backed up with real data. They are not simply opinions of the authors. ■

2.1 INTRODUCTION
This book is all about spreadsheet modeling. By the time you are finished, you’ll have seen some reasonably complex—and realistic—models. Many of you will also be transformed into Excel “power” users. However, we don’t want to start too quickly or assume too much background on your part. For practice in getting up to speed with basic Excel features, we have included an Excel tutorial in the CD-ROM that accompanies this book. (See the Excel
Tutorial.docx file.) You can work through this tutorial at your own speed and cover the topics you need help with. Even if you have used Excel extensively, give this tutorial a look.
You might be surprised how some of the tips can improve your productivity.
Second, this chapter provides an introduction to Excel modeling and illustrates some interesting and relatively simple models. The chapter also covers the modeling process and includes some of the less well known, but particularly helpful, Excel tools that are available. These tools include data tables, Goal Seek, lookup tables, and auditing commands.
Keep in mind, however, that our objective is not the same as that of the many “how-to”
Excel books on the market. Specifically, we are not teaching Excel just for its many interesting features. Rather, we plan to use these features to provide insights into real business problems. In short, Excel is a problem-solving tool, not an end in itself, in this book.

2.2 BASIC SPREADSHEET MODELING:
CONCEPTS AND BEST PRACTICES
Most mathematical models, including spreadsheet models, involve inputs, decision variables, and outputs. The inputs have given fixed values, at least for the purposes of the model. The decision variables are those a decision maker controls. The outputs are the ultimate values of interest; they are determined by the inputs and the decision variables. For example, suppose a manager must place an order for a certain seasonal product. This product will go out of date fairly soon, so this is the only order that will be made for the product.
The inputs are the fixed cost of the order; the unit variable cost of each item ordered; the price charged for each item sold; the “salvage” value for each item, if any, left in inventory after the product has gone out of date; and the demand for the product. The decision variable is the number of items to order. Finally, the key output is the profit (or loss) from the

24

Chapter 2 Introduction to Spreadsheet Modeling

Some inputs, such as demand in this example, contain a considerable degree of uncertainty. In some cases, as in
Example 2.4 later in this chapter, we model this uncertainty explicitly. product. You can also break this output into the outputs that contribute to it: the total ordering cost, the revenue from sales, and the salvage value from leftover items. You certainly have to calculate these outputs to obtain profit.
Spreadsheet modeling is the process of entering the inputs and decision variables into a spreadsheet and then relating them appropriately, by means of formulas, to obtain the outputs. After you’ve done this, you can then proceed in several directions. You might want to perform a sensitivity analysis to see how one or more outputs change as selected inputs or decision variables change. You might want to find the values of the decision variable(s) that minimize or maximize a particular output, possibly subject to certain constraints. You might also want to create charts that show graphically how certain parameters of the model are related.
These operations are illustrated with several examples in this chapter. Getting all the spreadsheet logic correct and producing useful results is a big part of the battle; however, we go further by stressing good spreadsheet modeling practices. You probaby won’t be developing spreadsheet models for your sole use; instead, you’ll be sharing them with colleagues or even a boss (or an instructor). The point is that other people will probably be reading and trying to make sense out of your spreadsheet models. Therefore, you must construct your spreadsheet models with readability in mind. Several features that can improve readability include the following:
■
■
■

■
■

■
■

A clear, logical layout to the overall model
Separation of different parts of a model, possibly across multiple worksheets
Clear headings for different sections of the model and for all inputs, decision variables, and outputs
Liberal use of range names
Liberal use of boldface, italics, larger font size, coloring, indentation, and other formatting features
Liberal use of cell comments
Liberal use of text boxes for assumptions and explanations

Obviously, the formulas and logic in any spreadsheet model must be correct; however, correctness will not take you very far if no one can understand what you’ve done. Much of the power of spreadsheets derives from their flexibility. A blank spreadsheet is like a big blank canvas waiting for you to insert useful data and formulas. Practically anything is allowed. However, you can abuse this power if you don’t have an overall plan for what should go where. Plan ahead before diving in, and if your plan doesn’t look good after you start filling in the spreadsheet, revise your plan.
The following example illustrates the process of building a spreadsheet model according to these guidelines. We build this model in stages. In the first stage, we build a model that is correct, but not very readable. At each subsequent stage, we modify the model to make it more readable. You do not need to go through each of these stages explicitly when you build your own models. You should strive for the final stage right away, at least after you get accustomed to the modeling process. The various stages are shown here simply for contrast.

EXAMPLE

2.1 O RDERING NCAA T-S HIRTS

I

t is March, and the annual NCAA Basketball Tournament is down to the final 4 teams.
Randy Kitchell is a t-shirt vendor who plans to order t-shirts with the names of the final
4 teams from a manufacturer and then sell them to the fans. The fixed cost of any order is
$750, the variable cost per t-shirt to Randy is $6, and Randy’s selling price is $10. However, this price will be charged only until a week after the tournament. After that time,

2.2 Basic Spreadsheet Modeling: Concepts and Best Practices

25

Randy figures that interest in the t-shirts will be low, so he plans to sell all remaining t-shirts, if any, at $4 each. His best guess is that demand for the t-shirts during the full-price period will be 1500. He is thinking about ordering 1400 t-shirts, but he wants to build a spreadsheet model that will let him experiment with the uncertain demand and his order quantity. How should he proceed?
Objective To build a spreadsheet model in a series of stages, all stages being correct but each stage being more readable and flexible than the previous stages.

Solution
The logic behind the model is simple. If demand is greater than the order quantity, Randy will sell all the t-shirts ordered for $10 each. However, if demand is less than the order quantity, Randy will sell as many t-shirts as are demanded at the $10 price and all leftovers at the $4 price. You can implement this logic in Excel with an IF function.
A first attempt at a spreadsheet model appears in Figure 2.1. (See the file TShirt Sales
Finished.xlsx, where we have built each stage on a separate worksheet.) You enter a possible demand in cell B3, a possible order quantity in cell B4, and then calculate the profit with the formula
=-750-6*B4+IF(B3>B4,10*B4,10*B3+4*(B4-B3))
This formula subtracts the fixed and variable costs and then adds the revenue according to the logic just described.
Figure 2.1
Base Model

1
2
3
4
5

A
B
NCAA t-shirt sales
Demand
Order
Profit

1500
1400
4850

Excel Function: IF
Excel’s IF function is probably already familiar to you, but it’s too important not to discuss.
It has the syntax =IF(condition,resultIfTrue,resultIfFalse). The condition is any expression that is either true or false. The two expressions resultIfTrue and resultIfFalse can be any expressions you would enter in a cell: numbers, text, or other Excel functions (including other IF functions). Note that if either expression is text, it must be enclosed in double quotes, such as

=IF(Score>=90,“A”,“B”)
Finally, Condition can be complex combinations of conditions, using the keywords AND or
OR. Then the syntax is, for example,

=IF(AND(Score1Order,
Selling_price*Order,Selling_price*Demand+Salvage_value*(Order-Demand))
This formula is admittedly more long-winded, but it’s certainly easier to read.
Figure 2.3
Model with Range
Names in Profit
Formula

1
2
3
4
5
6
7
8
9
10

A
NCAA t-shirt sales
Fixed order cost
Variable cost
Selling price
Discount price
Demand
Order
Profit

B

C

D

$750
$6
$10
$4

E

Range names used
Demand
Discount_price
Fixed_order_cost
Order
Selling_price
Variable_cost

1500
1400
$4,850

F

='Model 3'!$B$8
='Model 3'!$B$6
='Model 3'!$B$3
='Model 3'!$B$9
='Model 3'!$B$5
='Model 3'!$B$4

Randy might like to have profit broken down into various costs and revenues (Figure 2.4), rather than one single profit cell. The formulas in cells B12, B13, B15, and B16 are straightforward, so they are not repeated here. You then accumulate these to get profit in cell B17 with the formula
=-(B12+B13)+(B15+B16)
Figure 2.4
Model with
Intermediate
Outputs

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17

A
NCAA t-shirt sales

B

C

Fixed order cost
Variable cost
Selling price
Discount price

$750
$6
$10
$4

Demand
Order

1500
1400

Costs
Fixed cost
Variable costs
Revenues
Full-price shirts
Discount-price shirts
Profit

D

Range names used
Demand
Discount_price
Fixed_order_cost
Order
Selling_price
Variable_cost

E

F

='Model 4'!$B$8
='Model 4'!$B$6
='Model 4'!$B$3
='Model 4'!$B$9
='Model 4'!$B$5
='Model 4'!$B$4

$750
$8,400
$14,000
$0
$4,850

1
Some people refer to such numbers buried in formulas as magic numbers because they just seem to appear out of nowhere. Avoid magic numbers!

2.2 Basic Spreadsheet Modeling: Concepts and Best Practices

27

Figure 2.5
Model with
Category Labels and
Color Coding

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22

A
NCAA t-shirt sales

B

Input variables
Fixed order cost
Variable cost
Selling price
Discount price

$750
$6
$10
$4

Uncertain variable
Demand

D

F

1400

Output variables
Costs
Fixed cost
Variable costs
Revenues
Full-price shirts
Discount-price shirts
Profit

Range names used
Demand
Discount_price
Fixed_order_cost
Order
Selling_price
Variable_cost

E

1500

Decision variable
Order

C

='Model 5'!$B$10
='Model 5'!$B$7
='Model 5'!$B$4
='Model 5'!$B$13
='Model 5'!$B$6
='Model 5'!$B$5

$750
$8,400
$14,000
$0
$4,850

Of course, range names could be used for these intermediate output cells, but it’s probably more work than it’s worth. You must always use some judgment when deciding how many range names to use.
If Randy’s assistant is presented with this model, how does she know at a glance which cells contain inputs or decision variables or outputs? Labels and/or color coding can help to distinguish these types. A blue/red/gray color-coding style has been applied in Figure 2.5, along with descriptive labels in boldface. The blue cells at the top are input cells, the red cell in the middle is a decision variable, and the gray cell at the bottom is the key output.2 There is nothing sacred about this particular convention.
Feel free to adopt your own convention and style,
F U N D A M E N TA L I N S I G H T but be sure to use it consistently.
The model in Figure 2.5 is still not the last
Spreadsheet Layout and Documentation word on this problem. As shown in later examples, you could create data tables to see how sensitive
If you want your spreadsheets to be used (and you profit is to the inputs, the demand, and the order want your stock in your company to rise), give a lot of quantity. You could also create charts to show any thought to your spreadsheet layout and then docunumerical results graphically. But this is enough for ment your work carefully. For layout, think about now. You can now see that the model in Figure 2.5 whether certain data are best oriented in rows or is much more readable and flexible than the original columns, whether your work is better placed in a model in Figure 2.1. ■ single sheet or in multiple sheets, and so on. For documentation, you can use descriptive labels and headings, color coding and borders, cell comments, and text boxes to make your spreadsheets more readable. It takes time and careful planning to design and then document your spreadsheet models, but the time is well spent. And if you come back in a few days to a spreadsheet model you developed and you can’t make heads or tails of it, don’t be afraid to redesign your work completely—from the ground up.

Because good spreadsheet style is so important, an appendix to this chapter is included that discusses a few tools for editing and documenting your spreadsheet models. Use these tools right away and as you progress through the book.
In the rest of this chapter, we discuss a number of interesting examples and introduce important modeling concepts (such as sensitivity analysis), important Excel features (such as data tables), and even some important business concepts (such as

2
For users of the previous edition, we find it easier in Excel 2007 to color the cells rather than place colored borders around them, so we adopt this convention throughout the book. This color convention shows up clearly in the Excel files that accompany the book. However, in this two-color book (shades of gray and blue), it is difficult to “see” the color-coding scheme. We recommend that you look not only at the figures in the book, but at the actual Excel files.

28

Chapter 2 Introduction to Spreadsheet Modeling

net present value). To get the most from these examples, follow along at your own PC, starting with a blank spreadsheet. It’s one thing to read about spreadsheet modeling; it’s quite another to actually do it!

2.3 COST PROJECTIONS
In this next example, a company wants to project its costs of producing products, given that material and labor costs are likely to increase through time. We build a simple model and then use Excel’s charting capabilities to obtain a graphical image of projected costs.

EXAMPLE

2.2

P ROJECTING

THE

C OSTS

OF

B OOKSHELVES

AT

WOODWORKS

T

he Woodworks Company produces a variety of custom-designed wood furniture for its customers. One favorite item is a bookshelf, made from either cherry or oak. The company knows that wood prices and labor costs are likely to increase in the future. Table 2.1 shows the number of board-feet and labor hours required for a bookshelf, the current costs per board-foot and labor hour, and the anticipated annual increases in these costs. (The top row indicates that either type of bookshelf requires 30 board-feet of wood and 16 hours of labor.) Build a spreadsheet model that enables the company to experiment with the growth rates in wood and labor costs so that a manager can see, both numerically and graphically, how the costs of the bookshelves vary in the next few years.

Table 2.1

Input Data for Manufacturing a Bookshelf

Resource
Required per bookshelf
Current unit cost
Anticipated annual cost increase

Cherry

Oak

Labor

30
$7.30
2.4%

30
$4.30
1.7%

16
$18.50
1.5%

Business Objectives3 To build a model that allows Woodworks to see, numerically and graphically, how its costs of manufacturing bookshelves increase in the future and to allow the company to answer what-if questions with this model.
Excel Objectives To learn good spreadsheet practices, to enable copying formulas with the careful use of relative and absolute addresses, and to create line charts from multiple series of data.

Solution
Listing the key variables in a table before developing the actual spreadsheet model is useful, so we’ll continue to do this in many later examples (see Table 2.2.) This practice forces you to examine the roles of the variables—which are inputs, which are decision variables, and which are outputs. Although the variables and their roles are fairly clear for this example, later examples will require more thought.
3
In later chapters, we simply list the “Objective” of each example as we did in Example 2.1 in this chapter. However, because this chapter has been written to enhance basic spreadsheet skills, we separate the business objectives from the Excel objectives.

2.3 Cost Projections

29

Table 2.2

Key Variables for the Bookshelf Manufacturing Example

Input variables
Output variables

Wood and labor requirements per bookshelf, current unit costs of wood and labor, anticipated annual increases in unit costs
Projected unit costs of wood and labor, projected total bookshelf costs

The reasoning behind the model is straightforward. You first project the unit costs for wood and labor into the future. Then for any year, you multiply the unit costs by the required numbers of board-feet and labor hours per bookshelf. Finally, you add the wood and labor costs to obtain the total cost of a bookshelf.

DEVELOPING THE SPREADSHEET MODEL
The completed spreadsheet model appears in Figure 2.6 and in the file Bookshelf
Costs.xlsx.4 We develop it with the following steps.
Figure 2.6
Bookshelf Cost
Model

Always enter input values in input cells and then refer to them in Excel formulas. Do not bury input values in formulas!

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

A
B
ProjecƟng bookshelf costs at Woodworks

C

D

E

F

G

H

I

J

K

Projected Bookshelf Costs

Inputs
Requirements per bookshelf
Board-feet required
Labor hours required

Cherry
30
16

Oak
30
16

Costs of wood
Current cost per board-foot
Projected annual increase

Cherry
$7.30
2.4%

Oak
$4.30
1.7%

$600.00
$550.00
$500.00
$450.00
Cherry

$400.00

Oak

$350.00

Cost of labor
Current cost per labor hour
Projected annual increase

$300.00

$18.50
1.5%

0

1

2

3

4

5

6

Years from Now

Projected costs
Years from now
0
1
2
3
4
5
6

Cost per board-foot
Cherry
Oak
$7.30
$4.30
$7.48
$4.37
$7.65
$4.45
$7.84
$4.52
$8.03
$4.60
$8.22
$4.68
$8.42
$4.76

Cost per hour
Labor
$18.50
$18.78
$19.06
$19.35
$19.64
$19.93
$20.23

Cost per bookshelf
Cherry
Oak
$515.00
$425.00
$524.70
$431.63
$534.58
$438.37
$544.67
$445.21
$554.96
$452.16
$565.45
$459.22
$576.15
$466.39

1 Inputs. You should usually enter the inputs for a model in the upper-left corner of a worksheet as you can see in the shaded ranges in Figure 2.6, using the data from Table 2.1.
We have used our standard convention of coloring inputs—the numbers from the statement of the problem—blue. You can develop your own convention, but the input cells should be distinguished in some way. Note that the inputs are grouped logically and are explained with appropriate labels. You should always document your spreadsheet model with informational labels. Also, note that by entering inputs explicitly in input cells, you can refer to them later with Excel formulas.
2 Design output table. Think ahead of time how you want to structure your outputs.
We created a table where there is a row for every year in the future (year 0 corresponds to the current year), there are three columns for projected unit costs (columns B–D), and there are two columns for projected total bookshelf costs (columns E–F). The headings reflect this design. Of course, this isn’t the only possible design, but it works well. The important point is that you should have some logical design in mind before diving in.
4

The CD-ROM accompanying this book includes templates and completed files for all examples in the book, where all of the latter have “Finished” appended to their file names. However, especially in this chapter, we suggest that you start with a blank spreadsheet and follow the step-by-step instructions on your own.

30

Chapter 2 Introduction to Spreadsheet Modeling

Always try to organize your spreadsheet model so that you can take advantage of copying formulas across multiple cells.

3 Projected unit costs of wood. The dollar values in the range B19:F25 are all calculated from Excel formulas. Although the logic in this example is straightforward, it is still important to have a strategy in mind before you enter formulas. In particular, you should design your spreadsheet so that you can enter a single formula and then copy it whenever possible. This saves work and avoids errors. For the costs per board-foot in columns B and
C, enter the formula
=B9
in cell B19 and copy it to cell C19. Then enter the general formula
=B19*(1+B$10)
in cell B20 and copy it to the range B20:C25. We assume you know the rules for absolute and relative addresses (dollar sign for absolute, no dollar sign for relative), but it takes some planning to use these so that copying is possible. Make sure you understand why we made row 10 absolute but column B relative.

Typing dollar signs in formulas for absolute references is inefficient, so press the F4 key instead.

Excel Tip: Relative and Absolute Addresses in Formulas
Relative and absolute addresses are used in Excel formulas to facilitate copying. A dollar sign next to a column or row address indicates that the address is absolute and will not change when copying. The lack of a dollar sign indicates that the address is relative and will change when copying. After you select a cell in a formula, you can press the F4 key repeatedly to cycle through the relative/absolute possibilities, for example, =B4 (both column and row relative), =$B$4 (both column and row absolute), =B$4 (column relative, row absolute), and =$B4 (column absolute, row relative).
4 Projected unit labor costs. To calculate projected hourly labor costs, enter the formula =B13 in cell D19. Then enter the formula
=D19*(1+B$14)
in cell D20 and copy it down column D.
5 Projected bookshelf costs. Each bookshelf cost is the sum of its wood and labor costs. By a careful use of absolute and relative addresses, you can enter a single formula for these costs—for all years and for both types of wood. To do this, enter the formula
=B$5*B19+B$6*$D19
in cell E19 and copy it to the range E19:F25. The idea here is that the units of wood and labor per bookshelf are always in rows 5 and 6, and the projected unit labor cost is always in column D, but all other references must be relative to allow copying.
6 Chart. A chart is an invaluable addition to any table of data, especially in the business world, so charting in Excel is a skill worth mastering. Although not everyone agrees, the many changes Microsoft made regarding charts in Excel 2007 help you create charts more efficiently and effectively. We illustrate some of the possibilities here, but we urge you to experiment with other possibilities on your own. Start by selecting the range
E18:F25—yes, including the labels in row 18. Next, click on the Line dropdown on the
Insert ribbon and select the Line with Markers type. (The various options in the Charts group replace the Chart Wizard in previous versions of Excel.) You instantly get the basic line chart you want, with one series for Cherry and another for Oak. Also, when the chart is selected (has a border around it), you see three new Chart Tools ribbons: Design, Layout, and Format. The most important button on any of these ribbons is the Select Data

2.3 Cost Projections

31

Figure 2.7
Select Data
Dialog Box

Figure 2.8
Dialog Box for
Changing
Horizontal
Axis Labels

The many chart options are easily accessible from the three new Chart Tools ribbons in Excel 2007.
Don’t be afraid to experiment with them to produce professional-looking charts.

32

button on the Design ribbon. It lets you choose the ranges of the data for charting in case
Excel’s default choices (which are based on the selected range when you create the chart) are wrong. Click on Select Data now to obtain the dialog box in Figure 2.7. On the left, you control the one or more series being charted; on the right, you control the data used for the horizontal axis. By selecting E18:F25, you have the series on the left correct, including the names of these series (Cherry and Oak), but if you didn’t, you could select one of the series and click on Edit to change it. The data on the horizontal axis is currently the default 1, 2, and so on. We want it to be the data in column A. So click on the Edit button on the right and select the range A19:A25. (See Figure 2.8.) Your chart is now correctly labeled and charts the correct data. Beyond this, you can experiment with various formatting options to make the chart even better. For example, we rescaled the vertical axis to start at $300 rather than $0 (right-click on the numbers on the vertical axis and select Format Axis, or look at the many options under the Axes dropdown on the Layout ribbon), and we added a chart title at the top and a title for the horizontal axis at the bottom (see buttons on the Labels group on the Layout ribbon). You can spend a lot of time fine-tuning charts—maybe even too much time—but professional-looking charts are definitely appreciated.

Chapter 2 Introduction to Spreadsheet Modeling

Using the Model for What-If Questions
The model in Figure 2.6 can now be used to answer any what-if questions. In fact, many models are
The Power of Charts built for the purpose of permitting experimentation
A chart is typically much more informative to a business with various scenarios. The important point is that manager than the table of numbers it is based on. Don’t the model has been built in such a way that a manunderestimate the power of Excel charts for getting ager can enter any desired values in the input cells, your points across, and include them in your spreadsheet and all the outputs, including the chart, update automodels whenever possible. However, be prepared to do matically. As a simple example, if the annual persome investigating on your own. Excel offers an abuncentage increases for wood costs are twice as high dance of chart types and chart options to choose from, as Woodworks anticipated, you can enter these and they are not all equally suited to telling your story. higher values in row 10 and immediately see the effect, as shown in Figure 2.9. By comparing bookshelf costs in this scenario to those in the original scenario, the projected cost in year 6 for cherry bookshelves, for example, increases by about 6.5%, from $576.15 to $613.80.

F U N D A M E N TA L I N S I G H T

Figure 2.9 Effect of Higher Increases in Wood Costs

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

A carefully constructed model—with no input numbers buried in formulas—allows a manager to answer many what-if questions with a few keystrokes.

A
B
ProjecƟng bookshelf costs at Woodworks

C

D

E

F

G

H

I

J

K

Projected Bookshelf Costs

Inputs
Requirements per bookshelf
Board-feet required
Labor hours required

Cherry
30
16

Oak
30
16

Costs of wood
Current cost per board-foot
Projected annual increase

Cherry
$7.30
4.8%

Oak
$4.30
3.4%

$650.00
$600.00
$550.00
$500.00
$450.00

Cherry

$400.00

Oak

$350.00

Cost of labor
Current cost per labor hour
Projected annual increase

$300.00

$18.50
1.5%

0

1

2

3

4

5

6

Years from Now

Projected costs
Years from now
0
1
2
3
4
5
6

Cost per board-foot
Cherry
Oak
$7.30
$4.30
$7.65
$4.45
$8.02
$4.60
$8.40
$4.75
$8.81
$4.92
$9.23
$5.08
$9.67
$5.26

Cost per hour
Labor
$18.50
$18.78
$19.06
$19.35
$19.64
$19.93
$20.23

Cost per bookshelf
Cherry
Oak
$515.00
$425.00
$529.95
$433.83
$545.48
$442.87
$561.59
$452.13
$578.34
$461.62
$595.73
$471.35
$613.80
$481.32

You should appreciate by now why burying input numbers inside Excel formulas is such a bad practice. For example, if you had buried the annual increases of wood costs from row 10 in the formulas in columns B and C, imagine how difficult it would be to answer the what-if question in the previous paragraph. You would first have to find and then change all the numbers in the formulas, which is a lot of work. Even worse, it’s likely to lead to errors. ■

2.4 BREAKEVEN ANALYSIS
Many business problems require you to find the appropriate level of some activity. This might be the level that maximizes profit (or minimizes cost), or it might be the level that allows a company to break even—no profit, no loss. We discuss a typical breakeven analysis in the following example.

2.4 Breakeven Analysis

33

EXAMPLE

2.3 B REAKEVEN A NALYSIS

AT

G REAT T HREADS

T

he Great Threads Company sells hand-knitted sweaters. The company is planning to print a catalog of its products and undertake a direct mail campaign. The cost of printing the catalog is $20,000 plus $0.10 per catalog. The cost of mailing each catalog (including postage, order forms, and buying names from a mail-order database) is $0.15. In addition, the company plans to include direct reply envelopes in its mailings and incurs $0.20 in extra costs for each direct mail envelope used by a respondent. The average size of a customer order is
$40, and the company’s variable cost per order (due primarily to labor and material costs) averages about 80% of the order’s value—that is, $32. The company plans to mail 100,000 catalogs. It wants to develop a spreadsheet model to answer the following questions:
1. How does a change in the response rate affect profit?
2. For what response rate does the company break even?
3. If the company estimates a response rate of 3%, should it proceed with the mailing?
4. How does the presence of uncertainty affect the usefulness of the model?
Business Objectives To create a model to determine the company’s profit and to see how sensitive the profit is to the response rate from the mailing.
Excel Objectives To learn how to work with range names, to learn how to answer whatif questions with one-way data tables, to introduce Excel’s Goal Seek tool, and to learn how to document and audit Excel models with cell comments and the auditing toolbar.

Solution
The key variables appear in Table 2.3. Note that we have designated all variables as input variables, decision variables, or output variables. Furthermore, there is typically a key output variable, in this case, profit, that is of most concern. (In the next few chapters, we refer to it as the “target” variable.) Therefore, we distinguish this key output variable from the other output variables that we calculate along the way.

Table 2.3

Key Variables in Great Threads Problem

Input variables
Decision variable
Key output variable
Other output variables

Adopt some layout and formatting conventions, even if they differ from ours, to make your spreadsheets readable and easy to follow.

34

Various unit costs, average order size, response rate
Number mailed
Profit
Number of responses, revenue, and cost totals

The logic for converting inputs and decision variable into outputs is straightforward.
After you do this, you can investigate how the response rate affects the profit with a sensitivity analysis.
The completed spreadsheet model appears in Figure 2.10. (See the file Breakeven
Analysis.xlsx.) First, note the clear layout of the model. The input cells are colored blue, they are separated from the outputs, headings are boldfaced, several headings are indented, numbers are formatted appropriately, and a list to the right spells out all range names we have used. (See the upcoming Excel Tip on how to create this list.) Also, following the convention we use throughout the book, the decision variable (number mailed) is colored red, and the bottom-line output (profit) is colored gray.

Chapter 2 Introduction to Spreadsheet Modeling

Figure 2.10 Great Threads Model

1
2
3
4
5
6
7
8
9
10
11
12
13

A
Great Threads direct mail model

B

Catalog inputs
Fixed cost of prinƟng
Variable cost of prinƟng mailing

$20,000
$0.25

Decision variable
Number mailed

100000

Order inputs
Average order
Variable cost per order

$4 0
$32.20

We refer to this as the
Create from Selection shortcut. If you like it, you can get the dialog box in Figure 2.11 even quicker: Press
Ctrl-Shift-F3.

C

D

Model of responses
Response rate
Number of responses
Model of revenue, costs, and profit
Total Revenue
Fixed cost of prinƟng
Total variable cost of prinƟng mailing
Total variable cost of orders
Total cost
Profit

E

8%
8000

$320,000
$20,000
$25,000
$257,600
$302,600
$17,400

F

G
Range names used
Average_order
Fixed_cost_of_prinƟng
Number_mailed
Number_of_responses
Profit
Response_rate
Total_cost
Total_Revenue
Variable_cost_of_prinƟng_mailing
Variable_cost_per_order

H

I

=Model!$B$11
=Model!$B$4
=Model!$B$8
=Model!$E$5
=Model!$E$13
=Model!$E$4
=Model!$E$12
=Model!$E$8
=Model!$B$5
=Model!$B$12

Excel Tip: Creating Range Names
To create a range name for a range of cells (which could be a single cell), highlight the cell(s), click in the Name Box just to the left of the Formula Bar, and type a range name. Alternatively, if a column of labels appears next to the cells to be range-named, you can use these labels as the range names. To do this, highlight the labels and the cells to be named
(for example, A4:B5 in Figure 2.10), select the Create from Selection item on the Formulas ribbon, and make sure the appropriate box in the resulting dialog box (see Figure 2.11) is checked. The labels in our example are to the left of the cells to be named, so the Left column box should be checked. This is a very quick way to create range names, and we did it for all range names in the example. In fact, by keeping your finger on the Ctrl key, you can select multiple ranges.5 After all your ranges are selected, you can sometimes create all your range names in one step. Note that if a label contains any “illegal” range-name characters, such as a space, the illegal characters are converted to underscores.

Figure 2.11
Range Name Create
Dialog Box

If you like this tip, you can perform it even faster: Press the F3 key to bring up the
Paste Name dialog box. Excel Tip: Pasting Range Names
Including a list of the range names in your spreadsheet is often useful. To do this, select a cell (such as cell G4 in Figure 2.10), select the Use in Formula dropdown from the Formulas ribbon, and then click on the Paste List option. You get a list of all range names and their cell addresses. However, if you change any of these range names (delete one, for example), the paste list does not update automatically; you have to create it again.
5

Many users apparently believe range names are more work than they are worth. This shortcut for creating range names remedies that problem.

2.4 Breakeven Analysis

35

DEVELOPING THE SPREADSHEET MODEL
To create this model, proceed through the following steps.
1 Headings and range names. We’ve named a lot of cells, more than you might want to name, but you’ll see their value when you create formulas. In general, we strongly support range names, but it is possible to go overboard. You can waste a lot of time naming ranges that do not really need to be named. Of course, you can use the Create from Selection shortcut described previously to speed up the process.6
2 Values of input variables and the decision variable. Enter these values and format them appropriately. As usual, we’ve used our blue/red/gray color-coding scheme. Note that the number mailed has been designated as a decision variable, not as an input variable (and it’s colored red, not blue). This is because the company gets to choose the value of this variable. Finally, note that some of the values have been combined in the statement of the problem. For example, the $32.20 in cell B12 is really 80% of the $40 average order size, plus the $0.20 per return envelope. To document this process, comments appear in a few cells, as shown in Figure 2.12.

Figure 2.12 Cell Comments in Model

1
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9
10
11
12
13

A
Great Threads direct mail model

B

Catalog inputs
Fixed cost of prinƟng
Variable cost of prinƟng mailing

$20,000
$0.25

Decision variable
Number mailed

C

D

100000

E

Model of responses
Includes $0.10 for
Response rate printing and $0.15 for
Number
mailing each catalog of responses

Model of revenue, costs, and profit
Total Revenue
Fixed cost of prinƟng
Total variable
Includes 80% of the average cost of prinƟng mailing
$40 $40 order size, otal variable cost of orders
T plus $0.20 per return envelope
$32.20
Total cost
Profit

Order inputs
Average order
Variable cost per order

8%
8000

$320,000
$20,000
$25,000
$257,600
$302,600
$17,400

F

G
Range names used
Trial value,Averdo e_order will ag sensitivity analysis on
Fixed_cost_of_prinƟng
Number_mailed
Number_of_responses
Profit
Response_rate
Total_cost
Total_Revenue
Variable_cost_of_prinƟng_mailing
Variable_cost_per_order

H

I

=Model!$B$11
=Model!$B$4
=Model!$B$8
=Model!$E$5
=Model!$E$13
=Model!$E$4
=Model!$E$12
=Model!$E$8
=Model!$B$5
=Model!$B$12

Excel Tip: Inserting Cell Comments
Inserting comments in cells is a great way to document your spreadsheet models without introducing excessive clutter. To enter a comment in a cell, right-click on the cell, select the
Insert Comment item, and type your comment. This creates a little red mark in the cell, indicating a comment, and you can see the comment by resting the mouse pointer over the cell. When a cell contains a comment, you can edit or delete the comment by right-clicking on the cell and selecting the appropriate item. If you want all the cell comments to be visible (for example, in a printout as in Figure 2.12), click on the Office button, then on Excel
Options, then on the Advanced link, and select the Comment & Indicator option from the
Display group. Note that the Indicator Only option is the default.
3 Model the responses. We have not yet specified the response rate to the mailing, so enter any reasonable value, such as 8%, in the Response_rate cell. We will perform sensitivity on this value later on. Then enter the formula
=Number_mailed*Response_rate
in cell E5. (Are you starting to see the advantage of range names?)
6

We heard of one company that does not allow any formulas in its corporate spreadsheets to include cell references; they must all reference range names. This is pretty extreme, but that company’s formulas are certainly easy to read!

36

Chapter 2 Introduction to Spreadsheet Modeling

4

Model the revenue, costs, and profits. Enter the formula

=Number_of_responses*Average_order in cell E8, enter the formulas
=Fixed_cost_of_printing
=Variable_cost_of_printing_mailing*Number_mailed and =Number_of_responses*Variable_cost_per_order in cells E9, E10, and E11, enter the formula
=SUM(E9:E11)
in cell E12, and enter the formula
=Total_revenue-Total_cost
in cell E13. These formulas should all be self-explanatory, especially because of the range names used.

Figure 2.13
Data Table for Profit

Forming a One-Way Data Table
Now that a basic model has been created, we can answer the questions posed by the company. For question 1, we form a one-way data table to show how profit varies with the response rate as shown in Figure 2.13. Data tables are used often in this book, so make sure you understand how to create them. We walk you through the procedure once or twice, but from then on, you are on your own. First, enter a sequence of trial values of the response rate in column A, and enter a “link” to profit in cell B17 with the formula =Profit. This cell is shaded for emphasis, but this isn’t necessary. (In general, other outputs could be part of the table, and they would be placed in columns C, D, and so on. There would be a link to each output in row 17.) Finally, highlight the entire table range, A17:B27, and select Data Table from the What-If Analysis dropdown on the Data ribbon to bring up the dialog box in Figure 2.14. Fill it in as shown to indicate that the only input, Response_rate, is listed along a column. (You can enter either a range name or a cell address in this dialog box.)
A
B
15 QuesƟon 1 - sensiƟvity of profit to response rate
16
Response rate
Profit
17
$17,400
18
1%
-$37,200
19
2%
-$29,400
20
3%
-$21,600
21
4%
-$13,800
22
5%
-$6,000
23
6%
$1,800
24
7%
$9,600
25
8%
$17,400
26
9%
$25,200
27
10%
$33,000

C

D

E

F

Profit versus Response Rate
$40,000
$20,000
Profit

Data tables are also called what-if tables.
They let you see what happens to selected outputs if selected inputs change.

$0
-$20,000

1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

-$40,000
-$60,000

Response Rate

Figure 2.14
Data Table Dialog
Box

2.4 Breakeven Analysis

37

When you click on OK, Excel substitutes each response rate value in the table into the
Response_rate cell, recalculates profit, and reports it in the table. For a final touch, we have created a scatter chart of the values in the data table. (To do this, highlight the A18:B27 range and select the default type of Scatter from the Insert ribbon. Then you can fix it up by adding titles, removing the legend, and so on to suit your taste.)

Excel Tool: One-Way Data Table
A one-way data table allows you to see how one or more output variables vary as a single input variable varies over a selected range of values. These input values can be arranged vertically in a column or horizontally in a row. We’ll explain only the vertical arrangement because it’s the most common. To create the table, enter the input values in a column range, such as A18:A27 of Figure 2.13, and enter links to one or more output cells in columns to the right and one row above the inputs, as in cell B17 of Figure 2.13. Then highlight the entire table, beginning with the upper-left blank cell (A17 in the figure), select Data Table from the What-If Analysis dropdown on the Data ribbon, and fill in the resulting dialog box as in Figure 2.14. Leave the Row Input cell blank and use the cell where the original value of the input variable lives as the Column Input cell. When you click on OK, each value in the left column of the table is substituted into the column input cell, the spreadsheet recalculates, and the resulting value of the output is placed in the table. Also, if you click anywhere in the body of the table (B18:B27 in the figure), you will see that Excel has entered
F U N D A M E N TA L I N S I G H T the =TABLE function to remind you that a data table lives here. Note that the column input cell
The Power of Data Tables must be on the same worksheet as the table itself; otherwise, Excel issues an error.
Many Excel users (most of them?) are unaware of data tables, but they shouldn’t be. Data tables are among the most powerful and useful tools Excel has to offer. After you have developed a model that relates inputs to outputs, you can then build data tables in a matter of seconds to see how the outputs vary as key inputs vary over some range. Data tables are the perfect means for answering a large number of whatif questions quickly and easily.

As the chart indicates, profit increases in a linear manner as the response rate varies. More specifically, each percentage point increase in the response rate increases profit by $7800. Here is the reasoning. Each percentage point increase in response rate results in 100,000(0.01) ϭ 1000 more orders. Each order yields a revenue of $40, on average, but incurs a variable cost of $32.20. The net gain in profit is $7.80 per order, or $7800 for 1000 orders.

USING GOAL SEEK
The purpose of the
Goal Seek tool is to solve one equation in one unknown. Here, we find the response rate that makes profit equal to 0.

38

From the data table, you can see that profit changes from negative to positive when the response rate is somewhere between 5% and 6%. Question 2 asks for the exact breakeven point. This could be found by trial and error, but it’s easier to use Excel’s Goal Seek tool.
Essentially, Goal Seek is used to solve a single equation in a single unknown. Here, the equation is Profit=0, and the unknown is the response rate. In Excel terminology, the unknown is called the changing cell because you can change it to make the equation true.
To implement Goal Seek, select Goal Seek from the What-If Analysis dropdown on the
Data ribbon and fill in the resulting dialog box as shown in Figure 2.15. (Range names or cell addresses can be used in the top and bottom boxes, but a number must be entered in the middle box.) After you click on OK, the Response_rate and Profit cells have values
5.77% and $0. In words, if the response rate is 5.77%, Great Threads breaks even. If the response rate is greater than 5.77%, the company makes money; if the rate is less than
5.77%, the company loses money. Of course, this assumes that the company mails
100,000 catalogs. If it sends more or fewer catalogs, the breakeven response rate will change. Chapter 2 Introduction to Spreadsheet Modeling

Figure 2.15
Goal Seek Dialog
Box

Excel Tool: Goal Seek
The purpose of the Goal Seek tool is to solve one equation in one unknown. Specifically,
Goal Seek allows you to vary a single input cell to force a single output cell to a selected value. To use it, select Goal Seek from the What-If Analysis dropdown on the Data ribbon and fill in the resulting dialog box in Figure 2.15. Enter a reference to the output cell in the Set cell box, enter the numeric value you want the output cell to equal in the
To value box, and enter a reference to the input cell in the By changing cell box. Note that Goal Seek sometimes stops when the Set cell is close, but not exactly equal to, the desired value. To improve Goal Seek’s accuracy, click on the Office button, then Excel
Options, and then the Formulas link. Then check the Enable iterative calculation box and reduce Maximum Change to any desired level of precision. We chose a precision level of 0.000001. For this level of precision, Goal Seek searches until profit is within
0.000001 of the desired value, $0.

Later chapters, especially Chapters 10 through 12, deal explicitly with uncertainty. Limitations of the Model
Question 3 asks whether the company should proceed with the mailing if the response rate is only 3%. From the data table (see Figure 2.13), the apparent answer is “no” because profit is negative. However, like many companies, we are taking a short-term view with this reasoning. We should realize that many customers who respond to direct mail will reorder in the future. The company nets $7.80 per order. If each of the respondents ordered two more times, say, the company would earn 3000($7.80)(2) = $46,800 more than appears in the model, and profit would then be positive. The moral is that managers must look at the long-term impact of their decisions. However, if you want to incorporate the long term explicitly into the model, you must build a more complex model.
Finally, question 4 asks about the impact of uncertainty in the model. Obviously, not all model inputs are known with certainty. For example, the size of an order is not always $40—it might range, say, from $10 to $100. When there is a high degree of uncertainty about model inputs, it makes little sense to talk about the profit level or the breakeven response rate. It makes more sense to talk about the probability that profit will have a certain value or the probability that the company will break even. You’ll see how this can be done in the following example and in many more such examples in
Chapters 10 through 12.

Using the Formula Auditing Tool
The model in this example is fairly small and simple. Even so, we can use a little-known
Excel feature to see how all the parts fit together. This is the Formula Auditing tool, which is available on the Formulas ribbon. (It was buried in a menu in previous versions of Excel, but is now much more prominent, as it should be, in Excel 2007) See Figure 2.16.

2.4 Breakeven Analysis

39

Figure 2.16
Formula Auditing
Toolbar
The Formula Auditing tool is indispensable for untangling the logic in a spreadsheet, especially if someone else developed it!

Figure 2.17
Dependents of
Number_of_
responses Cell

Figure 2.18
Precedents of
Total_revenue Cell

The Trace Precedents and Trace Dependents buttons are probably the most useful buttons in this group. To see which formulas have direct links to the Number_of_ responses cell, select this cell and click on the Trace Dependents button. Arrows are drawn to each cell that directly depends on the number of responses, as shown in Figure 2.17. Alternatively, to see which cells are used to create the formula in the Total_ revenue cell, select this cell and click on the Trace Precedents button. Now you see that the Average_order and Number_of_responses cells are used directly to calculate revenue, as shown in Figure 2.18. Using these two buttons, you can trace your logic (or someone else’s logic) as far backward or forward as you like. When you are finished, just click on the Remove Arrows button.

1
2
3
4
5
6
7
8
9
10
11
12
13

1
2
3
4
5
6
7
8
9
10
11
12
13

A
Great Threads direct mail model

B

Catalog inputs
Fixed cost of prinƟng
Variable cost of prinƟng mailing

$20,000
$0.25

Decision variable
Number mailed

100000

Order inputs
Average order
Variable cost per order

C

$40
$32.20

A
Great Threads direct mail model

B

Catalog inputs
Fixed cost of prinƟng
Variable cost of prinƟng mailing

$20,000
$0.25

Decision variable
Number mailed

100000

Order inputs
Average order
Variable cost per order

$40
$32.20

D

Model of responses
Response rate
Number of responses

E

8%
8000

Model of revenue, costs, and profit
Total Revenue
Fixed cost of prinƟng
Total variable cost of prinƟng mailing
Total variable cost of orders
Total cost
Profit

C

$320,000
$20,000
$25,000
$257,600
$302,600
$17,400

D

E

Model of responses
Response rate
Number of responses
Model of revenue, costs, and profit
Total Revenue
Fixed cost of prinƟng
Total variable cost of prinƟng mailing
Total variable cost of orders
Total cost
Profit

8%
8000

$320,000
$20,000
$25,000
$257,600
$302,600
$17,400

Excel Tool: Formula Auditing Toolbar
The formula auditing toolbar allows you to see dependents of a selected cell (which cells have formulas that reference this cell) or precedents of a given cell (which cells are referenced in this cell’s formula). In fact, you can even see dependents or precedents that reside on a different worksheet. In this case, the auditing arrows appear as dashed lines and point to a small spreadsheet icon. By double-clicking on the dashed line, you can see a list of dependents or precedents on other worksheets. These tools are especially

40

Chapter 2 Introduction to Spreadsheet Modeling

useful for understanding how someone else’s spreadsheet works. Unlike in previous versions of Excel, the Formula Auditing tools in Excel 2007 are clearly visible on the
Formulas ribbon. ■

MODELING ISSUES
You can place charts on the same sheet as the underlying data or on separate chart sheets.The choice is a matter of personal preference. Is the spreadsheet layout in Figure 2.12 the best possible layout? This question is not too crucial because this model is so small. However, we have put all the inputs together (usually a good practice), and we have put all the outputs together in a logical order. You might want to put the answers to questions 1 and 2 on separate sheets, but with such a small model, it is arguably better to keep everything on a single sheet. We generally use separate sheets only when things start getting bigger and more complex.
One final issue is the placement of the chart. From the Chart Tools Design ribbon, you can click on the Move Chart button to select whether you want to place the chart on the worksheet (“floating” above the cells) or on a separate chart sheet that has no rows or columns. This choice depends on your personal preference—neither choice is necessarily better than the other—but for this small model, we favor keeping everything on a single sheet. Finally, we could have chosen the number mailed, rather than the response rate, as the basis for a sensitivity analysis. When running a sensitivity analysis, it’s typically based on an uncertain input variable, such as the response rate, or a decision variable that the decision maker controls. And, of course, there is no limit to the number of data tables you can create for a particular model. ■

PROBLEMS to 150,000 in increments of 10,000. Does it appear, from the results you see here, that there is an
“optimal” number to mail, from all possible values, that will maximize profit? Write a concise memo to management about your results.

Solutions for problems whose numbers appear within a color box can be found in the Student Solutions Files.
Order your copy today at the Winston/Albright product
Web site, academic.cengage.com/decisionsciences/ winston. Skill-Building Problems
1.

In the Great Threads model, the range E9:E11 does not have a range name. Open your completed Excel file and name this range Costs. Then look at the formula in cell E12. It does not automatically use the new range name. Modify the formula so that it does. Then click on cell G4 and paste the new list of range names over the previous list.

2.

The sensitivity analysis in the Great Threads example was on the response rate. Suppose now that the response rate is known to be 8%, and the company wants to perform a sensitivity analysis on the number mailed. After all, this is a variable under direct control of the company. Create a one-way data table and a corresponding XY chart of profit versus the number mailed, where the number mailed varies from 80,000

3.

Continuing the previous problem, use Goal Seek for each value of number mailed (once for 80,000, once for 90,000, and so on). For each, find the response rate that allows the company to break even. Then chart these values, where the number mailed is on the horizontal axis, and the breakeven response rate is on the vertical axis. Explain the behavior in this chart in a brief memo to management.

Skill-Extending Problem
4.

As the Great Threads problem is now modeled, if all inputs remain fixed except for the number mailed, profit will increase indefinitely as the number mailed increases. This hardly seems realistic—the company could become infinitely rich! Discuss realistic ways to modify the model so that this unrealistic behavior is eliminated. 2.4 Breakeven Analysis

41

2.5 ORDERING WITH QUANTITY DISCOUNTS
AND DEMAND UNCERTAINTY
In the following example, we again attempt to find the appropriate level of some activity: how much of a product to order when customer demand for the product is uncertain. Two important features of this example are the presence of quantity discounts and the explicit use of probabilities to model uncertain demand. Except for these features, the problem is very similar to the one discussed in Example 2.1.

EXAMPLE

2.4 O RDERING

WITH

Q UANTITY D ISCOUNTS

AT

S AM ’ S B OOKSTORE

S

am’s Bookstore, with many locations across the United States, places orders for all of the latest books and then distributes them to its individual bookstores. Sam’s needs a model to help it order the appropriate number of any title. For example, Sam’s plans to order a popular new hardback novel, which it will sell for $30. It can purchase any number of this book from the publisher, but due to quantity discounts, the unit cost for all books it orders depends on the number ordered. Specifically, if the number ordered is less than
1000, the unit cost is $24. After each 1000, the unit cost drops: to $23 for at least 1000 copies, to $22.25 for at least 2000, to $21.75 for at least 3000, and to $21.30 (the lowest possible unit cost) for at least 4000. For example, if Sam’s orders 2500 books, its total cost is $22.25(2500) ϭ $55,625. Sam’s is very uncertain about the demand for this book—it estimates that demand could be anywhere from 500 to 4500. Also, as with most hardback novels, this one will eventually come out in paperback. Therefore, if Sam’s has any hardbacks left when the paperback comes out, it will put them on sale for $10, at which price, it believes all leftovers will be sold. How many copies of this hardback novel should Sam’s order from the publisher?
Business Objectives To create a model to determine the company’s profit, given fixed values of demand and the order quantity, and then to model the demand uncertainty explicitly and to choose the expected profit maximizing (“best”) order quantity.
Excel Objectives To learn how to build in complex logic with IF formulas, to get online help about Excel functions with the fx button, to learn how to use lookup functions, to see how two-way data tables allow you to answer more extensive what-if questions, and to introduce Excel’s SUMPRODUCT function.

Solution
The key variables for this model appear in Table 2.4. Our primary modeling tasks are (1) to show how any combination of demand and order quantity determines the number of units sold, both at the regular price and at the leftover sale price, and (2) to calculate the total ordering cost for any order quantity. After we accomplish these tasks, we can model the uncertainty of demand explicitly and then choose the “best” order quantity.

Table 2.4

Key Variables for Sam’s Bookstore Problem

Input variables
Uncertain variable
Decision variable
Key output variable
Other output variables

42

Unit prices, table of unit costs specifying quantity discount structure
Demand
Order quantity
Profit
Units sold at each price, revenue, and cost totals

Chapter 2 Introduction to Spreadsheet Modeling

We first develop a spreadsheet model to calculate Sam’s profit for any order quantity and any possible demand. Then we perform a sensitivity analysis to see how profit depends on these two quantities. Finally, we indicate one possible method Sam’s might use to choose the “best” order quantity.

DEVELOPING THE SPREADSHEET MODEL
Whenever the term trial value is used for an input or a decision variable, you can be fairly sure that we will follow up with a data table or (in later chapters) by running
Solver to optimize.

The profit model appears in Figure 2.19. (See the file Quantity Discounts.xlsx.) Note that the order quantity and demand in the Order_quantity and Demand cells are “trial” values.
(Comments are in these cells as a reminder of this.) You can put any values in these cells, just to test the logic of the model. The Order_quantity cell is colored red because the company can choose its value. In contrast, the Demand cell is colored green here and in later chapters to indicate that an input value is uncertain and is being treated explicitly as such.
Also, note that a table is used to indicate the quantity discounts cost structure. You can use the following steps to build the model.

Figure 2.19 Sam’s Profit Model

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

A
B
Ordering decision with quanƟty discounts
Inputs
Unit cost - see table to right
Regular price
LeŌover price

C

$ 30
$10

Decision variable
Order quantity

2500

Uncertain quanƟty
Demand
Demand

D

E

F

Quantity discount structure
At least Unit cost
0
$24.00
1000
$23.00
2000
$22.25
3000
$21.75
4000
$21.30

2000
2000

Profit model
Units sold at regular price
Units sold at leŌover price
Revenue
Cost
Profit

G

H
I
J
Range names used:
Cost
CostLookup
Demand
Leftover_price
Order_quantity
Probabilities
Profit
Regular_price
Revenue
Units_sold_at_leŌover_price
Units_sold_at_regular_price
Units_sold_at_regular_price

K
=Model!$B$18
l!$B$18
=Model!$D$5:$E$9
=Model!$B$12
=Model!$B$6
=Model!$B$9
=Model!$B$35:$J$35
=Model!$B$19
=Model!$B$5
=Model!$B$17
=Model!$B$16
=Model!$B$15
Model!$B$15

2000
500
$65,000
$55,625
$9,375

1 Inputs and range names. Enter all inputs and name the ranges as indicated in columns H and I. Note that we used the Create from Selection shortcut to name all ranges except for CostLookup and Probabilities. For these latter two, we highlighted the ranges and entered the names in the Name Box—the “manual” method. (Why the difference? To use the Create from Selection shortcut, you must have appropriate labels in adjacent cells.
Sometimes this is simply not convenient.)
2 Revenues. The company can sell only what it has, and it sells any leftovers at the discounted sale price. Therefore, enter the formulas
=MIN(Order_quantity,Demand)
=IF(Order_quantity>Demand, Order_quantity-Demand,0) and =Units_sold_at_regular_price*Regular_price
+Units_sold_at_leftover_price*Leftover_price
in cells B15, B16, and B17. The logic in the first two of these cells is necessary to account correctly for the cases when the order quantity is greater than demand and when it’s less than or equal to demand. Note that you could use the following equivalent alternative to the
IF function:
=MAX(Order_quantity-Demand,0)

2.5 Ordering with Quantity Discounts and Demand Uncertainty

43

Excel Tool: fx Button
If you want to learn more about how an Excel function operates, use the fx button next to the
Formula bar. This is called the Insert Function button, although some people call it the
Function Wizard. If you’ve already entered a function, such as an IF function, in a cell and then click on the fx button, you get help on this function. If you select an empty cell and then click on the fx button, you can choose a function to get help on.

3 Total ordering cost. Depending on the order quantity, we find the appropriate unit cost from the unit cost table and multiply it by the order quantity to obtain the total ordering cost. This could be accomplished with a complex nested IF formula, but a much better way is to use the VLOOKUP function. Specifically, enter the formula
=VLOOKUP(Order_quantity,CostLookup,2)*Order_quantity
in cell B18. The VLOOKUP part of this formula says to compare the order quantity to the first (leftmost) column of the table in the CostLookup range and return the corresponding value in the second column (because the last argument is 2). In general, there are two important things to remember about lookup tables: (1) The leftmost column is always the column used for comparison, and (2) the entries in this column must be arranged in increasing order from top to bottom.

Excel Function: VLOOKUP
The VLOOKUP function acts like a tax table, where you look up the tax corresponding to your adjusted gross income from a table of incomes and taxes. To use it, first create a vertical lookup table, with values to use for comparison listed in the left column of the table in increasing order and corresponding output values in as many columns to the right as you like. (See the CostLookup range in Figure 2.19 for an example.) Then the VLOOKUP function typically takes three arguments: (1) the value you want to compare to the values in the left column, (2) the lookup table range, and (3) the index of the column you want the returned value to come from, where the index of the left column is 1, the index of the next column is 2, and so on. (See online help for an optional fourth argument, which is either TRUE or FALSE. The default is TRUE.) Here, “compare” means to scan down the leftmost column of the table and find the last entry less than or equal to the first argument. There is also an
HLOOKUP function that works exactly the same way, except that the lookup table is arranged in rows, not columns.
4

Profit. Calculate the profit with the formula

=Revenue-Cost

A two-way data table allows you to see how a single output varies as two inputs vary simultaneously. 44

Two-Way Data Table
The next step is to create a two-way data table for profit as a function of the order quantity and demand (see Figure 2.20). To create this table, first enter a link to the profit with the formula =Profit in cell A22, and enter possible order quantities and possible demands in column A and row 22, respectively. (We used the same values for both order quantity and demand, from 500 to 4500 in increments of 500. This is not necessary— we could let demand change in increments of 100 or even 1—but it’s reasonable. Perhaps Sam’s is required by the publisher to order in multiples of 500.) Then select Data
Table from the What-If Analysis dropdown on the Data ribbon, and enter the Demand cell as the Row Input cell and the Order_quantity cell as the Column Input cell (see Figure 2.21).

Chapter 2 Introduction to Spreadsheet Modeling

Figure 2.20 Profit as a Function of Order Quantity and Demand
A
B
C
D
E
F
21 Data table of profit as a funcƟon of order quanƟty (along side) and demand (along top) p q y( g
)
( g p)
22
$9,375
500
1000
1500
2000
2500
23
500
$3,000
$3,000
$3,000
$3,000
$3,000
24
1000
-$3,000
$7,000
$7,000
$7,000
$7,000
25
1500
-$9,500
$500 $10,500 $10,500 $10,500
26
2000
-$14,500
-$4,500
$5,500 $15,500 $15,500
27
2500
-$20,625 -$10,625
-$625
$9,375 $19,375
28
3000
-$25,250 -$15,250
-$5,250
$4,750 $14,750
29
3500
-$31,125 -$21,125 -$11,125
-$1,125
$8,875
30
4000
-$35,200 -$25,200 -$15,200
$35 200 $25 200 $15 200
-$5,200
$5 200
$4,800
$4 800
31
4500
-$40,850 -$30,850 -$20,850 -$10,850
-$850

G
3000
$3,000
$7,000
$10,500
$15,500
$19,375
$24,750
$18,875
$14,800
$14 800
$9,150

H
3500
$3,000
$7,000
$10,500
$15,500
$19,375
$24,750
$28,875
$24,800
$24 800
$19,150

I
4000
$3,000
$7,000
$10,500
$15,500
$19,375
$24,750
$28,875
$34,800
$34 800
$29,150

J
4500
$3,000
$7,000
$10,500
$15,500
$19,375
$24,750
$28,875
$34,800
$34 800
$39,150

Figure 2.21
Dialog Box for
Two-Way Data Table

Excel Tool: Two-Way Data Table
A two-way data table allows you to see how a single output cell varies as you vary two input cells. (Unlike a one-way data table, only a single output cell can be chosen.) To create this type of table, enter a reference to the output cell in the top-left corner of the table, enter possible values of the two inputs below and to the right of this corner cell, and highlight the entire table. Then select Data Table from the What-If Analysis dropdown on the
Data ribbon, and enter references to the cells where the original two input variables live.
The Row Input cell corresponds to the values along the top row of the table, and the Column Input cell corresponds to the values along the left column of the table. When you click on OK, Excel substitutes each pair of input values into these two input cells, recalculates the spreadsheet, and enters the corresponding output value in the table. By clicking on any cell in the body of the table, you can see that Excel also enters the function =TABLE as a reminder that the cell is part of a data table.

This is actually a preview of decision making under uncertainty.To calculate an expected profit, you multiply each profit by its probability and add the products.We cover this topic in depth in
Chapter 10.

The resulting data table shows that profit depends heavily on both order quantity and demand and (by scanning across rows) how higher demands lead to larger profits. But which order quantity Sam’s should select is still unclear. Remember that Sam’s has complete control over the order quantity (it can choose the row of the data table), but it has no direct control over demand (it cannot choose the column).
The ordering decision depends not only on which demands are possible, but on which demands are likely to occur. The usual way to express this information is with a set of probabilities that sum to 1. Suppose Sam’s estimates these as the values in row
35 of Figure 2.22. These estimates are probably based on other similar books it has sold in the past. The most likely demands are 2000 and 2500, with other values on both sides less likely. You can use these probabilities to find an expected profit for each order quantity. This expected profit is a weighted average of the profits in any row in the data table, using the probabilities as the weights. The easiest way to do this is to enter the formula
=SUMPRODUCT(B23:J23,Probabilities)

2.5 Ordering with Quantity Discounts and Demand Uncertainty

45

Figure 2.22 Comparison of Expected Profits
C
1000
1000
0.05

D
1500
1500
0.15

E

F

2000
2000
0.25

G

2500
2500
0.25

H

3000
3000
0.15

I

J

3500
3500
4000
4000
0.07
0.04
Sum of probabiliƟes -->

K
4500
4500
0.015
1

Expected Profit versus Order QuanƟty
Order 2000 to maximize the expected profit.
Expected Profit

A
B
33 Model of expected demands
34 Demand
500
500
35 Probability
0.025
36
37
Order quanƟty Expected profit
38
500
$3,000
39
1000
$6,750
40
1500
$9,500
41
2000
$12,250
42
2500
$11,375
43
3000
$9,500
44
$4,875
3500
45
4000
$1,350
46
4500
-$4,150
47
48
49
50
51
52

$14,000
$12,000
$10,000
$8,000
$6,000
$4,000
$2,000
$0
-$2,000
-$4,000
-$6,000

500

1000

1500

2000

2500

3000

3500

4000

4500

Order Quanity

in cell B38 and copy it down to cell B46. You also create a bar chart of these expected profits, as shown in Figure 2.22. (Excel refers to these as column charts. The height of each bar is the expected profit for that particular order quantity.)

Excel Function: SUMPRODUCT
The SUMPRODUCT function takes two range arguments, which must be exactly the same size and shape, and it sums the products of the corresponding values in these two ranges.
For example, the formula =SUMPRODUCT(A10:B11,E12:F13) is a shortcut for a formula involving the sum of 4 products: =A10*E12+A11*E13+B10*F12+B11*F13. This is an extremely useful function, especially when the ranges involved are large, and it’s used repeatedly throughout this book. (Actually, the SUMPRODUCT function can have more than two range arguments, all of the same size and shape, but the most common use of
SUMPRODUCT is when just two ranges are involved.)

The largest of the expected profits, $12,250, corresponds to an order quantity of 2000, so we would recommend that Sam’s order 2000 copies of the book. This does not guarantee that Sam’s will make a profit of $12,250—the actual profit depends on the eventual demand—but it represents a reasonable way to proceed in the face of uncertain demand.
You’ll learn much more about making decisions under uncertainty and the expected value criterion in Chapter 10. ■

PROBLEMS
Skill-Building Problems
5.

The spreadsheet model for Sam’s Bookstore contains a two-way data table for profit versus order quantity and demand. Experiment with Excel’s chart types to create a chart that shows this information graphically in an intuitive format. (Choose the format you would choose to give a presentation to your boss.)

46

Chapter 2 Introduction to Spreadsheet Modeling

6.

In some ordering problems, like the one for Sam’s
Bookstore, whenever demand exceeds existing inventory, the excess demand is not lost but is filled by expedited orders—at a premium cost to the company.
Change Sam’s model to reflect this behavior. Assume that the unit cost of expediting is $40, well above the highest “regular” unit cost.

7.

In the Sam’s Bookstore problem, the quantity discount structure is such that all the units ordered have the same unit cost. For example, if the order quantity is
2500, then each unit costs $22.25. Sometimes the quantity discount structure is such that the unit cost for the first so many items is one value, the unit cost for the next so many units is a slightly lower value, and so on. Modify the model so that Sam’s pays
$24 for units 1 to 1500, $23 for units 1501 to 2500, and $22 for units 2501 and above. For example, the total cost for an order quantity of 2750 is
1500(24) ϩ 1000(23) ϩ 250(22). (Hint: Use IF functions, not VLOOKUP.)

there is a list of possible demands, those currently in row 34. Then insert a new row below row 11 that lists the probabilities of these demands. Next, in the rows below the Profit Model label, calculate the units sold, revenue, cost, and profit for each demand. For example, the quantities in column C will be for the second possible demand. Finally, use SUMPRODUCT to calculate expected profit below the Profit row.
9.

Skill-Extending Problems
8.

The current spreadsheet model essentially finds the expected profit in several steps. It first finds the profit in cell B19 for a fixed value of demand. Then it uses a data table to find the profit for each of several demands, and finally it uses SUMPRODUCT to find the expected profit. Modify the model so that expected profit is found directly, without a data table. To do this, change row 11 so that instead of a single demand,

Continuing Problem 6, create a two-way data table for expected profit with order quantity along the side and unit expediting cost along the top. Allow the order quantity to vary from 500 to 4500 in increments of
500, and allow the unit expediting cost to vary from
$36 to $45 in increments of $1. Each column of this table will allow you to choose a “best” order quantity for a given unit expediting cost. How does this best order quantity change as the unit expediting cost increases? Write up your results in a concise memo to management. (Hint: You have to modify the existing spreadsheet model so that there is a cell for expected profit that changes automatically when you change either the order quantity or the unit expediting cost.
See Problem 8 for guidelines.)

2.6 ESTIMATING THE RELATIONSHIP BETWEEN
PRICE AND DEMAND
The following example illustrates a very important modeling concept: estimating relationships between variables by curve fitting. You’ll study this topic in much more depth in the regression discussion in Chapter 16, but we can illustrate the ideas at a relatively low level by taking advantage of some of Excel’s useful features.

EXAMPLE

2.5 E STIMATING S ENSITIVITY
AT THE L INKS C OMPANY

OF

D EMAND

TO

P RICE

The Links Company sells its golf clubs at golf outlet stores throughout the United States.
The company knows that demand for its clubs varies considerably with price. In fact, the price has varied over the past 12 months, and the demand at each price level has been observed. The data are in the data sheet of the file Golf Club Demand.xlsx (see Figure 2.23.)
For example, during the past month, when the price was $390, 6800 sets of clubs were sold.
(The demands in column C are in hundreds of units. The cell comment in cell C3 reminds you of this.) The company wants to estimate the relationship between demand and price and then use this estimated relationship to answer the following questions:
1. Assuming the unit cost of producing a set of clubs is $250 and the price must be a multiple of $10, what price should Links charge to maximize its profit?
2. How does the optimal price depend on the unit cost of producing a set of clubs?
3. Is the model an accurate representation of reality?

2.6 Estimating the Relationship Between Price and Demand

47

Figure 2.23
Demand and Price
Data for Golf Clubs

A
B
1 Demand for golf clubs
2
Month
Price
3
1
450
4
2
300
5
3
440
6
4
360
7
5
290
8
6
450
9
7
340
10
8
370
11
9
500
12
10
490
13
11
430
14
12
390
15

C

Demand
45
103
49
86
125
52
87
68
45
44
58
68

Business Objectives To estimate the relationship between demand and price, and to use this relationship to find the optimal price to charge.
Excel Objectives formatting. To illustrate Excel’s trendline tool, and to illustrate conditional

Solution
This example is divided into two parts: estimating the relationship between price and demand, and creating the profit model.

Estimating the Relationship Between Price and Demand
A scatterplot of demand versus price appears in Figure 2.24. (This can be created in the usual way with Excel’s Scatter chart.) Obviously, demand decreases as price increases, but we want to quantify this relationship. Therefore, after creating this chart, select the More
Trendline Options from the Trendline dropdown on the Chart Tools Layout ribbon to bring up the dialog box in Figure 2.25. This allows you to superimpose several different curves

Figure 2.24

130

Scatterplot of
Demand Versus
Price

120
110
100
90
80
70
60
50
40
280

48

320

Chapter 2 Introduction to Spreadsheet Modeling

360

400

440

480

520

(including a straight line) onto the scatterplot. We consider three possibilities, the linear, power, and exponential curves, defined by the following general equations (where y and x, a general output and a general input, correspond to demand and price for this example):
■
■
■

Linear: y ϭ a ϩ bx
Power: y ϭ axb
Exponential: y ϭ aebx

Before proceeding, we need to describe some general properties of these three functions because of their widespread applicability. The linear function is the easiest. Its graph is a straight line. When x changes by 1 unit, y changes by b units. The constant a is called the intercept, and b is called the slope.
The power function is a curve except in the special case where the exponent b is 1.
(Then it is a straight line.) The shape of this curve depends primarily on the exponent b. If b Ͼ 1, y increases at an increasing rate as x increases. If 0 Ͻ b Ͻ 1, y increases, but at a decreasing rate, as x increases. Finally, if b Ͻ 0, y decreases as x increases. An important property of the power curve is that when x changes by 1%, y changes by a constant percentage, and this percentage is approximately equal to b%. For example, if y ϭ 100xϪ2.35, then every 1% increase in x leads to an approximate 2.35% decrease in y.
The exponential function also represents a curve whose shape depends primarily on the constant b in the exponent. If b Ͼ 0, y increases as x increases; if b Ͻ 0, y decreases as x increases. An important property of the exponential function is that if x changes by 1 unit, y changes by a constant percentage, and this percentage is approximately equal to
100 ϫ b%. For example, if y ϭ 100eϪ0.014x, then whenever x increases by 1 unit, y decreases by approximately 1.4%. Here e is the special number 2.7182 . . . , and e to any power can be calculated in Excel with the EXP function. For example, you can calculate eϪ0.014 with the formula =EXP(-0.014).
Returning to the example, if you superimpose any of these curves on the scatterplot of demand versus price, Excel chooses the best-fitting curve of that type. Better yet, if you
Figure 2.25
More Trendline
Options Dialog Box

2.6 Estimating the Relationship Between Price and Demand

49

check the Display Equation on Chart option, you see the equation of this best-fitting curve.
Doing this for each type of curve, we obtain the results in Figures 2.26, 2.27, and 2.28. (The equations might not appear exactly as in the figures. However, they can be resized and reformatted to appear as shown.)

Figure 2.26
Best-Fitting
Straight Line

Linear Fit
130
120

y = -0.3546x + 211.3147

110
100
90
80
70
60
50
40
280

320

360

Figure 2.27
Best-Fitting
Power Curve

400

440

480

520

480

520

Power Fit
130
120

y = 5871064.2031x-1.9082

110
100
90
80
70
60
50
40
280

50

320

Chapter 2 Introduction to Spreadsheet Modeling

360

400

440

Figure 2.28

Best-Fitting Exponential Curve

ExponenƟal Fit
130
120
110

y = 466.5101e-0.0049x

100
90
80
70
60
50
40
280

320

360

400

440

480

520

Each of these curves provides the best-fitting member of its “family” to the demand/price data, but which of these three is best overall? We can answer this question by finding the mean absolute percentage error (MAPE) for each of the three curves. To do so, for any price in the data set and any of the three curves, we first predict demand by substituting the given price into the equation for the curve. The predicted demand is typically not the same as the observed demand, so we calculate the absolute percentage error (APE) with the general formula:
ΗObserved demand Ϫ Predicted demandΗ
APE ϭ ᎏᎏᎏᎏᎏ
Observed demand

(2.1)

Then we average these APE values for any curve to get its MAPE. The curve with the smallest MAPE is the best fit overall.
The calculations appear in Figure 2.29. After (manually) entering the parameters of the equations from the scatterplots into column B, proceed as follows.
1 Predicted demands. Substitute observed prices into the linear, power, and exponential functions to obtain the predicted demands in columns E, F, and G. Specifically, enter the formulas
=$B$19+$B$20*B4
=$B$22*B4^$B$23 and =$B$25*EXP($B$26*B4) in cells E19, F19, and G19, and copy them down their respective columns.

2.6 Estimating the Relationship Between Price and Demand

51

Figure 2.29

17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32

Finding the Best-Fitting Curve Overall

A
B
C
Parameters of best-fiƫng curves
Linear
Intercept
211.31
Slope
-0.3546
Power
Constant
5871064
Exponent
-1.9082
Exponential
Constant
466.51
Exponent
-0.00491

D

E
Linear
51.74
104.93
55.29
83.65
108.48
51.74
90.75
80.11
34.01
37.56
58.83
73.02

F
Prediction
Power
50.80
110.12
53.02
77.76
117.48
5 0 .8 0
86.73
73.80
41.55
43.18
55.40
66.75

G
Exponential
51.20
106.94
53.78
79.65
112.32
51.20
87.87
75.84
40.06
42.07
56.49
68.74
MAPE

H

I
J
Absolute percentage error
Linear
Power Exponential
14.98%
12.89%
13.78%
1.87%
6.91%
3.83%
12.83%
8.21%
9.75%
2.73%
9.58%
7.38%
13.22%
6.01%
10.14%
0.50%
2.31%
1.53%
4.31%
0.32%
1.00%
17.81%
8.53%
11.52%
24.42%
7.67%
10.99%
14.65%
1.86%
4.38%
1.43%
4.48%
2.61%
7.38%
1.84%
1.09%
9.68%

5.88%

6.50%

2 Average percentage errors. Apply equation (2.1) to calculate APEs in columns H, I, and J. Specifically, enter the general formula

=ABS($C4-E19)͞$C4 in cell H19 and copy it to the range H19:J30. (Do you see why column C is made absolute?
Remember that this is where the observed demands are stored.)
3 MAPE. Average the APEs in each column with the AVERAGE function to obtain the
MAPEs in row 32.
Evidently, the power curve provides the best fit, with a MAPE of 5.88%. In other words, its predictions are off, on average, by 5.88%. This power curve predicts that each 1% increase in price leads to an approximate 1.9% decrease in demand. (Economists would call this relationship elastic—demand is quite sensitive to price.)

DEVELOPING THE PROFIT MODEL
Now we move to the profit model, using the best-fitting power curve to predict demand from price. The key variables appear in Table 2.5. Note there is now one input variable, unit variable cost, and one decision variable, unit price. (The red background for the decision variable distinguishes it as such.) The profit model is straightforward to develop using the following steps (see Figure 2.30).

Table 2.5

Key Variables for Golf Club Problem

Input variable
Decision variable
Key output variable
Other output variables

52

Unit cost to produce
Unit price
Profit
Predicted demand, total revenue, total cost

Chapter 2 Introduction to Spreadsheet Modeling

Figure 2.30

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17

Profit Model

A
B
C
D
Profit model using best fiƫng power curve for esƟmaƟng demand model, E

Parameters of best-fiƫng power curve (from EsƟmaƟon sheet)
Constant
5871064
Exponent
-1.9082
Monetary inputs y p
Unit cost to produce

$250

Decision variable
Unit price (trial value)

$400

Profit model
Predicted demand
P di t d d d Total revenue
Total cost
Profit

63.601
63 601
$25,441
$15,900
$9,540

1 Predicted demand. Calculate the predicted demand in cell B14 with the formula
=B4*B11^B5
This uses the power function we estimated earlier.
2 Revenue, cost, profit. Enter the following formulas in cells B15, B16, and B17:
=B11*B14
=B8*B14 and =B15-B16
Here we assume that the company produces exactly enough sets of clubs to meet customer demand. Maximizing Profit To see which price maximizes profit, we build the data table shown in Figure 2.31. Here, the column input cell is B11 and the “linking” formula in cell B25 is =B17. The corresponding chart (a line chart) shows that profit first increases and then decreases. You can find the maximum profit and corresponding price in at least three ways. First, you can attempt to read them from the chart. Second, you can scan down the data table for the maximum profit, which is shown in the figure.
The following Excel Tip describes a third method that uses some of Excel’s more powerful features.

Excel Tip: Conditional Formatting
We colored cell B53 in Figure 2.31 because it corresponds to the maximum profit in the column, but Excel’s Conditional Formatting tool can do this for you—automatically.7 This tool was completely revised in Excel 2007 and is not only more prominent (on the Home ribbon) but much easier to use. To color the maximum profit, select the range of profits, B26:B75, select the Conditional Formatting dropdown, then Top/Bottom Rules, and then Top 10
Items to bring up the dialog box in Figure 2.32. By asking for the top 1 item, we automatically color the maximum value in the range. You should experiment with the many other
Conditional Formatting options. This is a great tool!
7

The value in cell B52 also appears to be the maximum, but to two decimals, it is slightly lower.

2.6 Estimating the Relationship Between Price and Demand

53

Profit as a
Function of Price

19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60

A
B
C
D
E
Maximum profit from data table below, with corresponding best unit price p , p g p Maximum profit
$10,409
Best price
$530
Data table for Profit as a funcƟon of Unit price
Unit price
Profit
$9,540
$260
$1,447
$270
$2,693
$280
$3,769
$3 769
$290
$4,699
$300
$5,506
$310
$6,207
$320
$6,815
$330
$7,345
$340
$7,805
$350
$8,206
$360
$8,554
$8,856
$370
$8 856
$380
$9,118
$390
$9,345
$400
$9,540
$410
$9,708
$420
$9,851
$430
$9,973
$440 $10,075
$450 $10,160
$
$ ,
$460 $10,230
$470 $10,286
$480 $10,330
$490 $10,363
$500 $10,387
$510 $10,402
$520 $10,409
$530 $10,409
$540 $10 403
$10,403
$550 $10,391
$560 $10,375
$570 $10,354
$580 $10,329
$590 $10,300
$600 $10,269

F

G

H

I

Profit versus Price
$12,000
$12 000
$10,000
$8,000
Profit

Figure 2.31

$6,000
$4,000
$2,000
$0

Price

Maximum profit

Figure 2.32
Conditional
Formatting
Dialog Box

What about the corresponding best price, shown in cell B21 of Figure 2.31? You could enter this manually, but wouldn’t it be nice if you could get Excel to find the maximum profit in the data table, determine the price in the cell to its left, and report it in cell B21, all automatically? Just enter the formula
=INDEX(A26:A75,MATCH(B20,B26:B75,0),1)
in cell B21, and the best price appears. This formula uses two Excel functions, MATCH and INDEX. MATCH compares the first argument (the maximum profit in cell B20) to the range specified in the second argument (the range of profits), and returns the index of the cell where a match appears. (The third argument, 0, specifies that we want an exact match.) In this case, the MATCH function returns 28 because the maximum profit is in the
28th cell of the profits range. Then the INDEX function is called effectively as
=INDEX(A26:A75,28,1). The first argument is the range of prices, the second is a row index, and the third is a column index. Very simply, this function says to return the value in the 28th row and first column of the prices range.
To find these functions, you can click on the fx button and examine the functions in the
Lookup & Reference category. After experimenting, we found that the INDEX and

54

Chapter 2 Introduction to Spreadsheet Modeling

MATCH combination solves the problem. You don’t have to memorize these functions, although this combination really does come in handy. Rather, you can often solve a problem by investigating some of Excel’s less well-known features. You don’t even need a manual—everything is in online help.

Sensitivity to Variable Cost
We now return to question 2 in the example: How does the best price change as the unit variable cost changes? We can answer this question with a two-way data table. Remember that this is a data table with two inputs—one along the left side and the other across the top row— and a single output. The two inputs for this problem are unit variable cost and unit price, and the single output is profit. The corresponding data table is in the range A83:F168, the top part of which appears in Figure 2.33. To develop this table, enter desired inputs in column A and row 83, enter the linking formula =B17 in cell A83 (it always goes in the top-left corner of a two-way data table), highlight the entire table, select Data Table from the What-If Analysis dropdown, and enter B8 as the Row Input cell and B11 as the Column Input cell.
Figure 2.33 Profit as a Function of Unit Cost and Unit Price
G

H

I

J

K

L

M

Maximum Profit versus Unit Cost
$20,000
Maximum Profit

A
B
C
D
E
F
77 Maximum profit for different unit costs (from data table below)
78 Unit cost
$150
$200
$250
$300
$350
79 Maximum profit
$16,552 $12,748 $10,409
$8,821
$7,669
80 Corresponding best price
$320
$420
$530
$630
$740
81
82 SensiƟvity of profit to unit cost (unit cost is along the top, unit price is along the side)
$9,540
$150
$200
$250
$300
$350
83
$160
$3,654 -$14,618 -$32,890
-$51,161
-$69,433
84
$170
$6,510
-$9,766 -$26,041
-$42,317
-$58,593
85
$180
$8,756
-$5,838 -$20,432
-$35,026
-$49,620
86
$190 $10,531
-$2,633 -$15,796
-$28,960
-$42,123
87
$200 $11,936
$0 -$11,936
-$23,872
-$35,808
88
$210 $13,050
$2,175
-$8,700
-$19,575
-$30,450
89
90
$220 $13,932
$3,980
-$5,971
-$15,922
-$25,873
91
$230 $14,627
$5,485
-$3,657
-$12,799
-$21,941
92
$240 $15,172
$6,743
-$1,686
-$10,115
-$18,543
93
$250 $15,594
$7,797
$0
-$7,797
-$15,594
94
$260 $15,917
$8,682
$1,447
-$5,788
-$13,023
95
$270 $16,157
$9,425
$2,693
-$4,039
-$10,772
96
$280 $16,330 $10,049
$3,769
-$2,512
-$8,793
97
$290 $16,447 $10,573
$4,699
-$1,175
-$7,049
98
$300 $16,518 $11,012
$5,506
$0
-$5,506
99
$310 $16,551 $11,379
$6,207
$1,034
-$4,138
$6,815
$1,947
-$2,921
100
$320 $16,552 $11,683
101
$330 $16 526 $11 935
$16,526 $11,935
$7,345
$7 345
$2,754
$2 754
-$1,836
$1 836
102
$340 $16,478 $12,142
$7,805
$3,469
-$867
103
$350 $16,412 $12,309
$8,206
$4,103
$0
104
$360 $16,331 $12,442
$8,554
$4,666
$778
105
$370 $16,237 $12,547
$8,856
$5,166
$1,476
106
$380 $16,132 $12,625
$9,118
$5,611
$2,104
107
$390 $16,020 $12,682
$9,345
$6,007
$2,670
108
$400 $15,900 $12,720
$9,540
$6,360
$3,180
109
$410 $15,775 $12,742
$9,708
$6,674
$3,640
$9,851
$6,954
$4,056
110
$420 $15,646 $12,748

$15,000
$10,000
$5,000
$0
$100

$150

$200

$250

$300

$350

Unit Cost

As before, you can scan the columns of the data table for the maximum profits and enter them (manually) in rows 79 and 80. (Alternatively, you can use the Excel features described in the previous Excel Tip to accomplish these tasks. Take a look at the finished version of the file for details. This file also explains how conditional formatting is used to color the maximum profit in each column of the table.) Then you can create a chart of maximum profit (or best price) versus unit cost. The chart in Figure 2.33 shows that the maximum profit decreases, but at a decreasing rate as the unit cost increases. Limitations of the Model
Question 3 asks you to step back from all these details and evaluate whether the model is realistic. First, there is no real reason to restrict golf club prices to multiples of $10. This
2.6 Estimating the Relationship Between Price and Demand

55

was only required so that we could use a data table to find the profit-maximizing price.
Ideally, we want to search over all possible prices to find the profit-maximizing price. Fortunately, Excel’s built-in Solver tool enables us to accomplish this task fairly easily. The problem of finding a profit-maximizing price is an example of an optimization model. In optimization models, we try to maximize or minimize a specified output cell by changing the values of the decision variable cells. Chapters 3 through 9 contain a detailed discussion of optimization models.
A second possible limitation of our model is the implicit assumption that price is the only factor that influences demand. In reality, other factors, such as advertising, the state of the economy, competitors’ prices, strength of competition, and promotional expenses, also influence demand. In Chapter 16, you’ll learn how to use multiple regression to analyze the dependence of one variable on two or more other variables. This technique allows you to incorporate other factors into the model for profit.
A final limitation of the model is that demand might not equal sales. For example, if actual demand for golf clubs during a year is 70,000 but the company’s annual capacity is only 50,000, the company would observe sales of only 50,000. This would cause it to underestimate actual demand, and the curve-fitting method would produce biased predictions. (Can you guess the probable effect on pricing decisions?)
As these comments indicate, most models are not perfect, but we have to start somewhere!

Other Modeling Issues
The layout of the Golf Club Demand.xlsx file is fairly straightforward. However, note that instead of a single sheet, there are two sheets, partly for logical purposes and partly to reduce clutter. There is one sheet for estimation of the demand function and the various scatterplots, and there is another for the profit model.
One last issue is the placement of the data tables for the sensitivity analysis. You might be inclined to put these on a separate Sensitivity sheet. However, Excel does not allow you to build a data table on one sheet that uses a row or column input cell from another sheet.
Therefore, you are forced to put the data tables on the same sheet as the profit model. ■

PROBLEMS
Skill-Building Problems
10. Suppose you have an extra 6 months of data on demands and prices, in addition to the data in the example. These extra data points are (350,84), (385,72),
(410,67), (400,62), (330,92), and (480,53). (The price is shown first and then the demand at that price.) After adding these points to the original data, use Excel’s
Trendline tool to find the best-fitting linear, power, and exponential trendlines. Finally, calculate the MAPE for each of these, based on all 18 months of data. Does the power curve still have the smallest MAPE?
11. Consider the power curve y ϭ 10000xϪ2.35. Calculate y when x ϭ 5; when x ϭ 10; and when x ϭ 20. For each of these values of x, find the percentage change in y

56

Chapter 2 Introduction to Spreadsheet Modeling

when x increases by 1%. That is, find the percentage change in y when x increases from 5 to 5.05; when it increases from 10 to 10.1; and when it increases from
20 to 20.2. Is this percentage change constant? What number is it very close to? Write a brief memo on what you have learned about power curves from these calculations. 12. Consider the exponential curve y ϭ 1000eϪ0.014x.
Calculate y when x ϭ 5; when x ϭ 10; and when x ϭ 20. For each of these values of x, find the percentage change in y when x increases by 1 unit.
That is, find the percentage change in y when x increases from 5 to 6; when it increases from 10 to 11; and when it increases from 20 to 21. Is this percentage

change constant? When expressed as a decimal, what number is it very close to? Write a brief memo on what you have learned about exponential curves from these calculations. Skill-Extending Problems

lowest MAPE. However, the exponential curve was not far behind. Rework the profit model using the exponential curve to relate demand to price. Write a brief memo indicating whether you get basically the same results as with the power curve or you get substantially different results.

13. In the profit model in this section, we used the power curve to relate demand and price because it has the

2.7 DECISIONS INVOLVING THE TIME VALUE OF MONEY
In many business situations, cash flows are received at different points in time, and a company must determine a course of action that maximizes the “value” of cash flows. Here are some examples:
■

Should a company buy a more expensive machine that lasts for 10 years or a less expensive machine that lasts for 5 years?

■

What level of plant capacity is best for the next 20 years?
A company must market one of several midsize cars. Which car should it market?

■

To make decisions when cash flows are received at different points in time, the key concept is that the later a dollar is received, the less valuable the dollar is. For example, suppose you can invest money at a 10% annual interest rate. Then $1.00 received now is essentially equivalent to $1.10 a year from now. The reason is that if you have $1.00 now, you can invest it and gain $0.10 in interest in one year. If r ϭ 0.10 is the interest rate (expressed as a decimal), we can write this as
$1.00 now ϭ $1.10 a year from now ϭ $1.00(1 ϩ r)

(2.2)

Dividing both sides of equation (2.2) by 1 ϩ r, we can rewrite it as
$1.00 ϫ 1͞(1 ϩ r)now ϭ $1.00 a year from now

(2.3)

The value 1͞(1 ϩ r) in equation (2.3) is called the discount factor, and it’s always less than 1. The quantity on the left, which evaluates to $0.909 for r ϭ 0.10, is called the present value of $1.00 received a year from now. The idea is that if you had $0.909 now, you could invest it at 10% and have it grow to $1.00 in a year.
In general, if money can be invested at annual rate r compounded each year, then $1 received t years from now has the same value as 1͞(1 ϩ r)t dollars received today—that is, the $1 is discounted by the discount factor raised to the t power. If you multiply a cash flow received t years from now by 1͞(1 ϩ r)t to obtain its present value, then the total of these present values over all years is called the net present value (NPV) of the cash flows. Basic financial theory states that projects with positive NPVs increase the value of the company, whereas projects with negative NPVs decrease the value of the company.
The rate r (usually called the discount rate) used by major corporations generally comes from some version of the capital asset pricing model. Most companies use a discount rate ranging from 10% to 20%. The following example illustrates how spreadsheet models and the time value of money can be used to make complex business decisions.
The discount factor is 1 divided by (1 plus the discount rate). To discount a cash flow that occurs t years from now, multiply it by the discount factor raised to the t power. The NPV is the sum of all discounted cash flows.

2.7 Decisions Involving the Time Value of Money

57

F U N D A M E N TA L I N S I G H T
The Time Value of Money
Money earned in the future is less valuable than money earned today, for the simple reason that money earned today can be invested to earn interest.
Similarly, costs incurred in the future are less “costly” than costs incurred today, which is why you don’t simply sum up revenues and costs in a multiperiod

EXAMPLE

2.6 C ALCULATING NPV

AT

model. You instead discount future revenues and costs to put them on an even ground with revenues and costs incurred today. The resulting sum of discounted cash flows is the net present value (NPV), and it forms the cornerstone of much of financial theory and applications.

A CRON

A

cron is a large drug company. At the current time, the beginning of year 0, Acron is trying to decide whether one of its new drugs, Niagra, is worth pursuing. Niagra is in the final stages of development and will be ready to enter the market in year 1. The final cost of development, to be incurred at the beginning of year 1, is $9.3 million. Acron estimates that the demand for Niagra will gradually grow and then decline over its useful lifetime of 20 years. Specifically, the company expects its gross margin (revenue minus cost) to be $1.2 million in year 1, then to increase at an annual rate of 10% through year 8, and finally to decrease at an annual rate of 5% through year 20. Acron wants to develop a spreadsheet model of its 20-year cash flows, assuming its cash flows, other than the initial development cost, are incurred at the ends of the respective years.8 Using an annual discount rate of 12% for the purpose of calculating NPV, the drug company wants to answer the following questions:
1. Is the drug worth pursuing, or should Acron abandon it now and not incur the $9.3 million development cost?
2. How do changes in the model inputs change the answer to question 1?
3. How realistic is the model?
Business Objectives To develop a model that calculates the NPV of Acron’s cash flows, to use this model to determine whether the drug should be developed further and then marketed, and to see how sensitive the answer to this question is to model parameters.
Excel Objectives To illustrate efficient selection and copying of large ranges and to learn
Excel’s NPV function.

Solution
The key variables in Acron’s problem appear in Table 2.6. The first two rows contain the inputs stated in the problem. We have made a judgment call as to which of these are known with some certainty and which are uncertain. Although we won’t do so in this chapter, a thorough study of Acron’s problem would treat this uncertainty explicitly, probably with simulation. For now, we accept the values given in the statement of the problem and leave the simulation for a later chapter.

8

To simplify the model, taxes are ignored.

58

Chapter 2 Introduction to Spreadsheet Modeling

Table 2.6

Key Variables for Acron’s Problem

Input variables

Development cost, first year gross margin, rate of increase during early years, years of growth, rate of decrease in later years, discount rate
NPV
Yearly gross margins

Key output variable
Other calculated variables

The model of Acron’s cash flows appears in Figure 2.34. As with many financial spreadsheet models that extend over a multiyear period, we enter “typical” formulas in the first year or two and then copy this logic down to all years. (In the previous edition, we made the years go across, not down. In that case, splitting the screen is useful so that you can see the first and last years of data. Splitting the screen is explained in the following
Excel Tip. The reason we modified the model to have the years go down, not across, is that it now fits easily on a screen, without needing to split the screen.)
Figure 2.34
Acron’s Model of
20-Year NPV

1
2
3
4
5
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9
10
11
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13
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15
16
17
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19
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24
25
26
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34

A
CalculaƟng NPV at Acron
Inputs
Development cost
Gross margin year 1
Rate of increase
Increase through year
Rate of decrease
Discount rate
Cash flows
End of year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
N PV

B

C

9.3
1.2
10%
8
5%
12 %

D
Range names used:
Development_cost
Discount_rate
Gross_margin_year_1
Gross_margin
Increase_through_year
Rate_of_decrease
Rate_of_increase

E

F

G

=Model!$B$4
=Model!$B$9
=Model!$B$5
=Model!$B$13:$B$32
=Model!$B$7
=Model!$B$8
=Model!$B$6

Gross margin
1.2000
1.3200
1.4520
1.5972
1.7569
1.9326
2.1259
2.3385
2.2215
2.1105
2.0049
1.9047
1.8095
1.7190
1.6330
1.5514
1.4738
1.4001
1.3301
1.2636
3.3003

Excel Tip: Splitting the Screen
To split the screen horizontally, drag the separator just to the right of the bottom scrollbar to the left. To split the screen vertically, drag the separator just above the right scrollbar downward. Drag either separator back to its original position to remove the split.

DEVELOPING THE SPREADSHEET MODEL
To create the model, complete the following steps. (See the file Calculating NPV.xlsx.)
1 Inputs and range names. Enter the given input data in the blue cells, and name the ranges as shown. As usual, note that the range names for cells B4 through B9 can be created all at once with the Create from Selection shortcut, as can the range name for the gross margins in column B. In the latter case, highlight the whole range B12:B32 and then use the Create from Selection shortcut.

2.7 Decisions Involving the Time Value of Money

59

2 Cash flows. Start by entering the formula
=Gross_margin_year_1
in cell B13 for the year 1 gross margin. Then enter the general formula
=IF(A140,$B$14*B38,0)
in cell B46 and copy it across. For shortage costs, enter the formula
=IF(B38=
>=
>=
>=
$15,000 $16,000 $18,000 $20,000 $21,000 $22,000 $24,000 $25,000 $30,000 $31,000 $31,000 $31,000

=Model!$B$20:$P$20
=Model!$B$22:$P$22
=Model!$B$12:$B$14
=Model!$B$16

Always document your spreadsheet conventions as clearly as possible. 180

The value in cell B16 is the money allocated to make the 2008 payment and buy bonds in 2008. It is both a changing cell and the target cell to minimize.

1 Inputs and range names. Enter the given data in the blue cells and name the ranges as indicated. Note that the bond costs in the range B5:B7 are entered as positive quantities.
Some financial analysts might prefer that they be entered as negative numbers, indicating outflows. It doesn’t really matter, however, as long as we are careful with spreadsheet formulas later on.
2 Money allocated and bonds purchased. As we discussed previously, the money allocated in 2008 and the numbers of bonds purchased are both decision variables, so enter any values for these in the Money_allocated and Bonds_purchased ranges. Note that the colorcoding convention is modified for the Money_allocated cell. Because it is both a changing cell and the target cell, we color it red but add a note to emphasize that it is the objective to maximize. Chapter 4 Linear Programming Models

3 Cash available to make payments. In 2008, the only cash available is the money initially allocated minus cash used to purchase bonds. Calculate this quantity in cell B20 with the formula
=Money_allocated-SUMPRODUCT(Bonds_purchased,B5:B7)
For all other years, the cash available comes from two sources: excess cash invested at the fixed interest rate the year before and payments from bonds. Calculate this quantity for 2009 in cell C20 with the formula
=(B20-B22)*(1+$B$9)+SUMPRODUCT(Bonds_purchased,C5:C7)
and copy it across row 20 for the other years.
As you see, this model is fairly straightforward to develop after you understand the role of the amount allocated in cell B16. However, we have often given this problem as an assignment to our students, and many fail to deal correctly with the amount allocated.
(They usually forget to make it a changing cell.) So make sure you understand what we have done and why we have done it this way.

SOLVER

USING SOLVER
The main Solver dialog box should be filled out as shown in Figure 4.36. As usual, the
Assume Linear Model and Assume Non-Negative options should be checked before optimizing. Again, notice that the Money_allocated cell is both the target cell and one of the changing cells.

Figure 4.36
Solver Dialog Box for the Pension
Model

Discussion of the Solution
The optimal solution appears in Figure 4.35. You might argue that the numbers of bonds purchased should be constrained to integer values. We tried this and the optimal solution changed very little: The optimal numbers of bonds to purchase changed to 74, 79, and 27, and the optimal money to allocate increased to $197,887. With this integer solution, shown in Figure 4.37, James sets aside $197,887 initially. Any less than this would not work—he couldn’t make enough from bonds to meet future pension payments. All but $20,387 of this initial allocation (see cell B20) is spent on bonds, and of the $20,387, $11,000 is used to make the 2008 pension payment. After this, the amounts in row 20, which are always sufficient to make the payments in row 22, are composed of returns from bonds and cash with interest from the previous year. Even more so than in previous examples, there’s no way to
“guess” this optimal solution. The timing of bond returns and the irregular pension payments make an optimization model an absolute necessity.

4.7 Financial Models

181

Figure 4.37 Optimal Integer Solution for the Pension Model
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Pension fund management
Costs (in 2008) and income (in other years) from bonds
2008
2009
Year
2010
$6 0
Bond 1
$980
$60
$970
$6 5
Bond 2
$ 65
Bond 3
$1,050
$75
$75
Interest rate

2011
$60
$65
$75

2012
$60
$65
$75

2013
$1,060
$65
$75

2014

2015

2016

2017

2018

$65
$75

$65
$75

$65
$75

$ 65
$75

$65
$75

2019
$1,065
$75

2020

2021

$75

$75

2022

$1,075

4%

Number of bonds (allowing fracƟonal values) to purchase in 2008
Bond 1
74.00
79.00
Bond 2
27.00
Bond 3
Money allocated

$197,887

Constraints to meet payments
Year
2008
2009
2010
$20,387 $21,363 $21,337
Amount available
>=
>=
>=
$11,000 $12,000 $14,000
Amount required
Range names used:
Amount_available
Amount_required
Bonds_purchased
Money_allocated

The value in cell B16 is the money allocated to make the 2008 payment and buy bonds in 2008. It is both a changing cell and the target cell to minimize.

ObjecƟve to minimize, also a changing cell

2011
2012
2013
20 14
2015
2016
2017
2018
2019
2020
2021
2022
$19,231 $16,000 $85,600 $77,464 $66,923 $54,919 $41,396 $25,252 $86,422 $60,704 $32,917 $31,019
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
$15,000 $16,000 $18,000 $20,000 $21,000 $22,000 $24,000 $25,000 $30,000 $31,000 $31,000 $31,000

=Model!$B$20:$P$20
=Model!$B$22:$P$22
=Model!$B$12:$B$14
=Model!$B$16

Sensitivity Analysis
Because the bond information and pension payments are evidently fixed, we see only one promising direction for sensitivity analysis: on the fixed interest rate in cell B9. We tried this, allowing this rate to vary from 2% to 6% in increments of 0.5%, and we kept track of the optimal changing cells, including the target cell. The results appear in Figure 4.38.
They indicate that as the interest rate increases, James can get by with fewer bonds of types
1 and 2, and he can allocate less money for the problem. The reason is that he is making more interest on excess cash.
Figure 4.38
Sensitivity to Fixed
Interest Rate

A
30 SensiƟvity to interest rate
31
2.0%
32
2.5%
33
3.0%
34
3.5%
35
4.0%
36
4.5%
37
5.0%
38
5.5%
39
6.0%
40

B

$B$12
77.12
76.24
75.37
74.53
73.69
72.88
72.09
71.30
70.54

C

D

$B$13
78.71
78.33
77.95
77.58
77.21
76.84
76.49
76.13
75.78

$B$14
28.84
28.84
28.84
28.84
28.84
28.84
28.84
28.84
28.84

E

$B$16
$202,010
$200,930
$199,863
$198,809
$197,768
$196,741
$195,727
$194,725
$193,737

■

A D D I T I O N A L A P P L I C AT I O N S
Using LP to Optimize Bond Portfolios
Many Wall Street firms buy and sell bonds. Rohn (1987) developed a bond selection model that maximizes profit from bond purchases and sales subject to constraints that minimize the firm’s risk exposure. The method used to model this situation is closely related to the method used to model the Barney-Jones problem. ■

182

Chapter 4 Linear Programming Models

PROBLEMS
Skill-Building Problems
30. Modify the Barney-Jones investment problem so that a minimum amount must be put into any investment, although this minimum can vary by investment. For example, the minimum amount for investment A might be $0, whereas the minimum amount for investment D might be $50,000. These minimum amounts should be inputs; you can make up any values you like. Run
Solver on your modified model.
31. In the Barney-Jones investment problem, increase the maximum amount allowed in any investment to
$150,000. Then run a one-way sensitivity analysis to the money market rate on cash. Capture one output variable: the maximum amount of cash ever put in the money market. You can choose any reasonable range for varying the money market rate.
32. We claimed that our model for Barney-Jones is generalizable. Try generalizing it to the case where there are two more potential investments, F and G. Investment
F requires a cash outlay in year 2 and returns $0.50 in each of the next 4 years for every dollar invested.
Investment G requires a cash outlay in year 3 and returns $0.75 in each of years 5, 6, and 7 for every dollar invested. Modify the model as necessary, making the objective the final cash after year 7.
33. In the Barney-Jones spreadsheet model, we ran investments across columns and years down rows. Many financial analysts seem to prefer the opposite. Modify the spreadsheet model so that years go across columns and investments go down rows. Run Solver to ensure that your modified model is correct! (There are two possible ways to do this, and you can experiment to see which you prefer. First, you could basically start over on a blank worksheet. Second, you could use
Excel’s TRANSPOSE function.)
34. In the pension fund problem, suppose there’s a fourth bond, bond 4. Its unit cost in 2006 is $1020, it returns coupons of $70 in years 2007 to 2010 and a payment of $1070 in 2011. Modify the model to incorporate this extra bond and reoptimize. Does the solution change— that is, should James purchase any of bond 4?
35. In the pension fund problem, suppose there is an upper limit of 60 on the number of bonds of any particular type that can be purchased. Modify the model to incorporate this extra constraint and then reoptimize.
How much more money does James need to allocate initially? 36. In the pension fund problem, suppose James has been asked to see how the optimal solution will change if the required payments in years 2011 to 2020 all

increase by the same percentage, where this percentage could be anywhere from 5% to 25%. Use an appropriate one-way SolverTable to help him out, and write a memo describing the results.
37. The pension fund model is streamlined, perhaps too much. It does all of the calculations concerning cash flows in row 20. James decides he would like to
“break these out” into several rows of calculations:
Beginning cash (for 2006, this is the amount allocated; for other years, it is the unused cash, plus interest, from the previous year), Amount spent on bonds
(positive in 2006 only), Amount received from bonds
(positive for years 2007 to 2020 only), Cash available for making pension fund payments, and (below the
Amount required row) Cash left over (amount invested in the fixed interest rate). Modify the model by inserting these rows, enter the appropriate formulas, and run Solver. You should obtain the same result, but you get more detailed information.

Skill-Extending Problems
38. Suppose the investments in the Barney-Jones problem sometimes require cash outlays in more than one year. For example, a $1 investment in investment
B might require $0.25 to be spent in year 1 and $0.75 to be spent in year 2. Does our model easily accommodate such investments? Try it with some cash outlay data you make up, run Solver, and interpret your results.
39. In the pension fund problem, if the amount of money initially is less than the amount found by Solver, then
James will not be able to meet all of the pension fund payments. Use the current model to demonstrate that this is true. To do so, enter a value less than the optimal value into cell B16. Then run Solver, but remove the Money_allocated cell as a changing cell and as the target cell. (If there is no target cell, Solver simply tries to find a solution that satisfies all of the constraints.) What do you find?
40. Continuing the previous problem in a slightly different direction, continue to use the Money_allocated cell as a changing cell, and add a constraint that it must be less than or equal to any value, such as $195,000, that is less than its current optimal value. With this constraint, James will again not be able to meet all of the pension fund payments. Create a new target cell to minimize the total amount of payments not met. The easiest way to do this is with IF functions. Unfortunately, this makes the model nonsmooth, and Solver might have trouble finding the optimal solution. Try it and see.

4.7 Financial Models

183

4.8 DATA ENVELOPMENT ANALYSIS (DEA)
The data envelopment analysis (DEA) method can be used to determine whether a university, hospital, restaurant, or other business is operating efficiently. Specifically, DEA can be used by inefficient organizations to benchmark efficient and “best-practice” organizations. According to Sherman and Ladino (1995):
Many managers of service organizations would describe benchmarking and best practice analysis as basic, widely accepted concepts already used in their businesses.
Closer examination indicates that the traditional techniques used to identify and promulgate best practices are not very effective, largely because the operations of these service organizations are too complex to allow them to identify best practices accurately. DEA provides an objective way to identify best practices in these service organizations and has consistently generated new insights that lead to substantial productivity gains that were not otherwise identifiable.
The following example illustrates DEA and is based on Callen (1991). See also Norton
(1994).

EXAMPLE

4.8 DEA

IN THE

H OSPITAL I NDUSTRY

C

onsider a group of three hospitals. To simplify matters, assume that each hospital
“converts” two inputs into three different outputs. (In a real DEA, there are typically many more inputs and outputs.) The two inputs used by each hospital are input 1 ϭ capital (measured by hundreds of hospital beds) input 2 ϭ labor (measured by thousands of labor hours used in a month)
The outputs produced by each hospital are output 1 ϭ hundreds of patient-days during month for patients under age 14 output 2 ϭ hundreds of patient-days during month for patients between 14 and 65 output 3 ϭ hundreds of patient-days for patients over 65
The inputs and outputs for these hospitals are given in Table 4.12. Which of these three hospitals is efficient in terms of using its inputs to produce outputs?

Table 4.12

Input and Output for the Hospital Example
Inputs

Outputs

1
Hospital 1
Hospital 2
Hospital 3

2

1

2

3

5
8
7

14
15
12

9
5
4

4
7
9

16
10
13

Objective To develop an LP spreadsheet model, using the DEA methodology, to determine whether each hospital is efficient in terms of using its inputs to produce its outputs.

184

Chapter 4 Linear Programming Models

WHERE DO THE NUMBERS COME FROM?
In a general DEA analysis, the organization’s inputs and outputs must first be defined. Then for each input or output, a unit of measurement must be selected. Neither of these is necessarily an easy task, because organizations such as hospitals, banks, and schools consume a variety of inputs and produce a variety of outputs that can be measured in alternative ways.
However, after the list of inputs and outputs has been chosen and units of measurement have been selected, we can use accounting data to find the required data, as in Table 4.12.

Solution
The idea is that when focusing on any particular hospital, we want to show it in the best possible light. That is, we want to value the inputs and outputs in such a way that this hospital looks as good as possible relative to the other hospitals. Specifically, to determine whether a hospital is efficient, we define a price per unit of each output and a cost per unit of each input. Then the efficiency of a hospital is defined to be
Value of hospital’s outputs
Efficiency of hospital ϭ ᎏᎏᎏ
Value of hospital’s inputs
The DEA approach uses the following four ideas to determine whether a hospital is efficient. ■

No hospital can be more than 100% efficient. Therefore, the efficiency of each hospital is constrained to be less than or equal to 1. To make this a linear constraint, we express it in this form:
Value of hospital’s outputs Յ Value of hospital’s inputs

■

■

■

When we are trying to determine whether a hospital is efficient, it simplifies matters to scale input prices so that the value of the hospital’s inputs equals 1. Any other value would suffice, but using 1 causes the efficiency of the hospital to be equal to the value of the hospital’s outputs.
If we are interested in evaluating the efficiency of a hospital, we should attempt to choose input costs and output prices that maximize this hospital’s efficiency. If the hospital’s efficiency equals 1, then the hospital is efficient; if the hospital’s efficiency is less than 1, then the hospital is inefficient.
All input costs and output prices must be nonnegative.

Putting these ideas together, the variables required for the DEA model are summarized in
Table 4.13. Note the reference to “selected hospital.” The model is actually analyzed three times, once for each hospital. So the selected hospital each time is the one currently in focus. Table 4.13

Variables and Constraints for the DEA Model

Input variables
Decision variables
(changing cells)
Objective (target cell)
Other calculated variables
Constraints

Inputs used, outputs produced for each hospital
Unit costs of inputs, unit prices of outputs for selected hospital
Total output value of selected hospital
Total input cost, total output value (for each hospital)
Total input cost must be greater than or equal to Total output value
(for each hospital)
Total cost for selected hospital must equal 1

4.8 Data Envelopment Analysis (DEA)

185

DEVELOPING THE SPREADSHEET MODEL
Figure 4.39 contains the DEA spreadsheet model used to determine the efficiency of hospital 1. (See the file Hospital DEA.xlsm.) To develop this model, proceed as follows.

Figure 4.39 DEA Model for Hospital 1
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22

B

C

D

E

F

G

H

DEA model for checking efficiency of a selected hospital
Selected hospital
Inputs used
Hospital 1
Hospital 2
Hospital 3
Unit costs of inputs

1
Input 1
5
8
7

Input 2
14
15
12

0.000

0.071

Constraints that input costs must cover output values
Hospital Input costs
1
1.000
>=
2
1.071
>=
3
0.857
>=

Outputs produced
Hospital 1
Hospital 2
Hospital 3
Unit prices of outputs

Output 1
9
5
4
0.0857

Output 2 Output 3
4
16
7
10
9
13
0.0571

I

J

Range names used p _
Input_costs
Output_values
Selected_hospital
Selected_hospital_input_cost
Selected_hospital_output_value
Unit_costs_of_inputs
Unit_prices_of_outputs

K

$ $ $ $
=Model!$B$14:$B$16
=Model!$D$14:$D$16
=Model!$B$3
=Model!$B$19
=Model!$B$22
=Model!$B$10:$C$10
=Model!$F$10:$H$10

0.000

Output values
1.000
0.829
0.857

Constraint that selected hospital's input cost must equal a nominal value of 1
Selected hospital input cost
1.000
=
1
Maximize selected hospital's output value (to see if it is 1, hence efficient)
Selected hospital output value
1.000

1 Input given data and name ranges. Enter the input and output information for each hospital in the ranges B6:C8 and F6:H8 and name the various ranges as indicated.
2 Selected hospital. Enter 1, 2, or 3 in cell B3, depending on which hospital you want to analyze. 3 Unit input costs and output prices. Enter any trial values for the input costs and output prices in the Unit_costs_of_inputs and Unit_prices_of_outputs ranges.
4 Total input costs and output values. In the Input_costs range, calculate the cost of the inputs used by each hospital. To do this, enter the formula
=SUMPRODUCT(Unit_costs_of_inputs,B6:C6)
in cell B14 for hospital 1, and copy this to the rest of the Input_costs range for the other hospitals. Similarly, calculate the output values by entering the formula
=SUMPRODUCT(Unit_prices_of_outputs,F6:H6)
in cell D14 and copying it to the rest of the Output_values range. Note that even though we are focusing on hospital 1’s efficiency, we still calculate input costs and output values for the other hospitals so that we have something to compare hospital 1 to.
5 Total input cost and output value for the selected hospital. In row 19, constrain the total input cost of the selected hospital to be 1 by entering the formula
=VLOOKUP(Selected_hospital,A14:B16,2)
in cell B19, and enter a 1 in cell D19. Similarly, enter the formula
=VLOOKUP(Selected_hospital,A14:D16,4)
in cell B22. (Make sure you understand how these VLOOKUP functions work.) Remember that because the selected hospital’s input cost is constrained to be 1, its output value in cell
B22 is automatically its efficiency.

186

Chapter 4 Linear Programming Models

SOLVER

USING SOLVER TO DETERMINE WHETHER HOSPITAL 1 IS EFFICIENT
To determine whether hospital 1 is efficient, use Solver as follows. (When you are finished, the Solver dialog box should appear as shown in Figure 4.40.)

Figure 4.40
Solver Dialog Box for the DEA Model

1 Objective. Select cell B22 as the target cell to maximize. Because the cost of hospital 1 inputs is constrained to be 1, this causes Solver to maximize the efficiency of hospital 1.
2 Changing cells. Choose the Unit_costs_of_inputs and Unit_prices_of_outputs ranges as the changing cells.
3 Selected hospital’s input cost constraint. Add the constraint Selected_hospital_ input_cost=1. This sets the total value of hospital 1’s inputs equal to 1.
4 Efficiency constraint. Add the constraint Input_costs>=Output_values. This ensures that no hospital is more than 100% efficient.
5 Specify nonnegativity and optimize. Under Solver Options, check the Assume Linear
Model and Assume Non-Negative options and then solve to obtain the optimal solution shown in Figure 4.39.
The 1 in cell B22 of this solution means that hospital 1 is efficient. In words, Solver has found a set of unit costs for the inputs and the unit prices for the outputs such that the total value of hospital 1’s outputs equals the total cost of its inputs.
Figure 4.41 DEA Model for Hospital 2
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22

B

C

D

E

F

G

H

DEA model for checking efficiency of a selected hospital
Selected hospital
Inputs used
Hospital 1
Hospital 2
Hospital 3
Unit costs of inputs

2
Input 1
5
8
7

Input 2
14
15
12

0.000

0.067

Constraints that input costs must cover output values
Hospital Input costs
1
0.933
>=
2
1.000
>=
3
0.800
>=

Outputs produced
Hospital 1
Hospital 2
Hospital 3
Unit prices of outputs

Output 1
9
5
4
0.0800

Output 2 Output 3
4
16
7
10
9
13
0.0533

I

J

Range names used p _
Input_costs
Output_values
Selected_hospital
Selected_hospital_input_cost
Selected_hospital_output_value
Unit_costs_of_inputs
Unit_prices_of_outputs

K

$ $ $ $
=Model!$B$14:$B$16
=Model!$D$14:$D$16
=Model!$B$3
=Model!$B$19
=Model!$B$22
=Model!$B$10:$C$10
=Model!$F$10:$H$10

0.000

Output values
0.933
0.773
0.800

Constraint that selected hospital's input cost must equal a nominal value of 1
Selected hospital input cost
1.000
=
1
Maximize selected hospital's output value (to see if it is 1, hence efficient)
Selected hospital output value
0.773

4.8 Data Envelopment Analysis (DEA)

187

Determining Whether Hospitals 2 and 3 Are Efficient
To determine whether hospital 2 is efficient, we simply replace the value in cell B3 by 2 and rerun Solver. The Solver settings do not need to be modified. (In fact, for your convenience, a button is included on the spreadsheet that runs a macro to run Solver.) The optimal solution appears in Figure 4.41. From the value of 0.773 in cell B22, we can see that hospital 2 is not efficient. Similarly, we can determine that hospital 3 is efficient by replacing the value in cell B3 by 3 and rerunning Solver (see Figure 4.42).
Figure 4.42 DEA Model for Hospital 3
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22

B

C

D

E

F

G

H

DEA model for checking efficiency of a selected hospital
Selected hospital
Inputs used
Hospital 1
Hospital 2
Hospital 3
Unit costs of inputs

3
Input 1
5
8
7

Input 2
14
15
12

0.000

0.083

Constraints that input costs must cover output values
Hospital Input costs
1
1.167
>=
2
1.250
>=
3
1.000
>=

Outputs produced
Hospital 1
Hospital 2
Hospital 3
Unit prices of outputs

Output 1
9
5
4
0.1000

Output 2 Output 3
4
16
7
10
9
13
0.0667

I

J

Range names used p _
Input_costs
Output_values
Selected_hospital
Selected_hospital_input_cost
Selected_hospital_output_value
Unit_costs_of_inputs
Unit_prices_of_outputs

K

$ $ $ $
=Model!$B$14:$B$16
=Model!$D$14:$D$16
=Model!$B$3
=Model!$B$19
=Model!$B$22
=Model!$B$10:$C$10
=Model!$F$10:$H$10

0.000

Output values
1.167
0.967
1.000

Constraint that selected hospital's input cost must equal a nominal value of 1
Selected hospital input cost
1.000
=
1
Maximize selected hospital's output value (to see if it is 1, hence efficient)
Selected hospital output value
1.000

In summary, we have found that hospitals 1 and 3 are efficient, but hospital 2 is inefficient. What Does It Mean to Be Efficient or Inefficient?
This idea of efficiency or inefficiency might still be a mystery, so let’s consider it further. A hospital is efficient if the inputs and outputs can be priced in such a way that this hospital gets out all of the value that it puts in. The pricing scheme depends on the hospital. Each hospital tries to price inputs and outputs to put its operations in the best possible light. In the example, hospital 1 attaches 0 prices to input 1 (hospital beds) and output 3 (patientdays for patients over 65), and it attaches positive prices to the rest. This makes hospital 1 look efficient. Hospital 3, which is also efficient, also attaches 0 prices to input 1 and output
3, but its prices for the others are somewhat different from hospital 1’s prices.
If DEA finds that a hospital is inefficient, then there is no pricing scheme where that hospital can recover its entire input costs in output values. Actually, it can be shown that if a hospital is inefficient, then a “combination” of the efficient hospitals can be found that uses no more inputs than the inefficient hospital, yet produces at least as much of each output as the inefficient hospital. In this sense, the hospital is inefficient.
To see how this combination can be found, consider the spreadsheet model in Figure 4.43. Begin by entering any positive weights in the Weights range. For any such weights (they don’t even need to sum to 1), consider the combination hospital as a fraction of hospital 1 and another fraction of hospital 3. For example, with the weights shown, the combination hospital uses about 26% of the inputs and produces about 26% of the outputs of hospital 1, and it uses about 66% of the inputs and produces about 66% of the outputs of hospital 3. When they are combined in row 6 with the SUMPRODUCT function [for example, the formula in cell D6 is =SUMPRODUCT(Weights,D4:D5)], we find the quantities of inputs this combination hospital uses and the quantities of outputs it produces. To

188

Chapter 4 Linear Programming Models

find weights where the combination hospital is better than hospital 2, we find any feasible solution to the inequalities indicated in rows 6 to 8 by using the Solver setup in Figure 4.44.
(The weights in Figure 4.43 do the job.) Note that there is no objective to maximize or minimize; all we want is a solution that satisfies the constraints. Furthermore, we know there is a feasible solution because we have already identified hospital 2 as being inefficient.

Figure 4.43
Illustrating How
Hospital 2 Is
Inefficient

A
1
2
3
4
5
6
7
8

B

C

D

E

F

G

H

I

Comparing combinaƟon of hospitals 1 and 3 to inefficient hospital 2

Hospital 1
Hospital 2
Combination
Hospital 2

Weights
0.2615
0.6615

Input 1
5
7
5. 938
=
>=
5
7
10

Figure 4.44
Solver Setup for
Finding an
Inefficiency

In reality, after DEA analysis identifies an organizational unit as being inefficient, this unit should consider benchmarking itself relative to the competition to see where it can make more efficient use of its inputs. ■

MODELING ISSUES
1. The ratio (input i cost)/(input j cost) can be interpreted as the marginal rate of substitution (at the optimal solution) of input i for input j. That is, the same level of outputs can be maintained if we decrease the use of input i by a small amount ∆ and increase the use of input j by [(input i cost)/(input j cost)]∆. For example, for hospital 2,
(input 2 cost/input 1 cost) ϭ 6700. This implies that if the use of input 2 decreases by a small amount ∆, hospital 2 can maintain its current output levels if the usage of input 1 increases by 6700∆.
2. The ratio (output i price)/(output j price) can be interpreted as the marginal rate of substitution (at the optimal solution) of output i for output j. That is, the same level of input usage can be maintained if we decrease the production of output i by a small amount ∆ and increase the production of output j by [(output i price)/(output j price)]∆. For example, for hospital 2, (output 2 price)/(output 1 price) ϭ 0.67. This implies that if the use of output 2 decreases by a small amount ∆, hospital 2 can maintain its current resource usage if the production of output 1 increases by 0.67∆. ■

4.8 Data Envelopment Analysis (DEA)

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A D D I T I O N A L A P P L I C AT I O N S
DEA for Evaluating School Bus Transportation
Sexton et al. (1994) used DEA to evaluate the efficiency of school bus transportation for the counties of North Carolina. For each county, they used two inputs: buses used and total operating expense. They used a single output: pupils transported per day. However, they noted a problem with “traditional” DEA. Consider two counties (county 1 and county 2) that use exactly the same inputs and produce the same outputs. Suppose that county 1 is very sparsely populated and county 2 is densely populated. Clearly, county 1 is transporting pupils more efficiently than county 2, but a DEA conducted by the method described will not show this. Realizing this, Sexton et al. developed a method to adjust the output of county 2 downward and the output of county 1 upward to compensate for this problem. The
North Carolina Department of Education penalized the inefficient counties by reducing their budgetary appropriations. Since the time DEA was performed, most counties have greatly increased their efficiency.

DEA in the Banking Industry
Sherman and Ladino (1995) discuss the use of DEA in identifying the most and least efficient branches in a banking firm with 33 branch banks. They found efficiency ratings that varied from 37% to 100%, with 23 of the 33 branches rated below 100% and 10 below
70%. Each of the inefficient branches was compared to a reference set of “best-practice” branches—efficient branches that offered the same types of services as the inefficient branch. This allowed them to make specific suggestions as to how the inefficient branches could improve. For example, they showed that branch 1 should be able to provide its current level and mix of services with 4.5 fewer customer-service personnel, 1.8 fewer sales service personnel, 0.3 fewer managers, $222,928 less in operating expenses, and 1304 fewer square feet. They also indicated the added amount of service that the inefficient branches could provide, in addition to resource savings, if these branches could become as efficient as the best-practice branches. For example, branch 1 could handle (per year) about
15,000 additional deposits, withdrawals, and checks cashed; 2000 added bank checks, bonds, and travelers’ checks; and 8 additional night deposits, while reducing the resources needed if it attained the efficiency level of the best-practice branches. See the May–June
1999 issue of Interfaces for more applications of DEA in the banking industry. ■

PROBLEMS
Skill-Building Problems
41. The Salem Board of Education wants to evaluate the efficiency of the town’s four elementary schools. The three outputs of the schools are
■ output 1 ϭ average reading score
■ output 2 ϭ average mathematics score
■ output 3 ϭ average self-esteem score
The three inputs to the schools are
■ input 1 ϭ average educational level of mothers
(defined by highest grade completed: 12 ϭ high school graduate; 16 ϭ college graduate, and so on)

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■
■

input 2 ϭ number of parent visits to school (per child) input 3 ϭ teacher-to-student ratio

The relevant information for the four schools is given in the file P04_41.xlsx. Determine which (if any) schools are inefficient.
42. Pine Valley Bank has three branches. You have been asked to evaluate the efficiency of each. The following inputs and outputs are to be used for the study:
■ input 1 ϭ labor hours used (hundreds per month)
■ input 2 ϭ space used (in hundreds of square feet)

■
■
■
■

input 3 ϭ supplies used per month (in dollars) output 1 ϭ loan applications per month output 2 ϭ deposits processed per month (in thousands) output 3 ϭ checks processed per month (in thousands) The relevant information is given in the file
P04_42.xlsx. Use these data to determine whether any bank branches are inefficient.
43. You have been asked to evaluate the efficiency of the
Port Charles Police Department. Three precincts are to be evaluated. The inputs and outputs for each precinct are as follows:
■ input 1 ϭ number of policemen
■ input 2 ϭ number of vehicles used

■
■

output 1 ϭ number of patrol units responding to service requests (thousands per year) output 2 ϭ number of convictions obtained each year (in hundreds)

You are given the data in the file P04_43.xlsx. Use this information to determine which precincts, if any, are inefficient.
44. You have been commissioned by Indiana University to evaluate the relative efficiency of four degree-granting units: Business, Education, Arts and Sciences, and
Health, Physical Education, and Recreation (HPER).
You are given the information in the file P04_44.xlsx.
Use DEA to identify all inefficient units.

4.9 CONCLUSION
In this chapter, we have developed LP spreadsheet models of many diverse situations.
Although there is no standard procedure that can be used to attack all problems, there are several keys to most spreadsheet optimization models:
■

■

■

■

■

Determine the changing cells, the cells that contain the values of the decision variables. These cells should contain the values the decision maker has direct control over, and they should determine all other outputs, either directly or indirectly. For example, in blending models, the changing cells should contain the amount of each input used to produce each output; in employee scheduling models, the changing cells should contain the number of employees who work each possible five-day shift. Set up the spreadsheet so that you can easily compute what you want to maximize or minimize (usually profit or cost). For example, in the aggregate planning model, a good way to compute total cost is to compute the monthly cost of operation in each row. Set up the spreadsheet so that the relationships between the cells in the spreadsheet and the problem constraints are readily apparent. For example, in the post office scheduling example, it is convenient to calculate the number of employees working each day of the week near the number of employees needed for each day of the week. Make your spreadsheet readable. Use descriptive labels, use range names liberally, use cell comments and text boxes for explanations, and think about your model layout before you dive in. This might not be too important for small, straightforward models, but it is crucial for large, complex models. Just remember that other people are likely to be examining your spreadsheet models.
LP models tend to fall into categories, but they are definitely not all alike. For example, a problem might involve a combination of the ideas discussed in the workforce scheduling, blending, and production process examples of this chapter.
Each new model presents new challenges, and you must be flexible and imaginative to meet these challenges. It takes practice and perseverance.

4.9 Conclusion

191

Summary of Key Management Science Terms
Term
Dual-objective model
Integer constraints
Multiple optimal solutions Heuristic
Nonsmooth problems
DEA (Data Envelopment
Analysis)

Explanation
Model with two competing objectives; usual strategy is to constrain one of them and optimize the other
Constraints that limit (some) changing cells to integer values Case where several solutions have the same optimal value of the objective
An “educated guess” solution, not guaranteed to be optimal but usually quick and easy to obtain
Nonlinear models with “sharp edges” or discontinuities that make them difficult to solve
Method for determining whether organizational units are efficient in terms of using their inputs to produce their outputs

Page
137
140
147
148
156
184

Summary of Key Excel Terms
Term
Range name shortcut Solver integer constraints Row, column sums shortcut

Explanation
Quick way to create range names, using labels in adjacent cells
Constraints on changing cells forcing them to be integers
Quick way of getting row and/or column sums from a table

Nonsmooth functions with
Solver

Avoid use of functions such as IF,
MIN, MAX, and ABS in Solver models; Solver can’t handle them predictably. Useful function for transferring column range to row range, or vice versa
Excel functions such as
TRANSPOSE that fill a whole range at once

TRANSPOSE function Array functions

Excel
Use Create from Selection on Formulas ribbon
Specify in Add Constraint dialog box with Solver
Highlight row under table and column to right of table, click on ͚ buttom

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135
140
153

156

Highlight result range, type
=TRANSPOSE(range), press
Ctrl+Shift+Enter
Highlight result range, type formula, press Ctrl+Shift+Enter

170

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PROBLEMS
Skill-Building Problems
45. During each 4-hour period, the Smalltown police force requires the following number of on-duty police officers: eight from midnight to 4 A.M.; seven from
4 A.M. to 8 A.M.; six from 8 A.M. to noon; six from noon to 4 P.M.; five from 4 P.M. to 8 P.M.; and four from
8 P.M. to midnight. Each police officer works two consecutive four-hour shifts.

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a. Determine how to minimize the number of police officers needed to meet Smalltown’s daily requirements. b. Use SolverTable to see how the number of police officers changes as the number of officers needed from midnight to 4 A.M. changes.
46. A bus company believes that it will need the following numbers of bus drivers during each of the next
5 years: 60 drivers in year 1; 70 drivers in year 2;

50 drivers in year 3; 65 drivers in year 4; 75 drivers in year 5. At the beginning of each year, the bus company must decide how many drivers to hire or fire.
It costs $4000 to hire a driver and $2000 to fire a driver. A driver’s salary is $10,000 per year. At the beginning of year 1, the company has 50 drivers.
A driver hired at the beginning of a year can be used to meet the current year’s requirements and is paid full salary for the current year.
a. Determine how to minimize the bus company’s salary, hiring, and firing costs over the next 5 years.
b. Use SolverTable to determine how the total number hired, total number fired, and total cost change as the unit hiring and firing costs each increase by the same percentage.
47. Shoemakers of America forecasts the following demand for the next 6 months: 5000 pairs in month 1;
6000 pairs in month 2; 5000 pairs in month 3; 9000 pairs in month 4; 6000 pairs in month 5; 5000 pairs in month 6. It takes a shoemaker 15 minutes to produce a pair of shoes. Each shoemaker works 150 hours per month plus up to 40 hours per month of overtime.
A shoemaker is paid a regular salary of $2000 per month plus $50 per hour for overtime. At the beginning of each month, Shoemakers can either hire or fire workers. It costs the company $1500 to hire a worker and $1900 to fire a worker. The monthly holding cost per pair of shoes is 3% of the cost of producing a pair of shoes with regular-time labor.
The raw materials in a pair of shoes cost $10. At the beginning of month 1, Shoemakers has 13 workers.
Determine how to minimize the cost of meeting (on time) the demands of the next 6 months.
48. NewAge Pharmaceuticals produces the drug NasaMist from four chemicals. Today, the company must produce 1000 pounds of the drug. The three active ingredients in NasaMist are A, B, and C. By weight, at least
8% of NasaMist must consist of A, at least 4% of B, and at least 2% of C. The cost per pound of each chemical and the amount of each active ingredient in 1 pound of each chemical are given in the file
P04_48.xlsx. At least 100 pounds of chemical 2 must be used.
a. Determine the cheapest way of producing today’s batch of NasaMist.
b. Use SolverTable to see how much the percentage of requirement of A is really costing NewAge. Let the percentage required vary from 6% to 12%.
49. You have decided to enter the candy business. You are considering producing two types of candies: Slugger candy and Easy Out candy, both of which consist solely of sugar, nuts, and chocolate. At present, you have in stock 10,000 ounces of sugar, 2000 ounces of nuts, and 3000 ounces of chocolate. The mixture used to make Easy Out candy must contain at least 20%

nuts. The mixture used to make Slugger candy must contain at least 10% nuts and 10% chocolate. Each ounce of Easy Out candy can be sold for $0.50, and each ounce of Slugger candy for $0.40.
a. Determine how you can maximize your revenue from candy sales.
b. Use SolverTable to determine how changes in the price of Easy Out change the optimal solution.
c. Use SolverTable to determine how changes in the amount of available sugar change the optimal solution. 50. Sunblessed Juice Company sells bags of oranges and cartons of orange juice. Sunblessed grades oranges on a scale of 1 (poor) to 10 (excellent). At present,
Sunblessed has 100,000 pounds of grade 9 oranges and 120,000 pounds of grade 6 oranges on hand. The average quality of oranges sold in bags must be at least 7, and the average quality of the oranges used to produce orange juice must be at least 8. Each pound of oranges that is used for juice yields a revenue of $1.50 and incurs a variable cost (consisting of labor costs, variable overhead costs, inventory costs, and so on) of
$1.05. Each pound of oranges sold in bags yields a revenue of $1.50 and incurs a variable cost of $0.70.
a. Determine how Sunblessed can maximize its profit.
b. Use SolverTable to determine how a change in the cost per bag of oranges changes the optimal solution. c. Use SolverTable to determine how a change in the amount of grade 9 oranges available affects the optimal solution.
d. Use SolverTable to determine how a change in the required average quality required for juice changes the optimal solution.
51. A bank is attempting to determine where its assets should be invested during the current year. At present,
$500,000 is available for investment in bonds, home loans, auto loans, and personal loans. The annual rates of return on each type of investment are known to be the following: bonds, 10%; home loans, 16%; auto loans, 13%; personal loans, 20%. To ensure that the bank’s portfolio is not too risky, the bank’s investment manager has placed the following three restrictions on the bank’s portfolio:
■ The amount invested in personal loans cannot exceed the amount invested in bonds.
■ The amount invested in home loans cannot exceed the amount invested in auto loans.
■ No more than 25% of the total amount invested can be in personal loans.
Help the bank maximize the annual return on its investment portfolio.
52. Young MBA Erica Cudahy can invest up to $15,000 in stocks and loans. Each dollar invested in stocks yields

4.9 Conclusion

193

$0.10 profit, and each dollar invested in a loan yields
$0.15 profit. At least 30% of all money invested must be in stocks, and at least $6000 must be in loans.
Determine how Erica can maximize the profit earned on her investments.
53. Bullco blends silicon and nitrogen to produce two types of fertilizers. Fertilizer 1 must be at least 40% nitrogen and sells for $70 per pound. Fertilizer 2 must be at least 70% silicon and sells for $40 per pound.
Bullco can purchase up to 8000 pounds of nitrogen at
$15 per pound and up to 10,000 pounds of silicon at
$10 per pound.
a. Assuming that all fertilizer produced can be sold, determine how Bullco can maximize its profit.
b. Use SolverTable to explore the effect on profit of changing the minimum percentage of nitrogen required in fertilizer 1.
c. Suppose the availabilities of nitrogen and silicon both increase by the same percentage from their current values. Use SolverTable to explore the effect of this change on profit.
54. Eli Daisy uses chemicals 1 and 2 to produce two drugs. Drug 1 must be at least 70% chemical 1, and drug 2 must be at least 60% chemical 2. Up to 4000 ounces of drug 1 can be sold at $6 per ounce; up to
3000 ounces of drug 2 can be sold at $5 per ounce.
Up to 4500 ounces of chemical 1 can be purchased at
$6 per ounce, and up to 4000 ounces of chemical 2 can be purchased at $4 per ounce. Determine how to maximize Daisy’s profit.
55. Hiland’s TV-Radio Store must determine how many
TVs and radios to keep in stock. A TV requires
10 square feet of floor space, whereas a radio requires
4 square feet; 5000 square feet of floor space is available. A TV sale results in an $80 profit, and a radio earns a profit of $20. The store stocks only TVs and radios. Marketing requirements dictate that at least
60% of all appliances in stock be radios. Finally, a TV ties up $200 in capital, and a radio $50. Hiland wants to have at most $60,000 worth of capital tied up at any time. a. Determine how to maximize Hiland’s profit.
b. Use SolverTable to explore how much profit the minimum percentage of radio requirement is costing Hiland’s.
c. Use SolverTable to explore how much profit the upper limit on capital being tied up is costing
Hiland’s.
56. Many Wall Street firms use LP models to select a desirable bond portfolio. The following is a simplified version of such a model. Solodrex is considering investing in four bonds; $1 million is available for investment. The expected annual return, the worstcase annual return on each bond, and the “duration” of each bond are given in the file P04_56.xlsx. (The

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Chapter 4 Linear Programming Models

duration of a bond is a measure of the bond’s sensitivity to interest rates.) Solodrex wants to maximize the expected return from its bond investments, subject to three constraints:
■ The worst-case return of the bond portfolio must be at least 8%.
■ The average duration of the portfolio must be at most 6. For example, a portfolio that invests
$600,000 in bond 1 and $400,000 in bond 4 has an average duration of [600,000(3) ϩ
400,000(9)]Ͳ1,000,000 ϭ 5.4.
■ Because of diversification requirements, at most
40% of the total amount invested can be invested in a single bond.
Determine how Solodrex can maximize the expected return on its investment.
57. Coalco produces coal at three mines and ships it to four customers. The cost per ton of producing coal, the ash and sulfur content (per ton) of the coal, and the production capacity (in tons) for each mine are given in the file P04_57.xlsx. The number of tons of coal demanded by each customer and the cost (in dollars) of shipping a ton of coal from a mine to each customer are also provided in this same file. The amount of coal shipped to each customer must contain at most 6% ash and at most 3.5% sulfur.
Show Coalco how to minimize the cost of meeting customer demands.
58. Furnco manufactures tables and chairs. A table requires 40 board feet of wood, and a chair requires
30 board feet of wood. Wood can be purchased at a cost of $1 per board foot, and 40,000 board feet of wood are available for purchase. It takes 2 hours of skilled labor to manufacture an unfinished table or an unfinished chair. Three more hours of skilled labor will turn an unfinished table into a finished table, and
2 more hours of skilled labor will turn an unfinished chair into a finished chair. A total of 6000 hours of skilled labor is available (and have already been paid for). All furniture produced can be sold at the following unit prices: an unfinished table, $70; a finished table, $140; an unfinished chair, $60; a finished chair, $110.
a. Determine how to maximize Furnco’s profit from manufacturing tables and chairs.
b. Use a two-way SolverTable to see how the numbers of unfinished products (both chairs and tables) sold depend on the selling prices of these unfinished products. Of course, neither unfinished selling price should be as large as the corresponding finished selling price.
59. A company produces three products, A, B, and C, and can sell these products in unlimited quantities at the following unit prices: A, $10; B, $56; C, $100.
Producing a unit of A requires 1 hour of labor; a unit

of B, 2 hours of labor plus 2 units of A; and a unit of
C, 3 hours of labor plus 1 unit of B. Any A that is used to produce B cannot be sold. Similarly, any B that is used to produce C cannot be sold. A total of 40 hours of labor is available. Determine how to maximize the company’s revenue.
60. Abotte Products produces three products, A, B, and C.
The company can sell up to 300 pounds of each product at the following prices (per pound): product
A, $10; product B, $12; product C, $20. Abotte purchases raw material at $5 per pound. Each pound of raw material can be used to produce either 1 pound of A or 1 pound of B. For a cost of $3 per pound processed, product A can be converted to 0.6 pound of product B and 0.4 pound of product C. For a cost of $2 per pound processed, product B can be converted to 0.8 pound of product C. Determine how
Abotte can maximize its profit.
61. Moneyco has $100,000 to invest at time 1 (the beginning of year 1). The cash flows associated with the five available investments are listed in the file
P04_61.xlsx. For example, every dollar invested in
A in year 1 yields $1.40 in year 4. In addition to these investments, Moneyco can invest as much money each year as it wants in CDs, which pay 6% interest.
The company wants to maximize its available cash in year 4. Assuming it can put no more than $50,000 in any investment, develop an LP model to help
Moneyco achieve its goal.
62. At the beginning of year 1, you have $10,000. Investments A and B are available; their cash flows are shown in the file P04_62.xlsx. Assume that any money not invested in A or B earns interest at an annual rate of 8%.
a. Determine how to maximize your cash on hand in year 4.
b. Use SolverTable to determine how a change in the year 3 yield for investment A changes the optimal solution to the problem.
c. Use SolverTable to determine how a change in the yield of investment B changes the optimal solution to the problem.
63. You now have $10,000, and the following investment plans are available to you during the next three years:
■ Investment A: Every dollar invested now yields
$0.10 a year from now and $1.30 three years from now. ■ Investment B: Every dollar invested now yields
$0.20 a year from now and $1.10 two years from now. ■ Investment C: Every dollar invested a year from now yields $1.50 three years from now.
During each year, you can place uninvested cash in money market funds that yield 6% interest per year.
However, you can invest at most $5000 in any one of

plans A, B, or C. Determine how to maximize your cash on hand three years from now.
64. Sunco processes oil into aviation fuel and heating oil.
It costs $65,000 to purchase each 1000 barrels of oil, which is then distilled and yields 500 barrels of aviation fuel and 500 barrels of heating oil. Output from the distillation can be sold directly or processed in the catalytic cracker. If sold after distillation without further processing, aviation fuel sells for
$80,000 per 1000 barrels, and heating oil sells for
$65,000 per 1000 barrels. It takes 1 hour to process
1000 barrels of aviation fuel in the catalytic cracker, and these 1000 barrels can be sold for $145,000. It takes 45 minutes to process 1000 barrels of heating oil in the cracker, and these 1000 barrels can be sold for
$125,000. Each day at most 20,000 barrels of oil can be purchased, and 8 hours of cracker time are available. Determine how to maximize Sunco’s profit.
65. All steel manufactured by Steelco must meet the following requirements: between 3.2% and 3.5% carbon; between 1.8% and 2.5% silicon; between 0.9% and 1.2% nickel; tensile strength of at least 45,000 pounds per square inch (psi). Steelco manufactures steel by combining two alloys. The cost and properties of each alloy are given in the file P04_69.xlsx. Assume that the tensile strength of a mixture of the two alloys can be determined by averaging the tensile strength of the alloys that are mixed together. For example, a
1 ton mixture that is 40% alloy 1 and 60% alloy 2 has a tensile strength of 0.4(42,000) ϩ 0.6(50,000).
Determine how to minimize the cost of producing a ton of steel.
66. Steelco manufactures two types of steel at three different steel mills. During a given month, each steel mill has 200 hours of blast furnace time available.
Because of differences in the furnaces at each mill, the time and cost to produce a ton of steel differ for each mill, as listed in the file P04_66.xlsx. Each month,
Steelco must manufacture at least 500 tons of steel 1 and 600 tons of steel 2. Determine how Steelco can minimize the cost of manufacturing the desired steel.
67. Based on Heady and Egbert (1964). Walnut Orchard has two farms that grow wheat and corn. Because of differing soil conditions, there are differences in the yields and costs of growing crops on the two farms.
The yields and costs are listed in the file P04_67.xlsx.
Each farm has 100 acres available for cultivation;
11,000 bushels of wheat and 7000 bushels of corn must be grown.
a. Determine a planting plan that will minimize the cost of meeting these requirements.
b. Use SolverTable to see how the total cost changes if the requirements for wheat and corn both change by the same percentage, where this percentage change can be as low as Ϫ50% or as high as ϩ50%.

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195

68. Candy Kane Cosmetics (CKC) produces Leslie
Perfume, which requires chemicals and labor. Two production processes are available. Process 1 transforms 1 unit of labor and 2 units of chemicals into
3 ounces of perfume. Process 2 transforms 2 units of labor and 3 units of chemicals into 5 ounces of perfume. It costs CKC $3 to purchase a unit of labor and $2 to purchase a unit of chemicals. Each year up to 20,000 units of labor and 35,000 units of chemicals can be purchased. In the absence of advertising, CKC believes it can sell 1000 ounces of perfume. To stimulate demand for Leslie, CKC can hire the lovely model Jenny Nelson. Jenny is paid $100 per hour.
Each hour Jenny works for the company is estimated to increase the demand for Leslie Perfume by 200 ounces. Each ounce of Leslie Perfume sells for $5.
Determine how CKC can maximize its profit.
69. Sunco Oil has refineries in Los Angeles and Chicago.
The Los Angeles refinery can refine up to 2 million barrels of oil per year, and the Chicago refinery up to
3 million. After the oil is refined, it’s shipped to two distribution points, Houston and New York City.
Sunco estimates that each distribution point can sell up to 5 million barrels per year. Because of differences in shipping and refining costs, the profit earned (in dollars) per million barrels of oil shipped depends on where the oil was refined and on the point of distribution. This information is listed in the file P04_65.xlsx.
Sunco is considering expanding the capacity of each refinery. Each million barrels of annual refining capacity that is added will cost $120,000 for the
Los Angeles refinery and $150,000 for the Chicago refinery. Determine how Sunco can maximize its profit (including expansion costs) over a 10-year period. 70. Feedco produces two types of cattle feed, both consisting totally of wheat and alfalfa. Feed 1 must contain at least 80% wheat, and feed 2 must contain at least 60% alfalfa. Feed 1 sells for $1.50 per pound, and feed 2 sells for $1.30 per pound. Feedco can purchase up to
1000 pounds of wheat at $0.50 per pound and up to
800 pounds of alfalfa at $0.40 per pound. Demand for each type of feed is unlimited. Determine how to maximize Feedco’s profit.
71. Carrington Oil produces gas 1 and gas 2 from two types of crude oil: crude 1 and crude 2. Gas 1 is allowed to contain up to 4% impurities, and gas 2 is allowed to contain up to 3% impurities. Gas 1 sells for $72 per barrel, whereas gas 2 sells for
$84 per barrel. Up to 4200 barrels of gas 1 and up to
4300 barrels of gas 2 can be sold. The cost per barrel of each crude, their availability, and the level of impurities in each crude are listed in the file
P04_71.xlsx. Before blending the crude oil into gas, any amount of each crude can be “purified” for a cost

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of $3.50 per barrel. Purification eliminates half of the impurities in the crude oil.
a. Determine how to maximize profit.
b. Use SolverTable to determine how an increase in the availability of crude 1 affects the optimal profit.
c. Use SolverTable to determine how an increase in the availability of crude 2 affects the optimal profit.
d. Use SolverTable to determine how a change in the price of gas 2 changes the optimal profit and the types of gas produced.
72. A company produces two products: A and B. Product
A sells for $11 per unit and product B sells for
$23 per unit. Producing a unit of product A requires
2 hours on assembly line 1 and 1 unit of raw material.
Producing a unit of product B requires 2 units of raw material, 1 unit of A, and 2 hours on assembly line 2.
On line 1, 1300 hours of time are available, and
500 hours are available on line 2. A unit of raw material can be bought (for $5 a unit) or produced
(at no cost) by using 2 hours of time on line 1.
a. Determine how to maximize profit.
b. The company will stop buying raw material when the price of raw material exceeds what value? (Use
SolverTable.)
73. Based on Thomas (1971). Toyco produces toys at two plants and sells in three regions. The current demands at these regions are given in the file P04_73.xlsx. Each plant can produce up to 2500 units. Each toy sells for $10, and the cost of producing and shipping a toy from a given plant to a region is also given in the file P04_73.xlsx. Toyco can advertise locally and nationally. Each $1 spent on a local ad raises sales in a region by 0.5 units, whereas each $1 spent advertising nationally increases sales in each region by 0.3 units.
a. Determine how Toyco can maximize its profit.
b. If sales stimulated by advertising exhibits diminishing returns, how would you change your model? 74. A bank needs exactly two employees working each hour from 9 A.M. to 5 P.M. Workers can work the shifts and are paid the wages listed in the file P04_74.xlsx.
For example, a worker working 9 A.M. to 2 P.M. is paid
$42.00. Find an assignment of workers that provides enough workers at minimum cost.
75. Based on Gatalla and Oearce (1974). Northwest
Airlines has determined that it needs the number of ticket agents during each hour of the day, as listed in the file P04_75.xlsx. Workers work nine-hour shifts, one hour of which is for lunch. The lunch hour can be either the fourth or fifth hour of their shift. What is the minimum number of workers needed by Northwest?
76. A rock company uses five types of rocks to fill four orders. The phosphate content, availability of each

type of rock, and the production cost per pound for each rock are listed in the file P04_76.xlsx, as well as the size of each order and the minimum and maximum phosphate percentage in each order. What is the cheapest way to fill the orders?
77. Autoco produces cars. Demand during each of the next 12 months is forecasted to be 945, 791, 364, 725,
268, 132, 160, 304, 989, 293, 279, and 794. Other relevant information is as follows:
■ Workers are paid $5000 per month.
■ It costs $500 to hold a car in inventory for a month.
The holding cost is based on each month’s ending inventory. ■ It costs $4000 to hire a worker.
■ It costs $20,000 to fire a worker.
■ Each worker can make up to 8 cars a month.
■ Workers are hired and fired at beginning of each month. ■ At the beginning of month 1 there are 500 cars in inventory and 60 workers.
How can the company minimize the cost of meeting demand for cars on time?
78. Oilco produces gasoline from five inputs. The cost, density, viscosity, and sulfur content, and the number of barrels available of each input are listed in the file
P04_78.xlsx. Gasoline sells for $72 per barrel. Gasoline can have a density of at most 0.98 units per barrel, a viscosity of at most 37 units per barrel, and a sulfur content of at most 3.7 units per barrel.
a. How can Oilco maximize its profit?
b. Describe how the optimal solution to the problem changes as the price of gasoline ranges from $65 to
$80 per barrel.
79. The HiTech company produces DVD players.
Estimated demand for the next 4 quarters is 5000;
10,000; 8000; and 2000. At the beginning of quarter 1,
HiTech has 60 workers. It costs $2000 to hire a worker and $4000 to fire a worker. Workers are paid $10,000 per quarter plus $80 for each unit they make during overtime. A new hire can make up to 60 units per quarter during regular-time, whereas a previously hired worker can make up to 90 units per quarter. Any worker can make up to 20 units per quarter during overtime. Each DVD player is sold for $160. It costs
$20 to hold a DVD player in inventory for a quarter.
Assume workers are hired and fired at the beginning of each quarter and that all of a quarter’s production is available to meet demand for that quarter. Initial inventory at the beginning of quarter 1 is 1000 DVD players. How can the company maximize its profit?
Assume that demand is lost if insufficient stock is available. That is, there is no backlogging of demand
(and there is no requirement that HiTech must satisfy all of its demand).

Skill-Extending Problems
80. MusicTech manufactures and sells a portable music device called an mTune (similar to an iPod). At beginning of month 1, the company has $70,000 and 6 employees. Each machine the company owns has the capacity to make up to 900 mTunes per month, and each worker can make up to 600 mTunes per month. The company cannot use more labor or machine capacity than is available in any given month. Also, the company wants to have a nonnegative cash balance at all points in time.
The company’s costs are the following:
■ Holding cost of $2 each month per mTune in ending inventory
■ Cost in month 1 of buying machines ($3000 per machine) ■ Raw material cost of $6 per mTune
■ Monthly worker wage of $3500
■ Hiring cost of $4000 per worker
■ Firing cost of $2000 per worker
In the absence of advertising, the monthly demands in months 1 through 6 are forecasted to be 5000, 8000,
7000, 4000, 5000, and 6000. However, MusicTech can increase demand each month by advertising. Every
$10 (up to a maximum of $10,000 per month) spent on advertising during a month increases demand for that month by 1 mTune. The devices are sold for $25 each. The sequence of events in any month is that the company buys machines (month 1 only), hires and fires workers, makes the mTunes, advertises, pays all costs for the month, and collects revenues for the month. Develop a model to maximize profit (total revenue minus total costs) earned during the next
6 months.
81. Assume we want to take out a $300,000 loan on a 20year mortgage with end-of-month payments. The annual rate of interest is 6%. Twenty years from now, we need to make a $40,000 ending balloon payment.
Because we expect our income to increase, we want to structure the loan so at the beginning of each year, our monthly payments increase by 2%.
a. Determine the amount of each year’s monthly payment. You should use a lookup table to look up each year’s monthly payment and to look up the year based on the month (e.g., month 13 is year 2, etc.). b. Suppose payment each month is to be same, and there is no balloon payment. Show that the monthly payment you can calculate from your spreadsheet matches the value given by the Excel
PMT function PMT(0.06͞12,240,Ϫ300000,0,0).
82. A graduated payment mortgage (GPM) enables the borrower to have lower payments earlier in the mortgage and increases payments later on. The

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assumption is the borrower’s income will increase over time so that it will be easier for the borrower to meet all payments. Suppose we borrow $60,000 on a 30-year monthly mortgage. We obtain a GPM where monthly payments increase 7.5% per year through year 5 and then remain constant from year 5 through year 30. For annual interest rates of 10%,
11%, 12%, 13%, and 14%, use Solver to find the amount of each year’s monthly payment.
83. Suppose you are planning for retirement. At the beginning of this year and each of the next 39 years, you plan to contribute some money to your retirement fund. Each year, you plan to increase your retirement contribution by $500. When you retire in 40 years, you plan to withdraw $100,000 at the beginning of each year for the next 20 years. You assume the following about the yields of your retirement investment portfolio:
■ During the first 20 years, your investments will earn 10% per year.
■ During all other years, your investments will earn
5% per year.
All contributions and withdrawals occur at the beginnings of the respective years.
a. Given these assumptions, what is the least amount of money you can contribute this year and still have enough to make your retirement withdrawals?
b. How does your answer change if inflation is 2% per year and your goal is to withdraw $100,000 per year (in today’s dollars) for 20 years?
84. Based on Brahms and Taylor (1999). Eli Lilly and
Pfizer are going to merge. Merger negotiations must settle the following issues:
■ What will the name of the merged corporation be?
■ Will corporate headquarters be in Indianapolis
(Lilly wants this) or New York (Pfizer wants this)?
■ Which company’s chairperson will be chairperson of the merged corporation?
■ Which company gets to choose the CEO?
■ On issue of layoffs, what percentage of each company’s view will prevail?
Brahms developed a remarkably simple method for the two adversaries to settle their differences. (This same method could be used to settle differences between other adversaries, such as a husband and wife in a divorce, Arab and Israel in Middle East, and so on.)
Each adversary allocates 100 points between all of the issues. These allocations are listed in the file
P04_84.xlsx. For example, Lilly believes headquarters is worth 25 points, whereas Pfizer thinks headquarters is only worth 10 points. Layoffs may be divided (for example, Lilly might get 70% of the say in layoffs and
Pfizer 30%), but on all other issues, only one company gets its way. The adjusted winner procedure says that the “best” way to make decisions on each issue is to:

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Give each adversary the same number of points.
Ensure that each company prefers its allocation to the allocation of its opponent.
Maximize the number of points received by either participant. Such a solution is equitable (because each party receives the same number of points) and is envy-free
(because neither side prefers what its opponent receives to what it receives). It can also be shown that the adjusted winner procedure yields a Pareto optimal solution. This means that no other allocation can make one player better off without making the other player worse off. Find the adjusted winner solution to the merger example. Also show that the adjusted winner solution for this example is Pareto optimal.
85. AdminaStar processes Medicare claims. At the beginning of month 1 they have a backlog of 40,000 difficult claims and 60,000 easy claims. The predicted claim volume for months 1 through 8 is listed in the file P04_85.xlsx. At the beginning of month 1,
AdminaStar has 70 experienced claim processors. Each month it can hire up to 10 trainees. At the end of each month, 5% of experienced employees quit, and 20% of trainees are fired. Each worker is available for 160 hours per month. The number of minutes needed by each worker to process each type of claim is also listed in the file P04_85.xlsx. AdminaStar wants ending inventory for months 2 through 7 to be no greater than
50,000 of each type of claim. All claims must be processed by the end of month 8. What is the minimum number of trainees that need to be hired during months
1 to 8? (Note: Trainees must be integers. Experienced workers will probably end up being fractional, but don’t worry about this.)
86. Based on Charnes and Cooper (1955). A small company is trying to determine employee salary based on following attributes: effectiveness, responsibility, initiative, experience, education, self expression, planning ability, intelligence, and the ability to get things done. Each of the company’s seven executives has been rated on each of these attributes, with the ratings shown in the file P04_86.xlsx.
The company wants to set each executive’s salary by multiplying a weight for each attribute by the executive’s score on each attribute. The salaries must satisfy the following constraints:
■ The salary of a lower-numbered executive must be at least as large as the salary of a higher-numbered executive. ■ Executive 1’s salary can be at most $160,000 and executive 7’s salary must be at least $40,000.
■ The salaries of executives 1, 5, and 7 should match
$160,000, $100,000, and $40,000, respectively, as closely as possible.
■ All attribute weights must be nonnegative.

Develop a method for setting salaries. [Hint: For executives 1, 5, and 7, define over and under changing cells and add a constraint such as Executive 5 salary ϩ (Amount executive 5 salary under $100,000)
Ϫ (Amount executive 5 salary over $100,000) ϭ
$100,000. Then the target cell to minimize is the sum of over and under changing cells for positions 1, 5, and 7. If you did not include the over and under changing cells, why would your model fail to be linear?] 87. During the next 4 quarters, Dorian Auto must meet
(on time) the following demands for cars: 4000 in quarter 1; 2000 in quarter 2; 5000 in quarter 3;
1000 in quarter 4. At the beginning of quarter 1, there are 300 autos in stock. The company has the capacity to produce at most 3000 cars per quarter.
At the beginning of each quarter, the company can change production capacity. It costs $100 to increase quarterly production capacity by one unit. For example, it would cost $10,000 to increase capacity from 3000 to 3100. It also costs $50 per quarter to maintain each unit of production capacity (even if it is unused during the current quarter). The variable cost of producing a car is $2000. A holding cost of $150 per car is assessed against each quarter’s ending inventory. At the end of quarter 4, plant capacity must be at least 4000 cars.
a. Determine how to minimize the total cost incurred during the next 4 quarters.
b. Use SolverTable to determine how much the total cost increases as the required capacity at the end of quarter 4 increases (from its current value of 4000).
88. The Internal Revenue Service (IRS) has determined that during each of the next 12 months it will need the numbers of supercomputers given in the file
P04_88.xlsx. To meet these requirements, the IRS rents supercomputers for a period of 1, 2, or 3 months.
It costs $1000 to rent a supercomputer for 1 month,
$1800 for 2 months, and $2500 for 3 months. At the beginning of month 1, the IRS has no supercomputers.
a. Determine the rental plan that meets the requirements for the next 12 months at minimum cost.
You can assume that fractional rentals are allowed.
Thus, if your solution says to rent 140.6 computers for one month, you can round this up to 141 or down to 140 without much effect on the total cost.
b. Suppose the monthly requirement increases anywhere from 10% to 50% each month. (Assume that whatever the percentage increase is, it is the same each month.) Use SolverTable to see whether the total rental cost increases by this same percentage. 89. You own a wheat warehouse with a capacity of 20,000 bushels. At the beginning of month 1, you have 6000 bushels of wheat. Each month, wheat can be bought and sold at the prices per 1000 bushels listed in the file

P04_89.xlsx. The sequence of events during each month is as follows:
■ You observe your initial stock of wheat.
■ You can sell any amount of wheat up to your initial stock at the current month’s selling price.
■ You can buy as much wheat as you want, subject to the limitation of warehouse size.
a. Determine how to maximize the profit earned over the next 10 months.
b. Use SolverTable to determine how a change in the capacity of the warehouse affects the optimal solution.
c. Use SolverTable to determine how simultaneous changes in the buying and selling price for month 6 affect the optimal solution.
90. You can calculate the risk index of an investment by taking the absolute values of percentage changes in the value of the investment for each year and averaging them. Suppose you are trying to determine what percentage of your money you should invest in T-bills, gold, and stocks. The file P04_90.xlsx lists the annual returns (percentage changes in value) for these investments for the years 1968 through 1988. Let the risk index of a portfolio be the weighted average of the risk indices of these investments, where the weights are the fractions of the portfolio assigned to the investments.
Suppose that the amount of each investment must be between 20% and 50% of the total invested. You would like the risk index of your portfolio to equal
0.15, and your goal is to maximize the expected return on your portfolio. Determine the maximum expected return on your portfolio, subject to the stated constraints. Use the average return earned by each investment during the years 1968 to 1988 as your estimate of expected return.
91. Based on Magoulas and Marinos-Kouris (1988). Oilco produces two products: regular and premium gasoline.
Each product contains 0.15 gram of lead per liter.
The two products are produced from these six inputs: reformate, fluid catalytic cracker gasoline (FCG), isomerate (ISO), polymer (POL), Methyl Tertiary
Butyl Ether (MTBE), and butane (BUT). Each input has four attributes: research octane number (RON),
Reid Vapor Pressure (RVP), ASTM volatility at
70 degrees Celsius, and ASTM volatility at 130 degrees Celsius. (ASTM is the American Society for
Testing and Materials.) The attributes and daily availability (in liters) of each input are listed in the file
P04_91.xlsx. The requirements for each output are also listed in this file. The daily demand (in thousands of liters) for each product must be met, but more can be produced if desired. The RON and ASTM requirements are minimums; the RVP requirement is a maximum. Regular gasoline sells for $0.754 per liter; premium gasoline for $0.819. Before each product is ready for sale, 0.15 gram per liter of lead must be

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removed. The cost of removing 0.1 gram per liter is
$0.213. At most, 38% of each type of gasoline can consist of FCG. How can Oilco maximize its daily profit? 92. Capsule Drugs manufactures two drugs: 1 and 2. The drugs are produced by blending two chemicals: 1 and
2. By weight, drug 1 must contain at least 65% chemical 1, and drug 2 must contain at least 55% chemical
1. Drug 1 sells for $6 per ounce, and drug 2 sells for
$4 per ounce. Chemicals 1 and 2 can be produced by one of two production processes. Running process 1 for an hour requires 7 ounces of raw material and
2 hours skilled labor, and it yields 3 ounces of each chemical. Running process 2 for an hour requires
5 ounces of raw material and 3 hours of skilled labor, and it yields 3 ounces of chemical 1 and 1 ounce of chemical 2. A total of 120 hours of skilled labor and
100 ounces of raw material are available. Determine how to maximize Capsule’s sales revenues.
93. Molecular Products produces 3 chemicals: B, C, and
D. The company begins by purchasing chemical A for a cost of $6 per 100 liters. For an additional cost of $3 and the use of 3 hours of skilled labor, 100 liters of
A can be transformed into 40 liters of C and 60 liters of B. Chemical C can either be sold or processed further. It costs $1 and 1 hour of skilled labor to process 100 liters of C into 60 liters of D and 40 liters of B. For each chemical, the selling price per 100 liters and the maximum amount (in 100s of liters) tha can be sold are listed in the file P04_93.xlsx.
A maximum of 200 labor hours is available.
Determine how Molecular can maximize its profit.
94. Bexter Labs produces three products: A, B, and C.
Bexter can sell up to 3000 units of product A, up to
2000 units of product B, and up to 2000 units of product C. Each unit of product C uses 2 units of A and
3 units of B and incurs $5 in processing costs. Products A and B are produced from either raw material 1 or raw material 2. It costs $6 to purchase and process 1 unit of raw material 1. Each processed unit of raw material 1 yields 2 units of A and 3 units of B. It costs $3 to purchase and process a unit of raw material 2. Each processed unit of raw material 2 yields 1 unit of A and 2 units of B. The unit prices for the products are
A, $5; B, $4; C, $25. The quality levels of each product are: A, 8; B, 7; C, 6. The average quality level of the units sold must be at least 7. Determine how to maximize Bexter’s profit.
95. Mondo Motorcycles is determining its production schedule for the next 4 quarters. Demands for motorcycles are forecasted to be 400 in quarter 1;
700 in quarter 2; 500 in quarter 3; 200 in quarter 4.
Mondo incurs four types of costs, as described here:
■ It costs Mondo $800 to manufacture each motorcycle. 200

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At the end of each quarter, a holding cost of $100 per motorcycle left in inventory is incurred.
When production is increased from one quarter to the next, a cost is incurred, primarily for training employees. If the increase in production is x motorcycles, the cost is $700x.
When production is decreased from one quarter to the next, a cost is incurred, primarily for severance pay and decreased morale. If the decrease in production is x motorcycles, the cost is $600x.

All demands must be met on time, and a quarter’s production can be used to meet demand for the current quarter (as well as future quarters). During the quarter immediately preceding quarter 1, 500
Mondos were produced. Assume that at the beginning of quarter 1, no Mondos are in inventory.
a. Determine how to minimize Mondo’s total cost during the next 4 quarters.
b. Use SolverTable to determine how Mondo’s optimal production schedule would be affected by a change in the cost of increasing production from one quarter to the next.
c. Use SolverTable to determine how Mondo’s optimal production schedule would be affected by a change in the cost of decreasing production from one quarter to the next.
96. Carco has a $1,500,000 advertising budget. To increase its automobile sales, the firm is considering advertising in newspapers and on television. The more
Carco uses a particular medium, the less effective each additional ad is. The file P04_96.xlsx lists the number of new customers reached by each ad. Each newspaper ad costs $1000, and each television ad costs $10,000.
At most, 30 newspaper ads and 15 television ads can be placed. How can Carco maximize the number of new customers created by advertising?
97. Broker Sonya Wong is currently trying to maximize her profit in the bond market. Four bonds are available for purchase and sale at the bid and ask prices shown in the file P04_97.xlsx. Sonya can buy up to 1000 units of each bond at the ask price or sell up to 1000 units of each bond at the bid price. During each of the next 3 years, the person who sells a bond will pay the owner of the bond the cash payments listed in the file
P04_97.xlsx. Sonya’s goal is to maximize her revenue from selling bonds minus her payment for buying bonds, subject to the constraint that after each year’s payments are received, her current cash position (due only to cash payments from bonds and not purchases or sales of bonds) is nonnegative. Note that her current cash position can depend on past coupons and that cash accumulated at the end of each year earns 5.25% annual interest. Determine how to maximize net profit from buying and selling bonds, subject to the constraints previously described. Why do you think

we limit the number of units of each bond that can be bought or sold?
98. Pear produces low-budget cars. Each car is sold for
$7900. The raw material in a car costs $5000. Labor time and robot time are needed to produce cars. A worker can do the needed labor on, at most, 100 cars per month; a robot can complete the needed work on, at most, 200 cars per month. Initially, Pear has 4 workers. Each worker receives a monthly salary of
$6000. It costs $2500 to hire a worker and $1000 to fire a worker. Hired workers are fully productive during the month they are hired. Robots must be bought at the beginning of month 1 at a cost of $15,000 per robot. The (assumed known) demand for cars is listed in the file P04_98.xlsx. At the end of each month, Pear incurs a holding cost of $200 per car.
How can Pear maximize the profit earned during the next 6 months?
99. The ZapCon Company is considering investing in three projects. If it fully invests in a project, the realized cash flows (in millions of dollars) will be as listed in the file P04_99.xlsx. For example, project 1 requires a cash outflow of $3 million today and returns $5.5 million 3 years from now. Today ZapCon has $2 million in cash. At each time point (0, 0.5, 1,
1.5, 2, and 2.5 years from today), the company can, if desired, borrow up to $2 million at 3.5% (per 6 months) interest. Leftover cash earns 3% (per 6 months) interest. For example, if after borrowing and investing at time 0, ZapCon has $1 million, it would receive $30,000 in interest at time 0.5 year. The company’s goal is to maximize cash on hand after cash flows 3 years from now are accounted for. What investment and borrowing strategy should it use?
Assume that the company can invest in a fraction of a project. For example, if it invests in 0.5 of project
3, it has, for example, cash outflows of –$1 million at times 0 and 0.5.
100. You are a CFA (chartered financial analyst). An overextended client has come to you because she needs help paying off her credit card bills. She owes the amounts on her credit cards listed in the file
P04_100.xlsx. The client is willing to allocate up to
$5000 per month to pay off these credit cards. All cards must be paid off within 36 months. The client’s goal is to minimize the total of all her payments. To solve this problem, you must understand how interest on a loan works. To illustrate, suppose the client pays
$5000 on Saks during month 1. Then her Saks balance at the beginning of month 2 is $20,000 Ϫ
[$5000 Ϫ 0.005(20,000)]. This follows because she incurs 0.005(20,000) in interest charges on her Saks card during month 1. Help the client solve her problem. After you have solved this problem, give an intuitive explanation of the solution found by Solver.

101. Aluminaca produces 100-foot-long, 200-foot-long, and 300-foot-long ingots for customers. This week’s demand for ingots is listed in the file P04_101.xlsx.
Aluminaca has four furnaces in which ingots can be produced. During 1 week, each furnace can be operated for 50 hours. Because ingots are produced by cutting up long strips of aluminum, longer ingots take less time to produce than shorter ingots. If a furnace is devoted completely to producing one type of ingot, the number it can produce in 1 week is listed in the file P04_101.xlsx. For example, furnace 1 could produce 350 300-foot ingots per week. The material in an ingot costs $10 per foot. A customer who wants a 100-foot or 200-foot ingot will accept an ingot of that length or longer. How can Aluminaca minimize the material costs incurred in meeting required weekly demands?
102. Each day, Eastinghouse produces capacitors during three shifts: 8 A.M. to 4 P.M., 4 P.M. to 12 A.M., and
12 A.M. to 8 A.M. The hourly salary paid to the employees on each shift, the price charged for each capacitor made during each shift, and the number of defects in each capacitor produced during a given shift are listed in the file P04_102.xlsx. The company can employ up to 25 workers, and each worker can be assigned to one of the three shifts. A worker produces 10 capacitors during a shift, but due to machinery limitations, no more than 10 workers can be assigned to any shift. Each capacitor produced can be sold, but the average number of defects per capacitor for the day’s production cannot exceed 3. Determine how Eastinghouse can maximize its daily profit.
103. During the next 3 months, Airco must meet (on time) the following demands for air conditioners: month 1,
300; month 2, 400; month 3, 500. Air conditioners can be produced in either New York or Los Angeles.
It takes 1.5 hours of skilled labor to produce an air conditioner in Los Angeles, and it takes 2 hours in
New York. It costs $400 to produce an air conditioner in Los Angeles, and it costs $350 in New York.
During each month, each city has 420 hours of skilled labor available. It costs $100 to hold an air conditioner in inventory for a month. At the beginning of month 1, Airco has 200 air conditioners in stock. Determine how Airco can minimize the cost of meeting air conditioner demands for the next
3 months.
104. Gotham City National Bank is open Monday through
Friday from 9 A.M. to 5 P.M. From past experience, the bank knows that it needs the numbers of tellers listed in the file P04_104.xlsx. Gotham City Bank hires two types of tellers. Full-time tellers work
9 A.M. to 5 P.M. five days a week, with one hour off each day for lunch. The bank determines when a fulltime employee takes his or her lunch hour, but each

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teller must go between 12 P.M. and 1 P.M. or between
1 P.M. and 2 P.M. Full-time employees are paid (including fringe benefits) $8 per hour, which includes payment for lunch hour. The bank can also hire parttime tellers. Each part-time teller must work exactly three consecutive hours each day. A part-time teller is paid $5 per hour and receives no fringe benefits. To maintain adequate quality of service, the bank has decided that, at most, five part-time tellers can be hired. Determine how to meet the bank’s teller requirements at minimum cost.
105. Based on Rothstein (1973). The Springfield City
Police Department employs 30 police officers. Each officer works 5 days per week. The crime rate fluctuates with the day of the week, so the number of police officers required each day depends on the day of the week, as follows: Saturday, 28; Sunday, 18;
Monday, 18; Tuesday, 24; Wednesday, 25; Thursday,
16; Friday, 21. The police department wants to schedule police officers to minimize the number whose days off are not consecutive. Determine how to accomplish this goal.
106. Based on Charnes and Cooper (1955). Alex Cornby makes his living buying and selling corn. On January
1, he has 500 tons of corn and $10,000. On the first day of each month, Alex can buy corn at the following prices per ton: January, $300; February, $350;
March, $400; April, $500. On the last day of each month, Alex can sell corn at the following prices per ton: January, $250; February, $400; March, $350;
April, $550. Alex stores his corn in a warehouse that can hold 1000 tons of corn. He must be able to pay cash for all corn at the time of purchase. Determine how Alex can maximize his cash on hand at the end of April.
107. City 1 produces 500 tons of waste per day, and city
2 produces 400 tons of waste per day. Waste must be incinerated at incinerator 1 or 2, and each incinerator can process up to 500 tons of waste per day. The cost to incinerate waste is $40 per ton at incinerator 1 and
$30 per ton at incinerator 2. Incineration reduces each ton of waste to 0.2 ton of debris, which must be dumped at one of two landfills. Each landfill can receive at most 200 tons of debris per day. It costs
$3 per mile to transport a ton of material (either debris or waste). Distances (in miles) between locations are listed in the file P04_107.xlsx. Determine how to minimize the total cost of disposing of the waste from both cities.
108. Based on Smith (1965). Silicon Valley Corporation
(Silvco) manufactures transistors. An important aspect of the manufacture of transistors is the melting of the element germanium (a major component of a transistor) in a furnace. Unfortunately, the melting process yields germanium of highly variable quality.

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Two methods can be used to melt germanium.
Method 1 costs $50 per transistor, and method 2 costs $70 per transistor. The qualities of germanium obtained by methods 1 and 2 are listed in the file
P04_108.xlsx. Silvco can refire melted germanium in an attempt to improve its quality. It costs $25 to refire the melted germanium for one transistor. The results of the refiring process are also listed in the file
P04_108.xlsx. For example, if grade 3 germanium is refired, half of the resulting germanium will be grade
3, and the other half will be grade 4. Silvco has sufficient furnace capacity to melt or refire germanium for at most 20,000 transistors per month. Silvco’s monthly demands are for 1000 grade 4 transistors,
2000 grade 3 transistors, 3000 grade 2 transistors, and 3000 grade 1 transistors. Determine how to minimize the cost of producing the needed transistors.
109. The Wild Turkey Company produces two types of turkey cutlets for sale to fast-food restaurants. Each type of cutlet consists of white meat and dark meat.
Cutlet 1 sells for $4 per pound and must consist of at least 70% white meat. Cutlet 2 sells for $3 per pound and must consist of at least 60% white meat. At most,
5000 pounds of cutlet 1 and 3000 pounds of cutlet 2 can be sold. The two types of turkey used to manufacture the cutlets are purchased from the GobbleGobble Turkey Farm. Each type 1 turkey costs $10 and yields 5 pounds of white meat and 2 pounds of dark meat. Each type 2 turkey costs $8 and yields 3 pounds of white meat and 3 pounds of dark meat.
Determine how to maximize Wild Turkey’s profit.
110. The production line employees at Grummins Engine work 4 days a week, 10 hours a day. Each day of the week, the following minimum numbers of line employees are needed: Monday through Friday,
70 employees; Saturday and Sunday, 30 employees.
Grummins employs 110 line employees. Determine how to maximize the number of consecutive days off received by these employees. For example, a worker who gets Sunday, Monday, and Wednesday off receives 2 consecutive days off.
111. Based on Lanzenauer et al. (1987). To process income tax forms, the IRS first sends each form through the data preparation (DP) department, where information is coded for computer entry. Then the form is sent to data entry (DE), where it is entered into the computer. During the next 3 weeks, the following quantities of forms will arrive: week 1,
40,000; week 2, 30,000; week 3, 60,000. All employees work 40 hours per week and are paid $500 per week. Data preparation of a form requires 15 minutes, and data entry of a form requires 10 minutes.
Each week, an employee is assigned to either data entry or data preparation. The IRS must complete processing all forms by the end of week 5 and wants

to minimize the cost of accomplishing this goal. Assume that all workers are full-time employees and that the IRS will have the same number of employees each week. Assume that all employees are capable of performing data preparation and data entry. Determine how many workers should be working and how the workers should allocate their hours during the next 5 weeks.
112. Based on Robichek et al. (1965). The Korvair
Department Store has $100,000 in available cash. At the beginning of each of the next 6 months, Korvair will receive revenues and pay bills as listed in the file P04_112.xlsx. It is clear that Korvair will have a short-term cash flow problem until the store receives revenues from the Christmas shopping season. To solve this problem, Korvair must borrow money.
At the beginning of July, the company takes out a
6-month loan. Any money borrowed for a 6-month period must be paid back at the end of December along with 9% interest (early payback does not reduce the total interest of the loan). Korvair can also meet cash needs through month-to-month borrowing.
Any money borrowed for a 1-month period incurs an interest cost of 4% per month. Determine how Korvair can minimize the cost of paying its bills on time.
113. Mackk Engine produces diesel trucks. New government emission standards have dictated that the average pollution emissions of all trucks produced in the next 3 years cannot exceed 10 grams per truck.
Mackk produces 2 types of trucks. Each type 1 truck sells for $20,000, costs $15,000 to manufacture, and emits 15 grams of pollution. Each type 2 truck sells for $17,000, costs $14,000 to manufacture, and emits
5 grams of pollution. Production capacity limits total truck production during each year to at most
320 trucks. The maximum numbers of each truck type that can be sold during each of the next 3 years are listed in the file P04_113.xlsx. Demand can be met from previous production or the current year’s production. It costs $2000 to hold 1 truck (of any type) in inventory for 1 year. Determine how Mackk can maximize its profit during the next 3 years.
114. Each hour from 10 A.M. to 7 P.M., Bank One receives checks and must process them. Its goal is to process all checks the same day they are received. The bank has 13 check processing machines, each of which can process up to 500 checks per hour. It takes one worker to operate each machine. Bank One hires both full-time and part-time workers. Full-time workers work 10 A.M. to 6 P.M., 11 A.M. to 7 P.M., or
12 P.M. to 8 P.M. and are paid $160 per day. Part-time workers work either 2 P.M. to 7 P.M. or 3 P.M. to 8 P.M. and are paid $75 per day. The numbers of checks received each hour are listed in the file P04_114.xlsx.
In the interest of maintaining continuity, Bank One believes that it must have at least 3 full-time workers

under contract. Develop a work schedule that processes all checks by 8 P.M. and minimizes daily labor costs.
115. Owens-Wheat uses 2 production lines to produce
3 types of fiberglass mat. The demand requirements
(in tons) for each of the next 4 months are shown in the file P04_115.xlsx. If it were dedicated entirely to the production of one product, a line 1 machine could produce either 20 tons of type 1 mat or 30 tons of type 2 mat during a month. Similarly, a line 2 machine could produce either 25 tons of type 2 mat or
28 tons of type 3 mat. It costs $5000 per month to operate a machine on line 1 and $5500 per month to operate a machine on line 2. A cost of $2000 is incurred each time a new machine is purchased, and a cost of $1000 is incurred if a machine is retired from service. At the end of each month, Owens would like to have at least 50 tons of each product in inventory.
At the beginning of month 1, Owens has 5 machines on line 1 and 8 machines on line 2. Assume the perton cost of holding either product in inventory for
1 month is $5.
a. Determine a minimum cost production schedule for the next 4 months.
b. There is an important aspect of this situation that cannot be modeled by linear programming. What is it? (Hint: If Owens makes product 1 and product 2 on line 1 during a month, is this as efficient as making just product 1 on line 1?)
116. Rylon Corporation manufactures Brute cologne and
Chanelle perfume. The raw material needed to manufacture each type of fragrance can be purchased for $60 per pound. Processing 1 pound of raw material requires 1 hour of laboratory time. Each pound of processed raw material yields 3 ounces of Regular
Brute cologne and 4 ounces of Regular Chanelle perfume. Regular Brute can be sold for $140 per ounce and Regular Chanelle for $120 per ounce.
Rylon also has the option of further processing Regular Brute and Regular Chanelle to produce Luxury
Brute, sold at $360 per ounce, and Luxury Chanelle, sold at $280 per ounce. Each ounce of Regular Brute processed further requires an additional 3 hours of laboratory time and a $40 processing cost and yields
1 ounce of Luxury Brute. Each ounce of Regular
Chanelle processed further requires an additional 2 hours of laboratory time and a $40 processing cost and yields 1 ounce of Luxury Chanelle. Each year,
Rylon has 6000 hours of laboratory time available and can purchase up to 4000 pounds of raw material.
a. Determine how Rylon can maximize its profit.
Assume that the cost of the laboratory hours is a fixed cost (so that it can be ignored for this problem). b. Suppose that 1 pound of raw material can be used to produce either 3 ounces of Brute or 4 ounces of

4.9 Conclusion

203

Chanelle. How does your answer to part a change? c. Use SolverTable to determine how a change in the price of Luxury Chanelle changes the optimal profit. d. Use SolverTable to determine how simultaneous changes in lab time and raw material availability change the optimal profit.
e. Use SolverTable to determine how a change in the lab time required to process Luxury Brute changes the optimal profit.
117. Sunco Oil has three different processes that can be used to manufacture various types of gasoline. Each process involves blending oils in the company’s catalytic cracker. Running process 1 for an hour costs
$20 and requires 2 barrels of crude oil 1 and 3 barrels of crude oil 2. The output from running process 1 for an hour is 2 barrels of gas 1 and 1 barrel of gas 2.
Running process 2 for an hour costs $30 and requires
1 barrel of crude 1 and 3 barrels of crude 2. The output from running process 2 for an hour is 3 barrels of gas 2. Running process 3 for an hour costs $14 and requires 2 barrels of crude 2 and 3 barrels of gas 2.
The output from running process 3 for an hour is
2 barrels of gas 3. Each month, 4000 barrels of crude
1, at $45 per barrel, and 7000 barrels of crude 2, at
$55 per barrel, can be purchased. All gas produced can be sold at the following per-barrel prices: gas 1,
$85; gas 2, $90; gas 3, $95. Determine how to maximize Sunco’s profit (revenues less costs). Assume that only 2500 hours of time on the catalytic cracker are available each month.
118. Flexco produces six products in the following manner. Each unit of raw material purchased yields 4 units of product 1, 2 units of product 2, and 1 unit of product 3. Up to 1200 units of product 1 can be sold, and up to 300 units of product 2 can be sold. Demand for products 3 and 4 is unlimited. Each unit of product 1 can be sold or processed further. Each unit of product 1 that is processed further yields 1 unit of product 4. Each unit of product 2 can be sold or processed further. Each unit of product 2 that is processed further yields 0.8 unit of product 5 and 0.3 unit of product 6.
Up to 1000 units of product 5 can be sold, and up to 800 units of product 6 can be sold. Up to 3000 units of raw material can be purchased at $6 per unit.
Leftover units of products 5 and 6 must be destroyed.
It costs $4 to destroy each leftover unit of product 5 and $3 to destroy each leftover unit of product 6.
Ignoring raw material purchase costs, the unit price and production cost for each product are listed in the file P04_118.xlsx. Determine a profit-maximizing production schedule for Flexco.
119. Each week, Chemco can purchase unlimited quantities of raw material at $6 per pound. Each pound of

204

Chapter 4 Linear Programming Models

purchased raw material can be used to produce either input 1 or input 2. Each pound of raw material can yield 2 ounces of input 1, requiring 2 hours of processing time and incurring $2 in processing costs.
Each pound of raw material can yield 3 ounces of input 2, requiring 2 hours of processing time and incurring $4 in processing costs. Two production processes are available. It takes 2 hours to run process
1, requiring 2 ounces of input 1 and 1 ounce of input
2. It costs $1 to run process 1. Each time process 1 is run, 1 ounce of product A and 1 ounce of liquid waste are produced. Each time process 2 is run requires 3 hours of processing time, 2 ounces of input 2, and 1 ounce of input 1. Each process 2 run yields 1 ounce of product B and 0.8 ounce of liquid waste. Process 2 incurs $8 in costs. Chemco can dispose of liquid waste in the Port Charles River or use the waste to produce product C or product D. Government regulations limit the amount of waste Chemco is allowed to dump into the river to 1000 ounces per week.
Each ounce of product C costs $4 to produce and sells for $11. Producing 1 ounce of product C requires 1 hour of processing time, 2 ounces of input 1, and 0.8 ounce of liquid waste. Each ounce of product
D costs $5 to produce and sells for $7. Producing 1 ounce of product D requires 1 hour of processing time, 2 ounces of input 2, and 1.2 ounces of liquid waste. At most, 5000 ounces of product A and 5000 ounces of product B can be sold each week, but weekly demand for products C and D is unlimited.
Product A sells for $18 per ounce and product B sells for $24 per ounce. Each week, 6000 hours of processing time are available. Determine how Chemco can maximize its weekly profit.
120. Bexter Labs produces three products: A, B, and C.
Bexter can sell up to 30 units of product A, up to 20 units of product B, and up to 20 units of product C.
Each unit of product C uses 2 units of A and 3 units of B and incurs $5 in processing costs. Products A and B are produced from either raw material 1 or raw material 2. It costs $6 to purchase and process 1 unit of raw material 1. Each processed unit of raw material 1 yields 2 units of A and 3 units of B. It costs $3 to purchase and process a unit of raw material 2.
Each processed unit of raw material 2 yields 1 unit of
A and 2 units of B. The unit prices for the products are A, $5; B, $4; C, $25. The quality levels of each product are A, 8; B, 7; C, 6. The average quality level of the units sold must be at least 7. Determine how to maximize Bexter’s profit.
121. Based on Franklin and Koenigsberg (1973). The city of Busville contains three school districts. The numbers of minority and nonminority students in each district are given in the file P04_121.xlsx. The local court has decided that each of the town’s two high schools (Cooley High and Walt Whitman High)

must have approximately the same percentage of minority students (within 5%) as the entire town.
The distances (in miles) between the school districts and the high schools are also given in the file P04_121.xlsx. Each high school must have an enrollment of 300 to 500 students. Determine an assignment of students to schools that minimizes the total distance students must travel to school.
122. Based on Carino and Lenoir (1988). Brady
Corporation produces cabinets. Each week, Brady requires 90,000 cubic feet of processed lumber. The company can obtain lumber in two ways. First, it can purchase lumber from an outside supplier and then dry it at the Brady kiln. Second, Brady can chop down trees on its land, cut them into lumber at its sawmill, and then dry the lumber at its kiln. The company can purchase grade 1 or grade 2 lumber.
Grade 1 lumber costs $3 per cubic foot and when dried yields 0.7 cubic foot of useful lumber. Grade 2 lumber costs $7 per cubic foot and when dried yields
0.9 cubic foot of useful lumber. It costs the company
$3 to chop down a tree. After being cut and dried, a log yields 0.8 cubic feet of lumber. Brady incurs costs of $4 per cubic foot of lumber it dries. It costs
$2.50 per cubic foot of logs sent through the sawmill.
Each week, the sawmill can process up to 35,000 cubic feet of lumber. Each week, up to 40,000 cubic feet of grade 1 lumber and up to 60,000 cubic feet of grade 2 lumber can be purchased. Each week,
40 hours of time are available for drying lumber.
The time it takes to dry 1 cubic foot of lumber is as follows: grade 1, 2 seconds; grade 2, 0.8 second; log, 1.3 seconds. Determine how Brady can minimize the weekly cost of meeting its demand for processed lumber.
123. Based on Dobson and Kalish (1988). Chandler
Enterprises produces two competing products, A and
B. The company wants to sell these products to two groups of customers, group 1 and group 2. The values each customer places on a unit of A and B are shown in the file P04_123.xlsx. Each customer will buy either product A or product B, but not both. A customer is willing to buy product A if she believes that the Premium of product A is greater than or equal to the Premium of product B and Premium of product A is greater than or equal to 0. Here, the
“premium” of a product is its value minus its price.
Similarly, a customer is willing to buy B if she believes the Premium of product B is greater than or equal to the Premium of product A and the Premium of product B is greater than or equal to 0. Group 1 has 1000 members, and group 2 has 1500 members.
Chandler wants to set prices for each product to ensure that group 1 members purchase product A and group 2 members purchase product B. Determine how Chandler can maximize its revenue.

124. Based on Robichek et al. (1965). At the beginning of month 1, Finco has $400 in cash. At the beginning of months 1, 2, 3, and 4, Finco receives certain revenues, after which it pays bills. (See the file
P04_124.xlsx.) Any money left over can be invested for 1 month at the interest rate of 0.1% per month; for 2 months at 0.5% per month; for 3 months at 1% per month; or for 4 months at 2% per month. Determine an investment strategy that maximizes cash on hand at the beginning of month 5.
125. During each 6-hour period of the day, the Bloomington Police Department needs at least the number of police officers shown in the file P04_125.xlsx. Police officers can be hired to work either 12 consecutive hours or 18 consecutive hours. Police officers are paid
$4 per hour for each of the first 12 hours they work in a day and $6 per hour for each of the next 6 hours they work in a day. Determine how to minimize the cost of meeting Bloomington’s daily police requirements.
126. Based on Glassey and Gupta (1978). A paper recycling plant processes box board, tissue paper, newsprint, and book paper into pulp that can be used to produce three grades of recycled paper (grades 1,
2, and 3). The prices per ton and the pulp contents of the four inputs are shown in the file P04_126.xlsx.
Two methods, de-inking and asphalt dispersion, can be used to process the four inputs into pulp. It costs $20 to de-ink a ton of any input. The process of de-inking removes 10% of the input’s pulp, leaving
90% of the original pulp. It costs $15 to apply asphalt dispersion to a ton of material. The asphalt dispersion process removes 20% of the input’s pulp.
At most, 3000 tons of input can be run through the asphalt dispersion process or the de-inking process.
Grade 1 paper can be produced only with newsprint or book paper pulp; grade 2 paper only with book paper, tissue paper, or box board pulp; and grade 3 paper only with newsprint, tissue paper, or box board pulp. To meet its current demands, the company needs 500 tons of pulp for grade 1 paper, 500 tons of pulp for grade 2 paper, and 600 tons of pulp for grade 3 paper. Determine how to minimize the cost of meeting the demands for pulp.
127. At the beginning of month 1, GE Capital has 50 million accounts. Of these, 40 million are paid up (0due), 4 million are 1 month overdue (1-due), 4 million are 2 months overdue (2-due), and 2 million are
3 months overdue (3-due). After an account is more than 3 months overdue, it’s written off as a bad debt.
For each overdue account, GE Capital can either phone the cardholder, send a letter, or do nothing.
A letter requires an average of 0.05 hour of labor, whereas a phone call requires an average of 0.10 hour of labor. Each month 500,000 hours of labor are available. We assume that the average amount of a monthly payment is $30. Thus, if a 2-due account

4.9 Conclusion

205

remains 2-due, it means that 1 month’s payment
($30) has been received, and if a 2-due account becomes 0-due, it means that 3 months’ payments
($90) have been received. On the basis of thousands of accounts, DMMs (Delinquency Movement
Matrices) shown in the file P04_127.xlsx have been estimated. For example, the top-left 0.60 entry in the first table means that 60% of all 1-due accounts that receive a letter become 0-due by the next month. The
0.10 and 0.30 values in this same row mean that 10% of all 1-due accounts remain 1-due after receiving a letter, and 30% of all 1-due accounts become 2-due after receiving a letter. Your goal is to determine how to allocate your workforce over the next 4 months to maximize the expected collection revenue received during that time. (Note: 0-due accounts are never contacted, which accounts for the lack of 0-due rows in the first two tables.)
128. It is February 15, 2006. Three bonds, as listed in the file P04_128.xlsx, are for sale. Each bond has a face value of $100. Every 6 months, starting 6 months from the current date and ending at the expiration date, each bond pays 0.5*(coupon rate)*(Face value).
At the expiration date the face value is paid. For example, the second bond pays
■ $2.75 on 8/15/06
■ $102.75 on 2/15/07
Given the current price structure, the question is whether there is a way to make an infinite amount of money. To answer this, we look for an arbitrage. An arbitrage exists if there is a combination of bond sales and purchases today that yields
■ a positive cash flow today
■ nonnegative cash flows at all future dates
If such a strategy exists, then it is possible to make an infinite amount of money. For example, if buying
10 units of bond 1 today and selling 5 units of bond 2 today yielded, say, $1 today and nothing at all future dates, then we could make $k by purchasing 10k units of bond 1 today and selling 5k units of bond 2 today. We could also cover all payments at future dates from money received on those dates. Clearly, we expect that bond prices at any point in time will be set so that no arbitrage opportunities exist.
a. Show that an arbitrage opportunity exists for the bonds in the file P04_128.xlsx. (Hint: Set up an
LP that maximizes today’s cash flow subject to constraints that cash flow at each future date is nonnegative. You should get a “no convergence” message from Solver.)
b. Usually bonds are bought at an ask price and sold at a bid price. Consider the same three bonds as before and suppose the ask and bid prices are as listed in the same file. Show that these bond prices admit no arbitrage opportunities.

206

Chapter 4 Linear Programming Models

Modeling Problems
129. You have been assigned to develop a model that can be used to schedule employees at a local fast-food restaurant. Assume that computer technology has advanced to the point where very large problems can be solved on a PC at the restaurant.
a. What data would you collect as inputs to your model? b. Describe in words several appropriate objective functions for your model.
c. Describe in words the constraints needed for your model. 130. You have been assigned to develop a model that can be used to schedule the nurses working in a maternity ward.
a. What data would you collect as inputs to your model? b. Describe in words several appropriate objective functions for your model.
c. Describe in words the constraints needed for your model. 131. Keefer Paper produces recycled paper from paper purchased from local offices and universities. The company sells three grades of paper: high-brightness paper, medium-brightness paper, and low-brightness paper. The high-brightness paper must have a brightness level of at least 90, the medium-brightness paper must have a brightness level of between 80 and
90, and the low-brightness paper must have a brightness level no greater than 80. Discuss how Keefer might use a blending model to maximize its profit.
132. In this chapter, we give you the cost of producing a product and other inputs that are used in the analysis.
Do you think most companies find it easy to determine the cost of producing a product? What difficulties might arise?
133. Discuss how the aggregate planning model could be extended to handle a company that produces several products on several types of machines.
What information would you need to model this type of problem?
134. A large CPA firm currently has 100 junior staff members and 20 partners. In the long run—say,
20 years from now—the firm would like to consist of 130 junior staff members and 20 partners. During a given year, 10% of all partners and 30% of all junior staff members leave the firm. The firm can control the number of hires each year and the fraction of junior employees who are promoted to partner each year. Can you develop a personnel strategy that would meet the CPA firm’s goals?

CASE

4.1 AMARCO, I NC . 10

audi Arabia is a kingdom in the Middle East with an area of 865,000 square miles, occupying about four-fifths of the Arabian Peninsula. With a population of about 10 million, this Muslim and Arab state is generally recognized as being formed in 1927 when
Ibn Sa’ud united the country and was acknowledged as the sovereign independent ruler. Summer heat is intense in the interior, reaching 124°F, but it is dry and tolerable in contrast to coastal regions and some highlands, which have high humidity during the summer. Winters (December through February) are cool, with the coldest weather occurring at high altitudes and in the far north. A minimum temperature recorded at at-Turayf in 1950 was 10°F, and it was accompanied by several inches of snow and an inch of ice on ponds. Average winter temperatures are 74°F at Jidda and 58°F at Riyadh (the capital city), which has an annual precipitation of 2.5 to 3 inches.
After oil was discovered in Bahrain in 1932, many companies turned to Saudi Arabia and started exploring. Thus, in 1937, the American Arabian Oil Company, Inc. (AMARCO), was formed as a joint venture between Standard Oil Company of California
(SOCAL) and the Government of Saudi Arabia to explore, produce, and market any petroleum found in the country. The year before, a geologist from
SOCAL had discovered a small quantity of oil in the
Eastern Province at Dammam Dome, on which the oil company town of Dhahran is now built. It was just beginning to be developed when another discovery was made—of what was to prove to be the largest oil field in the world. Called the Ghamar field, it would start Saudi Arabia on the road to becoming a highly developed country in just a generation. Located about 50 miles inland from the western shores of the
Persian Gulf, the Ghamar field is a structural accumulation along 140 miles of a north–south anticline. The productive area covers approximately 900 square miles, and the vertical oil column is about 1,300 feet.
It is generally considered to have recoverable reserves of about 75 billion barrels of oil. Total proven reserves in Saudi Arabia are estimated at more than
500 billion barrels, enough for more than a hundred years of production.

S

10
This case was written by William D. Whisler, California State
University, Hayward.

Since 1950, Saudi Arabia has experienced greater and more rapid changes than it had in the several preceding centuries. For example, during this time, as skilled nationals became available, more and more of the exploration, drilling, refining, and other production activities came under the control of the country.
SOCAL was left primarily with the marketing and transportation functions outside the country.
During the 1960s, AMARCO increased its profitability substantially by hiring Dr. George
Dantzig, then of the University of California, as a consultant. He supervised the development and implementation of LP models to optimize the production of different types of crude oils, their refining, and the marketing of some of their principal products. As a result of this effort, an operations research (OR) department was started in the company with the responsibility of continuing to review the firm’s operations to find other areas where costs might be decreased or profits increased by applications of OR.
Now attention is being focused on another aspect of one of the company’s small California refinery operations: the production of three types of aviation gasoline from the Saudi Arabian crude oil available. Recently, the marketing of petroleum products to the airline industry has become a rather substantial portion of AMARCO’s business. As shown in Figure 4.45, the three aviation gasolines, A,
B, and C, are made by blending four feedstocks:
Alkylate, Catalytic Cracked Gasoline, Straight Run
Gasoline, and Isopentane.
In Table 4.14,TEL stands for tetraethyl lead, which is measured in units of milliliters per gallon (ml/gal). Thus, a TEL of 0.5 means there is
0.5 milliliter of tetraethyl lead per gallon of feedstock. Table 4.14 shows that TEL does influence the octane number but does not influence the Reid Vapor Pressure.
Each type of aviation gasoline has a maximum permissible Reid Vapor Pressure of 7. Aviation gasoline A has a TEL level of 0.5 ml/gal and has a minimum octane number of 80. The TEL level of aviation gasolines B and C is 4 ml/gal, but the former has a minimum octane number of 91, whereas the latter has a minimum of 100.

Case 4.1 AMARCO, Inc.

207

Assume that all feedstocks going into aviation gasoline A are leaded at a TEL level of 0.5 ml/gal and that those going into aviation gasolines B and C are leaded at a TEL level of 4 ml/gal. Table 4.15 gives the

aviation gasoline data. A final condition is that marketing requires that the amount of aviation gas
A produced be at least as great as the amount of aviation gas B.

Figure 4.45
The Production of
Aviation Gasoline

Aviation
Gas B

Aviation
Gas A

Aviation
Gas C

Alkylate

Catalytic
Cracked
Gasoline

Straight
Run
Gasoline

Isopentane

Refinery

Crude Oil

Table 4.14

Stock Availabilitiesa
Feedstock
Straight
Run
Gasoline

Alkylate

Characteristic
Reid Vapor Pressure
Octane Number
If TEL is 0.5
If TEL is 4.0
Available (Bbl/day)
Value ($/Bbl)

Catalytic
Cracked
Gasoline

5

8

4

20

94
107.5
14,000
17.00

83
93
13,000
14.50

74
87
14,000
13.50

95
108
11,000
14.00

a

Isopentane

Some of the data in this case have been adapted from Walter W. Garvin, Introduction to Linear Programming (New York:
McGraw-Hill, 1960), Chapter 5.

Table 4.15

Aviation Gasoline Data
Aviation Gasoline

Characteristic
Minimum requirements (Bbl/day)
Price ($/Bbl)

208

A

B

C

12,000
15.00

13,000
16.00

12,000
16.50

Chapter 4 Linear Programming Models

Questions
1. AMARCO’s planners want to determine how the three grades of aviation gasoline should be blended from the available input streams so that the specifications are met and the income is maximized. Develop an LP spreadsheet model of the company’s problem.
2. Solve the linear programming model formulated in Question 1.

7.

The following questions should be attempted only after
Questions 1 and 2 have been answered correctly.
3. Suppose that a potential supply shortage of
Saudi Arabian petroleum products exists in the near future due to possible damage to
AMARCO’s oil production facilities from Iraqi attacks. This could cause the prices of the three types of aviation gasolines to double
(while the values of the stocks remain the same, because they are currently on hand).
How would this affect the refinery’s operations? If, after current stocks are exhausted, additional quantities must be obtained at values double those given in Table 4.14, how might AMARCO’s plans be affected?
4. Suppose that because of the new Iraqi crisis, the supply of alkylate is decreased by 1,800 bbl/day, catalytic cracked gas is decreased by
2,000 bbl/day, and straight run gasoline is decreased by 5,000 bbl/day. How does this affect AMARCO’s operations?
5. AMARCO is considering trying to fill the aviation gasoline shortage created by the new Iraqi crisis by increasing its own production. If additional quantities of alkylate, catalytic cracked gasoline, straight run gasoline, and isopentane are available, should they be processed? If so, how much of them should be processed, and how do their values affect the situation?
6. Due to the uncertainty about both the U.S. economy and the world economy resulting from the Iraqi crisis, AMARCO’s economists are considering doing a new market research study to reestimate the minimum requirement forecasts. With the economy continually weakening, it is felt that demand will decrease,

8.

9.

10.

possibly drastically, in the future. However, because such marketing research is expensive, management is wondering whether it would be worthwhile. That is, do changes in the minimum requirements have a significant effect on
AMARCO’s operations? What is the change in profit from an increase or a decrease in the minimum requirements? Over what ranges of demand do these profit changes apply?
Suppose that the Middle East crisis ends and a flood of oil fills the marketplace, causing the prices of aviation gasoline to drop to $10.00,
$11.00, and $11.50, respectively, for A, B, and C.
How would this affect the company’s plans?
Suppose that the U.S. government is considering mandating the elimination of lead from aviation gasoline to decrease air pollution. This law would be based on new technology that allows jet engines to burn unleaded gasoline efficiently at any octane level. Thus, there would no longer be any need for constraints on octane level.
How would such a new law affect AMARCO?
The Environmental Protection Agency is proposing regulations to decrease air pollution. It plans to improve the quality of aviation gasolines by decreasing the requirement on Reid Vapor
Pressure from 7 to 6. Management is concerned about this regulation and wonders how it might affect AMARCO’s profitability. Analyze and make a recommendation.
The Marketing Department indicates that
AMARCO will be able to increase its share of the market substantially with a new contract being negotiated with a new customer. The difficulty is that this contract will require that the amount of aviation gas A plus the amount of
B must be at least as great as the amount of C produced. Because aviation gasolines A and B are least profitable of the three, this could cause a big decrease in profit for the company. However, marketing indicates that this is a short-run view, because the “large” increase in market share with the concomitant long-run profit increases will more than offset the “temporary small decrease” in profits because of the additional restriction.What do you recommend? Why? ■
Case 4.1 AMARCO, Inc.

209

CASE

4.2 A MERICAN O FFICE S YSTEMS , I NC . 11

merican Office Systems, Inc., was established by the late R. J. Miller, Sr., in 1939. It started as an office supply story in Mountain View, California, and expanded slowly over the years into the manufacture of small office equipment, overhead projectors, and bookkeeping machines. In the 1950s, computers started eroding its market for bookkeeping machines, so the company diversified into the copy machine market. However, it never captured a large market share because bigger firms such as Xerox, Canon,
Sharp, and A. B. Dick were so firmly entrenched.
A few years ago, American Office Systems’ engineering staff developed an adapter that links a standard copy machine to personal computers, allowing a copy machine to be used as a laser printer, scanner, and fax. The adapters show great promise for both home and office use. However, the company is not well known by either the financial community or the copy machine market, principally due to its small size and rather lackluster record, so it could secure only $15 million in initial financial backing for the adapters. The $15 million was used to finance the construction of a small production facility and of administrative offices in 1994, and in 1995 production and sales began. Two versions of the adapter exist, one for IBM-compatible computers and one for
Macintosh computers. The former sells for $175 and the latter for $200.
At the beginning of December 1995, Dr. R. J.
Miller, II, President, convened a meeting about the coming year’s plans for the adapters. Rob Olsen,Vice
President of Production, argued that production facilities should be expanded: “Until we have sufficient capacity to produce the adapters,” he said,“there is no use advertising.” Sue Williams, Director of Marketing, replied,“On the contrary, without any demand for the adapters, there is no reason to produce them. We need to focus on advertising first.” J.T. Howell, the
Comptroller, pointed out that Olsen and Williams were talking about the situation as if it only involved a decision between production and marketing: “Yes, funds need to be allocated between production and advertising. However, more important than both is the cash flow difficulty that the company has been

A

11
This case was written by William D. Whisler, California State
University, Hayward.

210

Chapter 4 Linear Programming Models

experiencing. As you know, it was only yesterday that, finally, I was able to secure a $750,000 line of credit for the coming year from Citibank. I might add that it is at a very favorable interest rate of 16%. This will partially solve our cash flow problems and it will have a big effect on both production and advertising decisions. In addition, there are financial and accounting factors that must be allowed for in any decision about the adapters.” Olsen interjected, “Wow, this is more complicated than I anticipated originally. Before we make a decision, I think we ought to use some modern management science techniques to be sure that all the relevant factors are considered. Last week
I hired Carlos Garcia from Stanford. He has a Master’s
Degree in Operations Research. I think this would be a good project for him.” However, Williams said that she thinks that an executive, judgmental decision would be much better.“Let’s not get carried away with any of the quantitative mumbo-jumbo that Rob is always suggesting. Besides, his studies always take too much time and are so technical that no one can understand them. We need a decision by the end of next week.” After listening to the discussion, Miller decided to appoint an executive action team to study the problem and make a recommendation at next week’s meeting.“Rob and Sue, I want both of you to document your arguments in more detail. J.T., be more precise with your comments about the cash flow, accounting, and financial problems. And, by the way
Rob, have Carlos look into a model to see if it might produce some insights.”
Most of the $15 million initial financing was used to build a five-story building in Mountain View, south of San Francisco. Although currently only about 90% complete, it is being used. The first floor contains the production and shipping facilities plus a small storage area. A larger warehouse, already owned by the company, is located across the street. The other four floors of the building are for the engineering department (second floor), a research lab (third floor), and administration (top two floors). The production facility operates two shifts per day and has a production capacity of 30 IBM adapters and 10 Macintosh adapters per hour. Olsen uses 20 production days per month in his planning. Usually there are a few more, but these are reserved for maintenance and repairs. The last stage of the initial construction will

be finished by the beginning of the fourth quarter, making the building 100% finished. This will increase the production capacity rates by 10%.
Howell normally does the company’s financial planning monthly, and he assumes that cash flows associated with all current operating expenses, sales revenues (taking collections into account), advertising costs, loans from the line of credit, investments of excess cash in short-term government securities, and so forth, occur at the end of the corresponding month. Because he needs information for the meeting next week, however, he decides to do a rough plan on a quarterly basis. This means that all the just mentioned cash flows, and so on, will be assumed to occur at the end of the quarter. After the meeting, when more time is available, the plan will be expanded to a monthly basis. To get started, one of his senior financial analysts prepares the list of quarterly fixed operating expenses shown in Table 4.16. In addition, the accounting department calculates that the variable costs of the adapters are $100 each for the IBM version and $110 each for the Macintosh version.
Table 4.16

Quarterly Fixed Operating Expenses

Expense
Administrative expense
Fixed manufacturing costs
Sales agents’ salaries
Depreciation

Cost
$1,500,000
750,000
750,000
100,000

At present, American Office Systems is experiencing a cash flow squeeze due to the large cash requirements of the startup of the adapter production, advertising, and sales costs. If excess cash is available in any quarter, however, Howell says that the company policy is to invest it in short-term government securities, such as treasury bills. He estimates that during the coming year these investments will yield a return of 6%.
Olsen asks Garcia to look into the production and inventory aspects of the situation first, because this area was his specialty at Stanford. Then he says that he wants him to think about a programming model that might integrate all components of the

problem—production, sales, advertising, inventory, accounting, and finance. A mixed-integer programming model appears to be the most appropriate; however, he asks Garcia to use linear programming as an approximation due to the time limitations and
Williams’s concern about his ideas always being too technical. “There will be more time after next week’s meeting to refine the model,” he says.
After discussions with Olsen and Williams,
Garcia feels that something needs to be done to help the company handle the uncertainty surrounding future sales of the adapters. He points out that it is impossible to guarantee that the company will never be out of stock. However, it is possible to decrease shortages so that any difficulties associated with them would be small and they would not cause major disruptions or additional management problems, such as excess time and cost spent expediting orders, and so forth. Thus, Garcia formulates an inventory model. To be able to solve the model, he has to check the inventory levels of the adapters currently on hand in the warehouse. From these quantities, he calculates that there will be 10,000 IBM and 5,000 Macintosh adapters on hand at the beginning of 1996. Based on the results of the model, he recommends that a simple rule of thumb be used: production plus the end-of-period inventory for the adapters should be at least 10% larger than the estimated sales for the next period. This would be a safety cushion to help prevent shortages of the adapters. In addition, to provide a smooth transition to 1997, the inventory level plus production at the end of the fourth quarter of 1996 should be at least twice the maximum expected sales for that quarter.
Garcia says that using these rules of thumb will minimize annual inventory costs. When explaining the inventory model to Olsen, Garcia emphasizes the importance of including inventory carrying costs as part of any analysis, even though such costs frequently are not out-of-pocket. He says that his analysis of data provided by the accounting department yielded a 1% per month inventory carry cost, and this is what he used in his model.
Sales during the first year (1995) for the adapters are shown in Table 4.17. Next year’s sales are

Case 4.2 American Office Systems, Inc.

211

uncertain. One reason for the uncertainty is that they depend on the advertising. To begin the analysis,
Williams asks her marketing research analyst, Debra
Lu, to estimate the maximum sales levels for the coming four quarters if no advertising is done. Since last year’s sales of both models showed a steady increase throughout the year, Lu projects a continuation of the trend. She forecasts that the company will be able to sell any number of adapters up to the maximum expected sales amounts shown in Table 4.17.

the third quarter. The remaining 2% are written off and sold to a collection agency for $0.50 on the dollar.

Table 4.18

Collections

Quarter

IBM Adapters

1
2
3

0.75
0.20
0.03

Macintosh Adapters
0.80
0.11
0.05

Questions
Table 4.17 1995 Adapter Sales and Maximum
Expected 1996 Sales
1995 Sales
IBM
Macintosh
Quarter Adapters Adapters
1
2
3
4

5,000
6,000
7,000
8,000

1,000
1,200
1,400
1,600

1996 Maximum
Expected Sales
IBM
Macintosh
Adapters Adapters
9,000
10,000
11,000
12,000

1,800
2,000
2,200
2,400

Miller suggests that advertising in magazines such as PC World and Home Office will increase consumer awareness of both the company and adapters. The next day, Williams has a meeting with several staff members of a San Francisco advertising agency. They show her recommendations for two types of ads (one for the IBM adapters and one for the Macintosh adapters), give her cost information, and the estimated effectiveness of an advertising campaign. Armed with this information and some data from Lu, Williams prepares a brief report for
Miller setting out her reasons for thinking that each
$10 spent on advertising will sell an additional IBM adapter; the same relationship holds true for the
Macintosh adapter.
Based on an analysis of 1995 sales and accounts receivable, the accounting department determines that collection experience is as shown in Table 4.18. For example, 75% of the IBM adapters sold in a quarter are paid for during the quarter, 20% are paid for during the following quarter, and 3% are paid for during

212

Chapter 4 Linear Programming Models

1. Suppose that you are Garcia. Develop an LP spreadsheet model of the situation to help the executive action team make a decision about how to allocate funds between production and advertising so that all the cash flow, financial, accounting, marketing, inventory, and production considerations are taken into account and
American Office Systems’ profits are maximized.
Use the data collected and the estimates made by the members of the executive action team.
2. Solve the LP model formulated in Question 1.
The executive action team has assembled to reconsider the plans for the adapters for the coming year. Garcia, who developed the LP model, concludes his presentation by saying,“As everyone can see, the model gives the optimal solution that maximizes profits. Since I have incorporated the estimates and assumptions that all of you made, clearly it is the best solution. No other alternative can give a higher profit.” Even Williams, who initially was skeptical of using quantitative models for making executive-level decisions, is impressed and indicates that she will go along with the results.
Miller says,“Good work, Carlos! This is a complex problem but your presentation made it all seem so simple.
However, remember that those figures you used were based on estimates made by all of us. Some were little better than guesses. What happens if they are wrong? In other words, your presentation has helped me get a handle on the problem we are facing, and I know that models are useful where hard, accurate, data exist. However, with all the uncertainty in our situation and the many rough estimates made, it seems to me that I will still have to make a

judgment call when it comes down to making a final decision. Also, there has been a new development. J.T. tells me that we might be able to get another $1 million line of credit from a Bahamian bank. It will take a while to work out the details and maybe it will cost us a little.
I am wondering if it is worth it. What would we do with the $1 million if we got it?” T. J. responds,“We really need the $1 million. But it is a drop in the bucket. My analysis shows that we really need another $8 million line of credit.” Analyze, as Garcia is going to do, the effect of uncertainty and errors on the results of Questions 1 and 2 by answering the following questions.They should be attempted only after Questions 1 and 2 have been answered correctly.
3. One area where assumptions were made is adapter price.
a. What happens if the prices for the adapters are a little weak and they decrease to $173 for the IBM version and $198 for the
Macintosh version? Does this make any difference? b. What about decreases to $172 and $197, respectively, for the IBM and Macintosh versions? Explain the answers in terms that
Miller will understand.
c. Suppose that American Office Systems can increase the price of the adapters to $180 and $205. How would this affect the original solution? 4. Another potential variable is adapter production cost. a. Suppose that an error was made in determining the costs of the adapters and that they really should have been $102 for the IBM version and $112 for the Macintosh version.
What is the effect of this error?
b. What about costs of $105 and $115? Explain the answers in terms that Miller will understand. 5. Howell notes that one of the contributing factors to American Office Systems’ cash squeeze is the slow collection of accounts receivable. He is considering adopting a new collection procedure recommended by a consulting company. It will

cost $100,000 and will change the collection rates to those given in Table 4.19.
a. Analyze the effect of this new collection policy and make a recommendation to Howell about whether to implement the new procedure. As before, any accounts receivable not collected by the end of the third quarter will be sold to a collection agency for $0.50 on the dollar.
b. Howell wonders whether switching to selling adapters for all cash is worth the effort. This would ameliorate the cash squeeze because it would eliminate not only the slow collections but also the use of the collection agency for accounts that remain unpaid after 9 months.
It would cost about $90,000 more than at present to implement the all-cash policy because the accounting system would need to be modified and personnel would have to be retrained. Analyze this possibility and make a recommendation to Howell.

Table 4.19

New Collections

Quarter

IBM Adapters

Macintosh Adapters

1
2
3

0.90
0.07
0.01

0.92
0.03
0.01

6. Yet another variable is advertising effectiveness.
a. Suppose that Williams overestimated the effectiveness of advertising. It now appears that $100 is needed to increase sales by one adapter. How will this affect the original solution? Explain the answer in terms that Miller will understand.
b. What happens if the required advertising outlay is $12.50 per additional adapter sold?
7. Suppose that the line of credit from Citibank that
Howell thought he had arranged did not work out because of the poor financial situation of the company. The company can obtain one for the same amount from a small local bank; however, the interest rate is much higher, 24%. Analyze how this change affects American Office Systems.

Case 4.2 American Office Systems, Inc.

213

8. The safety cushion for inventory is subject to revision. a. Suppose that Garcia finds a bug in his original inventory model. Correcting it results in a safety cushion of 15% instead of the 10% he suggested previously. Determine whether this is important.
b. What if the error is 20%? Explain the answers in terms that Miller will understand.
9. Production capacity is scheduled to increase by
10% in the fourth quarter.
a. Suppose that Miller is advised by the construction company that the work will not be finished until the following year. How will this delay affect the company’s plans?
b. In addition to the delay in part a, suppose that an accident in the production facility damages some of the equipment so that the capacity is decreased by 10% in the fourth quarter. Analyze how this will affect the original solution.

214

Chapter 4 Linear Programming Models

10. Williams is worried about the accuracy of
Lu’s 1996 maximum expected sales forecasts. If errors in these forecasts have a big effect on the company profits, she is thinking about hiring a
San Francisco marketing research firm to do a more detailed analysis. They would charge
$50,000 for a study. Help Williams by analyzing what would happen if Lu’s forecasts are in error by 1,000 for IBM adapters and 200 for Macintosh adapters each quarter. Should she hire the marketing research firm?
11. a. To determine whether the extra $1 million line of credit is needed, analyze its effect on the original solution given in Question 2.
b. To fully understand the ramifications of the extra $1,000,000 line of credit, redo (1) Question 3b, (2) Question 4b, (3) Question 6a, and
(4) Question 8b. Summarize your results.
c. What about Howell’s claim that an extra
$8,000,000 line of credit is necessary? Use that adjustment and redo Question 6a. ■

CASE

4.3 L AKEFIELD C ORPORATION ’ S O IL T RADING D ESK

akefield Corporation’s oil trading desk buys and sells oil products (crude oil and refined fuels), options, and futures in international markets. The trading desk is responsible for buying raw material for Lakefield’s refining and blending operations and for selling final products. In addition to trading for the company’s operations, the desk also takes speculative positions. In speculative trades, the desk attempts to profit from its knowledge and information about conditions in the global oil markets.
One of the traders, Lisa Davies, is responsible for transactions in the cash market (as opposed to the futures or options markets). Lisa has been trading for several years and has seen the prices of oil-related products fluctuate tremendously. Figure 4.46 shows the prices of heating oil #2 and unleaded gasoline from January 1986 through July 1992. Although excessive volatility of oil prices is undesirable for most businesses, Lakefield’s oil trading desk often makes substantial profits in periods of high volatility.
The prices of various oil products tend to move together over long periods of time. Because finished oil products are refined from crude oil, the prices of

all finished products tend to rise if the price of crude increases. Because finished oil products are not perfect substitutes, the prices of individual products do not move in lockstep. In fact, over short time periods, the price movements of two products can have a low correlation. For example, in late 1989 and early
1990, there was a severe cold wave in the northeastern United States. The price of heating oil rose from $0.60 per gallon to over $1 per gallon. In the same time period, the price of gasoline rose just over
$0.10 per gallon.
Lisa Davies believes that some mathematical analysis might be helpful to spot trading opportunities in the cash markets. The next section provides background about a few important characteristics of fuel oils, along with a discussion of the properties of blended fuels and some implications for pricing.

L

Characteristics of Hydrocarbon Fuels
The many varieties of hydrocarbon fuels include heating oil, kerosene, gasoline, and diesel oil. Each type of fuel has many characteristics, for example, heat content, viscosity, freeze point, luminosity,

120

Figure 4.46
Price of Heating
Oil #2 and Unleaded
Gasoline
Price (cents/gallon)

100

80

60

40

20
Jan-86

Jan-87

Jan-88

Jan-89

Jan-90

Jan-91

Jan-92

Date

Heating Oil #2

Unleaded Gasoline

Case 4.3 Lakefield Corporation’s Oil Trading Desk

215

volatility (speed of vaporization), and so on. The relative importance of each characteristic depends on the intended use of the fuel. For example, octane rating is one of the most important characteristics of gasoline. Octane is a measure of resistance to ignition under pressure. An engine burning lowoctane fuel is susceptible to “engine knock,” which reduces its power output. Surprisingly, octane rating is more important than heat content for gasoline. In contrast, the most important characteristic of kerosene jet fuel is its heat content, but viscosity is also important. High viscosity fuels do not flow as smoothly through fuel lines.
For the types of fuels Lisa Davies usually trades, the most important characteristics are density, viscosity, sulfur content, and flash point, which are described next.When trading and blending other fuels, characteristics besides these four are important to consider.
Density The density of a substance is its mass per unit volume (e.g., grams per cubic centimeter). The density of water is 1 g/cc. A related measure is
American Petroleum Institute gravity (API), which is measured in degrees. API is related to density by
141.5
API ϭ ᎏᎏ Ϫ 131.5
D
where D is density measured in g/cc. Water has an
API of 10°. Note that density and API are inversely related. The specifications for kerosene jet fuel are nearly identical for all civilian airlines worldwide.
Kerosene jet fuel should have an API gravity between
37° and 51. Diesel fuel and heating oil are required to have an API not less than 30. API is important for controlling the flow of fuel in a combustion engine. It can also be used to limit the concentration of heavy hydrocarbon compounds in the fuel.
Viscosity Viscosity refers to the resistance of a liquid to flow. A highly viscous liquid, such as ketchup or molasses, does not pour easily. Viscosity is measured by the amount of time a specified volume of liquid takes to flow through a tube of a certain diameter. It is commonly measured in units of centistokes (hundredths of stokes). Most fuel

216

Chapter 4 Linear Programming Models

specifications place upper limits on viscosity. Less viscous fuel flows easily through lines and atomizes easily for efficient combustion. More viscous fuels must be heated initially to reduce viscosity.
Sulfur Content The content of sulfur is measured in percentage of total sulfur by weight. For example, a fuel with 2% sulfur content has 2 grams of sulfur for every 100 grams of fuel. Sulfur causes corrosion and abrasion of metal surfaces. Low sulfur content is important for maintaining the proper operation of equipment.
Flash Point The flash point of a substance is the lowest temperature at which the substance ignites when exposed to a flame. The product description of kerosene jet fuel from the American Society for
Testing and Materials specifies a flash point of at least
100°F. The New York Mercantile Exchange futures contract for heating oil #2 specifies a flash point of at least 130°F. Flash-point restrictions are often prescribed for safety reasons.
Table 4.20 gives a description of some fuels and their prices on a given day. In Table 4.20, the units of viscosity are centistokes, sulfur is given in percentage by weight, and flash point is in degrees Fahrenheit.
For convenience, all prices in Table 4.20 are given in dollars per barrel. In practice, the prices of heating oil, gasoline, and kerosene jet fuel are typically quoted in cents per gallon. (There are 42 gallons in a barrel.) Blending Fuels
Because hydrocarbon fuels are made of similar compounds and have similar characteristics, a certain degree of substitutability exists among fuels. Different fuels can also be blended to form a new fuel.
Next we describe how the characteristics of the individual fuels combine in the blended fuel.
Sulfur combines linearly by weight. This means, for example, that mixing equal weights of a 1% sulfur oil with a 3% sulfur oil produces a 2% sulfur oil. To a close approximation, sulfur combines linearly by volume (because the densities of oils are not very different). That is, combining 0.5 barrel of 1% sulfur oil with 0.5 barrel of 3% sulfur oil gives 1 barrel of very nearly 2% sulfur oil.

Table 4.20

Description of Available Fuels
Fuel 1
1% Sulfur
Fuel Oil

API
Viscosity
Sulfur
Flash point
Price

Fuel 3
0.7% Sulfur
Fuel Oil

10.50
477.00
1.00
140.00
16.08

10.50
477.00
3.00
140.00
13.25

10.50
477.00
0.70
140.00
17.33

Fuel 7
0.5% Vacuum
Gas Oil

API
Viscosity
Sulfur
Flash point
Price

Fuel 2
3% Sulfur
Fuel Oil

Fuel 8
Straight Run
(Low Sulfur)

25.00
25.00
0.50
200.00
21.46

21.00
212.00
0.30
250.00
21.00

Fuel 5
1% Vacuum
Gas Oil

Fuel 6
2% Vacuum
Gas Oil

34.00
3.50
0.20
130.00
24.10

25.00
25.00
1.00
200.00
20.83

25.00
25.00
2.00
200.00
20.10

Fuel 9
Straight Run
(High Sulfur)

Fuel 10
Kerosene
Jet Fuel

Fuel 11
Diesel
Fuel

Fuel 12

17.00
212.00
2.75
250.00
20.00

46.000
1.500
0.125
123.000
25.520

35.00
2.50
0.20
150.00
24.30

Ϫ4.50
261.00
2.37
109.00
11.50

In general, to say that a certain property of oil combines linearly (by volume) means the following:
Suppose xj barrels of oil j (for j ϭ 1, 2, . . . , n) are blended together to form one barrel of oil; that is, n ͚jϭ1xj ϭ 1. Also suppose that cj is the measure of the property of oil j. Then if the property combines linearly, the measure of the property for the blended n oil is a linear combination of the cj’s; that is, ͚jϭ1cjxj.
API gravity does not combine linearly, but density does combine linearly. For example, consider blending 0.5 barrel of oil that has a density of 0.8 g/cc with 0.5 barrel of oil with a density of 1.2 g/cc.
The resulting barrel of oil has a density of 1.0
(ϭ 0.8[0.5] ϩ 1.2[0.5]). The 0.8 g/cc density oil has an API of 45.38 , and the 1.2 g/cc density oil has an
API of Ϫ13.58°. If API combined linearly, the blended barrel of oil would have an API of 15.90°(ϭ 45.38
[0.5] Ϫ13.58 [0.5]). However, an API of 15.90 corresponds to a density of 0.96 g/cc, not 1.0 g/cc.12
Viscosity, measured in centistokes, does not combine linearly. However, chemical engineers have
12

To convert API to density, use D ϭ 141.5͞(API ϩ 131.5).

Fuel 4
Heating
Oil

Slurry

determined that viscosity can be transformed to another measure, called linear viscosity, which
(nearly) combines linearly.13 Similarly, flash points measured in degrees Fahrenheit do not combine linearly. But chemical engineers defined a new measure, termed linear flash point, which does combine linearly.14 Table 4.21 summarizes the properties of the 12 fuels measured in units that combine linearly.

Implications for Pricing
Sulfur in oil is a contaminant. Therefore, oil with a low sulfur content is more valuable than oil with a higher sulfur content, all other characteristics being equal. This relationship can be seen in Table 4.20 by comparing the prices of fuels 1, 2, and 3 and fuels 5,
13

Let vs represent viscosity measured in centistokes. Then linear viscosity, denoted v, is defined v ϭ ln(ln[vs ϩ 0.08]).
14
Let fp denote flash point measured in degrees Fahrenheit. Then linear flash point is defined f ϭ 1042(fp ϩ 460)Ϫ14.286. Empirical analysis of oil blending data confirms that the measure f combines nearly linearly.

Case 4.3 Lakefield Corporation’s Oil Trading Desk

217

Table 4.21 Properties of Available Fuels Measured in Units That Combine Linearly
Fuel 1
1% Sulfur
Fuel Oil

Density
Linear visc.
Sulfur
Linear flash
Price

Fuel 3
0.7% Sulfur
Fuel Oil

Fuel 4
Heating
Oil

Fuel 5
1% Vacuum
Gas Oil

Fuel 6
2% Vacuum
Gas Oil

0.996
1.819
1.000
204.800
16.080

0.996
1.819
3.000
204.800
13.250

0.996
1.819
0.700
204.800
17.330

0.855
0.243
0.200
260.400
24.100

0.904
1.170
1.000
52.500
20.830

0.904
1.170
2.000
52.500
20.100

Fuel 7
0.5% Vacuum
Gas Oil

Density
Linear visc.
Sulfur
Linear flash
Price

Fuel 2
3% Sulfur
Fuel Oil

Fuel 8
Straight Run
(Low Sulfur)

Fuel 9
Straight Run
(High Sulfur)

Fuel 10
Kerosene
Jet Fuel

Fuel 11
Diesel
Fuel

Fuel 12

0.904
1.170
0.500
52.500
21.460

0.928
1.678
0.300
18.500
21.000

0.953
1.678
2.750
18.500
20.000

0.797
Ϫ.782
0.125
308.800
25.520

0.850
Ϫ.054
0.200
161.700
24.300

1.114
1.716
2.370
437.000
11.500

6, and 7. Lower-density oils are generally preferred to higher-density oils, because energy per unit mass is higher for low-density fuels, which reduces the weight of the fuel. Lower-viscosity oils are preferred because they flow more easily through fuel lines than oils with higher viscosities. High flash points are preferred for safety reasons. However, because flash point and linear flash point are inversely related, this means that oils with lower linear flash point are preferred to oils with higher linear flash point.
That fuels can be blended cheaply to form new fuels affects price as well. For example, fuel 2 and fuel
3 from Table 4.20 can be blended to form a fuel with the same API, viscosity, sulfur, and flash point as fuel
1. In particular, 0.1304 barrel of fuel 2 and 0.8696 barrel of fuel 3 can be blended to form 1 barrel of a new fuel, which, in terms of the four main characteristics, is identical to fuel 1. Because the cost of blending is small, prices combine nearly linearly. The cost to create the blended fuel is $16.80 per barrel
($16.80 ϭ 0.1304[13.25] ϩ 0.8696[17.33]). If the price of fuel 1 were greater than $16.80, say $17.10,
Lisa Davies could create an arbitrage. She could buy fuels 2 and 3 in the appropriate proportions,

218

Chapter 4 Linear Programming Models

Slurry

Lakefield Corporation could blend them together, and Lisa could sell the blend at the price of fuel 1.
The profit would be $0.30 per barrel minus any blending and transaction costs. However, the actual price of fuel 1 is $16.08, so this plan does not represent an arbitrage opportunity.
The no-arbitrage pricing principle is simply a generalization of the previous example. No arbitrage means that the price of any fuel must be less than or equal to the cost of any blend of fuels of equal or better quality. As mentioned earlier, better means larger API, lower viscosity, lower sulfur content, and higher flash point. In terms of linear properties, better means lower density, lower linear viscosity, lower sulfur content, and lower linear flash point.
Any number of fuels (not just two) can be blended together. Lisa Davies would like to develop a system that automatically checks the no-arbitrage pricing condition for all of the fuels. If the condition is violated, she would like to know the appropriate amounts of the fuels to buy to create the arbitrage, the profit per barrel of the blended fuel, and the characteristics of the blended fuel.

Questions
1. Suppose that 0.3 barrel of fuel 2, 0.3 barrel of fuel 3, and 0.4 barrel of fuel 4 are blended together.What is the cost of the blended fuel?
What are the (linear) properties of the blended fuel (i.e., density, linear viscosity, sulfur content, and linear flash point)?
2. Using the data from Table 4.21, check whether any of the fuels violate the no-arbitrage pricing

condition. If no fuel violates the condition, which fuel’s price comes the closest to the no-arbitrage upper bound? If there is a violation, give the explicit recipe.
3. What modifications would you make to the analysis to account for blending costs?
4. What would be the important issues or steps involved in creating a real system for this problem? ■

Case 4.3 Lakefield Corporation’s Oil Trading Desk

219

4.4 F OREIGN C URRENCY T RADING

CASE

aily trading volume in the foreign exchange markets often exceeds $1 trillion. Participants trade in the spot currency markets, forward markets, and futures markets. In addition, currency options, currency swaps, and other derivative contracts are traded. For simplicity, this case focuses on the spot currency market only. A spot currency transaction is simply an agreement to buy some amount of one currency using another currency.15 For example, a
British company might need to pay a Japanese supplier 150 million yen. Suppose that the spot yen/pound rate is 195.07. Then the British company could use the spot currency market to buy 150 million yen at a cost of 768,954.7 (ϭ150,000,000͞
195.07) British pounds. A sample of today’s crosscurrency spot rates is given in Table 4.22. (See also the file Currency Rates.xlsx.)
To continue the example, suppose the company canceled the order from the supplier and wanted to convert the 150 million yen back into British pounds.
From Table 4.22, the pound/yen spot rate is
0.005126. So the company could use the 150 million yen to buy 768,900 (ϭ150,000,000 ϫ 0.005126) pounds. Note that the 768,900 pounds is less than the original 768,954.7 pounds. The difference is the result of the bid-offer spread: The price to buy yen
(the bid price) is greater than the price to sell yen
(the offer price). The bid-offer spread represents a transaction cost to the company.

D

Table 4.22

Occasionally, market prices may become “out of line” in the sense that there are arbitrage opportunities. In this context, arbitrage means that there is a set of spot currency transactions that creates positive wealth but does not require any funds to initiate—that is, it is a “money pump.” When such pure arbitrage opportunities exist, supply and demand forces will generally move prices to eliminate the opportunities. Hence, it is desirable to quickly identify arbitrage opportunities when they do exist and to take advantage of them to the greatest extent possible. Questions
1. Formulate a decision model to determine whether there are any arbitrage opportunities with the spot currency rates given in Table 4.22.
Note that an arbitrage opportunity could involve several currencies. If there is an arbitrage opportunity, your model should specify the exact set of transactions to achieve it.
2. Find the cross-currency rates in a recent paper— for example, in the Wall Street Journal—or on the
Web at http://www.oanda.com/convert/classic.
Check the numbers for an arbitrage opportunity.
If you find one, do you think it represents a real arbitrage opportunity? Why or why not? ■

Cross-Currency Spot Rates
To
US Dollar

From

US Dollar
British Pound
Euro
Japanese Yen
Brazilian Real

British Pound

Euro

Japanese Yen

Brazilian Real

1
1.6491
1.1317
0.008455
0.34374

0.60639
1
0.6861
0.005126
0.20836

0.88363
1.45751
1
0.007476
0.30349

118.27
195.07
133.77
1
40.62285

2.9092
4.79931
3.29496
0.02462
1

15
A spot transaction agreed to today is settled (i.e., the money changes hands) two business days from today. By contrast, a threemonth forward transaction agreed to today is settled (approximately) three months from today.

220

Chapter 4 Linear Programming Models

CHAPTER

Network Models

© FRANK RUMPENHORST/DPA/Landov

5

RESTRUCTURING BASF NORTH AMERICA’S
DISTRIBUTION SYSTEM quick look through Interfaces, the journal that chronicles management science success stories from real applications, indicates that many of these success stories involve network optimization, the subject of this chapter. A typical example appears in Sery et al. (2001). The authors describe their efforts to restructure BASF North America’s distribution system. The BASF Group, with headquarters in Germany, is one of the world’s leading chemical companies, with annual sales over $30 billion and more than 100,000 employees worldwide. BASF offers a variety of chemical and chemical-based products to customers in Europe, the NAFTA region,
South America, and Asia. You probably know the company from its catchy slogan,“We don’t make a lot of the products you buy. We make a lot of the products you buy better.” Its diverse product mix includes chemicals, polymers, automotive coatings, colors, dyes, pharmaceuticals, nylon fibers, and agricultural products.
In the mid-1990s, BASF examined its distribution of packaged goods in the North America region and discovered that it shipped 1.6 billion pounds of finished goods annually to customers from a network of 135 locations at an annual cost, including transportation and warehousing, of nearly
$100 million. The majority (86) of the 135 locations were distribution

A

221

centers (DCs), although almost a billion pounds were shipped directly from plants to customers. Unfortunately, there had never been any systematic attempt to optimize this network configuration; it had just evolved over the years. The authors of the study were asked to make recommendations that would (1) decrease logistics costs and (2) increase customer service, defined as the percentage of shipments that reach the customer on the same day or the next day. (This percentage was about 77% before the study.) The authors developed a linear programming model that, when implemented, was able to
(1) reduce the number of DCs from 86 to 12; (2) reduce the annual transport, facility, and inventory carrying costs by 6%; (3) achieve a one-time 9% improvement in cash flows from a reduction in the working capital tied up in inventory; and (4) increase the customer service measure to 90%. The redesign worked so well that BASF later developed similar models for its
European, Scandinavian, and Far East distribution systems.
The article’s description of the study is a virtual textbook example of the modeling process described in Chapter 1 of this book. The problem was first identified as follows:“Define the optimal number and location of warehouses and the corresponding material flows needed to meet anticipated customer demand and required delivery service times at the lowest overall cost.” The project team next performed the arduous task of collecting the various demands and costs required for the optimization model. Although we try to indicate “Where Do the Numbers Come From?” in the examples in this book, the authors of the study describe just how difficult data collection can be, particularly when the data is stored in a variety of legacy systems that use a wide range of data definitions. Next, the authors developed a verbal statement of the model, including all assumptions they made, which was then translated in a straightforward manner into the network optimization model itself. The next step was to build a decision support system to implement the model. This user-friendly system allowed BASF management to become comfortable with the model (and learn to trust it) by running it repeatedly under different scenarios to answer all sorts of what-if questions.
Finally, the model’s recommendations were used to redesign the distribution system in North America, and an honest evaluation of its effects—reduced costs and increased customer service—was made. ■

5.1 INTRODUCTION
Many important optimization models have a natural graphical network representation. In this chapter, we discuss some specific examples of network models. There are several reasons for distinguishing network models from other LP models:
■

■

222

The network structure of these models allows us to represent them graphically in a way that is intuitive to users. We can then use this graphical representation as an aid in the spreadsheet model development. In fact, for a book at this level, the best argument for singling out network problems for special consideration is the fact that they can be represented graphically.
Many companies have real problems, often extremely large, that can be represented as network models. In fact, many of the best management science success stories have involved large network models. For example, Delta Airlines developed a

Chapter 5 Network Models

■

network model to schedule its entire fleet of passenger airplanes. A few other real applications of network-based models are listed throughout the chapter, but the list is by no means exhaustive. A quick scan of the articles in the Interfaces journal indicates that there are probably more network-based applications reported than any other type.
Specialized solution techniques have been developed specifically for network models. Although we do not discuss the details of these solution techniques—and they are not implemented in Excel’s Solver—they are important in real-world applications because they allow companies to solve huge problems that could not be solved by the usual LP algorithms.

5.2 TRANSPORTATION MODELS
In many situations, a company produces products at locations called origins and ships these products to customer locations called destinations. Typically, each origin has a limited amount that it can ship, and each customer destination must receive a required quantity of the product. Spreadsheet optimization models can be used to determine the minimum-cost shipping plan for satisfying customer demands.
For now, we assume that the only possible shipments are those directly from an origin to a destination. That is, no shipments between origins or between destinations are possible. This problem—generally called the transportation problem—has been studied extensively in management science. In fact, it was one of the first management science models developed, more than a half century ago. The following is a typical example of a small transportation problem.

EXAMPLE

5.1 S HIPPING C ARS F ROM P LANTS

TO

R EGIONS

OF THE

C OUNTRY

T

he Grand Prix Automobile Company manufactures automobiles in three plants and then ships them to four regions of the country. The plants can supply the amounts listed in the right column of Table 5.1. The customer demands by region are listed in the bottom row of this table, and the unit costs of shipping an automobile from each plant to each region are listed in the middle of the table. Grand Prix wants to find the lowest-cost shipping plan for meeting the demands of the four regions without exceeding the capacities of the plants. Table 5.1

Input Data for Grand Prix Example
Region 1

Plant 1
Plant 2
Plant 3
Demand

Region 2

Region 3

Region 4

Capacity

131
250
178
450

218
116
132
200

266
263
122
300

120
278
180
300

450
600
500

Objective To develop a spreadsheet optimization model that finds the least-cost way of shipping the automobiles from plants to regions that stays within plant capacities and meets regional demands.

5.2 Transportation Models

223

WHERE DO THE NUMBERS COME FROM?
A typical transportation problem requires three sets of numbers: capacities (or supplies), demands (or requirements), and unit shipping (and possibly production) costs:
■

■

■

The capacities indicate the most each plant can supply in a given amount of time—a month, say—under current operating conditions. In some cases, it might be possible to increase the “base” capacities by using overtime, for example. In such cases, we could modify the model to determine the amounts of additional capacity to use (and pay for).
The customer demands are typically estimated from some type of forecasting model
(as discussed in Chapter 16). The forecasts are often based on historical customer demand data.
The unit shipping costs come from a transportation cost analysis—how much does it really cost to send a single automobile from any plant to any region? This is not an easy question to answer, and it requires an analysis of the best mode of transportation (railroad, plane, ship, or truck). However, companies typically have the required data. Actually, the unit “shipping” cost can also include the unit production cost at each plant. However, if this cost is the same across all plants, as we are tacitly assuming here, it can be omitted from the model.

Solution
The variables and constraints required for this model are listed in Table 5.2. We must know the amounts sent out of the plants and the amounts sent to the regions. However, these aggregate quantities are not directly the decision variables. The company must decide exactly the number of autos to send from each plant to each region—a shipping plan.

Table 5.2

Variables and Constraints for the Transportation Model

Input variables
Decision variables
(changing cells)
Objective (target cell)
Other calculated variables
Constraints

In a transportation problem, all flows go from left to right— from origins to destinations. More complex network structures are discussed in Section 5.4.
The common feature of models in this chapter is that they can be represented graphically, as in
Figure 5.1.

224

Plant capacities, regional demands, unit shipping costs
Number of autos sent from each plant to each region
Total shipping cost
Number sent out of each plant, number sent to each region
Number sent out of each plant must be less than or equal to
Plant capacity
Number sent to each region must be greater than or equal to
Region demand

Representing as a Network Model
A network diagram of this model appears in Figure 5.1. This diagram is typical of network models. It consists of nodes and arcs. A node, indicated by a circle, generally represents a geographical location. In this case, the nodes on the left correspond to plants, and the nodes on the right correspond to regions. An arc, indicated by an arrow, generally represents a route for getting a product from one node to another. Here, the arcs all go from a plant node to a region node—from left to right.
The problem data fit nicely on such a diagram. The capacities are placed next to the plant nodes, the demands are placed next to the region nodes, and the unit shipping costs are placed on the arcs. The decision variables are usually called flows. They represent the

Chapter 5 Network Models

amounts shipped on the various arcs. Sometimes (although not in this problem), there are upper limits on the flows on some or all of the arcs. These upper limits are called arc capacities, and they can also be shown on the diagram.1

Figure 5.1

Region 1

Plant 1

Region 3

218
266

300

300

250

120
600

200

Region 4

450

450

Region 2

131

Network
Representation
of Grand Prix
Problem

116

Plant 2
263
132
122

178
500

Plant 3

180

278

DEVELOPING THE SPREADSHEET MODEL
The spreadsheet model appears in Figure 5.2. (See the file Transportation 1.xlsx.) To develop this model, perform the following steps.
Figure 5.2 Grand Prix Transportation Model

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

A
B
Grand Prix transportaƟon model

C

D

E

F

G

H

Unit shipping costs
To
Region 1
$131
$250
$178

Region 2
$218
$116
$132

Region 3
$266
$263
$122

Region 4
$120
$278
$180

Shipping plan, and constraints on supply and demand
To
Region 1 Region 2
From
Plant 1
150
0
Plant 2
100
200
Plant 3
200
0
Total received
450
200
>=
>=
Demand
450
200

Region 3
0
0
300
300
>=
300

Region 4 Total shipped
300
450
0
300
0
500
300
>=
300

From

Plant 1
Plant 2
Plant 3

=
300

From

Plant 1
Plant 2
Plant 3

Original demands

Region 4 Total shipped
300
450
0
300
0
500
300
>=
300

ObjecƟve to minimize
Total cost
$176,050
SensiƟvity of total cost to percentage change in each of the demands
$B$23
-20%
$130,850
-15%
$140,350
-10%
$149,850
-5%
$162,770
0%
$176,050
5%
$189,330
10%
$202,610
15%
$215,890
20%
$229,170
25% Not feasible
30% Not feasible

Chapter 5 Network Models

Input for SolverTable
% change

=
300

Tax rate
30%
35%
22%