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R E V I S E D

T H I R T E E N T H

E D I T I O N

AN INTRODUCTION TO

MANAGEMENT SCIENCE QUANTITATIVE APPROACHES
TO DECISION MAKING
David R. Anderson
University of Cincinnati

Dennis J. Sweeney
University of Cincinnati

Thomas A. Williams
Rochester Institute of Technology

Jeffrey D. Camm
University of Cincinnati

Kipp Martin
University of Chicago

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

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An Introduction to Management Science: Quantitative Approaches to Decision Making, Revised Thirteenth Edition David R. Anderson, Dennis J. Sweeney, Thomas A. Williams, Jeffrey D. Camm, & Kipp Martin VP/Editorial Director: Jack W. Calhoun Publisher: Joe Sabatino Senior Acquisitions Editor: Charles McCormick, Jr. Developmental Editor: Maggie Kubale Editorial Assistant: Courtney Bavaro Marketing Communications Manager: Libby Shipp Marketing Manager: Adam Marsh Content Project Manager: Jacquelyn K Featherly Media Editor: Chris Valentine Manufacturing Coordinator: Miranda Klapper Production House/Compositor: MPS Limited, a Macmillan Company Senior Art Director: Stacy Jenkins Shirley Internal Designer: Michael Stratton/Chris Miller Design Cover Designer: Craig Ramsdell Cover Images: © Getty Images/GlowImages

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Dedication To My Parents Ray and Ilene Anderson DRA To My Parents James and Gladys Sweeney DJS To My Parents Phil and Ann Williams TAW To My Wife Karen Camm JDC To My Wife Gail Honda KM

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Brief Contents

Preface xxv About the Authors xxix Introduction 1 Chapter 1 Chapter 2 An Introduction to Linear Programming 28 Linear Programming: Sensitivity Analysis and Chapter 3 Interpretation of Solution 92 Linear Programming Applications in Marketing, Chapter 4 Finance, and Operations Management 153 Advanced Linear Programming Applications 214 Chapter 5 Distribution and Network Models 255 Chapter 6 Chapter 7 Integer Linear Programming 317 Nonlinear Optimization Models 365 Chapter 8 Project Scheduling: PERT/CPM 412 Chapter 9 Inventory Models 453 Chapter 10 Waiting Line Models 502 Chapter 11 Chapter 12 Simulation 542 Decision Analysis 602 Chapter 13 Multicriteria Decisions 659 Chapter 14 Time Series Analysis and Forecasting 703 Chapter 15 Markov Processes 761 Chapter 16 Chapter 17 Linear Programming: Simplex Method On Website Simplex-Based Sensitivity Analysis and Duality Chapter 18 On Website Solution Procedures for Transportation and Chapter 19 Assignment Problems On Website Minimal Spanning Tree On Website Chapter 20 Dynamic Programming On Website Chapter 21 Appendixes 787 Appendix A Building Spreadsheet Models 788 Appendix B Areas for the Standard Normal Distribution 815 Appendix C Values of e λ 817 Appendix D References and Bibliography 818 Appendix E Self-Test Solutions and Answers to Even-Numbered Problems 820 Index 853

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Contents

Preface xxv About the Authors xxix

Chapter 1 Introduction 1
1.1 Problem Solving and Decision Making 3 1.2 Quantitative Analysis and Decision Making 4 1.3 Quantitative Analysis 6 Model Development 7 Data Preparation 10 Model Solution 11 Report Generation 12 A Note Regarding Implementation 12 1.4 Models of Cost, Revenue, and Profit 14 Cost and Volume Models 14 Revenue and Volume Models 15 Profit and Volume Models 15 Breakeven Analysis 16 1.5 Management Science Techniques 16 Methods Used Most Frequently 18 Summary 19 Glossary 19 Problems 20 Case Problem Scheduling a Golf League 23 Appendix 1.1 Using Excel for Breakeven Analysis 24

Chapter 2 An Introduction to Linear Programming 28
2.1 A Simple Maximization Problem 30 Problem Formulation 31 Mathematical Statement of the Par, Inc., Problem 33 2.2 Graphical Solution Procedure 35 A Note on Graphing Lines 44 Summary of the Graphical Solution Procedure for Maximization Problems 46 Slack Variables 47 2.3 Extreme Points and the Optimal Solution 48 2.4 Computer Solution of the Par, Inc., Problem 50 Interpretation of Computer Output 51

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2.5 A Simple Minimization Problem 52 Summary of the Graphical Solution Procedure for Minimization Problems 54 Surplus Variables 55 Computer Solution of the M&D Chemicals Problem 56 2.6 Special Cases 57 Alternative Optimal Solutions 57 Infeasibility 58 Unbounded 60 2.7 General Linear Programming Notation 62 Summary 64 Glossary 65 Problems 66 Case Problem 1 Workload Balancing 82 Case Problem 2 Production Strategy 83 Case Problem 3 Hart Venture Capital 84 Appendix 2.1 Solving Linear Programs with LINGO 85 Appendix 2.2 Solving Linear Programs with Excel 87

Chapter 3 Linear Programming: Sensitivity Analysis and Interpretation of Solution 92
3.1 Introduction to Sensitivity Analysis 94 3.2 Graphical Sensitivity Analysis 95 Objective Function Coefficients 95 Right-Hand Sides 100 3.3 Sensitivity Analysis: Computer Solution 103 Interpretation of Computer Output 103 Cautionary Note on the Interpretation of Dual Values 106 The Modified Par, Inc., Problem 106 3.4 Limitations of Classical Sensitivity Analysis 110 Simultaneous Changes 111 Changes in Constraint Coefficients 112 Nonintuitive Dual Values 112 3.5 The Electronic Communications Problem 116 Problem Formulation 117 Computer Solution and Interpretation 118 Summary 122 Glossary 123 Problems 123 Case Problem 1 Product Mix 145 Case Problem 2 Investment Strategy 146 Case Problem 3 Truck Leasing Strategy 147 Appendix 3.1 Sensitivity Analysis with Excel 148 Appendix 3.2 Sensitivity Analysis with LINGO 150

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Chapter 4 Linear Programming Applications in
Marketing, Finance, and Operations Management 153
4.1 Marketing Applications 154 Media Selection 155 Marketing Research 158 4.2 Financial Applications 161 Portfolio Selection 161 Financial Planning 164 4.3 Operations Management Applications 168 A Make-or-Buy Decision 168 Production Scheduling 172 Workforce Assignment 179 Blending Problems 183 Summary 188 Problems 189 Case Problem 1 Planning an Advertising Campaign 202 Case Problem 2 Phoenix Computer 203 Case Problem 3 Textile Mill Scheduling 204 Case Problem 4 Workforce Scheduling 205 Case Problem 5 Duke Energy Coal Allocation 207 Appendix 4.1 Excel Solution of Hewlitt Corporation Financial Planning Problem 210

Chapter 5 Advanced Linear Programming
Applications 214
5.1 Data Envelopment Analysis 215 Evaluating the Performance of Hospitals 216 Overview of the DEA Approach 216 DEA Linear Programming Model 217 Summary of the DEA Approach 222 5.2 Revenue Management 223 5.3 Portfolio Models and Asset Allocation 229 A Portfolio of Mutual Funds 229 Conservative Portfolio 230 Moderate Risk Portfolio 232 5.4 Game Theory 236 Competing for Market Share 236 Identifying a Pure Strategy Solution 238 Identifying a Mixed Strategy Solution 239 Summary 247 Glossary 247 Problems 248

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Chapter 6 Distribution and Network Models 255
6.1 Transportation Problem 256 Problem Variations 260 A General Linear Programming Model 262 6.2 Assignment Problem 263 Problem Variations 266 A General Linear Programming Model 267 6.3 Transshipment Problem 268 Problem Variations 274 A General Linear Programming Model 274 6.4 Shortest-Route Problem 276 A General Linear Programming Model 279 6.5 Maximal Flow Problem 279 6.6 A Production and Inventory Application 283 Summary 286 Glossary 287 Problems 288 Case Problem 1 Solutions Plus 305 Case Problem 2 Distribution System Design 306 Appendix 6.1 Excel Solution of Transportation, Assignment, and Transshipment Problems 308

Chapter 7 Integer Linear Programming 317
7.1 Types of Integer Linear Programming Models 319 7.2 Graphical and Computer Solutions for an All-Integer Linear Program 321 Graphical Solution of the LP Relaxation 322 Rounding to Obtain an Integer Solution 322 Graphical Solution of the All-Integer Problem 323 Using the LP Relaxation to Establish Bounds 323 Computer Solution 324 7.3 Applications Involving 0-1 Variables 325 Capital Budgeting 325 Fixed Cost 326 Distribution System Design 329 Bank Location 334 Product Design and Market Share Optimization 337 7.4 Modeling Flexibility Provided by 0-1 Integer Variables 341 Multiple-Choice and Mutually Exclusive Constraints 341 k out of n Alternatives Constraint 342 Conditional and Corequisite Constraints 342 A Cautionary Note About Sensitivity Analysis 344

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Summary 344 Glossary 345 Problems 346 Case Problem 1 Textbook Publishing 357 Case Problem 2 Yeager National Bank 358 Case Problem 3 Production Scheduling with Changeover Costs 359 Appendix 7.1 Excel Solution of Integer Linear Programs 360 Appendix 7.2 LINGO Solution of Integer Linear Programs 361

Chapter 8 Nonlinear Optimization Models 365
8.1 A Production Application—Par, Inc., Revisited 367 An Unconstrained Problem 367 A Constrained Problem 368 Local and Global Optima 371 Dual Values 374 8.2 Constructing an Index Fund 374 8.3 Markowitz Portfolio Model 379 8.4 Blending: The Pooling Problem 382 8.5 Forecasting Adoption of a New Product 387 Summary 392 Glossary 392 Problems 393 Case Problem 1 Portfolio Optimization with Transaction Costs 402 Case Problem 2 CAFE Compliance in the Auto Industry 405 Appendix 8.1 Solving Nonlinear Problems with LINGO 408 Appendix 8.2 Solving Nonlinear Problems with Excel Solver 409

Chapter 9 Project Scheduling: PERT/CPM 412
9.1 Project Scheduling with Known Activity Times 413 The Concept of a Critical Path 414 Determining the Critical Path 416 Contributions of PERT/CPM 420 Summary of the PERT/CPM Critical Path Procedure 421 9.2 Project Scheduling with Uncertain Activity Times 422 The Daugherty Porta-Vac Project 423 Uncertain Activity Times 423 The Critical Path 425 Variability in Project Completion Time 428 9.3 Considering Time-Cost Trade-Offs 431 Crashing Activity Times 432 Linear Programming Model for Crashing 434

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Summary 436 Glossary 437 Problems 438 Case Problem R. C. Coleman 448 Appendix 9.1 Using Microsoft Office Project 450

Chapter 10 Inventory Models 453
10.1 Economic Order Quantity (EOQ) Model 454 The How-Much-to-Order Decision 459 The When-to-Order Decision 460 Sensitivity Analysis for the EOQ Model 461 Excel Solution of the EOQ Model 462 Summary of the EOQ Model Assumptions 463 10.2 Economic Production Lot Size Model 464 Total Cost Model 465 Economic Production Lot Size 467 10.3 Inventory Model with Planned Shortages 467 10.4 Quantity Discounts for the EOQ Model 472 10.5 Single-Period Inventory Model with Probabilistic Demand 474 Johnson Shoe Company 475 Nationwide Car Rental 479 10.6 Order-Quantity, Reorder Point Model with Probabilistic Demand 480 The How-Much-to-Order Decision 481 The When-to-Order Decision 482 10.7 Periodic Review Model with Probabilistic Demand 484 More Complex Periodic Review Models 487 Summary 488 Glossary 489 Problems 491 Case Problem 1 Wagner Fabricating Company 498 Case Problem 2 River City Fire Department 499 Appendix 10.1 Development of the Optimal Order Quantity (Q*) Formula for the EOQ Model 500 Appendix 10.2 Development of the Optimal Lot Size (Q*) Formula for the Production Lot Size Model 501

Chapter 11 Waiting Line Models 502
11.1 Structure of a Waiting Line System 504 Single-Channel Waiting Line 504 Distribution of Arrivals 504 Distribution of Service Times 506

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11.2

11.3

11.4 11.5 11.6 11.7

11.8

11.9

Queue Discipline 507 Steady-State Operation 507 Single-Channel Waiting Line Model with Poisson Arrivals and Exponential Service Times 508 Operating Characteristics 508 Operating Characteristics for the Burger Dome Problem 509 Managers’ Use of Waiting Line Models 510 Improving the Waiting Line Operation 510 Excel Solution of Waiting Line Model 511 Multiple-Channel Waiting Line Model with Poisson Arrivals and Exponential Service Times 512 Operating Characteristics 513 Operating Characteristics for the Burger Dome Problem 515 Some General Relationships for Waiting Line Models 517 Economic Analysis of Waiting Lines 519 Other Waiting Line Models 520 Single-Channel Waiting Line Model with Poisson Arrivals and Arbitrary Service Times 521 Operating Characteristics for the M/G/1 Model 521 Constant Service Times 523 Multiple-Channel Model with Poisson Arrivals, Arbitrary Service Times, and No Waiting Line 524 Operating Characteristics for the M/G/k Model with Blocked Customers Cleared 524 Waiting Line Models with Finite Calling Populations 526 Operating Characteristics for the M/M/1 Model with a Finite Calling Population 527 Summary 529 Glossary 531 Problems 531 Case Problem 1 Regional Airlines 539 Case Problem 2 Office Equipment, Inc. 540

Chapter 12 Simulation 542
12.1 Risk Analysis 545 PortaCom Project 545 What-If Analysis 545 Simulation 547 Simulation of the PortaCom Project 554 12.2 Inventory Simulation 558 Butler Inventory Simulation 561 12.3 Waiting Line Simulation 563 Hammondsport Savings Bank ATM Waiting Line 563 Customer Arrival Times 564

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Customer Service Times 565 Simulation Model 565 Hammondsport Savings Bank ATM Simulation 569 Simulation with Two ATMs 570 Simulation Results with Two ATMs 572 12.4 Other Simulation Issues 574 Computer Implementation 574 Verification and Validation 575 Advantages and Disadvantages of Using Simulation 575 Summary 576 Glossary 577 Problems 578 Case Problem 1 Tri-State Corporation 585 Case Problem 2 Harbor Dunes Golf Course 587 Case Problem 3 County Beverage Drive-Thru 589 Appendix 12.1 Simulation with Excel 590 Appendix 12.2 Simulation Using Crystal Ball 597

Chapter 13 Decision Analysis 602
13.1 Problem Formulation 604 Influence Diagrams 605 Payoff Tables 605 Decision Trees 606 13.2 Decision Making Without Probabilities 607 Optimistic Approach 607 Conservative Approach 607 Minimax Regret Approach 608 13.3 Decision Making with Probabilities 610 Expected Value of Perfect Information 613 13.4 Risk Analysis and Sensitivity Analysis 615 Risk Analysis 615 Sensitivity Analysis 616 13.5 Decision Analysis with Sample Information 620 Influence Diagram 620 Decision Tree 621 Decision Strategy 623 Risk Profile 627 Expected Value of Sample Information 629 Efficiency of Sample Information 630 13.6 Computing Branch Probabilities 630 Summary 634 Glossary 635 Problems 637 Case Problem 1 Property Purchase Strategy 651

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Case Problem 2 Lawsuit Defense Strategy 652 Appendix 13.1 Decision Analysis with Treeplan 653

Chapter 14 Multicriteria Decisions 659
14.1 Goal Programming: Formulation and Graphical Solution 660 Developing the Constraints and the Goal Equations 661 Developing an Objective Function with Preemptive Priorities 663 Graphical Solution Procedure 664 Goal Programming Model 667 14.2 Goal Programming: Solving More Complex Problems 668 Suncoast Office Supplies Problem 668 Formulating the Goal Equations 669 Formulating the Objective Function 670 Computer Solution 671 14.3 Scoring Models 674 14.4 Analytic Hierarchy Process 679 Developing the Hierarchy 680 14.5 Establishing Priorities Using AHP 680 Pairwise Comparisons 681 Pairwise Comparison Matrix 682 Synthesization 684 Consistency 685 Other Pairwise Comparisons for the Car Selection Problem 687 14.6 Using AHP to Develop an Overall Priority Ranking 688 Summary 690 Glossary 690 Problems 691 Case Problem EZ Trailers, Inc. 700 Appendix 14.1 Scoring Models with Excel 701

Chapter 15 Time Series Analysis and Forecasting 703
15.1 Time Series Patterns 705 Horizontal Pattern 705 Trend Pattern 707 Seasonal Pattern 709 Trend and Seasonal Pattern 710 Cyclical Pattern 713 Selecting a Forecasting Method 713 15.2 Forecast Accuracy 713 15.3 Moving Averages and Exponential Smoothing 717 Moving Averages 717 Weighted Moving Averages 720 Exponential Smoothing 721

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15.4 Trend Projection 726 Linear Trend 726 Nonlinear Trend 730 15.5 Seasonality 733 Seasonality Without Trend 734 Seasonality and Trend 737 Models Based on Monthly Data 739 Summary 740 Glossary 741 Problems 741 Case Problem 1 Forecasting Food and Beverage Sales 751 Case Problem 2 Forecasting Lost Sales 751 Appendix 15.1 Forecasting with Excel Data Analysis Tools 753 Appendix 15.2 Forecasting with Excel Solver 754 Appendix 15.3 Forecasting with LINGO 759

Chapter 16 Markov Processes 761
16.1 Market Share Analysis 763 16.2 Accounts Receivable Analysis 771 Fundamental Matrix and Associated Calculations 772 Establishing the Allowance for Doubtful Accounts 774 Summary 776 Glossary 776 Problems 777 Case Problem Dealer’s Absorbing State Probabilities in Blackjack 781 Appendix 16.1 Matrix Notation and Operations 782 Appendix 16.2 Matrix Inversion with Excel 785

Chapter 17 Linear Programming: Simplex Method On Website
17.1 An Algebraic Overview of the Simplex Method 17-2 Algebraic Properties of the Simplex Method 17-3 Determining a Basic Solution 17-3 Basic Feasible Solution 17-4 17.2 Tableau Form 17-5 17.3 Setting up the Initial Simplex Tableau 17-7 17.4 Improving the Solution 17-10 17.5 Calculating the Next Tableau 17-12 Interpreting the Results of an Iteration 17-15 Moving Toward a Better Solution 17-15 Interpreting the Optimal Solution 17-18 Summary of the Simplex Method 17-19

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17.6 Tableau Form: The General Case 17-20 Greater-Than-or-Equal-to Constraints 17-20 Equality Constraints 17-24 Eliminating Negative Right-Hand-Side Values 17-25 Summary of the Steps to Create Tableau Form 17-26 17.7 Solving a Minimization Problem 17-27 17.8 Special Cases 17-29 Infeasibility 17-29 Unboundedness 17-31 Alternative Optimal Solutions 17-32 Degeneracy 17-33 Summary 17-35 Glossary 17-36 Problems 17-37

Chapter 18 Simplex-Based Sensitivity Analysis and Duality
On Website
18.1 Sensitivity Analysis with the Simplex Tableau 18-2 Objective Function Coefficients 18-2 Right-Hand-Side Values 18-6 Simultaneous Changes 18-13 18.2 Duality 18-14 Economic Interpretation of the Dual Variables 18-16 Using the Dual to Identify the Primal Solution 18-18 Finding the Dual of Any Primal Problem 18-18 Summary 18-20 Glossary 18-21 Problems 18-21

Chapter 19 Solution Procedures for Transportation and Assignment Problems On Website
19.1 Transportation Simplex Method: A Special-Purpose Solution Procedure 19-2 Phase I: Finding an Initial Feasible Solution 19-2 Phase II: Iterating to the Optimal Solution 19-7 Summary of the Transportation Simplex Method 19-17 Problem Variations 19-17 19.2 Assignment Problem: A Special-Purpose Solution Procedure 19-18 Finding the Minimum Number of Lines 19-21 Problem Variations 19-21 Glossary 19-25 Problems 19-26

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Chapter 20 Minimal Spanning Tree On Website
A Minimal Spanning Tree Algorithm 20-2 Glossary 20-5 Problems 20-5

Chapter 21 Dynamic Programming On Website
21.1 21.2 21.3 21.4 A Shortest-Route Problem 21-2 Dynamic Programming Notation 21-6 The Knapsack Problem 21-10 A Production and Inventory Control Problem 21-16 Summary 21-20 Glossary 21-21 Problems 21-22 Case Problem Process Design 21-26

Appendixes 787

Appendix A Appendix B Appendix C Appendix D Appendix E
Index 853

Building Spreadsheet Models 788 Areas for the Standard Normal Distribution 815 Values of e λ 817

References and Bibliography 818 Self-Test Solutions and Answers to Even-Numbered Problems 820

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Preface

We are very excited to publish the revised thirteenth edition of a text that has been a leader in the field for over 20 years. The purpose of this revised thirteenth edition, as with previous editions, is to provide undergraduate and graduate students with a sound conceptual understanding of the role that management science plays in the decision-making process. The text describes many of the applications where management science is used successfully. Former users of this text have told us that the applications we describe have led them to find new ways to use management science in their organizations. An Introduction to Management Science is applications oriented and continues to use the problem-scenario approach that is a hallmark of every edition of the text. Using the problem-scenario approach, we describe a problem in conjunction with the management science model being introduced. The model is then solved to generate a solution and recommendation to management. We have found that this approach helps to motivate the student by not only demonstrating how the procedure works, but also how it contributes to the decision-making process. From the very first edition we have been committed to the challenge of writing a textbook that would help make the mathematical and technical concepts of management science understandable and useful to students of business and economics. Judging from the responses from our teaching colleagues and thousands of students, we have successfully met the challenge. Indeed, it is the helpful comments and suggestions of many loyal users that have been a major reason why the text is so successful. Throughout the text we have utilized generally accepted notation for the topic being covered so those students who pursue study beyond the level of this text should be comfortable reading more advanced material. To assist in further study, a references and bibliography section is included at the back of the book.

CHANGES IN THE REVISED THIRTEENTH EDITION
The thirteenth edition of Management Science is a major revision. We are very excited about it and want to tell you about some of the changes we have made and why. In addition to the major revisions described in the remainder of this section, this revised edition of the thirteenth edition has been updated to incorporate Microsoft® Office Excel® 2010. This involves some changes in the user interface of Excel and major changes in the interface and functionality of Excel Solver. The Solver in Excel 2010 is more reliable than in previous editions and offers new alternatives such as a multistart option for difficult nonlinear problems.

New Member of the ASWM Team
Prior to getting into the content changes, we want to announce that we are adding a new member to the ASWM author team. His name is Jeffrey Camm. Jeff received his Ph.D. from Clemson University. He has been at the University of Cincinnati since 1984, and has been a visiting scholar at Stanford University and a visiting professor of business administration at the Tuck School of Business at Dartmouth College. Jeff has published over 30 papers in the general area of optimization applied to problems in operations management. At the University of Cincinnati, he was named the Dornoff Fellow of Teaching Excellence and

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he was the 2006 recipient of the INFORMS Prize for the Teaching of Operations Research Practice. He currently serves as editor-in-chief of Interfaces, and is on the editorial board of INFORMS Transactions on Education. We welcome Jeff to the new ASWCM team and expect the new ideas from Jeff will make the text even better in the years to come. In preparing this thirteenth edition, we have been careful to maintain the overall format and approach of the previous edition. However, based on our classroom experiences and suggestions from users of previous editions, a number of changes have been made to enhance the text.

Made the Book Less Reliant on Specific Software
The first eight chapters on optimization no longer use output from The Management Scientist software. All figures illustrating computer output are generic and are totally independent of software selection. This provides flexibility for the instructor. In addition, we provide appendices that describe how to use Excel Solver and LINGO. For every model illustrated in the text we have both Excel and LINGO files available at the website. Prior users of The Management Scientist wishing to upgrade to similar software should consider using LINGO. This will be an easy transition and LINGO is far more flexible than The Management Scientist. The documented LINGO models (not available in MS 12e), available at the website, will aide in the transition. Excel Solver and LINGO have an advantage over The Management Scientist in that they do not require the user to move all variables to the left-hand side of the constraint. This eliminates the need to algebraically manipulate the model and allows the student to enter the model in the computer in its more natural form. For users wishing to use The Management Scientist, it will continue to be available on the website for the text.

New Appendix A: Building Spreadsheet Models
This appendix will prove useful to professors and students wishing to solve optimization models with Excel Solver. The appendix also contains a section on the principles of good spreadsheet modeling and a section on auditing tips. Exercises are also provided.

Chapter 15 Thoroughly Revised
Chapter 15, Times Series Analysis and Forecasting, has been thoroughly revised. The revised chapter is more focused on time series data and methods. A new section on forecast accuracy has been added and there is more emphasis on curve fitting. A new section on nonlinear trend has been added. In order to better integrate this chapter with the text, we show how finding the best parameter values in forecasting models is an application of optimization, and illustrate with Excel Solver and LINGO.

New Project Management Software
In Chapter 9, Project Scheduling: PERT/CPM, we added an appendix on Microsoft Office Project. This popular software is a valuable aid for project management and is software that the student may well encounter on the job. This software is available on the CD that is packaged with every new copy of the text.

Chapter 3 Significantly Revised
We significantly revised Chapter 3, Linear Programming: Sensitivity Analysis and Interpretation of Solution. The material is now presented in a more up-to-date fashion and emphasizes the ease of using software to analyze optimization models.

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New Management Science in Action, Cases, and Problems
Management Science in Action is the name of the short summaries that describe how the material covered in a chapter has been used in practice. In this edition you will find numerous Management Science in Action vignettes, cases, and homework problems.

Other Content Changes
A variety of other changes, too numerous to mention individually, have been made throughout the text in responses to suggestions of users and our students.

COMPUTER SOFTWARE INTEGRATION
We have been careful to write the text so that it is not dependent on any particular software package. But, we have included materials that facilitate using our text with several of the more popular software packages. The following software and files are available on the website for the text: • • LINGO trial version, LINGO and Excel Solver models for every optimization model presented in the text, Microsoft® Excel worksheets for most of the examples used throughout the text, TreePlanTM Excel add-in for decision analysis and manual.

Microsoft Project is provided on the CD that is packaged with every new copy of the text.

• •

FEATURES AND PEDAGOGY
We have continued many of the features that appeared in previous editions. Some of the important ones are noted here.

Annotations
Annotations that highlight key points and provide additional insights for the student are a continuing feature of this edition. These annotations, which appear in the margins, are designed to provide emphasis and enhance understanding of the terms and concepts being presented in the text.

Notes and Comments
At the end of many sections, we provide Notes and Comments designed to give the student additional insights about the statistical methodology and its application. Notes and Comments include warnings about or limitations of the methodology, recommendations for application, brief descriptions of additional technical considerations, and other matters.

Self-Test Exercises
Certain exercises are identified as self-test exercises. Completely worked-out solutions for those exercises are provided in an appendix at the end of the text. Students can attempt the self-test exercises and immediately check the solution to evaluate their understanding of the concepts presented in the chapter.

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ACKNOWLEDGMENTS
We owe a debt to many of our academic colleagues and friends for their helpful comments and suggestions during the development of this and previous editions. Our associates from organizations who supplied several of the Management Science in Action vignettes make a major contribution to the text. These individuals are cited in a credit line associated with each vignette. We are also indebted to our senior acquisitions editor, Charles McCormick, Jr.; our marketing communications manager, Libby Shipp; our developmental editor, Maggie Kubale; our content project manager, Jacquelyn K Featherly; our media editor, Chris Valentine; and others at Cengage Business and Economics for their counsel and support during the preparation of this text. We also wish to thank Lynn Lustberg, Project Manager at MPS Content Services for her help in manuscript preparation.

David R. Anderson Dennis J. Sweeney Thomas A. Williams Jeffrey D. Camm Kipp Martin

About the Authors

David R. Anderson. David R. Anderson is Professor Emeritus of Quantitative Analysis in the College of Business Administration at the University of Cincinnati. Born in Grand Forks, North Dakota, he earned his B.S., M.S., and Ph.D. degrees from Purdue University. Professor Anderson has served as Head of the Department of Quantitative Analysis and Operations Management and as Associate Dean of the College of Business Administration. In addition, he was the coordinator of the College’s first Executive Program. At the University of Cincinnati, Professor Anderson has taught introductory statistics for business students as well as graduate-level courses in regression analysis, multivariate analysis, and management science. He has also taught statistical courses at the Department of Labor in Washington, D.C. He has been honored with nominations and awards for excellence in teaching and excellence in service to student organizations. Professor Anderson has coauthored ten textbooks in the areas of statistics, management science, linear programming, and production and operations management. He is an active consultant in the field of sampling and statistical methods. Dennis J. Sweeney. Dennis J. Sweeney is Professor Emeritus of Quantitative Analysis and Founder of the Center for Productivity Improvement at the University of Cincinnati. Born in Des Moines, Iowa, he earned a B.S.B.A. degree from Drake University and his M.B.A. and D.B.A. degrees from Indiana University, where he was an NDEA Fellow. During 1978–79, Professor Sweeney worked in the management science group at Procter & Gamble; during 1981–82, he was a visiting professor at Duke University. Professor Sweeney served as Head of the Department of Quantitative Analysis and as Associate Dean of the College of Business Administration at the University of Cincinnati. Professor Sweeney has published more than thirty articles and monographs in the area of management science and statistics. The National Science Foundation, IBM, Procter & Gamble, Federated Department Stores, Kroger, and Cincinnati Gas & Electric have funded his research, which has been published in Management Science, Operations Research, Mathematical Programming, Decision Sciences, and other journals. Professor Sweeney has coauthored ten textbooks in the areas of statistics, management science, linear programming, and production and operations management. Thomas A. Williams. Thomas A. Williams is Professor Emeritus of Management Science in the College of Business at Rochester Institute of Technology. Born in Elmira, New York, he earned his B.S. degree at Clarkson University. He did his graduate work at Rensselaer Polytechnic Institute, where he received his M.S. and Ph.D. degrees. Before joining the College of Business at RIT, Professor Williams served for seven years as a faculty member in the College of Business Administration at the University of Cincinnati, where he developed the undergraduate program in information systems and then served as its coordinator. At RIT he was the first chairman of the Decision Sciences Department. He teaches courses in management science and statistics, as well as graduate courses in regression and decision analysis.

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About the Authors

Professor Williams is the coauthor of eleven textbooks in the areas of management science, statistics, production and operations management, and mathematics. He has been a consultant for numerous Fortune 500 companies and has worked on projects ranging from the use of data analysis to the development of large-scale regression models.

Jeffrey D. Camm. Jeffrey D. Camm is Professor of Quantitative Analysis and Head of the Department of Quantitative Analysis and Operations Management at the University of Cincinnati. Dr. Camm earned a Ph.D. in management science from Clemson University and a B.S. in mathematics from Xavier University. He has been at the University of Cincinnati since 1984, has been a visiting scholar at Stanford University, and a visiting professor of business administration at the Tuck School of Business at Dartmouth College. Dr. Camm has published over 30 papers in the general area of optimization applied to problems in operations management and his research has been funded by the Air Force Office of Scientific Research, the Office of Naval Research, and the U.S. Department of Energy. He was named the Dornoff Fellow of Teaching Excellence by the University of Cincinnati College of Business and he was the 2006 recipient of the INFORMS Prize for the Teaching of Operations Research Practice. He currently serves as editor-in-chief of Interfaces, and is on the editorial board of INFORMS Transactions on Education. Kipp Martin. Kipp Martin is Professor of Operations Research and Computing Technology at the Booth School of Business, University of Chicago. Born in St. Bernard, Ohio, he earned a B.A. in mathematics, an MBA, and a Ph.D. in management science from the University of Cincinnati. While at the University of Chicago, Professor Martin has taught courses in management science, operations management, business mathematics, and information systems. Research interests include incorporating Web technologies such as XML, XSLT, XQuery, and Web Services into the mathematical modeling process; the theory of how to construct good mixed integer linear programming models; symbolic optimization; polyhedral combinatorics; methods for large scale optimization; bundle pricing models; computing technology; and database theory. Professor Martin has published in INFORMS Journal of Computing, Management Science, Mathematical Programming, Operations Research, The Journal of Accounting Research, and other professional journals. He is also the author of The Essential Guide to Internet Business Technology (with Gail Honda) and Large Scale Linear and Integer Optimization.

CHAPTER
Introduction
CONTENTS 1.1 1.2 1.3 PROBLEM SOLVING AND DECISION MAKING 1.4 MODELS OF COST, REVENUE, AND PROFIT Cost and Volume Models Revenue and Volume Models Profit and Volume Models Breakeven Analysis MANAGEMENT SCIENCE TECHNIQUES Methods Used Most Frequently

1

QUANTITATIVE ANALYSIS AND DECISION MAKING

QUANTITATIVE ANALYSIS Model Development Data Preparation Model Solution Report Generation A Note Regarding Implementation

1.5

2

Chapter 1

Introduction

According to Irv Lustig of IBM ILOG, Inc., solution methods developed today are 10,000 times faster than the ones used 15 years ago.

Management science, an approach to decision making based on the scientific method, makes extensive use of quantitative analysis. A variety of names exists for the body of knowledge involving quantitative approaches to decision making; in addition to management science, two other widely known and accepted names are operations research and decision science. Today, many use the terms management science, operations research, and decision science interchangeably. The scientific management revolution of the early 1900s, initiated by Frederic W. Taylor, provided the foundation for the use of quantitative methods in management. But modern management science research is generally considered to have originated during the World War II period, when teams were formed to deal with strategic and tactical problems faced by the military. These teams, which often consisted of people with diverse specialties (e.g., mathematicians, engineers, and behavioral scientists), were joined together to solve a common problem by utilizing the scientific method. After the war, many of these team members continued their research in the field of management science. Two developments that occurred during the post–World War II period led to the growth and use of management science in nonmilitary applications. First, continued research resulted in numerous methodological developments. Probably the most significant development was the discovery by George Dantzig, in 1947, of the simplex method for solving linear programming problems. At the same time these methodological developments were taking place, digital computers prompted a virtual explosion in computing power. Computers enabled practitioners to use the methodological advances to solve a large variety of problems. The computer technology explosion continues, and personal computers can now be used to solve problems larger than those solved on mainframe computers in the 1990s. As stated in the Preface, the purpose of the text is to provide students with a sound conceptual understanding of the role that management science plays in the decision-making process. We also said that the text is applications oriented. To reinforce the applications nature of the text and provide a better understanding of the variety of applications in which management science has been used successfully, Management Science in Action articles are presented throughout the text. Each Management Science in Action article summarizes an application of management science in practice. The first Management Science in Action in this chapter, Revenue Management at American Airlines, describes one of the most significant applications of management science in the airline industry.

MANAGEMENT SCIENCE IN ACTION
REVENUE MANAGEMENT AT AMERICAN AIRLINES*
One of the great success stories in management science involves the work done by the operations research (OR) group at American Airlines. In 1982, Thomas M. Cook joined a group of 12 operations research analysts at American Airlines. Under Cook’s guidance, the OR group quickly grew to a staff of 75 professionals who developed models and conducted studies to support senior management decision making. Today the OR group is called Sabre and employs 10,000 professionals worldwide. One of the most significant applications developed by the OR group came about because of the deregulation of the airline industry in the late

1970s. As a result of deregulation, a number of low-cost airlines were able to move into the market by selling seats at a fraction of the price charged by established carriers such as American Airlines. Facing the question of how to compete, the OR group suggested offering different fare classes (discount and full fare) and in the process created a new area of management science referred to as yield or revenue management. The OR group used forecasting and optimization techniques to determine how many seats to sell at a discount and how many seats to hold for full fare. Although the initial implementation was relatively crude, the group continued to improve

1.1

Problem Solving and Decision Making

3

the forecasting and optimization models that drive the system and to obtain better data. Tom Cook counts at least four basic generations of revenue management during his tenure. Each produced in excess of $100 million in incremental profitability over its predecessor. This revenue management system at American Airlines generates nearly $1 billion annually in incremental revenue.

Today, virtually every airline uses some sort of revenue management system. The cruise, hotel, and car rental industries also now apply revenue management methods, a further tribute to the pioneering efforts of the OR group at American Airlines and its leader, Thomas M. Cook.
*Based on Peter Horner, “The Sabre Story,” OR/MS Today (June 2000).

1.1

PROBLEM SOLVING AND DECISION MAKING
Problem solving can be defined as the process of identifying a difference between the actual and the desired state of affairs and then taking action to resolve the difference. For problems important enough to justify the time and effort of careful analysis, the problemsolving process involves the following seven steps:
1. 2. 3. 4. 5. 6. 7. Identify and define the problem. Determine the set of alternative solutions. Determine the criterion or criteria that will be used to evaluate the alternatives. Evaluate the alternatives. Choose an alternative. Implement the selected alternative. Evaluate the results to determine whether a satisfactory solution has been obtained.

Decision making is the term generally associated with the first five steps of the problemsolving process. Thus, the first step of decision making is to identify and define the problem. Decision making ends with the choosing of an alternative, which is the act of making the decision. Let us consider the following example of the decision-making process. For the moment assume that you are currently unemployed and that you would like a position that will lead to a satisfying career. Suppose that your job search has resulted in offers from companies in Rochester, New York; Dallas, Texas; Greensboro, North Carolina; and Pittsburgh, Pennsylvania. Thus, the alternatives for your decision problem can be stated as follows:
1. 2. 3. 4. Accept the position in Rochester. Accept the position in Dallas. Accept the position in Greensboro. Accept the position in Pittsburgh.

The next step of the problem-solving process involves determining the criteria that will be used to evaluate the four alternatives. Obviously, the starting salary is a factor of some importance. If salary were the only criterion of importance to you, the alternative selected as “best” would be the one with the highest starting salary. Problems in which the objective is to find the best solution with respect to one criterion are referred to as single-criterion decision problems. Suppose that you also conclude that the potential for advancement and the location of the job are two other criteria of major importance. Thus, the three criteria in your decision problem are starting salary, potential for advancement, and location. Problems that involve more than one criterion are referred to as multicriteria decision problems. The next step of the decision-making process is to evaluate each of the alternatives with respect to each criterion. For example, evaluating each alternative relative to the

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Chapter 1

Introduction

TABLE 1.1 DATA FOR THE JOB EVALUATION DECISION-MAKING PROBLEM Starting Salary $48,500 $46,000 $46,000 $47,000 Potential for Advancement Average Excellent Good Average Job Location Average Good Excellent Good

Alternative 1. Rochester 2. Dallas 3. Greensboro 4. Pittsburgh

starting salary criterion is done simply by recording the starting salary for each job alternative. Evaluating each alternative with respect to the potential for advancement and the location of the job is more difficult to do, however, because these evaluations are based primarily on subjective factors that are often difficult to quantify. Suppose for now that you decide to measure potential for advancement and job location by rating each of these criteria as poor, fair, average, good, or excellent. The data that you compile are shown in Table 1.1. You are now ready to make a choice from the available alternatives. What makes this choice phase so difficult is that the criteria are probably not all equally important, and no one alternative is “best” with regard to all criteria. Although we will present a method for dealing with situations like this one later in the text, for now let us suppose that after a careful evaluation of the data in Table 1.1, you decide to select alternative 3; alternative 3 is thus referred to as the decision. At this point in time, the decision-making process is complete. In summary, we see that this process involves five steps: 1. 2. 3. 4. 5. Define the problem. Identify the alternatives. Determine the criteria. Evaluate the alternatives. Choose an alternative.

Note that missing from this list are the last two steps in the problem-solving process: implementing the selected alternative and evaluating the results to determine whether a satisfactory solution has been obtained. This omission is not meant to diminish the importance of each of these activities, but to emphasize the more limited scope of the term decision making as compared to the term problem solving. Figure 1.1 summarizes the relationship between these two concepts.

1.2

QUANTITATIVE ANALYSIS AND DECISION MAKING
Consider the flowchart presented in Figure 1.2. Note that it combines the first three steps of the decision-making process under the heading of “Structuring the Problem” and the latter two steps under the heading “Analyzing the Problem.” Let us now consider in greater detail how to carry out the set of activities that make up the decision-making process. Figure 1.3 shows that the analysis phase of the decision-making process may take two basic forms: qualitative and quantitative. Qualitative analysis is based primarily on the manager’s judgment and experience; it includes the manager’s intuitive “feel” for the problem and is more an art than a science. If the manager has had experience with similar

1.2

Quantitative Analysis and Decision Making

5

FIGURE 1.1 THE RELATIONSHIP BETWEEN PROBLEM SOLVING AND DECISION MAKING
Define the Problem Identify the Alternatives Determine the Criteria Problem Solving Evaluate the Alternatives Choose an Alternative Implement the Decision Evaluate the Results Decision Decision Making

problems or if the problem is relatively simple, heavy emphasis may be placed upon a qualitative analysis. However, if the manager has had little experience with similar problems, or if the problem is sufficiently complex, then a quantitative analysis of the problem can be an especially important consideration in the manager’s final decision. When using the quantitative approach, an analyst will concentrate on the quantitative facts or data associated with the problem and develop mathematical expressions that

FIGURE 1.2 AN ALTERNATE CLASSIFICATION OF THE DECISION-MAKING PROCESS
Structuring the Problem Define the Problem Identify the Alternatives Determine the Criteria Analyzing the Problem Evaluate the Alternatives Choose an Alternative

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Chapter 1

Introduction

FIGURE 1.3 THE ROLE OF QUALITATIVE AND QUANTITATIVE ANALYSIS
Analyzing the Problem Qualitative Analysis Summary and Evaluation Quantitative Analysis Make the Decision

Structuring the Problem Define the Problem Identify the Alternatives Determine the Criteria

Quantitative methods are especially helpful with large, complex problems. For example, in the coordination of the thousands of tasks associated with landing Apollo 11 safely on the moon, quantitative techniques helped to ensure that more than 300,000 pieces of work performed by more than 400,000 people were integrated smoothly.

describe the objectives, constraints, and other relationships that exist in the problem. Then, by using one or more quantitative methods, the analyst will make a recommendation based on the quantitative aspects of the problem. Although skills in the qualitative approach are inherent in the manager and usually increase with experience, the skills of the quantitative approach can be learned only by studying the assumptions and methods of management science. A manager can increase decision-making effectiveness by learning more about quantitative methodology and by better understanding its contribution to the decision-making process. A manager who is knowledgeable in quantitative decision-making procedures is in a much better position to compare and evaluate the qualitative and quantitative sources of recommendations and ultimately to combine the two sources in order to make the best possible decision. The box in Figure 1.3 entitled “Quantitative Analysis” encompasses most of the subject matter of this text. We will consider a managerial problem, introduce the appropriate quantitative methodology, and then develop the recommended decision. In closing this section, let us briefly state some of the reasons why a quantitative approach might be used in the decision-making process: 1. The problem is complex, and the manager cannot develop a good solution without the aid of quantitative analysis. 2. The problem is especially important (e.g., a great deal of money is involved), and the manager desires a thorough analysis before attempting to make a decision. 3. The problem is new, and the manager has no previous experience from which to draw. 4. The problem is repetitive, and the manager saves time and effort by relying on quantitative procedures to make routine decision recommendations.

Try Problem 4 to test your understanding of why quantitative approaches might be needed in a particular problem.

1.3

QUANTITATIVE ANALYSIS
From Figure 1.3, we see that quantitative analysis begins once the problem has been structured. It usually takes imagination, teamwork, and considerable effort to transform a rather general problem description into a well-defined problem that can be approached via quantitative analysis. The more the analyst is involved in the process of structuring the problem,

1.3

Quantitative Analysis

7

the more likely the ensuing quantitative analysis will make an important contribution to the decision-making process. To successfully apply quantitative analysis to decision making, the management scientist must work closely with the manager or user of the results. When both the management scientist and the manager agree that the problem has been adequately structured, work can begin on developing a model to represent the problem mathematically. Solution procedures can then be employed to find the best solution for the model. This best solution for the model then becomes a recommendation to the decision maker. The process of developing and solving models is the essence of the quantitative analysis process.

Model Development
Models are representations of real objects or situations and can be presented in various forms. For example, a scale model of an airplane is a representation of a real airplane. Similarly, a child’s toy truck is a model of a real truck. The model airplane and toy truck are examples of models that are physical replicas of real objects. In modeling terminology, physical replicas are referred to as iconic models. A second classification includes models that are physical in form but do not have the same physical appearance as the object being modeled. Such models are referred to as analog models. The speedometer of an automobile is an analog model; the position of the needle on the dial represents the speed of the automobile. A thermometer is another analog model representing temperature. A third classification of models—the type we will primarily be studying—includes representations of a problem by a system of symbols and mathematical relationships or expressions. Such models are referred to as mathematical models and are a critical part of any quantitative approach to decision making. For example, the total profit from the sale of a product can be determined by multiplying the profit per unit by the quantity sold. If we let x represent the number of units sold and P the total profit, then, with a profit of $10 per unit, the following mathematical model defines the total profit earned by selling x units: P = 10x
(1.1)

The purpose, or value, of any model is that it enables us to make inferences about the real situation by studying and analyzing the model. For example, an airplane designer might test an iconic model of a new airplane in a wind tunnel to learn about the potential flying characteristics of the full-size airplane. Similarly, a mathematical model may be used to make inferences about how much profit will be earned if a specified quantity of a particular product is sold. According to the mathematical model of equation (1.1), we would expect selling three units of the product (x 3) would provide a profit of P 10(3) $30. In general, experimenting with models requires less time and is less expensive than experimenting with the real object or situation. A model airplane is certainly quicker and less expensive to build and study than the full-size airplane. Similarly, the mathematical model in equation (1.1) allows a quick identification of profit expectations without actually requiring the manager to produce and sell x units. Models also have the advantage of reducing the risk associated with experimenting with the real situation. In particular, bad designs or bad decisions that cause the model airplane to crash or a mathematical model to project a $10,000 loss can be avoided in the real situation. The value of model-based conclusions and decisions is dependent on how well the model represents the real situation. The more closely the model airplane represents the real

8
Herbert A. Simon, a Nobel Prize winner in economics and an expert in decision making, said that a mathematical model does not have to be exact; it just has to be close enough to provide better results than can be obtained by common sense.

Chapter 1

Introduction

airplane, the more accurate the conclusions and predictions will be. Similarly, the more closely the mathematical model represents the company’s true profit-volume relationship, the more accurate the profit projections will be. Because this text deals with quantitative analysis based on mathematical models, let us look more closely at the mathematical modeling process. When initially considering a managerial problem, we usually find that the problem definition phase leads to a specific objective, such as maximization of profit or minimization of cost, and possibly a set of restrictions or constraints, such as production capacities. The success of the mathematical model and quantitative approach will depend heavily on how accurately the objective and constraints can be expressed in terms of mathematical equations or relationships. A mathematical expression that describes the problem’s objective is referred to as the objective function. For example, the profit equation P 10x would be an objective function for a firm attempting to maximize profit. A production capacity constraint would be necessary if, for instance, 5 hours are required to produce each unit and only 40 hours of production time are available per week. Let x indicate the number of units produced each week. The production time constraint is given by

5x … 40

(1.2)

The value of 5x is the total time required to produce x units; the symbol indicates that the production time required must be less than or equal to the 40 hours available. The decision problem or question is the following: How many units of the product should be scheduled each week to maximize profit? A complete mathematical model for this simple production problem is Maximize P = 10x objective function subject to (s.t.) 5x … 40 f constraints x Ú 0 The x 0 constraint requires the production quantity x to be greater than or equal to zero, which simply recognizes the fact that it is not possible to manufacture a negative number of units. The optimal solution to this model can be easily calculated and is given by x 8, with an associated profit of $80. This model is an example of a linear programming model. In subsequent chapters we will discuss more complicated mathematical models and learn how to solve them in situations where the answers are not nearly so obvious. In the preceding mathematical model, the profit per unit ($10), the production time per unit (5 hours), and the production capacity (40 hours) are environmental factors that are not under the control of the manager or decision maker. Such environmental factors, which can affect both the objective function and the constraints, are referred to as uncontrollable inputs to the model. Inputs that are controlled or determined by the decision maker are referred to as controllable inputs to the model. In the example given, the production quantity x is the controllable input to the model. Controllable inputs are the decision alternatives specified by the manager and thus are also referred to as the decision variables of the model. Once all controllable and uncontrollable inputs are specified, the objective function and constraints can be evaluated and the output of the model determined. In this sense, the output of the model is simply the projection of what would happen if those particular

1.3

Quantitative Analysis

9

FIGURE 1.4 FLOWCHART OF THE PROCESS OF TRANSFORMING MODEL INPUTS INTO OUTPUT
Uncontrollable Inputs (Environmental Factors)

Controllable Inputs (Decision Variables)

Mathematical Model

Output (Projected Results)

environmental factors and decisions occurred in the real situation. A flowchart of how controllable and uncontrollable inputs are transformed by the mathematical model into output is shown in Figure 1.4. A similar flowchart showing the specific details of the production model is shown in Figure 1.5. As stated earlier, the uncontrollable inputs are those the decision maker cannot influence. The specific controllable and uncontrollable inputs of a model depend on the particular problem or decision-making situation. In the production problem, the production time available (40) is an uncontrollable input. However, if it were possible to hire more employees or use overtime, the number of hours of production time would become a controllable input and therefore a decision variable in the model. Uncontrollable inputs can either be known exactly or be uncertain and subject to variation. If all uncontrollable inputs to a model are known and cannot vary, the model is referred to as a deterministic model. Corporate income tax rates are not under the influence of the manager and thus constitute an uncontrollable input in many decision models. Because these rates are known and fixed (at least in the short run), a mathematical model with corporate income tax rates as the only uncontrollable input would be a deterministic

FIGURE 1.5 FLOWCHART FOR THE PRODUCTION MODEL
Uncontrollable Inputs 10 Profit per Unit ($) 5 Production Time per Unit (Hours) 40 Production Capacity (Hours)

Value for the Production Quantity (x = 8) Controllable Input

Max 10 (8) s.t. 5 (8) ≤ 40 8 ≥ 0 Mathematical Model

Profit = 80 Time Used = 40 Output

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Chapter 1

Introduction

model. The distinguishing feature of a deterministic model is that the uncontrollable input values are known in advance. If any of the uncontrollable inputs are uncertain and subject to variation, the model is referred to as a stochastic or probabilistic model. An uncontrollable input to many production planning models is demand for the product. A mathematical model that treats future demand—which may be any of a range of values—with uncertainty would be called a stochastic model. In the production model, the number of hours of production time required per unit, the total hours available, and the unit profit were all uncontrollable inputs. Because the uncontrollable inputs were all known to take on fixed values, the model was deterministic. If, however, the number of hours of production time per unit could vary from 3 to 6 hours depending on the quality of the raw material, the model would be stochastic. The distinguishing feature of a stochastic model is that the value of the output cannot be determined even if the value of the controllable input is known because the specific values of the uncontrollable inputs are unknown. In this respect, stochastic models are often more difficult to analyze.

Data Preparation
Another step in the quantitative analysis of a problem is the preparation of the data required by the model. Data in this sense refer to the values of the uncontrollable inputs to the model. All uncontrollable inputs or data must be specified before we can analyze the model and recommend a decision or solution for the problem. In the production model, the values of the uncontrollable inputs or data were $10 per unit for profit, 5 hours per unit for production time, and 40 hours for production capacity. In the development of the model, these data values were known and incorporated into the model as it was being developed. If the model is relatively small and the uncontrollable input values or data required are few, the quantitative analyst will probably combine model development and data preparation into one step. In these situations the data values are inserted as the equations of the mathematical model are developed. However, in many mathematical modeling situations, the data or uncontrollable input values are not readily available. In these situations the management scientist may know that the model will need profit per unit, production time, and production capacity data, but the values will not be known until the accounting, production, and engineering departments can be consulted. Rather than attempting to collect the required data as the model is being developed, the analyst will usually adopt a general notation for the model development step, and then a separate data preparation step will be performed to obtain the uncontrollable input values required by the model. Using the general notation

c = profit per unit a = production time in hours per unit b = production capacity in hours the model development step of the production problem would result in the following general model: Max s.t.

cx ax … b x Ú 0

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Quantitative Analysis

11

A separate data preparation step to identify the values for c, a, and b would then be necessary to complete the model. Many inexperienced quantitative analysts assume that once the problem has been defined and a general model developed, the problem is essentially solved. These individuals tend to believe that data preparation is a trivial step in the process and can be easily handled by clerical staff. Actually, this assumption could not be further from the truth, especially with large-scale models that have numerous data input values. For example, a small linear programming model with 50 decision variables and 25 constraints could have more than 1300 data elements that must be identified in the data preparation step. The time required to prepare these data and the possibility of data collection errors will make the data preparation step a critical part of the quantitative analysis process. Often, a fairly large database is needed to support a mathematical model, and information systems specialists may become involved in the data preparation step.

Model Solution
Once the model development and data preparation steps are completed, we can proceed to the model solution step. In this step, the analyst will attempt to identify the values of the decision variables that provide the “best” output for the model. The specific decision-variable value or values providing the “best” output will be referred to as the optimal solution for the model. For the production problem, the model solution step involves finding the value of the production quantity decision variable x that maximizes profit while not causing a violation of the production capacity constraint. One procedure that might be used in the model solution step involves a trial-and-error approach in which the model is used to test and evaluate various decision alternatives. In the production model, this procedure would mean testing and evaluating the model under various production quantities or values of x. Note, in Figure 1.5, that we could input trial values for x and check the corresponding output for projected profit and satisfaction of the production capacity constraint. If a particular decision alternative does not satisfy one or more of the model constraints, the decision alternative is rejected as being infeasible, regardless of the objective function value. If all constraints are satisfied, the decision alternative is feasible and a candidate for the “best” solution or recommended decision. Through this trial-and-error process of evaluating selected decision alternatives, a decision maker can identify a good—and possibly the best—feasible solution to the problem. This solution would then be the recommended decision for the problem. Table 1.2 shows the results of a trial-and-error approach to solving the production model of Figure 1.5. The recommended decision is a production quantity of 8 because the feasible solution with the highest projected profit occurs at x 8. Although the trial-and-error solution process is often acceptable and can provide valuable information for the manager, it has the drawbacks of not necessarily providing the best solution and of being inefficient in terms of requiring numerous calculations if many decision alternatives are tried. Thus, quantitative analysts have developed special solution procedures for many models that are much more efficient than the trial-and-error approach. Throughout this text, you will be introduced to solution procedures that are applicable to the specific mathematical models that will be formulated. Some relatively small models or problems can be solved by hand computations, but most practical applications require the use of a computer. Model development and model solution steps are not completely separable. An analyst will want both to develop an accurate model or representation of the actual problem situation and to be able to find a solution to the model. If we approach the model development

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Chapter 1

Introduction

TABLE 1.2 TRIAL-AND-ERROR SOLUTION FOR THE PRODUCTION MODEL OF FIGURE 1.5 Decision Alternative (Production Quantity) x
0 2 4 6 8 10 12

Projected Profit
0 20 40 60 80 100 120

Total Hours of Production
0 10 20 30 40 50 60

Feasible Solution? (Hours Used ◊ 40)
Yes Yes Yes Yes Yes No No

Try Problem 8 to test your understanding of the concept of a mathematical model and what is referred to as the optimal solution to the model.

step by attempting to find the most accurate and realistic mathematical model, we may find the model so large and complex that it is impossible to obtain a solution. In this case, a simpler and perhaps more easily understood model with a readily available solution procedure is preferred even if the recommended solution is only a rough approximation of the best decision. As you learn more about quantitative solution procedures, you will have a better idea of the types of mathematical models that can be developed and solved. After a model solution is obtained, both the management scientist and the manager will be interested in determining how good the solution really is. Even though the analyst has undoubtedly taken many precautions to develop a realistic model, often the goodness or accuracy of the model cannot be assessed until model solutions are generated. Model testing and validation are frequently conducted with relatively small “test” problems that have known or at least expected solutions. If the model generates the expected solutions, and if other output information appears correct, the go-ahead may be given to use the model on the full-scale problem. However, if the model test and validation identify potential problems or inaccuracies inherent in the model, corrective action, such as model modification and/or collection of more accurate input data, may be taken. Whatever the corrective action, the model solution will not be used in practice until the model has satisfactorily passed testing and validation.

Report Generation
An important part of the quantitative analysis process is the preparation of managerial reports based on the model’s solution. In Figure 1.3, we see that the solution based on the quantitative analysis of a problem is one of the inputs the manager considers before making a final decision. Thus, the results of the model must appear in a managerial report that can be easily understood by the decision maker. The report includes the recommended decision and other pertinent information about the results that may be helpful to the decision maker.

A Note Regarding Implementation
As discussed in Section 1.2, the manager is responsible for integrating the quantitative solution with qualitative considerations in order to make the best possible decision. After completing the decision-making process, the manager must oversee the implementation

1.3

Quantitative Analysis

13

and follow-up evaluation of the decision. The manager should continue to monitor the contribution of the model during the implementation and follow-up. At times this process may lead to requests for model expansion or refinement that will cause the management scientist to return to an earlier step of the quantitative analysis process. Successful implementation of results is of critical importance to the management scientist as well as the manager. If the results of the quantitative analysis process are not correctly implemented, the entire effort may be of no value. It doesn’t take too many unsuccessful implementations before the management scientist is out of work. Because implementation often requires people to do things differently, it often meets with resistance. People want to know, “What’s wrong with the way we’ve been doing it?” and so on. One of the most effective ways to ensure successful implementation is to include users throughout the modeling process. A user who feels a part of identifying the problem and developing the solution is much more likely to enthusiastically implement the results. The success rate for implementing the results of a management science project is much greater for those projects characterized by extensive user involvement. The Management Science in Action, Quantitative Analysis at Merrill Lynch, discusses some of the reasons behind the success Merrill Lynch realized from using quantitative analysis.

MANAGEMENT SCIENCE IN ACTION
QUANTITATIVE ANALYSIS AT MERRILL LYNCH*
Merrill Lynch, a brokerage and financial services firm with more than 56,000 employees in 45 countries, serves its client base through two business units. The Merrill Lynch Corporate and Institutional Client Group serves more than 7000 corporations, institutions, and governments. The Merrill Lynch Private Client Group (MLPC) serves approximately 4 million households, as well as 225,000 small to mid-sized businesses and regional financial institutions, through more than 14,000 financial consultants in 600-plus branch offices. The management science group, established in 1986, has been part of MLPC since 1991. The mission of this group is to provide high-end quantitative analysis to support strategic management decisions and to enhance the financial consultant–client relationship. The management science group has successfully implemented models and developed systems for asset allocation, financial planning, marketing information technology, database marketing, and portfolio performance measurement.Although technical expertise and objectivity are clearly important factors in any analytical group, the management science group attributes much of its success to communications skills, teamwork, and consulting skills. Each project begins with face-to-face meetings with the client. A proposal is then prepared to outline

the background of the problem, the objectives of the project, the approach, the required resources, the time schedule, and the implementation issues. At this stage, analysts focus on developing solutions that provide significant value and are easily implemented. As the work progresses, frequent meetings keep the clients up to date. Because people with different skills, perspectives, and motivations must work together for a common goal, teamwork is essential. The group’s members take classes in team approaches, facilitation, and conflict resolution. They possess a broad range of multifunctional and multidisciplinary capabilities and are motivated to provide solutions that focus on the goals of the firm. This approach to problem solving and the implementation of quantitative analysis has been a hallmark of the management science group. The impact and success of the group translates into hard dollars and repeat business. The group received the annual Edelman award given by the Institute for Operations Research and the Management Sciences for effective use of management science for organizational success.

*Based on Russ Labe, Raj Nigam, and Steve Spence, “Management Science at Merrill Lynch Private Client Group,” Interfaces 29, no. 2 (March/April 1999): 1–14.

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Chapter 1

Introduction

NOTES AND COMMENTS
1. Developments in computer technology have increased the availability of management science techniques to decision makers. Many software packages are now available for personal computers. Microsoft Excel, and LINGO are widely used in management science courses and in industry. 2. Various chapter appendices provide step-bystep instructions for using Excel and LINGO to solve problems in the text. Microsoft Excel has become the most used analytical modeling software in business and industry. We recommend that you read Appendix A, Building Spreadsheet Models, located in the back of this text.

1.4

MODELS OF COST, REVENUE, AND PROFIT
Some of the most basic quantitative models arising in business and economic applications are those involving the relationship between a volume variable—such as production volume or sales volume—and cost, revenue, and profit. Through the use of these models, a manager can determine the projected cost, revenue, and/or profit associated with an established production quantity or a forecasted sales volume. Financial planning, production planning, sales quotas, and other areas of decision making can benefit from such cost, revenue, and profit models.

Cost and Volume Models
The cost of manufacturing or producing a product is a function of the volume produced. This cost can usually be defined as a sum of two costs: fixed cost and variable cost. Fixed cost is the portion of the total cost that does not depend on the production volume; this cost remains the same no matter how much is produced. Variable cost, on the other hand, is the portion of the total cost that is dependent on and varies with the production volume. To illustrate how cost and volume models can be developed, we will consider a manufacturing problem faced by Nowlin Plastics. Nowlin Plastics produces a variety of compact disc (CD) storage cases. Nowlin’s bestselling product is the CD-50, a slim, plastic CD holder with a specially designed lining that protects the optical surface of the disc. Several products are produced on the same manufacturing line, and a setup cost is incurred each time a changeover is made for a new product. Suppose that the setup cost for the CD-50 is $3000. This setup cost is a fixed cost that is incurred regardless of the number of units eventually produced. In addition, suppose that variable labor and material costs are $2 for each unit produced. The cost-volume model for producing x units of the CD-50 can be written as (1.3)

C(x) = 3000 + 2x where x = production volume in units C(x) = total cost of producing x units

1.4

Models of Cost, Revenue, and Profit

15

Once a production volume is established, the model in equation (1.3) can be used to compute the total production cost. For example, the decision to produce x 1200 units would result in a total cost of C(1200) 3000 2(1200) $5400. Marginal cost is defined as the rate of change of the total cost with respect to production volume. That is, it is the cost increase associated with a one-unit increase in the production volume. In the cost model of equation (1.3), we see that the total cost C(x) will increase by $2 for each unit increase in the production volume. Thus, the marginal cost is $2. With more complex total cost models, marginal cost may depend on the production volume. In such cases, we could have marginal cost increasing or decreasing with the production volume x.

Revenue and Volume Models
Management of Nowlin Plastics will also want information on the projected revenue associated with selling a specified number of units. Thus, a model of the relationship between revenue and volume is also needed. Suppose that each CD-50 storage unit sells for $5. The model for total revenue can be written as

R(x) = 5x where (1.4)

x = sales volume in units R(x) = total revenue associated with selling x units Marginal revenue is defined as the rate of change of total revenue with respect to sales volume. That is, it is the increase in total revenue resulting from a one-unit increase in sales volume. In the model of equation (1.4), we see that the marginal revenue is $5. In this case, marginal revenue is constant and does not vary with the sales volume. With more complex models, we may find that marginal revenue increases or decreases as the sales volume x increases.

Profit and Volume Models
One of the most important criteria for management decision making is profit. Managers need to be able to know the profit implications of their decisions. If we assume that we will only produce what can be sold, the production volume and sales volume will be equal. We can combine equations (1.3) and (1.4) to develop a profit-volume model that will determine the total profit associated with a specified production-sales volume. Total profit, denoted P(x), is total revenue minus total cost; therefore, the following model provides the total profit associated with producing and selling x units:

P(x) = R(x) - C(x) = 5x - (3000 + 2x) = - 3000 + 3x

(1.5)

Thus, the profit-volume model can be derived from the revenue-volume and cost-volume models.

16

Chapter 1

Introduction

Breakeven Analysis
Using equation (1.5), we can now determine the total profit associated with any production volume x. For example, suppose that a demand forecast indicates that 500 units of the product can be sold. The decision to produce and sell the 500 units results in a projected profit of

P(500) = - 3000 + 3(500) = -1500
In other words, a loss of $1500 is predicted. If sales are expected to be 500 units, the manager may decide against producing the product. However, a demand forecast of 1800 units would show a projected profit of

P(1800) = - 3000 + 3(1800) = 2400
This profit may be enough to justify proceeding with the production and sale of the product. We see that a volume of 500 units will yield a loss, whereas a volume of 1800 provides a profit. The volume that results in total revenue equaling total cost (providing $0 profit) is called the breakeven point. If the breakeven point is known, a manager can quickly infer that a volume above the breakeven point will result in a profit, whereas a volume below the breakeven point will result in a loss. Thus, the breakeven point for a product provides valuable information for a manager who must make a yes/no decision concerning production of the product. Let us now return to the Nowlin Plastics example and show how the total profit model in equation (1.5) can be used to compute the breakeven point. The breakeven point can be found by setting the total profit expression equal to zero and solving for the production volume. Using equation (1.5), we have

P(x) = - 3000 + 3x = 0 3x = 3000 x = 1000
Try Problem 12 to test your ability to determine the breakeven point for a quantitative model.

With this information, we know that production and sales of the product must be greater than 1000 units before a profit can be expected. The graphs of the total cost model, the total revenue model, and the location of the breakeven point are shown in Figure 1.6. In Appendix 1.1 we also show how Excel can be used to perform a breakeven analysis for the Nowlin Plastics production example.

1.5

MANAGEMENT SCIENCE TECHNIQUES
In this section we present a brief overview of the management science techniques covered in this text. Over the years, practitioners have found numerous applications for the following techniques:

Linear Programming Linear programming is a problem-solving approach developed for situations involving maximizing or minimizing a linear function subject to linear constraints that limit the degree to which the objective can be pursued. The production model developed in Section 1.3 (see Figure 1.5) is an example of a simple linear programming model. Integer Linear Programming Integer linear programming is an approach used for problems that can be set up as linear programs, with the additional requirement that some or all of the decision variables be integer values.

1.5

Management Science Techniques

17

FIGURE 1.6 GRAPH OF THE BREAKEVEN ANALYSIS FOR NOWLIN PLASTICS

Revenue and Cost ($)

10,000 8000 6000 4000 2000 0 Fixed Cost Loss 400 800

Total Revenue R (x) = 5x

Profit Total Cost C(x) = 3000 + 2 x

Breakeven Point = 1000 Units 1200 1600 Production Volume 2000 x

Distribution and Network Models A network is a graphical description of a problem consisting of circles called nodes that are interconnected by lines called arcs. Specialized solution procedures exist for these types of problems, enabling us to quickly solve problems in such areas as transportation system design, information system design, and project scheduling. Nonlinear Programming Many business processes behave in a nonlinear manner. For example, the price of a bond is a nonlinear function of interest rates; the quantity demanded for a product is usually a nonlinear function of the price. Nonlinear programming is a technique that allows for maximizing or minimizing a nonlinear function subject to nonlinear constraints.

Project Scheduling: PERT/CPM In many situations, managers are responsible for planning, scheduling, and controlling projects that consist of numerous separate jobs or tasks performed by a variety of departments, individuals, and so forth. The PERT (Program Evaluation and Review Technique) and CPM (Critical Path Method) techniques help managers carry out their project scheduling responsibilities.

Waiting-Line or Queueing Models Waiting-line or queueing models have been developed to help managers understand and make better decisions concerning the operation of systems involving waiting lines. Simulation Simulation is a technique used to model the operation of a system. This technique employs a computer program to model the operation and perform simulation computations. Decision Analysis Decision analysis can be used to determine optimal strategies in situations involving several decision alternatives and an uncertain or risk-filled pattern of events. Goal Programming Goal programming is a technique for solving multicriteria decision problems, usually within the framework of linear programming.

Inventory Models Inventory models are used by managers faced with the dual problems of maintaining sufficient inventories to meet demand for goods and, at the same time, incurring the lowest possible inventory holding costs.

Analytic Hierarchy Process This multicriteria decision-making technique permits the inclusion of subjective factors in arriving at a recommended decision.

18

Chapter 1

Introduction

Forecasting Forecasting methods are techniques that can be used to predict future aspects of a business operation. Markov Process Models Markov process models are useful in studying the evolution of certain systems over repeated trials. For example, Markov processes have been used to describe the probability that a machine, functioning in one period, will function or break down in another period.

Methods Used Most Frequently
Our experience as both practitioners and educators has been that the most frequently used management science techniques are linear programming, integer programming, network models (including transportation and transshipment models), and simulation. Depending upon the industry, the other methods in the preceding list are used more or less frequently. Helping to bridge the gap between the manager and the management scientist is a major focus of the text. We believe that the barriers to the use of management science can best be removed by increasing the manager’s understanding of how management science can be applied. The text will help you develop an understanding of which management science techniques are most useful, how they are used, and, most importantly, how they can assist managers in making better decisions. The Management Science in Action, Impact of Operations Research on Everyday Living, describes some of the many ways quantitative analysis affects our everyday lives.

MANAGEMENT SCIENCE IN ACTION
IMPACT OF OPERATIONS RESEARCH ON EVERYDAY LIVING*
Mark Eisner, associate director of the School of Operations Research and Industrial Engineering at Cornell University, once said that operations research “is probably the most important field nobody’s ever heard of.” The impact of operations research on everyday living over the past 20 years is substantial. Suppose you schedule a vacation to Florida and use Orbitz to book your flights. An algorithm developed by operations researchers will search among millions of options to find the cheapest fare. Another algorithm will schedule the flight crews and aircraft used by the airline. If you rent a car in Florida, the price you pay for the car is determined by a mathematical model that seeks to maximize revenue for the car rental firm. If you do some shopping on your trip and decide to ship your purchases home using UPS, another algorithm tells UPS which truck to put the packages on, the route the truck should follow, and where the packages

should be placed on the truck to minimize loading and unloading time. If you enjoy watching college basketball, operations research plays a role in which games you see. Michael Trick, a professor at the Tepper School of Business at Carnegie-Mellon, designed a system for scheduling each year’s Atlantic Coast Conference men’s and women’s basketball games. Even though it might initially appear that scheduling 16 games among the nine men’s teams would be easy, it requires sorting through hundreds of millions of possible combinations of possible schedules. Each of those possibilities entails some desirable and some undesirable characteristics. For example, you do not want to schedule too many consecutive home games, and you want to ensure that each team plays the same number of weekend games.

*Based on Virginia Postrel, “Operations Everything,” The Boston Globe, June 27, 2004.

NOTES AND COMMENTS
The Institute for Operations Research and the Management Sciences (INFORMS) and the Decision Sciences Institute (DSI) are two professional societies that publish journals and newsletters dealing with current research and applications of operations research and management science techniques.

Glossary

19

SUMMARY
This text is about how management science may be used to help managers make better decisions. The focus of this text is on the decision-making process and on the role of management science in that process. We discussed the problem orientation of this process and in an overview showed how mathematical models can be used in this type of analysis. The difference between the model and the situation or managerial problem it represents is an important point. Mathematical models are abstractions of real-world situations and, as such, cannot capture all the aspects of the real situation. However, if a model can capture the major relevant aspects of the problem and can then provide a solution recommendation, it can be a valuable aid to decision making. One of the characteristics of management science that will become increasingly apparent as we proceed through the text is the search for a best solution to the problem. In carrying out the quantitative analysis, we shall be attempting to develop procedures for finding the “best” or optimal solution.

GLOSSARY
Problem solving The process of identifying a difference between the actual and the desired state of affairs and then taking action to resolve the difference. Decision making The process of defining the problem, identifying the alternatives, determining the criteria, evaluating the alternatives, and choosing an alternative. Single-criterion decision problem A problem in which the objective is to find the “best” solution with respect to just one criterion.

Multicriteria decision problem A problem that involves more than one criterion; the objective is to find the “best” solution, taking into account all the criteria. Decision The alternative selected. Iconic model Model A representation of a real object or situation.

Analog model Although physical in form, an analog model does not have a physical appearance similar to the real object or situation it represents. Mathematical model Mathematical symbols and expressions used to represent a real situation. Constraints
Restrictions or limitations imposed on a problem.

A physical replica, or representation, of a real object.

Objective function A mathematical expression that describes the problem’s objective. Uncontrollable inputs The environmental factors or inputs that cannot be controlled by the decision maker. Decision variable
Another term for controllable input.

Controllable inputs The inputs that are controlled or determined by the decision maker. Deterministic model A model in which all uncontrollable inputs are known and cannot vary.

Stochastic (probabilistic) model A model in which at least one uncontrollable input is uncertain and subject to variation; stochastic models are also referred to as probabilistic models.

20

Chapter 1

Introduction

Infeasible solution A decision alternative or solution that does not satisfy one or more constraints. Feasible solution A decision alternative or solution that satisfies all constraints. Fixed cost The portion of the total cost that does not depend on the volume; this cost remains the same no matter how much is produced. Marginal cost
The rate of change of the total cost with respect to volume. The volume at which total revenue equals total cost.

Optimal solution The specific decision-variable value or values that provide the “best” output for the model.

Variable cost The portion of the total cost that is dependent on and varies with the volume. Marginal revenue Breakeven point
The rate of change of total revenue with respect to volume.

PROBLEMS
2. List and discuss the steps of the decision-making process. 1. Define the terms management science and operations research.

3. Discuss the different roles played by the qualitative and quantitative approaches to managerial decision making. Why is it important for a manager or decision maker to have a good understanding of both of these approaches to decision making?

4. A firm just completed a new plant that will produce more than 500 different products, using more than 50 different production lines and machines. The production scheduling decisions are critical in that sales will be lost if customer demands are not met on time. If no individual in the firm has experience with this production operation and if new production schedules must be generated each week, why should the firm consider a quantitative approach to the production scheduling problem? 5. What are the advantages of analyzing and experimenting with a model as opposed to a real object or situation? 6. Suppose that a manager has a choice between the following two mathematical models of a given situation: (a) a relatively simple model that is a reasonable approximation of the real situation, and (b) a thorough and complex model that is the most accurate mathematical representation of the real situation possible. Why might the model described in part (a) be preferred by the manager? 7. Suppose you are going on a weekend trip to a city that is d miles away. Develop a model that determines your round-trip gasoline costs. What assumptions or approximations are necessary to treat this model as a deterministic model? Are these assumptions or approximations acceptable to you? 8. Recall the production model from Section 1.3: Max s.t. 10x 5x … 40 x Ú 0 Suppose the firm in this example considers a second product that has a unit profit of $5 and requires 2 hours of production time for each unit produced. Use y as the number of units of product 2 produced.

Problems

21

9. Suppose we modify the production model in Section 1.3 to obtain the following mathematical model: Max s.t. 10x ax … 40 x Ú 0

a. b. c. d. e.

Show the mathematical model when both products are considered simultaneously. Identify the controllable and uncontrollable inputs for this model. Draw the flowchart of the input-output process for this model (see Figure 1.5). What are the optimal solution values of x and y? Is the model developed in part (a) a deterministic or a stochastic model? Explain.

10. A retail store in Des Moines, Iowa, receives shipments of a particular product from Kansas City and Minneapolis. Let x = number of units of the product received from Kansas City y = number of units of the product received from Minneapolis

where a is the number of hours of production time required for each unit produced. With a 5, the optimal solution is x 8. If we have a stochastic model with a 3, a 4, a 5, or a 6 as the possible values for the number of hours required per unit, what is the optimal value for x? What problems does this stochastic model cause?

11. For most products, higher prices result in a decreased demand, whereas lower prices result in an increased demand. Let d = annual demand for a product in units p = price per unit Assume that a firm accepts the following price-demand relationship as being realistic: d = 800 - 10p

a. Write an expression for the total number of units of the product received by the retail store in Des Moines. b. Shipments from Kansas City cost $0.20 per unit, and shipments from Minneapolis cost $0.25 per unit. Develop an objective function representing the total cost of shipments to Des Moines. c. Assuming the monthly demand at the retail store is 5000 units, develop a constraint that requires 5000 units to be shipped to Des Moines. d. No more than 4000 units can be shipped from Kansas City, and no more than 3000 units can be shipped from Minneapolis in a month. Develop constraints to model this situation. e. Of course, negative amounts cannot be shipped. Combine the objective function and constraints developed to state a mathematical model for satisfying the demand at the Des Moines retail store at minimum cost.

where p must be between $20 and $70. a. How many units can the firm sell at the $20 per-unit price? At the $70 per-unit price? b. Show the mathematical model for the total revenue (TR), which is the annual demand multiplied by the unit price.

22

Chapter 1

Introduction

13. Micromedia offers computer training seminars on a variety of topics. In the seminars each student works at a personal computer, practicing the particular activity that the instructor is presenting. Micromedia is currently planning a two-day seminar on the use of Microsoft Excel in statistical analysis. The projected fee for the seminar is $300 per student. The cost for the conference room, instructor compensation, lab assistants, and promotion is $4800. Micromedia rents computers for its seminars at a cost of $30 per computer per day. a. Develop a model for the total cost to put on the seminar. Let x represent the number of students who enroll in the seminar. b. Develop a model for the total profit if x students enroll in the seminar. c. Micromedia has forecasted an enrollment of 30 students for the seminar. How much profit will be earned if their forecast is accurate? d. Compute the breakeven point. 14. Eastman Publishing Company is considering publishing a paperback textbook on spreadsheet applications for business. The fixed cost of manuscript preparation, textbook design, and production setup is estimated to be $80,000. Variable production and material costs are estimated to be $3 per book. Demand over the life of the book is estimated to be 4000 copies. The publisher plans to sell the text to college and university bookstores for $20 each. a. What is the breakeven point? b. What profit or loss can be anticipated with a demand of 4000 copies? c. With a demand of 4000 copies, what is the minimum price per copy that the publisher must charge to break even? d. If the publisher believes that the price per copy could be increased to $25.95 and not affect the anticipated demand of 4000 copies, what action would you recommend? What profit or loss can be anticipated?

12. The O’Neill Shoe Manufacturing Company will produce a special-style shoe if the order size is large enough to provide a reasonable profit. For each special-style order, the company incurs a fixed cost of $1000 for the production setup. The variable cost is $30 per pair, and each pair sells for $40. a. Let x indicate the number of pairs of shoes produced. Develop a mathematical model for the total cost of producing x pairs of shoes. b. Let P indicate the total profit. Develop a mathematical model for the total profit realized from an order for x pairs of shoes. c. How large must the shoe order be before O’Neill will break even?

c. Based on other considerations, the firm’s management will only consider price alternatives of $30, $40, and $50. Use your model from part (b) to determine the price alternative that will maximize the total revenue. d. What are the expected annual demand and the total revenue corresponding to your recommended price?

15. Preliminary plans are under way for the construction of a new stadium for a major league baseball team. City officials have questioned the number and profitability of the luxury corporate boxes planned for the upper deck of the stadium. Corporations and selected individuals may buy the boxes for $100,000 each. The fixed construction cost for the upperdeck area is estimated to be $1,500,000, with a variable cost of $50,000 for each box constructed. a. What is the breakeven point for the number of luxury boxes in the new stadium? b. Preliminary drawings for the stadium show that space is available for the construction of up to 50 luxury boxes. Promoters indicate that buyers are available and that all 50 could be sold if constructed. What is your recommendation concerning the construction of luxury boxes? What profit is anticipated?

Case Problem

Scheduling a Golf League

23

16. Financial Analysts, Inc., is an investment firm that manages stock portfolios for a number of clients. A new client is requesting that the firm handle an $80,000 portfolio. As an initial investment strategy, the client would like to restrict the portfolio to a mix of the following two stocks:

Stock Oil Alaska Southwest Petroleum

Price/ Share $50 $30

Maximum Estimated Annual Return/Share $6 $4

Possible Investment $50,000 $45,000

Let x = number of shares of Oil Alaska y = number of shares of Southwest Petroleum a. Develop the objective function, assuming that the client desires to maximize the total annual return. b. Show the mathematical expression for each of the following three constraints: (1) Total investment funds available are $80,000. (2) Maximum Oil Alaska investment is $50,000. (3) Maximum Southwest Petroleum investment is $45,000.

17. Models of inventory systems frequently consider the relationships among a beginning inventory, a production quantity, a demand or sales, and an ending inventory. For a given production period j, let sj-1 xj dj sj = = = = ending inventory from the previous period (beginning inventory for period j ) production quantity in period j demand in period j ending inventory for period j

Note: Adding the x 0 and y 0 constraints provides a linear programming model for the investment problem. A solution procedure for this model will be discussed in Chapter 2.

a. Write the mathematical relationship or model that describes how these four variables are related. b. What constraint should be added if production capacity for period j is given by Cj? c. What constraint should be added if inventory requirements for period j mandate an ending inventory of at least Ij?

Case Problem

SCHEDULING A GOLF LEAGUE

Chris Lane, the head professional at Royal Oak Country Club, must develop a schedule of matches for the couples’ golf league that begins its season at 4:00 P.M. tomorrow. Eighteen couples signed up for the league, and each couple must play every other couple over the course of the 17-week season. Chris thought it would be fairly easy to develop a

24

Chapter 1

Introduction

schedule, but after working on it for a couple of hours, he has been unable to come up with a schedule. Because Chris must have a schedule ready by tomorrow afternoon, he asked you to help him. A possible complication is that one of the couples told Chris that they may have to cancel for the season. They told Chris they will let him know by 1:00 P.M. tomorrow whether they will be able to play this season.

Managerial Report
Prepare a report for Chris Lane. Your report should include, at a minimum, the following items: 1. A schedule that will enable each of the 18 couples to play every other couple over the 17-week season. 2. A contingency schedule that can be used if the couple that contacted Chris decides to cancel for the season.

Appendix 1.1

USING EXCEL FOR BREAKEVEN ANALYSIS

In Section 1.4 we introduced the Nowlin Plastics production example to illustrate how quantitative models can be used to help a manager determine the projected cost, revenue, and/or profit associated with an established production quantity or a forecasted sales volume. In this appendix we introduce spreadsheet applications by showing how to use Microsoft Excel to perform a quantitative analysis of the Nowlin Plastics example. Refer to the worksheet shown in Figure 1.7. We begin by entering the problem data into the top portion of the worksheet. The value of 3000 in cell B3 is the fixed cost, the value

FIGURE 1.7 FORMULA WORKSHEET FOR THE NOWLIN PLASTICS PRODUCTION EXAMPLE
A 1 Nowlin Plastics 2 3 Fixed Cost 4 5 Variable Cost Per Unit 6 7 Selling Price Per Unit 8 9 10 Models 11 12 Production Volume 13 14 Total Cost 15 16 Total Revenue 17 18 Total Profit (Loss) B 3000 2 5

800 =B3+B5*B12 =B7*B12 =B16-B14

Appendix 1.1

Using Excel for Breakeven Analysis

25

of 2 in cell B5 is the variable labor and material costs per unit, and the value of 5 in cell B7 is the selling price per unit. As discussed in Appendix A, whenever we perform a quantitative analysis using Excel, we will enter the problem data in the top portion of the worksheet and reserve the bottom portion for model development. The label “Models” in cell A10 helps to provide a visual reminder of this convention. Cell B12 in the models portion of the worksheet contains the proposed production volume in units. Because the values for total cost, total revenue, and total profit depend upon the value of this decision variable, we have placed a border around cell B12 and screened the cell for emphasis. Based upon the value in cell B12, the cell formulas in cells B14, B16, and B18 are used to compute values for total cost, total revenue, and total profit (loss), respectively. First, recall that the value of total cost is the sum of the fixed cost (cell B3) and the total variable cost. The total variable cost—the product of the variable cost per unit (cell B5) and the production volume (cell B12)—is given by B5*B12. Thus, to compute the value of total cost we entered the formula =B3+B5*B12 in cell B14. Next, total revenue is the product of the selling price per unit (cell B7) and the number of units produced (cell B12), which is entered in cell B16 as the formula =B7*B12. Finally, the total profit (or loss) is the difference between the total revenue (cell B16) and the total cost (cell B14). Thus, in cell B18 we have entered the formula =B16-B14. The worksheet shown in Figure 1.8 shows the formulas used to make these computations; we refer to it as a formula worksheet. To examine the effect of selecting a particular value for the production volume, we entered a value of 800 in cell B12. The worksheet shown in Figure 1.8 shows the values obtained by the formulas; a production volume of 800 units results in a total cost of $4600, a total revenue of $4000, and a loss of $600. To examine the effect of other production volumes, we only need to enter a different value into cell B12. To examine the

FIGURE 1.8 SOLUTION USING A PRODUCTION VOLUME OF 800 UNITS FOR THE NOWLIN PLASTICS PRODUCTION EXAMPLE
A 1 Nowlin Plastics 2 3 Fixed Cost 4 5 Variable Cost Per Unit 6 7 Selling Price Per Unit 8 9 10 Models 11 12 Production Volume 13 14 Total Cost 15 16 Total Revenue 17 18 Total Profit (Loss) B $3,000 $2 $5

WEB

file

Nowlin

800 $4,600 $4,000 $600

26

Chapter 1

Introduction

effect of different costs and selling prices, we simply enter the appropriate values in the data portion of the worksheet; the results will be displayed in the model section of the worksheet. In Section 1.4 we illustrated breakeven analysis. Let us now see how Excel’s Goal Seek tool can be used to compute the breakeven point for the Nowlin Plastics production example.

Determining the Breakeven Point Using Excel’s Goal Seek Tool
The breakeven point is the production volume that results in total revenue equal to total cost and hence a profit of $0. One way to determine the breakeven point is to use a trial-anderror approach. For example, in Figure 1.8 we saw that a trial production volume of 800 units resulted in a loss of $600. Because this trial solution resulted in a loss, a production volume of 800 units cannot be the breakeven point. We could continue to experiment with other production volumes by simply entering different values into cell B12 and observing the resulting profit or loss in cell B18. A better approach is to use Excel’s Goal Seek tool to determine the breakeven point. Excel’s Goal Seek tool allows the user to determine the value for an input cell that will cause the value of a related output cell to equal some specified value (called the goal). In the case of breakeven analysis, the “goal” is to set Total Profit to zero by “seeking” an appropriate value for Production Volume. Goal Seek will allow us to find the value of production volume that will set Nowlin Plastics’ total profit to zero. The following steps describe how to use Goal Seek to find the breakeven point for Nowlin Plastics: Step 1. Select the Data tab at the top of the Ribbon Step 2. Select What-If Analysis in the Data Tools group Step 3. Select Goal Seek in What-if Analysis Step 4. When the Goal Seek dialog box appears: Enter B18 in the Set cell box Enter 0 in the To value box Enter B12 in the By changing cell box Click OK

FIGURE 1.9 GOAL SEEK DIALOG BOX FOR THE NOWLIN PLASTICS PRODUCTION EXAMPLE

Appendix 1.1

Using Excel for Breakeven Analysis

27

FIGURE 1.10 BREAKEVEN POINT FOUND USING EXCEL’S GOAL SEEK TOOL FOR THE NOWLIN PLASTICS PRODUCTION EXAMPLE
A 1 Nowlin Plastics 2 3 Fixed Cost 4 5 Variable Cost Per Unit 6 7 Selling Price Per Unit 8 9 10 Models 11 12 Production Volume 13 14 Total Cost 15 16 Total Revenue 17 18 Total Profit (Loss) B $3,000 $2 $5

1000 $5,000 $5,000 $0

The completed Goal Seek dialog box is shown in Figure 1.9, and the worksheet obtained after selecting OK is shown in Figure 1.10. The Total Profit in cell B18 is zero, and the Production Volume in cell B12 has been set to the breakeven point of 1000.

CHAPTER

2
2.5 A SIMPLE MINIMIZATION PROBLEM Summary of the Graphical Solution Procedure for Minimization Problems Surplus Variables Computer Solution of the M&D Chemicals Problem SPECIAL CASES Alternative Optimal Solutions Infeasibility Unbounded

An Introduction to Linear Programming
CONTENTS 2.1 A SIMPLE MAXIMIZATION PROBLEM Problem Formulation Mathematical Statement of the Par, Inc., Problem

2.2

GRAPHICAL SOLUTION PROCEDURE A Note on Graphing Lines Summary of the Graphical Solution Procedure for Maximization Problems Slack Variables EXTREME POINTS AND THE OPTIMAL SOLUTION COMPUTER SOLUTION OF THE PAR, INC., PROBLEM Interpretation of Computer Output

2.6

2.3 2.4

2.7

GENERAL LINEAR PROGRAMMING NOTATION

Chapter 2

An Introduction to Linear Programming

29

Linear programming is a problem-solving approach developed to help managers make decisions. Numerous applications of linear programming can be found in today’s competitive business environment. For instance, Eastman Kodak uses linear programming to determine where to manufacture products throughout their worldwide facilities, and GE Capital uses linear programming to help determine optimal lease structuring. Marathon Oil Company uses linear programming for gasoline blending and to evaluate the economics of a new terminal or pipeline. The Management Science in Action, Timber Harvesting Model at MeadWestvaco Corporation, provides another example of the use of linear programming. Later in the chapter another Management Science in Action illustrates how the Hanshin Expressway Public Corporation uses linear programming for traffic control on an urban toll expressway in Osaka, Japan. To illustrate some of the properties that all linear programming problems have in common, consider the following typical applications:

1. A manufacturer wants to develop a production schedule and an inventory policy that will satisfy sales demand in future periods. Ideally, the schedule and policy will enable the company to satisfy demand and at the same time minimize the total production and inventory costs. 2. A financial analyst must select an investment portfolio from a variety of stock and bond investment alternatives. The analyst would like to establish the portfolio that maximizes the return on investment. 3. A marketing manager wants to determine how best to allocate a fixed advertising budget among alternative advertising media such as radio, television, newspaper, and magazine. The manager would like to determine the media mix that maximizes advertising effectiveness. 4. A company has warehouses in a number of locations throughout the United States. For a set of customer demands, the company would like to determine how much each warehouse should ship to each customer so that total transportation costs are minimized.

MANAGEMENT SCIENCE IN ACTION
TIMBER HARVESTING MODEL AT MEADWESTVACO CORPORATION*
MeadWestvaco Corporation is a major producer of premium papers for periodicals, books, commercial printing, and business forms. The company also produces pulp and lumber, designs and manufactures packaging systems for beverage and other consumables markets, and is a world leader in the production of coated board and shipping containers. Quantitative analyses at MeadWestvaco are developed and implemented by the company’s Decision Analysis Department. The department assists decision makers by providing them with analytical tools of quantitative methods as well as personal analysis and recommendations. MeadWestvaco uses quantitative models to assist with the long-range management of the company’s timberland. Through the use of large-scale linear programs, timber harvesting plans are developed to cover a substantial time horizon. These models consider wood market conditions,

mill pulpwood requirements, harvesting capacities, and general forest management principles. Within these constraints, the model arrives at an optimal harvesting and purchasing schedule based on discounted cash flow. Alternative schedules reflect changes in the various assumptions concerning forest growth, wood availability, and general economic conditions. Quantitative methods are also used in the development of the inputs for the linear programming models. Timber prices and supplies as well as mill requirements must be forecast over the time horizon, and advanced sampling techniques are used to evaluate land holdings and to project forest growth. The harvest schedule is then developed using quantitative methods.

*Based on information provided by Dr. Edward P. Winkofsky.

30
Linear programming was initially referred to as “programming in a linear structure.” In 1948 Tjalling Koopmans suggested to George Dantzig that the name was much too long; Koopmans suggested shortening it to linear programming. George Dantzig agreed and the field we now know as linear programming was named.

Chapter 2

An Introduction to Linear Programming

These examples are only a few of the situations in which linear programming has been used successfully, but they illustrate the diversity of linear programming applications. A close scrutiny reveals one basic property they all have in common. In each example, we were concerned with maximizing or minimizing some quantity. In example 1, the manufacturer wanted to minimize costs; in example 2, the financial analyst wanted to maximize return on investment; in example 3, the marketing manager wanted to maximize advertising effectiveness; and in example 4, the company wanted to minimize total transportation costs. In all linear programming problems, the maximization or minimization of some quantity is the objective. All linear programming problems also have a second property: restrictions, or constraints, that limit the degree to which the objective can be pursued. In example 1, the manufacturer is restricted by constraints requiring product demand to be satisfied and by the constraints limiting production capacity. The financial analyst’s portfolio problem is constrained by the total amount of investment funds available and the maximum amounts that can be invested in each stock or bond. The marketing manager’s media selection decision is constrained by a fixed advertising budget and the availability of the various media. In the transportation problem, the minimum-cost shipping schedule is constrained by the supply of product available at each warehouse. Thus, constraints are another general feature of every linear programming problem.

2.1

A SIMPLE MAXIMIZATION PROBLEM
Par, Inc., is a small manufacturer of golf equipment and supplies whose management has decided to move into the market for medium- and high-priced golf bags. Par’s distributor is enthusiastic about the new product line and has agreed to buy all the golf bags Par produces over the next three months. After a thorough investigation of the steps involved in manufacturing a golf bag, management determined that each golf bag produced will require the following operations: 1. 2. 3. 4. Cutting and dyeing the material Sewing Finishing (inserting umbrella holder, club separators, etc.) Inspection and packaging

The director of manufacturing analyzed each of the operations and concluded that if the company produces a medium-priced standard model, each bag will require ⁷⁄₁₀ hour in the cutting and dyeing department, ¹⁄₂ hour in the sewing department, 1 hour in the finishing department, and ¹⁄₁₀ hour in the inspection and packaging department. The more expensive deluxe model will require 1 hour for cutting and dyeing, ⁵⁄₆ hour for sewing, ²⁄₃ hour for finishing, and ¹⁄₄ hour for inspection and packaging. This production information is summarized in Table 2.1. Par’s production is constrained by a limited number of hours available in each department. After studying departmental workload projections, the director of manufacturing estimates that 630 hours for cutting and dyeing, 600 hours for sewing, 708 hours for finishing, and 135 hours for inspection and packaging will be available for the production of golf bags during the next three months. The accounting department analyzed the production data, assigned all relevant variable costs, and arrived at prices for both bags that will result in a profit contribution1 of $10 for
1 From an accounting perspective, profit contribution is more correctly described as the contribution margin per bag; for example, overhead and other shared costs have not been allocated.

2.1

A Simple Maximization Problem

31

TABLE 2.1 PRODUCTION REQUIREMENTS PER GOLF BAG Production Time (hours) Department Cutting and Dyeing Sewing Finishing Inspection and Packaging Standard Bag ⁷⁄₁₀ ¹⁄₂ 1 ¹⁄₁₀ Deluxe Bag 1 ⁵ ⁄₆ ²⁄₃ ¹⁄₄

It is important to understand that we are maximizing profit contribution, not profit. Overhead and other shared costs must be deducted before arriving at a profit figure.

every standard bag and $9 for every deluxe bag produced. Let us now develop a mathematical model of the Par, Inc., problem that can be used to determine the number of standard bags and the number of deluxe bags to produce in order to maximize total profit contribution.

Problem Formulation
Problem formulation, or modeling, is the process of translating the verbal statement of a problem into a mathematical statement. Formulating models is an art that can only be mastered with practice and experience. Even though every problem has some unique features, most problems also have common features. As a result, some general guidelines for model formulation can be helpful, especially for beginners. We will illustrate these general guidelines by developing a mathematical model for the Par, Inc., problem. Understand the Problem Thoroughly We selected the Par, Inc., problem to introduce linear programming because it is easy to understand. However, more complex problems will require much more thinking in order to identify the items that need to be included in the model. In such cases, read the problem description quickly to get a feel for what is involved. Taking notes will help you focus on the key issues and facts. Describe the Objective
The objective is to maximize the total contribution to profit.

Describe Each Constraint Four constraints relate to the number of hours of manufacturing time available; they restrict the number of standard bags and the number of deluxe bags that can be produced. Constraint 2: Number of hours of sewing time used must be less than or equal to the number of hours of sewing time available. Constraint 4: Number of hours of inspection and packaging time used must be less than or equal to the number of hours of inspection and packaging time available. Constraint 1: Number of hours of cutting and dyeing time used must be less than or equal to the number of hours of cutting and dyeing time available.

Constraint 3: Number of hours of finishing time used must be less than or equal to the number of hours of finishing time available.

Define the Decision Variables The controllable inputs for Par, Inc., are (1) the number of standard bags produced, and (2) the number of deluxe bags produced. Let S = number of standard bags D = number of deluxe bags
In linear programming terminology, S and D are referred to as the decision variables.

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Chapter 2

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Write the Objective in Terms of the Decision Variables Par’s profit contribution comes from two sources: (1) the profit contribution made by producing S standard bags, and (2) the profit contribution made by producing D deluxe bags. If Par makes $10 for every standard bag, the company will make $10S if S standard bags are produced. Also, if Par makes $9 for every deluxe bag, the company will make $9D if D deluxe bags are produced. Thus, we have
Total Profit Contribution = 10 S + 9D Because the objective—maximize total profit contribution—is a function of the decision variables S and D, we refer to 10S + 9D as the objective function. Using “Max” as an abbreviation for maximize, we write Par’s objective as follows: Max 10S + 9D

Write the Constraints in Terms of the Decision Variables Constraint 1: a
Hours of cutting and Hours of cutting and b … a b dyeing time used dyeing time available

Every standard bag Par produces will use ⁷⁄₁₀ hour cutting and dyeing time; therefore, the total number of hours of cutting and dyeing time used in the manufacture of S standard bags is ⁷⁄₁₀ S. In addition, because every deluxe bag produced uses 1 hour of cutting and dyeing time, the production of D deluxe bags will use 1D hours of cutting and dyeing time. Thus, the total cutting and dyeing time required for the production of S standard bags and D deluxe bags is given by Total hours of cutting and dyeing time used
The units of measurement on the left-hand side of the constraint must match the units of measurement on the right-hand side.

⁷⁄₁₀ S

1D

The director of manufacturing stated that Par has at most 630 hours of cutting and dyeing time available. Therefore, the production combination we select must satisfy the requirement 1D 630 (2.1)

⁷⁄₁₀ S

Constraint 2: a
Hours of sewing Hours of sewing b … a b time used time available

From Table 2.1, we see that every standard bag manufactured will require ¹⁄₂ hour for sewing, and every deluxe bag will require ⁵⁄₆ hour for sewing. Because 600 hours of sewing time are available, it follows that
¹⁄₂ S ⁵⁄₆ D

600

(2.2)

2.1

A Simple Maximization Problem

33

Constraint 3: a
Hours of finishing Hours of finishing b … a b time used time available

Every standard bag manufactured will require 1 hour for finishing, and every deluxe bag will require ²⁄₃ hour for finishing. With 708 hours of finishing time available, it follows that

1S

²⁄₃ D

708

(2.3)

Constraint 4: a
Hours of inspection and Hours of inspection and b … a b packaging time used packaging time available

Every standard bag manufactured will require ¹⁄₁₀ hour for inspection and packaging, and every deluxe bag will require ¹⁄₄ hour for inspection and packaging. Because 135 hours of inspection and packaging time are available, it follows that 135 (2.4)

¹⁄₁₀ S

¹⁄₄ D

We have now specified the mathematical relationships for the constraints associated with the four departments. Have we forgotten any other constraints? Can Par produce a negative number of standard or deluxe bags? Clearly, the answer is no. Thus, to prevent the decision variables S and D from having negative values, two constraints,

S Ú 0

and

D Ú 0

(2.5)

must be added. These constraints ensure that the solution to the problem will contain nonnegative values for the decision variables and are thus referred to as the nonnegativity constraints. Nonnegativity constraints are a general feature of all linear programming problems and may be written in the abbreviated form:
Try Problem 24(a) to test your ability to formulate a mathematical model for a maximization linear programming problem with less-than-or-equal-to constraints.

S, D Ú 0

Mathematical Statement of the Par, Inc., Problem
The mathematical statement or mathematical formulation of the Par, Inc., problem is now complete. We succeeded in translating the objective and constraints of the problem into a

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Chapter 2

An Introduction to Linear Programming

set of mathematical relationships referred to as a mathematical model. The complete mathematical model for the Par problem is as follows: Max 10S
⁷⁄₁₀ S ¹⁄₂ S ¹⁄₁₀ S

subject to (s.t.)

9D 630 Cutting and dyeing

1D

1S

⁵⁄₆ D ²⁄₃ D ¹⁄₄ D

600 Sewing

708 Finishing

S, D

0

135 Inspection and packaging

(2.6)

Try Problem 1 to test your ability to recognize the types of mathematical relationships that can be found in a linear program.

Our job now is to find the product mix (i.e., the combination of S and D) that satisfies all the constraints and, at the same time, yields a value for the objective function that is greater than or equal to the value given by any other feasible solution. Once these values are calculated, we will have found the optimal solution to the problem. This mathematical model of the Par problem is a linear programming model, or linear program. The problem has the objective and constraints that, as we said earlier, are common properties of all linear programs. But what is the special feature of this mathematical model that makes it a linear program? The special feature that makes it a linear program is that the objective function and all constraint functions are linear functions of the decision variables. Mathematical functions in which each variable appears in a separate term and is raised to the first power are called linear functions. The objective function (10S 9D) is linear because each decision variable appears in a separate term and has an exponent of 1. The amount of production time required in the cutting and dyeing department (⁷⁄₁₀ S 1D) is also a linear function of the decision variables for the same reason. Similarly, the functions on the left-hand side of all the constraint inequalities (the constraint functions) are linear functions. Thus, the mathematical formulation of this problem is referred to as a linear program. Linear programming has nothing to do with computer programming. The use of the word programming here means “choosing a course of action.” Linear programming involves choosing a course of action when the mathematical model of the problem contains only linear functions.

NOTES AND COMMENTS
1. The three assumptions necessary for a linear programming model to be appropriate are proportionality, additivity, and divisibility. Proportionality means that the contribution to the objective function and the amount of resources used in each constraint are proportional to the value of each decision variable. Additivity means that the value of the objective function and the total resources used can be found by summing the objective function contribution and the resources used for all decision variables. Divisibility means that the decision variables are continuous. The divisibility assumption plus the nonnegativity constraints mean that decision variables can take on any value greater than or equal to zero. 2. Management scientists formulate and solve a variety of mathematical models that contain an objective function and a set of constraints. Models of this type are referred to as mathematical programming models. Linear programming models are a special type of mathematical programming model in that the objective function and all constraint functions are linear.

2.2

Graphical Solution Procedure

35

2.2

GRAPHICAL SOLUTION PROCEDURE
A linear programming problem involving only two decision variables can be solved using a graphical solution procedure. Let us begin the graphical solution procedure by developing a graph that displays the possible solutions (S and D values) for the Par problem. The graph (Figure 2.1) will have values of S on the horizontal axis and values of D on the vertical axis. Any point on the graph can be identified by the S and D values, which indicate the position of the point along the horizontal and vertical axes, respectively. Because every point (S, D) corresponds to a possible solution, every point on the graph is called a solution point. The solution point where S 0 and D 0 is referred to as the origin. Because S and D must be nonnegative, the graph in Figure 2.1 only displays solutions where S 0 and D 0. Earlier, we saw that the inequality representing the cutting and dyeing constraint is
⁷⁄₁₀ S

1D

630

To show all solution points that satisfy this relationship, we start by graphing the solution points satisfying the constraint as an equality. That is, the points where ⁷⁄₁₀ S 1D 630. Because the graph of this equation is a line, it can be obtained by identifying two points that satisfy the equation and then drawing a line through the points. Setting S 0 and solving for D, we see that the point (S 0, D 630) satisfies the equation. To find a second point satisfying this equation, we set D 0 and solve for S. By doing so, we obtain

FIGURE 2.1 SOLUTION POINTS FOR THE TWO-VARIABLE PAR, INC., PROBLEM
D 1200 1000 Number of Deluxe Bags 800 600 400 200 A solution point with S = 400 and D = 300 (400, 300) A solution point with S = 200 and D = 800 (200, 800)

0

200

400

600

800

1000

1200

S

Number of Standard Bags

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Chapter 2

An Introduction to Linear Programming

⁷⁄₁₀ S

D

1(0) 630, or S 900. Thus, a second point satisfying the equation is (S 900, 0). Given these two points, we can now graph the line corresponding to the equation
⁷⁄₁₀ S

1D

630

This line, which will be called the cutting and dyeing constraint line, is shown in Figure 2.2. We label this line “C & D” to indicate that it represents the cutting and dyeing constraint line. Recall that the inequality representing the cutting and dyeing constraint is
⁷⁄₁₀ S

1D

630

Can you identify all of the solution points that satisfy this constraint? Because all points on the line satisfy ⁷⁄₁₀ S 1D 630, we know any point on this line must satisfy the constraint. But where are the solution points satisfying ⁷⁄₁₀ S 1D 630? Consider two solution points: (S 200, D 200) and (S 600, D 500). You can see from Figure 2.2 that the first solution point is below the constraint line and the second is above the constraint line. Which of these solutions will satisfy the cutting and dyeing constraint? For the point (S 200, D 200), we see that
⁷⁄₁₀ S

1D

⁷⁄₁₀ (200)

1(200)

340

FIGURE 2.2 THE CUTTING AND DYEING CONSTRAINT LINE
D 1200 1000 800 (0, 630) 600 400 200

Number of Deluxe Bags

7/ 10

S+

1D

(600, 500)
= 63 0

(200, 200)

C

&

D

(900, 0) 1000 1200 1400 S

0

200

400

600

800

Number of Standard Bags

2.2

Graphical Solution Procedure

37

Because the 340 hours is less than the 630 hours available, the (S 200, D 200) production combination, or solution point, satisfies the constraint. For the point (S 600, D 500), we have
⁷⁄₁₀ S

1D

⁷⁄₁₀ (600)

1(500)

920

Can you graph a constraint line and find the solution points that are feasible? Try Problem 2.

The 920 hours is greater than the 630 hours available, so the (S 600, D 500) solution point does not satisfy the constraint and is thus not feasible. If a solution point is not feasible for a particular constraint, then all other solution points on the same side of that constraint line are not feasible. If a solution point is feasible for a particular constraint, then all other solution points on the same side of the constraint line are feasible for that constraint. Thus, one has to evaluate the constraint function for only one solution point to determine which side of a constraint line is feasible. In Figure 2.3 we indicate all points satisfying the cutting and dyeing constraint by the shaded region. We continue by identifying the solution points satisfying each of the other three constraints. The solutions that are feasible for each of these constraints are shown in Figure 2.4. Four separate graphs now show the feasible solution points for each of the four constraints. In a linear programming problem, we need to identify the solution points that satisfy all the constraints simultaneously. To find these solution points, we can draw all four constraints on one graph and observe the region containing the points that do in fact satisfy all the constraints simultaneously.

FIGURE 2.3 FEASIBLE SOLUTIONS FOR THE CUTTING AND DYEING CONSTRAINT, REPRESENTED BY THE SHADED REGION
D 1200 1000 800 600 400 200

Number of Deluxe Bags

Cutting and Dyeing Constraint (C & D)
7/ 10

S+

1D

=6

30

C

&

D

0

200

400

600

800

1000

1200

1400

S

Number of Standard Bags

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Chapter 2

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FIGURE 2.4 FEASIBLE SOLUTIONS FOR THE SEWING, FINISHING, AND INSPECTION AND PACKAGING CONSTRAINTS, REPRESENTED BY THE SHADED REGIONS
D 1200 1000 Number of Deluxe Bags 800 600 400 200 0 200 400 600 800 Number of Deluxe Bags (0, 720) Sewing Constraint
1/ 2S

D 1200 1000 800 600 400 200 0 200 400
1S

(0, 1062)

Finishing Constraint

+
2 /3

D

+

=7

5/ 6D

08

=6 00

i Fin

Se wi ng
1000 1200

ng shi

(1200, 0) 1400 S

(708, 0) 800 1000 1200 1400 S

600

Number of Standard Bags D 1200 1000 Number of Deluxe Bags 800 600 400 200 0 200 400 600 (0, 540)
1/ 10 S

Number of Standard Bags

+ 1/

Inspection and Packaging Constraint (I & P)
=1 35

4D

I&

P

(1350, 0) 1200 1400 S

800

1000

Number of Standard Bags

Try Problem 7 to test your ability to find the feasible region given several constraints.

The graphs in Figures 2.3 and 2.4 can be superimposed to obtain one graph with all four constraints. This combined-constraint graph is shown in Figure 2.5. The shaded region in this figure includes every solution point that satisfies all the constraints simultaneously. Solutions that satisfy all the constraints are termed feasible solutions, and the shaded region is called the feasible solution region, or simply the feasible region. Any solution point on the boundary of the feasible region or within the feasible region is a feasible solution point. Now that we have identified the feasible region, we are ready to proceed with the graphical solution procedure and find the optimal solution to the Par, Inc., problem. Recall that the optimal solution for a linear programming problem is the feasible solution that provides

2.2

Graphical Solution Procedure

39

FIGURE 2.5 COMBINED-CONSTRAINT GRAPH SHOWING THE FEASIBLE REGION FOR THE PAR, INC., PROBLEM
D 1200 1000 i Fin

Number of Deluxe Bags

800 600 400 200

Se wi ng

Feasible Region

ng shi

C

&

D

I&

P

0

200

400

600

800

1000

1200

1400

S

Number of Standard Bags

the best possible value of the objective function. Let us start the optimizing step of the graphical solution procedure by redrawing the feasible region on a separate graph. The graph is shown in Figure 2.6. One approach to finding the optimal solution would be to evaluate the objective function for each feasible solution; the optimal solution would then be the one yielding the largest value. The difficulty with this approach is the infinite number of feasible solutions; thus, because one cannot possibly evaluate an infinite number of feasible solutions, this trial-and-error procedure cannot be used to identify the optimal solution. Rather than trying to compute the profit contribution for each feasible solution, we select an arbitrary value for profit contribution and identify all the feasible solutions (S, D) that yield the selected value. For example, which feasible solutions provide a profit contribution of $1800? These solutions are given by the values of S and D in the feasible region that will make the objective function

10S + 9D = 1800
This expression is simply the equation of a line. Thus, all feasible solution points (S, D) yielding a profit contribution of $1800 must be on the line. We learned earlier in this section how to graph a constraint line. The procedure for graphing the profit or objective function line is the same. Letting S 0, we see that D must be 200; thus, the solution

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Chapter 2

An Introduction to Linear Programming

FIGURE 2.6 FEASIBLE REGION FOR THE PAR, INC., PROBLEM
D 600

Number of Deluxe Bags

400

200

Feasible Region

0

200

400 Number of Standard Bags

600

800

S

point (S 0, D 200) is on the line. Similarly, by letting D 0, we see that the solution point (S 180, D 0) is also on the line. Drawing the line through these two points identifies all the solutions that have a profit contribution of $1800. A graph of this profit line is presented in Figure 2.7. Because the objective is to find the feasible solution yielding the largest profit contribution, let us proceed by selecting higher profit contributions and finding the solutions yielding the selected values. For instance, let us find all solutions yielding profit contributions of $3600 and $5400. To do so, we must find the S and D values that are on the following lines:

10S + 9D = 3600 and 10S + 9D = 5400
Using the previous procedure for graphing profit and constraint lines, we draw the $3600 and $5400 profit lines as shown on the graph in Figure 2.8. Although not all solution points on the $5400 profit line are in the feasible region, at least some points on the line are, and it is therefore possible to obtain a feasible solution that provides a $5400 profit contribution. Can we find a feasible solution yielding an even higher profit contribution? Look at Figure 2.8, and see what general observations you can make about the profit lines already drawn. Note the following: (1) the profit lines are parallel to each other, and (2) higher

2.2

Graphical Solution Procedure

41

FIGURE 2.7 $1800 PROFIT LINE FOR THE PAR, INC., PROBLEM
D 600

Number of Deluxe Bags

400

(0, 200) 200

Profit Line 10S + 9D = 1800

(180, 0) 0 200 400 Number of Standard Bags 600 800 S

FIGURE 2.8 SELECTED PROFIT LINES FOR THE PAR, INC., PROBLEM
D 600

Number of Deluxe Bags

400
10 S + 9D 10 S + = 0 54

200

9D

0

10 S + 9D = 18

= 0 36 0 00

0

200

400 Number of Standard Bags

600

800

S

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Chapter 2

An Introduction to Linear Programming

profit lines are obtained as we move farther from the origin. These observations can also be expressed algebraically. Let P represent total profit contribution. The objective function is

P

10S

9D

Solving for D in terms of S and P, we obtain 9D D 10S ¹⁰⁄₉ S

P ¹⁄₉ P

(2.7)

Equation (2.7) is the slope-intercept form of the linear equation relating S and D. The coefficient of S, ¹⁰⁄₉ , is the slope of the line, and the term ¹⁄₉ P is the D intercept (i.e., the value of D where the graph of equation (2.7) crosses the D axis). Substituting the profit contributions of P 1800, P 3600, and P 5400 into equation (2.7) yields the following slope-intercept equations for the profit lines shown in Figure 2.8: For P 1800,

D
For P 3600,

¹⁰⁄₉ S

200

D
For P 5400,

¹⁰⁄₉ S

400

D
Can you graph the profit line for a linear program? Try Problem 6.

¹⁰⁄₉ S

600

The slope ( ¹⁰⁄₉ ) is the same for each profit line because the profit lines are parallel. Further, we see that the D intercept increases with larger profit contributions. Thus, higher profit lines are farther from the origin. Because the profit lines are parallel and higher profit lines are farther from the origin, we can obtain solutions that yield increasingly larger values for the objective function by continuing to move the profit line farther from the origin in such a fashion that it remains parallel to the other profit lines. However, at some point we will find that any further outward movement will place the profit line completely outside the feasible region. Because solutions outside the feasible region are unacceptable, the point in the feasible region that lies on the highest profit line is the optimal solution to the linear program. You should now be able to identify the optimal solution point for this problem. Use a ruler or the edge of a piece of paper, and move the profit line as far from the origin as you can. What is the last point in the feasible region that you reach? This point, which is the optimal solution, is shown graphically in Figure 2.9. The optimal values of the decision variables are the S and D values at the optimal solution. Depending on the accuracy of the graph, you may or may not be able to determine the exact S and D values. Based on the graph in Figure 2.9, the best we can do is conclude that the optimal production combination consists of approximately 550 standard bags (S) and approximately 250 deluxe bags (D).

2.2

Graphical Solution Procedure

43

FIGURE 2.9 OPTIMAL SOLUTION FOR THE PAR, INC., PROBLEM
D 600

ne Li it of 8 Pr 766 um = im D ax 9 M S+ 10

Number of Deluxe Bags

400

Optimal Solution

200

0

200

400 Number of Standard Bags

600

800

S

A closer inspection of Figures 2.5 and 2.9 shows that the optimal solution point is at the intersection of the cutting and dyeing and the finishing constraint lines. That is, the optimal solution point is on both the cutting and dyeing constraint line
⁷⁄₁₀ S

1D

630

(2.8)

and the finishing constraint line 1S
²⁄₃ D

708

(2.9)

Thus, the optimal values of the decision variables S and D must satisfy both equations (2.8) and (2.9) simultaneously. Using equation (2.8) and solving for S gives
⁷⁄₁₀ S

630

1D

or

S

900

¹⁰⁄₇ D

(2.10)

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Chapter 2

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Substituting this expression for S into equation (2.9) and solving for D provides the following: 1(900 ¹⁰⁄₇ D) ²⁄₃ D 900 ¹⁰⁄₇ D ²⁄₃ D 900 ³⁰⁄₂₁D ¹⁴⁄₂₁ D ¹⁶⁄₂₁ D 708 708 708 192 192 252 ¹⁶⁄₂₁

D
Using D

252 in equation (2.10) and solving for S, we obtain

S

900

900

¹⁰⁄ ₇ (252)

360

540

Although the optimal solution to the Par, Inc., problem consists of integer values for the decision variables, this result will not always be the case.

The exact location of the optimal solution point is S 540 and D 252. Hence, the optimal production quantities for Par, Inc., are 540 standard bags and 252 deluxe bags, with a resulting profit contribution of 10(540) 9(252) $7668. For a linear programming problem with two decision variables, the exact values of the decision variables can be determined by first using the graphical solution procedure to identify the optimal solution point and then solving the two simultaneous constraint equations associated with it.

A Note on Graphing Lines
Try Problem 10 to test your ability to use the graphical solution procedure to identify the optimal solution and find the exact values of the decision variables at the optimal solution.

An important aspect of the graphical method is the ability to graph lines showing the constraints and the objective function of the linear program. The procedure we used for graphing the equation of a line is to find any two points satisfying the equation and then draw the line through the two points. For the Par, Inc., constraints, the two points were easily found by first setting S 0 and solving the constraint equation for D. Then we set D 0 and solved for S. For the cutting and dyeing constraint line
⁷⁄₁₀ S

1D

630

this procedure identified the two points (S 0, D 630) and (S 900, D 0). The cutting and dyeing constraint line was then graphed by drawing a line through these two points. All constraints and objective function lines in two-variable linear programs can be graphed if two points on the line can be identified. However, finding the two points on the line is not always as easy as shown in the Par, Inc., problem. For example, suppose a company manufactures two models of a small handheld computer: the Assistant (A) and the Professional (P). Management needs 50 units of the Professional model for its own salesforce, and expects sales of the Professional to be at most one-half of the sales of the Assistant. A constraint enforcing this requirement is

P or 50

¹⁄₂ A

2P - 100 … A

2.2

Graphical Solution Procedure

45

or

2P - A … 100
Using the equality form and setting P 0, we find the point (P 0, A 100) is on the constraint line. Setting A 0, we find a second point (P 50, A 0) on the constraint line. If we have drawn only the nonnegative (P 0, A 0) portion of the graph, the first point (P 0, A 100) cannot be plotted because A 100 is not on the graph. Whenever we have two points on the line but one or both of the points cannot be plotted in the nonnegative portion of the graph, the simplest approach is to enlarge the graph. In this example, the point (P 0, A 100) can be plotted by extending the graph to include the negative A axis. Once both points satisfying the constraint equation have been located, the line can be drawn. The constraint line and the feasible solutions for the constraint 2P A 100 are shown in Figure 2.10. As another example, consider a problem involving two decision variables, R and T. Suppose that the number of units of R produced had to be at least equal to the number of units of T produced. A constraint enforcing this requirement is

R Ú T or R - T Ú 0 FIGURE 2.10 FEASIBLE SOLUTIONS FOR THE CONSTRAINT 2P
A 300

A

100

200

0

2P

100

–A

(50, 0) 100 200 300 P

(0, –100) –100

=1

00

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Chapter 2

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Can you graph a constraint line when the origin is on the constraint line? Try Problem 5.

To find all solutions satisfying the constraint as an equality, we first set R 0 and solve for T. This result shows that the origin (T 0, R 0) is on the constraint line. Setting T 0 and solving for R provides the same point. However, we can obtain a second point on the line by setting T equal to any value other than zero and then solving for R. For instance, setting T 100 and solving for R, we find that the point (T 100, R 100) is on the line. With the two points (R 0, T 0) and (R 100, T 100), the constraint line R T 0 and the feasible solutions for R T 0 can be plotted as shown in Figure 2.11.

Summary of the Graphical Solution Procedure for Maximization Problems
For additional practice in using the graphical solution procedure, try Problem 24(b), 24(c), and 24(d).

As we have seen, the graphical solution procedure is a method for solving two-variable linear programming problems such as the Par, Inc., problem. The steps of the graphical solution procedure for a maximization problem are summarized here:

1. Prepare a graph of the feasible solutions for each of the constraints. 2. Determine the feasible region by identifying the solutions that satisfy all the constraints simultaneously. 3. Draw an objective function line showing the values of the decision variables that yield a specified value of the objective function. 4. Move parallel objective function lines toward larger objective function values until further movement would take the line completely outside the feasible region. 5. Any feasible solution on the objective function line with the largest value is an optimal solution. 0

FIGURE 2.11 FEASIBLE SOLUTIONS FOR THE CONSTRAINT R
T 300

T

200
T = 0

(100, 100) 100 (0, 0) 0

R

–

100

200

300

R

2.2

Graphical Solution Procedure

47

Slack Variables
In addition to the optimal solution and its associated profit contribution, Par’s management will probably want information about the production time requirements for each production operation. We can determine this information by substituting the optimal solution values (S 540, D 252) into the constraints of the linear program.

Constraint Cutting and dyeing Sewing Finishing Inspection and packaging

Hours Required for S 540 and D 252 ⁷⁄₁₀(540) 1(252) 630 ¹⁄₂(540) ⁵⁄₆(252) 480 1(540) ²⁄₃(252) 708 ¹⁄₁₀(540) ¹⁄₄(252) 117

Hours Available 630 600 708 135

Unused Hours 0 120 0 18

Can you identify the slack associated with a constraint? Try Problem 24(e).

Thus, the complete solution tells management that the production of 540 standard bags and 252 deluxe bags will require all available cutting and dyeing time (630 hours) and all available finishing time (708 hours), while 600 480 120 hours of sewing time and 135 117 18 hours of inspection and packaging time will remain unused. The 120 hours of unused sewing time and 18 hours of unused inspection and packaging time are referred to as slack for the two departments. In linear programming terminology, any unused capacity for a constraint is referred to as the slack associated with the constraint. Often variables, called slack variables, are added to the formulation of a linear programming problem to represent the slack, or idle capacity. Unused capacity makes no contribution to profit; thus, slack variables have coefficients of zero in the objective function. After the addition of four slack variables, denoted S1, S2, S3, and S4, the mathematical model of the Par, Inc., problem becomes Max s.t. 10S
⁷⁄₁₀ S ¹⁄₂ S

9D 1D
⁵⁄₆ D

0S1 1S1

0S2 1S2 0

0S3

0S4 630 600 708 135

²⁄₃D 1S ¹⁄₁₀ S ¹⁄₄D S, D, S1, S2, S3, S4
Can you write a linear program in standard form? Try Problem 18.

1S3

1S4

Whenever a linear program is written in a form with all constraints expressed as equalities, it is said to be written in standard form. Referring to the standard form of the Par, Inc., problem, we see that at the optimal solution (S 540 and D 252), the values for the slack variables are

Constraint Cutting and dyeing Sewing Finishing Inspection and packaging

Value of Slack Variable S1 0 S2 120 S3 0 S4 18

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Could we have used the graphical solution to provide some of this information? The answer is yes. By finding the optimal solution point on Figure 2.5, we can see that the cutting and dyeing and the finishing constraints restrict, or bind, the feasible region at this point. Thus, this solution requires the use of all available time for these two operations. In other words, the graph shows us that the cutting and dyeing and the finishing departments will have zero slack. On the other hand, the sewing and the inspection and packaging constraints are not binding the feasible region at the optimal solution, which means we can expect some unused time or slack for these two operations. As a final comment on the graphical analysis of this problem, we call your attention to the sewing capacity constraint as shown in Figure 2.5. Note, in particular, that this constraint did not affect the feasible region. That is, the feasible region would be the same whether the sewing capacity constraint were included or not, which tells us that enough sewing time is available to accommodate any production level that can be achieved by the other three departments. The sewing constraint does not affect the feasible region and thus cannot affect the optimal solution; it is called a redundant constraint.

NOTES AND COMMENTS
1. In the standard-form representation of a linear programming model, the objective function coefficients for slack variables are zero. This zero coefficient implies that slack variables, which represent unused resources, do not affect the value of the objective function. However, in some applications, unused resources can be sold and contribute to profit. In such cases, the corresponding slack variables become decision variables representing the amount of unused resources to be sold. For each of these variables, a nonzero coefficient in the objective function would reflect the profit associated with selling a unit of the corresponding resource. 2. Redundant constraints do not affect the feasible region; as a result, they can be removed from a linear programming model without affecting the optimal solution. However, if the linear programming model is to be re-solved later, changes in some of the data might make a previously redundant constraint a binding constraint. Thus, we recommend keeping all constraints in the linear programming model even though at some point in time one or more of the constraints may be redundant.

2.3

EXTREME POINTS AND THE OPTIMAL SOLUTION
Suppose that the profit contribution for Par’s standard golf bag is reduced from $10 to $5 per bag, while the profit contribution for the deluxe golf bag and all the constraints remain unchanged. The complete linear programming model of this new problem is identical to the mathematical model in Section 2.1, except for the revised objective function: Max 5S + 9D How does this change in the objective function affect the optimal solution to the Par, Inc., problem? Figure 2.12 shows the graphical solution of this new problem with the revised objective function. Note that without any change in the constraints, the feasible region does not change. However, the profit lines have been altered to reflect the new objective function. By moving the profit line in a parallel manner toward higher profit values, we find the optimal solution as shown in Figure 2.12. The values of the decision variables at this point

2.3

Extreme Points and the Optimal Solution

49

FIGURE 2.12 OPTIMAL SOLUTION FOR THE PAR, INC., PROBLEM WITH AN OBJECTIVE FUNCTION OF 5S 9D
D 600

Optimal Solution (S = 300, D = 420)

Number of Deluxe Bags

400

(0, 300)

Ma xi

mu

mP rof it L

200

5S +

ine

: 5S

9D =

+9

D

270

=5

0

280

(540, 0)
0 200 400 600 800 S

Number of Standard Bags

are S 300 and D 420. The reduced profit contribution for the standard bag caused a change in the optimal solution. In fact, as you may have suspected, we are cutting back the production of the lower-profit standard bags and increasing the production of the higherprofit deluxe bags. What observations can you make about the location of the optimal solutions in the two linear programming problems solved thus far? Look closely at the graphical solutions in Figures 2.9 and 2.12. Notice that the optimal solutions occur at one of the vertices, or “corners,” of the feasible region. In linear programming terminology, these vertices are referred to as the extreme points of the feasible region. The Par, Inc., feasible region has five vertices, or five extreme points (see Figure 2.13). We can now formally state our observation about the location of optimal solutions as follows:
For additional practice in identifying the extreme points of the feasible region and determining the optimal solution by computing and comparing the objective function value at each extreme point, try Problem 13.

The optimal solution to a linear program can be found at an extreme point of the feasible region.2 This property means that if you are looking for the optimal solution to a linear program, you do not have to evaluate all feasible solution points. In fact, you have to consider

2 We will discuss in Section 2.6 the two special cases (infeasibility and unboundedness) in linear programming that have no optimal solution, and for which this statement does not apply.

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FIGURE 2.13 THE FIVE EXTREME POINTS OF THE FEASIBLE REGION FOR THE PAR, INC., PROBLEM
D 600

5 4

Number of Deluxe Bags

400

200

Feasible Region

3

0

1

200

400 Number of Standard Bags

600

2

800

S

only the feasible solutions that occur at the extreme points of the feasible region. Thus, for the Par, Inc., problem, instead of computing and comparing the profit contributions for all feasible solutions, we can find the optimal solution by evaluating the five extreme-point solutions and selecting the one that provides the largest profit contribution. Actually, the graphical solution procedure is nothing more than a convenient way of identifying an optimal extreme point for two-variable problems.

2.4
In January 1952 the first successful computer solution of a linear programming problem was performed on the SEAC (Standards Eastern Automatic Computer). The SEAC, the first digital computer built by the National Bureau of Standards under U.S. Air Force sponsorship, had a 512-word memory and magnetic tape for external storage.

COMPUTER SOLUTION OF THE PAR, INC., PROBLEM
Computer programs designed to solve linear programming problems are now widely available. After a short period of familiarization with the specific features of the package, users are able to solve linear programming problems with few difficulties. Problems involving thousands of variables and thousands of constraints are now routinely solved with computer packages. Some of the leading commercial packages include CPLEX, Gurobi, LINGO, MOSEK, Risk Solver for Excel, and Xpress-MP. Packages are also available for free download. A good example is Clp (COIN-OR linear programming). The solution to Par, Inc. is shown in Figure 2.14. The authors have chosen to make this book flexible and not rely on a specific linear programming package. Hence, the output in Figure 2.14 is generic and is not an actual printout from a particular software package. The output provided in Figure 2.14 is typical of most linear programming packages. We use this output format throughout the text. At the website for this course two linear programming packages are provided. A description of the packages is provided in the appendices. In Appendix 2.1 we show how to solve the Par, Inc., problem using LINGO. In Appendix 2.2 we

2.4

Computer Solution of the Par, Inc., Problem

51

FIGURE 2.14 THE SOLUTION FOR THE PAR, INC., PROBLEM
Optimal Objective Value = Variable -------------S D Constraint -------------1 2 3 4 Variable ---------S D Constraint ---------1 2 3 4 7668.00000 Reduced Cost ----------------0.00000 0.00000 Dual Value ----------------4.37500 0.00000 6.93750 0.00000 Allowable Decrease ---------3.70000 2.33333 Allowable Decrease ---------134.40000 120.00000 128.00000 18.00000

Value --------------540.00000 252.00000 Slack/Surplus --------------0.00000 120.00000 0.00000 18.00000 Allowable Increase ---------3.50000 5.28571 Allowable Increase ---------52.36364 Infinite 192.00000 Infinite

WEB
Par

file

Objective Coefficient -----------10.00000 9.00000 RHS Value -----------630.00000 600.00000 708.00000 135.00000

show how to formulate a spreadsheet model for the Par, Inc., problem and use Excel Solver to solve the problem.

Interpretation of Computer Output
Let us look more closely at the output in Figure 2.14 and interpret the computer solution provided for the Par, Inc., problem. The optimal solution to this problem will provide a profit of $7668. Directly below the objective function value, we find the values of the decision variables at the optimal solution. We have S 540 standard bags and D 252 deluxe bags as the optimal production quantities. Recall that the Par, Inc., problem had four less-than-or-equal-to constraints corresponding to the hours available in each of four production departments. The information shown in the Slack/Surplus column provides the value of the slack variable for each of the departments. This information is summarized here:

Constraint Number 1 2 3 4

Constraint Name Cutting and dyeing Sewing Finishing Inspection and packaging

Slack 0 120 0 18

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From this information, we see that the binding constraints (the cutting and dyeing and the finishing constraints) have zero slack at the optimal solution. The sewing department has 120 hours of slack or unused capacity, and the inspection and packaging department has 18 hours of slack or unused capacity. The rest of the output in Figure 2.14 can be used to determine how changes in the input data impact the optimal solution. We shall defer discussion of reduced costs, dual values, allowable increases and decreases of objective function coefficients and right-hand-side values until Chapter 3, when we study the topic of sensitivity analysis.

NOTES AND COMMENTS
Linear programming solvers are now a standard feature of most spreadsheet packages. In Appendix 2.2 we show how spreadsheets can be used to solve linear programs by using Excel to solve the Par, Inc., problem.

2.5

A SIMPLE MINIMIZATION PROBLEM
M&D Chemicals produces two products that are sold as raw materials to companies manufacturing bath soaps and laundry detergents. Based on an analysis of current inventory levels and potential demand for the coming month, M&D’s management specified that the combined production for products A and B must total at least 350 gallons. Separately, a major customer’s order for 125 gallons of product A must also be satisfied. Product A requires 2 hours of processing time per gallon and product B requires 1 hour of processing time per gallon. For the coming month, 600 hours of processing time are available. M&D’s objective is to satisfy these requirements at a minimum total production cost. Production costs are $2 per gallon for product A and $3 per gallon for product B. To find the minimum-cost production schedule, we will formulate the M&D Chemicals problem as a linear program. Following a procedure similar to the one used for the Par, Inc., problem, we first define the decision variables and the objective function for the problem. Let

A = number of gallons of product A B = number of gallons of product B
With production costs at $2 per gallon for product A and $3 per gallon for product B, the objective function that corresponds to the minimization of the total production cost can be written as Min 2 A + 3B Next, consider the constraints placed on the M&D Chemicals problem. To satisfy the major customer’s demand for 125 gallons of product A, we know A must be at least 125. Thus, we write the constraint

1A Ú 125
For the combined production for both products, which must total at least 350 gallons, we can write the constraint

1A + 1B Ú 350

2.5

A Simple Minimization Problem

53

Finally, for the limitation of 600 hours on available processing time, we add the constraint

2 A + 1B … 600
After adding the nonnegativity constraints (A, B program for the M&D Chemicals problem: Min s.t. 0), we arrive at the following linear

2 A + 3B Ú 125 Demand for product A 1A 1 A + 1B Ú 350 Total production 2 A + 1B … 600 Processing time A, B Ú 0

Because the linear programming model has only two decision variables, the graphical solution procedure can be used to find the optimal production quantities. The graphical solution procedure for this problem, just as in the Par problem, requires us to first graph the constraint lines to find the feasible region. By graphing each constraint line separately and then checking points on either side of the constraint line, the feasible solutions for each constraint can be identified. By combining the feasible solutions for each constraint on the same graph, we obtain the feasible region shown in Figure 2.15.

FIGURE 2.15 THE FEASIBLE REGION FOR THE M&D CHEMICALS PROBLEM
B 600 500 400
1A 0 35 = tion B c + 1 rodu P

c Pro ing ess e Tim

A = 125

Gallons of Product B

300 200 100

2A B= +1 600

0

100

200

300

400

500

600

A

Gallons of Product A

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To find the minimum-cost solution, we now draw the objective function line corresponding to a particular total cost value. For example, we might start by drawing the line 2A 3B 1200. This line is shown in Figure 2.16. Clearly, some points in the feasible region would provide a total cost of $1200. To find the values of A and B that provide smaller total cost values, we move the objective function line in a lower left direction until, if we moved it any farther, it would be entirely outside the feasible region. Note that the objective function line 2A 3B 800 intersects the feasible region at the extreme point A 250 and B 100. This extreme point provides the minimum-cost solution with an objective function value of 800. From Figures 2.15 and 2.16, we can see that the total production constraint and the processing time constraint are binding. Just as in every linear programming problem, the optimal solution occurs at an extreme point of the feasible region.

Summary of the Graphical Solution Procedure for Minimization Problems
Can you use the graphical solution procedure to determine the optimal solution for a minimization problem? Try Problem 31.

The steps of the graphical solution procedure for a minimization problem are summarized here: 1. Prepare a graph of the feasible solutions for each of the constraints. 2. Determine the feasible region by identifying the solutions that satisfy all the constraints simultaneously.

FIGURE 2.16 GRAPHICAL SOLUTION FOR THE M&D CHEMICALS PROBLEM
B 600 500 Gallons of Product B 400 300 200 100
2A 2A

+3 B=

12

00

Optimal Solution (A = 250, B = 100)
0 100 200

+3

B=

80

0

300

400

500

600

A

Gallons of Product A

2.5

A Simple Minimization Problem

55

3. Draw an objective function line showing the values of the decision variables that yield a specified value of the objective function. 4. Move parallel objective function lines toward smaller objective function values until further movement would take the line completely outside the feasible region. 5. Any feasible solution on the objective function line with the smallest value is an optimal solution.

Surplus Variables
The optimal solution to the M&D Chemicals problem shows that the desired total production of A B 350 gallons has been achieved by using all available processing time of 2A 1B 2(250) 1(100) 600 hours. In addition, note that the constraint requiring that product A demand be met has been satisfied with A 250 gallons. In fact, the production of product A exceeds its minimum level by 250 125 125 gallons. This excess production for product A is referred to as surplus. In linear programming terminology, any excess quantity corresponding to a constraint is referred to as surplus. Recall that with a constraint, a slack variable can be added to the left-hand side of the inequality to convert the constraint to equality form. With a constraint, a surplus variable can be subtracted from the left-hand side of the inequality to convert the constraint to equality form. Just as with slack variables, surplus variables are given a coefficient of zero in the objective function because they have no effect on its value. After including two surplus variables, S1 and S2, for the constraints and one slack variable, S3, for the constraint, the linear programming model of the M&D Chemicals problem becomes Min s.t.

2 A + 3B + 0S1 + 0S2 + 0S3 - 1S1 1A = 125 1 A + 1B - 1S2 = 350 + 1S3 = 600 2 A + 1B A, B, S1, S2, S3 Ú 0

Try Problem 35 to test your ability to use slack and surplus variables to write a linear program in standard form.

All the constraints are now equalities. Hence, the preceding formulation is the standardform representation of the M&D Chemicals problem. At the optimal solution of A 250 and B 100, the values of the surplus and slack variables are as follows:

Constraint Demand for product A Total production Processing time

Value of Surplus or Slack Variables S1 125 S2 0 S3 0

Refer to Figures 2.15 and 2.16. Note that the zero surplus and slack variables are associated with the constraints that are binding at the optimal solution—that is, the total production and processing time constraints. The surplus of 125 units is associated with the nonbinding constraint on the demand for product A. In the Par, Inc., problem all the constraints were of the type, and in the M&D Chemicals problem the constraints were a mixture of and types. The number and types of constraints encountered in a particular linear programming problem depend on

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Try Problem 34 to practice solving a linear program with all three constraint forms.

the specific conditions existing in the problem. Linear programming problems may have some constraints, some constraints, and some constraints. For an equality constraint, feasible solutions must lie directly on the constraint line. An example of a linear program with two decision variables, G and H, and all three constraint forms is given here: Min s.t.

2G + 2H 1G + 3H … 12 3G + 1H Ú 13 1G - 1H = 3 G, H Ú 0

The standard-form representation of this problem is Min s.t.

2G + 2H + 0S1 + 0S2 1G + 3H + 1S1 = 12 3G + 1H - 1S2 = 13 1G - 1H = 3 G, H, S1, S2 Ú 0

The standard form requires a slack variable for the constraint and a surplus variable for the constraint. However, neither a slack nor a surplus variable is required for the third constraint because it is already in equality form. When solving linear programs graphically, it is not necessary to write the problem in its standard form. Nevertheless, you should be able to compute the values of the slack and surplus variables and understand what they mean, because the values of slack and surplus variables are included in the computer solution of linear programs. A final point: The standard form of the linear programming problem is equivalent to the original formulation of the problem. That is, the optimal solution to any linear programming problem is the same as the optimal solution to the standard form of the problem. The standard form has not changed the basic problem; it has only changed how we write the constraints for the problem.

Computer Solution of the M&D Chemicals Problem
The optimal solution to M&D is given in Figure 2.17. The computer output shows that the minimum-cost solution yields an objective function value of $800. The values of the decision variables show that 250 gallons of product A and 100 gallons of product B provide the minimum-cost solution. The Slack/Surplus column shows that the constraint corresponding to the demand for product A (see constraint 1) has a surplus of 125 units. This column tells us that production of product A in the optimal solution exceeds demand by 125 gallons. The Slack/Surplus values are zero for the total production requirement (constraint 2) and the processing time limitation (constraint 3), which indicates that these constraints are binding at the optimal solution. We will discuss the rest of the computer output that appears in Figure 2.17 in Chapter 3 when we study the topic of sensitivity analysis.

2.6

Special Cases

57

FIGURE 2.17 THE SOLUTION FOR THE M&D CHEMICALS PROBLEM
Optimal Objective Value = Variable -------------A B Constraint -------------1 2 3 Variable ---------A B Constraint ---------1 2 3 800.00000 Reduced Cost ----------------0.00000 0.00000 Dual Value ----------------0.00000 4.00000 -1.00000 Allowable Decrease ---------Infinite 1.00000 Allowable Decrease ---------Infinite 50.00000 125.00000

Value --------------250.00000 100.00000 Slack/Surplus --------------125.00000 0.00000 0.00000 Allowable Increase ---------1.00000 Infinite Allowable Increase ---------125.00000 125.00000 100.00000

WEB
M&D

file

Objective Coefficient -----------2.00000 3.00000 RHS Value -----------125.00000 350.00000 600.00000

2.6

SPECIAL CASES
In this section we discuss three special situations that can arise when we attempt to solve linear programming problems.

Alternative Optimal Solutions
From the discussion of the graphical solution procedure, we know that optimal solutions can be found at the extreme points of the feasible region. Now let us consider the special case in which the optimal objective function line coincides with one of the binding constraint lines on the boundary of the feasible region. We will see that this situation can lead to the case of alternative optimal solutions; in such cases, more than one solution provides the optimal value for the objective function. To illustrate the case of alternative optimal solutions, we return to the Par, Inc., problem. However, let us assume that the profit for the standard golf bag (S) has been decreased to $6.30. The revised objective function becomes 6.3S 9D. The graphical solution of this problem is shown in Figure 2.18. Note that the optimal solution still occurs at an extreme point. In fact, it occurs at two extreme points: extreme point 4 (S 300, D 420) and extreme point 3 (S 540, D 252). The objective function values at these two extreme points are identical; that is, 6.3S 9D 6.3(300) 9(420) 5670

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FIGURE 2.18 PAR, INC., PROBLEM WITH AN OBJECTIVE FUNCTION OF 6.3S (ALTERNATIVE OPTIMAL SOLUTIONS)
D 600

9D

5 4 (300, 420)

Number of Deluxe Bags

400

6.3 S
200

+9 D

=3 78 0

3 (540, 252)

6.3 S

+9

D

=5 67 0
800 S

0

1

200

400

600

2

Number of Standard Bags

and

Furthermore, any point on the line connecting the two optimal extreme points also provides an optimal solution. For example, the solution point (S 420, D 336), which is halfway between the two extreme points, also provides the optimal objective function value of A linear programming problem with alternative optimal solutions is generally a good situation for the manager or decision maker. It means that several combinations of the decision variables are optimal and that the manager can select the most desirable optimal solution. Unfortunately, determining whether a problem has alternative optimal solutions is not a simple matter. 6.3S 9D 6.3(420) 9(336) 5670

6.3S

9D

6.3(540)

9(252)

5670

Infeasibility
Problems with no feasible solution do arise in practice, most often because management’s expectations are too high or because too many restrictions have been placed on the problem.

Infeasibility means that no solution to the linear programming problem satisfies all the constraints, including the nonnegativity conditions. Graphically, infeasibility means that a feasible region does not exist; that is, no points satisfy all the constraints and the nonnegativity conditions simultaneously. To illustrate this situation, let us look again at the problem faced by Par, Inc. Suppose that management specified that at least 500 of the standard bags and at least 360 of the deluxe bags must be manufactured. The graph of the solution region may now be constructed to reflect these new requirements (see Figure 2.19). The shaded area in the lower

2.6

Special Cases

59

FIGURE 2.19 NO FEASIBLE REGION FOR THE PAR, INC., PROBLEM WITH MINIMUM PRODUCTION REQUIREMENTS OF 500 STANDARD AND 360 DELUXE BAGS
D 600 Points Satisfying Minimum Production Requirements Minimum S

Number of Deluxe Bags

400

Minimum D

200

Points Satisfying Departmental Constraints

0

200

400 Number of Standard Bags

600

800

S

left-hand portion of the graph depicts those points satisfying the departmental constraints on the availability of time. The shaded area in the upper right-hand portion depicts those points satisfying the minimum production requirements of 500 standard and 360 deluxe bags. But no points satisfy both sets of constraints. Thus, we see that if management imposes these minimum production requirements, no feasible region exists for the problem. How should we interpret infeasibility in terms of this current problem? First, we should tell management that given the resources available (i.e., production time for cutting and dyeing, sewing, finishing, and inspection and packaging), it is not possible to make 500 standard bags and 360 deluxe bags. Moreover, we can tell management exactly how much of each resource must be expended to make it possible to manufacture 500 standard and 360 deluxe bags. Table 2.2 shows the minimum amounts of resources that must be available, the amounts currently available, and additional amounts that would be required to accomplish this level of production. Thus, we need 80 more hours for cutting and dyeing, 32 more hours for finishing, and 5 more hours for inspection and packaging to meet management’s minimum production requirements. If, after reviewing this information, management still wants to manufacture 500 standard and 360 deluxe bags, additional resources must be provided. Perhaps by hiring another person to work in the cutting and dyeing department, transferring a person from elsewhere in the plant to work part-time in the finishing department, or having the sewing people help out periodically with the inspection and packaging, the resource requirements can be met. As you can see, many possibilities are available for corrective management action, once we discover the lack of a feasible solution. The important thing to realize is that linear programming

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TABLE 2.2 RESOURCES NEEDED TO MANUFACTURE 500 STANDARD BAGS AND 360 DELUXE BAGS Minimum Required Resources (hours) ⁷⁄₁₀(500) 1(360) 710 ¹⁄₂(500) ⁵⁄₆(360) 550 1(500) ²⁄₃(360) 740 ¹⁄₁₀(500) ¹⁄₄(360) 140 Available Resources (hours) 630 600 708 135 Additional Resources Needed (hours) 80 None 32 5

Operation Cutting and dyeing Sewing Finishing Inspection and packaging

analysis can help determine whether management’s plans are feasible. By analyzing the problem using linear programming, we are often able to point out infeasible conditions and initiate corrective action. Whenever you attempt to solve a problem that is infeasible using either LINGO or Excel Solver, you will get an error message indicating that the problem is infeasible. In this case you know that no solution to the linear programming problem will satisfy all constraints, including the nonnegativity conditions. Careful inspection of your formulation is necessary to try to identify why the problem is infeasible. In some situations, the only reasonable approach is to drop one or more constraints and re-solve the problem. If you are able to find an optimal solution for this revised problem, you will know that the constraint(s) that was omitted, in conjunction with the others, is causing the problem to be infeasible.

Unbounded
The solution to a maximization linear programming problem is unbounded if the value of the solution may be made infinitely large without violating any of the constraints; for a minimization problem, the solution is unbounded if the value may be made infinitely small. This condition might be termed managerial utopia; for example, if this condition were to occur in a profit maximization problem, the manager could achieve an unlimited profit. However, in linear programming models of real problems, the occurrence of an unbounded solution means that the problem has been improperly formulated. We know it is not possible to increase profits indefinitely. Therefore, we must conclude that if a profit maximization problem results in an unbounded solution, the mathematical model doesn’t represent the real-world problem sufficiently. Usually, what has happened is that a constraint has been inadvertently omitted during problem formulation. As an illustration, consider the following linear program with two decision variables, X and Y: Max s.t.

20X + 10Y 1X Ú 2 1Y … 5 X, Y Ú 0

In Figure 2.20 we graphed the feasible region associated with this problem. Note that we can only indicate part of the feasible region because the feasible region extends indefinitely in the direction of the X axis. Looking at the objective function lines in Figure 2.20, we see that the solution to this problem may be made as large as we desire. That is, no matter what solution we pick, we will always be able to reach some feasible solution with a larger value. Thus, we say that the solution to this linear program is unbounded.

2.6

Special Cases

61

FIGURE 2.20 EXAMPLE OF AN UNBOUNDED PROBLEM
Y 20

15

10 Objective function increases without limit. 5 Feasible Region X

0

5

10

15

20

20X

20X

Can you recognize whether a linear program involves alternative optimal solutions or infeasibility, or is unbounded? Try Problems 42 and 43.

Whenever you attempt to solve a problem that is unbounded using either LINGO or Excel Solver you will get a message telling you that the problem is unbounded. Because unbounded solutions cannot occur in real problems, the first thing you should do is to review your model to determine whether you incorrectly formulated the problem. In many cases, this error is the result of inadvertently omitting a constraint during problem formulation.

NOTES AND COMMENTS
1. Infeasibility is independent of the objective function. It exists because the constraints are so restrictive that no feasible region for the linear programming model is possible. Thus, when you encounter infeasibility, making changes in the coefficients of the objective function will not help; the problem will remain infeasible. 2. The occurrence of an unbounded solution is often the result of a missing constraint. However, a change in the objective function may cause a previously unbounded problem to become bounded with an optimal solution. For example, the graph in Figure 2.20 shows an unbounded solution for the objective function Max 20X 10Y. However, changing the objective function to Max 20X 10Y will provide the optimal solution X 2 and Y 0 even though no changes have been made in the constraints.

20X 0Y +1 =8 0

0Y +1 =1 60

= 0Y +1 240

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2.7

GENERAL LINEAR PROGRAMMING NOTATION
In this chapter we showed how to formulate linear programming models for the Par, Inc., and M&D Chemicals problems. To formulate a linear programming model of the Par, Inc., problem we began by defining two decision variables: S number of standard bags and D number of deluxe bags. In the M&D Chemicals problem, the two decision variables were defined as A number of gallons of product A and B number of gallons of product B. We selected decision-variable names of S and D in the Par, Inc., problem and A and B in the M&D Chemicals problem to make it easier to recall what these decision variables represented in the problem. Although this approach works well for linear programs involving a small number of decision variables, it can become difficult when dealing with problems involving a large number of decision variables. A more general notation that is often used for linear programs uses the letter x with a subscript. For instance, in the Par, Inc., problem, we could have defined the decision variables as follows:

x1 = number of standard bags x 2 = number of deluxe bags
In the M&D Chemicals problem, the same variable names would be used, but their definitions would change:

x1 = number of gallons of product A x 2 = number of gallons of product B
A disadvantage of using general notation for decision variables is that we are no longer able to easily identify what the decision variables actually represent in the mathematical model. However, the advantage of general notation is that formulating a mathematical model for a problem that involves a large number of decision variables is much easier. For instance, for a linear programming model with three decision variables, we would use variable names of x1, x2, and x3; for a problem with four decision variables, we would use variable names of x1, x2, x3, and x4, and so on. Clearly, if a problem involved 1000 decision variables, trying to identify 1000 unique names would be difficult. However, using the general linear programming notation, the decision variables would be defined as x1, x2, x3, . . . , x1000. To illustrate the graphical solution procedure for a linear program written using general linear programming notation, consider the following mathematical model for a maximization problem involving two decision variables: Max s.t.

3x1 +

2x 2

2x1 + 2x 2 … 8 1x1 + 0.5x 2 … 3 x1, x 2 Ú 0
We must first develop a graph that displays the possible solutions (x1 and x2 values) for the problem. The usual convention is to plot values of x1 along the horizontal axis and values of x2 along the vertical axis. Figure 2.21 shows the graphical solution for this two-variable problem. Note that for this problem the optimal solution is x1 2 and x2 2, with an objective function value of 10.

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63

FIGURE 2.21 GRAPHICAL SOLUTION OF A TWO-VARIABLE LINEAR PROGRAM WITH GENERAL NOTATION
7
x2

6

5

Co nst rain t2

4

Co

ns tra

3

in t1
Optimal Solution x1 = 2, x2 = 2

2

Optimal Value = 3x1 + 2x2 = 10

1

3x1 + 2x2 = 6

0

1

2

3

4

5

x1

Using general linear programming notation, we can write the standard form of the preceding linear program as follows: Max s.t.

3x1 +

2x 2 + 0s1 + 0s2 = 8 1s2 = 3

2x1 + 2x 2 + 1s1 1x1 + 0.5x 2 + x1, x 2, s1, s2 Ú 0
Thus, at the optimal solution x1 s2 0. 2 and x2

2; the values of the slack variables are s1

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SUMMARY
We formulated linear programming models for two problems: the Par, Inc., maximization problem and the M&D Chemicals minimization problem. For both problems we showed a graphical solution procedure and provided a computer solution to the problem in a generic solution table. In formulating a mathematical model of these problems, we developed a general definition of a linear programming model. A linear programming model is a mathematical model with the following characteristics: 1. A linear objective function that is to be maximized or minimized 2. A set of linear constraints 3. Variables that are all restricted to nonnegative values

Slack variables may be used to write less-than-or-equal-to constraints in equality form and surplus variables may be used to write greater-than-or-equal-to constraints in equality form. The value of a slack variable can usually be interpreted as the amount of unused resource, whereas the value of a surplus variable indicates the amount over and above some stated minimum requirement. When all constraints have been written as equalities, the linear program has been written in its standard form. If the solution to a linear program is infeasible or unbounded, no optimal solution to the problem can be found. In the case of infeasibility, no feasible solutions are possible, whereas, in the case of an unbounded solution, the objective function can be made infinitely large for a maximization problem and infinitely small for a minimization problem. In the case of alternative optimal solutions, two or more optimal extreme points exist, and all the points on the line segment connecting them are also optimal. This chapter concludes with a section showing how to write a linear program using general linear programming notation. The Management Science in Action, Using Linear Programming for Traffic Control, provides another example of the widespread use of linear programming. In the next two chapters we will see many more applications of linear programming.

MANAGEMENT SCIENCE IN ACTION
USING LINEAR PROGRAMMING FOR TRAFFIC CONTROL*
The Hanshin Expressway was the first urban toll expressway in Osaka, Japan. Although in 1964 its length was only 2.3 kilometers, today it is a largescale urban expressway network of 200 kilometers. The Hanshin Expressway provides service for the Hanshin (Osaka-Kobe) area, the second-most populated area in Japan. An average of 828,000 vehicles use the expressway each day, with daily traffic sometimes exceeding 1 million vehicles. In 1990, the Hanshin Expressway Public Corporation started using an automated traffic control system in order to maximize the number of vehicles flowing into the expressway network. The automated traffic control system relies on two control methods: (1) limiting the number of cars that enter the expressway at each entrance ramp; and (2) providing drivers with up-to-date and accurate

traffic information, including expected travel times and information about accidents. The approach used to limit the number of vehicles depends upon whether the expressway is in a normal or steady state of operation, or whether some type of unusual event, such as an accident or a breakdown, has occurred. In the first phase of the steady-state case, the Hanshin system uses a linear programming model to maximize the total number of vehicles entering the system, while preventing traffic congestion and adverse effects on surrounding road networks. The data that drive the linear programming model are collected from detectors installed every 500 meters along the expressway and at all entrance and exit ramps. Every five minutes the real-time data collected from the detectors are used to update the model coefficients, and a new linear program

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computes the maximum number of vehicles the expressway can accommodate. The automated traffic control system has been successful. According to surveys, traffic control decreased the length of congested portions of the expressway by 30% and the duration by 20%. It

proved to be extremely cost effective, and drivers consider it an indispensable service.
*Based on T. Yoshino, T. Sasaki, and T. Hasegawa, “The Traffic-Control System on the Hanshin Expressway,” Interfaces (January/February 1995): 94–108.

GLOSSARY
Constraint An equation or inequality that rules out certain combinations of decision variables as feasible solutions. Problem formulation The process of translating the verbal statement of a problem into a mathematical statement called the mathematical model. Decision variable A controllable input for a linear programming model. Nonnegativity constraints A set of constraints that requires all variables to be nonnegative.

Mathematical model A representation of a problem where the objective and all constraint conditions are described by mathematical expressions. Linear programming model A mathematical model with a linear objective function, a set of linear constraints, and nonnegative variables. Linear program Another term for linear programming model. Linear functions Mathematical expressions in which the variables appear in separate terms and are raised to the first power. Feasible solution A solution that satisfies all the constraints. Feasible region The set of all feasible solutions. Slack variable A variable added to the left-hand side of a less-than-or-equal-to constraint to convert the constraint into an equality. The value of this variable can usually be interpreted as the amount of unused resource. Standard form A linear program in which all the constraints are written as equalities. The optimal solution of the standard form of a linear program is the same as the optimal solution of the original formulation of the linear program. Redundant constraint A constraint that does not affect the feasible region. If a constraint is redundant, it can be removed from the problem without affecting the feasible region. Extreme point Graphically speaking, extreme points are the feasible solution points occurring at the vertices or “corners” of the feasible region. With two-variable problems, extreme points are determined by the intersection of the constraint lines. Surplus variable A variable subtracted from the left-hand side of a greater-than-orequal-to constraint to convert the constraint into an equality. The value of this variable can usually be interpreted as the amount over and above some required minimum level. Alternative optimal solutions The case in which more than one solution provides the optimal value for the objective function. Infeasibility The situation in which no solution to the linear programming problem satisfies all the constraints. Unbounded If the value of the solution may be made infinitely large in a maximization linear programming problem or infinitely small in a minimization problem without violating any of the constraints, the problem is said to be unbounded.

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PROBLEMS
1. Which of the following mathematical relationships could be found in a linear programming model, and which could not? For the relationships that are unacceptable for linear programs, state why. a. 1A 2B 70 b. 2A 2B 50 c. 1A 2B2 10 d. 3 2A 2B 15 e. 1A 1B 6 f. 2A 5B 1AB 25 2. Find the solutions that satisfy the following constraints: a. 4A 2B 16 b. 4A 2B 16 c. 4A 2B 16

3. Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints: a. 3A 2B 18 b. 12A 8B 480 c. 5A 10B 200

4. Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints: a. 3A 4B 60 b. 6A 5B 60 c. 5A 2B 0 5. Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints: a. A 0.25 (A B) b. B 0.10 (A B) c. A 0.50 (A B) 4B, and

7. Identify the feasible region for the following set of constraints: 0.5A + 0.25B Ú 30 1A + 5B Ú 250 0.25A + 0.5B … 50 A, B Ú 0 8. Identify the feasible region for the following set of constraints: 2A - 1B … 0 -1A + 1.5B … 200 A, B Ú 0 9. Identify the feasible region for the following set of constraints: 3A - 2B Ú 0 2A - 1B … 200 1A … 150 A, B Ú 0

6. Three objective functions for linear programming problems are 7A 10B, 6A 4A 7B. Show the graph of each for objective function values equal to 420.

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10. For the linear program

Max s.t.

2A + 3B 1A + 2B … 6 5A + 3B … 15 A, B Ú 0

11. Solve the following linear program using the graphical solution procedure: Max s.t. 5A + 5B … 100 1B … 80 2A + 4B … 400 A, B Ú 0 1A 12. Consider the following linear programming problem: Max s.t. 3A + 3B 2A + 4B … 12 6A + 4B … 24 A, B Ú 0

find the optimal solution using the graphical solution procedure. What is the value of the objective function at the optimal solution?

13. Consider the following linear program: Max s.t.

a. Find the optimal solution using the graphical solution procedure. b. If the objective function is changed to 2A 6B, what will the optimal solution be? c. How many extreme points are there? What are the values of A and B at each extreme point? 1A + 2B … 5 1B … 4 2A + 2B = 12 A, B Ú 0 1A

14. RMC, Inc., is a small firm that produces a variety of chemical products. In a particular production process, three raw materials are blended (mixed together) to produce two products: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of ²⁄₅ ton of material 1 and ³⁄₅ of material 3. A ton of solvent base is a mixture of ¹⁄₂ ton of material 1, ¹⁄₅ ton of material 2, and ³⁄₁₀ ton of material 3. After deducting relevant costs, the profit contribution is $40 for every ton of fuel additive produced and $30 for every ton of solvent base produced.

a. Show the feasible region. b. What are the extreme points of the feasible region? c. Find the optimal solution using the graphical procedure.

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RMC’s production is constrained by a limited availability of the three raw materials. For the current production period, RMC has available the following quantities of each raw material: Raw Material Material 1 Material 2 Material 3 Amount Available for Production 20 tons 5 tons 21 tons

15. Refer to the Par, Inc., problem described in Section 2.1. Suppose that Par’s management encounters the following situations: a. The accounting department revises its estimate of the profit contribution for the deluxe bag to $18 per bag. b. A new low-cost material is available for the standard bag, and the profit contribution per standard bag can be increased to $20 per bag. (Assume that the profit contribution of the deluxe bag is the original $9 value.) c. New sewing equipment is available that would increase the sewing operation capacity to 750 hours. (Assume that 10A 9B is the appropriate objective function.) If each of these situations is encountered separately, what is the optimal solution and the total profit contribution? 16. Refer to the feasible region for Par, Inc., problem in Figure 2.13. 5 a. Develop an objective function that will make extreme point 4 the optimal extreme point. b. What is the optimal solution for the objective function you selected in part (a)? c. What are the values of the slack variables associated with this solution? 17. Write the following linear program in standard form: Max s.t. 5A + 2B 1A - 2B … 420 2A + 3B … 610 6A - 1B … 125 A, B Ú 0 18. For the linear program Max s.t. 4A + 1B 10A + 2B … 30 3A + 2B … 12 2A + 2B … 10 A, B Ú 0

Assuming that RMC is interested in maximizing the total profit contribution, answer the following: a. What is the linear programming model for this problem? b. Find the optimal solution using the graphical solution procedure. How many tons of each product should be produced, and what is the projected total profit contribution? c. Is there any unused material? If so, how much? d. Are any of the constraints redundant? If so, which ones?

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19. Given the linear program

a. Write this problem in standard form. b. Solve the problem using the graphical solution procedure. c. What are the values of the three slack variables at the optimal solution? Max s.t.

3A + 4B -1A + 2B … 8 1A + 2B … 12 2A + 1B … 16 A, B Ú 0

20. For the linear program

a. Write the problem in standard form. b. Solve the problem using the graphical solution procedure. c. What are the values of the three slack variables at the optimal solution? Max s.t.

3A + 2B A + B Ú 3A + 4B … A Ú A - B … A, B Ú 0 4 24 2 0

21. Consider the following linear program: Max s.t.

a. Write the problem in standard form. b. Solve the problem. c. What are the values of the slack and surplus variables at the optimal solution?

2A + 3B 5A + 5B … 400 -1A + 1B … 10 1A + 3B Ú 90 A, B Ú 0 Constraint 1 Constraint 2 Constraint 3

22. Reiser Sports Products wants to determine the number of All-Pro (A) and College (C) footballs to produce in order to maximize profit over the next four-week planning horizon. Constraints affecting the production quantities are the production capacities in three departments: cutting and dyeing; sewing; and inspection and packaging. For

Figure 2.22 shows a graph of the constraint lines. a. Place a number (1, 2, or 3) next to each constraint line to identify which constraint it represents. b. Shade in the feasible region on the graph. c. Identify the optimal extreme point. What is the optimal solution? d. Which constraints are binding? Explain. e. How much slack or surplus is associated with the nonbinding constraint?

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FIGURE 2.22 GRAPH OF THE CONSTRAINT LINES FOR EXERCISE 21
B 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90 100 A

the four-week planning period, 340 hours of cutting and dyeing time, 420 hours of sewing time, and 200 hours of inspection and packaging time are available. All-Pro footballs provide a profit of $5 per unit and College footballs provide a profit of $4 per unit. The linear programming model with production times expressed in minutes is as follows: Max s.t. 5A + 4C 12A + 6C … 20,400 Cutting and dyeing 9A + 15C … 25,200 Sewing 6A + 6C … 12,000 Inspection and packaging A, C Ú 0

A portion of the graphical solution to the Reiser problem is shown in Figure 2.23. a. Shade the feasible region for this problem. b. Determine the coordinates of each extreme point and the corresponding profit. Which extreme point generates the highest profit? c. Draw the profit line corresponding to a profit of $4000. Move the profit line as far from the origin as you can in order to determine which extreme point will provide the optimal solution. Compare your answer with the approach you used in part (b). d. Which constraints are binding? Explain. e. Suppose that the values of the objective function coefficients are $4 for each All-Pro model produced and $5 for each College model. Use the graphical solution procedure to determine the new optimal solution and the corresponding value of profit.

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FIGURE 2.23 PORTION OF THE GRAPHICAL SOLUTION FOR EXERCISE 22
C 3500 3000 Number of College Footballs 2500 2000 1500 1000 500

0

500

1000

1500

2000

2500

3000

A

Number of All-Pro Footballs

23. Embassy Motorcycles (EM) manufacturers two lightweight motorcycles designed for easy handling and safety. The EZ-Rider model has a new engine and a low profile that make it easy to balance. The Lady-Sport model is slightly larger, uses a more traditional engine, and is specifically designed to appeal to women riders. Embassy produces the engines for both models at its Des Moines, Iowa, plant. Each EZ-Rider engine requires 6 hours of manufacturing time and each Lady-Sport engine requires 3 hours of manufacturing time. The Des Moines plant has 2100 hours of engine manufacturing time available for the next production period. Embassy’s motorcycle frame supplier can supply as many EZ-Rider frames as needed. However, the Lady-Sport frame is more complex and the supplier can only provide up to 280 Lady-Sport frames for the next production period. Final assembly and testing requires 2 hours for each EZ-Rider model and 2.5 hours for each Lady-Sport model. A maximum of 1000 hours of assembly and testing time are available for the next production period. The company’s accounting department projects a profit contribution of $2400 for each EZ-Rider produced and $1800 for each LadySport produced. a. Formulate a linear programming model that can be used to determine the number of units of each model that should be produced in order to maximize the total contribution to profit. b. Solve the problem graphically. What is the optimal solution? c. Which constraints are binding?

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24. Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcher’s model. The firm has 900 hours of production time available in its cutting and sewing department, 300 hours available in its finishing department, and 100 hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the following table: Production Time (hours) Model Regular model Catcher’s model Cutting and Sewing 1
³⁄₂

Finishing
¹⁄₂ ¹⁄₃

Packaging and Shipping
¹⁄₈ ¹⁄₄

Profit/Glove $5 $8

25. George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund. Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at least 30% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%. a. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives. b. Solve the problem using the graphical solution procedure.

Assuming that the company is interested in maximizing the total profit contribution, answer the following: a. What is the linear programming model for this problem? b. Find the optimal solution using the graphical solution procedure. How many gloves of each model should Kelson manufacture? c. What is the total profit contribution Kelson can earn with the given production quantities? d. How many hours of production time will be scheduled in each department? e. What is the slack time in each department?

26. The Sea Wharf Restaurant would like to determine the best way to allocate a monthly advertising budget of $1000 between newspaper advertising and radio advertising. Management decided that at least 25% of the budget must be spent on each type of media, and that the amount of money spent on local newspaper advertising must be at least twice the amount spent on radio advertising. A marketing consultant developed an index that measures audience exposure per dollar of advertising on a scale from 0 to 100, with higher values implying greater audience exposure. If the value of the index for local newspaper advertising is 50 and the value of the index for spot radio advertising is 80, how should the restaurant allocate its advertising budget in order to maximize the value of total audience exposure? a. Formulate a linear programming model that can be used to determine how the restaurant should allocate its advertising budget in order to maximize the value of total audience exposure. b. Solve the problem using the graphical solution procedure.

27. Blair & Rosen, Inc. (B&R), is a brokerage firm that specializes in investment portfolios designed to meet the specific risk tolerances of its clients. A client who contacted B&R this past week has a maximum of $50,000 to invest. B&R’s investment advisor decides to recommend a portfolio consisting of two investment funds: an Internet fund and a Blue

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28. Tom’s, Inc., produces various Mexican food products and sells them to Western Foods, a chain of grocery stores located in Texas and New Mexico. Tom’s, Inc., makes two salsa products: Western Foods Salsa and Mexico City Salsa. Essentially, the two products have different blends of whole tomatoes, tomato sauce, and tomato paste. The Western Foods Salsa is a blend of 50% whole tomatoes, 30% tomato sauce, and 20% tomato paste. The Mexico City Salsa, which has a thicker and chunkier consistency, consists of 70% whole tomatoes, 10% tomato sauce, and 20% tomato paste. Each jar of salsa produced weighs 10 ounces. For the current production period, Tom’s, Inc., can purchase up to 280 pounds of whole tomatoes, 130 pounds of tomato sauce, and 100 pounds of tomato paste; the price per pound for these ingredients is $0.96, $0.64, and $0.56, respectively. The cost of the spices and the other ingredients is approximately $0.10 per jar. Tom’s, Inc., buys empty glass jars for $0.02 each, and labeling and filling costs are estimated to be $0.03 for each jar of salsa produced. Tom’s contract with Western Foods results in sales revenue of $1.64 for each jar of Western Foods Salsa and $1.93 for each jar of Mexico City Salsa. a. Develop a linear programming model that will enable Tom’s to determine the mix of salsa products that will maximize the total profit contribution. b. Find the optimal solution.

Chip fund. The Internet fund has a projected annual return of 12%, whereas the Blue Chip fund has a projected annual return of 9%. The investment advisor requires that at most $35,000 of the client’s funds should be invested in the Internet fund. B&R services include a risk rating for each investment alternative. The Internet fund, which is the more risky of the two investment alternatives, has a risk rating of 6 per thousand dollars invested. The Blue Chip fund has a risk rating of 4 per thousand dollars invested. For example, if $10,000 is invested in each of the two investment funds, B&R’s risk rating for the portfolio would be 6(10) 4(10) 100. Finally, B&R developed a questionnaire to measure each client’s risk tolerance. Based on the responses, each client is classified as a conservative, moderate, or aggressive investor. Suppose that the questionnaire results classified the current client as a moderate investor. B&R recommends that a client who is a moderate investor limit his or her portfolio to a maximum risk rating of 240. a. What is the recommended investment portfolio for this client? What is the annual return for the portfolio? b. Suppose that a second client with $50,000 to invest has been classified as an aggressive investor. B&R recommends that the maximum portfolio risk rating for an aggressive investor is 320. What is the recommended investment portfolio for this aggressive investor? Discuss what happens to the portfolio under the aggressive investor strategy. c. Suppose that a third client with $50,000 to invest has been classified as a conservative investor. B&R recommends that the maximum portfolio risk rating for a conservative investor is 160. Develop the recommended investment portfolio for the conservative investor. Discuss the interpretation of the slack variable for the total investment fund constraint.

29. AutoIgnite produces electronic ignition systems for automobiles at a plant in Cleveland, Ohio. Each ignition system is assembled from two components produced at AutoIgnite’s plants in Buffalo, New York, and Dayton, Ohio. The Buffalo plant can produce 2000 units of component 1, 1000 units of component 2, or any combination of the two components each day. For instance, 60% of Buffalo’s production time could be used to produce component 1 and 40% of Buffalo’s production time could be used to produce component 2; in this case, the Buffalo plant would be able to produce 0.6(2000) 1200 units of component 1 each day and 0.4(1000) 400 units of component 2 each day. The Dayton plant can produce 600 units of component 1, 1400 units of component 2, or any combination of the two components each day. At the end of each day, the component production at Buffalo and Dayton is sent to Cleveland for assembly of the ignition systems on the following workday.

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30. A financial advisor at Diehl Investments identified two companies that are likely candidates for a takeover in the near future. Eastern Cable is a leading manufacturer of flexible cable systems used in the construction industry, and ComSwitch is a new firm specializing in digital switching systems. Eastern Cable is currently trading for $40 per share, and ComSwitch is currently trading for $25 per share. If the takeovers occur, the financial advisor estimates that the price of Eastern Cable will go to $55 per share and ComSwitch will go to $43 per share. At this point in time, the financial advisor has identified ComSwitch as the higher risk alternative. Assume that a client indicated a willingness to invest a maximum of $50,000 in the two companies. The client wants to invest at least $15,000 in Eastern Cable and at least $10,000 in ComSwitch. Because of the higher risk associated with ComSwitch, the financial advisor has recommended that at most $25,000 should be invested in ComSwitch. a. Formulate a linear programming model that can be used to determine the number of shares of Eastern Cable and the number of shares of ComSwitch that will meet the investment constraints and maximize the total return for the investment. b. Graph the feasible region. c. Determine the coordinates of each extreme point. d. Find the optimal solution. 31. Consider the following linear program: Min s.t.

a. Formulate a linear programming model that can be used to develop a daily production schedule for the Buffalo and Dayton plants that will maximize daily production of ignition systems at Cleveland. b. Find the optimal solution.

3A + 4B 1A + 3B Ú 6 1A + 1B Ú 4 A, B Ú 0

32. Identify the three extreme-point solutions for the M&D Chemicals problem (see Section 2.5). Identify the value of the objective function and the values of the slack and surplus variables at each extreme point. 33. Consider the following linear programming problem: Min s.t. A + A + 2A + 3A + -2A + A, B 2B 4B B 1.5B 6B Ú 0 … Ú … Ú 21 7 21 0

Identify the feasible region and find the optimal solution using the graphical solution procedure. What is the value of the objective function?

a. Find the optimal solution using the graphical solution procedure and the value of the objective function. b. Determine the amount of slack or surplus for each constraint. c. Suppose the objective function is changed to max 5A + 2B. Find the optimal solution and the value of the objective function.

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34. Consider the following linear program: Min s.t. 2A + 2B 1A + 3B … 12 3A + 1B Ú 13 1A - 1B = 3 A, B Ú 0 a. Show the feasible region. b. What are the extreme points of the feasible region? c. Find the optimal solution using the graphical solution procedure. Min s.t.

35. For the linear program

6A + 4B 2A + 1B Ú 12 1A + 1B Ú 10 1B … 4 A, B Ú 0

36. As part of a quality improvement initiative, Consolidated Electronics employees complete a three-day training program on teaming and a two-day training program on problem solving. The manager of quality improvement has requested that at least 8 training programs on teaming and at least 10 training programs on problem solving be offered during the next six months. In addition, senior-level management has specified that at least 25 training programs must be offered during this period. Consolidated Electronics uses a consultant to teach the training programs. During the next quarter, the consultant has 84 days of training time available. Each training program on teaming costs $10,000 and each training program on problem solving costs $8000. a. Formulate a linear programming model that can be used to determine the number of training programs on teaming and the number of training programs on problem solving that should be offered in order to minimize total cost. b. Graph the feasible region. c. Determine the coordinates of each extreme point. d. Solve for the minimum cost solution.

a. Write the problem in standard form. b. Solve the problem using the graphical solution procedure. c. What are the values of the slack and surplus variables?

37. The New England Cheese Company produces two cheese spreads by blending mild cheddar cheese with extra sharp cheddar cheese. The cheese spreads are packaged in 12-ounce containers, which are then sold to distributors throughout the Northeast. The Regular blend contains 80% mild cheddar and 20% extra sharp, and the Zesty blend contains 60% mild cheddar and 40% extra sharp. This year, a local dairy cooperative offered to provide up to 8100 pounds of mild cheddar cheese for $1.20 per pound and up to 3000 pounds of extra sharp cheddar cheese for $1.40 per pound. The cost to blend and package the cheese spreads, excluding the cost of the cheese, is $0.20 per container. If each container of Regular is sold for $1.95 and each container of Zesty is sold for $2.20, how many containers of Regular and Zesty should New England Cheese produce?

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38. Applied Technology, Inc. (ATI), produces bicycle frames using two fiberglass materials that improve the strength-to-weight ratio of the frames. The cost of the standard grade material is $7.50 per yard and the cost of the professional grade material is $9.00 per yard. The standard and professional grade materials contain different amounts of fiberglass, carbon fiber, and Kevlar as shown in the following table:

Fiberglass Carbon fiber Kevlar

Standard Grade 84% 10% 6%

Professional Grade 58% 30% 12%

39. Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client’s needs. For a new client, Innis has been authorized to invest up to $1.2 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%. The client wants to minimize risk subject to the requirement that the annual income from the investment be at least $60,000. According to Innis’s risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3; the higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis’s client also specified that at least $300,000 be invested in the money market fund. a. Determine how many units of each fund Innis should purchase for the client to minimize the total risk index for the portfolio. b. How much annual income will this investment strategy generate? c. Suppose the client desires to maximize annual return. How should the funds be invested?

ATI signed a contract with a bicycle manufacturer to produce a new frame with a carbon fiber content of at least 20% and a Kevlar content of not greater than 10%. To meet the required weight specification, a total of 30 yards of material must be used for each frame. a. Formulate a linear program to determine the number of yards of each grade of fiberglass material that ATI should use in each frame in order to minimize total cost. Define the decision variables and indicate the purpose of each constraint. b. Use the graphical solution procedure to determine the feasible region. What are the coordinates of the extreme points? c. Compute the total cost at each extreme point. What is the optimal solution? d. The distributor of the fiberglass material is currently overstocked with the professional grade material. To reduce inventory, the distributor offered ATI the opportunity to purchase the professional grade for $8 per yard. Will the optimal solution change? e. Suppose that the distributor further lowers the price of the professional grade material to $7.40 per yard. Will the optimal solution change? What effect would an even lower price for the professional grade material have on the optimal solution? Explain.

40. Photo Chemicals produces two types of photographic developing fluids. Both products cost Photo Chemicals $1 per gallon to produce. Based on an analysis of current inventory levels and outstanding orders for the next month, Photo Chemicals’ management specified that at least 30 gallons of product 1 and at least 20 gallons of product 2 must be produced during the next two weeks. Management also stated that an existing inventory of highly perishable raw material required in the production of both fluids must be used within the

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41. Southern Oil Company produces two grades of gasoline: regular and premium. The profit contributions are $0.30 per gallon for regular gasoline and $0.50 per gallon for premium gasoline. Each gallon of regular gasoline contains 0.3 gallons of grade A crude oil and each gallon of premium gasoline contains 0.6 gallons of grade A crude oil. For the next production period, Southern has 18,000 gallons of grade A crude oil available. The refinery used to produce the gasolines has a production capacity of 50,000 gallons for the next production period. Southern Oil’s distributors have indicated that demand for the premium gasoline for the next production period will be at most 20,000 gallons. a. Formulate a linear programming model that can be used to determine the number of gallons of regular gasoline and the number of gallons of premium gasoline that should be produced in order to maximize total profit contribution. b. What is the optimal solution? c. What are the values and interpretations of the slack variables? d. What are the binding constraints? 42. Does the following linear program involve infeasibility, unbounded, and/or alternative optimal solutions? Explain. Max s.t. 4A + 8B 2A + 2B … 10 -1A + 1B Ú 8 A, B Ú 0 43. Does the following linear program involve infeasibility, unbounded, and/or alternative optimal solutions? Explain. Max s.t. 1A + 1B 8A + 6B Ú 24 2B Ú 4 A, B 0 44. Consider the following linear program: Max s.t. 1A + 1B 5A + 3B … 15 3A + 5B … 15 A, B Ú 0

next two weeks. The current inventory of the perishable raw material is 80 pounds. Although more of this raw material can be ordered if necessary, any of the current inventory that is not used within the next two weeks will spoil—hence, the management requirement that at least 80 pounds be used in the next two weeks. Furthermore, it is known that product 1 requires 1 pound of this perishable raw material per gallon and product 2 requires 2 pounds of the raw material per gallon. Because Photo Chemicals’ objective is to keep its production costs at the minimum possible level, the firm’s management is looking for a minimum cost production plan that uses all the 80 pounds of perishable raw material and provides at least 30 gallons of product 1 and at least 20 gallons of product 2. What is the minimum cost solution?

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45. Consider the following linear program: Max s.t.

a. What is the optimal solution for this problem? b. Suppose that the objective function is changed to 1A + 2B. Find the new optimal solution.

1A - 2B -4A + 3B … 3 1A - 1B … 3 A, B Ú 0

a. b. c. d.

46. The manager of a small independent grocery store is trying to determine the best use of her shelf space for soft drinks. The store carries national and generic brands and currently has 200 square feet of shelf space available. The manager wants to allocate at least 60% of the space to the national brands and, regardless of the profitability, allocate at least 10% of the space to the generic brands. How many square feet of space should the manager allocate to the national brands and the generic brands under the following circumstances? a. The national brands are more profitable than the generic brands. b. Both brands are equally profitable. c. The generic brand is more profitable than the national brand. 47. Discuss what happens to the M&D Chemicals problem (see Section 2.5) if the cost per gallon for product A is increased to $3.00 per gallon. What would you recommend? Explain. 48. For the M&D Chemicals problem in Section 2.5, discuss the effect of management’s requiring total production of 500 gallons for the two products. List two or three actions M&D should consider to correct the situation you encounter.

Graph the feasible region for the problem. Is the feasible region unbounded? Explain. Find the optimal solution. Does an unbounded feasible region imply that the optimal solution to the linear program will be unbounded?

49. PharmaPlus operates a chain of 30 pharmacies. The pharmacies are staffed by licensed pharmacists and pharmacy technicians. The company currently employs 85 full-time equivalent pharmacists (combination of full time and part time) and 175 full-time equivalent technicians. Each spring management reviews current staffing levels and makes hiring plans for the year. A recent forecast of the prescription load for the next year shows that at least 250 full-time equivalent employees (pharmacists and technicians) will be required to staff the pharmacies. The personnel department expects 10 pharmacists and 30 technicians to leave over the next year. To accommodate the expected attrition and prepare for future growth, management stated that at least 15 new pharmacists must be hired. In addition, PharmaPlus’s new service quality guidelines specify no more than two technicians per licensed pharmacist. The average salary for licensed pharmacists is $40 per hour and the average salary for technicians is $10 per hour. a. Determine a minimum-cost staffing plan for PharmaPlus. How many pharmacists and technicians are needed? b. Given current staffing levels and expected attrition, how many new hires (if any) must be made to reach the level recommended in part (a)? What will be the impact on the payroll?

50. Expedition Outfitters manufactures a variety of specialty clothing for hiking, skiing, and mountain climbing. Its management decided to begin production on two new parkas designed

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79

51. English Motors, Ltd. (EML), developed a new all-wheel-drive sports utility vehicle. As part of the marketing campaign, EML produced a video tape sales presentation to send to both owners of current EML four-wheel-drive vehicles as well as to owners of four-wheeldrive sports utility vehicles offered by competitors; EML refers to these two target markets as the current customer market and the new customer market. Individuals who receive the new promotion video will also receive a coupon for a test drive of the new EML model for one weekend. A key factor in the success of the new promotion is the response rate, the percentage of individuals who receive the new promotion and test drive the new model. EML estimates that the response rate for the current customer market is 25% and the response rate for the new customer market is 20%. For the customers who test drive the new model, the sales rate is the percentage of individuals that make a purchase. Marketing research studies indicate that the sales rate is 12% for the current customer market and 20% for the new customer market. The cost for each promotion, excluding the test drive costs, is $4 for each promotion sent to the current customer market and $6 for each promotion sent to the new customer market. Management also specified that a minimum of 30,000 current customers should test drive the new model and a minimum of 10,000 new customers should test drive the new model. In addition, the number of current customers who test drive the new vehicle must be at least twice the number of new customers who test drive the new vehicle. If the marketing budget, excluding test drive costs, is $1.2 million, how many promotions should be sent to each group of customers in order to maximize total sales?

for use in extremely cold weather: the Mount Everest Parka and the Rocky Mountain Parka. The manufacturing plant has 120 hours of cutting time and 120 hours of sewing time available for producing these two parkas. Each Mount Everest Parka requires 30 minutes of cutting time and 45 minutes of sewing time, and each Rocky Mountain Parka requires 20 minutes of cutting time and 15 minutes of sewing time. The labor and material cost is $150 for each Mount Everest Parka and $50 for each Rocky Mountain Parka, and the retail prices through the firm’s mail order catalog are $250 for the Mount Everest Parka and $200 for the Rocky Mountain Parka. Because management believes that the Mount Everest Parka is a unique coat that will enhance the image of the firm, they specified that at least 20% of the total production must consist of this model. Assuming that Expedition Outfitters can sell as many coats of each type as it can produce, how many units of each model should it manufacture to maximize the total profit contribution?

52. Creative Sports Design (CSD) manufactures a standard-size racket and an oversize racket. The firm’s rackets are extremely light due to the use of a magnesium-graphite alloy that was invented by the firm’s founder. Each standard-size racket uses 0.125 kilograms of the alloy and each oversize racket uses 0.4 kilograms; over the next two-week production period only 80 kilograms of the alloy are available. Each standard-size racket uses 10 minutes of manufacturing time and each oversize racket uses 12 minutes. The profit contributions are $10 for each standard-size racket and $15 for each oversize racket, and 40 hours of manufacturing time are available each week. Management specified that at least 20% of the total production must be the standard-size racket. How many rackets of each type should CSD manufacture over the next two weeks to maximize the total profit contribution? Assume that because of the unique nature of their products, CSD can sell as many rackets as they can produce.

53. Management of High Tech Services (HTS) would like to develop a model that will help allocate their technicians’ time between service calls to regular contract customers and new customers. A maximum of 80 hours of technician time is available over the two-week planning period. To satisfy cash flow requirements, at least $800 in revenue (per technician) must be generated during the two-week period. Technician time for regular customers generates $25 per hour. However, technician time for new customers only

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54. Jackson Hole Manufacturing is a small manufacturer of plastic products used in the automotive and computer industries. One of its major contracts is with a large computer company and involves the production of plastic printer cases for the computer company’s portable printers. The printer cases are produced on two injection molding machines. The M-100 machine has a production capacity of 25 printer cases per hour, and the M-200 machine has a production capacity of 40 cases per hour. Both machines use the same chemical material to produce the printer cases; the M-100 uses 40 pounds of the raw material per hour and the M-200 uses 50 pounds per hour. The computer company asked Jackson Hole to produce as many of the cases during the upcoming week as possible; it will pay $18 for each case Jackson Hole can deliver. However, next week is a regularly scheduled vacation period for most of Jackson Hole’s production employees; during this time, annual maintenance is performed for all equipment in the plant. Because of the downtime for maintenance, the M-100 will be available for no more than 15 hours, and the M-200 will be available for no more than 10 hours. However, because of the high setup cost involved with both machines, management requires that, each machine must be operated for at least 5 hours. The supplier of the chemical material used in the production process informed Jackson Hole that a maximum of 1000 pounds of the chemical material will be available for next week’s production; the cost for this raw material is $6 per pound. In addition to the raw material cost, Jackson Hole estimates that the hourly cost of operating the M-100 and the M-200 are $50 and $75, respectively. a. Formulate a linear programming model that can be used to maximize the contribution to profit. b. Find the optimal solution. 55. The Kartick Company is trying to determine how much of each of two products to produce over the coming planning period. There are three departments, A, B and C, with limited labor hours available in each department. Each product must be processed by each department and the per-unit requirements for each product, labor hours available, and per-unit profit are as shown below.

generates an average of $8 per hour because in many cases a new customer contact does not provide billable services. To ensure that new customer contacts are being maintained, the technician time spent on new customer contacts must be at least 60% of the time spent on regular customer contacts. Given these revenue and policy requirements, HTS would like to determine how to allocate technician time between regular customers and new customers so that the total number of customers contacted during the two-week period will be maximized. Technicians require an average of 50 minutes for each regular customer contact and 1 hour for each new customer contact. a. Develop a linear programming model that will enable HTS to allocate technician time between regular customers and new customers. b. Find the optimal solution.

Labor required in each department Department A B C Profit Contribution Product (hrs./unit) Product 1 Product 2 1.00 0.30 0.30 0.12 0.15 0.56
$33.00 $24.00

Labor Hours Available 100 36 50

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A linear program for this situation is as follows:

Let Maximize

x1 x2
s.t.

the amount of product 1 to produce the amount of product 2 to produce 24 x2 .30 x2 100 Department A Department C Department B

33 x1 1.0 x1

x1, x2

.15 x1

.30 x1

0

.56 x2

.12 x2

50

36

Mr. Kartick (the owner) used trial and error with a spreadsheet model to arrive at a solution. His proposed solution is x1 75 and x2 60, as shown below in Figure 2.24. He said he felt his proposed solution is optimal. Is his solution optimal? Without solving the problem, explain why you believe this solution is optimal or not optimal.

FIGURE 2.24 MR. KARTICK’S TRIAL-AND-ERROR MODEL

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56. Assume you are given a minimization linear program that has an optimal solution. The problem is then modified by changing an equality constraint in the problem to a less-thanor-equal-to constraint. Is it possible that the modified problem is infeasible? Answer yes or no and justify.

57. Assume you are given a minimization linear program that has an optimal solution. The problem is then modified by changing a greater-than-or-equal-to constraint in the problem to a less-than-or-equal-to constraint. Is it possible that the modified problem is infeasible? Answer yes or no and justify.

58. A consultant was hired to build an optimization model for a large marketing research company. The model is based on a consumer survey that was taken in which each person was asked to rank 30 new products in descending order based on their likelihood of purchasing the product. The consultant was assigned the task of building a model that selects the minimum number of products (which would then be introduced into the marketplace) such that the first, second, and third choice of every subject in the survey is included in the list of selected products. While building a model to figure out which products to introduce, the consultant’s boss walked up to her and said: “Look, if the model tells us we need to introduce more than 15 products, then add a constraint which limits the number of new products to 15 or less. It’s too expensive to introduce more than 15 new products.” Evaluate this statement in terms of what you have learned so far about constrained optimization models.

Case Problem 1

WORKLOAD BALANCING

Digital Imaging (DI) produces photo printers for both the professional and consumer markets. The DI consumer division recently introduced two photo printers that provide color prints rivaling those produced by a professional processing lab. The DI-910 model can produce a 4" 6" borderless print in approximately 37 seconds. The more sophisticated and faster DI-950 can even produce a 13" 19" borderless print. Financial projections show profit contributions of $42 for each DI-910 and $87 for each DI-950. The printers are assembled, tested, and packaged at DI’s plant located in New Bern, North Carolina. This plant is highly automated and uses two manufacturing lines to produce the printers. Line 1 performs the assembly operation with times of 3 minutes per DI-910 printer and 6 minutes per DI-950 printer. Line 2 performs both the testing and packaging operations. Times are 4 minutes per DI-910 printer and 2 minutes per DI-950 printer. The shorter time for the DI-950 printer is a result of its faster print speed. Both manufacturing lines are in operation one 8-hour shift per day.

Managerial Report
Perform an analysis for Digital Imaging in order to determine how many units of each printer to produce. Prepare a report to DI’s president presenting your findings and recommendations. Include (but do not limit your discussion to) a consideration of the following:

1. The recommended number of units of each printer to produce to maximize the total contribution to profit for an 8-hour shift. What reasons might management have for not implementing your recommendation? 2. Suppose that management also states that the number of DI-910 printers produced must be at least as great as the number of DI-950 units produced. Assuming that the objective is to maximize the total contribution to profit for an 8-hour shift, how many units of each printer should be produced?

Case Problem 2

Production Strategy

83

For each solution that you develop, include a copy of your linear programming model and graphical solution in the appendix to your report.

3. Does the solution you developed in part (2) balance the total time spent on line 1 and the total time spent on line 2? Why might this balance or lack of it be a concern to management? 4. Management requested an expansion of the model in part (2) that would provide a better balance between the total time on line 1 and the total time on line 2. Management wants to limit the difference between the total time on line 1 and the total time on line 2 to 30 minutes or less. If the objective is still to maximize the total contribution to profit, how many units of each printer should be produced? What effect does this workload balancing have on total profit in part (2)? 5. Suppose that in part (1) management specified the objective of maximizing the total number of printers produced each shift rather than total profit contribution. With this objective, how many units of each printer should be produced per shift? What effect does this objective have on total profit and workload balancing?

Case Problem 2

PRODUCTION STRATEGY

Better Fitness, Inc. (BFI), manufactures exercise equipment at its plant in Freeport, Long Island. It recently designed two universal weight machines for the home exercise market. Both machines use BFI-patented technology that provides the user with an extremely wide range of motion capability for each type of exercise performed. Until now, such capabilities have been available only on expensive weight machines used primarily by physical therapists. At a recent trade show, demonstrations of the machines resulted in significant dealer interest. In fact, the number of orders that BFI received at the trade show far exceeded its manufacturing capabilities for the current production period. As a result, management decided to begin production of the two machines. The two machines, which BFI named the BodyPlus 100 and the BodyPlus 200, require different amounts of resources to produce. The BodyPlus 100 consists of a frame unit, a press station, and a pec-dec station. Each frame produced uses 4 hours of machining and welding time and 2 hours of painting and finishing time. Each press station requires 2 hours of machining and welding time and 1 hour of painting and finishing time, and each pec-dec station uses 2 hours of machining and welding time and 2 hours of painting and finishing time. In addition, 2 hours are spent assembling, testing, and packaging each BodyPlus 100. The raw material costs are $450 for each frame, $300 for each press station, and $250 for each pec-dec station; packaging costs are estimated to be $50 per unit. The BodyPlus 200 consists of a frame unit, a press station, a pec-dec station, and a legpress station. Each frame produced uses 5 hours of machining and welding time and 4 hours of painting and finishing time. Each press station requires 3 hours machining and welding time and 2 hours of painting and finishing time, each pec-dec station uses 2 hours of machining and welding time and 2 hours of painting and finishing time, and each legpress station requires 2 hours of machining and welding time and 2 hours of painting and finishing time. In addition, 2 hours are spent assembling, testing, and packaging each Body-Plus 200. The raw material costs are $650 for each frame, $400 for each press station, $250 for each pec-dec station, and $200 for each leg-press station; packaging costs are estimated to be $75 per unit. For the next production period, management estimates that 600 hours of machining and welding time, 450 hours of painting and finishing time, and 140 hours of assembly, testing,

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and packaging time will be available. Current labor costs are $20 per hour for machining and welding time, $15 per hour for painting and finishing time, and $12 per hour for assembly, testing, and packaging time. The market in which the two machines must compete suggests a retail price of $2400 for the BodyPlus 100 and $3500 for the BodyPlus 200, although some flexibility may be available to BFI because of the unique capabilities of the new machines. Authorized BFI dealers can purchase machines for 70% of the suggested retail price. BFI’s president believes that the unique capabilities of the BodyPlus 200 can help position BFI as one of the leaders in high-end exercise equipment. Consequently, he has stated that the number of units of the BodyPlus 200 produced must be at least 25% of the total production.

Managerial Report
Analyze the production problem at Better Fitness, Inc., and prepare a report for BFI’s president presenting your findings and recommendations. Include (but do not limit your discussion to) a consideration of the following items: 1. What is the recommended number of BodyPlus 100 and BodyPlus 200 machines to produce? 2. How does the requirement that the number of units of the BodyPlus 200 produced be at least 25% of the total production affect profits? 3. Where should efforts be expended in order to increase profits?

Include a copy of your linear programming model and graphical solution in an appendix to your report.

Case Problem 3

HART VENTURE CAPITAL

Hart Venture Capital (HVC) specializes in providing venture capital for software development and Internet applications. Currently HVC has two investment opportunities: (1) Security Systems, a firm that needs additional capital to develop an Internet security software package, and (2) Market Analysis, a market research company that needs additional capital to develop a software package for conducting customer satisfaction surveys. In exchange for Security Systems stock, the firm has asked HVC to provide $600,000 in year 1, $600,000 in year 2, and $250,000 in year 3 over the coming three-year period. In exchange for their stock, Market Analysis has asked HVC to provide $500,000 in year 1, $350,000 in year 2, and $400,000 in year 3 over the same three-year period. HVC believes that both investment opportunities are worth pursuing. However, because of other investments, they are willing to commit at most $800,000 for both projects in the first year, at most $700,000 in the second year, and $500,000 in the third year. HVC’s financial analysis team reviewed both projects and recommended that the company’s objective should be to maximize the net present value of the total investment in Security Systems and Market Analysis. The net present value takes into account the estimated value of the stock at the end of the three-year period as well as the capital outflows that are necessary during each of the three years. Using an 8% rate of return, HVC’s financial analysis team estimates that 100% funding of the Security Systems project has a net present value of $1,800,000, and 100% funding of the Market Analysis project has a net present value of $1,600,000. HVC has the option to fund any percentage of the Security Systems and Market Analysis projects. For example, if HVC decides to fund 40% of the Security Systems

Appendix 2.1

Solving Linear Programs with LINGO

85

project, investments of 0.40($600,000) $240,000 would be required in year 1, 0.40($600,000) $240,000 would be required in year 2, and 0.40($250,000) $100,000 would be required in year 3. In this case, the net present value of the Security Systems project would be 0.40($1,800,000) $720,000. The investment amounts and the net present value for partial funding of the Market Analysis project would be computed in the same manner.

Managerial Report
Perform an analysis of HVC’s investment problem and prepare a report that presents your findings and recommendations. Include (but do not limit your discussion to) a consideration of the following items: 1. What is the recommended percentage of each project that HVC should fund and the net present value of the total investment? 2. What capital allocation plan for Security Systems and Market Analysis for the coming three-year period and the total HVC investment each year would you recommend? 3. What effect, if any, would HVC’s willingness to commit an additional $100,000 during the first year have on the recommended percentage of each project that HVC should fund? 4. What would the capital allocation plan look like if an additional $100,000 is made available? 5. What is your recommendation as to whether HVC should commit the additional $100,000 in the first year?

Provide model details and relevant computer output in a report appendix.

Appendix 2.1
LINGO is a product of LINDO Systems. It was developed by Linus E. Schrage and Kevin Cunningham at the University of Chicago.

SOLVING LINEAR PROGRAMS WITH LINGO

In this appendix we describe how to use LINGO to solve the Par, Inc., problem. When you start LINGO, two windows are immediately displayed. The outer or main frame window contains all the command menus and the command toolbar. The smaller window is the model window; this window is used to enter and edit the linear programming model you want to solve. The first item we enter into the model window is the objective function. Recall that the objective function for the Par, Inc., problem is Max 10S 9D. Thus, in the first line of the LINGO model window, we enter the following expression: MAX = 10*S + 9*D; Note that in LINGO the symbol * is used to denote multiplication and that the objective function line ends with a semicolon. In general, each mathematical expression (objective function and constraints) in LINGO is terminated with a semicolon. Next, we press the enter key to move to a new line. The first constraint in the Par, Inc., problem is 0.7S 1D 630. Thus, in the second line of the LINGO model window we enter the following expression:

0.7*S + 1*D 6 = 630
Note that LINGO interprets the symbol as . Alternatively, we could enter instead of . As was the case when entering the objective function, a semicolon is required at the

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end of the first constraint. Pressing the enter key moves us to a new line as we continue the process by entering the remaining constraints as shown here: 0.5*S 1*S
⁵⁄₆*D ²⁄₃*D

600 135 708

0.1*S

0.25*D

The model window will now appear as follows: 0.7*S 1*S MAX 1*D 10*S 9*D 600 708 630

0.5*S 0.1*S

⁵⁄₆*D ²⁄₃*D

0.25*D

135

When entering a fraction into LINGO it is not necessary to convert the fraction into an equivalent or rounded decimal number. For example, simply enter the fraction ²⁄₃ into LINGO as ²⁄₃ and do not worry about converting to a decimal or how many decimal places to use. Enter ⁷⁄₁₀ either as ⁷⁄₁₀ or .7. Let LINGO act as a calculator for you. LINGO is very flexible about the format of an equation and it is not necessary to have the variables on the left hand side of an equation and the constant term on the right. For example, 0.7*S could also be entered as 0.7*S 630 1*D 1*D 630

This feature will be very useful later when writing models in a clear and understandable form. Finally, note that although we have expressly included a coefficient of 1 on the variable D above, this is not necessary. In LINGO, 1*D and D are equivalent. If you make an error in entering the model, you can correct it at any time by simply positioning the cursor where you made the error and entering the necessary correction. To solve the model, select the Solve command from the LINGO menu or press the Solve button on the toolbar at the top of the main frame window. LINGO will begin the solution process by determining whether the model conforms to all syntax requirements. If the LINGO model doesn’t pass these tests, you will be informed by an error message. If LINGO does not find any errors in the model input, it will begin to solve the model. As part of the solution process, LINGO displays a Solver Status window that allows you to monitor the progress of the solver. LINGO displays the solution in a new window titled “Solution Report.” The output that appears in the Solution Report window for the Par, Inc., problem is shown in Figure 2.25. The first part of the output shown in Figure 2.25 indicates that an optimal solution has been found and that the value of the objective function is 7668. We see that the optimal solution is S 540 and D 252, and that the slack variables for the four constraints (rows 2–5) are 0, 120, 0, and 18. We will discuss the use of the information in the Reduced Cost column and the Dual Price column in Chapter 3 when we study the topic of sensitivity analysis.

Appendix 2.2

Solving Linear Programs with Excel

87

FIGURE 2.25 PAR, INC., SOLUTION REPORT USING LINGO
Global optimal solution found. Objective value: Total solver iterations: Variable -------------S D Row -------------1 2 3 4 5 Value --------------540.0000 252.0000 Slack or Surplus ---------------7668.000 0.000000 120.0000 0.000000 18.00000

7668.000 2 Reduced Cost ----------------0.000000 0.000000 Dual Price ----------------1.000000 4.375000 0.000000 6.937500 0.000000

Appendix 2.2

SOLVING LINEAR PROGRAMS WITH EXCEL

In this appendix we will use an Excel worksheet to solve the Par, Inc., linear programming problem. We will enter the problem data for the Par problem in the top part of the worksheet and develop the linear programming model in the bottom part of the worksheet.

Formulation
Whenever we formulate a worksheet model of a linear program, we perform the following steps: Step 1. Step 2. Step 3. Step 4. Enter the problem data in the top part of the worksheet. Specify cell locations for the decision variables. Select a cell and enter a formula for computing the value of the objective function. Select a cell and enter a formula for computing the left-hand side of each constraint. Step 5. Select a cell and enter a formula for computing the right-hand side of each constraint.

The formula worksheet that we developed for the Par, Inc., problem using these five steps is shown in Figure 2.26. Note that the worksheet consists of two sections: a data section and a model section. The four components of the model are screened, and the cells reserved for the decision variables are enclosed in a boldface box. Figure 2.26 is called a formula worksheet because it displays the formulas that we have entered and not the values computed from those formulas. In a moment we will see how Excel’s Solver is used to find the optimal solution to the Par, Inc., problem. But first, let’s review each of the preceding steps as they apply to the Par, Inc., problem. Step 1. Enter the problem data in the top part of the worksheet. Cells B5:C8 show the production requirements per unit for each product. Note that in cells C6 and C7, we have entered the exact fractions. That is, in cell C6 we have entered ⁵⁄₆ and in cell C7 we have entered ²⁄₃.

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FIGURE 2.26 FORMULA WORKSHEET FOR THE PAR, INC., PROBLEM
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 Par, Inc. Production Time Standard Deluxe 0.7 1 0.5 0.83333 1 0.66667 0.1 0.25 10 9 B C D

Operation Cutting and Dyeing Sewing Finishing Inspection and Packaging Profit Per Bag

Time Available 630 600 708 135

Model Decision Variables Standard Deluxe Bags Produced Maximize Total Profit =B9*B16+C9*C16 Hours Used (LHS) =B5*B16+C5*C16 =B6*B16+C6*C16 =B7*B16+C7*C16 =B8*B16+C8*C16 Hours Available (RHS) =D5 =D6 =D7 =D8

Constraints Cutting and Dyeing Sewing Finishing Inspection and Packaging