An-Introduction-to-Management-Science-Anderson-15e-2019.pdf

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Copy Cop Co Coop opy py pyrrig righ ri iig t 20 019 19 Ce enga ng ge Learn rn ning ing. ng. ng. All Allll R Al Rig Ri iig gh htts hts ts RRese Re es esse errved rv ved v ed d. Ma May not not be cop piied ed, sca ed e s nned n , orr dup du du uppllica ica ica ic cated, ted tte e ed d d,, in in w who whol hole olle o o orr iin nppart par art rt. WCN CN 02-20 -20 200-203 3  An Introduction to Management Science Quantitative Approaches to Decision Making Fifteenth Edition David R. Anderson Dennis J. Sweeney Thomas A. Williams University of Cincinnati University of Cincinnati Rochester Institute of Technology Jeffrey D. Camm James J. Cochran Wake Forest University University of Alabama Michael J. Fry Jeffrey W. Ohlmann University of Cincinnati University of Iowa Australia Brazil Mexico Singapore United Kingdom United States Multi-media study-tools such as videos, games, flashcards, crossword puzzles, and more allow you to review and check your understanding of key concepts to help you prepare for quizzes and exams. The MindTap eReader is fully optimized for the iPad, provides note-taking and highlighting capabilities, and features an online text-to-speech application that vocalizes the content - providing a fun reading experience. Flashcards — use the MindTap eReader’s pre-made flashcards or make your own. Then print the cards and get to work. CengageNOW Users Achieve Higher Grades Treatment Control Final Course Grades “I love the check your work option. 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For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. ,PSRUWDQW1RWLFH0HGLDFRQWHQWUHIHUHQFHGZLWKLQWKHSURGXFWGHVFULSWLRQRUWKHSURGXFW WH[WPD\QRWEHDYDLODEOHLQWKHH%RRNYHUVLRQ  An Introduction to Management Science: © 2019, 2016 Cengage Learning, Inc. Quantitative Approaches to Decision Making, Fifteenth Edition Unless otherwise noted, all content is © Cengage David R. Anderson, Dennis J. Sweeney, Thomas A. Williams, Jeffrey D. Camm, ALL RIGHTS RESERVED. No part of this work covered by the copyright herein James J. Cochran, Michael J. Fry, may be reproduced or distributed in any form or by any means, except as Jeffrey W. Ohlmann permitted by U.S. copyright law, without the prior written permission of the copyright owner. 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Printed in the United States of America Print Number: 01 Print Year: 2017  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 Parents Randall and Jeannine Camm JDC To My Wife Teresa JJC To My Parents Mike and Cynthia Fry MJF To My Parents Willis and Phyllis Ohlmann JWO Brief Contents Preface xxi About the Authors xxv Chapter 1 Introduction 1 Chapter 2 An Introduction to Linear Programming 27 Chapter 3 Linear Programming: Sensitivity Analysis and Interpretation of Solution 84 Chapter 4 Linear Programming Applications in Marketing, Finance, and Operations Management 139 Chapter 5 Advanced Linear Programming Applications 195 Chapter 6 Distribution and Network Models 234 Chapter 7 Integer Linear Programming 291 Chapter 8 Nonlinear Optimization Models 336 Chapter 9 Project Scheduling: PERT/CPM 381 Chapter 10 Inventory Models 417 Chapter 11 Waiting Line Models 461 Chapter 12 Simulation 497 Chapter 13 Decision Analysis 543 Chapter 14 Multicriteria Decisions 613 Chapter 15 Time Series Analysis and Forecasting 654 Chapter 16 Markov Processes On Website Chapter 17 Linear Programming: Simplex Method On Website Chapter 18 Simplex-Based Sensitivity Analysis and Duality On Website Chapter 19 Solution Procedures for Transportation and Assignment Problems On Website Chapter 20 Minimal Spanning Tree On Website Chapter 21 Dynamic Programming On Website Appendices 711 Appendix A Building Spreadsheet Models 712 Appendix B Areas for the Standard Normal Distribution 741 Appendix C Values of e2 743 Appendix D References and Bibliography 744 Appendix E Self-Test Solutions and Answers to Even-Numbered Exercises On Website Index 747 Contents Preface xxi About the Authors xxv 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 9 Model Solution 10 Report Generation 12 A Note Regarding Implementation 12 1.4 Models of Cost, Revenue, and Profit 13 Cost and Volume Models 13 Revenue and Volume Models 14 Profit and Volume Models 14 Breakeven Analysis 14 1.5 Management Science Techniques 15 Methods Used Most Frequently 16 Summary 18 Glossary 18 Problems 19 Case Problem Scheduling a Golf League 23 Appendix 1.1 Using Excel for Breakeven Analysis 24 Chapter 2 An Introduction to Linear Programming 27 2.1 A Simple Maximization Problem 29 Problem Formulation 29 Mathematical Statement of the Par, Inc., Problem 32 2.2 Graphical Solution Procedure 33 A Note on Graphing Lines 41 Summary of the Graphical Solution Procedure for Maximization Problems 43 Slack Variables 44 2.3 Extreme Points and the Optimal Solution 45 2.4 Computer Solution of the Par, Inc., Problem 46 Interpretation of Computer Output 47 2.5 A Simple Minimization Problem 48 Summary of the Graphical Solution Procedure for Minimization Problems 50 x Contents Surplus Variables 50 Computer Solution of the M&D Chemicals Problem 52 2.6 Special Cases 53 Alternative Optimal Solutions 53 Infeasibility 54 Unbounded 56 2.7 General Linear Programming Notation 57 Summary 58 Glossary 60 Problems 61 Case Problem 1 Workload Balancing 75 Case Problem 2 Production Strategy 76 Case Problem 3 Hart Venture Capital 77 Appendix 2.1 Solving Linear Programs with Excel Solver 78 Appendix 2.2 Solving Linear Programs with LINGO 82 Chapter 3 Linear Programming: Sensitivity Analysis and Interpretation of Solution 84 3.1 Introduction to Sensitivity Analysis 86 3.2 Graphical Sensitivity Analysis 86 Objective Function Coefficients 87 Right-Hand Sides 91 3.3 Sensitivity Analysis: Computer Solution 94 Interpretation of Computer Output 94 Cautionary Note on the Interpretation of Dual Values 96 The Modified Par, Inc., Problem 97 3.4 Limitations of Classical Sensitivity Analysis 100 Simultaneous Changes 101 Changes in Constraint Coefficients 102 Nonintuitive Dual Values 103 3.5 The Electronic Communications Problem 105 Problem Formulation 106 Computer Solution and Interpretation 107 Summary 110 Glossary 111 Problems 112 Case Problem 1 Product Mix 131 Case Problem 2 Investment Strategy 132 Case Problem 3 Truck Leasing Strategy 133 Appendix 3.1 Sensitivity Analysis with Excel Solver 133 Appendix 3.2 Sensitivity Analysis with LINGO 136 Chapter 4 Linear Programming Applications in Marketing, Finance, and Operations Management 139 4.1 Marketing Applications 140 Media Selection 140 Marketing Research 143 Contents xi 4.2 Financial Applications 146 Portfolio Selection 146 Financial Planning 149 4.3 Operations Management Applications 153 A Make-or-Buy Decision 153 Production Scheduling 157 Workforce Assignment 163 Blending Problems 166 Summary 171 Problems 171 Case Problem 1 Planning an Advertising Campaign 184 Case Problem 2 Schneider’s Sweet Shop 185 Case Problem 3 Textile Mill Scheduling 186 Case Problem 4 Workforce Scheduling 187 Case Problem 5 Duke Energy Coal Allocation 189 Appendix 4.1 Excel Solution of Hewlitt Corporation Financial Planning Problem 191 Chapter 5 Advanced Linear Programming Applications 195 5.1 Data Envelopment Analysis 196 Evaluating the Performance of Hospitals 197 Overview of the DEA Approach 197 DEA Linear Programming Model 198 Summary of the DEA Approach 203 5.2 Revenue Management 203 5.3 Portfolio Models and Asset Allocation 209 A Portfolio of Mutual Funds 210 Conservative Portfolio 210 Moderate Risk Portfolio 213 5.4 Game Theory 216 Competing for Market Share 216 Identifying a Pure Strategy Solution 219 Identifying a Mixed Strategy Solution 219 Summary 226 Glossary 226 Problems 227 Chapter 6 Distribution and Network Models 234 6.1 Supply Chain Models 235 Transportation Problem 235 Problem Variations 240 A General Linear Programming Model 241 Transshipment Problem 242 Problem Variations 245 A General Linear Programming Model 247 6.2 Assignment Problem 248 Problem Variations 251 A General Linear Programming Model 252 xii Contents 6.3 Shortest-Route Problem 253 A General Linear Programming Model 256 6.4 Maximal Flow Problem 257 6.5 A Production and Inventory Application 260 Summary 263 Glossary 264 Problems 265 Case Problem 1 Solutions Plus 281 Case Problem 2 Supply Chain Design 282 Appendix 6.1 Excel Solution of Transportation, Transshipment, and Assignment Problems 284 Chapter 7 Integer Linear Programming 291 7.1 Types of Integer Linear Programming Models 293 7.2 Graphical and Computer Solutions for an All-Integer Linear Program 295 Graphical Solution of the LP Relaxation 295 Rounding to Obtain an Integer Solution 295 Graphical Solution of the All-Integer Problem 297 Using the LP Relaxation to Establish Bounds 297 Computer Solution 298 7.3 Applications Involving 0-1 Variables 298 Capital Budgeting 299 Fixed Cost 300 Distribution System Design 302 Bank Location 305 Product Design and Market Share Optimization 309 7.4 Modeling Flexibility Provided by 0-1 Integer Variables 313 Multiple-Choice and Mutually Exclusive Constraints 313 k out of n Alternatives Constraint 313 Conditional and Corequisite Constraints 314 A Cautionary Note About Sensitivity Analysis 315 Summary 316 Glossary 316 Problems 317 Case Problem 1 Textbook Publishing 327 Case Problem 2 Yeager National Bank 328 Case Problem 3 Production Scheduling with Changeover Costs 329 Case Problem 4 Applecore Children’s Clothing 329 Appendix 7.1 Excel Solution of Integer Linear Programs 331 Appendix 7.2 LINGO Solution of Integer Linear Programs 334 Chapter 8 Nonlinear Optimization Models 336 8.1 A Production Application—Par, Inc., Revisited 338 An Unconstrained Problem 338 A Constrained Problem 339 Contents xiii Local and Global Optima 341 Dual Values 344 8.2 Constructing an Index Fund 345 8.3 Markowitz Portfolio Model 349 8.4 Blending: The Pooling Problem 352 8.5 Forecasting Adoption of a New Product 356 Summary 361 Glossary 361 Problems 362 Case Problem 1 Portfolio Optimization with Transaction Costs 370 Case Problem 2 CAFE Compliance in the Auto Industry 373 Appendix 8.1 Solving Nonlinear Problems with Excel Solver 375 Appendix 8.2 Solving Nonlinear Problems with LINGO 378 Chapter 9 Project Scheduling: PERT/CPM 381 9.1 Project Scheduling Based on Expected Activity Times 382 The Concept of a Critical Path 383 Determining the Critical Path 385 Contributions of PERT/CPM 389 Summary of the PERT/CPM Critical Path Procedure 390 9.2 Project Scheduling Considering Uncertain Activity Times 391 The Daugherty Porta-Vac Project 391 Uncertain Activity Times 391 The Critical Path 394 Variability in Project Completion Time 395 9.3 Considering Time–Cost Trade-Offs 399 Crashing Activity Times 400 Linear Programming Model for Crashing 402 Summary 404 Glossary 404 Problems 405 Case Problem 1 R. C. Coleman 414 Appendix 9.1 Finding Cumulative Probabilities for Normally Distributed Random Variables 416 Chapter 10 Inventory Models 417 10.1 Economic Order Quantity (EOQ) Model 418 The How-Much-to-Order Decision 422 The When-to-Order Decision 423 Sensitivity Analysis for the EOQ Model 424 Excel Solution of the EOQ Model 425 Summary of the EOQ Model Assumptions 426 10.2 Economic Production Lot Size Model 427 Total Cost Model 427 Economic Production Lot Size 429 xiv Contents 10.3 Inventory Model with Planned Shortages 430 10.4 Quantity Discounts for the EOQ Model 434 10.5 Single-Period Inventory Model with Probabilistic Demand 436 Neiman Marcus 437 Nationwide Car Rental 440 10.6 Order-Quantity, Reorder Point Model with Probabilistic Demand 441 The How-Much-to-Order Decision 443 The When-to-Order Decision 443 10.7 Periodic Review Model with Probabilistic Demand 445 More Complex Periodic Review Models 448 Summary 449 Glossary 449 Problems 450 Case Problem 1 Wagner Fabricating Company 457 Case Problem 2 River City Fire Department 458 Appendix 10.1 Development of the Optimal Order Quantity (Q) Formula for the EOQ Model 459 Appendix 10.2 Development of the Optimal Lot Size (Q*) Formula for the Production Lot Size Model 460 Chapter 11 Waiting Line Models 461 11.1 Structure of a Waiting Line System 463 Single-Server Waiting Line 463 Distribution of Arrivals 463 Distribution of Service Times 464 Queue Discipline 465 Steady-State Operation 465 11.2 Single-Server Waiting Line Model with Poisson Arrivals and Exponential Service Times 466 Operating Characteristics 466 Operating Characteristics for the Burger Dome Problem 467 Managers’ Use of Waiting Line Models 468 Improving the Waiting Line Operation 468 Excel Solution of Waiting Line Model 469 11.3 Multiple-Server Waiting Line Model with Poisson Arrivals and Exponential Service Times 470 Operating Characteristics 471 Operating Characteristics for the Burger Dome Problem 472 11.4 Some General Relationships for Waiting Line Models 475 11.5 Economic Analysis of Waiting Lines 476 11.6 Other Waiting Line Models 478 11.7 Single-Server Waiting Line Model with Poisson Arrivals and Arbitrary Service Times 479 Operating Characteristics for the M/G/1 Model 479 Constant Service Times 480 Contents xv 11.8 Multiple-Server Model with Poisson Arrivals, Arbitrary Service Times, and No Waiting Line 481 Operating Characteristics for the M/G/k Model with Blocked Customers Cleared 481 11.9 Waiting Line Models with Finite Calling Populations 483 Operating Characteristics for the M/M/1 Model with a Finite Calling Population 483 Summary 486 Glossary 487 Problems 487 Case Problem 1 Regional Airlines 494 Case Problem 2 Office Equipment, Inc. 495 Chapter 12 Simulation 497 12.1 What-If Analysis 499 Sanotronics 499 Base-Case Scenario 499 Worst-Case Scenario 500 Best-Case Scenario 500 12.2 Simulation of Sanotronics Problem 500 Use of Probability Distributions to Represent Random Variables 501 Generating Values for Random Variables with Excel 502 Executing Simulation Trials with Excel 506 Measuring and Analyzing Simulation Output 507 12.3 Inventory Simulation 510 Simulation of the Butler Inventory Problem 512 12.4 Waiting Line Simulation 514 Black Sheep Scarves 515 Customer (Scarf) Arrival Times 515 Customer (Scarf) Service (Inspection) Times 515 Simulation Model 516 Simulation of Black Sheep Scarves 519 Simulation with Two Quality Inspectors 520 Simulation Results with Two Quality Inspectors 521 12.5 Simulation Considerations 523 Verification and Validation 523 Advantages and Disadvantages of Using Simulation 524 Summary 524 Summary of Steps for Conducting a Simulation Analysis 525 Glossary 525 Problems 526 Case Problem 1 Four Corners 532 Case Problem 2 Harbor Dunes Golf Course 534 Case Problem 3 County Beverage Drive-Thru 535 Appendix 12.1 Probability Distributions for Random Variables 537 Appendix 12.2 Simulation with Analytic Solver 540 xvi Contents Chapter 13 Decision Analysis 543 13.1 Problem Formulation 545 Influence Diagrams 545 Payoff Tables 546 Decision Trees 546 13.2 Decision Making Without Probabilities 547 Optimistic Approach 548 Conservative Approach 548 Minimax Regret Approach 549 13.3 Decision Making with Probabilities 550 Expected Value of Perfect Information 553 13.4 Risk Analysis and Sensitivity Analysis 554 Risk Analysis 554 Sensitivity Analysis 555 13.5 Decision Analysis with Sample Information 559 Influence Diagram 559 Decision Tree 560 Decision Strategy 562 Risk Profile 564 Expected Value of Sample Information 568 Efficiency of Sample Information 568 13.6 Computing Branch Probabilities with Bayes’ Theorem 568 13.7 Utility Theory 572 Utility and Decision Analysis 574 Utility Functions 577 Exponential Utility Function 580 Summary 582 Glossary 582 Problems 584 Case Problem 1 Property Purchase Strategy 597 Case Problem 2 Lawsuit Defense Strategy 599 Case Problem 3 Rob’s Market 600 Case Problem 4 College Softball Recruiting 601 Appendix 13.1 Decision Trees with Analytic Solver 602 Chapter 14 Multicriteria Decisions 613 14.1 Goal Programming: Formulation and Graphical Solution 614 Developing the Constraints and the Goal Equations 615 Developing an Objective Function with Preemptive Priorities 616 Graphical Solution Procedure 617 Goal Programming Model 620 14.2 Goal Programming: Solving More Complex Problems 621 Suncoast Office Supplies Problem 621 Formulating the Goal Equations 622 Formulating the Objective Function 623 Computer Solution 624 Contents xvii 14.3 Scoring Models 626 14.4 Analytic Hierarchy Process 630 Developing the Hierarchy 631 14.5 Establishing Priorities Using AHP 631 Pairwise Comparisons 632 Pairwise Comparison Matrix 633 Synthesization 635 Consistency 636 Other Pairwise Comparisons for the Car Selection Problem 637 14.6 Using AHP to Develop an Overall Priority Ranking 639 Summary 641 Glossary 642 Problems 642 Case Problem 1 EZ Trailers, Inc. 651 Appendix 14.1 Scoring Models with Excel 652 Chapter 15 Time Series Analysis and Forecasting 654 15.1 Time Series Patterns 656 Horizontal Pattern 656 Trend Pattern 657 Seasonal Pattern 660 Trend and Seasonal Pattern 660 Cyclical Pattern 660 Selecting a Forecasting Method 662 15.2 Forecast Accuracy 663 15.3 Moving Averages and Exponential Smoothing 668 Moving Averages 668 Weighted Moving Averages 671 Exponential Smoothing 672 15.4 Linear Trend Projection 675 15.5 Seasonality 679 Seasonality Without Trend 679 Seasonality with Trend 682 Models Based on Monthly Data 684 Summary 685 Glossary 685 Problems 686 Case Problem 1 Forecasting Food and Beverage Sales 693 Case Problem 2 Forecasting Lost Sales 694 Appendix 15.1 Forecasting with Excel Data Analysis Tools 695 Appendix 15.2 Using the Excel Forecast Sheet 703 Chapter 16 Markov Processes 16-1 On Website 16.1 Market Share Analysis 16-2 16.2 Accounts Receivable Analysis 16-10 Fundamental Matrix and Associated Calculations 16-11 Establishing the Allowance for Doubtful Accounts 16-12 xviii Contents Summary 16-14 Glossary 16-15 Problems 16-15 Case Problem 1 Dealer’s Absorbing State Probabilities in Blackjack 16-20 Appendix 16.1 Matrix Notation and Operations 16-21 Appendix 16.2 Matrix Inversion with Excel 16-24 Chapter 17 Linear Programming: Simplex Method 17-1 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 17-6 17.3 Setting up the Initial Simplex Tableau 17-7 17.4 Improving the Solution 17-9 17.5 Calculating the Next Tableau 17-11 Interpreting the Results of an Iteration 17-13 Moving Toward a Better Solution 17-14 Summary of the Simplex Method 17-16 17.6 Tableau Form: The General Case 17-17 Greater-Than-or-Equal-to Constraints 17-17 Equality Constraints 17-21 Eliminating Negative Right-Hand-Side Values 17-22 Summary of the Steps to Create Tableau Form 17-22 17.7 Solving a Minimization Problem 17-24 17.8 Special Cases 17-26 Infeasibility 17-26 Unboundedness 17-27 Alternative Optimal Solutions 17-28 Degeneracy 17-29 Summary 17-31 Glossary 17-32 Problems 17-33 Chapter 18 Simplex-Based Sensitivity Analysis and Duality 18-1 On Website 18.1 Sensitivity Analysis with the Simplex Tableau 18-2 Objective Function Coefficients 18-2 Right-Hand-Side Values 18-6 18.2 Duality 18-12 Economic Interpretation of the Dual Variables 18-14 Using the Dual to Identify the Primal Solution 18-16 Finding the Dual of Any Primal Problem 18-16 Contents xix Summary 18-18 Glossary 18-18 Problems 18-19 Chapter 19 Solution Procedures for Transportation and Assignment Problems 19-1 On Website 19.1 Transportation Simplex Method: A Special-Purpose Solution Procedure 19-2 Phase I: Finding an Initial Feasible Solution 19-3 Phase II: Iterating to the Optimal Solution 19-6 Summary of the Transportation Simplex Method 19-14 Problem Variations 19-16 19.2 Assignment Problem: A Special-Purpose Solution Procedure 19-17 Finding the Minimum Number of Lines 19-19 Problem Variations 19-20 Glossary 19-23 Problems 19-24 Chapter 20 Minimal Spanning Tree 20-1 On Website 20.1 A Minimal Spanning Tree Algorithm 20-2 Glossary 20-5 Problems 20-5 Case Problem Hinds County Realty Partners, Inc. 20-7 Chapter 21 Dynamic Programming 21-1 On Website 21.1 A Shortest-Route Problem 21-2 21.2 Dynamic Programming Notation 21-6 21.3 The Knapsack Problem 21-9 21.4 A Production and Inventory Control Problem 21-15 Summary 21-19 Glossary 21-20 Problems 21-20 Case Problem Process Design 21-24 Appendices 711 Appendix A Building Spreadsheet Models 712 Appendix B Areas for the Standard Normal Distribution 741 Appendix C Values of e2 743 Appendix D References and Bibliography 744 Appendix E Self-Test Solutions and Answers to Even-Numbered Exercises On Website Index 747 Preface We are very excited to publish the fifteenth edition of a text that has been a leader in the field for over 25 years. The purpose of this fifteenth 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 manage- ment science in their organizations. An Introduction to Management Science: Quantiative Approaches to Decision Making, 15e 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 demonstrating not only how the procedure works, but also how it contributes to the decision-making process. From the 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 un- derstandable 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 chal- lenge. 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 cov- ered 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 FIFTEENTH EDITION We are very excited about the changes in the fifteenth edition of Management Science and want to explain them and why they were made. Many changes have been made throughout the text in response to suggestions from instructors and students. While we cannot list all these changes, we highlight the more significant revisions. Updated Chapter 12: Simulation The most substantial content change in this latest edition involves the coverage of simula- tion. We maintain an intuitive introduction by continuing to use the concepts of best-, worst-, and base-case scenarios, but we have added a more elaborate treatment of uncertainty by using Microsoft Excel to develop spreadsheet simulation models. Within the chapter, we explain how to construct a spreadsheet simulation model using only native Excel function- ality. The chapter also includes two new appendices. The first appendix describes several probability distributions commonly used in simulation and how to generate values from these distributions using native Excel commands. In the second appendix, we introduce an Excel add-in, Analytic Solver, which facilitates the construction and analysis of spreadsheet simulation models. Nine new problems are introduced, and several others have been updated to reflect the new simulation coverage. xxii Preface Other Content Changes A variety of other changes have been made throughout the text. Appendices 4.1 and 7.1 have been updated to reflect changes to Solver in Microsoft Excel 2016. An appendix has been added to Chapter 15 that discusses the Forecast Tool in Microsoft Excel 2016. In addition to updating Appendix A for Microsoft Excel 2016, we have added a section on conducting a what-if analysis using Data Tables and Goal Seek. Management Science in Action The Management Science in Action vignettes describe how the material covered in a chapter is used in practice. Some of these are provided by practitioners. Others are based on articles from publications such as Interfaces and OR/MS Today. We updated the text with nine new Management Science in Action vignettes in this edition. Cases and Problems The quality of the problems and case problems is an important feature of the text. In this edi- tion we have updated over 15 problems and added 3 new case problems. COMPUTER SOFTWARE INTEGRATION To make it easy for new users of Excel Solver or LINGO, we provide both Excel and LINGO files with the model formulation for every optimization problem that appears in the body of the text. The model files are well-documented and should make it easy for the user to under- stand the model formulation. The text is updated for Microsoft Excel 2016, but Excel 2010 and later versions allow all problems to be solved using the standard version of Excel Solver. For LINGO users, the text has been updated for LINGO 16.0. 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 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.  Preface xxiii 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. ANCILLARY TEACHING AND LEARNING MATERIALS For Students Print and online resources are available to help the student work more efficiently. LINGO. A link to download an educational version of the LINGO software is avail- able on the student website at www.cengagebrain.com. Analytic Solver. If using Analytic Solver with this text, you can receive the latest An- alytic Solver license at Frontline Systems—[email protected] or 775-831-0300. For Instructors Instructor support materials are available to adopters from the Cengage Learning customer ser- vice line at 800-423-0563 or through www.cengage.com. Instructor resources are available on the Instructor Companion Site, which can be found and accessed at login.cengage.com, including: Solutions Manual. The Solutions Manual, prepared by the authors, includes solu- tions for all problems in the text. Solutions to Case Problems. These are also prepared by the authors and contain solutions to all case problems presented in the text. PowerPoint Presentation Slides. The presentation slides contain a teaching outline that incorporates figures to complement instructor lectures. Test Bank. Cengage Learning Testing Powered by Cognero is a flexible, online sys- tem that allows you to: author, edit, and manage test bank content from multiple Cengage Learning solutions, create multiple test versions in an instant, deliver tests from your LMS, your classroom or wherever you want. The Test Bank is also available in Microsoft Word. CengageNOWv2 CengageNOWv2 is a powerful course management and online homework tool that pro- vides robust instructor control and customization to optimize the learning experience and meet desired outcomes. CengageNOWv2 features author-written homework from the textbook, integrated eBook, assessment options, and course management tools, including gradebook. For more information about instructor resources, please contact your Cengage Learning Consultant. xxiv Preface ACKNOWLEDGMENTS We owe a debt to many of our colleagues and friends whose names appear below 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. Art Adelberg Dan Matthews CUNY Queens College Trine University Joseph Bailey Aravind Narasipur University of Maryland Chennai Business School Ike C. Ehie Nicholas W. Twigg Kansas State University Coastal Carolina University John K. Fielding Julie Ann Stuart Williams University of Northwestern Ohio University of West Florida Subodha Kumar Mays Business School Texas A&M University We are also indebted to our Senior Product Team Manager, Joe Sabatino; our Senior Product Man- ager, Aaron Arnsparger; our Senior Marketing Manager, Nate Anderson; our Content Developer, Anne Merrill; our Senior Digital Content Designer, Brandon Foltz; our Content Project Manager, Jean Buttrom; and others at Cengage for their counsel and support during the preparation of this text. David R. Anderson Dennis J. Sweeney Thomas A. Williams Jeffrey D. Camm James J. Cochran Michael J. Fry Jeffrey W. Ohlmann About the Authors David R. Anderson. David R. Anderson is Professor 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. Profes- sor 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 excel- lence in teaching and excellence in service to student organizations. Professor Anderson has coauthored 10 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 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 30 articles and monographs in the area of management science and statistics. The National Science Foundation, IBM, Procter & Gam- ble, Federated Department Stores, Kroger, and Cincinnati Gas & Electric have funded his research, which has been published in Management Science, Operations Research, Math- ematical Programming, Decision Sciences, and other journals. Professor Sweeney has coauthored 10 textbooks in the areas of statistics, management science, linear programming, and production and operations management. Thomas A. Williams. Thomas A. Williams is Professor of Management Science in the Col- lege 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 Cincin- nati, 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 regres- sion and decision analysis. Professor Williams is the coauthor of 11 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. xxvi About the Authors Jeffrey D. Camm. Jeffrey D. Camm is the Inmar Presidential Chair and Associate Dean of Analytics in the School of Business at Wake Forest University. Born in Cincinnati, Ohio, he holds a B.S. from Xavier University (Ohio) and a Ph.D. from Clemson University. Prior to joining the faculty at Wake Forest, he was on the faculty of the University of Cincinnati. He has also 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 marketing. He has published his research in Science, Management Science, Operations Research, Interfaces, and other professional jour- nals. Dr. Camm was named the Dornoff Fellow of Teaching Excellence at the University of Cincinnati and he was the 2006 recipient of the INFORMS Prize for the Teaching of Opera- tions Research Practice. A firm believer in practicing what he preaches, he has served as an operations research consultant to numerous companies and government agencies. From 2005 to 2010 he served as editor-in-chief of Interfaces. James J. Cochran. James J. Cochran is Professor of Applied Statistics and the Rogers- Spivey Faculty Fellow in the Department of Information Systems, Statistics, and Manage- ment Science at The University of Alabama. Born in Dayton, Ohio, he holds a B.S., an M.S., and an M.B.A. from Wright State University and a Ph.D. from the University of Cincinnati. He has been a visiting scholar at Stanford University, Universidad de Talca, the University of South Africa, and Pole Universitaire Leonard de Vinci. Professor Cochran has published over 30 papers in the development and application of operations research and statistical methods. He has published his research in Management Science, The American Statistician, Communications in Statistics - Theory and Methods, European Journal of Operational Research, Journal of Combinatorial Optimization, and other professional journals. He was the 2008 recipient of the INFORMS Prize for the Teach- ing of Operations Research Practice and the 2010 recipient of the Mu Sigma Rho Statistical Education Award. Professor Cochran was elected to the International Statistics Institute in 2005 and named a Fellow of the American Statistical Association in 2011. He received the Founders Award from the American Statistical Association in 2014 and he received the Karl E. Peace Award for Outstanding Statistical Contributions for the Betterment of Society from the American Statistical Association in 2015. A strong advocate for effective operations re- search and statistics education as a means of improving the quality of applications to real problems, Professor Cochran has organized and chaired teaching effectiveness workshops in Montevideo, Uruguay; Cape Town, South Africa; Cartagena, Colombia; Jaipur, India; Buenos Aires, Argentina; Nairobi, Kenya; Buea, Cameroon; Osijek, Croatia; Kathmandu, Nepal; Havana, Cuba; and Ulaanbaatar, Mongolia. He has served as an operations research or statistical consultant to numerous companies and not-for-profit organizations. From 2007 to 2012 Professor Cochran served as editor-in-chief of INFORMS Transactions on Educa- tion, and he is on the editorial board of several journals including Interfaces, Significance, and ORiON. Michael J. Fry. Michael J. Fry is Professor and Head of the Department of Operations, Business Analytics, and Information Systems in the Carl H. Lindner College of Business at the University of Cincinnati. Born in Killeen, Texas, he earned a B.S. from Texas A&M University, and M.S.E. and Ph.D. degrees from the University of Michigan. He has been at the University of Cincinnati since 2002, where he has been named a Lindner Research Fellow and has served as Assistant Director and Interim Director of the Center for Business Analytics. He has also been a visiting professor at the Samuel Curtis Johnson Graduate School of Management at Cornell University and the Sauder School of Business at the University of British Columbia. Professor Fry has published over 20 research papers in journals such as Operations Research, M&SOM, Transportation Science, Naval Research Logistics, IIE Transactions, and Interfaces. His research interests are in applying quantitative management methods to About the Authors xxvii the areas of supply chain analytics, sports analytics, and public-policy operations. He has worked with many different organizations for his research, including Dell, Inc., Copeland Corporation, Starbucks Coffee Company, Great American Insurance Group, the Cincinnati Fire Department, the State of Ohio Election Commission, the Cincinnati Bengals, and the Cincinnati Zoo. In 2008, he was named a finalist for the Daniel H. Wagner Prize for Excel- lence in Operations Research Practice, and he has been recognized for both his research and teaching excellence at the University of Cincinnati. Jeffrey W. Ohlmann. Jeffrey W. Ohlmann is Associate Professor of Management Sciences and Huneke Research Fellow in the Tippie College of Business at the University of Iowa. Born in Valentine, Nebraska, he earned a B.S. from the University of Nebraska, and M.S. and Ph.D. degrees from the University of Michigan. He has been at the University of Iowa since 2003. Professor Ohlmann’s research on the modeling and solution of decision-making prob- lems has produced over twenty research papers in journals such as Operations Research, Mathematics of Operations Research, INFORMS Journal on Computing, Transportation Science, the European Journal of Operational Research, and Interfaces. He has collaborated with companies such as Transfreight, LeanCor, Cargill, the Hamilton County Board of Elec- tions, and three National Football League franchises. Due to the relevance of his work to industry, he was bestowed the George B. Dantzig Dissertation Award and was recognized as a finalist for the Daniel H. Wagner Prize for Excellence in Operations Research Practice.  An Introduction to Management Science Quantitative Approaches to Decision Making Fifteenth Edition  CHAPTER 1 Introduction CONTENTS 1.1 PROBLEM SOLVING AND 1.4 MODELS OF COST, REVENUE, DECISION MAKING AND PROFIT 1.2 QUANTITATIVE ANALYSIS Cost and Volume Models AND DECISION MAKING Revenue and Volume Models Profit and Volume Models 1.3 QUANTITATIVE ANALYSIS Breakeven Analysis Model Development Data Preparation 1.5 MANAGEMENT SCIENCE Model Solution TECHNIQUES Report Generation Methods Used Most Frequently A Note Regarding Implementation APPENDIX 1.1 USING EXCEL FOR BREAKEVEN ANALYSIS 2 Chapter 1 Introduction Management science, an approach to decision making based on the scientific method, makes extensive use of quantitative analysis. A variety of names exist for the body of knowledge involving quantitative approaches to decision making; in addition to manage- ment 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 develop- ment 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 prob- lems. The computer technology explosion continues; smart phones, tablets, and other mobile-computing devices can now be used to solve problems larger than those solved on mainframe computers in the 1990s. More recently, the explosive growth of data from sources such as smart phones and other personal-electronic devices provide access to much more data today than ever before. Addi- tionally, the Internet allows for easy sharing and storage of data, providing extensive access to a variety of users to the necessary inputs to management-science models. 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 application 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 AT&T Park, describes one of the most important applications of management science in the sports and entertainment industry. MANAGEMENT SCIENCE IN ACTION REVENUE MANAGEMENT AT AT&T PARK* Imagine the difficult position Russ Stanley, Vice of the product and uses operations research to President of Ticket Services for the San Francisco determine if the price should be changed to reflect Giants, found himself facing late in the 2010 base- these conditions. As the scheduled takeoff date for ball season. Prior to the season, his organization a flight nears, the cost of a ticket increases if seats adopted a dynamic approach to pricing its tickets for the flight are relatively scarce. On the other similar to the model successfully pioneered by hand, the airline discounts tickets for an approach- Thomas M. Cook and his operations research group ing flight with relatively few ticketed passengers. at American Airlines. Stanley desparately wanted Through the use of optimization to dynamically set the Giants to clinch a playoff birth, but he didn’t ticket prices, American Airlines generates nearly want the team to do so too quickly. $1 billion annually in incremental revenue. When dynamically pricing a good or service, an The management team of the San Francisco organization regularly reviews supply and demand Giants recognized similarities between their primary  1.1 Problem Solving and Decision Making 3 product (tickets to home games) and the primary “Our hearts are in our stomachs; we’re pacing product sold by airlines (tickets for flights) and watching these games.” adopted a similar revenue management system. Does revenue management and operations If a particular Giants’ game is appealing to fans, research work? Today, virtually every airline tickets sell quickly and demand far exceeds supply uses some sort of revenue-management system, as the date of the game approaches; under these and the cruise, hotel, and car rental industries conditions fans will be willing to pay more and the also now apply revenue-management methods. Giants charge a premium for the ticket. Similarly, As for the Giants, Stanley said dynamic pric- tickets for less attractive games are discounted to ing provided a 7% to 8% increase in revenue per reflect relatively low demand by fans. This is why seat for Giants’ home games during the 2010 Stanley found himself in a quandary at the end of season. Coincidentally, the Giants did win the the 2010 baseball season. The Giants were in the National League West division on the last day of middle of a tight pennant race with the San Diego the season and ultimately won the World Series. Padres that effectively increased demand for tick- Several professional sports franchises are now ets to Giants’ games, and the team was actually looking to the Giants’ example and considering scheduled to play the Padres in San Francisco for implementation of similar dynamic ticket-pricing the last three games of the season. While Stan- systems. ley certainly wanted his club to win its division and reach the Major League Baseball playoffs, he *Based on Peter Horner, “The Sabre Story,” OR/MS also recognized that his team’s revenues would Today (June 2000); Ken Belson, “Baseball Tickets Too be greatly enhanced if it didn’t qualify for the Much? Check Back Tomorrow,” NewYork Times.com playoffs until the last day of the season. “I guess (May 18, 2009); and Rob Gloster, “Giants Quadruple financially it is better to go all the way down to the Price of Cheap Seats as Playoffs Drive Demand,” last game,” Stanley said in a late season interview. Bloomberg Business-week (September 30, 2010). 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 problem-solving process involves the following seven steps: 1. Identify and define the problem. 2. Determine the set of alternative solutions. 3. Determine the criterion or criteria that will be used to evaluate the alternatives. 4. Evaluate the alternatives. 5. Choose an alternative. 6. Implement the selected alternative. 7. 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 problem-solving 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 compa- nies 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. Accept the position in Rochester. 2. Accept the position in Dallas. 3. Accept the position in Greensboro. 4. 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 4 Chapter 1 Introduction TABLE 1.1 DATA FOR THE JOB EVALUATION DECISION-MAKING PROBLEM Starting Potential for Job Alternative Salary Advancement Location 1. Rochester $48,500 Average Average 2. Dallas $46,000 Excellent Good 3. Greensboro $46,000 Good Excellent 4. Pittsburgh $47,000 Average Good 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 start- ing 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 care- ful 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. Define the problem. 2. Identify the alternatives. 3. Determine the criteria. 4. Evaluate the alternatives. 5. Choose an alternative. Note that missing from this list are the last two steps in the problem-solving process: imple- menting the selected alternative and evaluating the results to determine whether a satisfac- tory 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 “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.  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 Decision the Making Criteria Problem Evaluate Solving the Alternatives Choose an Alternative Implement Decision the Decision Evaluate the Results FIGURE 1.2 AN ALTERNATE CLASSIFICATION OF THE DECISION-MAKING PROCESS Structuring the Problem Analyzing the Problem Define Identify Determine Evaluate Choose the the the the an Problem Alternatives Criteria Alternatives Alternative 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 problems or if the problem is relatively simple, heavy emphasis may be placed upon a qual- itative 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 quantita- tive facts or data associated with the problem and develop mathematical expressions that 6 Chapter 1 Introduction FIGURE 1.3 THE ROLE OF QUALITATIVE AND QUANTITATIVE ANALYSIS Analyzing the Problem Qualitative Analysis Structuring the Problem Define Identify Determine Summary Make the the the and the Problem Alternatives Criteria Evaluation Decision Quantitative Analysis 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 quan- titative 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: Try Problem 4 to test your 1. The problem is complex, and the manager cannot develop a good solution without understanding of why quan- the aid of quantitative analysis. titative approaches might 2. The problem is especially important (e.g., a great deal of money is involved), and the be needed in a particular problem. 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 quan- titative procedures to make routine decision recommendations. 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 prob- lem, 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 scien- tist 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  1.3 Quantitative Analysis 7 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 5 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 characteris- tics 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 5 3) would provide a profit of P 5 10(3) 5 $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 deci- sions that cause the model airplane to crash or a mathematical model to project a $10,000 loss can be avoided in the real situation. Herbert A. Simon, a Nobel The value of model-based conclusions and decisions is dependent on how well the Prize winner in economics model represents the real situation. The more closely the model airplane represents the real and an expert in deci- airplane, the more accurate the conclusions and predictions will be. Similarly, the more sion making, said that a mathematical model does closely the mathematical model represents the company’s true profit–volume relationship, not have to be exact; it just the more accurate the profit projections will be. has to be close enough to Because this text deals with quantitative analysis based on mathematical models, let us provide better results than look more closely at the mathematical modeling process. When initially considering a mana- can be obtained by common gerial problem, we usually find that the problem definition phase leads to a specific objec- sense. tive, 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 5 10x would be an objective function 8 Chapter 1 Introduction for a firm attempting to maximize profit. A production capacity constraint would be neces- sary if, for instance, 5 hours are required to produce each unit and only 40 hours of produc- tion 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 5 10x objective function subject to ss.t.d 5x # 40 x$0 6 constraints 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 5 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 ($10), the production time per unit (5 hours), and the production capacity (40 hours) are environmental factors that are not under the control of the manager or decision maker. Such environmental factors, which can affect both the objective function and the constraints, are referred to as uncontrollable inputs to the model. Inputs that are completely controlled or determined by the decision maker are referred to as controllable inputs to the model. In the example given, the production quantity x is the controllable input to the model. Controllable inputs are the decision alternatives specified by the manager and thus are also referred to as the decision variables of the model. Once all controllable and uncontrollable inputs are specified, the objective function and constraints can be evaluated and the output of the model determined. In this sense, the output of the model is simply the projection of what would happen if those particular envi- ronmental factors and decisions occurred in the real situation. A flowchart of how control- lable and uncontrollable inputs are transformed by the mathematical model into output is shown in Figure 1.4. A similar flowchart showing the specific details of the production model is shown in Figure 1.5. FIGURE 1.4 FLOWCHART OF THE PROCESS OF TRANSFORMING MODEL INPUTS INTO OUTPUT Uncontrollable Inputs (Environmental Factors) Controllable Mathematical Output Inputs Model (Projected Results) (Decision Variables)  1.3 Quantitative Analysis 9 FIGURE 1.5 FLOWCHART FOR THE PRODUCTION MODEL Uncontrollable Inputs 10 Profit per Unit ($) 5 Production Time per Unit (Hours) 40 Production Capacity (Hours) Max 10 (8) Value for Profit = $80 s.t. the Production 5 (8) # 40 Time Used = 40 Quantity (x = 8) 8 $0 Hours Controllable Mathematical Output Input Model 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 prob- lem 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 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 to the decision maker, the model is referred to as a stochastic or probabilistic model. An uncontrollable input to many produc- tion 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 deter- ministic. 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 10 Chapter 1 Introduction 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 equa- tions 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 5 profit per unit a 5 production time in hours per unit b 5 production capacity in hours the model development step of the production problem would result in the following general model: Max cx s.t. ax # b x$0 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 has been 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 data- base 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 vari- ous 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 produc- tion capacity constraint. If a particular decision alternative does not satisfy one or more of  1.3 Quantitative Analysis 11 TABLE 1.2 TRIAL-AND-ERROR SOLUTION FOR THE PRODUCTION MODEL OF FIGURE 1.5 Decision Alternative Total Feasible (Production Quantity) Projected Hours of Solution? x Profit Production (Hours Used # 40) 0 0 0 Yes 2 20 10 Yes 4 40 20 Yes 6 60 30 Yes 8 80 40 Yes 10 100 50 No 12 120 60 No the model constraints, the decision alternative is rejected as being infeasible, regardless of