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Nagraj (Raju) Balakrishnan, Barry Render, Ralph M. Stair, Chuck Munson
Managerial Decision Modeling

Nagraj (Raju) Balakrishnan, Barry Render,
Ralph M. Stair, Chuck Munson
Managerial
Decision Modeling
Business Analytics with Spreadsheets
Fourth Edition
PRESS

ISBN 978-1-5015-1510-1
e-ISBN (PDF) 978-1-5015-0620-8
e-ISBN (EPUB) 978-1-5015-0631-4
Library of Congress Cataloging-in-Publication Data
A CIP catalog record for this book has been applied for at the Library of Congress.
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available on the Internet at http://dnb.dnb.de.
© 2017 Walter de Gruyter Inc., Boston/Berlin
Printing and binding: CPI books GmbH, Leck
♾ Printed on acid-free paper
Printed in Germany
www.degruyter.com

To my children, Nitin and Nandita, and most of all to my darling wife, Meena,
my rock – N.B.
To Donna, Charlie, and Jesse – B.R.
To Ken Ramsing and Alan Eliason – R.M.S.
To my wife Kim and my sons Christopher and Mark
for their unwavering support and encouragement – C.M

Acknowledgments
The authors would like to thank Howard Weiss for his outstanding job in developing
ExcelModules, and for adding a new Mac version. Raju Balakrishnan would like to
thank his daughter Nandita for her diligent help in reviewing and revising many of the
end-of-chapter exercises, Nesreen El-Rayes for her research in updating the Decision
Modeling In Action boxes, and Megan Seccombe for her valuable assistance with the
revised figures and Excel screenshots.
Several people worked very hard to bring the book through the publication
process. We would like to gratefully acknowledge the outstanding help provided by
Jeffrey Pepper, Megan Lester, Caitlyn Nardozzi, John Woolsey, Angie MacAllister, and
Mark Watanabe who worked on the book for De|G PRESS. Thank you all!
It is no secret that unlike courses in functional areas such as finance, marketing,
and accounting, decision modeling courses always face an uphill battle in getting stu-
dents interested and excited about the material (despite its increased value in today’s
business world). We hope that this book will be an ally to all in this endeavor.

About the Authors
Raju Balakrishnan serves as Dean of the College of Business (COB) at the Univer-
sity of Michigan-Dearborn. Prior to that, he was Senior Associate Dean in the College
of Business and Behavioral Science at Clemson University, where he served on the
faculty and administration for nearly 19 years. He has also served on the faculty at
Tulane University and has taught in the Executive MBA program at Tulane and the
University of Georgia.
Dr. Balakrishnan earned his Ph.D. in Management from Purdue University, M.S.
in Mechanical Engineering from the University of Kentucky, and a B.E. with honors in
Mechanical Engineering from the University of Madras in India.
He has published extensively in leading academic journals such as Decision
Sciences, Production and Operations Management, European Journal of Operatio-
nal Research, IIE Transactions, and Computers & Operations Research. His teaching
interests include spreadsheet-based decision modeling, statistics, and operations
management. He serves as senior departmental editor of Production and Operations
Management and served as secretary of the Production and Operations Management
Society during 2006–08. He has been recognized multiple times for teaching excel-
lence at both Clemson and Tulane, and has received several research awards inclu-
ding best paper awards from the Decision Sciences Institute and the Institute of Indus-
trial Engineers.
Barry Render is the Charles Harwood Professor of Management Science Emeritus
at the Crummer Graduate School of Business at Rollins College. He received his M.S.
in Operations Research and his Ph.D. in Quantitative Analysis at the University of
Cincinnati. He previously taught at George Washington University, the University of
New Orleans, Boston University, and George Mason University, where he held the GM
Foundation Professorship in Decision Sciences and was Chair of the Decision Science
Department. Dr. Render also worked in the aerospace industry.
Professor Render has co-authored ten textbooks with Pearson/Prentice Hall,
including Quantitative Analysis for Management, Operations Management, Princip-
les of Operations Management, Service Management, Introduction to Management
Science, and has published more than one hundred articles on a variety of manage-
ment topics.
Dr. Render has also been honored as an AACSB Fellow and named as a Senior
Fulbright Scholar in 1982 and in 1993. He was twice vice-president of the Decision
Science Institute Southeast Region and served as Software Review Editor for Decision
Line from 1989 to 1995. He was Editor of the New York Times Operations Management
special issues from 1996 to 2001. Finally, Professor Render has been involved in con-
sulting for for many organizations, including NASA; the FBI; the U.S. Navy; Fairfax
County, Virginia; and C&P Telephone.
Before retiring in 2009, he taught operations management courses in Rollins
College’s MBA programs. In 1995 and in 2009 he was named as that school’s Professor

x About The Authors
of the Year, and in 1996 was selected by Roosevelt University to receive the St. Claire
Drake Award for Outstanding Scholarship.
Ralph Stair is Professor Emeritus of Management Information Systems in the College
of Business at Florida State University. He received a B.S. in Chemical Engineering
from Purdue University and an MBA from Tulane University. He received his Ph.D. in
operations management from the University of Oregon.
He has taught at the University of Oregon, the University of Washington, the Uni-
versity of New Orleans, and Florida State University. He has twice taught in Florida
State University’s Study Abroad Program in London.
Dr. Stair has published numerous articles and books, has funded a student scho-
larship at St. Johns Northwestern Military Academy, and endowed a faculty prize in
innovative education at Florida State University.
Chuck Munson is a Professor of Operations Management and Ph.D. Program Director
in the Carson College of Business at Washington State University. His received his
BSBA summa cum laude in finance, along with his MSBA and Ph.D. in operations
management, from Washington University in St. Louis. For two years, he served as
Associate Dean for Graduate Programs in Business at Washington State U. He also
worked for three years as a financial analyst for Contel Telephone Corporation.
Professor Munson serves as a senior editor for Production and Operations Manage-
ment, and he serves on the editorial review board of four other journals. He has pub-
lished more than 25 articles in such journals as Production and Operations Manage-
ment, IIE Transactions, Decision Sciences, Naval Research Logistics, European Journal
of Operational Research, Journal of the Operational Research Society, International
Journal of Production Economics, and Annals of Operations Research. He is a coauthor
of Operations Management: Sustainability and Supply Chain Management (12 ed.). He
is also editor of The Supply Chain Management Casebook: Comprehensive Coverage
and Best Practices in SCM, and he has co-authored the research monograph Quantity
Discounts: An Overview and Practical Guide for Buyers and Sellers.
Dr. Munson teaches Business Modeling with Spreadsheets, along with opera-
tions management courses at the undergraduate, MBA, and Ph.D. levels at Washing-
ton State U. His major awards include winning the Sahlin Faculty Excellence Award
for Instruction, the top teaching award at Washington State University 2016); being
a Founding Board Member of the Washington State University President’s Teaching
Academy (2004); winning the WSU College of Business Outstanding Teaching Award
(2001 and 2015), Research Award (2004), and Service Award (2009 and 2013); and
being named the WSU MBA Professor of the Year (2000 and 2008).

Contents
Chapter 1: Introduction to Managerial Decision Modeling 1
1.1 What is Decision Modeling? 2
1.2 Types of Decision Models 3
Deterministic Models 3
Probabilistic Models 4
Quantitative versus Qualitative Data 5
Using Spreadsheets in Decision Modeling 5
1.3 Steps Involved in Decision Modeling 6
Step 1: Formulation 7
Step 2: Solution 9
Step 3: Interpretation and Sensitivity Analysis 10
1.4 Spreadsheet Example of a Decision Model: Tax Computation 11
1.5 Spreadsheet Example of a Decision Model: Break-Even Analysis 16
Using Goal Seek to Find the Break-Even Point 18
1.6 Possible Problems in Developing Decision Models 21
Defining the Problem 21
Developing a Model 22
Acquiring Input Data 22
Developing a Solution 23
Testing the Solution 23
Analyzing the Results 23
1.7 Implementation—Not Just the Final Step 24
1.8 Summary 24
1.9 Exercises 27
Chapter 2: Linear Programming Models: Graphical and Computer Methods 33
2.1 Introduction 34
2.2 Developing a Linear Programming Model 35
Formulation 35
Solution 35
Interpretation and Sensitivity Analysis 36
Properties of a Linear Programming Model 36
Basic Assumptions of a Linear Programming Model 37
2.3 Formulating a Linear Programming Problem 38
Linear Programming Example: Flair Furniture Company 38
Decision Variables 39
The Objective Function 39
Constraints 40
Nonnegativity Constraints and Integer Values 41
Guidelines for Developing a Correct LP Model 41

xii Contents
2.4
Graphical Solution of a Linear Programming Problem with
Two Variables 43
Graphical Representation of Constraints 43
Painting Time Constraint 46
Feasible Region 47
Identifying an Optimal Solution by Using Level Lines 48
Identifying an Optimal Solution by Using All Corner Points 51
Comments on Flair Furniture’s Optimal Solution 52
Extension to Flair Furniture’s LP Model 52
2.5 A Minimization Linear Programming Problem 54
Holiday Meal Turkey Ranch 55
Graphical Solution of the Holiday Meal Turkey Ranch Problem 56
2.6 Special Situations in Solving Linear Programming Problems 58
Redundant Constraints 58
Infeasibility 59
Alternate Optimal Solutions 60
Unbounded Solution 61
2.7
Setting Up and Solving Linear Programming Problems Using Excel’s Solver
62
Using Solver to Solve the Flair Furniture Problem 63
The Objective Cell 65
Creating Cells for Constraint RHS Values 67
Entering Information in Solver 68
Using Solver to Solve Flair Furniture Company’s Modified Problem 76
Using Solver to Solve the Holiday Meal Turkey Ranch Problem 77
2.8
Algorithmic Solution Procedures for Linear Programming Problems 79
2.9 Summary 80
2.10 Exercises 85
Chapter 3:
Linear Programming Modeling Applications with Computer Analyses in
Excel 101
3.1 Using Linear Programming to Solve Real-World Problems 102
3.2 Manufacturing Applications 103
Product Mix Problem 103
Make-Buy Decision Problem 108
3.3 Marketing Applications 112
Media Selection Problem 112
Marketing Research Problem 113
3.4 Finance Applications 118
Portfolio Selection Problem 118
Alternate Formulations of the Portfolio Selection Problem 121
3.5 Employee Staffing Applications 123
Labor Planning Problem 123

Contents xiii
Extensions to the Labor Planning Problem 127
Assignment Problem 127
3.6 Transportation Applications 127
Vehicle Loading Problem 127
Expanded Vehicle Loading Problem—Allocation Problem 132
Transportation Problem 133
3.7 Blending Applications 134
Diet Problem 134
Blending Problem 136
3.8 Multiperiod Applications 141
Production Scheduling Problem 141
Sinking Fund Problem 147
3.9 Summary 151
3.10 Exercises 153
Chapter 4: Linear Programming Sensitivity Analysis 181
4.1 Importance of Sensitivity Analysis 182
Why Do We Need Sensitivity Analysis? 182
4.2 Sensitivity Analysis Using Graphs 183
Types of Sensitivity Analysis 185
Impact of Changes in an Objective Function Coefficient 185
Impact of Changes in a Constraint’s Right-Hand-Side Value 187
4.3 Sensitivity Analysis Using Solver Reports 193
Solver Reports 194
Sensitivity Report 195
Impact of Changes in a Constraint’s RHS Value 196
Impact of Changes in an Objective Function Coefficient 198
4.4 Sensitivity Analysis for a Larger Maximization Example 200
Anderson Home Electronics Example 200
Some Questions We Want Answered 203
4.5 Analyzing Simultaneous Changes by Using the 100% Rule 206
Simultaneous Changes in Constraint RHS Values 206
Simultaneous Changes in OFC Values 207
4.6 Pricing Out New Variables 207
Anderson’s Proposed New Product 207
4.7 Sensitivity Analysis for a Minimization Example 211
Burn-Off Diet Drink Example 211
Burn-Off’s Excel Solution 212
Answering Sensitivity Analysis Questions for Burn-Off 213
4.8 Summary 216
4.9 Exercises 218

xiv Contents
Chapter 5: Transportation, Assignment, and Network Models 239
5.1 Types of Network Models 239
Transportation Model 240
Transshipment Model 240
Assignment Model 240
Maximal-Flow Model 241
Shortest-Path Model 241
Minimal-Spanning Tree Model 241
Implementation Issues 241
5.2 Characteristics of Network Models 242
Types of Arcs 242
Types of Nodes 243
Common Characteristics 243
5.3 Transportation Model 244
LP Formulation for Executive Furniture’s Transportation Model 246
Solving the Transportation Model Using Excel 247
Unbalanced Transportation Models 249
An Application of the Transportation Model: Facility Location 251
5.4 Transportation Models with Max-Min and Min-Max Objectives 252
5.5 Transshipment Model 256
Executive Furniture Company Example—Revisited 256
LP Formulation for Executive Furniture’s Transshipment Model 256
Lopez Custom Outfits—A Larger Transshipment Example 258
LP Formulation for Lopez Custom Outfits Transshipment Model 259
5.6 Assignment Model 262
Fix-It Shop Example 263
Solving Assignment Models 264
LP Formulation for Fix-It Shop’s Assignment Model 266
5.7 Maximal-Flow Model 268
Road System in Waukesha, Wisconsin 268
LP Formulation for Waukesha Road System’s Maximal-Flow Model 269
5.8 Shortest-Path Model 272
Ray Design Inc. Example 273
LP Formulation for Ray Design Inc.’s Shortest-Path Model 274
5.9 Minimal-Spanning Tree Model 276
Lauderdale Construction Company Example 276
5.10 Summary 279
5.11 Exercises 282
Chapter 6: Integer, Goal, and Nonlinear Programming Models 303
6.1 Models That Relax Linear Programming Conditions 304

Contents xv
Integer Programming Models 304
Goal Programming Models 305
Nonlinear Programming Models 305
6.2 Models with General Integer Variables 305
Harrison Electric Company 306
Using Solver to Solve Models with General Integer Variables 309
Solver Options 313
Should We Include Integer Requirements in a Model? 315
6.3 Models with Binary Variables 317
Portfolio Selection at Simkin and Steinberg 317
Set-Covering Problem at Sussex County 322
6.4 Mixed Integer Models: Fixed-Charge Problems 325
Locating a New Factory for Hardgrave Machine Company 326
6.5 Goal Programming Models 331
Goal Programming Example: Wilson Doors Company 331
Solving Goal Programming Models with Weighted Goals 335
Solving Goal Programming Models with Ranked Goals 338
Comparing the Two Approaches for Solving GP Models 344
6.6 Nonlinear Programming Models 344
Why Are NLP Models Difficult to Solve? 345
Solving Nonlinear Programming Models Using Solver 347
Computational Procedures for Nonlinear Programming Problems 354
6.7 Summary 354
6.8 Exercises 357
Chapter 7: Project Management 383
7.1 Planning, Scheduling, and Controlling Projects 384
Phases in Project Management 384
Use of Software Packages in Project Management 387
7.2 Project Networks 387
Identifying Activities 388
Identifying Activity Times and Other Resources 389
Project Management Techniques: PERT and CPM 389
Project Management Example: General Foundry, Inc. 391
Drawing the Project Network 392
7.3 Determining the Project Schedule 394
Forward Pass 396
Backward Pass 398
Calculating Slack Time and Identifying the Critical Path(s) 399
Total Slack Time versus Free Slack Time 401
7.4 Variability in Activity Times 402
PERT Analysis 403

xvi Contents
Probability of Project Completion 406
Determining Project Completion Time for a Given Probability 408
Variability in Completion Time of Noncritical Paths 409
7.5 Managing Project Costs and Other Resources 410
Planning and Scheduling Project Costs: Budgeting Process 410
Monitoring and Controlling Project Costs 413
Managing Other Resources 415
7.6 Project Crashing 417
Crashing General Foundry’s Project (Hand Calculations) 418
Crashing General Foundry’s Project Using Linear Programming 421
Using Linear Programming to Determine Earliest and Latest Starting Times
424
7.7 Summary 425
7.8 Exercises 429
Chapter 8: Decision Analysis 449
8.1 What is Decision Analysis? 450
8.2 The Five Steps in Decision Analysis 450
Thompson Lumber Company Example 451
8.3 Types of Decision-Making Environments 453
8.4 Decision Making Under Uncertainty 454
Maximax Criterion 455
Maximin Criterion 455
Criterion of Realism (Hurwicz) 456
Equally Likely (Laplace) Criterion 457
Minimax Regret Criterion 457
Using Excel to Solve Decision-Making Problems under Uncertainty 458
8.5 Decision Making Under Risk 461
Expected Monetary Value 461
Expected Opportunity Loss 462
Expected Value of Perfect Information 463
Using Excel to Solve Decision-Making Problems under Risk 464
8.6 Decision Trees 466
Folding Back a Decision Tree 467
8.7 Decision Trees for Multistage Decision-Making Problems 469
A Multistage Decision-Making Problem for Thompson Lumber 469
Expanded Decision Tree for Thompson Lumber 470
Folding Back the Expanded Decision Tree for Thompson Lumber 472
Expected Value of Sample Information 474
8.8 Estimating Probability Values Using Bayesian Analysis 475
Calculating Revised Probabilities 476
Potential Problems in Using Survey Results 478
8.9 Utility Theory 478

Contents xvii
Measuring Utility and Constructing a Utility Curve 479
Utility as a Decision-Making Criterion 483
8.10 Summary 485
8.11 Exercises 488
Chapter 9: Queuing Models 509
9.1 The Importance of Queuing Theory 510
Approaches for Analyzing Queues 510
9.2 Queuing System Costs 511
9.3 Characteristics of a Queuing System 513
Arrival Characteristics 513
Queue Characteristics 516
Service Facility Characteristics 516
Measuring the Queue’s Performance 519
Kendall’s Notation for Queuing Systems 520
Variety of Queuing Models Studied Here 520
9.4 M/M/1 Queuing System 521
Assumptions of the M/M/1 Queuing Model 521
Operating Characteristic Equations for an M/M/1 Queuing System 522
Arnold’s Muffler Shop Example 523
Using ExcelModules for Queuing Model Computations 524
Cost Analysis of the Queuing System 527
Increasing the Service Rate 528
9.5 M/M/s Queuing System 529
Operating Characteristic Equations for an M/M/s Queuing System 530
Arnold’s Muffler Shop Revisited 531
9.6 M/D/1 Queuing System 533
Operating Characteristic Equations for an M/D/1 Queuing System 534
Garcia-Golding Recycling, Inc. 535
9.7 M/G/1 Queuing System 536
Operating Characteristic Equations for an M/G/1 Queuing System 537
Meetings with Professor Crino 537
Using Excel’s Goal Seek to Identify Required Model Parameters 539
9.8 M/M/S/∞/N Queuing System 540
Operating Characteristic Equations for the Finite Population Queuing
System 542
Department of Commerce Example 543
9.9 More Complex Queuing Systems 546
9.10 Summary 547
9.11 Exercises 551

xviii Contents
Chapter 10: Simulation Modeling 565
10.1 Why Create a Simulation? 566
Simulation Basics 566
Advantages and Disadvantages of Simulation 568
10.2 Monte Carlo Simulation 569
Step 1: Establish a Probability Distribution for Each Variable 570
Step 2: Simulate Values from the Probability Distributions 571
Step 3: Repeat the Process for a Series of Replications 573
10.3 Role of Computers in Simulation 574
Types of Simulation Software Packages 575
Random Generation from Some Common Probability Distributions Using
Excel 575
10.4 Simulation Model to Compute Expected Profit 582
Setting Up the Model 583
Replication by Copying the Model 585
Replication Using Data Table 586
10.5 Simulation Model of an Inventory Problem 591
Simkin’s Hardware Store 591
Computation of Costs 596
Using Scenario Manager to Include Decisions in a Simulation Model 598
10.6 Simulation Model of a Queuing Problem 601
Denton Savings Bank 601
10.7 Simulation Model of a Revenue Management Problem 605
Judith’s Airport Limousine Service 605
Replicating the Model Using Data Table and Scenario Manager 608
10.8 Other Types of Simulation Models 610
Operational Gaming 610
Systems Simulation 610
10.9 Summary 611
10.10 Exercises 615
Chapter 11: Forecasting Models 647
11.1 What is Forecasting? 648

Contents xix
11.2 Types of Forecasts 649
Qualitative Models 650
Time-Series Models 650
Causal Models 650
11.3 Qualitative Forecasting Models 650
11.4 Measuring Forecast Error 651
11.5 Basic Time-Series Forecasting Models 652
Components of a Time Series 653
Stationary and Nonstationary Time-Series Data 654
Moving Averages 654
Using ExcelModules for Forecasting Model Computations 655
Weighted Moving Averages 659
Exponential Smoothing 664
11.6 Trend and Seasonality in Time-Series Data 668
Linear Trend Analysis 668
Scatter Chart 669
Least-Squares Procedure for Developing a Linear Trend Line 672
Seasonality Analysis 676
11.7 Decomposition of a Time Series 678
Multiplicative Decomposition Example: Sawyer Piano House 678
Using ExcelModules for Multiplicative Decomposition 679
11.8 Causal Forecasting Models: Simple and Multiple Regression 684
Causal Simple Regression Model 684
Causal Simple Regression Using ExcelModules 686
CausalSimple Regression Using Excel’s AnalysisToolPak(Data Analysis) 692
Causal Multiple Regression Model 696
Causal Multiple Regression Using ExcelModules 696
Causal Multiple Regression Using Excel’s Analysis ToolPak (Data Analysis)
700
11.9 Summary 705
11.10 Exercises 710
Appendix A: Probability Concepts and Applications 731
A.1 Fundamental Concepts 731
Types of Probability 732
A.2 Mutually Exclusive and Collectively Exhaustive Events 733
Adding Mutually Exclusive Events 734
Law of Addition for Events that Are Not Mutually Exclusive 735
A.3 Statistically Independent Events 736
A.4 Statistically Dependent Events 737
A.5 Revising Probabilities with Bayes’ Theorem 740
General Form of Bayes’ Theorem 741
A.6 Further Probability Revisions 742

xx Contents
A.7 Random Variables 743
A.8 Probability Distributions 745
Probability Distribution of a Discrete Random Variable 745
Expected Value of a Discrete Probability Distribution 747
Variance of a Discrete Probability Distribution 747
Probability Distribution of a Continuous Random Variable 748
A.9 The Normal Distribution 750
Area under the Normal Curve 751
Using the Standard Normal Table 752
Haynes Construction Company Example 753
A.10 The Exponential Distribution 756
A.11 The Poisson Distribution 757
A.12 Summary 758
A.13 Exercises 760
Appendix B: Useful Excel 2016 Commands and Procedures for Installing
ExcelModules 767
1B.1 Introduction 767
B.2 Getting Started 767
Organization of a Worksheet 768
Navigating through a Worksheet 769
B.3 The Ribbon, Toolbars, and Tabs 769
Excel Help 774
B.4 Working with Worksheets 775
B.5 Using Formulas and Functions 775
Copying Formulas 779
Errors in Using Formulas and Functions 779
B.6 Printing Worksheets 780
B.7 Excel Options and Add-Ins 781
B.8 ExcelModules 784
Installing ExcelModules 784
Running ExcelModules 784
ExcelModules Help and Options 786
Appendix C: Areas Under The Standard Normal Curve 787
Appendix D: Brief Solutions to All Odd-Numbered End-Of-Chapter
Problems 789
Index 795

Preface
In recent years, the use of spreadsheets to teach decision modeling (alternatively referred to
as business analytics, management science, operations research, and quantitative analysis)
has become standard practice in many business programs. This emphasis has revived inte-
rest in the field significantly, and several books have attempted to discuss spreadsheet-based
decision modeling. However, some of these books have become too spreadsheet oriented,
focusing more on the spreadsheet commands to use than on the underlying decision model.
Other books have maintained their algorithmic approach to decision modeling, adding
spreadsheet instructions almost as an afterthought. In the fourth edition of Managerial
Decision Modeling: Business Analytics with Spreadsheets, we have continued to build on our
success with the first three editions in trying to achieve the perfect balance between the
decision modeling process and the use of spreadsheets to set up and solve decision models.
In so doing, the book not only serves the needs of students but those of professionals who
wish to use the techniques presented here. In keeping with the growing emphasis on busi-
ness analytics and the use of many of the decision modeling techniques in this field, we have
retitled the book.
It is important that books that support decision modeling try to combine the power
to logically model and analyze diverse decision-making scenarios with software-based
solution procedures. Therefore, this edition continues to focus on teaching the reader the
skills needed to apply decision models to different kinds of organizational decision-making
situations. The discussions are very application oriented and software based, with a view
toward how a manager can effectively apply the models learned here to improve the decis-
ion-making process. The target audiences for this book are students in undergraduate and
graduate level introductory decision modeling courses in business and engineering schools
and professionals who need to use the content delivered in this book every day. However,
this book will also be useful in other introductory courses that cover some of the core decis-
ion modeling topics, such as linear programming, network modeling, project management,
decision analysis, and simulation.
Although the emphasis in this edition continues to be on using spreadsheets for deci-
sion modeling, the book remains, at heart, a decision modeling book. That is, while we use
spreadsheets as a tool to quickly set up and solve decision models, our aim is not to teach
students how to blindly use a spreadsheet without understanding how and why it works.
To accomplish this, we discuss the fundamental concepts, assumptions, and limitations
behind each decision modeling technique, show how each decision model works, and
illustrate the real-world usefulness of each technique with many applications from both
for-profit and not-for-profit organizations.
Basic knowledge of algebra and Excel are the only prerequisites. For your convenience, we
have included brief introductions to Excel 2016 and probability in the appendices.
This book’s chapters, supplements, and software package cover virtually every major
topic in the decision modeling field and are arranged to provide a distinction between
techniques that deal with deterministic environments and those that deal with probabili-
stic environments. We have included more material than most instructors can cover in a
typical first course. We hope that the resulting flexibility of topic selection is appreciated
by instructors who need to tailor their courses to different audiences and curricula.

Overall Approach
While writing this fourth edition, we have continued to adhere to certain themes that have
worked very well in the first three editions:
–
– First, we have tried to separate the discussion of each decision modeling technique
into three distinct issues:
1. Formulation or problem setup
2. Model solution
3. Interpretation of the results and what-if analysis
In this three-step framework, steps 1 and 3 (formulation and interpretation) call upon
the manager’s expertise. Mastering these steps now will give readers a competitive advan-
tage later, in the marketplace, when it is necessary to make business decisions.
–
– Second, that most business and engineering professionals or students are not develo-
pers. Hence, to deal with step 2 (model solution), we have fully integrated Excel into our
discussions so that readers can take full advantage of the wide availability and accepta-
bility of spreadsheet-based software for decision modeling techniques.
Excel is a very important part of what would be considered the two main topics in
any basic decision modeling book: linear programming and simulation. However, we
recognize that some topics are not well suited for spreadsheet-based software, such as
project management, where Excel is generally not the best choice.
–
– Third, we try to ensure that readers focus on what they are doing and why they are
doing it, rather than just mechanically learning which Excel formula to use, or button
to press. To facilitate this, we also briefly discuss the steps and rationale of the solu-
tion process in many cases.
–
– Finally, we note that most of the students in decision modeling courses are likely to
specialize in other functional areas, such as finance, marketing, accounting, opera-
tions, and human resources. In addition, we expect that a wide array of professionals
will find the book a best solution. We therefore try to integrate decision modeling
techniques with problems drawn from these different areas so that readers can reco-
gnize the importance of what they are learning and the potential benefits of using
decision modeling in real-world settings. In addition, we have included summaries of
selected articles from journals such as Interfaces that discuss the actual application of
decision modeling techniques to real-world problems.
Features in This Book
The features of the first three editions of this book that have been well received as effective
aids to the learning process have been updated and expanded in this fourth edition.
In creating this edition, we not only updated the content, we analyzed how we could
best present the content from a learning point of view. Readers benefit from being able to
deep dive into a chapter filled with examples and exercises, carefully explained, so that
they can master the content. But readers also need quick review whether it be for a student
cramming for a test or a professional wanting to recall something not used in a while. So,
we created a Summary section that includes an overview of the chapter, then dozens of
detailed Key Points, backed up with a Glossary of the terms used in the chapter, highligh-
ted in red in the text as they appear.
xxii Preface

We hope that the features listed below will continue to elp readers better understand
the material:
–
– Consistent layout and format for creating effective Excel models—The consistent layout
and format for creating spreadsheet models for all linear, integer, goal, and nonlinear
programming problems is best suited to the beginner in using these types of decision
models.
–
– Functional use of color—We have standardized the use of colors so that the various
components of the models are easily identifiable.
–
– Excel Notes and Excel Extra boxes—We have added separate Excel Notes boxes to
provide simple Excel tips to make the spreadsheet usage as easy and error-free as pos-
sible. In addition, in each chapter, we have provided an Excel Extra box that illustrate
advanced Excel techniques or commands.
–
– Description of the algebraic formulation and its spreadsheet implementation for all
examples—For each model, we first discuss the algebraicformulationsothat thereader
can understand the logic and rationale behind the decision model. The spreadsheet
implementation then closely follows for ease of understanding.
–
– Numerous screen captures of Excel outputs, with detailed callouts—We have included
numerous screen captures of Excel files with detailed callouts explaining the impor-
tant entries and components of the model. Excel files are located at degruyter.com/
view/product/486941 and, for your convenience, the callouts are shown as comments
on appropriate cells in these Excel files.
–
– Ability to teach topics without the use of additional software—Several topics can be
studied using only Excel’s standard built-in add-ins and commands. For example, we
have discussed how Excel’s Data Table and Scenario Manager procedures can be used
to analyze and replicate even large simulation models.
–
– Extensive discussion of linear programming sensitivity analysis, using the Solver report—
The discussion of linear programming sensitivity analysis in this book is more com-
prehensive than that in any competing book.
–
– Decision Modeling In Action boxes—These boxes summarize published articles that illust-
rate how real-world organizations have used decision models to solve problems.
–
– ExcelModules—This software package from Professor Howard Weiss of Temple Uni-
versity solves problems in queuing models (Chapter 9), forecasting models (Chapter
11), and inventory control models (in an optional Chapter 12 found online and down-
loadable from the Companion Website). Readers can see the power of this software
package in modeling and solving problems in these chapters. ExcelModules is menu
driven and easy to use, and it is available at degruyter.com/view/product/486941. A
Mac version of the program is also available for the first time.
Major Changes in the FOURTH Edition
We have made the following major changes in this fourth edition—All spreadsheet appli-
cations have been fully updated to Excel 2016. The software program ExcelModules that
accompanies this book has also been updated to suit Excel 2016 as well as 32-bit and 64-bit
systems. In addition a Mac version of this software is now available.
–
– Significant number of new end-of-chapter exercises—We have added at least eight new
exercises in each chapter. On average, there are now more than 45 end-of-chapter
exercises per chapter.
Preface xxiii

xxiv Preface
–
– More challenging chapter examples and end-of-chapter exercises—Many of the chapter
examples and end-of-chapter exercises have been revised to make them more current,
rigorous, and better suited to a computer-based solution environment inviting readers
to modify the Excel models contained in the chapters to incorporate new constraints
or conditions. This requires readers to first thoroughly understand the original Excel
models before attempting to modify them.
–
– New Excel Extra Boxes—In addition to the Excel Notes boxes that provide quick tips on
Excel commands and procedures relevant to the topic being discussed, we have added
new Excel Extra boxes. These boxes illustrate advanced Excel techniques and com-
mands including descriptions of cell comments; locking cells; data validation; drop-
down lists; linked charts; VBA for user interaction; sorting; identifying the owner of
a max or min search; hiding rows, columns, sheets, and formulas; automating with
macros; conditional formatting; and scroll bars and other form controls.
–
– Updated Decision Modeling In Action boxes—Decision Modeling In Action boxes illust-
rate the use of decision modeling in real-world scenarios. Many of these examples are
from recent issues of Interfaces.
–
– Streamlined network problem formulations—We have modified the algebraic formula-
tion and Excel implementation of certain network problems in Chapters 5 and 6 to
provide a more streamlined presentation.
–
– Excel functions—We have added an extensive list of common Excel functions for your
reference in Appendix B.
–
– Better introductions—Set expectations in each chapter.
Companion Website
The following items can be downloaded at degruyter.com/view/product/486941:
1. Data Files—Excel files for all examples discussed in the book. (For easy reference, the
relevant file names are printed below the titles of the corresponding figures at appro-
priate places in the book.)
2. Online Chapter—The electronic-only Chapter 12: Inventory Control Models (PDF).
3. ExcelModulesSoftware—Thisprogramsolvesproblemsandexamplesinthequeuingmodels
(Chapter 9), forecasting models (Chapter 11), and the downloadable inventory control
models (Chapter 12) chapters in this book. Available in both Windows and Mac versions.
4. Solutions to End-of-Chapter Exercises—Detailed Excel solutions for all end-of-chapter
exercises. Access is available to faculty adopters only.
Raju Balakrishnan
313-593-5462 (phone)
rajub@umich.edu (e-mail)
Barry Render
brender@rollins.edu (e-mail)
Ralph Stair
ralphmstair@cs.com (e-mail)
Chuck Munson
509-335-3076 (phone)
munson@wsu.edu (e-mail)

Chapter 1
Introduction to Managerial Decision Modeling
Do you frequently struggle to make decisions? Some people argue incessantly over
relatively trivial choices, such as where to go for lunch. More life-altering decisions,
such as where to go to college, whether or not to take that new job offer, or whether or
not to say yes to a marriage proposal, can hound us for days and keep us up at night.
And when we make a wrong decision, the regret that we feel can haunt us for weeks,
months, or even years.
Managers face similar dilemmas as they struggle to make the best decisions for
their respective organizations. Great decisions can lead to millions of extra dollars for
the company and personal promotions or bonuses for the decision maker. Poor deci-
sions can lead to huge financial losses for the company and potential job loss for the
decision maker. People make many decisions on a personal level and even for their
companies based on “gut feel.” That may be the best approach for some decisions.
But for many decisions, decision-making tools can provide tremendous guidance by
illustrating the pros and cons of various alternatives. This is the essence of decision
modeling.
We begin this chapter by defining decision modeling and then delineating the
two major types of decision models. Next, we discuss the three major steps involved
in decision modeling. The vast majority of the models covered in this book are quanti-
tative in nature. Fortunately, these generally do not require the skills of a professional
mathematician to set up or solve. Most of these models require only standard algebra
and arithmetic, along with little bit of statistical background. An important reason
for this is that we let the computer do most of the “heavy mathematical lifting” for
us. And while many specialized computer optimization packages exist to solve large-
scale decision models, it turns out that the standard spreadsheet, Microsoft Excel,
can solve many types of modeling problems of reasonable size. We focus exclusively
in this book on using Excel to solve the models that we present.
Most managers around the globe have Excel on their computers, and many use
Excel frequently. As such, Excel can be a great tool for modeling because co-workers
may be more comfortable with spreadsheets than they would be with unfamiliar spe-
cialized programs. Excel has allowed the “common manager” to use and even build
his or her own decision models without the need to hire a specialist.
We introduce two models in this chapter that illustrate the standard approach for
modeling with Excel that we will use for the rest of the book. The first illustrates how
to compute estimated income taxes, and the second utilizes the Excel feature Goal
Seek to perform a simple profit break-even analysis for a small firm. We conclude
Chapter 1 by describing certain pitfalls that may arise in the modeling process and
some challenges with implementation.
DOI 10.1515/9781501506208-001

2 Chapter 1: Introduction to Managerial Decision Modeling
Chapter Objectives
After completing this chapter, you will be able to:
1. Define decision model and describe the importance of such models.
2. Understand the two types of decision models: deterministic and probabilistic
models.
3. Understand the steps involved in developing decision models in practical
situations.
4. Understand the use of spreadsheets in developing decision models.
5. Discuss possible problems in developing decision models.
1.1 What is Decision Modeling?
Although there are several definitions of decision modeling, we define it here as a sci-
entific approach to managerial decision making. Alternatively, we can define it as the
development of a model (usually mathematical) of a real-world problem scenario or
environment. The resulting model typically should be such that the decision-making
process is not affected by personal bias, whim, emotions, or guesswork. This model
can then be used to provide insights into the solution of the managerial problem.
Decision modeling is also commonly referred to as quantitative analysis, management
science, or operations research. In this book, we prefer the term decision modeling
because we will discuss all modeling techniques in a managerial decision-making
context.
You may have heard about the explosion of “big data” or “data analytics” in the
business world. The increasing power of technology to collect massive amounts of
data from customers and other sources, along with never-ending comments appea-
ring in social media, have opened possibilities for companies that were heretofore
unimaginable. Just imagine the amount of data being collected daily by companies
such as Google, Facebook, Twitter, and Yahoo, and the wealth of useful information
contained in that data. The term “analytics” is being used in many ways, but at its core
it describes transforming data into information, hopefully leading to sound business
decisions. This is exactly what decision modeling encompasses. In fact, “data ana-
lytics” is now considered by many to be synonymous with decision modeling, quan-
titative analysis, management science, and operations research. Firms are searching
for employees with these skills like never before. If you can master the skills, you will
be highly valued by the marketplace.
Organizations such as American Airlines, United Airlines, IBM, Google, UPS,
FedEx, and ATT frequently use decision modeling to help solve complex problems.
Although mathematical tools have been in existence for thousands of years, the
formal study and application of quantitative (or mathematical) decision modeling
techniques to practical decision making is largely a product of the twentieth century.
The decision modeling techniques studied here have been applied successfully to an
increasingly wide variety of complex problems in business, government, health care,

Types of Decision Models 3
education, and many other areas. Many such successful uses are discussed throug-
hout this book.
It isn’t enough, though, just to know the mathematical details of how a particular
decision modeling technique can be set up and solved. It is equally important to be
familiar with the limitations, assumptions, and specific applicability of the model.
The correct use of decision modeling techniques usually results in solutions that are
timely, accurate, flexible, economical, reliable, easy to understand, and easy to use.
1.2 Types of Decision Models
Decision models can be broadly classified into two categories, based on the type and
nature of the decision-making problem environment under consideration: (1) deter-
ministic models and (2) probabilistic models. We define each type in the following
sections.
Deterministic Models
Deterministic models assume that all the relevant input data values are known with
certainty; that is, they assume that all the information needed for modeling a decision-
making problem environment is available, with fixed and known values. An example
of such a model is the case of Dell Corporation, which makes several different types
of PC products (e.g., desktops, laptops), all of which compete for the same resources
(e.g., labor, hard disks, chips, working capital). Dell knows the specific amounts of
each resource required to make one unit of each type of PC, based on the PC’s design
specifications. Further, based on the expected selling price and cost prices of various
resources, Dell knows the expected profit contribution per unit of each type of PC. In
such an environment, if Dell decides on a specific production plan, it is a simple task
to compute the quantity required of each resource to satisfy that production plan. For
example, if Dell plans to ship 50,000 units of a specific laptop model, and each unit
includes a pair of 8.0 GB DDR4 memory chips, then Dell will need 100,000 units of
these memory chips. Likewise, it is easy to compute the total profit that will be rea-
lized by this production plan (assuming that Dell can sell all the laptops it makes).
Perhaps the most common and popular deterministic modeling technique is
linear programming (LP). In Chapter 2, we first discuss how small LP models can be
set up and solved. We extend our discussion of LP in Chapter 3 to more complex pro-
blems drawn from a variety of business disciplines. In Chapter 4, we study how the
solution to LP models produces, as a byproduct, a great deal of information useful
for managerial interpretation of the results. Finally, in Chapters 5 and 6, we study a
few extensions to LP models. These include several different network flow models
(Chapter 5), as well as integer, nonlinear, and multi-objective (goal) programming
models (Chapter 6).

As we demonstrate during our study of deterministic models, a variety of impor-
tant managerial decision-making problems can be set up and solved using these
techniques.
Probabilistic Models
In contrast to deterministic models, probabilistic models (also called stochastic
models) assume that some input data values are not known with certainty. That is,
they assume that the values of some important variables will not be known before
decisions are made. It is therefore important to incorporate this “ignorance” into the
model. An example of this type of model is the decision of whether to start a new busi-
ness venture. As we have seen with the high variability in the stock market during the
past several years, the success of such ventures is uncertain. However, investors (e.g.,
venture capitalists, founders) have to make decisions regarding this type of venture
based on their expectations of future performance. Clearly, such expectations are not
guaranteed to occur. In recent years, we have seen several examples of firms that have
yielded (or are likely to yield) great rewards to their investors (e.g., Google, Facebook,
Twitter) and others that have either failed (e.g., eToys.com, Pets.com) or been much
more modest in their returns.
Another example of probabilistic modeling to which students may be able to relate
easily is their choice of a major when they enter college. Clearly, there is a great deal of
uncertainty regarding several issues in this decision-making problem: the student’s
aptitude for a specific major, his or her actual performance in that major, the employ-
ment situation in that major in four years, etc. Nevertheless, a student must choose a
major early in his or her college career. Recollect your own situation. In all likelihood,
you used your own assumptions (or expectations) regarding the future to evaluate the
various alternatives (i.e., you developed a “model” of the decision-making problem).
These assumptions may have been the result of information from various sources,
such as parents, friends, and guidance counselors. The important point to note here
is that none of this information is guaranteed, and no one can predict with 100%
accuracy what exactly will happen in the future. Therefore, decisions made with this
information, while well thought out and well intentioned, may still turn out not to be
the best choices. For example, how many of your friends changed majors during their
college careers?
Because their results are not guaranteed, does this mean that probabilistic deci-
sion models are of limited value? As we will see later in this book, the answer is an
emphatic no. Probabilistic modeling techniques provide a structured approach for
managers to incorporate uncertainty into their models and to evaluate decisions
under alternate expectations regarding this uncertainty. They do so by using proba-
bilities on the “random,” or unknown, variables. Probabilistic modeling techniques
discussed in this book include decision analysis (Chapter 8), queuing (Chapter 9),
simulation (Chapter 10), and forecasting (Chapter 11). Two other techniques, project
management (Chapter 7) and inventory control (Chapter 12), include aspects of both

Types of Decision Models 5
deterministic and probabilistic modeling. For each modeling technique, we discuss
what kinds of criteria can be used when there is uncertainty and how to use these
models to identify the preferred decisions.
Because uncertainty plays a vital role in probabilistic models, some knowledge of
basic probability and statistical concepts is useful. Appendix A provides a brief over-
view of this topic. It should serve as a good refresher while studying these modeling
techniques.
Quantitative versus Qualitative Data
Any decision modeling process starts with data. Like raw material for a factory, these
data are manipulated or processed into information valuable to people making deci-
sions. This processing and manipulating of raw data into meaningful information is
the heart of decision modeling.
In dealing with a decision-making problem, managers may have to consider
both qualitative and quantitative factors. For example, suppose we are considering
several different investment alternatives, such as certificates of deposit, the stock
market, and real estate. We can use quantitative factors, such as rates of return, finan-
cial ratios, and cash flows, in our decision model to guide our ultimate decision. In
addition to these factors, however, we may also wish to consider qualitative factors,
such as pending state and federal legislation, new technological breakthroughs, and
the outcome of an upcoming election. It can be difficult to quantify these qualitative
factors.
Due to the presence (and relative importance) of qualitative factors, the role of
quantitative decision modeling in the decision-making process can vary. When there
is a lack of qualitative factors, and when the problem, model, and input data remain
reasonably stable and steady over time, the results of a decision model can automate
the decision-making process. For example, some companies use quantitative inven-
tory models to determine automatically when to order additional new materials and
how much to order. In most cases, however, decision modeling is an aid to the decis-
ion-making process. The results of decision modeling should be combined with other
(qualitative) information while making decisions in practice.
Using Spreadsheets in Decision Modeling
In keeping with the ever-increasing presence of technology in modern times, compu-
ters have become an integral part of the decision modeling process in today’s busi-
ness environments. Until the early 1990s, many of the modeling techniques discussed
here required specialized software packages in order to be solved using a computer.
However, spreadsheet packages such as Microsoft Excel have become increasingly
capable of setting up and solving most of the decision modeling techniques com-
monly used in practical situations. For this reason, the current trend in many college
courses on decision modeling focuses on spreadsheet-based instruction. In keeping

with this trend, we discuss the role and use of spreadsheets (specifically Microsoft
Excel) during our study of the different decision modeling techniques presented here.
In addition to discussing the use of some of Excel’s built-in functions and procedu-
res (e.g., Goal Seek, Data Table, Chart Wizard), we also discuss a few add-ins for Excel.
The Data Analysis and Solver add-ins come standard with Excel. A custom add-in
called ExcelModules is included on the Companion Website and used in Chapter 9
(Queuing Models), Chapter 11 (Forecasting Models), and the online Chapter 12 (Inven-
tory Control Models).
Because a knowledge of basic Excel commands and procedures facilitates under-
standing the techniques and concepts discussed here, we recommend reading Appendix
B, which provides a brief overview of the Excel features that are most useful in decision
modeling. In addition, at appropriate places throughout this book, we discuss several
Excel functions and procedures specific to each decision modeling technique.
Decision Modeling In Action
IBM Uses Decision Modeling to Improve the Productivity of Its Sales Force
IBM is a well-known multinational computer technology, software, and services company with
more than 380,000 employees and revenue of more than $79 billion. A majority of IBM’s revenue
comes from services, including outsourcing, consulting, and systems integration.
Recognizing that improving the efficiency and productivity of this large sales force can be an
effective operational strategy to drive revenue growth and manage expenses, IBM Research
developed two broad decision modeling initiatives to explore this issue. The first initiative
provides a set of analytical models designed to identify new sales opportunities at existing
IBM accounts and at noncustomer companies. The second initiative allocates sales resources
optimally based on field-validated analytical estimates of future revenue opportunities in market
segments. IBM estimates the revenue impact of these two initiatives to be in the several hund-
reds of millions of dollars each year.
Source: Based on R. Lawrence et al. “Operations Research Improves Sales Force Productivity at
IBM,” Interfaces 40, 1 (January-February 2010): 33–46.
1.3 Steps Involved in Decision Modeling
Regardless of the size and complexity of the decision-making problem at hand, the
decision modeling process involves three distinct steps: (1) formulation, (2) solution,
and (3) interpretation. Figure 1.1 provides a schematic overview of these steps, along
with the components, or parts, of each step. We discuss each of these steps in the
following sections.

Steps Involved in Decision Modeling 7
Figure 1.1: The Decision Modeling Approach
It is important to note that it is common to have an iterative process between these
three steps before obtaining the final solution. For example, testing the solution (see
Figure 1.1) might reveal that the model is incomplete or that some of the input data
are being measured incorrectly. This means that the formulation needs to be revised.
That, in turn, causes all the subsequent steps to be changed.
Step 1: Formulation
Formulation is the process by which each aspect of a problem scenario is translated
and expressed in terms of a mathematical model. This is perhaps the most important
and challenging step in decision modeling because the results of a poorly formulated
problem will almost surely be incorrect. It is also in this step that the decision maker’s
ability to analyze a problem rationally comes into play. Even the most sophisticated
software program will not automatically formulate a problem. The aim in formulation
is to ensure that the mathematical model completely addresses all the issues relevant

to the problem at hand. Formulation can be further classified into three parts: (1) defi-
ning the problem, (2) developing a model, and (3) acquiring input data.
Defining the Problem The first part in formulation (and in decision modeling) is to
develop a clear, concise statement of the problem. This statement gives direction and
meaning to all the parts that follow it.
In many cases, defining the problem is perhaps the most important, and the
most difficult, part. It is essential to go beyond just the symptoms of the problem
at hand and identify the true causes behind it. One problem may be related to other
problems, and solving a problem without regard to its related problems may actually
worsen the situation. Thus, it is important to analyze how the solution to one problem
affects other problems or the decision-making environment in general. Experience
has shown that poor problem definition is a major reason for failure of management
science groups to serve their organizations well.
When a problem is difficult to quantify, it may be necessary to develop specific,
measurable objectives. For example, say a problem is defined as inadequate health care
delivery in a hospital. The objectives might be to increase the number of beds, reduce
the average number of days a patient spends in the hospital, increase the physician-to-
patient ratio, and so on. When objectives are used, however, the real problem should be
kept in mind. It is important to avoid obtaining specific and measurable objectives that
may not solve the real problem.
Developing a Model Once we select the problem to be analyzed, the next part is to
develop a decision model. Even though you might not be aware of it, you have been
using models most of your life. For example, you may have developed the following
model about friendship: Friendship is based on reciprocity, an exchange of favors.
Hence, if you need a favor, such as a small loan, your model would suggest that you
ask a friend.
Of course, there are many other types of models. An architect may make a phy-
sical model of a building he or she plans to construct. Engineers develop scale
models of chemical plants, called pilot plants. An analog model, e.g., a thermome-
ter measuring temperature or an oil dipstick signaling the level of oil remaining in
a car, represents a phenomenon but does not look like it. A schematic model is a
picture or drawing of reality. Automobiles, lawn mowers, circuit boards, typewri-
ters, and numerous other devices have schematic models (drawings and pictures)
that reveal how these devices work.
What sets decision modeling apart from other modeling techniques is that the
models we develop here are mathematical. A mathematical model is a set of mathe-
matical relationships. In most cases, these relationships are expressed as equations
and inequalities, as they are in a spreadsheet model that computes sums, averages,
or standard deviations.

Steps Involved in Decision Modeling 9
Although there is considerable flexibility in the development of models, most
of the models presented here contain one or more variables and parameters. A vari-
able, as the name implies, is a measurable quantity that may vary or that is subject
to change. Variables can be controllable or uncontrollable. A controllable variable
is also called a decision variable. An example is how many inventory items to order.
A problem parameter is a measurable quantity that is inherent in the problem, such
as the cost of placing an order for more inventory items. In most cases, variables are
unknown quantities, whereas parameters (or input data) are known quantities.
All models should be developed carefully. They should be solvable, realistic, and
easy to understand and modify, and the required input data should be obtainable. A
model developer must be careful to include the appropriate amount of detail for the
model to be solvable yet realistic.
Acquiring Input Data Once we have developed a model, we must obtain the input
data to be used in the model. Obtaining accurate data is essential because even if the
model is a perfect representation of reality, improper data will result in misleading
results. This situation is called garbage in, garbage out (GIGO). For larger problems,
collecting accurate data can be one of the most difficult aspects of decision modeling.
Several sources can be used in collecting data. In some cases, company reports
and documents can be used to obtain the necessary data. Another source is interviews
with employees or other persons related to the firm. These individuals can sometimes
provide excellent information, and their experience and judgment can be invaluable.
A production supervisor, for example, might be able to tell you with a great degree of
accuracy the amount of time it takes to manufacture a particular product. Sampling
and direct measurement provide other sources of data for the model. You may need
to know how many pounds of a raw material are used in producing a new photoche-
mical product. This information can be obtained by going to the plant and actually
measuring the amount of raw material being used. In other cases, statistical sampling
procedures can be used to obtain data.
Step 2: Solution
The solution step is when the mathematical expressions resulting from the formula-
tion process are solved to identify the optimal solution. Until the mid-1990s, typical
courses in decision modeling focused a significant portion of their attention on this
step because it was the most difficult aspect of studying the modeling process. As
stated earlier, thanks to computer technology, the focus today has shifted away from
the detailed steps of the solution process and toward the availability and use of soft-
ware packages. The solution step can be further classified into two parts: (1) develo-
ping a solution and (2) testing the solution.
Developing a Solution Developing a solution involves manipulating the model to
arrive at the best (or optimal) solution to the problem. In some cases, this may require

that a set of mathematical expressions be solved to determine the best decision. In
other cases, you can use a trial-and-error method, trying various approaches and
picking the one that results in the best decision. For some problems, you may wish to
try all possible values for the variables in the model to arrive at the best decision; this
is called complete enumeration. For problems that are quite complex and difficult,
you may be able to use an algorithm. An algorithm consists of a series of steps or
procedures that we repeat until we find the best solution. Regardless of the approach,
the accuracy of the solution depends on the accuracy of the input data and the
decision model itself.
Testing the Solution Before a solution can be analyzed and implemented, it must be
tested completely. Because the solution depends on the input data and the model,
both require testing. There are several ways to test input data. One is to collect
additional data from a different source and use statistical tests to compare these new
data with the original data. If there are significant differences, more effort is required
to obtain accurate input data. If the data are accurate but the results are inconsistent
with the problem, the model itself may not be appropriate. In this case, the model
should be checked to make sure that it is logical and represents the real situation.
Step 3: Interpretation and Sensitivity Analysis
Assuming that the formulation is correct and has been successfully implemented and
solved, how does a manager use the results? Here again, the decision maker’s exper-
tise is called upon because it is up to him or her to recognize the implications of the
presented results. We discuss this step in two parts: (1) analyzing the results and sen-
sitivity analysis and (2) implementing the results.
Analyzing the Results and Sensitivity Analysis Analyzing the results starts with
determining the implications of the solution. In most cases, a solution to a problem
will result in some kind of action or change in the way an organization is operating.
The implications of these actions or changes must be determined and analyzed before
the results are implemented.
Because a model is only an approximation of reality, the sensitivity of the solution
to changes in the model and input data is an important part of analyzing the results.
This type of analysis is called sensitivity, post-optimality, or what-if analysis. Sensi-
tivity analysis is used to determine how much the solution will change if there are
changes in the model or the input data. When the optimal solution is very sensitive
to changes in the input data and the model specifications, additional testing must be
performed to make sure the model and input data are accurate and valid.
The importance of sensitivity analysis cannot be overemphasized. Because input
data may not always be accurate or model assumptions may not be completely appro-
priate, sensitivity analysis can become an important part of decision modeling.

Spreadsheet Example of a Decision Model: Tax Computation 11
Implementing the Results The final part of interpretation is to implement the results.
This can be much more difficult than one might imagine. Even if the optimal solution
will result in millions of dollars in additional profits, if managers resist the new
solution, the model is of no value. Experience has shown that numerous decision
modeling teams have failed in their efforts because they have failed to implement a
good, workable solution properly.
After the solution has been implemented, it should be closely monitored. Over
time, there may be numerous changes that call for modifications of the original solu-
tion. A changing economy, fluctuating demand, and model enhancements requested
by managers and decision makers are examples of changes that might require an ana-
lysis to be modified.
1.4 Spreadsheet Example of a Decision Model: Tax Computation
Now that we have discussed what a decision model is, let us develop a simple model
for a real-world situation we all face each year: paying taxes. Sue and Robert Miller,
a newly married couple, will be filing a joint tax return for the first time this year.
Because both work as independent contractors (Sue is an interior decorator, and Rob
is a painter), their projected income is subject to some variability. However, because
their earnings are not taxed at the source, they know that they have to pay estimated
income taxes on a quarterly basis, based on their estimated taxable income for the
year. To help calculate this tax, the Millers would like to set up a spreadsheet-based
decision model. Assume that they have the following information available:
–
– Their only source of income is from their jobs.
–
– They would like to put away 7% of their total income in a retirement account, up
to a maximum of $8,000. Any amount they put in that account can be deducted
from their total income for tax purposes.
–
– They are entitled to a personal exemption of $4,050 each. This means that they
can deduct $8,100 (= 2 × $4,050) from their total income for tax purposes.
–
– The standard deduction for married couples filing taxes jointly this year is
$12,700. This means that $12,700 of their income is free from any taxes and can be
deducted from their total income.
–
– They do not anticipate having any other deductions from their income for tax
purposes.
–
– The tax brackets for this year are 10% for the first $18,650 of taxable income, 15%
between $18,651 and $75,900, and 25% between $75,901 and $153,100. The Millers
don’t believe that tax brackets beyond $153,100 are relevant for them this year.

EXCEL NOTES
–
– The Companion Website for this book, at degruyter.com/view/product/486941, contains the
Excel file for each sample problem discussed here. The relevant file/sheet name is shown
below the title of the corresponding figure in this book.
–
– In each of our Excel layouts, for clarity, we color code the cells as follows:
• Variable input cells, in which we enter specific values for the variables in the problem, are
shaded yellow.
• Output cells, which show the results of our analysis, are shaded green.
–
– We have used callouts to annotate the screenshots in this book to highlight important issues
in the decision model.
–
– Wherever necessary, many of these callouts are also included as comments in the Excel files
themselves, making it easier for you to understand the logic behind each model.
Figure 1.2 shows the formulas that we can use to develop a decision model for the
Millers. Just as we have done for this Excel model (and all other models in this book),
we strongly recommend that you get in the habit of using descriptive titles, labels,
and comments in any decision model you create. The reason for this is very simple: In
many real-world settings, decision models that you create are likely to be passed on to
others. In such cases, the use of comments will help them understand your thought
process. Perhaps an appropriate question you should always ask yourself is, “Will I
understand this model a year or two after I first write it?” If appropriate labels and
comments are included in the model, the answer should always be yes.
In Figure 1.2, the known problem parameter values (i.e., constants) are shown
in the box labeled Known Parameters. Rather than use these known constant values
directly in the formulas, we recommend that you develop the habit of entering each
known value into a cell and then using that cell reference in the formulas. In addi-
tion to being more “elegant,” this way of modeling has the advantage of making any
future changes to these values easy. This is one of the most important Excel practices
to implement! Many expensive spreadsheet mistakes are made in companies because
numbers are hard-coded into formulas in multiple places throughout spreadsheets—
and these spreadsheets originally may have been developed by managers long retired
from the firm. When formulas reference easily identifiable cells for all their parame-
ters, users can be confident that parameter changes need to be made only once and
that all formulas will be correctly updated.
Cells B13 and B14 denote the only two variable data entries in this decision model:
Sue’s and Rob’s estimated incomes for this year. When we enter values for these two
variables, the results are computed in cells B17:B26 and presented in the box labeled
“Tax Computation.”

This box shows
all the known
input parameter
values.
This box shows the
two input variables.
Minimun of (7% of
total income $8,000)
Maximum of
(0, taxable income)
15% tax between $18,651
and $75,900. This tax is
calculated only if taxable
income exceeds $18,650.
25% tax between $75,901 and
$153,100. This tax is calculated
only if taxable income exceeds
$75,900.
10% tax up to $18,650
Figure 1.2: Formula View of Excel Layout for the Millers’ Tax Computation
File: Figure 1.2.xlsx; Sheet: Figure 1.2
Cell B17 shows the total income. The MIN function is used in cell B18 to specify the
tax-deductible retirement contribution as the smaller value of 7% of total income and
$8,000. Cells B19 and B20 set the personal exemptions and the standard deduction,
respectively. The net taxable income is shown in cell B21, and the MAX function is
used here to ensure that this amount is never below zero. The taxes payable at the
10%, 15%, and 25% rates are then calculated in cells B22, B23, and B24, respectively.
In each of these cells, the MIN function is used to ensure that only the incremental
taxable income is taxed at a given rate. (For example, in cell B23, only the portion
of taxable income above $18,650 is taxed at the 15% rate, up to an upper limit of
$75,900.) The IF function is used in cells B23 and B24 to check whether the taxable
income exceeds the lower limit for the 15% and 25% tax rates, respectively. If the
taxable income does not exceed the relevant lower limit, the IF function sets the tax
payable at that rate to zero. Finally, the total tax payable is computed in cell B25, and
the estimated quarterly tax is computed in cell B26.

EXCEL EXTRA
Cell Comments
The most useful spreadsheets are self-documenting. You can create descriptive callout boxes
similar to those describing certain cells in Figure 1.2 by using the built-in Excel feature that allows
you to document cell entries with comment boxes. Any cell can have an associated comment box.
Similar to how messages appear in Word or Excel when you place the mouse over certain command
buttons, your cell comment will appear when the user places the mouse over that particular
cell. A small red triangle in the upper right-hand side of the cell indicates that the cell contains a
comment. Alternatively, you can set the comment box to be displayed at all times. Comment boxes
can be resized by editing the comment and dragging one of the eight squares on the outside of the
box as desired. Font size and comment contents can also be changed by editing the comment.
–
– To create a comment: Right click on the cell and select Insert Comment [then enter text]
–
– To change or resize: Right click on the cell and select Edit Commen
–
– To show constantly: Right click on the cell and select Show/Hide Comments
–
– To remove “Show Comment:” Right click on the cell and select Hide Comment
–
– To remove the comment completely: Right click on the cell and select Delete Comment
–
– Three options for printing comments:
Click Page Layout|Print Titles|Sheet|Comments
(1) (None)—none will be printed (default)
(2) At end of sheet—all comments will be printed
(3) As displayed on sheet—only comments set to “show“ are printed on the spreadsheet
Now that we have developed this decision model, how can the Millers actually use
it? Suppose Sue estimates her income this year at $65,000, and Rob estimates his at
$60,000. We enter these values into cells B13 and B14, respectively. The decision model
immediately lets us know that the Millers have a taxable income of $96,200 and that
they should pay estimated taxes of $3,881.88 each quarter. These input values, and
the resulting computations, are shown in Figure 1.3. We can use this decision model in
a similar fashion with any other estimated income values for Sue and Rob.

Estimated income
Total income of $125,000
has been reduced to
taxable income of only
$96,200.
The Millers should pay
$3881.88 in estimated
taxes each quarter.
Figure 1.3: Excel Decision Model for the Millers’ Tax Computation
Using Decision Modeling to Combat Spread of Hepatitis B Virus in the United
States and China
Hepatitis B is a vaccine-preventable viral disease that is a major public health problem, parti-
cularly among Asian populations. Left untreated, it can lead to death from cirrhosis and liver
cancer. More than 350 million people are chronically infected with the hepatitis B virus (HBV)
worldwide. In the United States (US), although about 10% of Asian and Pacific Islanders are
chronically infected, about two-thirds of them are unaware of their infection. In China, HBV infec-
tion is a leading cause of death.
During several years of work conducted at the Asian Liver Center at Stanford University, the authors
used combinations of decision modeling techniques to analyze the cost effectiveness of various
intervention schemes to combat the spread of the disease in the US and China. The results of these
analyses have helped change US public health policy on hepatitis B screening, and they have
helped encourage China to enact legislation to provide free vaccination for millions of children.

These policies are an important step in eliminating health disparities and ensuring that millions
of people can now receive the hepatitis B vaccination they need. The Global Health Coordinator
of the Asian Liver Center states that this research “has been incredibly important to accelerating
policy changes to improve health related to HBV.”
Source: Based on D. W. Hutton, M. L. Brandeau, and S. K. So. “Doing Good with Good OR: Sup-
porting Cost-Effective Hepatitis B Interventions,” Interfaces 41, 3 (May-June 2011): 289–300.
Observe that the decision model we have developed for the Millers‘ example does
not optimize the decision in any way. That is, the model simply computes the estima-
ted taxes for a given income level. It does not, for example, determine whether these
taxes can be reduced in some way through better tax planning. Later in this book, we
discuss decision models that not only help compute the implications of a specified
decision, but also help identify the optimal decision, based on some objective or goal.
1.5
Spreadsheet Example of a Decision Model: Break-Even
Analysis
Let’s now develop another decision model—this one to compute the total profit for a
firm, as well as the associated break-even point. We know that profit is simply the dif-
ference between revenue and expense. In most cases, we can express revenue as the
selling price per unit multiplied by the number of units sold. Likewise, we can express
expense as the sum of the total fixed and variable costs. In turn, the total variable
cost is the variable cost per unit multiplied by the number of units sold. Thus, we can
express profit using the following mathematical expression:
Profit = (Selling price per unit) × (Number of units) − (Fixed cost)
− (Variable cost per unit) × (Number of units) (1–1)
Let’s use Bill Pritchett’s clock repair shop to demonstrate the creation of a deci-
sion model to calculate profit and the associated break-even point. Bill’s company,
Pritchett’s Precious Time Pieces, buys, sells, and repairs old clocks and clock parts.
Bill sells rebuilt springs for a unit price of $10. The fixed cost of the equipment to
build the springs is $1,000. The variable cost per unit is $5 for spring material. If we
represent the number of springs (units) sold as the variable X, we can restate the profit
as follows:
Profit = $10X − $1,000 − $5X
Figure 1.4 shows the formulas used in developing the decision model for Bill
Pritchett’s example. Cells B4, B5, and B6 show the known problem parameter values—
namely, revenue per unit, fixed cost, and variable cost per unit, respectively. Cell B9
is the lone variable in the model, and it represents the number of units sold (i.e., X).
Using these entries, the total revenue, total variable cost, total cost, and profit are

Spreadsheet Example of a Decision Model: Break-Even Analysis 17
computed in cells B12, B14, B15, and B16, respectively. For example, if we enter a value
of 1,000 units for X in cell B9, the profit is calculated as $4,000 in cell B16, as shown
in Figure 1.5.
Figure 1.4: Formula View of Excel Layout for Pritchett’s Precious Time Pieces
Figure 1.5: Excel Decision Model for Pritchett’s Precious Time Pieces

In addition to computing the profit, decision makers are often interested in the break-
even point (BEP). The BEP is the number of units sold that will result in total revenue
equaling total costs (i.e., profit is $0). We can determine the BEP analytically by
setting profit equal to $0 and solving for X in Bill Pritchett’s profit expression. That is
0 = (Selling price per unit) × (Number of units) − (Fixed cost)
− (Variable cost per unit) × (Number of units)
which can be mathematically rewritten as
Break-even point (BEP) = Fixed cost ⁄ (Selling price per unit
− Variable cost per unit) (1–2)
For Bill Pritchett’s example, we can compute the BEP as $1,000/($10 − $5) = 200
springs. The BEP in dollars (which we denote as BEP$
) can then be computed as
BEP$
= Fixed cost + Variable costs × BEP (1–3)
For Bill Pritchett’s example, we can compute BEP$
as $1,000 + $5 × 200 = $2,000.
Using Goal Seek to Find the Break-Even Point
While the preceding analytical computations for BEP and BEP$
are fairly simple, an
advantage of using computer-based models is that many of these results can be cal-
culated automatically. For example, we can use a procedure in Excel called Goal Seek
to calculate the BEP and BEP$
values in the decision model shown in Figure 1.5. The
Goal Seek procedure allows us to specify a desired value for a target cell. This target
cell should contain a formula that involves a different cell, called the changing cell.
Once we specify the target cell, its desired value, and the changing cell in Goal Seek,
the procedure automatically manipulates the changing cell value to try and make the
target cell achieve its desired value.
In our case, we want to manipulate the value of the number of units X (in cell B9
of Figure 1.5) such that the profit (in cell B16 of Figure 1.5) takes on a value of zero.
That is, cell B16 is the target cell, its desired value is zero, and cell B9 is the changing
cell. Observe that the formula of profit in cell B16 is a function of the value of X in cell
B9 (see Figure 1.4).
Figure 1.6 shows how the Goal Seek procedure is implemented in Excel. As shown
in Figure 1.6 (a), we invoke Goal Seek by clicking the Data tab on Excel’s main menu
bar, followed by the What-If Analysis button (found in the Data Tools group within
the Data tab), and then finally on Goal Seek. The window shown in Figure 1.6 (b) is
displayed. We specify cell B16 in the Set cell box, a desired value of zero for this cell
in the To value box, and cell B9 in the By changing cell box. When we now click OK,
the Goal Seek Status window shown in Figure 1.6 (c) is displayed, indicating that the
target of $0 profit has been achieved. Cell B9 shows the resulting BEP value of 200
units. The corresponding BEP$
value of $2,000 is shown in cell B15.

Spreadsheet Example of a Decision Model: Break-Even Analysis 19
Figure 1.6: Using Excel’s Goal Seek to Compute the Break-Even Point For Pritchett’s Precious Time
Pieces
Observe that we can use Goal Seek to compute the sales level needed to obtain any
desired profit. For example, see if you can verify that in order to get a profit of $10,000,
Bill Pritchett would have to sell 2,200 springs. We will use the Goal Seek procedure
again in Chapter 9.

EXCEL NOTES
–
– Bear in mind that, for more complicated functions, if an equation has multiple roots, Goal
Seek will return only one of those roots. The answer may represent the closest root to the
starting point in the changing cell, so choose carefully. Try this experiment. Insert the number
5 into cell A1 and the formula =A1^2 into cell A2. Invoke Goal Seek, where the Set cell is A2,
To value is 36, and By changing cell is A1. You should get an answer of 6 in cell A1. Now insert
the number −3 into cell A1 and repeat the same Goal Seek procedure. You should get an
answer of −6 in cell A1. If your answer was supposed to represent, for example, a nonnega-
tive production quantity, that could be a problem!
–
– Goal Seek should be able to indicate if there is no solution that will give you the target value
that you are seeking. Try the above experiment again by setting the target value as −49. Goal
Seek should search for a short time and then return the message, “Goal Seeking with cell A2
may not have found a solution” (because no number that is squared would produce a nega-
tive number). Goal Seek will place a value from the last iteration into cell A1, but that number
is meaningless and does not produce the desired result. Thus, always check the message in
the Goal Seek Status box carefully.
Operations Research Increases Revenues at HP
After the internet revolution, Hewlett Packard (HP) decided to change its marketing strategy and
enter the online consumer sales business. This was done through the creation of an “HPDirect.
com” portal that enables direct customers or retailers to buy HP products online. HP had some
initial problems in the e-commerce value chain process, and the firm faced fierce competition
from Dell’s online portal.
To help HP increase the accuracy of predicting the number of customers who are willing to buy its
products, which products will be bought, through which channels, and when, the data scientists
at HP Global Analytics (GA) used mathematical programming, Bayesian modeling, regression
analysis, and time-series forecasting models to develop solutions for customer acquisition,
development, and retention. The operations-research-based solutions that were implemented
helped increase the annual portal traffic by 2.6%, resulting in $44 million in incremental sales.
In addition, average order size increased by 15%, resulting in $63 million in additional revenues
and savings of $2 million through cost reduction from better inventory management.
Source: Based on S. Subramanian, D. Hill, and P. Dhore. “Hewlett Packard: Delivering Profitable
Growth for HPDirect.com Using Operations Research,” Interfaces 43, 1 (January – February 2013):
48–61.

Possible Problems in Developing Decision Models 21
1.6 Possible Problems in Developing Decision Models
We present the decision modeling approach as a logical and systematic means of
tackling decision-making problems. Even when these steps are followed carefully,
however, many difficulties can hurt the chances of implementing solutions to real-
world problems. We now look at problems that can occur during each of the steps of
the decision modeling approach.
Defining the Problem
In the worlds of business, government, and education, problems are, unfortunately, not
easily identified. Decision analysts typically face four roadblocks in defining a problem.
We use an application, inventory analysis, throughout this section as an example.
Conflicting Viewpoints Analysts often may have to consider conflicting viewpoints in
defining a problem. For example, in inventory problems, financial managers usually
believe that inventory is too high because inventory represents cash not available
for other investments. In contrast, sales managers often believe that inventory is too
low because high levels may be needed to fill unexpected orders. If analysts adopt
either of these views as the problem definition, they have essentially accepted one
manager’s perception. They can, therefore, expect resistance from the other manager
when the “solution” emerges. So, it’s important to consider both points of view before
stating the problem.
Impact on Other Departments Problems do not exist in isolation and are not owned
by just one department of a firm. For example, inventory is closely tied with cash flows
and various production problems. A change in ordering policy can affect cash flows
and upset production schedules to the point that savings on inventory are exceeded
by increased financial and production costs. The problem statement should therefore
be as broad as possible and include inputs from all concerned departments.
Beginning Assumptions People often tend to state problems in terms of solutions.
For example, the statement that inventory is too low implies a solution: that its levels
should be raised. An analyst who starts off with this assumption will likely find that
inventory should be raised! From an implementation perspective, a “good” solution
to the right problem is much better than an “optimal” solution to the wrong problem.
Solution Outdated Even if a problem has been specified correctly at present, it can
change during the development of the model. In today’s rapidly changing business
environment, especially with the amazing pace of technological advances, it is
not unusual for problems to change virtually overnight. The analyst who presents
solutions to problems that no longer exist can’t expect credit for providing timely help.

Discovering Diverse Content Through
Random Scribd Documents

CALIFORNIA CAR WINDOWS.
Lark songs ringing to Heaven,
Earth light clear as the sky;
Air like the breath of a greenhouse
With the greenhouse roof on high.
Flowers to see till you’re weary,
To travel in hours and hours;
Ranches of gold and purple,
Counties covered with flowers!
A rainbow, a running rainbow,
That flies at our side for hours;
A ribbon, a broidered ribbon,
A rainbow ribbon of flowers.

LIMITS.
On sand—loose sand and shifting—
On sand—dry sand and drifting—
The city grows to the west;
Not till its border reaches
The ocean-beaten beaches
Will it rest.
On hills—steep hills and lonely,
That stop at cloudland only—
The city climbs to the sky;
Not till the souls who make it
Touch the clear light and take it,
Will it die.

You start
From the town’s hot heart
To ride up Powell Street.
Hotel and theatre and crowding shops,
And Market’s cabled stream that never stops,
And the mixed hurrying beat
Of countless feet—
Take a front seat.
Before you rise
Six terraced hills, up to the low-hung skies;
Low where across the hill they seem to lie,
And then—how high!
Up you go slowly. To the right
A wide square, green and bright.
Above that green a broad façade,
Strongly and beautifully made,
In warm clear color standeth fair and true
Against the blue.
Only, above, two purple domes rise bold,
Twin-budded spires, bright-tipped with balls of gold.
Past that, and up you glide,
Up, up, till, either side,
Wide earth and water stretch around—away—
The straits, the hills, and the low-lying, wide-spread, dusky bay.
Great houses here,
Dull, opulent, severe.
Dives’ gold birds on guarding lamps a-wing—
Dead gold, that may not sing!
Fair on the other side
Smooth, steep-laid sweeps of turf and green boughs waving wide.
This is the hilltop’s crown.
Below you, down
In blurred, dim streets, the market quarter lies,
Foul, narrow, torn with cries
Of tortured things in cages, and the smell
Of daily bloodshed rising; that is hell.
But up here on the crown of Powell Street
The air is sweet;
A d th i f l t b d

And the green swaying mass of eucalyptus bends
Like hands of friends,
To gladden you despite the mansions’ frown.
Then you go down.
Down, down, and round the turns to lower grades;
Lower in all ways; darkening with the shades
Of poverty, old youth, and unearned age,
And that quick squalor which so blots the page
Of San Francisco’s beauty,—swift decay
Chasing the shallow grandeur of a day.
Here, like a noble lady of lost state,
Still calmly smiling at encroaching fate,
Amidst the squalor, rises Russian Hill,—
Proud, isolated, lonely, lovely still.
So on you glide.
Till the blue straits lie wide
Before you; purple mountains loom across,
And islands green as moss;
With soft white fog-wreaths drifting, drifting through
To comfort you;
And light, low-singing waves that tell you reach
The end,—North Beach.

A strange day—bright and still;
Strange for the stillness here,
For the strong trade-winds blow
With such a steady sweep it seems like rest,
Forever steadily across the crest
Of Russian Hill.
Still now and clear,—
So clear you count the houses spreading wide
In the fair cities on the farther side
Of our broad bay;
And brown Goat Island lieth large between,
Its brownness brightening into sudden green
From rains of yesterday.
Blue? Blue above of Californian sky,
Which has no peer on earth for its pure flame;
Bright blue of bay and strait spread wide below,
And, past the low, dull hills that hem it so,—
Blue as the sky, blue as the placid bay,—
Blue mountains far away.
Thanks this year for the early rains that came
To bless us, meaning Summer by and by.
This is our Spring-in-Autumn, making one
The Indian Summer tenderness of sun—
Its hazy stillness, and soft far-heard sound—
And the sweet riot of abundant spring,
The greenness flaming out from everything,
The sense of coming gladness in the ground.
From this high peace and purity look down;
Between you and the blueness lies the town.
Under those huddled roofs the heart of man
Beats warmer than this brooding day,
Spreads wider than the hill-rimmed bay,
And throbs to tenderer life, were it but seen,
Than all this new-born, all-enfolding green!
Within that heart lives still

Within that heart lives still
All that one guesses, dreams, and sees—
Sitting in sunlight, warm, at ease—
From this high island,—Russian Hill.

Again!
Another day of rain!
It has rained for years.
It never clears.
The clouds come down so low
They drag and drip
Across each hill-top’s tip.
In progress slow
They blow in from the sea
Eternally;
Hang heavily and black,
And then roll back;
And rain and rain and rain,
Both drifting in and drifting out again.
They come down to the ground,
These clouds, where the ground is high;
And, lest the weather fiend forget
And leave one hidden spot unwet,
The fog comes up to the sky!
And all our pavement of planks and logs
Reeks with the rain and steeps in the fogs
Till the water rises and sinks and presses
Into your bonnets and shoes and dresses;
And every outdoor-going dunce
Is wet in forty ways at once.
Wet?
It’s wetter than being drowned.
Dark?
Such darkness never was found
Since first the light was made. And cold?
O come to the land of grapes and gold,
Of fruit and flowers and sunshine gay,
When the rainy season’s under way!
And they tell you calmly, evermore,
They never had such rain before!
What’s that you say? Come out?

What s that you say? Come out?
Why, see that sky!
Oh, what a world! so clear! so high!
So clean and lovely all about;
The sunlight burning through and through,
And everything just blazing blue.
And look! the whole world blossoms again
The minute the sunshine follows the rain.
Warm sky—earth basking under—
Did it ever rain, I wonder?

The flowing waves of our warm sea
Roll to the beach and die,
But the soul of the waves forever fills
The curving crests of our restless hills
That climb so wantonly.
Up and up till you look to see
Along the cloud-kissed top
The great hill-breakers curve and comb
In crumbling lines of falling foam
Before they settle and drop.
Down and down, with the shuddering sweep
Of the sea-wave’s glassy wall,
You sink with a plunge that takes your breath,
A thrill that stirreth and quickeneth,
Like the great line steamer’s fall.
We have laid our streets by the square and line,
We have built by the line and square;
But the strong hill-rises arch below
And force the houses to curve and flow
In lines of beauty there.
And off to the north and east and south,
With wildering mists between,
They ring us round with wavering hold,
With fold on fold of rose and gold,
Violet, azure, and green.

CITY’S BEAUTY.
Fair, oh, fair are the hills uncrowned,
Only wreathed and garlanded
With the soft clouds overhead,
With the waving streams of rain;
Fair in golden sunlight drowned,
Bathed and buried in the bright
Warm luxuriance of light,—
Fair the hills without a stain.
Fairer far the hills should stand
Crownèd with a city’s halls,
With the glimmer of white walls,
With the climbing grace of towers;
Fair with great fronts tall and grand,
Stately streets that meet the sky,
Lovely roof-lines, low and high,—
Fairer for the days and hours.
Woman’s beauty fades and flies,
In the passing of the years,
With the falling of the tears,
With the lines of toil and stress;
City’s beauty never dies,—
Never while her people know
How to love and honor so
Her immortal loveliness.

TWO SKIES.
FROM ENGLAND.
They have a sky in Albion,
At least they tell me so;
But she will wear a veil so thick,
And she does have the sulks so quick,
And weeps so long and slow,
That one can hardly know.
Yes, there’s a sky in Albion.
She’s shown herself of late.
And where it was not white or gray,
It was quite bluish—in a way;
But near and full of weight,
Like an overhanging plate!
Our sky in California!
Such light the angels knew,
When the strong, tender smile of God
Kindled the spaces where they trod,
And made all life come true!
Deep, soundless, burning blue!

WINDS AND LEAVES.
FROM ENGLAND.
Wet winds that flap the sodden leaves!
Wet leaves that drop and fall!
Unhappy, leafless trees the wind bereaves!
Poor trees and small!
All of a color, solemn in your green;
All of a color, sombre in your brown;
All of a color, dripping gray between
When leaves are down!
O for the bronze-green eucalyptus spires
Far-flashing up against the endless blue!
Shifting and glancing in the steady fires
Of sun and moonlight too.
Dark orange groves! Pomegranate hedges bright,
And varnished fringes of the pepper trees!
And O that wind of sunshine! Wind of light!
Wind of Pacific seas!

ON THE PAWTUXET.
Broad and blue is the river, all bright in the sun;
The little waves sparkle, the little waves run;
The birds carol high, and the winds whisper low;
The boats beckon temptingly, row upon row;
Her hand is in mine as I help her step in.
Please Heaven, this day I shall lose or shall win—
Broad and blue is the river.
Cool and gray is the river, the sun sinks apace,
And the rose-colored twilight glows soft in her face.
In the midst of the rose-color Venus doth shine,
And the blossoming wild grapes are sweeter than wine;
Tall trees rise above us, four bridges are past,
And my stroke’s running slow as the current runs fast—
Cool and gray is the river.
Smooth and black is the river, no sound as we float
Save the soft-lapping water in under the boat.
The white mists are rising, the moon’s rising too,
And Venus, triumphant, rides high in the blue.
I hold the shawl round her, her hand is in mine,
And we drift under grape-blossoms sweeter than wine—
Smooth and black is the river.

A MOONRISE.
The heavy mountains, lying huge and dim,
With uncouth outline breaking heaven’s brim;
And while I watched and waited, o’er them soon,
Cloudy, enormous, spectral, rose the moon.

THEIR GRASS!
A PROTEST FROM CALIFORNIA.

They say we have no grass!
To hear them talk
You’d think that grass could walk
And was their bosom friend,—no day to pass
Between them and their grass.
“No grass!” they say who live
Where hot bricks give
The hot stones all their heat and back again,—
A baking hell for men.
“O, but,” they haste to say, “we have our parks,
Where fat policemen check the children’s larks;
And sign to sign repeats as in a glass,
‘Keep off the grass!’
We have our cities’ parks and grass, you see!”
Well—so have we!
But ’tis the country that they sing of most. “Alas,”
They sing, “for our wide acres of soft grass!—
To please us living and to hide us dead—”
You’d think Walt Whitman’s first was all they read!
You’d think they all went out upon the quiet
Nebuchadnezzar to outdo in diet!
You’d think they found no other green thing fair,
Even its seed an honor in their hair!
You’d think they had this bliss the whole year round,—
Evergreen grass!—and we, ploughed ground!
But come now, how does earth’s pet plumage grow
Under your snow?
Is your beloved grass as softly nice
When packed in ice?
For six long months you live beneath a blight,—
No grass in sight.
You bear up bravely. And not only that,
But leave your grass and travel; and thereat
We marvel deeply, with slow western mind,
Wondering within us what these people find
Among our common oranges and palms

Among our common oranges and palms
To tear them from the well-remembered charms
Of their dear vegetable. But still they come,
Frost-bitten invalids! to our bright home,
And chide our grasslessness! Until we say,
“But if you hate it so, why come? Why stay?
Just go away!
Go to—your grass!”

THE PROPHETS.
Time was we stoned the Prophets. Age on age,
When men were strong to save, the world hath slain them.
People are wiser now; they waste no rage—
The Prophets entertain them!

There was once a little animal,
No bigger than a fox,
And on five toes he scampered
Over Tertiary rocks.
They called him Eohippus,
And they called him very small,
And they thought him of no value—
When they thought of him at all;
For the lumpish old Dinoceras
And Coryphodon so slow
Were the heavy aristocracy
In days of long ago.
Said the little Eohippus,
“I am going to be a horse!
And on my middle finger-nails
To run my earthly course!
I’m going to have a flowing tail!
I’m going to have a mane!
I’m going to stand fourteen hands high
On the psychozoic plain!”
The Coryphodon was horrified,
The Dinoceras was shocked;
And they chased young Eohippus,
But he skipped away and mocked.
Then they laughed enormous laughter,
And they groaned enormous groans,
And they bade young Eohippus
Go view his father’s bones.
Said they, “You always were as small
And mean as now we see,
And that’s conclusive evidence
That you’re always going to be.
What! Be a great, tall, handsome beast,
With hoofs to gallop on?
Why! You’d have to change your nature!”
Said the Loxolophodon.
They considered him disposed of,
And retired with gait serene;

And retired with gait serene;
That was the way they argued
In “the early Eocene.”
There was once an Anthropoidal Ape,
Far smarter than the rest,
And everything that they could do
He always did the best;
So they naturally disliked him,
And they gave him shoulders cool,
And when they had to mention him
They said he was a fool.
Cried this pretentious Ape one day,
“I’m going to be a Man!
And stand upright, and hunt, and fight,
And conquer all I can!
I’m going to cut down forest trees,
To make my houses higher!
I’m going to kill the Mastodon!
I’m going to make a fire!”
Loud screamed the Anthropoidal Apes
With laughter wild and gay;
They tried to catch that boastful one,
But he always got away.
So they yelled at him in chorus,
Which he minded not a whit;
And they pelted him with cocoanuts,
Which didn’t seem to hit.
And then they gave him reasons
Which they thought of much avail,
To prove how his preposterous
Attempt was sure to fail.
Said the sages, “In the first place,
The thing cannot be done!
And, second, if it could be,
It would not be any fun!
And, third, and most conclusive,
And admitting no reply,

You would have to change your nature!
We should like to see you try!”
They chuckled then triumphantly,
These lean and hairy shapes,
For these things passed as arguments
With the Anthropoidal Apes.
There was once a Neolithic Man,
An enterprising wight,
Who made his chopping implements
Unusually bright.
Unusually clever he,
Unusually brave,
And he drew delightful Mammoths
On the borders of his cave.
To his Neolithic neighbors,
Who were startled and surprised,
Said he, “My friends, in course of time,
We shall be civilized!
We are going to live in cities!
We are going to fight in wars!
We are going to eat three times a day
Without the natural cause!
We are going to turn life upside down
About a thing called gold!
We are going to want the earth, and take
As much as we can hold!
We are going to wear great piles of stuff
Outside our proper skins!
We are going to have Diseases!
And Accomplishments!! And Sins!!!”
Then they all rose up in fury
Against their boastful friend,
For prehistoric patience
Cometh quickly to an end.
Said one, “This is chimerical!
Utopian! Absurd!”
Said another, “What a stupid life!
Too dull upon my word!”

Too dull, upon my word!
Cried all, “Before such things can come,
You idiotic child,
You must alter Human Nature!”
And they all sat back and smiled.
Thought they, “An answer to that last
It will be hard to find!”
It was a clinching argument
To the Neolithic Mind!

The garden beds I wandered by
One bright and cheerful morn,
When I found a new-fledged butterfly
A-sitting on a thorn,
A black and crimson butterfly,
All doleful and forlorn.
I thought that life could have no sting
To infant butterflies,
So I gazed on this unhappy thing
With wonder and surprise,
While sadly with his waving wing
He wiped his weeping eyes.
Said I, “What can the matter be?
Why weepest thou so sore?
With garden fair and sunlight free
And flowers in goodly store—”
But he only turned away from me
And burst into a roar.
Cried he, “My legs are thin and few
Where once I had a swarm!
Soft fuzzy fur—a joy to view—
Once kept my body warm,
Before these flapping wing-things grew,
To hamper and deform!”
At that outrageous bug I shot
The fury of mine eye;
Said I, in scorn all burning hot,
In rage and anger high,
“You ignominious idiot!
Those wings are made to fly!”
“I do not want to fly,” said he,
“I only want to squirm!”
And he drooped his wings dejectedly,
But still his voice was firm;
“I do not want to be a fly!

I do not want to be a fly!
I want to be a worm!”
O yesterday of unknown lack!
To-day of unknown bliss!
I left my fool in red and black,
The last I saw was this,—
The creature madly climbing back
Into his chrysalis.

I was climbing up a mountain-path
With many things to do,
Important business of my own,
And other people’s too,
When I ran against a Prejudice
That quite cut off the view.
My work was such as could not wait,
My path quite clearly showed,
My strength and time were limited,
I carried quite a load;
And there that hulking Prejudice
Sat all across the road.
So I spoke to him politely,
For he was huge and high,
And begged that he would move a bit
And let me travel by.
He smiled, but as for moving!—
He didn’t even try.
And then I reasoned quietly
With that colossal mule:
My time was short—no other path—
The mountain winds were cool.
I argued like a Solomon;
He sat there like a fool.
Then I flew into a passion,
I danced and howled and swore.
I pelted and belabored him
Till I was stiff and sore;
He got as mad as I did—
But he sat there as before.
And then I begged him on my knees;
I might be kneeling still
If so I hoped to move that mass
Of obdurate ill-will—
As well invite the monument

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Managerial Decision Modeling 4th Edition Nagraj Balakrishnan

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