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Each student is required to design an efficient portfolio frontier consisting of at least four companies (each from a different S &P 500 sector) using historical annual return data over the last fifteen years. A brief discussion of why the student selected the specific sectors and a specific company within that sector must be included in the assignment. Chapter 8 of Ragsdale shows a step by step approach for building the portfolio and drawing the efficient frontier diagram using Excel Solver.

Each student must only a single Word document for this assignment.

The attached file is the PDF of the reference book and the link will help you to learn about S&P sectors

https://corporatefinanceinstitute.com/resources/kn…

; and you can find historical annual return data from

https://finance.yahoo.com/quote/AAPL/history

; and the analysis template for AAPL is shown in the screenshot.

Cliff T.
Ragsdale
& Decision Analysis 5e
REVISED
A Practical Introduction to Management Science
Contents
1. Introduction to Modeling and Decision Analysis 1
Introduction 1
The Modeling Approach to Decision Making 3
Characteristics and Benefits of Modeling 3
Mathematical Models 4
Categories of Mathematical Models 6
The Problem-Solving Process 7
Anchoring and Framing Effects 9
Good Decisions vs. Good Outcomes 11
Summary 11
References 12
The World of Management Science 12
Questions and Problems 14
Case 14
2. Introduction to Optimization and Linear Programming 17
Introduction 17
Applications of Mathematical Optimization 17
Characteristics of Optimization Problems 18
Expressing Optimization Problems Mathematically 19
Decisions 19
Constraints 19
Objective 20
Mathematical Programming Techniques 20
An Example LP Problem 21
Formulating LP Models 21
Steps in Formulating an LP Model 21
Summary of the LP Model for the Example Problem 23
The General Form of an LP Model 23
Solving LP Problems: An Intuitive Approach 24
Solving LP Problems: A Graphical Approach 25
Plotting the First Constraint 26 Plotting the Second Constraint 26 Plotting the Third
Constraint 27 The Feasible Region 28 Plotting the Objective Function 29 Finding the
Optimal Solution Using Level Curves 30 Finding the Optimal Solution by Enumerating
the Corner Points 32 Summary of Graphical Solution to LP Problems 32
Understanding How Things Change 33
Special Conditions in LP Models 34
Alternate Optimal Solutions 34
Infeasibility 38
Summary 39
viii
Redundant Constraints 35
Unbounded Solutions 37
Contents
ix
References 39
Questions and Problems 39
Case 44
3. Modeling and Solving LP Problems in a Spreadsheet 45
Introduction 45
Solving LP Problems in a Spreadsheet 46
The Steps in Implementing an LP Model in a Spreadsheet 46
A Spreadsheet Model for the Blue Ridge Hot Tubs Problem 48
Organizing the Data 49 Representing the Decision Variables 49 Representing the
Objective Function 49 Representing the Constraints 50 Representing the Bounds on the
Decision Variables 50
How Solver Views the Model 51
Using Solver 53
Deï¬ning the Set (or Target) Cell 54 Deï¬ning the Variable Cells 56 Deï¬ning the
Constraint Cells 56 Deï¬ning the Nonnegativity Conditions 58 Reviewing the Model 59
Options 59 Solving the Model 59
Goals and Guidelines for Spreadsheet Design 61
Make vs. Buy Decisions 63
Deï¬ning the Decision Variables 63 Deï¬ning the Objective Function 64 Deï¬ning the
Constraints 64 Implementing the Model 64 Solving the Model 66 Analyzing the
Solution 66
An Investment Problem 67
Deï¬ning the Decision Variables 68 Deï¬ning the Objective Function 68 Deï¬ning the
Constraints 69 Implementing the Model 69 Solving the Model 71 Analyzing the
Solution 72
A Transportation Problem 72
Deï¬ning the Decision Variables 72 Deï¬ning the Objective Function 73 Deï¬ning the
Constraints 73 Implementing the Model 74 Heuristic Solution for the Model 76
Solving the Model 76 Analyzing the Solution 77
A Blending Problem 78
Deï¬ning the Decision Variables 79 Deï¬ning the Objective Function 79 Deï¬ning the
Constraints 79 Some Observations About Constraints, Reporting, and Scaling 80
Rescaling the Model 81 Implementing the Model 82 Solving the Model 83 Analyzing
the Solution 84
A Production and Inventory Planning Problem 85
Deï¬ning the Decision Variables 85 Deï¬ning the Objective Function 86 Deï¬ning the
Constraints 86 Implementing the Model 87 Solving the Model 89 Analyzing the
Solution 90
A Multi-Period Cash Flow Problem 91
Deï¬ning the Decision Variables 91 Deï¬ning the Objective Function 92 Deï¬ning the
Constraints 92 Implementing the Model 94 Solving the Model 96 Analyzing the
Solution 96 Modifying The Taco-Viva Problem to Account for Risk (Optional) 98
Implementing the Risk Constraints 100 Solving the Model 101 Analyzing the
Solution 102
x
Contents
Data Envelopment Analysis 102
Deï¬ning the Decision Variables 103 Deï¬ning the Objective 103 Deï¬ning the constraints
103 Implementing the Model 104 Solving the Model 106 Analyzing the Solution 111
Summary 112
References 113
The World of Management Science 113
Questions and Problems 114
Cases 130
4. Sensitivity Analysis and the Simplex Method 136
Introduction 136
The Purpose of Sensitivity Analysis 136
Approaches to Sensitivity Analysis 137
An Example Problem 137
The Answer Report 138
The Sensitivity Report 140
Changes in the Objective Function Coefï¬cients 140
A Note About Constancy 142 Alternate Optimal Solutions 143 Changes in the RHS
Values 143 Shadow Prices for Nonbinding Constraints 144 A Note About Shadow
Prices 144 Shadow Prices and the Value of Additional Resources 146 Other Uses of
Shadow Prices 146 The Meaning of the Reduced Costs 147 Analyzing Changes in
Constraint Coefï¬cients 149 Simultaneous Changes in Objective Function Coefï¬cients 150
A Warning About Degeneracy 151
The Limits Report 151
The Sensitivity Assistant Add-in (Optional) 152
Creating Spider Tables and Plots 153
Creating a Solver Table 155
The Simplex Method (Optional) 158
Creating Equality Constraints Using Slack Variables 158
Finding the Best Solution 162
Basic Feasible Solutions 159
Summary 162
References 162
The World of Management Science 163
Questions and Problems 164
Cases 171
5. Network Modeling 177
Introduction 177
The Transshipment Problem 177
Characteristics of Network Flow Problems 177 The Decision Variables for Network Flow
Problems 179 The Objective Function for Network Flow Problems 179 The Constraints
for Network Flow Problems 180 Implementing the Model in a Spreadsheet 181
Analyzing the Solution 182
The Shortest Path Problem 184
An LP Model for the Example Problem 186 The Spreadsheet Model and Solution 186
Network Flow Models and Integer Solutions 188
Contents
The Equipment Replacement Problem 189
The Spreadsheet Model and Solution 190
Transportation/Assignment Problems 193
Generalized Network Flow Problems 194
Formulating an LP Model for the Recycling Problem 195 Implementing the Model 196
Analyzing the Solution 198 Generalized Network Flow Problems and Feasibility 199
Maximal Flow Problems 201
An Example of a Maximal Flow Problem 201
The Spreadsheet Model and Solution 203
Special Modeling Considerations 205
Minimal Spanning Tree Problems 208
An Algorithm for the Minimal Spanning Tree Problem 209
Problem 209
Solving the Example
Summary 210
References 210
The World of Management Science 211
Questions and Problems 212
Cases 227
6. Integer Linear Programming 232
Introduction 232
Integrality Conditions 232
Relaxation 233
Solving the Relaxed Problem 233
Bounds 235
Rounding 236
Stopping Rules 239
Solving ILP Problems Using Solver 240
Other ILP Problems 243
An Employee Scheduling Problem 243
Deï¬ning the Decision Variables 244 Deï¬ning the Objective Function 245 Deï¬ning the
Constraints 245 A Note About the Constraints 245 Implementing the Model 246
Solving the Model 247 Analyzing the Solution 247
Binary Variables 248
A Capital Budgeting Problem 249
Deï¬ning the Decision Variables 249 Deï¬ning the Objective Function 250 Deï¬ning the
Constraints 250 Setting Up the Binary Variables 250 Implementing the Model 250
Solving the Model 251 Comparing the Optimal Solution to a Heuristic Solution 253
Binary Variables and Logical Conditions 253
The Fixed-Charge Problem 254
Deï¬ning the Decision Variables 255 Deï¬ning the Objective Function 255 Deï¬ning the
Constraints 256 Determining Values for â€œBig Mâ€ 256 Implementing the Model 257
Solving the Model 259 Analyzing the Solution 260
Minimum Order/Purchase Size 261
Quantity Discounts 261
Formulating the Model 262
The Missing Constraints 262
xi
xii
Contents
A Contract Award Problem 262
Formulating the Model: The Objective Function and Transportation Constraints 263
Implementing the Transportation Constraints 264 Formulating the Model: The Side
Constraints 265 Implementing the Side Constraints 266 Solving the Model 267
Analyzing the Solution 268
The Branch-and-Bound Algorithm (Optional) 268
Branching 269 Bounding 272
of B&B Example 274
Branching Again 272
Bounding Again 272
Summary
Summary 274
References 275
The World of Management Science 276
Questions and Problems 276
Cases 291
7. Goal Programming and Multiple Objective Optimization 296
Introduction 296
Goal Programming 296
A Goal Programming Example 297
Deï¬ning the Decision Variables 298 Deï¬ning the Goals 298 Deï¬ning the Goal
Constraints 298 Deï¬ning the Hard Constraints 299 GP Objective Functions 300
Deï¬ning the Objective 301 Implementing the Model 302 Solving the Model 303
Analyzing the Solution 303 Revising the Model 304 Trade-offs: The Nature of GP 305
Multiple Objective Optimization 307
An MOLP Example 309
Deï¬ning the Decision Variables 309 Deï¬ning the Objectives 310 Deï¬ning the
Constraints 310 Implementing the Model 310 Determining Target Values for the
Objectives 311 Summarizing the Target Solutions 313 Determining a GP Objective 314
The MINIMAX Objective 316 Implementing the Revised Model 317
Solving the Model 318
Comments on MOLP 320
Summary 321
References 321
The World of Management Science 321
Questions and Problems 322
Cases 334
8. Nonlinear Programming & Evolutionary Optimization 339
Introduction 339
The Nature of NLP Problems 339
Solution Strategies for NLP Problems 341
Local vs. Global Optimal Solutions 342
Economic Order Quantity Models 344
Contents
Implementing the Model 347 Solving the Model 348
Comments on the EOQ Model 349
xiii
Analyzing the Solution 349
Location Problems 350
Deï¬ning the Decision Variables 351 Deï¬ning the Objective 351 Deï¬ning the
Constraints 352 Implementing the Model 352 Solving the Model and Analyzing the
Solution 353 Another Solution to the Problem 354 Some Comments About the Solution
to Location Problems 354
Nonlinear Network Flow Problem 355
Deï¬ning the Decision Variables 356 Deï¬ning the Objective 356 Deï¬ning the
Constraints 357 Implementing the Model 357 Solving the Model and Analyzing
the Solution 360
Project Selection Problems 360
Deï¬ning the Decision Variables 361 Deï¬ning the Objective Function 361 Deï¬ning
the Constraints 362 Implementing the Model 362 Solving the Model 364
Optimizing Existing Financial Spreadsheet Models 365
Implementing the Model 365 Optimizing the Spreadsheet Model 367 Analyzing
the Solution 368 Comments on Optimizing Existing Spreadsheets 368
The Portfolio Selection Problem 368
Deï¬ning the Decision Variables 370 Deï¬ning the Objective 370 Deï¬ning the
Constraints 371 Implementing the Model 371 Analyzing the Solution 373
Handling Conï¬‚icting Objectives in Portfolio Problems 374
Sensitivity Analysis 376
Lagrange Multipliers 378
Solver Options for Solving NLPs 379
Evolutionary Algorithms 380
Beating the Market 382
A Spreadsheet Model for the Problem 382
Solution 384
Solving the Model 383
Analyzing the
The Traveling Salesperson Problem 385
A Spreadsheet Model for the Problem 386
Solution 387
Solving the Model 387
Summary 389
References 389
The World of Management Science 389
Questions and Problems 390
Cases 404
9. Regression Analysis 409
Introduction 409
An Example 409
Regression Models 411
Simple Linear Regression Analysis 412
Defining â€œBest Fitâ€ 413
Solving the Problem Using Solver 414
Solving the Problem Using the Regression Tool 417
Evaluating the Fit 419
Analyzing the
xiv
Contents
The R2 Statistic 421
Making Predictions 422
The Standard Error 423 Prediction Intervals for New Values of Y 423
Intervals for Mean Values of Y 425 A Note About Extrapolation 426
Conï¬dence
Statistical Tests for Population Parameters 426
Analysis of Variance 427
Statistical Tests 430
Assumptions for the Statistical Tests 427
Introduction to Multiple Regression 430
A Multiple Regression Example 431
Selecting the Model 433
Models with One Independent Variable 433 Models with Two Independent Variables
434 Inï¬‚ating R2 436 The Adjusted-R2 Statistic 437 The Best Model with Two
Independent Variables 437 Multicollinearity 437 The Model with Three Independent
Variables 438
Making Predictions 439
Binary Independent Variables 440
Statistical Tests for the Population Parameters 440
Polynomial Regression 441
Expressing Nonlinear Relationships Using Linear Models 442
Regression 446
Summary of Nonlinear
Summary 446
References 447
The World of Management Science 447
Questions and Problems 448
Cases 454
10. Discriminant Analysis 459
Introduction 459
The Two-Group DA Problem 460
Group Locations and Centroids 460 Calculating Discriminant Scores 461 The
Classiï¬cation Rule 465 Reï¬ning the Cutoff Value 466 Classiï¬cation Accuracy 467
Classifying New Employees 468
The k-Group DA Problem 469
Multiple Discriminant Analysis 471
Distance Measures 472
Summary 477
References 477
The World of Management Science 478
Questions and Problems 478
Cases 481
11. Time Series Forecasting 485
Introduction 485
Time Series Methods 486
Measuring Accuracy 486
Stationary Models 487
MDA Classiï¬cation 474
Contents
Moving Averages 488
Forecasting with the Moving Average Model 490
Weighted Moving Averages 492
Forecasting with the Weighted Moving Average Model 493
Exponential Smoothing 494
Forecasting with the Exponential Smoothing Model 496
Seasonality 498
Stationary Data with Additive Seasonal Effects 500
Forecasting with the Model 502
Stationary Data with Multiplicative Seasonal Effects 504
Forecasting with the Model 507
Trend Models 507
An Example 507
Double Moving Average 508
Forecasting with the Model 510
Double Exponential Smoothing (Holtâ€™s Method) 511
Forecasting with Holtâ€™s Method 513
Holt-Winterâ€™s Method for Additive Seasonal Effects 514
Forecasting with Holt-Winterâ€™s Additive Method 517
Holt-Winterâ€™s Method for Multiplicative Seasonal Effects 518
Forecasting with Holt-Winterâ€™s Multiplicative Method 521
Modeling Time Series Trends Using Regression 522
Linear Trend Model 523
Forecasting with the Linear Trend Model 525
Quadratic Trend Model 526
Forecasting with the Quadratic Trend Model 528
Modeling Seasonality with Regression Models 528
Adjusting Trend Predictions with Seasonal Indices 529
Computing Seasonal Indices 530
Seasonal Indices 532
Forecasting with Seasonal Indices 531
Reï¬ning the
Seasonal Regression Models 534
The Seasonal Model 535
Forecasting with the Seasonal Regression Model 536
Crystal Ball Predictor 538
Using CB Predictor 538
Combining Forecasts 544
Summary 544
References 545
The World of Management Science 545
Questions and Problems 546
Cases 554
12. Introduction to Simulation Using Crystal Ball 559
Introduction 559
Random Variables and Risk 559
xv
xvi
Contents
Why Analyze Risk? 560
Methods of Risk Analysis 560
Best-Case/Worst-Case Analysis 561
What-If Analysis 562
Simulation 562
A Corporate Health Insurance Example 563
A Critique of the Base Case Model 565
Spreadsheet Simulation Using Crystal Ball 565
Starting Crystal Ball 566
Random Number Generators 566
Discrete vs. Continuous Random Variables 569
Preparing the Model for Simulation 570
Deï¬ning Assumptions for the Number of Covered Employees 572 Deï¬ning
Assumptions for the Average Monthly Claim per Employee 574 Deï¬ning Assumptions
for the Average Monthly Claim per Employee 575
Running the Simulation 576
Selecting the Output Cells to Track 576 Selecting the Number of Iterations 577
Determining the Sample Size 577 Running the Simulation 578
Data Analysis 578
The Best Case and the Worst Case 579 The Distribution of the Output Cell 579 Viewing
the Cumulative Distribution of the Output Cells 580 Obtaining Other Cumulative
Probabilities 581
Incorporating Graphs and Statistics into a Spreadsheet 581
The Uncertainty of Sampling 581
Constructing a Conï¬dence Interval for the True Population Mean 583 Constructing a
Conï¬dence Interval for a Population Proportion 584 Sample Sizes and Conï¬dence
Interval Widths 585
The Benefits of Simulation 585
Additional Uses of Simulation 586
A Reservation Management Example 587
Implementing the Model 587
Using the Decision Table Tool 589
An Inventory Control Example 595
Implementing the Model 596 Replicating the Model 600 Optimizing the Model 601
Comparing the Original and Optimal Ordering Policies 603
A Project Selection Example 604
A Spreadsheet Model 605
Solutions 609
Solving the Problem with OptQuest 607
A Portfolio Optimization Example 611
A Spreadsheet Model 612
Solving the Problem with OptQuest 615
Summary 616
References 617
The World of Management Science 617
Questions and Problems 618
Cases 632
Considering Other
Contents
xvii
13. Queuing Theory 641
Introduction 641
The Purpose of Queuing Models 641
Queuing System Configurations 642
Characteristics of Queuing Systems 643
Arrival Rate 644
Service Rate 645
Kendall Notation 647
Queuing Models 647
The M/M/s Model 648
An Example 649 The Current Situation 650 Adding a Server 650 Economic
Analysis 651
The M/M/s Model with Finite Queue Length 652
The Current Situation 653
Adding a Server 653
The M/M/s Model with Finite Population 654
An Example 655
The Current Situation 655
The M/G/1 Model 658
The Current Situation 659
Adding the Automated Dispensing Device 659
The M/D/1 Model 661
Simulating Queues and the Steady-state Assumption 662
Summary 663
References 663
The World of Management Science 663
Questions and Problems 665
Cases 671
14. Project Management 673
Introduction 673
An Example 673
Creating the Project Network 674
A Note on Start and Finish Points 676
CPM: An Overview 677
The Forward Pass 678
The Backward Pass 680
Determining the Critical Path 682
A Note on Slack 683
Project Management Using Spreadsheets 684
Important Implementation Issue 688
Gantt Charts 688
Project Crashing 691
An LP Approach to Crashing 691 Determining the Earliest Crash Completion Time 693
Implementing the Model 694 Solving the Model 695 Determining a Least Costly Crash
Schedule 696 Crashing as an MOLP 698
xviii
Contents
PERT: An Overview 699
The Problems with PERT 700
Implications 702
Simulating Project Networks 702
An Example 702 Generating Random Activity Times 702
Running the Simulation 704 Analyzing the Results 706
Implementing the Model 704
Microsoft Project 707
Summary 710
References 710
The World of Management Science 710
Questions and Problems 711
Cases 720
15. Decision Analysis 724
Introduction 724
Good Decisions vs. Good Outcomes 724
Characteristics of Decision Problems 725
An Example 725
The Payoff Matrix 726
Decision Alternatives 727
States of Nature 727
The Payoff Values 727
Decision Rules 728
Nonprobabilistic Methods 729
The Maximax Decision Rule 729
Decision Rule 731
The Maximin Decision Rule 730
The Minimax Regret
Probabilistic Methods 733
Expected Monetary Value 733
Expected Regret 735
Sensitivity Analysis 736
The Expected Value of Perfect Information 738
Decision Trees 739
Rolling Back a Decision Tree 740
Using TreePlan 742
Adding Branches 743 Adding Event Nodes 744 Adding the Cash Flows 748
Determining the Payoffs and EMVs 748 Other Features 749
Multistage Decision Problems 750
A Multistage Decision Tree 751
Developing A Risk Proï¬le 753
Sensitivity Analysis 754
Spider Charts and Tornado Charts 755
Strategy Tables 758
Using Sample Information in Decision Making 760
Conditional Probabilities 761
The Expected Value of Sample Information 762
Computing Conditional Probabilities 763
Bayesâ€™s Theorem 765
Utility Theory 766
Utility Functions 766 Constructing Utility Functions 767 Using Utilities to Make
Decisions 770 The Exponential Utility Function 770 Incorporating Utilities
in TreePlan 771
Multicriteria Decision Making 772
Contents
The Multicriteria Scoring Model 773
The Analytic Hierarchy Process 777
Pairwise Comparisons 777 Normalizing the Comparisons 779 Consistency 780
Obtaining Scores for the Remaining Criteria 781 Obtaining Criterion Weights 782
Implementing the Scoring Model 783
Summary 783
References 784
The World of Management Science 785
Questions and Problems 786
Cases 796
Index 801
xix
Chapter 1
Introduction to Modeling
and Decision Analysis
1.0 Introduction
This book is titled Spreadsheet Modeling and Decision Analysis: A Practical Introduction to
Management Science, so letâ€™s begin by discussing exactly what this title means. By the
very nature of life, all of us must continually make decisions that we hope will solve
problems and lead to increased opportunities for ourselves or the organizations for
which we work. But making good decisions is rarely an easy task. The problems faced
by decision makers in todayâ€™s competitive, fast-paced business environment are often
extremely complex and can be addressed by numerous possible courses of action. Evaluating these alternatives and choosing the best course of action represents the essence of
decision analysis.
During the past decade, millions of business people discovered that one of the most
effective ways to analyze and evaluate decision alternatives involves using electronic
spreadsheets to build computer models of the decision problems they face. A computer
model is a set of mathematical relationships and logical assumptions implemented in a
computer as a representation of some real-world decision problem or phenomenon.
Today, electronic spreadsheets provide the most convenient and useful way for business
people to implement and analyze computer models. Indeed, most business people
probably would rate the electronic spreadsheet as their most important analytical tool
apart from their brain! Using a spreadsheet model (a computer model implemented via
a spreadsheet), a business person can analyze decision alternatives before having to
choose a speciï¬c plan for implementation.
This book introduces you to a variety of techniques from the ï¬eld of management science that can be applied in spreadsheet models to assist in the decision-analysis process.
For our purposes, we will deï¬ne management science as a ï¬eld of study that uses computers, statistics, and mathematics to solve business problems. It involves applying the
methods and tools of science to management and decision making. It is the science of
making better decisions. Management science is also sometimes referred to as operations research or decision science. See Figure 1.1 for a summary of how management science has been applied successfully in several real-world situations.
In the not too distant past, management science was a highly specialized ï¬eld that
generally could be practiced only by those who had access to mainframe computers and
who possessed an advanced knowledge of mathematics and computer programming
languages. However, the proliferation of powerful personal computers (PCs) and the
development of easy-to-use electronic spreadsheets have made the tools of management science far more practical and available to a much larger audience. Virtually
1
2
Chapter 1
Introduction to Modeling and Decision Analysis
FIGURE 1.1
Examples of
successful
management
science
applications
Home Runs in Management Science
Over the past decade, scores of operations research and management science
projects saved companies millions of dollars. Each year, the Institute For Operations Research and the Management Sciences (INFORMS) sponsors the Franz
Edelman Awards competition to recognize some of the most outstanding OR/MS
projects during the past year. Here are some of the â€œhome runsâ€ from the 2004
Edelman Awards (described in Interfaces, Vol. 31, No. 1, Januaryâ€“February, 2005).
â€¢ At the turn of the century, Motorola faced a crisis due to economic conditions
in its marketplaces; the company needed to reduce costs dramatically and
quickly. A natural target was its purchases of goods and services, as these expenses account for more than half of Motorolaâ€™s costs. Motorola decided to create an Internet-based system to conduct multi-step negotiations and auctions
for supplier negotiation. The system can handle complex bids and constraints,
such as bundled bids, volume-based discounts, and capacity limits. In addition, it can optimize multi-product, multi-vendor awards subject to these constraints and nonlinear price schedules. Beneï¬ts: In 2003, Motorola used this
system to source 56 percent of its total spending, with 600 users and a total savings exceeding \$600 million.
â€¢ Waste Management is the leading company in North America in the wastecollection industry. The company has a ï¬‚eet of over 26,000 vehicles for collecting
waste from nearly 20 million residential customers, plus another two million
commercial customers. To improve trash collection and make its operations more
efï¬cient, Waste Management implemented a vehicle-routing application to optimize its collection routes. Beneï¬ts: The successful deployment of this system
brought beneï¬ts including the elimination of nearly 1,000 routes within one year
of implementation and an estimated annual savings of \$44 million.
â€¢ Hong Kong has the worldâ€™s busiest port. Its largest terminal operator, Hong
Kong International Terminals (HIT), has the busiest container terminal in the
world serving over 125 ships per week, with 10 berths at which container ships
dock, and 122 yard cranes to move containers around the 227 acres of storage
yard. Thousands of trucks move containers into and out of the storage yard
each day. HIT implemented a decision-support system (with several embedded decision models and algorithms) to guide its operational decisions concerning the number and deployment of trucks for moving containers, the assignment of yard cranes, and the storage locations for containers. Beneï¬ts: The
cumulative effect of this system has led to a 35 percent reduction in container
handling costs, a 50 percent increase in throughput, and a 30 percent improvement in vessel turnaround time.
â€¢ The John Deere Company sells lawn equipment, residential and commercial
mowers, and utility tractors through a network of 2,500 dealers, supported by ï¬ve
Deere warehouses. Each dealer stocks about 100 products, leading to approximately 250,000 product-stocking locations. Furthermore, demand is quite seasonal
and stochastic. Deere implemented a system designed to optimize large-scale
multi-echelon, non-stationary stochastic inventory systems. Deere runs the
system each week to obtain recommended stocking levels for each product for
each stocking location for each week over a 26-week planning horizon. Beneï¬ts:
The impact of the application has been remarkable, leading to an inventory reduction of nearly one billion dollars and improving customer-service levels.
Characteristics and Benefits of Modeling
3
everyone who uses a spreadsheet today for model building and decision making is a
practitioner of management scienceâ€”whether they realize it or not.
1.1 The Modeling Approach
to Decision Making
The idea of using models in problem solving and decision analysis is really not new, and
certainly is not tied to the use of computers. At some point, all of us have used a modeling approach to make a decision. For example, if you ever have moved into a dormitory,
apartment, or house, you undoubtedly faced a decision about how to arrange the furniture in your new dwelling. There probably were several different arrangements to consider. One arrangement might give you the most open space but require that you build
a loft. Another might give you less space but allow you to avoid the hassle and expense
of building a loft. To analyze these different arrangements and make a decision, you did
not build the loft. You more likely built a mental model of the two arrangements, picturing what each looked like in your mindâ€™s eye. Thus, a simple mental model is sometimes all that is required to analyze a problem and make a decision.
For more complex decision problems, a mental model might be impossible or insufï¬cient, and other types of models might be required. For example, a set of drawings or
blueprints for a house or building provides a visual model of the real-world structure.
These drawings help illustrate how the various parts of the structure will ï¬t together
when it is completed. A road map is another type of visual model because it assists a driver in analyzing the various routes from one location to another.
You probably also have seen car commercials on television showing automotive engineers using physical models or scale models to study the aerodynamics of various
car designs, to ï¬nd the shape that creates the least wind resistance and maximizes fuel
economy. Similarly, aeronautical engineers use scale models of airplanes to study the
ï¬‚ight characteristics of various fuselage and wing designs. And civil engineers might
use scale models of buildings and bridges to study the strengths of different construction techniques.
Another common type of model is a mathematical model, which uses mathematical
relationships to describe or represent an object or decision problem. Throughout this
book we will study how various mathematical models can be implemented and analyzed on computers using spreadsheet software. But before we move to an in-depth
discussion of spreadsheet models, letâ€™s look at some of the more general characteristics
and beneï¬ts of modeling.
1.2 Characteristics and Benefits
of Modeling
Although this book focuses on mathematical models implemented in computers via
spreadsheets, the examples of non-mathematical models given earlier are worth discussing a bit more because they help illustrate several important characteristics and
beneï¬ts of modeling in general. First, the models mentioned earlier are usually simpliï¬ed versions of the object or decision problem they represent. To study the aerodynamics
of a car design, we do not need to build the entire car complete with engine and stereo.
Such components have little or no effect on aerodynamics. So, although a model is often
4
Chapter 1
Introduction to Modeling and Decision Analysis
a simpliï¬ed representation of reality, the model is useful as long as it is valid. A valid
model is one that accurately represents the relevant characteristics of the object or decision problem being studied.
Second, it is often less expensive to analyze decision problems using a model. This is
especially easy to understand with respect to scale models of big-ticket items such as
cars and planes. Besides the lower ï¬nancial cost of building a model, the analysis of a
model can help avoid costly mistakes that might result from poor decision making. For
example, it is far less costly to discover a ï¬‚awed wing design using a scale model of an
aircraft than after the crash of a fully loaded jetliner.
Frank Brock, former executive vice president of the Brock Candy Company, related
the following story about blueprints his company prepared for a new production facility. After months of careful design work he proudly showed the plans to several of his
production workers. When he asked for their comments, one worker responded, â€œItâ€™s a
ï¬ne looking building, Mr. Brock, but that sugar valve looks like itâ€™s about twenty feet
away from the steam valve.â€ â€œWhatâ€™s wrong with that?â€ asked Brock. â€œWell, nothing,â€
said the worker, â€œexcept that I have to have my hands on both valves at the same time!â€1
Needless to say, it was far less expensive to discover and correct this â€œlittleâ€ problem
using a visual model before pouring the concrete and laying the pipes as originally
planned.
Third, models often deliver needed information on a more timely basis. Again, it is
relatively easy to see that scale models of cars or airplanes can be created and analyzed
more quickly than their real-world counterparts. Timeliness is also an issue when vital
data will not become available until later. In these cases, we might create a model to help
predict the missing data to assist in current decision making.
Fourth, models are frequently helpful in examining things that would be impossible
to do in reality. For example, human models (crash dummies) are used in crash tests to
see what might happen to an actual person if a car were to hit a brick wall at a high
speed. Likewise, models of DNA can be used to visualize how molecules ï¬t together.
Both of these are difï¬cult, if not impossible, to do without the use of models.
Finally, and probably most important, models allow us to gain insight and understanding about the object or decision problem under investigation. The ultimate purpose of using models is to improve decision making. As you will see, the process of
building a model can shed important light and understanding on a problem. In some
cases, a decision might be made while building the model as a previously misunderstood element of the problem is discovered or eliminated. In other cases, a careful analysis of a completed model might be required to â€œget a handleâ€ on a problem and gain the
insights needed to make a decision. In any event, the insight gained from the modeling
process ultimately leads to better decision making.
1.3 Mathematical Models
As mentioned earlier, the modeling techniques in this book differ quite a bit from scale
models of cars and planes, or visual models of production plants. The models we will
build use mathematics to describe a decision problem. We use the term â€œmathematicsâ€
in its broadest sense, encompassing not only the most familiar elements of math, such as
algebra, but also the related topic of logic.
1
Colson, Charles and Jack Eckerd, Why America Doesnâ€™t Work (Denver, Colorado: Word Publishing, 1991), 146â€“147.
Mathematical Models
5
Now, letâ€™s consider a simple example of a mathematical model:
PROFIT = REVENUE âˆ’ EXPENSES
1.1
Equation 1.1 describes a simple relationship between revenue, expenses, and proï¬t.
It is a mathematical relationship that describes the operation of determining proï¬tâ€”or
a mathematical model of proï¬t. Of course, not all models are this simple, but taken piece
by piece, the models we will discuss are not much more complex than this one.
Frequently, mathematical models describe functional relationships. For example, the
mathematical model in equation 1.1 describes a functional relationship between revenue, expenses, and proï¬t. Using the symbols of mathematics, this functional relationship is represented as:
PROFIT = f(REVENUE, EXPENSES)
1.2
In words, the previous expression means â€œproï¬t is a function of revenue and
expenses.â€ We also could say that proï¬t depends on (or is dependent on) revenue and
expenses. Thus, the term PROFIT in equation 1.2 represents a dependent variable,
whereas REVENUE and EXPENSES are independent variables. Frequently, compact
symbols (such as A, B, and C) are used to represent variables in an equation such as 1.2.
For instance, if we let Y, X1, and X2 represent PROFIT, REVENUE, and EXPENSES, respectively, we could rewrite equation 1.2 as follows:
Y = f(X1, X2 )
1.3
The notation f(.) represents the function that deï¬nes the relationship between the dependent variable Y and the independent variables X1 and X2. In the case of determining
PROFIT from REVENUE and EXPENSES, the mathematical form of the function f(.) is
quite simple: f(X1, X2 ) = X1 âˆ’ X2. However, in many other situations we will model, the
form of f(.) is quite complex and might involve many independent variables. But regardless of the complexity of f(.) or the number of independent variables involved,
many of the decision problems encountered in business can be represented by models
that assume the general form,
Y = f(X1, X2, . . . , Xk)
1.4
In equation 1.4, the dependent variable Y represents some bottom-line performance
measure of the problem we are modeling. The terms X1, X2, . . . , Xk represent the different independent variables that play some role or have some effect in determining the
value of Y. Again, f(.) is the function (possibly quite complex) that speciï¬es or describes
the relationship between the dependent and independent variables.
The relationship expressed in equation 1.4 is very similar to what occurs in most
spreadsheet models. Consider a simple spreadsheet model to calculate the monthly
payment for a car loan, as shown in Figure 1.2.
The spreadsheet in Figure 1.2 contains a variety of input cells (for example, purchase
price, down payment, trade-in, term of loan, annual interest rate) that correspond
conceptually to the independent variables X1, X2, . . . , Xk in equation 1.4. Similarly, a
variety of mathematical operations are performed using these input cells in a manner
analogous to the function f(.) in equation 1.4. The results of these mathematical operations determine the value of some output cell in the spreadsheet (for example, monthly
payment) that corresponds to the dependent variable Y in equation 1.4. Thus, there is a
direct correspondence between equation 1.4 and the spreadsheet in Figure 1.2. This type
of correspondence exists for most of the spreadsheet models in this book.
6
Chapter 1
Introduction to Modeling and Decision Analysis
FIGURE 1.2
Example of a
model
1.4 Categories of Mathematical Models
Not only does equation 1.4 describe the major elements of mathematical or spreadsheet
models, but it also provides a convenient means for comparing and contrasting the
three categories of modeling techniques presented in this bookâ€”Prescriptive Models,
Predictive Models, and Descriptive Models. Figure 1.3 summarizes the characteristics
and techniques associated with each of these categories.
In some situations, a manager might face a decision problem involving a very precise,
well-deï¬ned functional relationship f(.) between the independent variables X1, X2, . . . ,
Xk and the dependent variable Y. If the values for the independent variables are under
FIGURE 1.3
Categories and
characteristics of
management
science modeling
techniques
Model Characteristics
Category
Form of f (.)
Values of Independent
Variables
Management Science
Techniques
Prescriptive
Models
known,
well-deï¬ned
known or under decision
makerâ€™s control
Predictive
Models
unknown,
ill-deï¬ned
known or under decision
makerâ€™s control
Descriptive
Models
known,
well-deï¬ned
unknown or uncertain
Linear Programming,
Networks, Integer
Programming, CPM,
Goal Programming,
EOQ, Nonlinear
Programming
Regression Analysis,
Time Series Analysis,
Discriminant Analysis
Simulation, Queuing,
PERT, Inventory Models
The Problem-Solving Process
7
the decision makerâ€™s control, the decision problem in these types of situations boils
down to determining the values of the independent variables X1, X2, . . ., Xk that produce
the best possible value for the dependent variable Y. These types of models are called
Prescriptive Models because their solutions tell the decision maker what actions to
take. For example, you might be interested in determining how a given sum of money
should be allocated to different investments (represented by the independent variables)
to maximize the return on a portfolio without exceeding a certain level of risk.
A second category of decision problems is one in which the objective is to predict or
estimate what value the dependent variable Y will take on when the independent variables X1, X2, . . ., Xk take on speciï¬c values. If the function f(.) relating the dependent and
independent variables is known, this is a very simple taskâ€”simply enter the speciï¬ed
values for X1, X2, . . ., Xk into the function f(.) and compute Y. In some cases, however, the
functional form of f(.) might be unknown and must be estimated for the decision maker
to make predictions about the dependent variable Y. These types of models are called
Predictive Models. For example, a real estate appraiser might know that the value of a
commercial property (Y) is inï¬‚uenced by its total square footage (X1) and age (X2),
among other things. However, the functional relationship f(.) that relates these variables
to one another might be unknown. By analyzing the relationship between the selling
price, total square footage, and age of other commercial properties, the appraiser might
be able to identify a function f(.) that relates these two variables in a reasonably accurate
manner.
The third category of models you are likely to encounter in the business world is
called Descriptive Models. In these situations, a manager might face a decision problem that has a very precise, well-deï¬ned functional relationship f(.) between the independent variables X1, X2, . . ., Xk and the dependent variable Y. However, there might
be great uncertainty as to the exact values that will be assumed by one or more of the
independent variables X1, X2, . . ., Xk. In these types of problems, the objective is to
describe the outcome or behavior of a given operation or system. For example, suppose
a company is building a new manufacturing facility and has several choices about the
type of machines to put in the new plant, and also various options for arranging the
machines. Management might be interested in studying how the various plant conï¬gurations would affect on-time shipments of orders (Y), given the uncertain number of
orders that might be received (X1) and the uncertain due dates (X2) that might be required by these orders.
1.5 The Problem-Solving Process
Throughout our discussion, we have said that the ultimate goal in building models is to
help managers make decisions that solve problems. The modeling techniques we will
study represent a small but important part of the total problem-solving process. To become an effective modeler, it is important to understand how modeling ï¬ts into the
entire problem-solving process.
Because a model can be used to represent a decision problem or phenomenon, we
might be able to create a visual model of the phenomenon that occurs when people
solve problemsâ€”what we call the problem-solving process. Although a variety of models could be equally valid, the one in Figure 1.4 summarizes the key elements of the
problem-solving process and is sufï¬cient for our purposes.
The ï¬rst step of the problem-solving process, identifying the problem, is also the
most important. If we do not identify the correct problem, all the work that follows will
amount to nothing more than wasted effort, time, and money. Unfortunately, identifying
8
FIGURE 1.4
Chapter 1
Identify
Problem
A visual model of
the problemsolving process
Introduction to Modeling and Decision Analysis
Formulate and
Implement Model
Analyze
Model
Test
Results
Implement
Solution
Unsatisfactory Results
the problem to solve is often not as easy as it seems. We know that a problem exists
when there is a gap or disparity between the present situation and some desired state of
affairs. However, we usually are not faced with a neat, well-deï¬ned problem. Instead,
we often ï¬nd ourselves facing a â€œmessâ€!2 Identifying the real problem involves gathering a lot of information and talking with many people to increase our understanding of
the mess. We must then sift through all this information and try to identify the root
problem or problems causing the mess. Thus, identifying the real problem (and not just
the symptoms of the problem) requires insight, some imagination, time, and a good bit
of detective work.
The end result of the problem-identiï¬cation step is a well-deï¬ned statement of the
problem. Simply deï¬ning a problem well will often make it much easier to solve.
Having identiï¬ed the problem, we turn our attention to creating or formulating a model
of the problem. Depending on the nature of the problem, we might use a mental model,
a visual model, a scale model, or a mathematical model. Although this book focuses on
mathematical models, this does not mean that mathematical models are always applicable or best. In most situations, the best model is the simplest model that accurately
reï¬‚ects the relevant characteristic or essence of the problem being studied.
We will discuss several different management science modeling techniques in this
book. It is important that you not develop too strong a preference for any one technique.
Some people have a tendency to want to formulate every problem they face as a model
that can be solved by their favorite management science technique. This simply will not
work.
As indicated in Figure 1.3, there are fundamental differences in the types of problems
a manager might face. Sometimes, the values of the independent variables affecting a
problem are under the managerâ€™s control; sometimes they are not. Sometimes, the form
of the functional relationship f(.) relating the dependent and independent variables is
well-deï¬ned, and sometimes it is not. These fundamental characteristics of the problem
should guide your selection of an appropriate management science modeling technique. Your goal at the model-formulation stage is to select a modeling technique that
ï¬ts your problem, rather than trying to ï¬t your problem into the required format of a
pre-selected modeling technique.
After you select an appropriate representation or formulation of your problem, the
next step is to implement this formulation as a spreadsheet model. We will not dwell on
the implementation process now because that is the focus of the remainder of this book.
After you verify that your spreadsheet model has been implemented accurately, the next
step in the problem-solving process is to use the model to analyze the problem it represents. The main focus of this step is to generate and evaluate alternatives that might lead
to a solution. This often involves playing out a number of scenarios or asking several
â€œWhat if?â€ questions. Spreadsheets are particularly helpful in analyzing mathematical
models in this manner. In a well-designed spreadsheet model, it should be fairly simple
to change some of the assumptions in the model to see what might happen in different
2
This characterization is borrowed from Chapter 5, James R. Evans, Creative Thinking in the Decision and Management Sciences (Cincinnati, Ohio: South-Western Publishing, 1991), 89â€“115.
Anchoring and Framing Effects
9
situations. As we proceed, we will highlight some techniques for designing spreadsheet
models that facilitate this type of â€œwhat if?â€ analysis. â€œWhat if?â€ analysis is also very appropriate and useful when working with nonmathematical models.
The end result of analyzing a model does not always provide a solution to the actual
problem being studied. As we analyze a model by asking various â€œWhat if?â€ questions,
it is important to test the feasibility and quality of each potential solution. The blueprints
that Frank Brock showed to his production employees represented the end result of his
analysis of the problem he faced. He wisely tested the feasibility and quality of this alternative before implementing it, and discovered an important ï¬‚aw in his plans. Thus, the
testing process can give important new insights into the nature of a problem. The testing
process is also important because it provides the opportunity to double-check the validity of the model. At times, we might discover an alternative that appears to be too good
to be true. This could lead us to ï¬nd that some important assumption has been left out of
the model. Testing the results of the model against known results (and simple common
sense) helps ensure the structural integrity and validity of the model. After analyzing the
model, we might discover that we need to go back and modify the model.
The last step of the problem-solving process, implementation, is often the most difï¬cult. By their very nature, solutions to problems involve people and change. For better
or for worse, most people resist change. However, there are ways to minimize the seemingly inevitable resistance to change. For example, it is wise, if possible, to involve anyone who will be affected by the decision in all steps of the problem-solving process. This
not only helps develop a sense of ownership and understanding of the ultimate solution, but it also can be the source of important information throughout the problemsolving process. As the Brock Candy story illustrates, even if it is impossible to include
those affected by the solution in all steps, their input should be solicited and considered
before a solution is accepted for implementation. Resistance to change and new systems
also can be eased by creating ï¬‚exible, user-friendly interfaces for the mathematical models that often are developed in the problem-solving process.
Throughout this book, we focus mostly on the model formulation, implementation,
analysis, and testing steps of the problem-solving process, summarized in Figure 1.4.
Again, this does not imply that these steps are more important than the others. If we do
not identify the correct problem, the best we can hope for from our modeling effort is
â€œthe right answer to the wrong question,â€ which does not solve the real problem. Similarly, even if we do identify the problem correctly and design a model that leads to a perfect solution, if we cannot implement the solution, then we still have not solved the
problem. Developing the interpersonal and investigative skills required to work with
people in deï¬ning the problem and implementing the solution are as important as the
mathematical modeling skills you will develop by working through this book.
1.6 Anchoring and Framing Effects
At this point, some of you reading this book are probably thinking it is better to rely on
subjective judgment and intuition rather than models when making decisions. Indeed,
most nontrivial decision problems involve some issues that are difï¬cult or impossible to
structure and analyze in the form of a mathematical model. These unstructurable aspects of a decision problem might require the use of judgment and intuition. However,
it is important to realize that human cognition is often ï¬‚awed and can lead to incorrect
judgments and irrational decisions. Errors in human judgment often arise because of
what psychologists term anchoring and framing effects associated with decision
problems.
10
Chapter 1
Introduction to Modeling and Decision Analysis
Anchoring effects arise when a seemingly trivial factor serves as a starting point (or
anchor) for estimations in a decision-making problem. Decision makers adjust their
estimates from this anchor but nevertheless remain too close to the anchor and usually
under-adjust. In a classic psychological study on this issue, one group of subjects were
asked to individually estimate the value of 1 2 3 4 5 6 7 8 (without using
a calculator). Another group of subjects were each asked to estimate the value of 8 7
6 5 4 3 2 1. The researchers hypothesized that the ï¬rst number presented (or
perhaps the product of the ï¬rst three or four numbers) would serve as a mental anchor.
The results supported the hypothesis. The median estimate of subjects shown the numbers in ascending sequence (1 2 3 . . .) was 512, whereas the median estimate of subjects shown the sequence in descending order (8 7 6 . . .) was 2,250. Of course, the
order of multiplication for these numbers is irrelevant and the product of both series is
the same: 40,320.
Framing effects refer to how a decision maker views or perceives the alternatives in a
decision problemâ€”often involving a win/loss perspective. The way a problem is
framed often inï¬‚uences the choices made by a decision maker and can lead to irrational
behavior. For example, suppose you have just been given \$1,000 but must choose one of
the following alternatives: (A1) Receive an additional \$500 with certainty, or (B1) Flip a
fair coin and receive an additional \$1,000 if heads occurs or \$0 additional is tails occurs.
Here, alternative A1 is a â€œsure winâ€ and is the alternative most people prefer. Now suppose you have been given \$2,000 and must choose one of the following alternatives: (A2)
Give back \$500 immediately, or (B2) Flip a fair coin and give back \$0 if heads occurs or
\$1,000 if tails occurs. When the problem is framed this way, alternative A2 is a â€œsure
lossâ€ and many people who previously preferred alternative A1 now opt for alternative B2 (because it holds a chance of avoiding a loss). However, Figure 1.5 shows a single decision tree for these two scenarios making it clear that, in both cases, the â€œAâ€
alternative guarantees a total payoff of \$1,500, whereas the â€œBâ€ alternative offers a 50%
chance of a \$2,000 total payoff and a 50% chance of a \$1,000 total payoff. (Decision trees
will be covered in greater detail in a later chapter.) A purely rational decision maker
should focus on the consequences of his or her choices and consistently select the same
alternative, regardless of how the problem is framed.
Whether we want to admit it or not, we are all prone to make errors in estimation due
to anchoring effects and may exhibit irrationality in decision making due to framing
effects. As a result, it is best to use computer models to do what they are best at (i.e.,
modeling structurable portions of a decision problem) and let the human brain do what
it is best at (i.e., dealing with the unstructurable portion of a decision problem).
FIGURE 1.5
Payoffs
\$1,500
Alternative A
Decision tree for
framing effects
Initial state
\$2,000
Alternative B
(Flip coin)
Tails (50%)
\$1,000
Summary
11
1.7 Good Decisions vs. Good Outcomes
The goal of the modeling approach to problem solving is to help individuals make good
decisions. But good decisions do not always result in good outcomes. For example, suppose the weather report on the evening news predicts a warm, dry, sunny day tomorrow. When you get up and look out the window tomorrow morning, suppose there is
not a cloud in sight. If you decide to leave your umbrella at home and subsequently get
soaked in an unexpected afternoon thundershower, did you make a bad decision? Certainly not. Unforeseeable circumstances beyond your control caused you to experience
a bad outcome, but it would be unfair to say that you made a bad decision. Good decisions sometimes result in bad outcomes. See Figure 1.6 for the story of another good decision having a bad outcome.
The modeling techniques presented in this book can help you make good decisions,
but cannot guarantee that good outcomes will always occur as a result of those decisions. Even when a good decision is made, luck often plays a role in determining
whether a good or bad outcome occurs. However, using a structured, modeling approach to decision making should produce good outcomes more frequently than making decisions in a more haphazard manner.
1.8 Summary
This book introduces you to a variety of techniques from the ï¬eld of management science that can be applied in spreadsheet models to assist in decision analysis and problem solving. This chapter discussed how spreadsheet models of decision problems can
be used to analyze the consequences of possible courses of action before a particular
alternative is selected for implementation. It described how models of decision problems differ in several important characteristics and how you should select a modeling
technique that is most appropriate for the type of problem being faced. Finally, it discussed how spreadsheet modeling and analysis ï¬t into the problem-solving process.
FIGURE 1.6
Andre-Francois Raffray thought he had a great deal in 1965 when he agreed to
pay a 90-year-old woman named Jeanne Calment \$500 a month until she died to
acquire her grand apartment in Arles, northwest of Marseilles in the south of
Franceâ€”a town Vincent Van Gogh once roamed. Buying apartments â€œfor lifeâ€ is
common in France. The elderly owner gets to enjoy a monthly income from the
buyer who gambles on getting a real estate bargainâ€”betting the owner doesnâ€™t
live too long. Upon the ownerâ€™s death, the buyer inherits the apartment regardless
of how much was paid. But in December of 1995, Raffray died at age 77, having
paid more than \$180,000 for an apartment he never got to live in.
On the same day, Calment, then the worldâ€™s oldest living person at 120, dined
on foie gras, duck thighs, cheese, and chocolate cake at her nursing home near the
sought-after apartment. And she does not need to worry about losing her \$500
monthly income. Although the amount Raffray already paid is twice the
apartmentâ€™s current market value, his widow is obligated to keep sending the
monthly check to Calment. If Calment also outlives her, then the Raffray children
will have to pay. â€œIn life, one sometimes makes bad deals,â€ said Calment of the
outcome of Raffrayâ€™s decision. (Source: The Savannah Morning News, 12/29/95.)
A good decision
outcome
12
Chapter 1
Introduction to Modeling and Decision Analysis
1.9 References
Edwards, J., P. Finlay, and J. Wilson. â€œThe role of the OR specialist in â€˜do it yourselfâ€™ spreadsheet development.â€ European Journal of Operational Research, vol. 127, no. 1, 2000.
Forgione, G. â€œCorporate MS Activities: An Update.â€ Interfaces, vol. 13, no. 1, 1983.
Hall, R. â€œWhatâ€™s So Scientiï¬c about MS/OR?â€ Interfaces, vol. 15, 1985.
Hastie, R. and R. M. Dawes. Rational Choice in an Uncertain World, Sage Publications, 2001.
Schrage, M. Serious Play, Harvard Business School Press, 2000.
Sonntag, C. and Grossman, T. â€œEnd-User Modeling Improves R&D Management at AgrEvo Canada, Inc.â€
Interfaces, vol. 29, no. 5, 1999.
THE WORLD OF MANAGEMENT SCIENCE
â€Business Analysts Trained in Management Science Can Be
a Secret Weapon in a CIOâ€™s Quest for Bottom-Line Results.â€
Efï¬ciency nuts. These are the people you see at cocktail parties explaining how the
host could disperse that crowd around the popular shrimp dip if he would divide
it into three bowls and place them around the room. As she draws the improved
trafï¬c ï¬‚ow on a paper napkin, you notice that her favorite word is â€œoptimizeâ€â€”a
tell-tale sign that she has studied the ï¬eld of â€œoperations researchâ€ or â€œmanagement scienceâ€ (also known as OR/MS).
OR/MS professionals are driven to solve logistics problems. This trait might
not make them the most popular people at parties, but it is exactly what todayâ€™s information systems (IS) departments need to deliver more business value. Experts
say that smart IS executives will learn to exploit the talents of these mathematical
wizards in their quest to boost a companyâ€™s bottom line.
According to Ron J. Ponder, chief information ofï¬cer (CIO) at Sprint Corp. in
Kansas City, Mo., and former CIO at Federal Express Corp., â€œIf IS departments had
more participation from operations research analysts, they would be building
much better, richer IS solutions.â€ As someone who has a Ph.D. in operations research and who built the renowned package-tracking systems at Federal Express,
Ponder is a true believer in OR/MS. Ponder and others say analysts trained in
OR/MS can turn ordinary information systems into money-saving, decisionsupport systems, and are ideally suited to be members of the business process
reengineering team. â€œIâ€™ve always had an operations research department reporting
to me, and itâ€™s been invaluable. Now Iâ€™m building one at Sprint,â€ says Ponder.
The Beginnings
OR/MS got its start in World War II, when the military had to make important decisions about allocating scarce resources to various military operations. One of the ï¬rst
business applications for computers in the 1950s was to solve operations research
problems for the petroleum industry. A technique called linear programming was
used to ï¬gure out how to blend gasoline for the right ï¬‚ash point, viscosity, and octane
in the most economical way. Since then, OR/MS has spread throughout business and
government, from designing efï¬cient drive-thru window operations for Burger King
Corp. to creating ultrasophisticated computerized stock trading systems.
A classic OR/MS example is the crew scheduling problem faced by all major airlines. How do you plan the itineraries of 8,000 pilots and 17,000 ï¬‚ight attendants
The World of Management Science
when there is an astronomical number of combinations of planes, crews, and cities?
The OR/MS analysts at United Airlines came up with a scheduling system called
Paragon that attempts to minimize the amount of paid time that crews spend waiting for ï¬‚ights. Their model factors in constraints such as labor agreement provisions and Federal Aviation Administration regulations, and is projected to save the
airline at least \$1 million a year.
OR/MS and IS
Somewhere in the 1970s, the OR/MS and IS disciplines went in separate directions.
â€œThe IS profession has had less and less contact with the operations research
folks . . . and IS lost a powerful intellectual driver,â€ says Peter G. W. Keen, executive
director of the International Center for Information Technologies in Washington,
D.C. However, many feel that now is an ideal time for the two disciplines to rebuild
some bridges.
Todayâ€™s OR/MS professionals are involved in a variety of IS-related ï¬elds, including inventory management, electronic data interchange, supply chain management, IT security, computer-integrated manufacturing, network management,
and practical applications of artiï¬cial intelligence. Furthermore, each side needs
something the other side has: OR/MS analysts need corporate data to plug into
their models, and the IS folks need to plug the OR/MS models into their strategic
information systems. At the same time, CIOs need intelligent applications that
enhance the bottom line and make them heroes with the chief executive ofï¬cer.
OR/MS analysts can develop a model of how a business process works now
and simulate how it could work more efï¬ciently in the future. Therefore, it makes
sense to have an OR/MS analyst on the interdisciplinary team that tackles business process reengineering projects. In essence, OR/MS professionals add more
value to the IS infrastructure by building â€œtools that really help decision makers
analyze complex situations,â€ says Andrew B. Whinston, director of the Center for
Information Systems Management at the University of Texas at Austin.
Although IS departments typically believe their job is done if they deliver accurate and timely information, Thomas M. Cook, president of American Airlines Decision Technologies, Inc. says that adding OR/MS skills to the team can produce
intelligent systems that actually recommend solutions to business problems. One
of the big success stories at Cookâ€™s operations research shop is a â€œyield managementâ€ system that decides how much to overbook and how to set prices for each
seat so that a plane is ï¬lled up and proï¬ts are maximized. The yield management
system deals with more than 250 decision variables and accounts for a signiï¬cant
amount of American Airlinesâ€™ revenue.
Where to Start
So how can the CIO start down the road toward collaboration with OR/MS analysts? If the company already has a group of OR/MS professionals, the IS department can draw on their expertise as internal consultants. Otherwise, the CIO can
simply hire a few OR/MS wizards, throw a problem at them, and see what happens. The payback may come surprisingly quickly. As one former OR/MS professional put it: â€œIf I couldnâ€™t save my employer the equivalent of my own salary in
the ï¬rst month of the year, then I wouldnâ€™t feel like I was doing my job.â€
Adapted from: Mitch Betts, â€œEfï¬ciency Einsteins,â€ ComputerWorld, March 22, 1993, p. 64.
13
14
Chapter 1
Introduction to Modeling and Decision Analysis
Questions and Problems
1. What is meant by the term decision analysis?
2. Deï¬ne the term computer model.
3. What is the difference between a spreadsheet model and a computer model?
4. Deï¬ne the term management science.
5. What is the relationship between management science and spreadsheet modeling?
6. What kinds of spreadsheet applications would not be considered management
science?
7. In what ways do spreadsheet models facilitate the decision-making process?
8. What are the beneï¬ts of using a modeling approach to decision making?
9. What is a dependent variable?
10. What is an independent variable?
11. Can a model have more than one dependent variable?
12. Can a decision problem have more than one dependent variable?
13. In what ways are prescriptive models different from descriptive models?
14. In what ways are prescriptive models different from predictive models?
15. In what ways are descriptive models different from predictive models?
16. How would you deï¬ne the words description, prediction, and prescription?
Carefully consider what is unique about the meaning of each word.
17. Identify one or more mental models you have used. Can any of them be expressed
mathematically? If so, identify the dependent and independent variables in your
model.
18. Consider the spreadsheet model shown in Figure 1.2. Is this model descriptive, predictive, or prescriptive in nature, or does it not fall into any of these categories?
19. What are the steps in the problem-solving process?
20. Which step in the problem-solving process do you think is most important? Why?
21. Must a model accurately represent every detail of a decision situation to be useful?
Why or why not?
22. If you were presented with several different models of a given decision problem,
which would you be most inclined to use? Why?
23. Describe an example in which business or political organizations may use anchoring
effects to inï¬‚uence decision making.
24. Describe an example in which business or political organizations may use framing
effects to inï¬‚uence decision making.
25. Suppose sharks have been spotted along the beach where you are vacationing with
a friend. You and your friend have been informed of the shark sightings and are
aware of the damage a shark attack can inï¬‚ict on human ï¬‚esh. You both decide (individually) to go swimming anyway. You are promptly attacked by a shark while
your friend has a nice time body surï¬ng in the waves. Did you make a good or bad
26. Describe an example in which a well-known business, political, or military leader
made a good decision that resulted in a bad outcome, or a bad decision that resulted
in a good outcome.
CASE 1.1
Patrickâ€™s luck had changed overnightâ€”but not his skill at mathematical reasoning. The
day after graduating from college he used the \$20 that his grandmother had given him
Case
15
as a graduation gift to buy a lottery ticket. He knew that his chances of winning the
lottery were extremely low and it probably was not a good way to spend this money.
But he also remembered from the class he took in management science that bad decisions sometimes result in good outcomes. So he said to himself, â€œWhat the heck? Maybe
this bad decision will be the one with a good outcome.â€ And with that thought, he
bought his lottery ticket.
The next day Patrick pulled the crumpled lottery ticket out of the back pocket of his
blue jeans and tried to compare his numbers to the winning numbers printed in the
paper. When his eyes ï¬nally came into focus on the numbers they also just about
popped out of his head. He had a winning ticket! In the ensuing days he learned that
his share of the jackpot would give him a lump sum payout of about \$500,000 after
taxes. He knew what he was going to do with part of the money: buy a new car, pay
off his college loans, and send his grandmother on an all-expenses-paid trip to Hawaii.
But he also knew that he couldnâ€™t continue to hope for good outcomes to arise from
more bad decisions. So he decided to take half of his winnings and invest it for his
retirement.
A few days later, Patrick was sitting around with two of his fraternity buddies, Josh
and Peyton, trying to ï¬gure out how much money his new retirement fund might be
worth in 30 years. They were all business majors in college and remembered from their
ï¬nance class that if you invest p dollars for n years at an annual interest rate of i percent
then in n years you would have p(1 + i)n dollars. So they ï¬gure that if Patrick invested
\$250,000 for 30 years in an investment with a 10% annual return then in 30 years he
would have \$4,362,351 (that is, \$250,000(1 + 0.10)30 ).
But after thinking about it a little more, they all agreed that it would be unlikely
for Patrick to find an investment that would produce a return of exactly 10%
each and every year for the next 30 years. If any of this money is invested in stocks
then some years the return might be higher than 10% and some years it would probably be lower. So to help account for the potential variability in the investment
returns Patrick and his friends came up with a plan; they would assume he could
find an investment that would produce an annual return of 17.5% seventy percent
of the time and a return (or actually a loss) of 7.5% thirty percent of the time.
Such an investment should produce an average annual return of 0.7(17.5%)
0.3( 7.5%) = 10%. Josh felt certain that this meant Patrick could still expect his
\$250,000 investment to grow to \$4,362,351 in 30 years (because \$250,000(1 0.10)30
= \$4,362,351).
After sitting quietly and thinking about it for a while, Peyton said that he thought
Josh was wrong. The way Peyton looked at it, Patrick should see a 17.5% return in 70%
of the 30 years (or 0.7(30) = 21 years) and a 7.5% return in 30% of the 30 years (or
0.3(30) = 9 years). So, according to Peyton, that would mean Patrick should have
\$250,000(1 + 0.175)21(1 0.075)9 = \$3,664,467 after 30 years. But thatâ€™s \$697,884 less
than what Josh says Patrick should have.
After listening to Peytonâ€™s argument, Josh said he thought Peyton was wrong
because his calculation assumes that the â€œgoodâ€ return of 17.5% would occur in each of
the ï¬rst 21 years and the â€œbadâ€ return of 7.5% would occur in each of the last 9 years.
But Peyton countered this argument by saying that the order of good and bad returns
does not matter. The commutative law of arithmetic says that when you add or multiply
numbers, the order doesnâ€™t matter (that is, X + Y = Y + X and X Y = Y X). So
Peyton says that because Patrick can expect 21 â€œgoodâ€ returns and 9 â€œbadâ€ returns and
it doesnâ€™t matter in what order they occur, then the expected outcome of the investment
should be \$3,664,467 after 30 years.
16
Chapter 1
Introduction to Modeling and Decision Analysis
Patrick is now really confused. Both of his friendsâ€™ arguments seem to make perfect
sense logicallyâ€”but they lead to such different answers, and they canâ€™t both be right.
What really worries Patrick is that he is starting his new job as a business analyst in
a couple of weeks. And if he canâ€™t reason his way to the right answer in a relatively
simple problem like this, what is he going to do when he encounters the more difï¬cult
problems awaiting him the business world? Now he really wishes he had paid more
attention in his management sciences class.
So what do you think? Who is right, Josh or Peyton? And more important, why?
Chapter 2
Introduction to Optimization
and Linear Programming
2.0 Introduction
Our world is ï¬lled with limited resources. The amount of oil we can pump out of the
earth is limited. The amount of land available for garbage dumps and hazardous waste is
limited and, in many areas, diminishing rapidly. On a more personal level, each of us has
a limited amount of time in which to accomplish or enjoy the activities we schedule each
day. Most of us have a limited amount of money to spend while pursuing these activities.
Businesses also have limited resources. A manufacturing organization employs a limited
number of workers. A restaurant has a limited amount of space available for seating.
Deciding how best to use the limited resources available to an individual or a business is a universal problem. In todayâ€™s competitive business environment, it is increasingly important to make sure that a companyâ€™s limited resources are used in the most
efï¬cient manner possible. Typically, this involves determining how to allocate the
resources in such a way as to maximize proï¬ts or minimize costs. Mathematical
programming (MP) is a ï¬eld of management science that ï¬nds the optimal, or most efï¬cient, way of using limited resources to achieve the objectives of an individual or a business. For this reason, mathematical programming often is referred to as optimization.
2.1 Applications of
Mathematical Optimization
To help you understand the purpose of optimization and the types of problems for
which it can be used, letâ€™s consider several examples of decision-making situations in
which MP techniques have been applied.
Determining Product Mix. Most manufacturing companies can make a variety of
products. However, each product usually requires different amounts of raw materials
and labor. Similarly, the amount of proï¬t generated by the products varies. The manager
of such a company must decide how many of each product to produce to maximize
proï¬ts or to satisfy demand at minimum cost.
Manufacturing. Printed circuit boards, like those used in most computers, often have
hundreds or thousands of holes drilled in them to accommodate the different electrical
components that must be plugged into them. To manufacture these boards, a computercontrolled drilling machine must be programmed to drill in a given location, then move
17
18
Chapter 2
Introduction to Optimization and Linear Programming
the drill bit to the next location and drill again. This process is repeated hundreds or
thousands of times to complete all the holes on a circuit board. Manufacturers of these
boards would beneï¬t from determining the drilling order that minimizes the total
distance the drill bit must be moved.
Routing and Logistics. Many retail companies have warehouses around the country
that are responsible for keeping stores supplied with merchandise to sell. The amount of
merchandise available at the warehouses and the amount needed at each store tends to
ï¬‚uctuate, as does the cost of shipping or delivering merchandise from the warehouses
to the retail locations. Large amounts of money can be saved by determining the least
costly method of transferring merchandise from the warehouses to the stores.
Financial Planning. The federal government requires individuals to begin withdrawing
money from individual retirement accounts (IRAs) and other tax-sheltered retirement programs no later than age 70.5. There are various rules that must be followed to avoid paying
penalty taxes on these withdrawals. Most individuals want to withdraw their money in a
manner that minimizes the amount of taxes they must pay while still obeying the tax laws.
Optimization Is Everywhere
Going to Disney World this summer? Optimization will be your ubiquitous companion, scheduling the crews and planes, pricing the airline tickets and hotel
rooms, even helping to set capacities on the theme park rides. If you use Orbitz to
book your ï¬‚ights, an optimization engine sifts through millions of options to ï¬nd
the cheapest fares. If you get directions to your hotel from MapQuest, another optimization engine ï¬gures out the most direct route. If you ship souvenirs home, an
optimization engine tells UPS which truck to put the packages on, exactly where on
the truck the packages should go to make them fastest to load and unload, and
what route the driver should follow to make his deliveries most efï¬ciently.
(Adapted from: V. Postrel, â€œOperation Everything,â€ The Boston Globe, June 27, 2004.)
2.2 Characteristics of
Optimization Problems
These examples represent just a few areas in which MP has been used successfully. We
will consider many other examples throughout this book. However, these examples
give you some idea of the issues involved in optimization. For instance, each example
involves one or more decisions that must be made: How many of each product should
be produced? Which hole should be drilled next? How much of each product should be
shipped from each warehouse to the various retail locations? How much money should
an individual withdraw each year from various retirement accounts?
Also, in each example, restrictions, or constraints, are likely to be placed on the alternatives available to the decision maker. In the ï¬rst example, when determining
the number of products to manufacture, a production manager probably is faced with a
limited amount of raw materials and a limited amount of labor. In the second example,
the drill never should return to a position where a hole has already been drilled. In the
Expressing Optimization Problems Mathematically
19
third example, there is a physical limitation on the amount of merchandise a truck can
carry from one warehouse to the stores on its route. In the fourth example, laws determine the minimum and maximum amounts that can be withdrawn from retirement
accounts without incurring a penalty. There might be many other constraints for these
examples. Indeed, it is not unusual for real-world optimization problems to have hundreds or thousands of constraints.
A ï¬nal common element in each of the examples is the existence of some goal or objective that the decision maker considers when deciding which course of action is best.
In the ï¬rst example, the production manager can decide to produce several different
product mixes given the available resources, but the manager probably will choose the
mix of products that maximizes proï¬ts. In the second example, a large number of possible drilling patterns can be used, but the ideal pattern probably will involve moving the
drill bit the shortest total distance. In the third example, there are numerous ways merchandise can be shipped from the warehouses to supply the stores, but the company
probably will want to identify the routing that minimizes the total transportation cost.
Finally, in the fourth example, individuals can withdraw money from their retirement
accounts in many ways without violating the tax laws, but they probably want to ï¬nd
the method that minimizes their tax liability.
2.3 Expressing Optimization
Problems Mathematically
From the preceding discussion, we know that optimization problems involve three elements: decisions, constraints, and an objective. If we intend to build a mathematical
model of an optimization problem, we will need mathematical terms or symbols to represent each of these three elements.
2.3.1 DECISIONS
The decisions in an optimization problem often are represented in a mathematical
model by the symbols X1, X2, . . . , Xn. We will refer to X1, X2, . . . , Xn as the decision
variables (or simply the variables) in the model. These variables might represent the
quantities of different products the production manager can choose to produce. They
might represent the amount of different pieces of merchandise to ship from a warehouse
to a certain store. They might represent the amount of money to be withdrawn from different retirement accounts.
The exact symbols used to represent the decision variables are not particularly important. You could use Z1, Z2, . . . , Zn or symbols like Dog, Cat, and Monkey to represent
the decision variables in the model. The choice of which symbols to use is largely a matter of personal preference and might vary from one problem to the next.
2.3.2 CONSTRAINTS
The constraints in an optimization problem can be represented in a mathematical model
in several ways. Three general ways of expressing the possible constraint relationships
in an optimization problem are:
A â€œless than or equal toâ€ constraint:
A â€œgreater than or equal toâ€ constraint:
f(X1, X2, . . . , Xn) â‰¤ b
f(X1, X2, . . . , Xn) â‰¥ b
An â€œequal toâ€ constraint:
f(X1, X2, . . . , Xn) = b
20
Chapter 2
Introduction to Optimization and Linear Programming
In each case, the constraint is some function of the decision variables that must be
less than or equal to, greater than or equal to, or equal to some speciï¬c value (represented above by the letter b). We will refer to f(X1, X2, . . . , Xn) as the left-hand-side (LHS)
of the constraint and to b as the right-hand-side (RHS) value of the constraint.
For example, we might use a â€œless than or equal toâ€ constraint to ensure that the total
labor used in producing a given number of products does not exceed the amount of
available labor. We might use a â€œgreater than or equal toâ€ constraint to ensure that the
total amount of money withdrawn from a personâ€™s retirement accounts is at least the
minimum amount required by the IRS. You can use any number of these constraints to
represent a given optimization problem depending on the requirements of the situation.
2.3.3 OBJECTIVE
The objective in an optimization problem is represented mathematically by an objective
function in the general format:
MAX (or MIN):
f(X1, X2, . . . , Xn)
The objective function identiï¬es some function of the decision variables that the decision maker wants to either MAXimize or MINimize. In our earlier examples, this function might be used to describe the total proï¬t associated with a product mix, the total
distance the drill bit must be moved, the total cost of transporting merchandise, or a
retireeâ€™s total tax liability.
The mathematical formulation of an optimization problem can be described in the
general format:
MAX (or MIN):
Subject to:
f0(X1, X2, . . . , Xn)
f1(X1, X2, . . . , Xn) â‰¤ b1
2.1
2.2
:
fk(X1, X2, . . . , Xn) â‰¥ bk
:
2.3
fm(X1, X2, . . . , Xn) = bm
2.4
This representation identiï¬es the objective function (equation 2.1) that will be
maximized (or minimized) and the constraints that must be satisï¬ed (equations 2.2
through 2.4). Subscripts added to the f and b in each equation emphasize that the
functions describing the objective and constraints can all be different. The goal in optimization is to ï¬nd the values of the decision variables that maximize (or minimize) the
objective function without violating any of the constraints.
2.4 Mathematical
Programming Techniques
Our general representation of an MP model is just thatâ€”general. You can use many
kinds of functions to represent the objective function and the constraints in an MP
model. Of course, you always should use functions that accurately describe the objective and constraints of the problem you are trying to solve. Sometimes, the functions in
a model are linear in nature (that is, they form straight lines or ï¬‚at surfaces); other times,
Formulating LP Models
21
they are nonlinear (that is, they form curved lines or curved surfaces). Sometimes, the
optimal values of the decision variables in a model must take on integer values (whole
numbers); other times, the decision variables can assume fractional values.
Given the diversity of MP problems that can be encountered, many techniques have
been developed to solve different types of MP problems. In the next several chapters, we
will look at these MP techniques and develop an understanding of how they differ and
when each should be used. We will begin by examining a technique called linear
programming (LP), which involves creating and solving optimization problems with
linear objective functions and linear constraints. LP is a very powerful tool that can be
applied in many business situations. It also forms a basis for several other techniques
discussed later and is, therefore, a good starting point for our investigation into the ï¬eld
of optimization.
2.5 An Example LP Problem
We will begin our study of LP by considering a simple example. You should not interpret this to mean that LP cannot solve more complex or realistic problems. LP has been
used to solve extremely complicated problems, saving companies millions of dollars.
However, jumping directly into one of these complicated problems would be like starting a marathon without ever having gone out for a jogâ€”you would get winded and
could be left behind very quickly. So weâ€™ll start with something simple.
Blue Ridge Hot Tubs manufactures and sells two models of hot tubs: the Aqua-Spa
and the Hydro-Lux. Howie Jones, the owner and manager of the company, needs to
decide how many of each type of hot tub to produce during his next production
cycle. Howie buys prefabricated ï¬berglass hot tub shells from a local supplier and
adds the pump and tubing to the shells to create his hot tubs. (This supplier has the
capacity to deliver as many hot tub shells as Howie needs.) Howie installs the same
type of pump into both hot tubs. He will have only 200 pumps available during his
next production cycle. From a manufacturing standpoint, the main difference between the two models of hot tubs is the amount of tubing and labor required. Each
Aqua-Spa requires 9 hours of labor and 12 feet of tubing. Each Hydro-Lux requires
6 hours of labor and 16 feet of tubing. Howie expects to have 1,566 production labor
hours and 2,880 feet of tubing available during the next production cycle. Howie
earns a proï¬t of \$350 on each Aqua-Spa he sells and \$300 on each Hydro-Lux he
sells. He is conï¬dent that he can sell all the hot tubs he produces. The question is,
how many Aqua-Spas and Hydro-Luxes should Howie produce if he wants to maximize his proï¬ts during the next production cycle?
2.6 Formulating LP Models
The process of taking a practical problemâ€”such as determining how many Aqua-Spas
and Hydro-Luxes Howie should produceâ€”and expressing it algebraically in the form
of an LP model is known as formulating the model. Throughout the next several chapters, you will see that formulating an LP model is as much an art as a science.
2.6.1 STEPS IN FORMULATING AN LP MODEL
There are some general steps you can follow to help make sure your formulation of a particular problem is accurate. We will walk through these steps using the hot tub example.
22
Chapter 2
Introduction to Optimization and Linear Programming
1. Understand the problem. This step appears to be so obvious that it hardly seems
worth mentioning. However, many people have a tendency to jump into a problem
and start writing the objective function and constraints before they really understand the problem. If you do not fully understand the problem you face, it is unlikely
that your formulation of the problem will be correct.
The problem in our example is fairly easy to understand: How many Aqua-Spas
and Hydro-Luxes should Howie produce to maximize his proï¬t, while using no
more than 200 pumps, 1,566 labor hours, and 2,880 feet of tubing?
2. Identify the decision variables. After you are sure you understand the problem,
you need to identify the decision variables. That is, what are the fundamental decisions that must be made to solve the problem? The answers to this question often
will help you identify appropriate decision variables for your model. Identifying the
decision variables means determining what the symbols X1, X2, . . . , Xn represent in
In our example, the fundamental decision Howie faces is this: How many AquaSpas and Hydro-Luxes should be produced? In this problem, we will let X1 represent
the number of Aqua-Spas to produce and X2 represent the number of Hydro-Luxes to
produce.
3. State the objective function as a linear combination of the decision variables.
After determining the decision variables you will use, the next step is to create the
objective function for the model. This function expresses the mathematical relationship between the decision variables in the model to be maximized or minimized.
In our example, Howie earns a proï¬t of \$350 on each Aqua-Spa (X1) he sells and
\$300 on each Hydro-Lux (X2) he sells. Thus, Howieâ€™s objective of maximizing the
proï¬t he earns is stated mathematically as:
MAX:
350X1 + 300X2
For whatever values might be assigned to X1 and X2, the previous function calculates the associated total proï¬t that Howie would earn. Obviously, Howie wants to
maximize this value.
4. State the constraints as linear combinations of the decision variables. As mentioned earlier, there are usually some limitations on the values that can be assumed
by the decision variables in an LP model. These restrictions must be identiï¬ed and
stated in the form of constraints.
In our example, Howie faces three major constraints. Because only 200 pumps are
available and each hot tub requires one pump, Howie cannot produce more than a
total of 200 hot tubs. This restriction is stated mathematically as:
1X1 + 1X2 â‰¤ 200
This constraint indicates that each unit of X1 produced (that is, each Aqua-Spa
built) will use one of the 200 pumps availableâ€”as will each unit of X2 produced (that
is, each Hydro-Lux built). The total number of pumps used (represented by 1X1
1X2) must be less than or equal to 200.
Another restriction Howie faces is that he has only 1,566 labor hours available
during the next production cycle. Because each Aqua-Spa he builds (each unit of X1)
requires 9 labor hours and each Hydro-Lux (each unit of X2) requires 6 labor hours,
the constraint on the number of labor hours is stated as:
9X1 6X2 â‰¤ 1,566
The General Form of an LP Model
23
The total number of labor hours used (represented by 9X1 6X2) must be less
than or equal to the total labor hours available, which is 1,566.
The ï¬nal constraint speciï¬es that only 2,880 feet of tubing is available for the next
production cycle. Each Aqua-Spa produced (each unit of X1) requires 12 feet of tubing, and each Hydro-Lux produced (each unit of X2) requires 16 feet of tubing. The
following constraint is necessary to ensure that Howieâ€™s production plan does not
use more tubing than is available:
12X1 16X2 â‰¤ 2,880
The total number of feet of tubing used (represented by 12X1 16X2) must be less
than or equal to the total number of feet of tubing available, which is 2,880.
5. Identify any upper or lower bounds on the decision variables. Often, simple upper
or lower bounds apply to the decision variables. You can view upper and lower
bounds as additional constraints in the problem.
In our example, there are simple lower bounds of zero on the variables X1 and X2
because it is impossible to produce a negative number of hot tubs. Therefore, the
following two constraints also apply to this problem:
X1 â‰¥ 0
X2 â‰¥ 0
Constraints like these are often referred to as nonnegativity conditions and are
quite common in LP problems.
2.7 Summary of the LP Model
for the Example Problem
The complete LP model for Howieâ€™s decision problem can be stated as:
MAX:
Subject to:
350X1 300X2
1X1 1X2 â‰¤ 200
9X1 6X2 â‰¤ 1,566
12X1 16X2 â‰¤ 2,880
1X1
â‰¥
0
1X2 â‰¥
0
2.5
2.6
2.7
2.8
2.9
2.10
In this model, the decision variables X1 and X2 represent the number of Aqua-Spas
and Hydro-Luxes to produce, respectively. Our goal is to determine the values for X1
and X2 that maximize the objective in equation 2.5 while simultaneously satisfying all
the constraints in equations 2.6 through 2.10.
2.8 The General Form of an LP Model
The technique of linear programming is so named because the MP problems to which it
applies are linear in nature. That is, it must be possible to express all the functions in an
24
Chapter 2
Introduction to Optimization and Linear Programming
LP model as some weighted sum (or linear combination) of the decision variables. So,
an LP model takes on the general form:
MAX (or MIN):
Subject to:
c1X1 c2X2 cnXn
a11X1 a12X2 a1nXn â‰¤ b1
:
ak1X1 ak2X2 aknXn â‰¥ bk
:
am1X1 am2X2 amnXn = bm
2.11
2.12
2.13
2.14
Up to this point, we have suggested that the constraints in an LP model represent
some type of limited resource. Although this is frequently the case, in later chapters you
will see examples of LP models in which the constraints represent things other than limited resources. The important point here is that any problem that can be formulated in
the above fashion is an LP problem.
The symbols c1, c2, . . . , cn in equation 2.11 are called objective function coefï¬cients
and might represent the marginal proï¬ts (or costs) associated with the decision variables X1, X2, . . . , Xn, respectively. The symbol aij found throughout equations 2.12
through 2.14 represents the numeric coefï¬cient in the ith constraint for variable Xj. The
objective function and constraints of an LP problem represent different weighted sums
of the decision variables. The bi symbols in the constraints, once again, represent values
that the corresponding linear combination of the decision variables must be less than or
equal to, greater than or equal to, or equal to.
You should now see a direct connection between the LP model we formulated for
Blue Ridge Hot Tubs in equations 2.5 through 2.10 and the general deï¬nition of an LP
model given in equations 2.11 through 2.14. In particular, note that the various symbols
used in equations 2.11 through 2.14 to represent numeric constants (that is, the cj, aij,
and bi) were replaced by actual numeric values in equations 2.5 through 2.10. Also,
note that our formulation of the LP model for Blue Ridge Hot Tubs did not require the
use of â€œequal toâ€ constraints. Different problems require different types of constraints,
and you should use whatever types of constraints are necessary for the problem at
hand.
2.9 Solving LP Problems:
An Intuitive Approach
After an LP model has been formulated, our interest naturally turns to solving it. But before we actually solve our example problem for Blue Ridge Hot Tubs, what do you think
is the optimal solution to the problem? Just by looking at the model, what values for X1
and X2 do you think would give Howie the largest proï¬t?
Following one line of reasoning, it might seem that Howie should produce as many
units of X1 (Aqua-Spas) as possible because each of these generates a proï¬t of \$350,
whereas each unit of X2 (Hydro-Luxes) generates a proï¬t of only \$300. But what is the
maximum number of Aqua-Spas that Howie could produce?
Howie can produce the maximum number of units of X1 by making no units of X2
and devoting all his resources to the production of X1. Suppose we let X2 = 0 in the
model in equations 2.5 through 2.10 to indicate that no Hydro-Luxes will be produced.
Solving LP Problems: A Graphical Approach
25
What then is the largest possible value of X1? If X2 = 0, then the inequality in equation 2.6 tells us:
X1 â‰¤ 200
2.15
So we know that X1 cannot be any greater than 200 if X2 = 0. However, we also have
to consider the constraints in equations 2.7 and 2.8. If X2 = 0, then the inequality in
equation 2.7 reduces to:
9X1 â‰¤ 1,566
2.16
If we divide both sides of this inequality by 9, we ï¬nd that the previous constraint is
equivalent to:
X1 â‰¤ 174
2.17
Now consider the constraint in equation 2.8. If X2 = 0, then the inequality in equation 2.8 reduces to:
12X1 â‰¤ 2,880
2.18
Again, if we divide both sides of this inequality by 12, we ï¬nd that the previous
constraint is equivalent to:
X1 â‰¤ 240
2.19
So, if X2 = 0, the three constraints in our model imposing upper limits on the value of
X1 reduce to the values shown in equations 2.15, 2.17, and 2.19. The most restrictive of
these constraints is equation 2.17. Therefore, the maximum number of units of X1 that
can be produced is 174. In other words, 174 is the largest value X1 can take on and still
satisfy all the constraints in the model.
If Howie builds 174 units of X1 (Aqua-Spas) and 0 units of X2 (Hydro-Luxes), he will
have used all of the labor that is available for production (9X1 = 1,566 if X1 = 174). However, he will have 26 pumps remaining (200 X1 = 26 if X1 = 174) and 792 feet of tubing remaining (2,880 12X1 = 792 if X1 = 174). Also, notice that the objective function
value (or total proï¬t) associated with this solution is:
\$350X1 \$300X2 = \$350 174 \$300 0 = \$60,900
From this analysis, we see that the solution X1 = 174, X2 = 0 is a feasible solution to the
problem because it satisï¬es all the constraints of the model. But is it the optimal solution?
In other words, is there any other possible set of values for X1 and X2 that also satisï¬es
all the constraints and results in a higher objective function value? As you will see, the
intuitive approach to solving LP problems that we have taken here cannot be trusted
because there actually is a better solution to Howieâ€™s problem.
2.10 Solving LP Problems:
A Graphical Approach
The constraints of an LP model deï¬ne the set of feasible solutionsâ€”or the feasible
regionâ€”for the problem. The difï¬culty in LP is determining which point or points in
the feasible region correspond to the best possible value of the objective function. For
simple problems with only two decision variables, it is fairly easy to sketch the feasible
region for the LP model and locate the optimal feasible point graphically. Because the
graphical approach can be used only if there are two decision variables, it has limited
practical use. However, it is an extremely good way to develop a basic understanding of
26
Chapter 2
Introduction to Optimization and Linear Programming
the strategy involved in solving LP problems. Therefore, we will use the graphical approach to solve the simple problem faced by Blue Ridge Hot Tubs. Chapter 3 shows how
to solve this and other LP problems using a spreadsheet.
To solve an LP problem graphically, ï¬rst you must plot the constraints for the problem and identify its feasible region. This is done by plotting the boundary lines of the constraints and identifying the points that will satisfy all the constraints. So, how do we do
this for our example problem (repeated below)?
MAX:
Subject to:
350X1 300X2
1X1 1X2 â‰¤ 200
9X1 6X2 â‰¤ 1,566
12X1 16X2 â‰¤ 2,880
1X1
â‰¥
0
1X2 â‰¥
0
2.20
2.21
2.22
2.23
2.24
2.25
2.10.1 PLOTTING THE FIRST CONSTRAINT
The boundary of the ï¬rst constraint in our model, which speciï¬es that no more than
200 pumps can be used, is represented by the straight line deï¬ned by the equation:
X1 X2 = 200
2.26
If we can ï¬nd any two points on this line, the entire line can be plotted easily by
drawing a straight line through these points. If X2 = 0, we can see from equation 2.26
that X1 = 200. Thus, the point (X1, X2) = (200, 0) must fall on this line. If we let X1 = 0,
from equation 2.26, it is easy to see that X2 = 200. So, the point (X1, X2) = (0, 200) also
must fall on this line. These two points are plotted on the graph in Figure 2.1 and connected to form the straight line representing equation 2.26.
Note that the graph of the line associated with equation 2.26 actually extends beyond
the X1 and X2 axes shown in Figure 2.1. However, we can disregard the points beyond
these axes because the values assumed by X1 and X2 cannot be negative (because we also
have the constraints given by X1 â‰¥ 0 and X2 â‰¥ 0).
The line connecting the points (0, 200) and (200, 0) in Figure 2.1 identiï¬es the points
(X1, X2) that satisfy the equality X1 X2 = 200. But recall that the ï¬rst constraint in the
LP model is the inequality X1 X2 â‰¤ 200. Thus, after plotting the boundary line of a constraint, we must determine which area on the graph corresponds to feasible solutions
for the original constraint. This can be done easily by picking an arbitrary point on
either side of (i.e., not on) the boundary line and checking whether it satisï¬es the original constraint. For example, the point (X1, X2) = (0, 0) is not on the boundary line of the
ï¬rst constraint and also satisï¬es the ï¬rst constraint. Therefore, the area of the graph on
the same side of the boundary line as the point (0, 0) c…
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