数学建模（英文版.第3版）

数学建模这门课程在数学及其在各个领域的应用之间架起一座桥梁。本书介绍了速个建模过程的原理，通过本书的学习，学生将有机会在以下建模活动中亲身实践，增强解决问题的能力；设计创意模型和经验模型、模型分析以及模型研究。

本书特点

论证了离散动态系统、离散优化和仿真等技术如何促进现代应用数学的发展

强调通过模型设计提高学生的创造性，展现模型构建的艺术特性，包括经验建模和仿真建模的思想

将数学建模方法与多样化建模和置信度建立等更具创造性的方面结合起来

在设计创意模型和经验模型、模型分析以及模型研究中融入个人项目和小组项目，并且包含大量的例子和习题随书光盘包含软件、附加的建模场景和项目以及过去数学建模竞赛的题目

本书特点

论证了离散动态系统、离散优化和仿真等技术如何促进现代应用数学的发展

强调通过模型设计提高学生的创造性，展现模型构建的艺术特性，包括经验建模和仿真建模的思想

将数学建模方法与多样化建模和置信度建立等更具创造性的方面结合起来

在设计创意模型和经验模型、模型分析以及模型研究中融入个人项目和小组项目，并且包含大量的例子和习题随书光盘包含软件、附加的建模场景和项目以及过去数学建模竞赛的题目

To facilitate an early initiation of the modeling experience, the first edition of this text was designed to be taught concurrently or immediately after an introductory business or engineering calculus course. In the second edition, we added chapters treating discrete dynamical systems, linear programming and numerical search methods, and an introduction to probabilistic modeling. Additionally, we expanded our introduction of simulation. In this edition we have included solution methods to some simple dynamical systems to reveal their long-term behavior. We have also added basic numerical solution methods to the chapters covering modeling with differential equations. The text has been reorganized into two parts: Part One,Discrete Modeling (Chapters 1-8), and Part Two, Continuous Modeling (Chapters 9-12). This organizational structure allows for teaching an entire modeling course based on Part One and which does not require the calculus. Part Two then addresses continuous models based on optimization and differential equations which can be presented concurrently with freshman calculus. The text gives students an opportunity to cover all phases of the mathematical modeling process. The new CD-ROM accompanying the text contains software, additional modeling scenarios and projects, and a link to past problems from the Mathematical Contest in Modeling. We thank Sol Garfunkel and the COMAP staff for preparing the CD and for their support of modeling activities that we refer to under Resource Materials below.

Goals and Orientation

The course continues to be a bridge between the study of mathematics and the applications of mathematics to various fields. The course affords the student an early opportunity to see how the pieces of an applied problem fit together. The student investigates meaningful and practical problems chosen from common experiences encompassing many academic disciplines, including the mathematical sciences, operations research, engineering, and the management and life sciences.

This text provides an introduction to the entire modeling process. The student will have occasions to practice the following facets of modeling and enhance their problem-solving capabilities:

1. Creative and Empirical Model Construction: Given a real-world scenario,the student learns to identify a problem, make assumptions and collect data, propose a model, test the assumptions, refine the model as necessary, fit the model to data if appropriate, and analyze the underlying mathematical structure of the model to appraise the sensitivity of the conclusions when the assumptions are not precisely met.

2. Model Analysis: Given a model, the student learns to work backward to uncover the implicit underlying assumptions, assess critically how well those as- sumptions fit the scenario at hand, and estimate the sensitivity of the conclusions when the assumptions are not precisely met.

3. Model Research: The student investigates a specific area to gain a deeper understanding of some behavior and learns to use what has already been created or discovered.

Student Background and Course Content Because our desire is to initiate the modeling experience as early as possible in the student's program, the only prerequisite for Chapters 9, 10, and 11 is a basic understanding of single-variable differential and integral calculus. Although some unfamiliar mathematical ideas are taught as part of the modeling process, the emphasis is on using mathematics already known by the students after completing high school. This emphasis is especially tree in Part One. The modeling course will then motivate students to study the more advanced courses such as linear algebra, differential equations, optimization and linear programming, numerical analysis, probability, and statistics. The power and utility of these subjects are intimated throughout the text.

Further, the scenarios and problems in the text are not designed for the application of a particular mathematical technique. Instead, they demand thoughtful ingenuity in using fundamental concepts to find reasonable solutions to "open-ended" problems. Certain mathematical techniques (such as Monte Carlo simulation, curve fitting, and dimensional analysis) are presented because often they are not formally covered at the undergraduate level. Instructors should find great flexibility in adapting the text to meet the particular needs of students through the problem assignments and student projects. We have used this material to teach courses to both undergraduate and graduate students, and even as a basis for faculty seminars.

Organization of the Text

The organization of the text is best understood with the aid of Figure 1. The first eight chapters constitute Part One and require only precalculus mathematics as a prerequisite. We begin with the idea of modeling change using simple finite difference equations. This approach is quite intuitive to the student and provides us with several concrete models to support our discussion of the modeling process in Chapter 2. There we classify models, analyze the modeling process, and construct several proportionality models or submodels which are then revisited in the next two chapters. In Chapter 3 the student is presented with three criteria for fitting a specific curve-type to a collected data set, with emphasis on the least-squares cfiteflon. Chapter 4 addresses the problem of capturing the trend of a collected set of data. In this empirical construction process, we begin with fitting simple oneterm models approximating collected data sets and progress to more sophisticated interpolating models, including polynomial smoothing models and cubic splines.

Simulation models are discussed in Chapter 5. An empirical model is fit to some collected data, and then Monte Carlo simulation is used to duplicate the behavior being investigated. The presentation motivates the eventual study of probability and statistics.

Chapter 6 provides an introduction to probabihsfic modeling. The topics of Markov processes, reliability, and linear regression are introduced, building on scenarios and analysis presented previously. Chapter 7 addresses the issue of findingthe best-fitting model using the other two criteria presented in Chapter 3. Linear programming is the method used for finding the "best" model for one of the criteria,and numerical search techniques can be used for the other. The chapter concludes with an introduction to numerical search methods including the dichotomous and golden section methods. Part One ends with Chapter 8, which is devoted to dimensional analysis, a topic of great importance in the physical sciences and engineering.a combination of graphing calculators and computers to be advantageous throughout the course. The use of a spreadsheet is beneficial in Chapters 1, 5, and 7, and the capability for graphical displays of data is enormously useful, even essential,whenever data is provided. Students will find computers useful, too, in transforming data, least-squares curve fitting, divided difference tables and cubic splines,programming simulation models, linear programming and numerical search methods, and numerical solutions to differential equations. The CD accompanying this text provides some basic technology tools that students and instructors can use as a foundation for modeling with technology. Several FORTRAN executable programs are provided to execute the methodologies presented in Chapter 4. Also included is a tutorial on the computer algebra system MAPLE and its use for this text.

Resource Materials

We have found material provided by the Consortium for Mathematics and Its Application (COMAP) to be outstanding and particularly well suited to the course we propose. Individual modules for the undergraduate classroom, UMAP Modules,may be used in a variety of ways. First, they may be used as instructional material to support several lessons. In this mode a student completes the self-study module by working through its exercises (the detailed solutions provided with the module can be conveniently removed before it is issued). Another option is to put together a block of instruction using one or more UMAP modules suggested in the projects sections of the text. The modules also provide excellent sources for "model research" because they cover a wide variety of applications of mathematics in many fields. In this mode, a student is given an appropriate module to research and is asked to complete and report on the module. Finally, the modules are excellent resources for scenarios for which students can practice model construction. In this mode the instructor writes a scenario for a student project based on an application addressed in a particular module and uses the module as background material, perhaps having the student complete the module at a later date. The CD accompanying the text contains most of the UMAPS referenced throughout. Information on the availability of newly developed interdisciplinary projects can be obtained by writing COMAP at the address given previously, calling COMAP at 1-800-772-6627,or electronically: order@comap.com

A great source of student-group projects are the Mathematical Contest in Modeling (MCM) and the Interdisciplinary Mathematical Contest in Modeling (IMCM). These projects can be taken from the link provided on the CD and tailored by the instructor to meet specific goals for their class. These are also good resources to prepare teams to compete in the MCM and IMCM contests currently sponsored by the National Security Agency (NSA) and COMAP. The contest is sponsored by COMAP with funding support from the National Security Agency, the Society of Industrial and Applied Mathematics, the Institute for Operations Research and the Management Sciences, and the Mathemafcal Association of America. Addi- tional information concerning the contest can be obtained by contacting COMAP, or visiting their website at www.comap.com.

Acknowledgments

It is always a pleasure to acknowledge individuals who have played a role in the development of a book. We are particularly grateful to B.G. (retired) Jack M. Pollin and Dr. Carroll Wilde for stimulating our interest in teaching modeling and for support and guidance in our careers. We're indebted to many colleagues for reading the first edition manuscript and suggesting modifications and problems: Rickey Kolb, John Kenelly, Robert Schmidt, Stan Leja, Bard Mansager, and especially Steve Maddox and Jim McNulty.

We are indebted to a number of individuals who authored or coauthored UMAP materials that support the text: David Cameron, Bfindell Horelick, Michael Jaye, Sinan Koont, Start Leja, Michael Wells, and Carroll Wilde. In addition, we thank Solomon Garfunkel and the entire COMAP staff for their cooperation on this project, especially Roland Cheyney for his help with the production of the CD that accompanies the text. We also thank Tom O'Neil and his students for their contributions to the CD and Tom's helpful suggestions in support of modeling activities.

The production of any mathematics text is a complex process and we have been especially fortunate in having a superb and creative production staff at Brooks/Cole. In particular, we express our thanks to Craig Barth, our editor for the first edition, Gary Ostedt, the second edition, and Gary Ostedt and Bob Pirtle, our editors for this edition. For this edition we are especially grateful to Tom Ziolkowski, our marketing manager; Tom Novack, our production editor; Merrill Peterson and Matrix Productions for production service; and Amy Moellefing for her superb copyediting anti typesetting. We are especially grateful to Wendy Fox for providing her drawing of the Cadet Chapel at West Point for the dedication page.

Finally, we are grateful to our wives--Judi Giordano, Gale Weir, and Wendy Fox--for their inspiration and support.

Frank R. Giordano

Maurice D. Weir

William P. Fox

Goals and Orientation

The course continues to be a bridge between the study of mathematics and the applications of mathematics to various fields. The course affords the student an early opportunity to see how the pieces of an applied problem fit together. The student investigates meaningful and practical problems chosen from common experiences encompassing many academic disciplines, including the mathematical sciences, operations research, engineering, and the management and life sciences.

This text provides an introduction to the entire modeling process. The student will have occasions to practice the following facets of modeling and enhance their problem-solving capabilities:

1. Creative and Empirical Model Construction: Given a real-world scenario,the student learns to identify a problem, make assumptions and collect data, propose a model, test the assumptions, refine the model as necessary, fit the model to data if appropriate, and analyze the underlying mathematical structure of the model to appraise the sensitivity of the conclusions when the assumptions are not precisely met.

2. Model Analysis: Given a model, the student learns to work backward to uncover the implicit underlying assumptions, assess critically how well those as- sumptions fit the scenario at hand, and estimate the sensitivity of the conclusions when the assumptions are not precisely met.

3. Model Research: The student investigates a specific area to gain a deeper understanding of some behavior and learns to use what has already been created or discovered.

Student Background and Course Content Because our desire is to initiate the modeling experience as early as possible in the student's program, the only prerequisite for Chapters 9, 10, and 11 is a basic understanding of single-variable differential and integral calculus. Although some unfamiliar mathematical ideas are taught as part of the modeling process, the emphasis is on using mathematics already known by the students after completing high school. This emphasis is especially tree in Part One. The modeling course will then motivate students to study the more advanced courses such as linear algebra, differential equations, optimization and linear programming, numerical analysis, probability, and statistics. The power and utility of these subjects are intimated throughout the text.

Further, the scenarios and problems in the text are not designed for the application of a particular mathematical technique. Instead, they demand thoughtful ingenuity in using fundamental concepts to find reasonable solutions to "open-ended" problems. Certain mathematical techniques (such as Monte Carlo simulation, curve fitting, and dimensional analysis) are presented because often they are not formally covered at the undergraduate level. Instructors should find great flexibility in adapting the text to meet the particular needs of students through the problem assignments and student projects. We have used this material to teach courses to both undergraduate and graduate students, and even as a basis for faculty seminars.

Organization of the Text

The organization of the text is best understood with the aid of Figure 1. The first eight chapters constitute Part One and require only precalculus mathematics as a prerequisite. We begin with the idea of modeling change using simple finite difference equations. This approach is quite intuitive to the student and provides us with several concrete models to support our discussion of the modeling process in Chapter 2. There we classify models, analyze the modeling process, and construct several proportionality models or submodels which are then revisited in the next two chapters. In Chapter 3 the student is presented with three criteria for fitting a specific curve-type to a collected data set, with emphasis on the least-squares cfiteflon. Chapter 4 addresses the problem of capturing the trend of a collected set of data. In this empirical construction process, we begin with fitting simple oneterm models approximating collected data sets and progress to more sophisticated interpolating models, including polynomial smoothing models and cubic splines.

Simulation models are discussed in Chapter 5. An empirical model is fit to some collected data, and then Monte Carlo simulation is used to duplicate the behavior being investigated. The presentation motivates the eventual study of probability and statistics.

Chapter 6 provides an introduction to probabihsfic modeling. The topics of Markov processes, reliability, and linear regression are introduced, building on scenarios and analysis presented previously. Chapter 7 addresses the issue of findingthe best-fitting model using the other two criteria presented in Chapter 3. Linear programming is the method used for finding the "best" model for one of the criteria,and numerical search techniques can be used for the other. The chapter concludes with an introduction to numerical search methods including the dichotomous and golden section methods. Part One ends with Chapter 8, which is devoted to dimensional analysis, a topic of great importance in the physical sciences and engineering.a combination of graphing calculators and computers to be advantageous throughout the course. The use of a spreadsheet is beneficial in Chapters 1, 5, and 7, and the capability for graphical displays of data is enormously useful, even essential,whenever data is provided. Students will find computers useful, too, in transforming data, least-squares curve fitting, divided difference tables and cubic splines,programming simulation models, linear programming and numerical search methods, and numerical solutions to differential equations. The CD accompanying this text provides some basic technology tools that students and instructors can use as a foundation for modeling with technology. Several FORTRAN executable programs are provided to execute the methodologies presented in Chapter 4. Also included is a tutorial on the computer algebra system MAPLE and its use for this text.

Resource Materials

We have found material provided by the Consortium for Mathematics and Its Application (COMAP) to be outstanding and particularly well suited to the course we propose. Individual modules for the undergraduate classroom, UMAP Modules,may be used in a variety of ways. First, they may be used as instructional material to support several lessons. In this mode a student completes the self-study module by working through its exercises (the detailed solutions provided with the module can be conveniently removed before it is issued). Another option is to put together a block of instruction using one or more UMAP modules suggested in the projects sections of the text. The modules also provide excellent sources for "model research" because they cover a wide variety of applications of mathematics in many fields. In this mode, a student is given an appropriate module to research and is asked to complete and report on the module. Finally, the modules are excellent resources for scenarios for which students can practice model construction. In this mode the instructor writes a scenario for a student project based on an application addressed in a particular module and uses the module as background material, perhaps having the student complete the module at a later date. The CD accompanying the text contains most of the UMAPS referenced throughout. Information on the availability of newly developed interdisciplinary projects can be obtained by writing COMAP at the address given previously, calling COMAP at 1-800-772-6627,or electronically: order@comap.com

A great source of student-group projects are the Mathematical Contest in Modeling (MCM) and the Interdisciplinary Mathematical Contest in Modeling (IMCM). These projects can be taken from the link provided on the CD and tailored by the instructor to meet specific goals for their class. These are also good resources to prepare teams to compete in the MCM and IMCM contests currently sponsored by the National Security Agency (NSA) and COMAP. The contest is sponsored by COMAP with funding support from the National Security Agency, the Society of Industrial and Applied Mathematics, the Institute for Operations Research and the Management Sciences, and the Mathemafcal Association of America. Addi- tional information concerning the contest can be obtained by contacting COMAP, or visiting their website at www.comap.com.

Acknowledgments

It is always a pleasure to acknowledge individuals who have played a role in the development of a book. We are particularly grateful to B.G. (retired) Jack M. Pollin and Dr. Carroll Wilde for stimulating our interest in teaching modeling and for support and guidance in our careers. We're indebted to many colleagues for reading the first edition manuscript and suggesting modifications and problems: Rickey Kolb, John Kenelly, Robert Schmidt, Stan Leja, Bard Mansager, and especially Steve Maddox and Jim McNulty.

We are indebted to a number of individuals who authored or coauthored UMAP materials that support the text: David Cameron, Bfindell Horelick, Michael Jaye, Sinan Koont, Start Leja, Michael Wells, and Carroll Wilde. In addition, we thank Solomon Garfunkel and the entire COMAP staff for their cooperation on this project, especially Roland Cheyney for his help with the production of the CD that accompanies the text. We also thank Tom O'Neil and his students for their contributions to the CD and Tom's helpful suggestions in support of modeling activities.

The production of any mathematics text is a complex process and we have been especially fortunate in having a superb and creative production staff at Brooks/Cole. In particular, we express our thanks to Craig Barth, our editor for the first edition, Gary Ostedt, the second edition, and Gary Ostedt and Bob Pirtle, our editors for this edition. For this edition we are especially grateful to Tom Ziolkowski, our marketing manager; Tom Novack, our production editor; Merrill Peterson and Matrix Productions for production service; and Amy Moellefing for her superb copyediting anti typesetting. We are especially grateful to Wendy Fox for providing her drawing of the Cadet Chapel at West Point for the dedication page.

Finally, we are grateful to our wives--Judi Giordano, Gale Weir, and Wendy Fox--for their inspiration and support.

Frank R. Giordano

Maurice D. Weir

William P. Fox

1 Modeling Change 1

Introduction 1

Example 1: Testing for Proportionality 2

1.1 Modeling Change with Difference Equations 4

Example I: A Savings Certificate 5

Example 2: Mortgaging a Home 6

1.2 Approximating Change with Difference Equations

Example 1: Growth of a Yeast Culture 9

Example 2: Growth of a Yeast Culture Revisited 10

Example 3: Spread of a Contagious Disease 12

Example 4: Decay of Digoxin in the Bloodstream 13

Example 5: Heating of a Cooled Object 14

1.3 Solutions to Dynamical Systems 18

Example 1: A Savings Certificate Revisited 18

Example 2: Sewage Treatment 21

Example 3: Prescription for Digoxin 25

Example 4: An Investment Annuity 26

Example 5: A Checking Account 28

Example 6: An Investment Annuity Revisited 30

1.4 Systems of Difference Equations 35

Example I: A Car Rental Company 35

Example 2: The Battle of Trafalgar 38

Example 3: Competitive Hunter Model--Spotted Owls and Hawks

Example 4: Voting Tendencies of the Political Parties 44

2 The Modeling Process, Proportionality, and Geometric Similarity Introduction 52

2.1 Mathematical Models 54

Example 1: Vehicular Stopping Distance 59

2.2 Modeling Using Proportionality 65

Example 1: Kepler's Third Law 67

2.3 Modeling Using Geometric Similarity 75

Example 1: Raindrops from a Motionless Cloud 77

Example 2: Modeling a Bass Fishing Derby 79

2.4 Automobile Gasoline Mileage 88

2.5 Body Weight and Height, Strength and Agility 91

Model Fitting 97

Introduction 97

3.1 Fitting Models to Data Graphically 101

3.2 Analytic Methods of Model Fitting 107

3.3 Applying the Least-Squares Criterion 114

3.4 Choosing a Best Model 119

Example 1: Vehicular Stopping Distance 122

4 Experimental Modeling 126

Introduction 126

4.1 Harvesting in the Chesapeake Bay and Other

One-Term Models 127

Example 1: Harvesting Bluefish 130

Example 2: Harvesting Blue Crabs 131

4.2 High-Order Polynomial Models 138

Example 1: Elapsed Time of a Tape Recorder 140

4.3 Smoothing: Low-Order Polynomial Models 146

Example 1: Elapsed Time of a Tape Recorder Revisited 147

Example 2: Elapsed Time of a Tape Recorder Revisited Again

Example 3: Vehicle Stopping Distance 153

Example 4: Growth of a Yeast Culture 155

4.4 Cubic Spline Models 159

Example 1: Vehicle Stopping Distance Revisited 167

5 Simulation Modeling 175

Introduction 175

5.1 Simulating Deterministic Behavior:Area Under a Curve

5.2 Generating Random Numbers 182

5.3 Simulating Probabilistic Behavior '186

5.4 Inventory Model: Gasoline and Consumer Demand 194

5.5 Queuing Models 205

Example 1: A Harbor System 205

Example 2: Morning Rush Hour 213

6 Discrete Probabilistic Modeling 217

Introduction 217

6.1 Probabilistic Modeling with Discrete Systems 217

Example 1: Rental Car Company Revisited 217

Example 2: Voting Tendencies 219

6.2 Modeling Component and System Reliability 223

Example 1: Series Systems 224

Example 2: Parallel Systems 224

Example 3: Series and Parallel Combinations 224

6.3 Linear Regression 227

Example 1: Ponderosa Pines 229

Example 2: The Bass Fishing Derby Revisited 231

7 Discrete Optimization Modeling 236

7.1 An Overview of Discrete Optimization Modeling 237

Example 1: Determining a Production Schedule 237

Example 2: Space Shuttle Cargo 240

Example 3: Approximation by a Piecewise Linear Function 241

7.2 Linear Programming h Geometric Solutions 250

Example 1: The Carpenter's Problem 251

Example 2: A Data-Fitting Problem 252

7.3 Linear Programming Ih Algebraic Solutions 259

Example 1: Solving the Carpenter's Problem Algebraically 261

7.4 Linear Programming III: The Simplex Method 263

Example 1: The Carpenter's Problem Revisited 268

Example 2: Using the Tableau Format 271

7.5 Linear Programming IV: Sensitivity Analysis 273

7.6 Numerical Search Methods 279

Example 1: Using the Dichotomous Search Method 282

Example 2: Using the Golden Section Search Method 285

Example 3: Model-Fitting Criterion Revisited 287

Example 4: Optimizing Industrial Flow 288

8 Dimensional Analysis and Similitude 292

Introduction 292

8.1 Dimensions as Products 295

Example 1: A Simple Pendulum 298

Example 2: Wind Force on a Van 301

8.2 The Process of Dimensional Analysis 304

Example l: Terminal Velocity of a Raindrop 309

Example 2: Automobile Gas Mileage Revisited 311

8.3 A Damped Pendulum 313

8.4 Examples Illustrating Dimensional Analysis 319

Example I : Explosion Analysis 319

Example 2: How Long Should You Roast a Turkey? 324

8.5 Similitude 330

Example I: Drag Force on a Submarine 331

9 Graphs of Functions as Models 336

9.1 An Arms Race 336

Example 1: Civil Defense 345

Example 2: Mobile Launching Pads 346

Example 3: Multiple Warheads 347

Example 4: MIRVs Revisited: Counting Warheads 348

9.2 Modeling an Arms Race in Stages 350

9.3 Managing Nonrenewable Resources: The Energy Crisis

9.4 Effects of Taxation on the Energy Crisis 359

9.5 A Gasoline Shortage and Taxation 364

10 Modeling with a Differential Equation 368

Introduction 368

10.1 Population Growth 371

10.2 Prescribing Drug Dosage

10.3 Braking Distance Revisited 391

10.4 Graphical Solutions of Autonomous Differential Equations

Example 1: Drawing a Phase Line and Sketching

Solution Curves 396

Example 2: Cooling Soup 399

Example 3: Logistic Growth Revisited 400

10.5 Numerical Approximation Methods 404

Example 1: Using Euler's Method 406

Example 2: A Savings Certificate Revisited 407

11 Modeling with Systems of

Differential Equations 412

Introduction 412

11.1 Graphical Solutions of Autonomous Systems of First-Order

Differential Equations 413

Example 1: A Linear Autonomous System 414

Example 2: A Nonlinear Autonomous System 415

11.2 A Competitive Hunter Model 419

11.3 A Predator-Prey Model 427

11.4 Two Military Examples 437

Example I: Lanchester Combat Models 437

Example 2: Economic Aspects of an Arms Race 444

11.5 Euler's Method for Systems of Differential Equations 450

Example 1: Using Euler' s Method for Systems 451

Example 2: A Trajectory and Solution Curves 452

12 Continuous Optimization Modeling 458

Introduction 458

12.1 An Inventory Problem: Minimizing the Cost of Delivery and

Storage 459

12.2 A Manufacturing Problem: Maximizing Profit in Producing

Competing Products 468

12.3 Constrained Continuous Optimization 474

Example l: An Oil Transfer Company 474

Example 2: A Space Shuttle Water Container 476

12.4 Managing Renewable Resources: The Fishing Industry 480

A Problems from the Mathematics Contest in

Modeling. 1985-2002 490

B An Elevator Simulation Algorithm s23

C The Revised Simplex Method

Index 535

Introduction 1

Example 1: Testing for Proportionality 2

1.1 Modeling Change with Difference Equations 4

Example I: A Savings Certificate 5

Example 2: Mortgaging a Home 6

1.2 Approximating Change with Difference Equations

Example 1: Growth of a Yeast Culture 9

Example 2: Growth of a Yeast Culture Revisited 10

Example 3: Spread of a Contagious Disease 12

Example 4: Decay of Digoxin in the Bloodstream 13

Example 5: Heating of a Cooled Object 14

1.3 Solutions to Dynamical Systems 18

Example 1: A Savings Certificate Revisited 18

Example 2: Sewage Treatment 21

Example 3: Prescription for Digoxin 25

Example 4: An Investment Annuity 26

Example 5: A Checking Account 28

Example 6: An Investment Annuity Revisited 30

1.4 Systems of Difference Equations 35

Example I: A Car Rental Company 35

Example 2: The Battle of Trafalgar 38

Example 3: Competitive Hunter Model--Spotted Owls and Hawks

Example 4: Voting Tendencies of the Political Parties 44

2 The Modeling Process, Proportionality, and Geometric Similarity Introduction 52

2.1 Mathematical Models 54

Example 1: Vehicular Stopping Distance 59

2.2 Modeling Using Proportionality 65

Example 1: Kepler's Third Law 67

2.3 Modeling Using Geometric Similarity 75

Example 1: Raindrops from a Motionless Cloud 77

Example 2: Modeling a Bass Fishing Derby 79

2.4 Automobile Gasoline Mileage 88

2.5 Body Weight and Height, Strength and Agility 91

Model Fitting 97

Introduction 97

3.1 Fitting Models to Data Graphically 101

3.2 Analytic Methods of Model Fitting 107

3.3 Applying the Least-Squares Criterion 114

3.4 Choosing a Best Model 119

Example 1: Vehicular Stopping Distance 122

4 Experimental Modeling 126

Introduction 126

4.1 Harvesting in the Chesapeake Bay and Other

One-Term Models 127

Example 1: Harvesting Bluefish 130

Example 2: Harvesting Blue Crabs 131

4.2 High-Order Polynomial Models 138

Example 1: Elapsed Time of a Tape Recorder 140

4.3 Smoothing: Low-Order Polynomial Models 146

Example 1: Elapsed Time of a Tape Recorder Revisited 147

Example 2: Elapsed Time of a Tape Recorder Revisited Again

Example 3: Vehicle Stopping Distance 153

Example 4: Growth of a Yeast Culture 155

4.4 Cubic Spline Models 159

Example 1: Vehicle Stopping Distance Revisited 167

5 Simulation Modeling 175

Introduction 175

5.1 Simulating Deterministic Behavior:Area Under a Curve

5.2 Generating Random Numbers 182

5.3 Simulating Probabilistic Behavior '186

5.4 Inventory Model: Gasoline and Consumer Demand 194

5.5 Queuing Models 205

Example 1: A Harbor System 205

Example 2: Morning Rush Hour 213

6 Discrete Probabilistic Modeling 217

Introduction 217

6.1 Probabilistic Modeling with Discrete Systems 217

Example 1: Rental Car Company Revisited 217

Example 2: Voting Tendencies 219

6.2 Modeling Component and System Reliability 223

Example 1: Series Systems 224

Example 2: Parallel Systems 224

Example 3: Series and Parallel Combinations 224

6.3 Linear Regression 227

Example 1: Ponderosa Pines 229

Example 2: The Bass Fishing Derby Revisited 231

7 Discrete Optimization Modeling 236

7.1 An Overview of Discrete Optimization Modeling 237

Example 1: Determining a Production Schedule 237

Example 2: Space Shuttle Cargo 240

Example 3: Approximation by a Piecewise Linear Function 241

7.2 Linear Programming h Geometric Solutions 250

Example 1: The Carpenter's Problem 251

Example 2: A Data-Fitting Problem 252

7.3 Linear Programming Ih Algebraic Solutions 259

Example 1: Solving the Carpenter's Problem Algebraically 261

7.4 Linear Programming III: The Simplex Method 263

Example 1: The Carpenter's Problem Revisited 268

Example 2: Using the Tableau Format 271

7.5 Linear Programming IV: Sensitivity Analysis 273

7.6 Numerical Search Methods 279

Example 1: Using the Dichotomous Search Method 282

Example 2: Using the Golden Section Search Method 285

Example 3: Model-Fitting Criterion Revisited 287

Example 4: Optimizing Industrial Flow 288

8 Dimensional Analysis and Similitude 292

Introduction 292

8.1 Dimensions as Products 295

Example 1: A Simple Pendulum 298

Example 2: Wind Force on a Van 301

8.2 The Process of Dimensional Analysis 304

Example l: Terminal Velocity of a Raindrop 309

Example 2: Automobile Gas Mileage Revisited 311

8.3 A Damped Pendulum 313

8.4 Examples Illustrating Dimensional Analysis 319

Example I : Explosion Analysis 319

Example 2: How Long Should You Roast a Turkey? 324

8.5 Similitude 330

Example I: Drag Force on a Submarine 331

9 Graphs of Functions as Models 336

9.1 An Arms Race 336

Example 1: Civil Defense 345

Example 2: Mobile Launching Pads 346

Example 3: Multiple Warheads 347

Example 4: MIRVs Revisited: Counting Warheads 348

9.2 Modeling an Arms Race in Stages 350

9.3 Managing Nonrenewable Resources: The Energy Crisis

9.4 Effects of Taxation on the Energy Crisis 359

9.5 A Gasoline Shortage and Taxation 364

10 Modeling with a Differential Equation 368

Introduction 368

10.1 Population Growth 371

10.2 Prescribing Drug Dosage

10.3 Braking Distance Revisited 391

10.4 Graphical Solutions of Autonomous Differential Equations

Example 1: Drawing a Phase Line and Sketching

Solution Curves 396

Example 2: Cooling Soup 399

Example 3: Logistic Growth Revisited 400

10.5 Numerical Approximation Methods 404

Example 1: Using Euler's Method 406

Example 2: A Savings Certificate Revisited 407

11 Modeling with Systems of

Differential Equations 412

Introduction 412

11.1 Graphical Solutions of Autonomous Systems of First-Order

Differential Equations 413

Example 1: A Linear Autonomous System 414

Example 2: A Nonlinear Autonomous System 415

11.2 A Competitive Hunter Model 419

11.3 A Predator-Prey Model 427

11.4 Two Military Examples 437

Example I: Lanchester Combat Models 437

Example 2: Economic Aspects of an Arms Race 444

11.5 Euler's Method for Systems of Differential Equations 450

Example 1: Using Euler' s Method for Systems 451

Example 2: A Trajectory and Solution Curves 452

12 Continuous Optimization Modeling 458

Introduction 458

12.1 An Inventory Problem: Minimizing the Cost of Delivery and

Storage 459

12.2 A Manufacturing Problem: Maximizing Profit in Producing

Competing Products 468

12.3 Constrained Continuous Optimization 474

Example l: An Oil Transfer Company 474

Example 2: A Space Shuttle Water Container 476

12.4 Managing Renewable Resources: The Fishing Industry 480

A Problems from the Mathematics Contest in

Modeling. 1985-2002 490

B An Elevator Simulation Algorithm s23

C The Revised Simplex Method

Index 535

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