Mathematical Modeling and Computer Simulation

by Maki, Daniel P.; Thompson, Maynard

• ISBN13: 9780534384784
• ISBN10: 0534384781
• Format: Hardcover
• Copyright: 2005-01-06
• Publisher: Cengage Learning

Summary

Learn to build and use mathematical models with MATHEMATICAL MODELING AND COMPUTER SIMULATION! Through the description of mathematical and computer models in a variety of situations, this mathematics text helps you learn that model building is a dynamic process involving simplification, approximation, abstraction, analysis, computation, and comparison. Case studies illustrate how the model building process is applied to real life situations arising in a variety of settings, including business, genetics, population biology, and social science. An appendix on student projects provides you with a selection of classroom-tested projects with hints and suggestions for organizing project work and communicating results.

Table of Contents

CHAPTER 1 Basic Principles	1	(24)
1.0 Overview of the Uses of the Term Model	1	(3)
1.1 The Process of Constructing Mathematical Models	4	(3)
1.2 Types of Mathematical Models and Some Practical Aspects of Model Building	7	(3)
Deterministic versus Stochastic Models	8	(1)
Implementing a Model	9	(1)
1.3 A Classic Example	10	(3)
Ptolemy	10	(1)
Copernicus and Kepler	11	(1)
Newton	12	(1)
1.4 Axiom Systems and Models	13	(7)
Axioms	14	(1)
Axiom Systems	15	(1)
Models and Formal Model Building	16	(4)
1.5 Simulation Models	20	(1)
1.6 Practical Aspects of Model Building	20	(5)
Intuitive Evaluations	21	(1)
Statistics for the Model-Building Process	21	(4)
CHAPTER 2 Model Building: Selected Case Studies	25	(81)
2.0 Introduction	25	(1)
2.1 Mendelian Genetics	25	(11)
Some Observations	26	(1)
A Real Model	27	(1)
A Mathematical Model	28	(8)
2.2 Models for Growth Processes	36	(14)
Choices for Growth Models	37	(5)
Discrete Logistic Model	42	(8)
2.3 Social Choice	50	(13)
2.4 Moving Mobile Homes	63	(7)
The Situation	63	(1)
A Real Model	64	(1)
A Mathematical Model	65	(5)
2.5 A Stratified Population Model	70	(8)
2.6 Simulation Models in Athletics, Marketing, and Population Studies	78	(8)
2.7 Waiting in Line Again!	86	(7)
The Setting	86	(1)
Assumptions Used in the Model	87	(1)
Comments on the Exponential Distribution	88	(1)
Predictions	89	(1)
Comments on Queues with Exponential Arrivals and Service	89	(1)
Remarks about Other Distributions	90	(3)
2.8 Estimating Parameters and Testing Hypotheses	93	(13)
Maximum Likelihood 93 Minimum Discrepancy	95	(2)
Testing Hypotheses and Comparing Models	97	(9)
CHAPTER 3 Markov Chains and Related Stochastic Models	106	(47)
3.0 Introduction	106	(1)
3.1 The Setting and Some Examples	106	(9)
Animal Ranges	106	(4)
The Effects of Group Structure on Small-Group Decision Making	110	(5)
3.2 Basic Properties of Markov Chains	115	(10)
The Markov Assumption	116	(2)
State Vectors	118	(1)
Multistep Transitions and the Sequence of State Vectors	119	(6)
3.3 Classification of Markov Chains and the Long-Range Behavior of Regular Markov Chains	125	(10)
3.4 Absorbing Chains and Applications to Ergodic Chains	135	(18)
Applications of Absorbing Chains to Ergodic Chains	141	(7)
Chapter Appendix: Mathematical Details	148	(5)
CHAPTER 4 Simulation Models	153	(59)
4.0 Introduction	153	(1)
4.1 The Simulation Process	153	(11)
4.2 Generating Values of Discrete Random Variables	164	(7)
Random Numbers Distributed as a Discrete Random Variable	166	(3)
A Simulation	169	(2)
4.3 Discrete Event Simulation	171	(14)
Simulating Markov Chains	178	(7)
4.4 Generating Values of Continuous Random Variables	185	(10)
Summary of the Method	188	(7)
4.5 Applications and Validation of Simulation Modeling	195	(17)
Estimating Customer Flow in a Retail Store	195	(3)
Meeting Demands in a Fitness Center	198	(3)
Modeling the Spread of a Communicable Disease	201	(6)
Verification and Validation of Simulation Models and Interpretation of Output	207	(5)
CHAPTER 5 Linear Programming Models	212	(54)
5.0 Introduction	212	(1)
5.1 Formulation of Linear Programming Problems	213	(10)
A Diet Problem	213	(2)
A Resource Allocation Problem	215	(2)
A Transportation Problem	217	(6)
5.2 Linear Programming Problems and Duality	223	(11)
Complementary Slackness	228	(1)
General Linear Programming Problems	229	(5)
5.3 Duality, Sensitivity, and Uncertainty	234	(11)
Changes in A, b, and c	237	(8)
5.4 An Example of Integer Programming: A Job Assignment Problem	245	(7)
The Problem	246	(1)
A Real Model	246	(1)
Comments on the Model and Its History	247	(5)
5.5 Network and Flows	252	(14)
Definitions and Notation	253	(3)
Maximal Flows and Minimal Cuts	256	(2)
Integer Transportation Problems	258	(2)
The Assignment Problem Revisited	260	(6)
APPENDIX Addendum for Students and Teachers on Projects and Presentations	266	(15)
A.0 Introduction	266	(1)
A.1 The Role of Projects and the Types Useful in Learning Model Building	266	(2)
A.2 Examples of Projects	268	(8)
Locating a Community College Campus (Weekend Project, 72 hours)	269	(2)
Construction of an Earthen Dam (Weekend Project, 72 hours)	271	(1)
Allocating Teachers (90-minute, In-Class Project)	272	(1)
A Pipeline Flow Problem (75-minute, In-Class Project)	273	(1)
An Irrigation Problem (90-minute, In-Class Project)	273	(2)
An Evacuation Plan for an Elementary School (75-minute, In-Class Project)	275	(1)
A Vaccination Problem (Long-Term, Several Weeks)	275	(1)
A Credit Union Scheduling Problem (Long-Term, Several Weeks)	276	(1)
A.3 Reports and Presentations	276	(1)
A.4 Evaluating Project Reports	277	(1)
A.5 Sources of Projects	278	(3)
Client-Driven Projects	278	(1)
Self-Initiated or Instructor-Initiated Projects	279	(1)
The Mathematical Contest in Modeling	280	(1)
Index	281