Breadcrumbs Section. Click here to navigate to respective pages.

Book

Book

# Advanced Mathematical Modeling with Technology

DOI link for Advanced Mathematical Modeling with Technology

Advanced Mathematical Modeling with Technology book

# Advanced Mathematical Modeling with Technology

DOI link for Advanced Mathematical Modeling with Technology

Advanced Mathematical Modeling with Technology book

#### Get Citation

## ABSTRACT

Mathematical modeling is both a skill and an art and must be practiced in order to maintain and enhance the ability to use those skills. Though the topics covered in this book are the typical topics of most mathematical modeling courses, this book is best used for individuals or groups who have already taken an introductory mathematical modeling course. *Advanced Mathematical Modeling with Technology* will be* *of interest to instructors and students offering courses focused on discrete modeling or modeling for decision making.

Each chapter begins with a problem to motivate the reader. The problem tells "what" the issue is or problem that needs to be solved. In each chapter, the authors apply the principles of mathematical modeling to that problem and present the steps in obtaining a model. The key focus is the mathematical model and the technology is presented as a method to solve that model or perform sensitivity analysis.

We have selected , where applicable to the content because of their wide accessibility. The authors utilize technology to build, compute, or implement the model and then analyze the it.

**Features:**

- MAPLE©, Excel©, and R© to support the mathematical modeling process.
- Excel templates, macros, and programs are available upon request from authors.
- Maple templates and example solution are also available.
- Includes coverage of mathematical programming.
- The power and limitations of simulations is covered.
- Introduces multi-attribute decision making (MADM) and game theory for solving problems.

The book provides an overview to the decision maker of the wide range of applications of quantitative approaches to aid in the decision making process, and present a framework for decision making.

Table of Contents

1. Perfect Partners: Mathematical Modeling and Technology

2. Review of Modeling with Discrete Dynamical Systems and Modeling Systems of DDS

3. Modeling with Differential Equations

4. Modeling System of Ordinary Differential Equation

5. Regression and Advanced Regression Methods and Models

6. Linear, Integer and Mixed Integer Programming

7. Nonlinear Optimization Methods

8. Multivariable Optimization

9. Simulation Models

10. Modeling Decision Making with Multi-Attribute Decision Modeling with Technology

11. Modeling with Game Theory

12. Appendix Using R

Index

Biographies

**Dr. William P. Fox** is currently a visiting professor of Computational Operations Research at the College of William and Mary. He is an emeritus professor in the Department of Defense Analysis at the Naval Postgraduate School and teaches a three-course sequence in mathematical modeling for decision making. He received his Ph.D. in Industrial Engineering from Clemson University. He has taught at the United States Military Academy for twelve years until retiring and at Francis Marion University where he was the chair of mathematics for eight years. He has many publications and scholarly activities including twenty plus books and one hundred and fifty journal articles.

**Colonel (R) Robert E. Burks, Jr., Ph.D**. is an Associate Professor in the Defense Analysis Department of the Naval Postgraduate School (NPS) and the Director of the NPS’ Wargaming Center. He holds a Ph.D. in Operations Research form the Air Force Institute of Technology. He is a retired logistics Army Colonel with more than thirty years of military experience in leadership, advanced analytics, decision modeling, and logistics operations who served as an Army Operations Research analyst at the Naval Postgraduate School, TRADOC Analysis Center, United States Military Academy, and the United States Army Recruiting Command.

**1. Perfect Partners: Mathematical Modeling and Technology.** 1.1. Overview of Some Real Big Problems and The Process of Mathematical Modeling. 1.2. The Modeling Process. 1.3. Illustrative Examples. 1.4. Technology. 1.5. Exercises. 1.6 Projects. 1.7. References and Suggested Future Readings.** 2. Review of Modeling with Discrete Dynamical Systems and Modeling Systems of DDS.** 2.1. Introduction and Review of Modeling with Discrete Dynamical Systems. 2.2. Equilibrium and Stability Values and Long-Term Behavior. 2.3. Introduction to Systems of Discrete Dynamical Systems. 2.4. Iteration and Graphical Solution. 2.5. Modeling of Predator - Prey Model, Sir Model, and Military Models. 2.6. Technology Examples for Discrete Dynamical Systems. 2.7. Exercises. 2.8. Projects. 2.9. References and Suggested Future Readings. **3. Modeling with Differential Equations.** 3.1. Applied First Order Models. 3.2. Slope Fields and Qualitative Assessments of Autonomous First Order ODE. 3.3 Analytical Solution to 1st Order ODEs. 3.4 Numerical Methods for Solutions to 1st Order Odes with Technology. 3.5. Technology Examples for Ordinary Differential Equations. 3.6. Exercises. 3.7. Projects. 3.8. References and Suggested Future Readings. **4. Modeling System of Ordinary Differential Equations.** 4.1. Introduction. 4.2. Applied Systems of Differential Equations. 4.3. Qualitative Assessment of Autonomous Systems of First Order Differential Equations. 4.4. Solving Homogeneous and Non-Homogeneous Systems. 4.5. Technology Examples for Systems of Ordinary Differential Equations. 4.6. Exercises. 4.7. Projects. 4.8. References and Suggested Future Readings. **5. Regression and Advanced Regression Methods and Models.** 5.1. Introduction. 5.2. Nonlinear Regression. 5.3. Technology Examples for Regression. 5.4. Logistics Regression Models. 5.5. Technology Examples for Poisson Regression. 5.6. Exercises. 5.7. Projects. 5.8. References and Suggested Future Readings. **6. Linear Integer and Mixed Integer Programming.** 6.1. Introduction. 6.2. formulating Linear Programming Problems. 6.3. Graphical Linear Programming. 6.4. Technology Examples for Linear Programming. 6.5. Linear Programming Case Study. 6.6. Sensitivity Analysis with Technology. 6.7. Exercises. 6.8. Projects. 6.9. References and Suggested Further Reading. **7. Nonlinear Optimization Methods.** 7.1. Introduction. 7.2. Unconstrained Single Variable Optimization and Basic Theory. 7.3. Models with Basic Applications of Max-Min Theory. 7.4. Technology Examples for Nonlinear Optimization. 7.5. Single Variable Numerical Search Techniques with Technology. 7.6. Exercises. 7.7. Projects. 7.8. References and Suggested Further Readings. **8. Multivariable Optimization.** 8.1. Introduction. 8.2. Unconstrained Optimization. 8.3. Multivariable Numerical Search Methods for Unconstrained Optimization. 8.4. Constrained Optimization. 8.5. inequality Constraints-Kuhn-Tucker (KTC) Necessary/Sufficient Conditions. 8.6. Technology Examples for Computational KTC. 8.7. Exercises. 8.8. Projects. 8.9. References and Suggested Reading. **9. Simulation Models.** 9.1. Introduction. 9.2. Random Number and Monte Carlo Simulation. 9.3. Probability and Monte Carlo Simulation Using Deterministic Behavior. 9.4. Deterministic Simulations in R and Maple. 9.5. Probability and Monte Carlo Simulation Using Probabilistic Behavior. 9.6. Applied Simulations and Queuing Models. 9.7. Exercises. 9.8. Projects. 9.9. References and Suggested Readings. **10. Modeling Decision Making with Multi-Attribute Decision Modeling with Technology.** 10.1. Introduction. 10.2. Weighting Methods. 10.3. Pairwise Comparison by Saaty (AHP). 10.4. Entropy Method. 10.5. Simple Additive Weights (Saw) Method. 10.6. Technique of Order Preference by Similarly to the Ideal Solution (TOPSIS). 10.7. Modeling of Ranking Units Using Data Envelopment Analysis (DEA) with Linear Programming. 10.8. Technology for Multi-Attribute Decision Making. 10.9. Exercises. 10.10. Projects. 10.11. References and Suggested Readings. **11. Modeling with Game Theory.** 11.1. Introduction to total Conflict (Zero-Sum) Games. 11.2. Finding Alternate Optimal Solutions in A Two Person Zero-Sum Game. 11.3. The Partial Conflict Game Analysis without Communication. 11.4. Methods to Obtain the Equalizing Strategies. 11.5. Nash Arbitration Method. 11.6. Illustrative Modeling Examples of Zero-Sum Games. 11.7. Partial Conflict Games Illustrative Examples. 11.8. Exercises. 11.9. Projects. 11.10. References and Suggested Readings. **12. Appendix Using R.** Index