Advanced Problem Solving Using Maple

Applied Mathematics, Operations Research, Business Analytics, and Decision Analysis

William P Fox, William Bauldry

• Publisher: Chapman and Hall/CRC
• Year: 2020
• Print ISBN: 9781138601871
• eBook ISBN: 9780429891311
• Topic: Computer Science
• Subtopic: Programming
• Index: Computer Science

Advanced Problem Solving Using Maple™: Applied Mathematics, Operations Research, Business Analytics, and Decision Analysis applies the mathematical modeling process by formulating, building, solving, analyzing, and criticizing mathematical models. Scenarios are developed within the scope of the problem-solving process.

The text focuses on discrete dynamical systems, optimization techniques, single-variable unconstrained optimization and applied problems, and numerical search methods. Additional coverage includes multivariable unconstrained and constrained techniques. Linear algebra techniques to model and solve problems such as the Leontief model, and advanced regression techniques including nonlinear, logistics, and Poisson are covered. Game theory, the Nash equilibrium, and Nash arbitration are also included.

Features:

• The text's case studies and student projects involve students with real-world problem solving
• Focuses on numerical solution techniques in dynamical systems, optimization, and numerical analysis
• The numerical procedures discussed in the text are algorithmic and iterative
• Maple is utilized throughout the text as a tool for computation and analysis
• All algorithms are provided with step-by-step formats

About the Authors:

William P. Fox is an emeritus professor in the Department of Defense Analysis at the Naval Postgraduate School. Currently, he is an adjunct professor, Department of Mathematics, the College of William and Mary. He received his PhD at Clemson University and has many publications and scholarly activities including twenty books and over one hundred and fifty journal articles.

William C. Bauldry, Prof. Emeritus and Adjunct Research Prof. of Mathematics at Appalachian State University, received his PhD in Approximation Theory from Ohio State. He has published many papers on pedagogy and technology, often using Maple, and has been the PI of several NSF-funded projects incorporating technology and modeling into math courses. He currently serves as Associate Director of COMAP's Math Contest in Modeling (MCM).

1

Introduction to Problem Solving and Maple

Objectives:

(1)Understand the nature of problem solving.

(2)Understand the use of Maple commands.

(3)Understand the Maple Applications Center and its uses.

1.1Problem Solving

What do we mean by problem solving? We interpret this as having a real problem whose understanding and solution requires quantitative analysis and one or more solution techniques using mathematics. To put into context, we say we need a well-defined problem. After we have a well-defined problem, we must brainstorm variables and assumptions that might impact the problem. We build or select a known model and choose a solution technique or combinations of techniques to obtain an answer. We solve and perform sensitivity analysis. We interpret all results, implement, and if necessary, refine the entire process.

In many ways, this process is very similar to the mathematical modeling processes described in other texts: Albright [A2011], Giordano et al. [GFH2013], and Meerscheart [M2007] to name a few. Readers may want to examine these texts for a more detailed approach. As a co-author of the Giordano text, my approach is most similar to the approach we describe in that text.

There are four- and five-step processes for problem solving. We present and describe a simple five-step method.

Step 1.Define and understand the problem.

Step 2.Develop strategies to solve the problem. This includes a problem-solving formulation including a methodology to obtain a solution. If data is available, examine the data, plot it, and look for patterns.

Step 3.Solve the problem formulated in Step 2.

Step 4.Perform a self-reflection of your process. You want to make sure the solution answers the problem from Step 1. You also want to ensure the results pass the “commonsense” test. If not go back to Step 2 and reformulate the strategy.

Step 5.If necessary, extend the problem.

We will not concentrate on the modeling portion but on the selection of the model and the solution technique processes including the use of technology in the solution process.

One key point is that our results must pass the “commonsense” test. For example, we were conducting spring-mass experiments in a classroom on the 3rd floor of our mathematics and science building. The simple purpose was to investigate Hooke’s Law. The springs were small and the weights varied from a fraction to about 50 grams. After the experiments, we asked the students to calculate the stretch of their spring if it were attached to a seat, and they sat on the seat. Every student found an answer, but none said the spring would most likely break long before it stretched that far.

Let’s preview a problem we will see in Volumes I and II. We have data for time (t) and an index (y) from [0, 100]. Our plot shows a negative linear trend. We compute the correlation which is −0.94, and is interpreted as a strong negative linear relationship. We use linear regression to build a regression equation...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Preface
• 1. Introduction to Problem Solving and Maple
• 2. Discrete Dynamical Models
• 3. Problem Solving with Single-Variable Optimization
• 4. Problem Solving with Multivariable Constrained and Unconstrained Optimization
• 5. Problem Solving with Linear Systems of Equations Using Linear Algebra Techniques
• 6. Review of Regression Models and Advanced Regression Models
• 7. Problem Solving with Game Theory
• 8. Introduction to Problem Solving with Multi-Attribute Decision Making
• Index