Interactive Operations Research with Maple: Methods and Models

Mahmut Parlar,
Birkhauser, Boston, ISBN 0-8176-4165-3

Imagine your broker hands you a hot stock tip. You're tempted to shift some of your current portfolio into the hot stock.

However, doing so would increase the risk of your portfolio and incur broker commissions. Is the potential improvement in your portfolio's return worth the added risk and costs? Or imagine you're in charge of placing clothing orders for a department store. You know that keeping too much clothing in inventory means higher storage costs, while running out of inventory means lost sales opportunities. In the face of uncertain customer demand, when, and in what quantities, should you place orders so that you maximise long-term profit on average? These scenarios are examples of problems of Operations Research (OR). Broadly defined, OR is the application of mathematics to determine the optimal use of limited resources (for example, money, time, equipment, space). Differential calculus is sufficient for solving single-variable optimisation problems with no constraints. However, when thousands of variables, constraints on those variables or random variables are present, calculus doesn't cut it, and OR must come into play.

There are heaps of university-level textbooks on OR, each describing techniques, algorithms and theorems for solving optimisa-tion problems. Dr. Mahmut Parlar's "Interactive Operations Research with Maple" is the first textbook that implements them in Maple worksheets. Each chapter introduces a particular OR tool, outlines the theory behind it, implements it in a Maple worksheet and solves an example problem. (The worksheets and examples may be viewed as HTML or downloaded from the author's website http://www. business.mcmaster.ca/msis/profs/parlar/ for free.)

For instance, Chapter 4 shows the power of the new simplex package in Maple 6 for entering and solving linear programs.

In Chapter 5 on nonlinear programming, Parlar uses Maple's built-in hessian() and definite() functions for automatically determining the positive definiteness of Hessian matrices. He then demonstrates Maple's highly versatile solve() function for solving the Kuhn-Tucker optimality conditions, the crux of nonlinear optimisation. As an illustration, he uses these three Maple functions to solve a variant of the stock allocation problem described above.

Chapter 8 on Inventory Models introduces classic techniques of efficient inventory control. Maple worksheets included in the chapter implement these techniques to solve the department store ordering problem described above.

This book is unique in that it is the only one that uses Maple's abilities to solve OR problems, many of which would be impossible to solve with other software. Many problems in stochastic processes, queueing theory, inventory, etc., are now easy to tackle using Maple's symbolic manipulation capabilities.