Mt/CS 410 Mathematical Modeling - Fall 2002

Assignment 10: Optimization Problems and Linear Programming Methods

November 12, 2002

Over the next several class periods we will consider a variety of optimization problems, and methods for studying this kind of problem. In response to suggestions that the workload for this course has been rather heavy (compared to some other classes), I will cover some of this material by lecture and class discussion with some homework exercises to be turned in. The material that would have been covered by Project 7 will be covered in class activities and homework exercises.

Chapters 8 and 9 are parallel chapters. Chapter 8 introduces a certain class problems (optimization problems), and shows us how to set these up. Chapter 9 gives us a variety of methods for studying these problems.

• Read Sections 8.1 and 8.2.

Optimization problems are problems in which we are to maximize or minimize something. Setting up such problems consists of

1. identifying the objective function, and
2. identifying the constraints.
Solving the optimization problem will require finding the maximum or minimum value of the objective function over the domain defined by the constraints.
• Work with your group on exercises from pages 270 - 275. We will do some of these as a class activity on Tuesday, November 12. One or two problems will be assigned to turn in.
• Read Section 8.3. This is an in-depth example of a fairly complex (and relatively ordinary) business problem. This is an example of a non-linear optimization problem ... and serves to illustrate how methods of calculus can be applied to this kind of problem.
• Work with your group on exercises from pages 282 - 284. We will do some of these as a class activity on Thursday, November 14. One or two problems will be assigned to turn in.
• Read Section 9.1. This section presents a geometric strategy for solving linear optimization problems. In class discussion, I will show how this problem is a variation on the problem of finding a maximum or minimum value for a function of two variables. We can use fairly simple geometric strategies since the system is assumed to be linear optimization problem.
• Work with your group on exercises from pages 310 - 311. We will do some of these as a class activity on Tuesday, November 19.
• Read Section 9.2. This section presents an algebraic strategy for solving linear optimization problems. This algebraic strategy what lies behind computer algorithms for solving this kind of problem by the simplex method. (The simplex method is presented in Section 9.3, but we will not cover that section in this class this semester.)
• As a class activity on Thursday, November 21, you will have a chance to work with your group on the same exercises from pages 310 - 311, this time using an algebraic solution strategy. One or two problems will be assigned to turn in.