Optimization in Practice with MATLAB® For Engineering Students and Professionals

Overview

In most previous chapters, continuous optimization problems were considered where the design variables were assumed to be continuous; that is, design variables assume real values within given ranges. In many practical engineering problems, the acceptable values of the design variables do not form a continuous set. These problems are referred to as discrete optimization problems. For example, the number of rivets required in a riveted joint has to be an integer (such as 1, 2, 3). Another example is when the feasible region of the design variable is a set of given discrete numbers, such as {6.25, 6.95, 7.65}, which may be the available standardized sizes of nuts. The basics of discrete optimization were introduced in Chapter 9, where some pertinent elementary methods were presented. This chapter introduces more advanced approaches. The reader is advised to first review Chapter 9 as preparation for the current chapter.

This chapter is organized as follows. The next section (Sec. 14.2) provides the problem classes, examples, and definition (along with the notion of computational complexity of the solution algorithms). Section 14.3 discusses the basics of some popular techniques used to solve integer programming problems, with examples. The methods studied will be: the exhaustive search method (Sec. 14.3.1), the graphical method (Sec. 14.3.2), the relaxation method (Sec. 14.3.3), the branch and bound method (Sec. 14.3.4), and the cutting plane method (Sec. 14.3.5). Popular current software options (Sec. 14.3.7) are also discussed. The chapter concludes with a summary in Sec. 14.4.

Problem Classes, Examples and Definition

This section presents discrete optimization problem classes, problem examples, and problem definition. Computational complexity of the solution algorithms is also briefly addressed in connection with the discrete optimization problem definition.