Chapter 9 - More on Linear Programming Methods for Optimum Design

Abstract

This chapter describes the theory that leads to the steps used in the problems related to the simplex method. Derivation of the simplex method is based on answering the two questions: which current nonbasic variable should become basic, and which current basic variable should become nonbasic. The main idea of bringing a nonbasic variable into the basic set is to improve the design, that is, to reduce the current value of the cost function. A clue to the desired improvement is obtained by examining the cost function expression. The cost coefficients of the nonbasic variables play a key role in the simplex method and are called the reduced or relative cost coefficients. They are used to identify a nonbasic variable that should become basic to reduce the current value of the cost function. Expressing the cost function in terms of the current nonbasic variables is a key step in the simplex method. A slightly different procedure can be used to solve linear programming problems having “≥ type” and equality constraints. The artificial variables are introduced into the problem as before. However, the artificial cost function is not used. Instead, the original cost function is augmented by adding to it the artificial variables multiplied by large positive constants. The additional terms act as penalties for having artificial variables in the problem. Since artificial variables are basic, they need to be eliminated from the cost function before the simplex method can be used to solve the preceding modified problem.