Chapter 8 - Linear Programming Methods for Optimum Design

Abstract

A standard model for linear programming (LP) problems is defined as minimization of a cost function subject to equality constraints and nonnegative design variables. Basic concepts and terminology related to LP problems are explained with a sample problem. The concept of basic solutions of a rectangular system of linear equations Ax = b is explained and basic and nonbasic variables are defined. The basic idea of the Simplex method and its steps are explained with all “less than or equal to” type constraints. Then the two-phase Simplex method for general LP problems is presented, explained, and illustrated. In Phase I, artificial variables are introduced for equality and “greater than or equal to” type constraints, and an artificial cost function is minimized to obtain a basic feasible solution for the original LP problem. In Phase II, the Simplex method continued to obtain an optimum solution for the LP problem. Postoptimality analysis for the problem is presented where changes to the constraint limits, ranging of the right side parameters and ranging of the cost coefficients are discussed and illustrated with examples.