Chapter 18 - Bounded variable problems

Abstract

This chapter focuses on the bounded variable problems. In linear programming problems, the variables are constrained by lower and upper bounds such as 1j ≤ xj ≤ uj. The lower bound 1j could be caused by contractual obligations or policy restrictions and the upper bound uj could be caused by capacity limitations, resource limitations or the size of the market. In the standard simplex method, a basic solution of the original system is defined as a solution obtained by setting (n–m) nonbasic variables equal to zero and solving the resulting m × m square system for the basic variables. A solution is said to be a basic feasible solution if all the variables are nonnegative and constitutes an extreme point solution. From the set of all extreme point solutions, the simplex method finds an optimal solution to the problem. The chapter also defines the basic solution for the bounded variable problem to be one for which (n – m) variables are set equal to either their lower or upper bounds (zero or u) and solves the resulting system of m equations (corresponding to the basic matrix).