Linear programming problems

This chapter provides equivalent formulations of the general linear programming problem and discusses some of its fundamental properties that make it possible to develop the most widely used method—the simplex method for solving the problem. The general linear programming problem is to find values of a set of variables x1, x2, .. xn that optimizes (maximizes or minimizes) a linear function. The linear programming problem can be presented in a variety of forms. It may be a problem of maximization or minimization under the conditions with ≤, =, and/or ≥ type of inequalities. Different aspects of development in linear programming are based on different forms of the problem. It can be easily shown that these different forms are equivalent to each other and the results attained with one form of the problem, therefore are valid for all types of linear programming problem. The chapter considers three equivalent formulations of the problem that are also equivalent to the general form: the canonical form that is primarily used in the development of the theory of duality; the standard form that is used in the development of the methods of computation; and the mixed form that contains both the conditions of equalities and inequalities, is used to represent some practical situations.,