Decision Variables In Linear Programming

Decision variables. In a linear program, the variables are a set of quantities to be determined for solving the problem; i.e., the problem is solved when the best values of the variables have been identified. The variables are sometimes called decision variables because the problem is to decide what value each variable should take. Usually, the variables represent the amount of a resource to use or the level of some activity. For instance, a variable might represent the number of acres to cut from a particular part of the forest during a given period. Regularly, one of the hardest and most crucial steps in formulating a problem as a linear program is to define the variables of the problem. Sometimes creative variable definition can be used

The objective of a linear programming problem will be to maximize or to minimize some numerical value. This value may be the expected net present value of a project or a forest property; or it may be the cost of a project; it could also be the amount of wood produced, the expected number of visitor-days at a park, the number of endangered species that will be saved, or the amount of a particular type of habitat to be maintained. Linear programming is an extremely general technique, and its applications are limited mainly by our imaginations and our ingenuity. The objective function indicates how each variable contributes to the value to be optimized in solving the problem. The coefficients of the objective function indicate the contribution to the value of the objective function of one unit of the corresponding variable. The objective function has to be linear in the decision variables, which means it must be the sum of constants times decision variables.
Constraints. Constraints define the possible values that the variables of a linear programming problem may take. They typically represent resource constraints, or the minimum or maximum level of some activity or

This formulation appears to be quite limited and restrictive; as we will see later, however, any linear programming problem can be transformed in canonical form. Therefore, the following discussion is valid for linear programs in general. Given any values for y_3 and y_4, the values of y_1 and y_2 are determined uniquely by the equalities. In fact, setting y3 = y4 = 0 immediately gives a feasible solution with y1 = 6 and y2 = 4. Such solutions will play a central role and are referred to as basic feasible solutions. In general, given a canonical form for any linear program, a basic feasible solution is given by setting the variable that appears once in constraint j, called the jth basic-variable, equal to the right-hand side of the jth constraint and by setting the remaining variables, called non-basic, all to zero. Collectively the basic variables are termed a basis. We have introduced the new terms such as canonical, basis, and basic variable in our discussion because these terms have been firmly established as part of linear programming vernacular. Canonical is a word used in many contexts in mathematics, as it is here, to mean ‘‘a special or standard representation of a problem or concept,’’ usually chosen to facilitate study of the problem or concept. Basis and basic are concepts in linear algebra; our use of these terms agrees with linear-algebra interpretations of the linear