What is the Objective Function?

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 much each variable contributes to the value to be optimized in the problem. The objective function takes the following general form:

where

ci = the objective function coefficient corresponding to the ith variable, and
Xi = the ith decision variable. \1

The coefficients of the objective function indicate the contribution to the value of the objective function of one unit of the corresponding variable. For example, if the objective function is to maximize the present value of a project, and Xi is the ith possible activity in the project, then ci (the objective function coefficient corresponding to Xi ) gives the net present value generated by one unit of activity i. As another example, if the problem is to minimize the cost of achieving some goal, Xi might be the amount of resource i used in achieving the goal. In this case, ci would be the cost of using one unit of resource i.

Note that the way the general objective function above has been written implies that each variable has a coefficient in the objective function. Of course, some variables may not contribute to the objective function. In this case, you can either think of the variable as having a coefficient of zero, or you can think of the variable as not being in the objective function at all.


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