Operations Research Models and Methods / Computation / Mathematical Programming /
Nonlinear Programming


Product Mix with Decreasing Marginal Profit

We modify the product mix example used to illustrate the linear programming add-in to incorporate nonlinear terms in the objective. Management has determined that the marginal profit for each product decreases as the number produced increases. A study determines that the profit for each product is approximated by the quadratic function given below. The table shows the coefficients for each product. With the same constraints on machine time, our goal is to find the product mix that maximizes profit.

 The graph at the left shows the profit for the first product as a function of production. The function starts at zero, rises to a maximum and then decreases. This is a concave function of production. The marginal profit, or the derivative of the profit function, is decreasing with production volume.

The mathematical programming model is the same as in the linear programming example except the objective function is now the sum of nonlinear terms. This is called a separable objective function because each term is a function of only one decision variable.

Specifying Model Characteristics

Updated 1/16/01
Operations Research Models and Methods

by Paul A. Jensen and Jon Bard, University of Texas, Copyright by the Authors