Models

Nonlinear Objective

 The Problem Unsuccessful Model Successful Model
 Nonlinear Objective The Problem

This section considers the important topic of modeling problems with nonlinear objective functions. When the functions are not concave (when maximizing), it is necessary to use binary variables as part of the model. In fact, this is an important application of integer programming.

Consider a 4-machine production system producing three products. The machines are labeled A through D, and the products are labeled P, Q and R. The table below gives the time required in minutes for each unit of product on each of the four machines. During a one week period, the machines each have 2400 minutes of capacity.

Production times on machine (minutes)

 P Q R Machine A 20 10 10 Machine B 12 28 16 Machine C 15 6 16 Machine D 10 15 0

The profit for sales of each product is a nonlinear function of the amount sold. The table below shows profits as a function of sales for the three products.  In each case, up to 100 units can be sold. A number in a cell is the profit per unit in the specified range.

Profit data for products

 Profit/unit, \$ P Q R Sales less than 30 60 40 20 Sales between 30 and 60 45 60 70 Sales between 60 and 100 35 65 20

When the unit profit is the same over a range of values, the resulting function is called a piecewise linear function. The functions are for the three products are illustrated below. The profit function for P is said to be a concave function because it the slope of the function decreases as the amount sold increases. The function for Q is a convex function because the slope is increasing as the amount increases. The function for R is both convex and concave because the slope begins at 20, rises to 70 and then decreases again to 20. We will see that the shape of the function has much to do with how it is modeled.

Operations Research Models and Methods
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by Paul A. Jensen