Models
Manufacturing Example
 Manufacturing Example Concave Objective: Product Revenue a Concave Function of Sales
 Linearity requires that the revenue for each unit of product sold remains constant over the range zero to the value given in Table 1 for maximum sales. In many situations, however, companies may reduce the unit price of a product to boost its sales. When such a policy is employed, the product's marginal revenue decreases with the amount sold. In economic terms, the marginal revenue is the derivative of the total revenue function with respect to sales volume. Call the marginal revenues for the three products, and assume that each is a linear function of P, Q and R, respectively.  In the following modification, the relevant objective function coefficients are defined so that the marginal revenue reduces to half of the original value at maximum sales. For product P, the original value of revenue is 90/unit and maximum sales is 100. We model the marginal revenue as 90 - (45/100)P. The first sale provides a revenue of \$90 while the last sale provides a revenue of \$45. The marginal revenues for the three products are decreasing linear functions of production volume. The total revenue associated with the products is the integral of the marginal values. With a linear marginal revenue, the total revenue is a concave quadratic function. This is called a quadratic, separable function because the highest order of the nonlinear terms is 2, and each term is a function of only one variable. We substitute this objective in the LP to form a nonlinear programming model. The linear constraints of the LP are still valid. Solution to the Concave Model By comparing the new solution given below with the one obtained for the linear model, we see that the profit is reduced, more of product Q is produced but less of products P and R.  The bottleneck is machine B which is used for the entire 2400 minutes it is available. In contrast to linear programming, this solution is not an extreme point even though the feasible region is polynomial (defined by linear constraints). This observation follows from the fact that all three product variables, all four raw material variables and three of the four machine usage slack variables, 10 variables in all, are not at extreme values. Because there are 8 structural constraints, an extreme point solution will have at most 8 variables falling strictly within their lower and upper bounds. We note that the objective function Z is a concave function of the decision variables P, Q and R. For a maximization problem, when the objective function is concave and the constraints form a convex feasible region, we can be sure that the solution obtained is a global maximum (i.e., no other feasible solution provides a greater objective value). We now illustrate what may happen when the objective function is convex rather than concave. A number of difficulties arise that are not readily apparent.

Operations Research Models and Methods
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by Paul A. Jensen