Successful Model

The solution to the problem of modeling the nonconcave profit functions is to add binary (taking the values 0 and 1) variables to the model. The new variables control the order that the pieces of the profit function are used in the solution. For product Q we need two new variables and modify the bound constraints for product Q as below.

From the bounds we see that if is 0, only is allowed to be greater than 0. If is 1 and is 0, is forced to 30 and is allowed to increase. If both and are 1, is forced to 30 and is forced to 30 and is allowed to increase.

The variables representing R are controlled by a single binary variable. The variable and revised bounds are shown below.

Only one variable is necessary because there is only one convex portion of the profit function. The complete Excel model with the optimum solution is illustrated below.

The optimum solution uses only the first segment of the objective for Q and the first two segments for R.

One concern might be the noninteger values for the production variables. This could be remedied by requiring P, Q and R to be integer. The mixed integer programming model with the first six variables integer has a slightly smaller objective than when the production variables were not restricted to be integer.