Example 7

EXAMPLE 7Adding Nonzero Minimums: beverages

Suppose that in Example 6, the profit for cranapple changes from 3 cents per gallon to 2 cents and the profit for appleberry changes from 4 cents per gallon to 5 cents. You can verify that this change moves the optimal production policy to the point (0, 50); no cranapple is produced. This result is not surprising: Appleberry is giving a higher profit and the policy is to produce as much of it as possible. But suppose the manufacturer wants to incorporate nonzero minimums into the linear-programming specifications so that there will always be both cranapple, , and appleberry, , produced. Specifically, the manufacturer decides that and are desirable minimums. Figure 4.12 is the mixture chart showing the new profit formula and the nonzero minimums, along with the unchanged rest of the beverage problem.

Figure 4.12: Figure 4.12 Mixture chart for Example 7.

The feasible region for beverages in Example 6 is shown in Figure 4.13a. The feasible region for beverages in this example is shown in Figure 4.13b. You can verify that, starting at the lower-left corner of the new feasible region and moving clockwise around its boundary, we have corner points (20, 10), (20, 40), (50, 25), and (60, 10). (One of those points was also a corner point of the old feasible region. Can you explain why?) Table 4.6 shows the evaluation of the profit formula at these corner points. For this modified problem, the optimal production policy is to produce 20 gallons of cranapple and 40 of appleberry, for a maximum profit of 240 cents.

Figure 4.13: Figure 4.13 Feasible region for Examples 6 and 7. (a) Zero minimums. (b) Nonzero minimums.

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Table 4.6: Table 4.6 Profit Evaluation for Beverages

Corner Point	Value of the Profit Formula:
(20, 10)
(20, 40)
(50, 25)
(60, 10)

One final note about this solution concerns the resources. The point (20, 40) is on the resource constraint line for the apple juice resource, so it represents using up all the available apple juice. We can see this by inserting the apple juice resource constraint: is true. However, (20, 40) is below the line for the cranberry juice resource, indicating that there will be slack, or leftover, amounts of cranberry juice. Specifically, substituting (20, 40) into the cranberry juice constraint gives , which is 60 quarts less than the 200 quarts available. The slack is 60 quarts of cranberry juice. Dealing with slack can be an important consideration for manufacturers. Can you see why?