EXAMPLE 6 Mixtures of Two Fruit Juices: Beverages

A juice manufacturer produces and sells two fruit beverages: 1 gallon of cranapple is made from 3 quarts of cranberry juice and 1 quart of apple juice; and 1 gallon of apple-berry is made from 2 quarts of apple juice and 2 quarts of cranberry juice. The manufacturer makes a profit of 3 cents on a gallon of cranapple and 4 cents on a gallon of appleberry. Today, there are 200 quarts of cranberry juice and 100 quarts of apple juice available. How many gallons of cranapple and how many gallons of appleberry should be produced to obtain the highest profit without exceeding available supplies? We use zeroes as “reality minimums.” The mixture chart for this problem is shown in Figure 4.8.

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For each resource, we develop a resource constraint reflecting the fact that the manufacturer cannot use more of that resource than is available. The number of quarts of cranberry juice needed for gallons of cranapple is . Similarly, quarts of cranberry are needed for making gallons of appleberry. So if the manufacturer makes gallons of cranapple and gallons of appleberry, then quarts of cranberry juice will be used. Because there are only 200 quarts of cranberry available, we get the cranberry resource constraint . Note that the numbers 3, 2, and 200 are all in the cranberry column. We get another resource constraint from the column for the apple juice resource: . We also have these minimum constraints: and .

Finally, we have the profit formula. Because is the profit from making units of cranapple and is the profit from making units of appleberry, we get the profit formula .

We summarize our analysis of the juice mixture problem. Maximize the profit formula, , given these constraints:

Remember, in a mixture problem, our job is to find a production policy (), that makes all the constraints true and maximizes the profit.

Figure 4.9a shows the result of graphing the constraint associated with the cranberry resource, while Figure 4.9b shows the result of graphing the constraint associated with the apple resource, taking into account that the amounts of these resources used cannot be negative. When these two diagrams are superimposed, we get the diagram in Figure 4.9c. Now, to carry out the pictorial method, we need to find the profits associated with the four corner points shown. This is done in Table 4.5.

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When we evaluate the profit formula at the four corner points, we see that the optimal production policy is to make 50 gallons of cranapple and 25 gallons of apple- berry, for a profit of 250 cents.