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# the Friendly family;LPP and sensitivity analysis

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On their farm, the Friendly family grows apples that they harvest each fall and make into three products-apple butter, applesauce, and apple jelly. They sell these three items at several local grocery stores, at craft fairs in the region, and at their own Friendly Farm Pumpkin Festival for two weeks in October. Their three primary resources are cooking time in their kitchen, their own labor time, and the apples. They have a total of 500 cooking hours available, and it requires 3.5 hours to cook a 10-gallon batch of apple butter, 5.2 hours to cook 10 gallons of applesauce, and 2.8 hours to cook 10 gallons of jelly. A 10-gallon batch of apple butter requires 1.2 hours of labor, a batch of sauce takes 0.8 hour, and a batch of jelly requires 1.5 hours. They have 240 hours of labor available during the fall. They produce about 6,500 apples each fall. A batch of apple butter requires 40 apples, a 10-gallon batch of applesauce requires 55 apples, and a batch of jelly requires 20 apples. After the products are canned, a batch of apple butter will generate \$190 in sales revenue, a batch of applesauce will generate a sales revenue of \$170, and a batch of jelly will generate sales revenue of \$155. The Friendly's want to know how many batches of apple butter, applesauce, and apple jelly to produce in order to maximize their revenues.

a. Formulate a linear programming model for this problem.
b. Solve the model using the computer.

2. A. If the Friendly's in problem 1 were to use leftover apples to feed livestock, which they esti¬mate is a cost savings that is worth \$0.08 per apple in revenue, how would this affect the model and solution?
B. Instead of feeding the leftover apples to the livestock, the Friendly's are thinking about producing apple cider. Cider will require 1.5 hours of cooking, 0.5 hour of labor, 60 apples per batch, and will sell for \$45 per batch. Should the Friendly's use all of their apples and produce cider along with their other three products?