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Linear Programming Model

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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 450 cooking hours available, and it requires 3.5 hours to cook a 10-galloon batch of apple butter, 5.2 hours to cook 10 gallons of apple sauce, 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 hours, and a batch of jelly requires 1.5 hours. They have 240 hours of labor available during fall. They produce 5000 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 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.

Formulate a linear programming model for this problem.

1. What is the optimal revenue?

2. How many batches of apple butter should Friendly's produce in order to maximize the revenue?

3. How many batches of applesauce should Friendly's produce in order to maximize the revenue?

4. How many batches of apple jelly should Friendly's produce in order to maximize the revenue?

5. What is the shadow price for cooking time?

6. What is the sensitivity range for the cooking time usage constraint right hand side coefficient?

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