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# Friendly Family apple products: linear programming model

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 2 weeks in October. Their three primary sources are cooking time in the 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. The Friendly family has 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 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 by using the computer either excel solver or QM for windows

#### Solution Preview

See the attached files.

a. Formulate a linear programming model for this problem.

Variables:
X1=Batches of apple butter produced
X2= Batches of apple sauce produced
X3= Batches of apple jelly produced

Objective Function:
Maximize: 190X1+170X2+155X3

Subject to
3.5X1+5.2X2+2.8X3<=500 - Cooking hours available
1.2X1+0.8X2+1.5X3<=240 - Own labor time ...

#### Solution Summary

The solution creates a linear programming model for Friendly Family apple products. The expert solve the linear programming model.

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