On their farm, the Friendly family grows apples that they harvest each fall and

Question

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 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. 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 Friendlys 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 (called mathematical or algebraic model) for this problem.  Showing procedures please.

Explanation / Answer

Given :

Hours of labor required for 10-gallon batch of apple butter =1.2 hrs

Hours of labor required for Batch of sauce = 0.8 hr

Hours of labor required for Batch of jelly = 1.5 hrs

Total cooking hours available = 500 hrs

Labor hours available= 240 hrs

Apples produced each fall= 6,500

Apples required for apple butter =40

Apples required for 10-gallon batch of applesauce = 55

Apples required for jelly= 20

Revenue generated by a batch of apple butter = $190

Revenue generated by a batch of applesauce = $170

Revenue generated by a batch of jelly = $155.

Assumptions:

i) No. of 10-gallon batches of apple butter = x

ii) No. of batches of applesauce = y

iii) No. of batches of apple jelly = z

iv) No cooking hours are parallel

Solution :

The amount of cooking time required = 3.5x + 5.2y + 2.8z.

Given max. 500 cooking hours available

1. Cooking Time Constraint :3.5x + 5.2y + 2.8z <= 500

Similarily , amount of labor required 1.2x + 0.8y + 1.5z and max labor hours= 240

2.Labor constraint :1.2x + 0.8y + 1.5z <= 240

3.Constraint due to apple supplies:40x + 55y + 20z <= 6500

4.Non-negativity constraint : x,y,z >= 0

Objective function= Maximize the revenue= 190x + 170y + 155z

Related Questions

Navigate

• Browse (All)
• Browse O
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.