A California grower has a 50-acre farm on which to plant strawberries and tomato

Question

A California grower has a 50-acre farm on which to plant strawberries and tomatoes. The grower has available 300 hours of labor per week and 800 tons of fertilizer, and he has contracted for shipping space for a maximum of 26 acres' worth of strawberries and 37 acres' worth of tomatoes. An acre of strawberries requires 10 hours of labor and 8 tons of fertilizer, whereas an acre of tomatoes requires 3 hours of labor and 20 tons of fertilizer. The profit from an acre of strawberries is $200, and the profit from an acre of tomatoes is $500. The farmer wants to know the number of acres of strawberries and tomatoes to plant to maximize profit. (a) Write down the objective function o the linear programming model for this problem. (b) Write down the constraints of the linear programming model for this problem.

Explanation / Answer

The Objective Function of this Linear Programming Problem is :

Objective: Maximise the Profit from the sale of Strawberries and tomatoes by choosing the no. of acres to be used for the production of the same.

Let X1 be the no. of acres used to produce the Strawberries.

X2 be the no. of acres used to produce the Tomatoes.

Profit from the Each Acre of Strawberry : $ 200 and Tomato : $ 500

Therfore Objective Function is : Maximize Z = $ 200 X1 + $ 500 X2

Constraints of Linear Programming is :

Constraint 1 -

Limit on the No. of Labor Hours Available:

Total No. of Labor Hours available is 300 Hours. So, we should produce the Total Production, we aim to subject to the Limit of 300 Hours. One Acre of Strawberry requires 10 Hours and an acre of tomato requires 3 Hours of Labor Hours.

So, Constraint 1 = Subject to 10 X1 + 3 X2 ? 300

Constraint 2 :

WE have a constraint of 800 tons of fertilizer available so, we have to produce the desired quantity by using the limited no. of tons available. one acre of strawberry requires 8 tons of fertilizer where as an acre of tomato requires 20 tons of fertilizer.

So, Constraint 2 = 8 X1 + 20 X2 ? 800

Constraint 3 :

California Grower has a total Farm of 50 Acres. So, our production is limited to 50 Acres.

So, Constraint 3 = X1 + X2 ? 50

Constraint 4 :

He has contracted shipping space of maximum 26 acres of strawberry and 37 acres of tomatoes. So, production shall be limited to such no. of acres.

Constraint 4 = X1 ? 26, X2 ? 37

In total,

Maximize Z = $ 200 X1 + $ 500 X2

Subject to:

10 X1 + 3 X2 ? 300

8 X1 + 20 X2 ? 800

X1 + X2 ? 50

X1 ? 26, X2 ? 37

X1 ? 0, X2 ? 0 (Non - Negativity Constraint)

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