Question 1 (25 pts): Walnut Orchard has two farms that grow wheat and corn. Beca

Question

Question 1 (25 pts): Walnut Orchard has two farms that grow wheat and corn. Because of different soil conditions, there are differences in the yields and cost of growing crops on the two farms. The yields and costs are shown in Table 1. Each farm has 100 acres available for cultivation; 11,000 bushels of wheat and 7000 bushels of corn must be grown. Formulate a LP( Linear Programming) problem that will minimize the cost of meeting these demands. Corn yield/acre (bushel) Cost/acre of corn (S) Wheat yield/acre (bushels) Cost/acre of wheat (S) Table 1 Farm 1 500 100 400 90 Farm 2 650 120 350 80

Explanation / Answer

Let c_1 be the number of acres in which corn is grown on farm 1 and c_2 be the number of acres in which corn is grown on farm 2 . Similarly let w_1 be the number of acres in which wheat is grown on farm 1 and w_2 be the number of acres in which wheat is grown on farm 2.
Then
The total corn yield = 500c_1 + 650c_2
The total cost of cultivating corn = 100 c_1 + 120 c_2
The total wheat yield = 400 w_1 + 350 w_2
The total cost of cultivating wheat = 90 w_1 + 80 w_2

So total cost = 90 w_1 + 80 w_2 + 100 c_1 + 120 c_2.
We are to minimize the cost while ensuring that 11,000 bushels of wheat and 7000 bushels of corn is grown.
It must also be noted that c_1 + w_1 and c_2 + w_2 = 100 must be less than or equal to 100.

So the corresponding LP would be
Minimize 90 w_1 + 80 w_2 + 100 c_1 + 120 c_2
Subject to
c_1 + w_1 <= 100;
c_2 + w_2 <= 100;
400 w_1 + 350 w_2 >= 11,000;
500c_1 + 650c_2 >= 7000.

On solving this LP using an online LP solver the following optimal solution was obtained
w_1 = 27.5 ; w_2 = 0; c_1 = 0; c_2 = 10.77 and the total cost = 3767.4$

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