Question

1. A rancher owns 500 acres of land which is suitable for growing sorghum, corn, and wheat. She expects net profits per acre from each crop to be: $55 for sorghum, $60 for corn, and $50 for wheat. She is able to supply 3,000 hours per year of on-farm labor. Additionally, she has 4,500 hours of tractor time available for the production of these three crops. The requirements of each crop are summarized below.

	Sorghum	Corn	Wheat
Land (acres)	1.0	1.0	1.0
Labor (hours)	0.4	0.2	0.3
Tractor (hours)	0.5	0.2	0.4

1. Write out the linear programming problem, including the objective function and all constraints. Please represent the problem and constraints in general equation form.

Answer 1

Identify decision variable

Land for Sorghum = S acre

Land for Corn = C acre

Land for Wheat = W acres

Write objective function

Rancher would want to maximize profit on the crop that she cultivates on the land.

Max (S * $55) + (C * $60) + (W * $50)

Write constraints

Total land for cultivation in 500 acres. so our land constraint equation is

Constraint 1: (S * 1.0) + (C * 1.0) + (W * 1.0) <= 500

     =>              S + C + W <= 500

She is able to supply 3000 hours per year of farm labor. This provides us the upper cap for labor hours constraint

Constraint 2: (S * 0.4) + (C * 0.2) + (W * 0.3) <= 3000

She has 4500 hours of tractor time. This provides us the upper cap for tractor hours constraint

Constraint 3: (S * 0.5) + (C * 0.2) + (W * 0.4) <= 4500

non negativity restriction

The values of S, C and W are greater than or equal to 0.

S, C, W >= 0