A farmer in Egypt owns 50 acres of land. He is going to plant each acre with cotton or corn. Each acre planted with cotton yields $400 profit; each with corn yields $200 profit. The labor and fertilizer used for each acre are given in the table below. Resources available include 150 workers and 200 tons of fertilizer.
Cotton
Corn
Labor (Workers)5
3
Fertilizer (Tons)6
2
(A) Formulate a linear programming model that will enable the farmer to determine the number of acres that should be planted cotton and/or corn in order to maximize his profit.
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(B) Find an optimal solution to the model in part (A) and determine the maximum profit.
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(C) Implement the model in part (A) in Excel Solver and obtain an answer report. Which constraints are binding on the optimal solution?
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(D) Obtain a sensitivity report for the model in part (A). How much should the farmer be willing to pay for an additional worker?
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(E) Suppose the farmer hires 10 additional workers. Can you use the sensitivity analysis obtained for part (D) to determine his expected profit? Would his planting plan change? Explain your answer.
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(F) Suppose the farmer now wants to hire 20 additional workers, instead of just 10. Can you use the sensitivity analysis obtained for part (D) to determine his expected profit? Explain your answer.
What will be an ideal response?
and
The linear programming model to maximize the profit from this farmer’s land is as follows:Maximize 
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Subject to:
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(B) The optimal solution is
, and the maximum profit is $12,000.
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(C) The constraints section of the answer report (below) shows that the labor constraint is binding (along with the nonnegativity constraint for corn).
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(D) The shadow price for labor (shown below) indicates that an additional worker would increase profit by $80. The farmer should be willing to pay up to this amount.
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(E) Using the shadow price for labor, his profit would increase by $80*10=$800. The ten extra workers would enable him to plant two additional acres of cotton.
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(F) We cannot use the sensitivity report in this instance, because the increase of 20 workers is beyond the allowable increase for the shadow price (up to 16.67 additional workers). We would need to re-run the model to determine the effect of adding 20 workers.
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