Rienzi Farms grows sugar cane and soybeans on its 500 acres of land. An acre of soybeans brings a $1000 contribution to overhead and profit; an acre of sugar cane has a contribution of $2000. Because of a government program no more than 200 acres may be planted in soybeans. During the planting season 1200 hours of planting time will be available. Each acre of soybeans requires 2 hours, while each acre of sugar cane requires 5 hours. The company seeks maximum contribution (profit) from its planting decision.
a. Algebraically state the decision variables, objective and constraints.
b. Plot the constraints
c. Solve graphically, using the corner-point method.

Question

Laura Cranford · Accepted Answer

Final Answer :

The problem statement is contained in the software panel below. The graphical and corner-point solutions are found in the second and third panels. The optimal solution is 200 acres in soybeans and 160 acres in sugar cane. There's not enough labour to plant all 500 acres when 200 acres are in soybeans.
Rienzi Farms Solution

200160520,000\begin{array} { | l | r | r | r | r | r | } \hline & \mathrm { X } 1 & \mathrm { X } 2 & & \text { RHS } & \text { Dual } \\\hline \text { Maximize } & 1,000 & 2,000 & & & \\\hline \text { Acres } & 1 & 1 . & \square & 500 & 0.\\\hline \text { Soybean } & & & & & \\\text { restriction } & 1 & 0 & \square & 200 & 200 \\\hline \text { Planting } & & & & & \\\text { labour } & 2 & 5 & \square& 1,200 & 400 \\\hline \text { Solution --- } & 200 & 160 & & 520,000 & \\\hline\end{array}

Corner Points

X1X2Z0002000200,0000240480,000200160520,000\begin{array} { | l | l | l| } \hline X 1 & X 2 & Z \\\hline 0 & 0 & 0 \\\hline 200 & 0 & 200,000 \\\hline 0 & 240 & 480,000 \\\hline 200 & 160 & 520,000 \\\hline\end{array}

$The problem statement is contained in the software panel below. The graphical and corner-point solutions are found in the second and third panels. The optimal solution is 200 acres in soybeans and 160 acres in sugar cane. There's not enough labour to plant all 500 acres when 200 acres are in soybeans. Rienzi Farms Solution \begin{array} { | l | r | r | r | r | r | } \hline & \mathrm { X } 1 & \mathrm { X } 2 & & \text { RHS } & \text { Dual } \\ \hline \text { Maximize } & 1,000 & 2,000 & & & \\ \hline \text { Acres } & 1 & 1 . & \square & 500 & 0.\\ \hline \text { Soybean } & & & & & \\ \text { restriction } & 1 & 0 & \square & 200 & 200 \\ \hline \text { Planting } & & & & & \\ \text { labour } & 2 & 5 & \square& 1,200 & 400 \\ \hline \text { Solution --- } & 200 & 160 & & 520,000 & \\ \hline \end{array} Corner Points \begin{array} { | l | l | l| } \hline X 1 & X 2 & Z \\ \hline 0 & 0 & 0 \\ \hline 200 & 0 & 200,000 \\ \hline 0 & 240 & 480,000 \\ \hline 200 & 160 & 520,000 \\ \hline \end{array}$

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