Answer to Question #245224 in Operations Research for AJBASA

Question

A farmer has 10 acres to plant in rice and corn. He has to plant at least 7 acres. However, he has only PhP1200 to spend and each acre of rice costs PhP200 to plant and each acre of corn costs PhP100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of rice and 2 hours to plant an acre of corn. If the profit is PhP500 per acre of rice and PhP300 per acre of corn, how many acres of each should be planted to maximize profits?

Answer 1

Suppose that the farmer plants $x$ acres of rice and $y$ acres of corn.

A farmer has 10 acres to plant in rice and corn

$$x+y\leq10$$

He has to plant at least 7 acres

$$x+y\geq7$$

He has only PhP1200 to spend

$$200x+100y\leq1200$$

The farmer has to get the planting done in 12 hours

$$x+2y\leq12$$

The object is to maximize profits

$$z=500x+300y$$

$$\begin{matrix} x+y\leq10 \\ x+y\geq7 \\ 2x+y\leq 12\\ x+2y\leq 12\\ x\geq0, y\geq0 \end{matrix}$$

$$\begin{matrix} y\leq10-x, 0\leq x\leq 10 \\ y\geq7-x, 0\leq x\leq 7 \\ y\leq 12-2x, 0\leq x\leq 6\\ y\leq6- \dfrac{1}{2}x,0\leq x\leq 12\\ \end{matrix}$$

$$7-x=6- \dfrac{1}{2}x$$ $$x=2, y=5, Point\ A(2, 5)$$

$$7-x=12-2x$$ $$x=5, y=2, Point\ C(5, 2)$$

$$6- \dfrac{1}{2}x=12-2x$$ $$x=4, y=4, Point\ B(4, 4)$$

Evaluate the objective function in all of the fleasible corner points

Point$\ A(2, 5)$

$$z(2,5)=500(2)+300(5)=2500$$

Point$\ B(4, 4)$

$$z(4,4)=500(4)+300(4)=3200$$

Point$\ C(5, 2)$

$$z(5,2)=500(5)+300(2)=3100$$

400 acres of rice and 400 acres of corn should be planted to maximize profits.

The maximum profit is $3200.