Question

Your farm encompasses 500 acres, and you are planning to grow soybeans, corn, and wheat in the coming planting season. Fertilizer costs per acre are: $5 for soybeans, $2 for corn, and $1 for wheat. You estimate that each acre of soybeans will require an average of 5 hours of labor per week, while tending to corn and wheat will each require an average of 2 hours per week. Based on past yields and current market prices, you estimate a profit of $10,500 for each acre of soybeans, $7,000 for each acre of corn, and $3,500 for each acre of wheat. You can afford to spend no more than $3,000 on fertilizer, and your farm laborers can supply 3,000 hours per week. How many acres of each crop should you plant to maximize total profits?

soybeans	acre(s)
corn	acre(s)
wheat	acre(s)

In this event, will you be using all the available labor?

Yes No    

Answer 1

Let x acres of soybeans, y acres of corn, z acres of wheat should be planted to maximize total profit.

Then by the given conditions we have,

Maximize W = 10500x+7000y+3500z

Subject to x+y+z 500

5x+2y+z 3000

5x+2y+2z 3000

x,y,z 0

Introducing slack variables we get,

Maximize W = 10500x+7000y+3500z+0a+0b+0c

Subject to x+y+z+a = 500

5x+2y+z+b = 3000

5x+2y+2z+c = 3000

x,y,z,a,b,c 0

COMPUTATION TABLE :

		cj	10500	7000	3500	0	0	0
cB	B	b	a1	a2	a3	a4	a5	a6
0	a4	500	1	1	1	1	0	0
0	a5	3000	5	2	1	0	1	0
0	a6	3000	5	2	2	0	0	1
		zj-cj	-10500	-7000	-3500	0	0	0
10500	a1	500	1	1	1	1	0	0
0	a5	500	0	-3	-4	-5	1	0
0	a6	500	0	-3	-3	-5	0	1
		zj-cj	0	3500	7000	10500	0	0

In the last tableau, we see that z_j-c_j 0.

Then the optimal solution is :

x = 500, y = 0, z = 0 and W_max = 10500*500+7000*0+3500*0 = 5250000.

Therefore, 500 acres of soybeans, 0 acres of corn, 0 acres of wheat should be planted to maximize total profit and the maximum profit is $5,250,000.

Here, i am using (500*5) hours ,i.e., 2500 hours of labor time instead of 3000 hours.

No, in this event, i am not using all available labor.