Question

1. Farmer planting corn and soybeans. The farmer has a 320 acre farm on which she plants two crops: corn and soya beans. For each acre of corn planted, her expenses are $50 and for each acre of soybeans planted, he expenses are $100. Each acre of corn requires 100 bushels of storage and yields a profit of $60; each acre of soybeans requires 40 bushels of storage and yields a profit of $90. If the total amount of storage space available is 19,200 bushels and the farmer has only $20,000 on hand; Required: a) What are the decision variables of this problem? b) Write the objective function of this problem c) What are the constraint of the problem; Write them; d) Write completed linear programming model for the problem

Answer 1

a) c, s

b) P(c, s) = 60c + 90s

c) c + s ≤ 320

50c + 100s ≤ 20000

100c + 40s ≤ 19200

d)
`c + s+s_1=320`

`50c + 100s+s_2=20000`

`100c + 40s+s_3=19200`

Explanation:

a) Let c represent the acres of corn planted and s represent the acres of soybeans planted.

the decision variables of this problem are c and s

a) Each acre of corn yields a profit of $60 and Each acre of soybeans yields a profit of $90

The objective function P(c, s) = 60c + 90s

c)

1) The farmer has a 320 acre farm on which she plants two crops: corn and soya beans. The constraint for this problem is:

c + s ≤ 320

2) her expenses are $50 and for each acre of soybeans planted, he expenses are $100. the farmer has only $20,000 on hand. The constraint is:

50c + 100s ≤ 20000

3) Each acre of corn requires 100 bushels of storage and each acre of soybeans requires 40 bushels of storage. the total amount of storage space available is 19,200 bushels, The constraint is:

100c + 40s ≤ 19200

d) We have a total of 3 equations. To balance the equations,we would introduce 4 slack variables:
`s_1,s_2,s_3`

`c + s+s_1=320`

`50c + 100s+s_2=20000`

`100c + 40s+s_3=19200`

Therefore solving these equations gives

c=140, s=130

P(c, s) = 60c + 90s = 60(140) + 90(130)= $20100