1. A farmer grows barley and corn on his 40-acre farm. Each acre planted with barley yields 5 bushels of barle

Question

A farmer grows barley and corn on his 40-acre farm. Each acre planted with barley yields 5 bushels of barley and each acre planted with corn yields 4 bushels. He can sell at most 120 bushels of barley and 110 bushels of corn. Barley sells for £40 per bushel and corn sells for £50 per bushel. To harvest an acre of barley requires 6 hours of labour and to harvest an acre of corn requires 10 hours of labour. A maximum of 300 hours of labour can be purchased at £10 per hour.

Let B = number of acres planted with barley; Let C = number of acres planted with corn; and L = number of hours of labour purchased.

a) Write down the formulation that underlies the solution.

Using the LP output given on the next page:

b) State the optimal solution.

c) Explain the meaning of the “dual price” for Row 4.

d) Explain the meaning of “reduced cost” of B.

e) By how much could the available land decrease before the current solution is n

Answer 1

B + C ≤ 40

6B + 10C ≤ 300

5B ≤ 120, B ≤ 24

4C ≤ 110, C ≤ 22.5

R = (40)(5)B + (50)(4)C

C = 6B + 10C

P = 194B + 190C

P = 194B + 190(40 - B)

P = 194B + 7600 - 190B

P ≤ 4B + 7600

P ≤ 4(24) + 7600

P ≤ 7,696

P = 194(40 - C) + 190C

P ≤ 7,760 - 194C + 190C

P ≤ 7,760 - 4C

P ≤ 7,760 - 4(22.5)

P ≤ 7670

The optimal solution is therefore 24 acres barley and 16 acres corn.