Beerco manufactures ale and beer from corn, hops, and malt. Currently, 40 lb of corn, 30 lb of hops, and 40 lb of malt are available. A barrel of ale sells for $40 and requires 1 lb of corn, 1 lb of hops, and 2 lb of malt. A barrel of beer sells for $50 and requires 2 lb of corn, 1 lb of hops, and 1 lb of malt. Beerco can sell all ale and beer that is produced. To maximize total sales revenue, Beerco should solve the following LP:
ALE = barrels of ale produced, and BEER = barrels of beer produced. An optimal tableau for this LP is shown in Table.
|
z |
Ale |
Beer |
s1 |
s2 |
s3 |
rhs |
|
1 |
0 |
0 |
20 |
0 |
10 |
1,200 |
|
0 |
0 |
1 |
|
0 |
|
|
|
0 |
0 |
0 |
|
1 |
|
|
|
0 |
1 |
0 |
|
0 |
|
|
a Write down the dual to Beerco’s LP and find its optimal solution.
b Find the range of values of the price of ale for which the current basis remains optimal.
c Find the range of values of the price of beer for which the current basis remains optimal.
d Find the range of values of the amount of available corn for which the current basis remains optimal.
e Find the range of values of the amount of available hops for which the current basis remains optimal.
f Find the range of values of the amount of available malt for which the current basis remains optimal.
g Suppose Beerco is considering manufacturing maltliquor. A barrel of malt liquor requires 0.5 lb of corn, 3lb of hops, and 3 lb of malt and sells for $50. Should Beerco manufacture any malt liquor?
h Suppose we express the Beerco constraints in ounces. Write down the new LP and its dual.
i What is the optimal solution to the dual of the new LP?
