Solve the below problem:
Q: A meat packing plant produces 480 hams, 400 pork bellies and 230 picnic hams every day; each of these prodcuts can be sold either fresh or smoked. The total number of hams, bellies and picnics that can be smoked during a normal working day is 420; in addition, up to 250 products can be smoked on overtime at a higher cost. The net profits are as follows:
|
|
Fresh
|
Smoked on regular time
|
Smoked (overtime)
|
|
Hams
|
$8
|
$14
|
$11
|
|
Bellies
|
$4
|
$12
|
$7
|
|
Picnics
|
$4
|
$13
|
$9
|
For example, the following schedule yields a total net profit of $9,965:
|
|
Fresh
|
Smoked on regular time
|
Smoked (overtime)
|
|
Hams
|
165
|
280
|
35
|
|
Bellies
|
295
|
70
|
35
|
|
Picnics
|
55
|
70
|
105
|
The objective is to find the schedule that maximizes the total net profit.
(a) Formulate as an LP problem in standard form.
(b) Verify using complementary slakness if the following schedule is optimal:
• Smoke all 400 bellies on regular time.
• Smoke 20 picnics on regular time and 210 on overtime. •
Smoke 40 hams on overtime and sell 440 fresh.
(c) For the optimal solution, all the bellies and the picnics are smoked. However, suf- ficiently drastic changes in market prices might provide an incentive to change this policy. Assume that the market price of fresh bellies increases by x dollars while all the other prices remain fixed at their original levels. How large would x have to be in order to make it profitable for the plant to sell fresh bellies? Ask and answer a similar question for picnics. How would the sales of small ammounts of fresh bellies and picnics affect the rest of the operation? What precisely do "small amounts" mean in this context?