The following exercise is a larger example of a local dairy market. There are three dairy cooperatives that produce the following amounts of raw milk on an annual basis: cooperative 1 (c1) produces 5 billion pounds of milk, cooperative 2 (c2) produces 4.808 billion pounds of milk, and cooperative 3 (c3) produces 12 billion pounds of milk. Each of the three cooperatives may sell any or all of their milk to two different cheese processors (p1 and p2) or to two different fluid milk bottlers ( f1 and f2). Assume that the cheese and fluid milk production functions are:
cheese = 1/10 raw milk, where cheese is in pounds, and every pound of raw milk produces 1/10 pound of cheese
fluid = 1/8.63 raw milk, where fluid is in pounds, and every pound of raw milk produces EXR1/8.63 gallons of fluid milk
Assume that both cheese processors and both fluid bottlers have unlimited capacity to handle and process any amount of raw milk. There are three supermarket chain stores that demand the following amounts of cheese and fluid milk each year: supermarket 1 (s1) needs 100 million pounds of cheese and 200 million gallons of milk, supermarket 2 (s2) needs 400 million pounds of cheese and 800 million gallons of milk, and super- market 3 (s3) needs 300 million pounds of cheese and 600 million gallons of milk. The following table summarizes the per unit transportation costs for raw milk and for cheese. cwt = 100 pounds.
|
From/To
|
(unit)
|
p1
|
p2
|
f1
|
f2
|
s1
|
s2
|
s3
|
|
c1
|
cwt
|
$0.50
|
$0.60
|
$0.95
|
$0.75
|
|
|
|
|
c2
|
cwt
|
$0.40
|
$0.30
|
$0.99
|
$0.89
|
|
|
|
|
c3
|
cwt
|
$0.87
|
$0.99
|
$0.75
|
$0.99
|
|
|
|
|
p1
|
lb
|
|
|
|
|
$0.02
|
$0.01
|
$0.03
|
|
p2
|
lb
|
|
|
|
|
$0.04
|
$0.06
|
$0.05
|
|
f1
|
gal
|
|
|
|
|
$0.34
|
$0.32
|
$0.41
|
|
f2
|
gal
|
|
|
|
|
$0.28
|
$0.29
|
$0.16
|
a. Write the LP tableau for this problem that minimizes total transportation costs of raw milk hauling from farm to the cheese-fluid plants and from cheese-fluid plants to the supermarkets.
b. Use Solver to find the optimal solution for this problem.
c. Summarize the optimal solution. What are the shipments from the cooperatives to the plants; from the plants to the supermarkets? Interpret the dual values (SPs) on the various nodes of the problem