There are three dairy cooperatives that produce the following amounts of raw milk on an annual basis: Cooperative 1 (c1) produces 8 billion pounds of milk, cooperative 2 (c2) produces 10 billion pounds of milk, and cooperative 3 (c3) produces 15 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 dealers ( f1 and f2), or to two different butter manufacturers (b1 and b2). Assume that the cheese, fluid milk, and butter production functions are characterized by the following:
1 pound of cheese requires 10 pounds of raw milk. 1 gallon of milk requires 5 pounds of raw milk.1 pound of butter requires 21 pounds of raw milk. Assume that the cheese processors, fluid dealers, and butter manufacturers have unlimited capacity to handle and process any amount of raw milk. There are two super- market chain stores that demand the following amounts of cheese, fluid milk, and butter each year: Supermarket 1 (s1) needs 100 million pounds of cheese, 200 million gallons of milk, and 150 million pounds of butter. Supermarket 2 (s2) needs 400 million poundsof cheese, 800 million gallons of milk, and 250 million pounds of butter. The following table summarizes the per unit transportation costs for raw milk and for cheese.
|
From/To
|
(unit)
|
p1
|
p2
|
f1
|
f2
|
b1
|
b2
|
s1
|
s2
|
|
c1
|
lb
|
$0.05
|
$0.06
|
$0.11
|
$0.08
|
$0.15
|
$0.05
|
|
|
|
c2
|
lb
|
$0.04
|
$0.03
|
$0.09
|
$0.08
|
$0.04
|
$0.07
|
|
|
|
c3
|
lb
|
$0.07
|
$0.12
|
$0.08
|
$0.10
|
$0.16
|
$0.04
|
|
|
|
p1
|
lb
|
|
|
|
|
|
|
$0.02
|
$0.01
|
|
p2
|
lb
|
|
|
|
|
|
|
$0.04
|
$0.06
|
|
f1
|
gal
|
|
|
|
|
|
|
$0.34
|
$0.32
|
|
f2
|
gal
|
|
|
|
|
|
|
$0.28
|
$0.29
|
|
b1
|
lb
|
|
|
|
|
|
|
$0.03
|
$0.02
|
|
b2
|
lb
|
|
|
|
|
|
|
$0.01
|
$0.03
|
a. Write the LP model for this problem that minimizes total transportation costs of hauling raw milk from the farm to the cheese, fluid, and butter plants and from the cheese, fluid, and butter plants to the supermarkets. Make sure to define all notation used in your model.
b. Show how you would modify the model in part a to account for the following capacity constraints: cheese plant 1 having a capacity of handling 3 billion pounds of milk, cheese plant 2 having a capacity of handling 5 billion pounds of milk, fluid plant 1 having a capacity of handling 4 billion pounds of milk, fluid plant 2 having a capacity of handling 3 billion pounds of milk, butter plant 1 hav- ing a capacity of handling 7 billion pounds of milk, and butter plant 2 having a capacity of handling 8 billion pounds of milk (for clarity, consider listing all of these separately instead of in paragraph form).