A dairy farmer's cows needs three nutrients (A, B, and C) to subsist and produce milk each day. Each cow must receive the equivalent of 100 units of nutrient A, 200 units of nutrient B, and 50 units of nutrient C in order to maximize milk output. The farmer can use any combination of three feeds (f1, f2, and f3) in meeting these minimum requirements. The local feed dealer sells all three feeds, which have the following cost per pound and nutrient equivalents (for A, B, and C) per pound.
Nutrient Content Per Pound
|
Feed
|
A
|
B
|
C
|
Cost ($/pound)
|
|
f1
|
5
|
22
|
3
|
0.25
|
|
f2
|
10
|
25
|
2
|
0.50
|
|
f3
|
7
|
12
|
5
|
0.27
|
Assume that the farmer's objective is to minimize the cost per cow of buying any com- bination of these three feeds that satisfies the daily nutrient requirement of the cows.
a. Solve this problem using the simplex method.
b. Your optimal solution should indicate that no amount of feed f2 should be pur- chased and fed to the farmer's cows. By how much should feed f2's current price of $0.50 per pound decrease in order for the optimal solution to change?
c. Your optimal solution should indicate that some amount of feed f3 should be pur- chased and fed to the farmer's cows. By how much should feed f3's current price of $0.27 per pound increase in order for the optimal solution to change?
d. What is the range of feasibility for the minimum requirement for nutrient A?
e. What is the range of feasibility for the minimum requirement for nutrient B?
f. What is the range of feasibility for the minimum requirement for nutrient C?
g. How would you interpret these ranges for this problem?
h. What are the SPs for each constraint of this problem? What is their economic interpretation for this problem?