The Navy has 9,000 pounds of material in Albany, Georgia that it wishes to ship to three installations: San Diego, Norfolk, and Pensacola. They require 4,000, 2,500, and 2,500 pounds, respectively. Government regulations require equal distribution of shipping among the three carriers.
The shipping costs per pound for truck, railroad, and airplane transit are shown below. Formulate and solve a linear program to determine the shipping arrangements (mode, destination, and quantity) that will minimize the total shipping cost.
Mode San Diego Norfolk Pensacola
Truck $12 $ 6 $ 5
Railroad 20 11 9
Airplane 30 26 28
1. Define the Decision Variables
We want to determine the pounds of material, xij , to be shipped by mode i to destination j. The following table summarizes the decision variables:
San Diego Norfolk Pensacola
Truck x11 x12 x13
Railroad x21 x22 x23
Airplane x31 x32 x33
2. Define the Objective Function
Minimize the total shipping cost.
Min: (shipping cost per pound for each mode per destination pairing) x (number of pounds shipped by mode per destination pairing).
Min: 12x11 + 6x12 + 5x13 + 20x21 + 11x22 + 9x23 + 30x31 + 26x32 + 28x33
3. Define the Constraints
Equal use of transportation modes:
(1) x11 + x12 + x13 = 3000
(2) x21 + x22 + x23 = 3000
(3) x31 + x32 + x33 = 3000
Destination material requirements:
(4) x11 + x21 + x31 = 4000
(5) x12 + x22 + x32 = 2500...