Stateline Shipping and Transport Company

Introduction The Stateline Shipping and Transport Company is negotiating a shipping contract with Polychem. Polychem would like Stateline to pick up and transport their waste products from their six plants to three waste disposal sites. There is a possibility that using the plants and waste disposal sites as intermediate shipping points could lower the overall shipping costs. A Stateline manager is trying to determine the shipping routes that will minimize Stateline’s total cost and if it would be cheaper to ship directly from the plants to the waste sites or use intermediate shipping points (Taylor, 2011, p. 273-274). The estimated shipping costs ($/barrel) from the six plants to the three waste disposal sites are shown in Table 1. Waste Disposal Site
Plant A. Whitewater B. Los Canos C. Duras
1. Kingsport $12 $15 $17
2. Danville 14 9 10
3. Macon 13 20 11
4. Selma 17 16 19
5. Columbus 7 14 12
6. Allentown 22 16 18
Table 1: Estimated Shipping Costs from Plant to Waste Disposal Site The waste generated by the plants represents the supply for the transportation model and is shown in Table 2. The model’s demand is the number of barrels each waste disposal site can accommodate in one week. This data is shown in Table 3.

Plant Waste per Week (barrels)
Waste Disposal Site Weekly Accommodation (barrels)
1. Kingsport 35 A. Whitewater 65
2. Danville 26 B. Los Canos 80
3. Macon 42 C. Duras 105
4. Selma 53 Total Demand 250
5. Columbus 29
6. Allentown 38
Total Supply 223
Table 2: Generated Waste (Supply) Table 3: Barrel Accommodation (Demand) The estimated shipping cost ($/barrel) between each plant is shown in Table 4 and between each waste disposal site is shown in Table 5. Plant
Plant 1. Kingsport 2. Danville 3. Macon 4. Selma 5. Columbus 6. Allentown
1. Kingsport $ - $ 6 $ 4 $ 9 $ 7 $ 8
2. Danville 6 - 11 10 12 7
3. Macon 5 11 - 3 7 15
4. Selma 9 10 3 - 3 16
5. Columbus 7 12 7 3 - 14
6. Allentown 8 7 15 16 14 -
Table 4: Estimated Shipping Costs from Plant to Plant Waste Disposal Site
Waste Disposal Site A. Whitewater B. Los Canos C. Duras
A. Whitewater $ - $ 12 $ 10
B. Los Canos 12 - 15
C. Duras 10 15 -
Table 5: Estimated Shipping Costs from Waste Disposal Site to Waste Disposal Site
Transportation Model The objective of this transportation model is to determine the shipping routes that minimize the total cost of waste product transportation from the six plants to the three waste disposal sites. The objective function Z represents the cost. The decision variables, XIJ, represents the waste transported from the I-th plant (I=1, 2, 3, 4, 5, 6) to the J-th waste disposal site (J=A, B, C). The linear programming model is defined as follows:
Objective Function
Z = $12X1A+15X1B+17X1C+14X2A+9X2B+10X2C+13X3A+20X3B+11X3C+17X4A+16X4B+19X4C+7X5A+14X5B+12X5C+22X6A+16X6B+18X6C Minimize Costs
Constraints
X1A + X1B + X1C = 35 Supply from Kingsport to Waste Disposal Sites
X2A + X2B + X2C = 26 Supply from Danville to Waste Disposal Sites
X3A + X3B + X3C = 42 Supply from Macon to Waste Disposal Sites
X4A + X4B + X4C = 53 Supply from Selma to Waste Disposal Sites
X5A + X5B + X5C = 29 Supply from Columbus to Waste Disposal Sites
X6A + X6B + X6C = 38 Supply from Allentown to Waste Disposal Sites
X1A + X2A + X3A +X4A + X5A + X6A