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Rachel Sundusky is the manager of the South-Atlantic office of the Stateline Shipping and Transport Company. She is in the process of negotiating a new shipping contract with Polychem, a company that manufacture s chemicals for industrial use. Polychem wants Stateline to pick up and transport waste products from six plants to four waste disposal sites. Rachel is very concerned about this proposed arrangement. The chemical wastes that will be hauled can be hazardous to humans and environment if they leak. In addition, a number of towns and communities in the region where the plants are located prohibit hazardous materials from being shipped through their municipal limits. Thus, not only will the shipment have to be handled carefully and transported at reduced speeds, Rachel has estimated the cost of shipping a barrel of waste from each of the six plants to each of the three waste disposal sites as shown in Table 1.

Table 1

Waste Disposal Sites

Plants Whitewater Los Canos Duras Kingsport $12 $15 $17 Danville 14 9 10 Macon 13 20 11 Selma 17 16 19

Columbus 7 14 12 Allentown 22 16 18

Each week the plants generate amounts of waste as shown in table 2.

Table 2

The three waste disposal sites at Whitewater, Los Canos and Duras can accommodate a maximum of 65, 80, and 105 barrel per week, respectively. In addition, to shipping directly from each of the six plants to one of the three waste disposal sites, Rachel is also considering using each of the plants and waste disposal sites as intermediate shipping points. Trucks would be able to drop a load at a plant or disposal site to be picked up and carried on to the final destination by another truck, and vice versa. Stateline would not incur and handling costs since Polychem has agreed to take care of all local handling of the waste materials at the plants and the waste disposal sites. In other Sheffield Hallam University Department Of Engineering and Mathematics

words, the only cost Stateline incurs is the actual transportation cost. So Rachel wants to be able to consider the possibility that it may be cheaper to drop and pick up loads at intermediate points rather than shipping them directly. Table 3 shows how much Rachel estimates the shipping costs per barrel between each of the six plants to be.

Table 3

• Plants Kingsport Danville Macon Selma Columbus Allentown
• Kingsport $- $6 $4 $9 $7 $8 Danville 6 – 11 10 12 7 Macon 5 11 – 3 7 15 Selma 9 10 3 – 3 16
• Columbus 7 12 7 3 – 14 Allentown 8 7 15 16 14 –
• The estimated shipping cost per barrel between each of the three disposal sites is shown in Table 4.

Table 4

Rachel wants to determine the shipping routes that will minimise Stateline’s total cost in order to develop a contract proposal to submit to Polychem for waste disposal. She particularly wants to know if it is cheaper to ship directly from the plants to the waste sites or if she should drop and pick up some loads at the various plants and waste sites. Develop a model to assist Rachel and solve the model to determine the optimal routes.