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Unformatted text preview: 1) The model for the transportation problem consists of 18 decision variables, representing the number of barrels of wastes transported from each of the 6 plants to each of the 3 waste disposal sites: ij x = Number of Barrels transported per week from plant i to the jth waste disposal site, where i = 1, 2, 3, 4, 5, 6 and j = A, B, C. The objective function of the manager is to minimize the total transportation cost for all shipments. Thus the objective function is the sum of the individual shipping costs from each plant to each waste disposal site: Minimize Z = 12 1 A x + 15 1 B x + 17 1 C x + 14 2 A x + 9 2 B x + 10 2 C x + 13 3 A x + 20 3 B x +11 3 C x +17 4 A x +16 4 B x +19 4 C x +7 5 A x +14 5 B x +12 5 C x +22 6 A x +16 6 B x +18 6 C x The constraints in the model are the number of barrels of wastes available per week at each plant and the number of barrels of wastes accommodated at each waste disposal site. There are 9 constraints one for each plant supply and one for each waste disposal sites demand. The six supply constraints are: 1 A x + 1 B x + 1 C x = 35 2 A x + 2 B x + 2 C x = 26 3 A x + 3 B x + 3 C x = 42 4 A x + 4 B x + 4 C x = 53 5 A x + 5 B x + 5 C x = 29 6 A x + 6 B x + 6 C x = 38 As an example, here the supply constraint 1 A x + 1 B x + 1 C x = 35 represents the number of barrels transported from the plant Kingsport to all the three waste disposal sites. The amount transported from Kingsport is limited to the 35 barrels available. The three demand constraints are: 1 A x + 2 A x + 3 A x + 4 A x + 5 A x + 6 A x 65 1 B x + 2 B x + 3 B x + 4 B x + 5 B x + 6 B x 80 1 C x + 2 C x + 3 C x + 4 C x + 5 C x + 6 C x 105 Here the demand constraint 1 A x + 2 A x + 3 A x + 4 A x + 5 A x + 6 A x 65 represents the number of barrels transported to the waste disposal site Whitewater from all the six plants. The barrel of wastes that can accommodate in the waste disposal site Whitewater is limited to 65 barrels. The demand constraints are inequalities because the total demand (65+80+105) = 250 exceeds the total supply (26+42+53+29+38) = 223. The linear programming model for the transportation problem is summarized as follows: Minimize Z = 12 1 A x + 15 1 B x + 17 1 C x + 14 2 A x + 9 2 B x + 10 2 C x + 13 3 A x + 20 3 B x +11 3 C x +17 4 A x +16 4 B x +19 4 C x +7 5 A x +14 5 B x +12 5 C x +22 6 A x +16 6 B x +18 6 C x Subject to 1 A x + 1 B x + 1 C x = 35 2 A x + 2 B x + 2 C x = 26 3 A x + 3 B x + 3 C x = 42 4 A x + 4 B x + 4 C x = 53 5 A x + 5 B x + 5 C x = 29 6 A x + 6 B x + 6 C x = 38 1 A x + 2 A x + 3 A x + 4 A x + 5 A x + 6 A x 65 1 B x + 2 B x + 3 B x + 4 B x + 5 B x + 6 B x 80 1 C x + 2 C x + 3 C x + 4 C x + 5 C x + 6 C x 105 ij...
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 Spring '11
 Dralexander

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