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Transportation - Transportation problems In D1 you studied...

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In D1 you studied linear programming problems and their solutions using the simplex algorithm. In this chapter and the next one the solutions of two special types of linear programming problems are considered by methods other than the simplex algorithm. These are transportation problems and assignment problems . Both of these problem types could be solved using the simplex algorithm, but the process would result in very large simplex tableaux and numerous simplex iterations. Because of the special characteristics of each problem, however, alternative solution methods requiring significantly less mathematical manipulation than the simplex method have been developed. 1.1 The transportation problem The transportation problem deals with the distribution of goods from several points of supply, such as factories, often known as sources , to a number of points of demand, such as warehouses, often known as destinations . Each source is able to supply a fixed number of units of the product, usually called the capacity or availability , and each destination has a fixed demand, usually known as the requirement . The objective is to schedule shipments from sources to destinations so that the total transport cost is a minimum. Example 1 A concrete company transports concrete from three plants, 1, 2 and 3, to three construction sites, A, B and C. The plants are able to supply the following numbers of tons per week: Plant Supply capacity) 1 300 2 300 3 100 Transportation problems 1
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The requirements of the sites, in numbers of tons per week, are: Construction site Demand requirement) A 200 B 200 C 300 The cost of transporting 1 ton of concrete from each plant to each site is shown in the figure below in pounds per ton. 1 2 3 A B 4 5 7 4 5 8 5 9 3 C Plants Sites For computational purposes it is convenient to put all the above information into a table, as in the simplex method. In this table each row represents a source and each column represents a destination . 1.2 Formulation as a linear programming problem Before proceeding with the solution of the transportation problem, using the method developed especially for it, we will show that it can be formulated as a linear programming problem. The decision variables x ij are the numbers of tons transported from plant i where i 1, 2, 3) to each site j where j A, B, C). A B C Supply availability) 1 4 3 8 300 2 7 5 9 300 3 4 5 5 100 Demand requirement) 200 200 300 To From Sites Plants 2 Transportation problems
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The objective function represents the total transportation cost £ Z . Each term in the objective function Z represents the cost of tonnage transported on one route. For example, for the route 2 ! C the term is 9 x 2C , that is: cost per ton 9) ± number of tons transported x 2C ) Hence the objective function is: Z 4 x 1A 3 x 1B 8 x 1C 7 x 2A 5 x 2B 9 x 2C 4 x 3A 5 x 3B 5 x 3C 1) Notice that in this problem the total supply is 300 300 100 700 and the total demand is 200 200 300 700.
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