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Unformatted text preview: Economics 146 – Fall, 2007 Transportation Problem Last revision: 11/14/07. The transportation problem that we have considered has a remarkably diverse range of applications. A feature of such problems is that there are usually very large numbers of variables and constraints, so that a use of the simplex algorithm may require a huge amount of computation. Several alternative algorithms have been developed with the goal of using the special structure of the transportation problem to significantly reduce the amount of computation. The Transportation Problem – Revisited. Various applications use different types of terminology to describe the sources and destinations for the transportation problem. We will use the terms plant and market generically for source and destination, respectively. We fix notation for the rest of this section. There are m plants and n markets, all dealing with the same commodity. The data is: • ( c ij : i = 1 , . . . , m ; j = 1 , . . . , n ) – where c ij is the cost of shipping one unit of the commodity from the i th plant to the j th market; • ( σ i : i = 1 , . . . , m ) – where σ i is the amount of units of the commodity supplied by the i th plant; • ( δ j : j = 1 , . . . , n ) – where δ j is the amount of units of the commodity demanded at the j th market We seek a shipping schedule ( x ij : i = 1 , . . . , m ; j = 1 , . . . , n ), where x ij is the number of commodity units to be shipped from the i th plant to the j th market, so as to minimize m X i =1 n X j =1 c ij x ij (1) subject to n X j =1 x ij ≤ σ i for all i = 1 , . . . , m (2) m X i =1 x ij ≥ δ j for all j = 1 , . . . , n (3) and x ij ≥ 0 for all i = 1 , . . . , m ; j = 1 , . . . , n (4) Basic Assumption 1. We assume that σ i ≥ , δ j ≥ , c ij ≥ 0 for all i = 1 , . . . , m ; j = 1 , . . . , n . Observe that the constraints of the problem imply that m X i =1 σ i ≥ m X i =1 n X j =1 x ij ≥ n X j =1 δ j (5) 1 Consequently, a necessary condition for a feasible solution to exist is that total supply be greater than or equal to total demand. Basic Assumption 2. We assume that ∑ m i =1 σ i ≥ ∑ n j =1 δ j . We could find solutions by rewriting this LP problem in standard form, converting the standard form to computational form, and using the simplex method. This would be very inefficient since the resulting computational problem has mn variables and m + n constraints, the constraint matrix has mostly 0’s in it, and the special structure of the problem is ignored....
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 Fall '07
 farmer
 Economics, Linear Programming, Optimization, basic feasible solution, cij xij

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