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# transportation problem

A special case of linear programming, dealing with the least-cost supply of goods from N origins to M destinations. If the amounts available at N are known, together with the demand at M and the transport costs between each N-M pair of locations, solution of the transportation problem yields the least-cost pattern of flows between the origins and destinations.

Solution of the problem is normally iterative, beginning with a feasible solution (i.e. one consistent with the supplies at all N points and the demands at all M) and converges upon the optimum (or one of the optima if there are more than one). The iteration procedure operates by establishing relative prices at the origins and destinations, and converges on an optimal set of prices by solving two problems simultaneously: (a) a primal problem, which involves establishing an optimal pattern of flows which minimizes transport costs; and (b) a dual problem, which establishes a pattern of prices that maximizes the value-added in transportation.

The basic procedure can be adapted to more complex situations. A capacitated network may restrict the flow on certain links in the system, for example, and a dummy (or dump) destination may be used to absorb (at nil transport costs) supplies available at, at least some, of the N origins which are additional to demand. Even where transport is not involved, as in the allocation of M plots of agricultural land to N crops with a minimum expenditure on fertilizers, it is possible to structure the problem in terms of the transportation model.

The technique has been applied in geography in two main ways. First, it has been used in attempts to explain flow and/or production patterns, to show that they are consistent with the model\'s least-cost basis; these are rarely successful, as \'real-world\' situations normally find more of the N-M links being used than solution of the transportation problem suggests. Secondly, it may be used to evaluate the relative efficiency of an actual flow pattern, but the latter usually involves more constraints and heterogeneities than can be accounted for in the model.Â (AMH)

Suggested Reading Hay, A.M. 1977: Linear programming: elementary geographical applications of the transportation problem. Norwich: Geo Books CATMOG 11.Â Taaffe, E.J. and Gauthier, H.L. 1973: Geography of transportation. Englewood Cliffs, NJ: Prentice-Hall.

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