Transportation problem

A linear-programming problem of the following form is called a transportation problem:

The variables {x _ij} represent a shipment of a homogeneous product from origin i to destination j, where the {a _i} are the amounts of the product to be shipped from the origins i, and the {b _j} are the amounts demanded by the destinations j. Here we restrict Σ_i a _i = Σ_j b _j. The problem can also be formulated with the origin constraints as ≥ inequalities and the destination constraints as ≤ inequalities, and the restriction that the total supply equal the total demand need not apply. It can be shown that if the {a _i} and {b _j} are integers, than an optimal basic feasible solution exists that is all integer. The transportation problem is a special network problem whose network representation is called a bipartite graph. The special case with m = n and all {a _i} and {b _j} equal to 1 is the assignment problem. A transportation problem can be solved by direct application of the simplex method, but due...