Multidimensional transportation problem

This problem is usually a transportation problem with a third index that refers to a product type. Here there are i origins, j destinations and k types of products available at the origins and demanded at the destinations. The variables x _ijk represent the amount of the kth product shipped from the ith origin to the jth destination. The constraint set is a set of linear balance equations, with the usual linear cost objective function. It is also a special form of the multi-commodity network-flow problem. Unlike the transportation problem, its optimal solution may not be integer-valued even if the network data are given as integers. The problem can also be defined with more than three indices. Multicommodity network flows; Transportation problem.