# Multidimensional transportation problem

**DOI:**https://doi.org/10.1007/1-4020-0611-X_651

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 *k*th product shipped from the *i*th origin to the *j*th 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.