Extreme points of general transportation polytopes

TL;DR

This paper characterizes the extreme points of a generalized class of transportation polytopes where row and column sums are bounded within specified ranges, extending previous results on transportation matrices.

Contribution

It introduces a broader class of transportation polytopes with bounded row and column sums and fully characterizes their extreme points.

Findings

The set of such matrices forms a convex polytope.

Extreme points are fully characterized for the bounded sum case.

Generalizes previous characterizations of transportation polytopes.

Abstract

Transportation matrices are $m\times n$ non-negative matrices whose row sums and row columns are equal to, or dominated above with given integral vectors $R$ and $C$. Those matrices belong to a convex polytope whose extreme points have been previously characterized. In this article, a more general set of non-negative transportation matrices is considered, whose row sums are bounded by two integral non-negative vectors $R_{min}$ and $R_{max}$ and column sums are bounded by two integral non-negative vectors $C_{min}$ and $C_{max}$. It is shown that this set is also a convex polytope whose extreme points are then fully characterized.