An axial four-index transportation problem and its algebraic characterizations

Abstract

In this study, we describe an axial four-index transportation problem of order 1 × m × n × 1, which is equivalent to a circularization network flow problem with m + n + 2 nodes and (m + 1)(n + 1) arcs. We then give some algebraic characterizations of the axial four-index transportation problem, and compare the results we obtained with some results on the planar four-index transportation problem of order 1 × m × n × 1. It is shown that these problems have common algebraic characterizations, and can be solved in terms of eigenvectors of the matrices J_m and J_n , where J_m is an m × m matrix whose entries are 1.

Keywords:

• Transportation problem
• Axial four-index transportation problem
• Circularization network flow problem