Construction and algebraic characterizations of a planar four-index transportation problem equivalent to a circularization network flow problem

Abstract

In this study, a circularization network flow problem with m + n + 2 nodes and (m + 1)(n + 1) arcs is described as a planar four-index transportation problem of order 1 × m × n × 1. Construction and several algebraic characterizations of the planar four-index transportation problem of order 1 × m × n × 1 are investigated using the generalized inverse and singular value decomposition of its coefficient matrix. The results are compared with some results we obtained on the transportation problem with m sources and n destinations. It is shown that these problems can be solved in terms of eigenvectors of the matrices J _m and J _n , where J _m is a m × m matrix whose entries are 1.

Keywords:

• Transportation problem
• Planar four-index transportation problem
• Circularization network flow problem
• Generalized inverses
• Singular value decomposition