CONSTRUCTION AND ALGEBRAIC CHARACTERIZATIONS OF A PLANAR FOUR-INDEX TRANSPORTATION PROBLEM EQUIVALENT TO A CIRCULARIZATION NETWORK FLOW PROBLEM

摘要

In this study, a circularizalion 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 x m x n x 1. Construction and several algebraic characterizations oi the planar four-index transportation problem of order 1 x m x n x I 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 x m matrix whose entries are 1.