A combinatorial approach to the problem of solvability of the three-index transportation problem (Q1090602)

We give a new approach to the three-index transportation problem, based on the consideration of the three-dimensional analog of the known problem regarding systems of distinct representatives. On the basis of necessary conditions for the existence of a covering of a three-dimensional matrix, we prove necessary conditions for the existence of an admissible solution of the problem T, which are a refinement of the known conditions of \textit{G. Smith} [Oper. Res. 21, 380-386 (1973; Zbl 0259.90021); Apl. Mat. 20, 432-435 (1975; Zbl 0323.90030)] and \textit{J. Moravek} and \textit{M. Vlach} [Oper. Res. 15, 542-545 (1967)] and \textit{K. B. Haley} [ibid. 15, 545-546 (1967)].