Easy transportation-like problems onK-dimensional arrays

Abstract

TheK-dimensional version of two transportation-like problems is posed and efficiently solved. The caseK=2 goes as follows: Given an (m,n)-matrixA of reals, a realm-vectoru, and a realn-vectorv, find a real (m,n)-matrixX minimizing

$$either f(X) = \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {(X_{ij} - A_{ij} )^2 or g(X)} = \mathop {\max }\limits_{ij} } \{ X_{ij} - A_{ij} \} ,$$

subject to linear constraints of transportation type, i.e.,

$$\begin{gathered} \sum\limits_{j = 1}^n {X_{ij} } = u_i , i = 1,...,m, \hfill \\ \sum\limits_{i = 1}^m {X_{ij} } = \upsilon _j , j = 1,...,n. \hfill \\\end{gathered} $$

In the generalization for anyK≥2, consideration of thef(X)-type of objective function yields an explicit formula for the optimal solution, whereas for theg(X)-type a one-pass algorithm is proposed.