The minimum weight t-composition of an integer

Abstract

Let p and t, p ≥ t, be positive integers. A t-composition of p is an ordered t-tuple of positive integers summing p. If T = (s_1, s_2, . . . , s_t) is a t-composition p and W is a (p − (t − 1)) × t matrix, then W(T) = w_(s_1)+ w_(s_2)2... + w_(s_k)k is called the weight of the t-composition T. We show that finding a minimum weight t-composition of p can be reduced to the determination of the shortest path in a certain digraph with O(tp) vertices. This study was motivated by a problem arising from the automobile industry, and the presented result is useful when dealing with huge location problems.