# Minimum weight t-composition of an integer

If $p \geq t$ are positive integers, a t-composition of p is an ordered t-tuple of positive integers summing p. If $T=(s_1, s_2, \dots, s_t)$ is a t-composition of p and W is a $p-(t-1) \times t$ matrix, call $W(T)= \sum_{k=1}^t w_{s_k k}$ the weight of the t-composition T. We show that finding a minimum weight t-composition of p reduces to the determination of a 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.