On a transport problem and monoids of non-negative integers

TL;DR

This paper investigates a transportation problem involving cars, leading to the study of a specific set of integers defined by inequalities, demonstrating it forms a submonoid of non-negative integers and providing algorithms for its computation.

Contribution

It introduces a new set related to a transportation problem, proves it forms a submonoid, and develops algorithms to compute this set.

Findings

The set T∪{0} is a submonoid of (N,+).

Algorithms for computing the set T are provided.

The problem connects transportation issues with algebraic structures.

Abstract

A problem about how to transport profitably a group of cars leads us to study the set $T$ formed by the integers $n$ such that the system of inequalities, with non-negative integer coefficients, $$a_1x_1 +\cdots+ a_px_p + \alpha \leq n \leq b_1x_1 +\cdots+ b_px_p - \beta$$ has at least one solution in ${\mathbb N}^p$. We will see that $T\cup\{0\}$ is a submonoid of $({\mathbb N},+)$. Moreover, we show algorithmic processes to compute $T$.