A min-max theorem for the minimum fleet-size problem

TL;DR

This paper establishes a min-max theorem linking the maximum number of incompatible trips to the minimum fleet size in fleet management, providing a theoretical foundation for optimizing vehicle allocation.

Contribution

It introduces a novel min-max theorem that connects trip incompatibility with fleet size, advancing the theoretical understanding of fleet-sizing problems.

Findings

Proves a min-max theorem for fleet-size and trip incompatibility.

Shows the equivalence between maximum incompatible trips and minimum fleet size.

Abstract

A retrospective fleet-sizing problem can be solved via bipartite matching, where a maximum cardinality matching corresponds to the minimum number of vehicles needed to cover all trips. We prove a min-max theorem on this minimum fleet-size problem: the maximum number of pairwise incompatible trips is equal to the minimum fleet size needed.