Aircraft routing: periodicity and complexity

TL;DR

This paper investigates the relationship between periodicity and solutions in aircraft routing, proving existence of certain periodic solutions under maintenance constraints and establishing NP-hardness for non-periodic cases.

Contribution

It demonstrates that periodic solutions of a specific form always exist under certain maintenance constraints and proves NP-hardness for non-periodic aircraft routing problems.

Findings

Periodic solutions exist when maintenance is required at most every four days.

NP-hardness is established for non-periodic aircraft routing.

Polynomial-time solvable case identified.

Abstract

The aircraft routing problem is one of the most studied problems of operations research applied to aircraft management. It involves assigning flights to aircraft while ensuring regular visits to maintenance bases. This paper examines two aspects of the problem. First, we explore the relationship between periodic instances, where flights are the same every day, and periodic solutions. The literature has implicitly assumed-without discussion-that periodic instances necessitate periodic solutions, and even periodic solutions in a stronger form, where every two airplanes perform either the exact same cyclic sequence of flights, or completely disjoint cyclic sequences. However, enforcing such periodicity may eliminate feasible solutions. We prove that, when regular maintenance is required at most every four days, there always exist periodic solutions of this form. Second, we consider the computational hardness of the problem. Even if many papers in this area refer to the NP-hardness of the aircraft routing problem, such a result is only available in the literature for periodic instances. We establish its NP-hardness for a non-periodic version. Polynomiality of a special but natural case is also proven.