Polyhedral study of a temporal rural postman problem: application in inspection of railway track without disturbing train schedules

TL;DR

This paper introduces a novel polyhedral approach and branch-and-cut algorithm for the temporal rural postman problem, optimizing railway track inspection scheduling without disrupting train schedules, demonstrated through a real-world case study.

Contribution

It develops a new cascaded graph formulation and facet-defining inequalities, significantly improving solution efficiency for the RPP-TU in railway inspection applications.

Findings

Reduced computation time by up to 48% using the proposed algorithm.

Achieved a 93% improvement over existing MILP solvers in a case study.

Effectively modeled temporal constraints in railway track inspection scheduling.

Abstract

The Rural Postman Problem with Temporal Unavailability (RPP-TU) is a variant of the Rural Postman Problem (RPP) specified for multi-agent planning over directed graphs with temporal constraints. These temporal constraints represent the unavailable time intervals for each arc during which agents cannot traverse the arc. Such arc unavailability scenarios occur in routing and scheduling of the instrumented wagons for inspection of railway tracks without disturbing the train schedules, i.e. the scheduled trains prohibit access to the signal blocks (sections of railway track separated by signals) for some finite interval of time. A three-index formulation for the RPP-TU is adopted from the literature. The three-index formulation has binary variables for describing the route information of the agents, and continuous non-negative variables to describe the schedules at pre-defined locations. A relaxation of the three-index formulation for RPP-TRU, referred to as Cascaded Graph Formulation (CGF), is investigated in this work. The CGF has attributes that simplify the polyhedral study of time-dependent arc routing problems like RPP-TRU. A novel branch-and-cut algorithm is proposed to solve the RPP-TU, where branching is performed over the service arcs. A family of facet-defining inequalities, derived from the polyhedral study, is used as cutting planes in the proposed branch-and-cut algorithm to reduce the computation time by up to $48\%$. Finally, an application of this work is showcased using a simulation case study of a railway inspection scheduling problem based on Kurla-Vashi-Thane suburban network in Mumbai, India. An improvement of $93\%$ is observed when compared to a Benders' decomposition based MILP solver from the literature.