Temporal Recurring Unavailabilities in Multi-agent Rural Postman Problem: Navigating railway tracks during availability time intervals

Time-dependent (or temporal) properties may arise in many network-based planning problems, particularly in the routing and scheduling of railway track inspection problems. The availability of tracks depends on the train schedules, maintenance possessions, etc. In the absence of side constraints, this routing and scheduling problem is formulated as a multi-agent rural postman problem on a temporal-directed network; where a given set of rail track sections must be visited while respecting the temporal attributes due to railway track unavailabilities. In this work, we adopt a three-index formulation for the multi-agent Rural Postman Problem with Temporal Recurring Unavailabilities (RPP-TRU) and frame it as a Mixed Integer Linear Programming (MILP) problem. In addition, we propose relevant theoretical studies for RPP-TRU to ensure the feasibility of the proposed optimization problem. Two approaches of an exact algorithm are proposed, based on Benders' decomposition framework, to address the disjunctive unavailability constraints occurring in its scheduling sub-problems, alongside the NP-Hard routing (master) problem. A polynomial-time algorithm is designed to address the scheduling sub-problem, while the NP-Hard master problem is solved using MILP toolbox. Comparison results with RPP (without temporal constraints) show a minor compromise with the spatial cost solution with significantly less delay, hence suitable for real-world routing and scheduling applications occurring in a shared network like railways. A simulation study on a part of the Mumbai suburban railway network demonstrates the working of the proposed methodology under a realistic setting.