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This paper compares different solution approaches for the multi-objective shortest path problem (MSPP) on multigraphs. Multigraphs as a modelling tool are able to capture different available trade-offs between objectives for a given section of a route. For this reason, they are increasingly popular in modelling transportation problems with multiple conflicting objectives (e.g., travel time and fuel consumption), such as time-dependent vehicle routing, multi-modal transportation planning, energy-efficient driving, and airport operations. The multigraph MSPP is more complex than the NP-hard simple graph MSPP. Therefore, approximate solution methods are often needed to find a good approximation of the true Pareto front in a given time budget. Evolutionary algorithms have been successfully applied for the simple graph MSPP. However, there has been limited investigation of their applications to the multigraph MSPP. Here, we extend the most popular genetic representations to the multigraph case and compare the achieved solution qualities. Two heuristic initialisation methods are also considered to improve the convergence properties of the algorithms. The comparison is based on a diverse set of problem instances, including both bi-objective and triple objective problems. We found that the metaheuristic approach with heuristic initialisation provides good solutions in shorter running times compared to an exact algorithm. The representations were all found to be competitive. The results are encouraging for future application to the time-constrained multigraph MSPP.

Routing and scheduling problems with increasingly realistic modelling approaches often entail the consideration of multiple objectives, time constraints, and modelling the system as a multigraph. The latter is required in multiple applications to represent alternative routes with different costs linking the same nodes. The detailed modelling approach increases computational complexity and may also lead to violation of the additivity property of costs. Therefore approximate solution methods become more suitable. This paper focuses on one particular real-world application, the Airport Ground Movement Problem, where both time constraints and parallel arcs are involved. We introduce a novel Memetic Algorithm for Routing in Multigraphs with Time constraints (MARMT) and present a comprehensive study on its different variants; these variants are based on diverse genetic representation methods. We propose a local search operator that provides significant improvements. Our results also show that the best variant of MARMT is consistently producing high quality results in shorter times compared to a state of the art enumerative algorithm. The algorithms are tested on real data. MARMT can be adapted for other applications with minor modifications, such as train operations or electric vehicle routing.

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