A Matheuristic approach for the Quickest Multicommodity k-Splittable Flow Problem

The literature on k-splittable flows, see Baier et al. (2002) Baier et al. (2005), provides evidence on how controlling the number of used paths enables practical applications of flows optimization in many real-world contexts. Such a modeling feature has never been integrated so far in Quickest Flows, a class of optimization problems suitable to cope with situations such as emergency evacuations, transportation planning and telecommunication systems, where one aims to minimize the makespan, i.e. the overall time needed to complete all the operations, see Pascoal et al. (2006) Pascoal et al. (2006). In order to bridge this gap, a novel optimization problem, the Quickest Multicommodity k-Splittable Flow Problem (QMCkSFP) is introduced in this paper. The problem seeks to minimize the makespan of transshipment operations for given demands of multiple commodities, while imposing restrictions on the maximum number of paths for each single commodity. The computational complexity of this problem is analyzed, showing its NP-hardness in the strong sense, and an original Mixed-Integer Programming formulation is detailed. We propose a matheuristic algorithm based on a hybridized Very Large-Scale Neighborhood Search that, utilizing the presented mathematical formulation, explores multiple search spaces to solve efficiently large instances of the QMCkSFP. High quality computational results obtained on benchmark test sets are presented and discussed, showing how the proposed matheuristic largely outperforms a state-of-the-art heuristic scheme frequently adopted in path-restricted flow problems.