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

Abstract

The literature on k-splittable flows, see Baier et al. (2002) [1], 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) [2]. In order to bridge this gap, a novel optimization problem, the Quickest Multicommodity k-Splittable Flow Problem (QM CkSF P) 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 N P-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 QM CkSF P. 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.

A branch and price algorithm to solve the Quickest Multicommodity k-splittable Flow Problem

“…The first contribution integrating k-splittable flow tools in dynamic networks flows involved a (3 + 2 √ 2)-approximation algorithm for the single commodity Dynamic k-Splittable Flow Problem with a continuous time parameter [38]. Recently, Melchiori and Sgalambro [40]…”

“…This can be observed, for instance, in distribution planning, where time-efficient goods dispatch has to be achieved by a limited number of fleet vehicles [32], in emergency transportation management, where an unsupervised evacuation can lead to fatal congestion episodes and thus drastically preclude its overall success [39], and in the telecommunication field, where cost-efficient use of the network for quick data packets transmission is crucial [6,29]. This novel combination of k-splittable and dynamic flows has rarely been addressed in the literature and only recently Melchiori and Sgalambro [40] provided a formal introduction of the Quickest Multicommodity k-splittable Flow Problem (QM CkSF P ). This strongly N P -hard problem explicitly accounts for a limited number (k) of paths to be allowed in flow routing in a dynamic network, combining the requirement of a quickest (dynamic) multicommodity flow with restrictions on the number of active paths on each distinct commodity.…”

“…the number of time steps required to accomplish the overall multicommodity routing, assuming that flow storage is not allowed. The three-index path-based M ILP formulation for the QM CkSF P as presented in the work of [40] follows, with the complete notation collected in the box.…”

“…Arc capacities at each point in time are stated by Constraints (6) and finally, bottleneck Constraints (7) force the amount of flow on a given path to be zero at every time instant if the path is not activated for flow transshipment. Proof of the strong N P -hardness of the QM CkSF P can be found in [40], where a reduction from the Minimum Cost kSFP is proposed.…”

“…The same approach has been successfully used in [40] for generating a pool of good and promising paths to be used throughout the matheuristic approach to obtain a good quality starting feasible solution. The columns of the initial RRM P are then accordingly populated by including all pathindependent y variables and those fractional variables x h pt associated with each identified path p and repeated for each discrete time step t ∈ T .…”