In the literature on Network Optimization, k-splittable flows were introduced to enhance modeling accuracy in cases where an upper bound on the number of supporting paths for each commodity needs to be imposed, thus extending the suitability of network flow tools for an increased number of practical applications. Such modeling feature has recently been extended to dynamic flows with the introduction of the novel strongly NP-hard Quickest Multicommodity k-splittable Flow Problem (QMCkFP). Such a flows over time problem asks for routing and scheduling of each commodity demand through at most k different paths in a dynamic network with arc capacities per time step, while minimizing the time required by the overall process. In this work, we propose the first exact algorithm for solving the QMCkSFP. The developed technique is a Branch and Price algorithm based on original relaxation, pricing and branching procedures. Linearization and variable substitution are used to obtain the relaxation problem from the path-based formulation of the QMCkSFP. The pricing problem is modeled as a Shortest Path Problem with Forbidden Paths with additional node-set resources on a time expansion of the original digraph and is solved via a tailored dynamic programming algorithm. Two branching rules are designed for restoring feasibility whenever k-splittable or binary variable domain constraints are violated. The results of an extensive batch of computational experiments conducted on small to medium-size reference instances are presented, showing a highly satisfactory performance of the proposed algorithm. The paper concludes with a discussion on further lines of research.

In transportation networks with limited capacities and travel times on the arcs, a class of problems attracting a growing scientific interest is represented by the optimal routing and scheduling of given amounts of flow to be transshipped from the origin points to the specific destinations in minimum time. Such problems are of particular concern to emergency transportation where evacuation plans seek to minimize the time evacuees need to clear the affected area and reach the safe zones. Flows over time approaches are among the most suitable mathematical tools to provide a modelling representation of these problems from a macroscopic point of view. Among them, the Quickest Path Problem (QPP), requires an origin-destination flow to be routed on a single path while taking into account inflow limits on the arcs and minimizing the makespan, namely, the time instant when the last unit of flow reaches its destination. In the context of emergency transport, the QPP represents a relevant modelling tool, since its solutions are based on unsplittable dynamic flows that can support the development of evacuation plans which are very easy to be correctly implemented, assigning one single evacuation path to a whole population. This way it is possible to prevent interferences, turbulence, and congestions that may affect the transportation process, worsening the overall clearing time. Nevertheless, the current state-of-the-art presents a lack of studies on multicommodity generalizations of the QPP, where network flows refer to various populations, possibly with different origins and destinations. In this paper we provide a contribution to fill this gap, by considering the Multicommodity Quickest Path Problem (MCQPP), where multiple commodities, each with its own origin, destination and demand, must be routed on a capacitated network with travel times on the arcs, while minimizing the overall makespan and allowing the flow associated to each commodity to be routed on a single path. For this optimization problem, we provide the first mathematical formulation in the scientific literature, based on mixed integer programming and encompassing specific features aimed at empowering the suitability of the arising solutions in real emergency transportation plans. A computational experience performed on a set of benchmark instances is then presented to provide a proof-of-concept for our original model and to evaluate the quality and suitability of the provided solutions together with the required computational effort. Most of the instances are solved at the optimum by a commercial MIP solver, fed with a lower bound deriving from the optimal makespan of a splittable-flow relaxation of the MCQPP.