Dynamic flow problems with bounded number of paths: models, algorithms and applications

In this thesis we investigate the concept of path limitations, expressed as the maximum number of paths that can be activated by each distinct commodity, hence secured by the so-called k-splittable flows, within the dynamic framework of Network Flow theory. In particular, we formally introduce the first flows over time problem explicitly encompassing path number restrictions, namely the Quickest Multicommodity k-Splittable Flow Problem (QMCkSFP). The problem asks for routing and scheduling each commodity demand through at most k different paths while minimizing the makespan of the overall process, i.e. the time required to accomplish the transshipment operations. We model the QMCkSFP in a dynamic digraph with time-independent structural features, considering flow traveling over a discretized time horizon, and motivated by practical applications, we force the transshipment operations to be performed only through elementary/loopless paths and without flow storage at intermediate nodes. In this setting, we provide a path-flow Mixed-Integer Linear Programming formulation for the problem that requires as an input the complete set of available paths for each distinct commodity. In detail, fractional variables are employed to track release of flows at each point in time for each commodity and path; a first set of binary variables allows for the identification and linear minimization of the makespan by recording arrival times at destination of each routed commodity demand, while a second set of binaries imposes k-splittable path limitations to each commodity. The formulation presents a number of columns which grows exponentially with the instance size, due to the presence of variables associated with the number of paths for each commodity multiplied by the number of instants in the discretized time horizon. The computational complexity of the introduced problem is formally investigated, proving its belongs to the class of strongly NP-hard problems by a reduction from the static Minimum Cost k-Splittable Flow Problem. When it comes to solution approaches, the first algorithm designed for the ad-hoc resolution of the QMCkSFP is an improvement matheuristic hybridizing a Very Large-scale Neighborhood Search (VLNS) with a mathematical programming strategy in its exploration routine. The method employs a state-of-the-art ranking procedure to generate a sufficiently large list of good and promising candidate paths to be used throughout its implementation. In particular, the initial solution is constructed by selecting for each commodity the k top-ranked paths from the generated list w.r.t. their transmission time, i.e. the time required to transship the entire commodity demand. The QMCkSFP path-flow formulation is restricted to these initial paths and solved to optimality via a MIP solver. Then, at each improvement iteration of the matheuristic, a different neighborhood is constructed by heuristically identifying for each commodity a collection of candidate paths from the list and it is explored to optimality by solving the related restricted path-flow formulation. The improvement search proceeds through a Variable Neighborhood Descent scheme by increasing the cardinality of the path collections to be identified till a time limit is reached. A thorough computational experience is conducted on the proposed model and matheuristic approach, starting by the resolution to optimality of a set of very reduced-size instances, making use of the path-based formulation fed with a complete enumeration of all the available paths. A tuning process is then carried out on the matheuristic's parameters for the identification of the optimal setting to be adopted in the evaluation of its performance. An exhaustive proof-of-concept for the correctness and effectiveness of the matheuristic resolution strategy is then conducted: to this aim, a comparison is performed on a set of small to medium-sized instances against the Quickest Multicommodity Flow Problem, i.e. the free-flow relaxation of the problem with no upper bounds on the number of usable paths. Finally, we test performances of our matheuristic against those of a Randomized Rounding-based algorithm on two different benchmark test sets, the first one collecting networks of large size w.r.t. number of nodes and arcs, and the second networks with an extremely high number of commodities. Our second algorithmic contribution is represented by an exact strategy for solving the QMCkSFP at the optimum. The developed technique falls within the Branch and Price paradigm and is based on original relaxation, pricing and branching procedures. In particular, the so-called Restricted Relaxed Master Problem (RRMP) is obtained by linearizing time-arrival binary variables and by substituting those related to k-splittable path restrictions from the path-based formulation of the problem. The same ranking algorithm employed in the matheuristic initialization phase is here adopted for the construction of the path-related columns to be considered at the root node of the Branch and Bound decisional tree. In particular, the best k paths w.r.t. the transmission time are generated for each commodity and all related fractional variables are included as initial columns. The Column Generation procedure is then applied to identify, through the resolution of the so-called pricing problem, new entering variables defined as pair (path,departure time), or to certify the optimality of the nodes' RRMPs. This is done on a slightly modified time-expansion of the original dynamic digraph where the time dimension is decoupled from the network structure to give rise to a novel equivalent static digraph. Two original branching rules are designed for restoring feasibility. A first one, performed whenever k-splittable flow constraints are violated, forces the usage of some paths and simultaneously forbids the activation of others over the discretized time horizon. Working on path-based variables, the generated branching cuts are included in RRMPs of the child nodes. A second rule implements a refined version of the standard 0-1 branching on fractional variables by selecting the candidate time-arrival variable at the highest time instant. In order to account for k-splittable-related branching cuts, the pricing problem is modeled as a Shortest Path Problem with Forbidden Paths where additional node-set resources are introduced to secure the generation of loopless paths. A tailored labelling algorithm based on dynamic programming is designed and employed to solve the pricing problem to optimality. All variables modeling the utilization of the optimal identified path at each point in time are included as new columns. The computational experience conducted on the designed Branch and Price algorithm begins by studying its correctness and computational performances when applied on a collection of small to medium-sized instances. As the final experiment, we employ the Branch and Price approach as a tool to assess the quality of the solutions obtained by the developed VLNS-based matheuristic. To this aim, we consider the same testbed of small to medium-sized instances used in the matheuristic experiments with our Branch and Price, feeding it with the matheuristic results as initial incumbent solutions, then measuring and comparing the residual optimality GAP achieved after the execution of the exact algorithm. The overall toolbox developed in this research lays the foundations for a number of refined and additional novel contributions within the dynamic k-splittable flow setting, whose preliminary guidelines are discussed in the closing chapter of this thesis.