José-Manuel Belenguer

A Branch-and-Cut method for the Capacitated Location-Routing Problem

Most of the time in a distribution system, depot location and vehicle routing are interdependent and recent researches have shown that the overall system cost may be excessive if routing decisions are ignored when locating depots. The location routing problem (LRP) overcomes this drawback by simultaneously tackling location and routing decisions. This paper presents two formulations of the location-routing problem with capacities on routes and depots and proposes an exact method based on a branch and cut approach using these formulations. The method is evaluated on two sets of randomly generated instances, and compared to heuristics and another lower bound

A cutting plane algorithm for the capacitated arc routing problem

The Capacitated Arc Routing Problem (CARP) consists of finding a set of minimum cost routes that service all the positive-demand edges of a given graph, subject to capacity restrictions.In this paper, we introduce some new valid inequalities for the CARP. We have designed and implemented a cutting plane algorithm for this problem based on these new inequalities and some other which were already known. Several identification algorithms have been developed for all these valid inequalities. This cutting plane algorithm has been applied to three sets of instances taken from the literature as well as to a new set of instances with real data, and the resulting lower bound was optimal in 47 out of the 87 instances tested. Furthermore, for all the instances tested, our algorithm outperformed all the existing lower bounding procedures for the CARP.

Separating capacity constraints in the CVRP using tabu search

Abstract Branch and Cut methods have shown to be very successful in the resolution of some hard combinatorial optimization problems. The success has been remarkable for the Symmetric Traveling Salesman Problem (TSP). The crucial part in the method is the cutting plane algorithm: the algorithm that looks for valid inequalities that cut off the current nonfeasible linear program (LP) solution. In turn this part relies on a good knowledge of the corresponding polyhedron and our ability to design algorithms that can identify violated valid inequalities. This paper deals with the separation of the capacity constraints for the Capacitated Vehicle Routing Problem (CVRP). Three algorithms are presented: a constructive algorithm, a randomized greedy algorithm and a very simple tabu search procedure. As far as we know this is the first time a metaheuristic is used in a separation procedure. The aim of this paper is to present this application. No advanced tabu technique is used. We report computational results with these heuristics on difficult instances taken from the literature as well as on some randomly generated instances. These algorithms were used in a Branch and Cut procedure that successfully solved to optimality large CVRP instances.

A Lower Bound for the Split Delivery Vehicle Routing Problem

In this paper we consider the Split Delivery Vehicle Routing Problem (SDVRP), a relaxation of the known Capacitated Vehicle Routing Problem (CVRP) in which the demand of any client can be serviced by more than one vehicle. We define a feasible solution of this problem, and we show that the convex hull of the associated incidence vectors is a polyhedron ( PSDVRP), whose dimension depends on whether a vehicle visiting a client must service, or not, at least one unit of the client demand. From a partial and linear description ofPSDVRP and a new family of valid inequalities, we develop a lower bound whose quality is exhibited in the computational results provided, which include the optimal resolution of some known instances--one of them with 50 clients. This instance is, as far as we know, the biggest one solved so far.

Lower and upper bounds for the mixed capacitated arc routing problem

This paper presents a linear formulation, valid inequalities, and a lower bounding procedure for the mixed capacitated arc routing problem (MCARP). Moreover, three constructive heuristics and a memetic algorithm are described. Lower and upper bounds have been compared on two sets of randomly generated instances. Computational results show that the average gaps between lower and upper bounds are 0.51% and 0.33%, respectively.

The Capacitated Arc Routing Problem: Valid Inequalities and Facets

In this paper we study the polyhedron associated with the Capacitated Arc Routing Problem (CARP) where a maximum number K of vehicles is available. We show that a subset of the facets of the CARP polyhedron depends only on the demands of the required edges and they can be derived from the study of the Generalized Assignment Problem (GAP). The conditions for a larger class of valid inequalities to define facets of the CARP polyhedron still depend on the properties of the GAP polyhedron. We introduce the special case of the CARP where all the required edges have unit demand (CARPUD) to avoid the number problem represented by the GAP. This allows us to make a polyhedral study in which the conditions for the inequalities to be facet inducing are easily verifiable. We give necessary and sufficient conditions for a variety of inequalities, which are valid for CARP, to be facet inducing for CARPUD.The resulting partial description of the polyhedron has been used to develop a cutting plane algorithm for the Capacitated Arc Routing Problem. The lower bound provided by this algorithm outperformed all the existing lower bounds for the CARP on a set of 34 instances taken from the literature.

Split-Delivery Capacitated Arc-Routing Problem: Lower Bound and Metaheuristic

This paper proposes lower and upper bounds for the split-delivery capacitated arc-routing problem (SDCARP), a variant of the capacitated arc-routing problem in which an edge can be serviced by several vehicles. Recent papers on related problems in node routing have shown that this policy can bring significant savings. It is also more realistic in applications such as urban refuse collection, where a vehicle can become full in the middle of a street segment. This work presents the first lower bound for the SDCARP, computed with a cutting plane algorithm and an evolutionary local search reinforced by a multistart procedure and a variable neighborhood descent. Tests on 126 instances show that the new metaheuristic outperforms on average a published memetic algorithm; achieves small deviations to the lower bound; and finds 44 optima, including 10 new ones.

A Branch and Cut method for the Capacitated Location-Routing Problem

Most of the time in a distribution system, depot location and vehicle routing are interdependent and recent researches have shown that the overall system cost may be excessive if routing decisions are ignored when locating depots. The Location Routing Problem (LRP) overcomes this drawback by simultaneously tackling location and routing decisions. This paper presents two formulations of the Location-Routing Problem with capacities on routes and depots and proposes an exact method based on a branch and cut approach using these formulations. The method is evaluated on two sets of randomly generated instances, and compared to heuristics and another lower bound.

A branch-and-cut algorithm for the single truck and trailer routing problem with satellite depots

In the single truck and trailer routing problem with satellite depots (STTRPSD), a truck with a detachable trailer based at a main depot must serve the demand of a set of customers accessible only by truck. Therefore, before serving the customers, it is necessary to detach the trailer in an appropriate parking place (called either a satellite depot or a trailer point) and transfer goods between the truck and the trailer. This problem has applications in milk collection for farms that cannot be reached using large vehicles. In this work we present an integer programming formulation of the STTRPSD. This formulation is tightened with several families of valid inequalities for which we have developed different (exact and heuristic) separation procedures. Using these elements, we have implemented a branch-and-cut algorithm for the solution of the STTRPSD. A computational experiment with published instances shows that the proposed branch-and-cut algorithm consistently solves problems with up to 50 customers and 10 satellite depots, and it has also been able to solve instances with up to 20 satellite depots and 100 clustered customers.