Enrique Benavent

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.

The Capacitated Arc Routing Problem: Lower bounds

In this paper, we consider the Capacitated Arc Routing Problem (CARP), in which a fleet of vehicles, based on a specified vertex (the depot) and with a known capacity Q, must service a subset of the edges of a graph, with minimum total cost and such that the load assigned to each vehicle does not exceed its capacity. New lower bounds are developed for this problem, producing at least as good results as the already existing ones. Three of the proposed lower bounds are obtained from the resolution of a minimum cost perfect matching problem. The fourth one takes into account the vehicle capacity and is computed using a dynamic programming algorithm. Computational results, in which these bounds are compared on a set of test problems, are included.

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.

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.

An Exact Algorithm for the Quadratic Assignment Problem on a Tree

The Tree QAP is a special case of the Quadratic Assignment Problem QAP where the nonzero flows form a tree. No condition is required for the distance matrix. This problem is NP-complete and is also a generalization of the Traveling Salesman Problem. In this paper, we present a branch-and-bound algorithm for the exact solution of the Tree QAP based on an integer programming formulation of the problem. The bounds are computed using a Lagrangian relaxation of this formulation. To solve the relaxed problem, we present a Dynamic Programming algorithm which is polynomially bounded. The obtained lower bound is very sharp and equals the optimum in many cases. This fact allows us to employ a reduction method to decrease the number of variables and leads to search-trees with a small number of nodes compared to those usually encountered in problems of this type. Computational results are given for problems with size up to 25.

Linear Programming Based Methods for Solving Arc Routing Problems

From the pioneering works of Dantzig, Edmonds and others, polyhedral (i.e. linear programming based) methods have been successfully applied to the resolution of many combinatorial optimization problems. See Junger, Reinelt & Rinaldi (1995) for an excellent survey on this topic. Roughly speaking, the method consists of trying to formulate the problem as a Linear Program and using the existing powerful methods of Linear Programming to solve it.

The Directed Rural Postman Problem with Turn Penalties

In this paper, we introduce a generalization of the directed rural postman problem including new features that can be encountered in practice when routes have to be operated on a street network: some turns are forbidden and other turns are allowed but with some penalties. This new problem is called the directed rural postman problem with turn penalties (DRPP-TP); we present some complexity results and three heuristics for the DRPP-TP: two of them are constructive, whereas the third one is an improvement heuristic. We also present a transformation of the DRPP-TP into an asymmetric traveling salesman problem (ATSP) that allows us to solve the problem exactly using an existing ATSP code. Computational results on a set of instances with up to 180 nodes and 666 arcs, are given.

Lower bounds and heuristics for the Windy Rural Postman Problem

In this paper we present several heuristic algorithms and a cutting-plane algorithm for the Windy Rural Postman Problem. This problem contains several important Arc Routing Problems as special cases and has very interesting real-life applications. Extensive computational experiments over different sets of instances are also presented.