Ángel Corberán

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.

Heuristic solutions to the problem of routing school buses with multiple objectives

In this paper we address the problem of routing school buses in a rural area. We approach this problem with a node routing model with multiple objectives that arise from conflicting viewpoints. From the point of view of cost, it is desirable to minimise the number of buses used to transport students from their homes to school and back. From the point of view of service, it is desirable to minimise the time that a given student spends en route. The current literature deals primarily with single-objective problems and the models with multiple objectives typically employ a weighted function to combine the objectives into a single one. We develop a solution procedure that considers each objective separately and search for a set of efficient solutions instead of a single optimum. Our solution procedure is based on constructing, improving and then combining solutions within the framework of the evolutionary approach known as scatter search. Experimental testing with real data is used to assess the merit of our proposed procedure.

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 polyhedral approach to the Rural Postman Problem

Abstract In this paper we study the polyhedron associated with the Rural Postman Problem (RPP). Because the RPP is NP-hard, we cannot expect to find a complete description of the rural postman polyhedron of a general graph, but a partial knowledge of such a description frequently proves to be useful for both theoretical and computational purposes. We have tried to characterize the facial structure of this unbounded full-dimensional polyhedron. Sets of valid inequalities inducing facets have been studied as well as their use in a cutting-plane algorithm. The application of this algorithm to a set of RPP instances taken from the literature and two instances of larger size taken from a real world graph is described. All these instances were solved to optimality.

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.

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.

The Rural Postman Problem on mixed graphs with turn penalties

In this paper we deal with a problem which generalizes the Rural Postman Problem defined on a mixed graph (MRPP). The generalization consists of associating a non-negative penalty to every turn as well as considering the existence of forbidden turns. This new problem fits real-world situations more closely than other simpler problems. A solution tour must traverse all the requiring service arcs and edges of the graph while not making forbidden turns. Its total cost will be the sum of the costs of the traversed arcs and edges together with the penalties associated with the turns done. The Mixed Rural Postman Problem with Turn Penalties (MRPPTP) consists of finding such a tour with a total minimum cost. We show that the new problem is NP-hard, even in some particular cases. In order to solve the MRPPTP, a polynomial transformation of the problem into the Asymmetric Traveling Salesman Problem (ATSP) can be done and then apply heuristics and exact methods for the ATSP. This transformation is summarized here and a specific heuristic algorithm, based on two recent heuristics for the MRPP, is also presented. Extensive computational experiments with more than 250 instances establish the effectiveness of our procedures.

Polyhedral results for a vehicle routing problem

Abstract The Vehicle Routing Problem is a well known, and hard, combinatorial problem, whose polyhedral structure has deserved little attention. In this paper we consider the particular case in which all the demands are equal (since in the general case the associated polytope may be empty). From a known formulation of the problem we obtain the dimension of the corresponding polytope and we study the facetial properties of every inequality in it.

Arc Routing: Problems, Methods, and Applications

This book provides a thorough and up-to-date discussion of arc routing by world-renowned researchers. Organized by problem type, the book offers a rigorous treatment of complexity issues, models, algorithms, and applications. Arc Routing: Problems, Methods, and Applications opens with a historical perspective of the field and is followed by three sections that cover complexity and the Chinese Postman and the Rural Postman problems; the Capacitated Arc Routing Problem and routing problems with min-max and profit maximization objectives; and important applications, including meter reading, snow removal, and waste collection. Audience: This book will be of interest to practitioners, researchers, and graduate students in operations research, engineering, logistics, mathematics, and business.

Heuristics for the mixed rural postman problem

Abstract The Rural Postman Problem on a mixed graph (MRPP) consists of finding a minimum cost tour which traverses, at least once, the arcs and edges of a given subset of the arcs and edges of the graph. This problem is known to be NP-hard. This paper presents two heuristic approaches to solve it. An approximate algorithm based on the resolution of some flow and matching problems and a tabu search implementation is presented. The tabu search algorithm seeks high-quality tours by means of a switching mechanism in an intensification phase and two levels of diversification. Computational results are presented to assess the merits of the method. Scope and purpose Routing Problems arise in several areas of distribution management and logistics and their practical significance has been well documented in the literature. Arc Routing Problems are routing problems where a single vehicle or a fleet of vehicles must service the arcs of a transportation network. Such problems occur in a variety of practical contexts, such as garbage collection, mail delivery, snow removal and school bus routing. In this paper we focus on the Mixed Rural Postman Problem (MRPP), an NP-hard problem that includes, as special cases, the well-known Chinese Postman Problem and the Rural Postman Problem. The MRPP fits real-world situations more closely than other simpler problems. In this paper we present heuristic procedures for the MRPP, where the effectiveness of the procedures is measured by the combination of solution quality and speed.