A guided local search heuristic for the capacitated arc routing problem

Abstract

This paper presents a new local search algorithm for the capacitated arc routing problem (CARP). The procedure uses single vehicle moves and moves that operate on two routes, both derived from a node routing context but properly adapted to work well for arc routing problems. We combine the algorithm with the meta-heuristic guided local search and further use the mechanisms of neighbor lists and edge marking to improve the solution quality and to save computation time. Experiments on standard benchmark problems from the literature show that our algorithm outperforms the existing heuristics for the CARP. On a set of new test problems, the local search approach consistently produces high quality solutions and often detects an optimal solution within limited computation time.

Introduction

In the capacitated arc routing problem (CARP) (Golden and Wong, 1981), we are given an undirected, connected graph G=(V,E) with vertex set V and edge set E. Every edge e∈E has a non-negative cost or length c_e and a non-negative demand for service q_e. The edges with positive demand make up the subset of the required edges E_R. Given a vehicle capacity Q, the CARP consists of finding a set of vehicle routes of minimum cost, such that every required edge is serviced by exactly one vehicle, each route starts and ends at a pre-specified vertex v₀ (the depot) and the total demand serviced by a route does not exceed the vehicle capacity Q. Golden and Wong (1981) show that the CARP is NP-hard. In fact, they prove that even finding a solution with a cost (counting only the deadheading) less than 5/3 times the optimal cost, is NP-hard.

Several, mostly heuristic, solution procedures for the CARP are reported in the literature. Surveys on these algorithms and the various applications of the CARP, as well as on other related arc routing problems, can be found in Assad and Golden (1995), Eiselt et al. (1995) and the recent book of Dror (2000). In this paper, we present a new local search algorithm for the CARP. In contrast with the extensive literature on local search approaches for node routing problems, only few local search heuristics are available for the CARP. In the context of winter gritting (salt spreading on roads in wintertime), Eglese (1994) presents a solution procedure for a multi-depot CARP with additional side constraints. In a first stage, an Eulerian graph is partitioned into small cycles and an initial solution is constructed by aggregating cycles into routes using a greedy saving heuristic. Next, within a simulated annealing framework, cycles are exchanged between the routes to improve the solution. The use of cycles to build routes in the CARP, is criticized in Eglese and Li (1994). Obviously, when the total demand serviced in the cycles is rather large compared with the truck capacity, it will be more difficult to construct good routings by aggregating cycles instead of individual edges. Another approach using cycles to build routings is given in Amberg et al. (2000). More efficient algorithms for the CARP include the tabu search procedures of Greistorfer, 1994, Greistorfer, 2000 and of Hertz et al. (2000); the latter, called CARPET, is one of the best procedures developed so far. The success of CARPET relies on a number of routines to perform edge removal, re-insertion and re-optimization operations. Some of the routines were originally developed to solve the undirected rural postman problem (URPP), Hertz et al. (1999). CARPET uses a neighborhood structure similar to that of TABUROUTE, a tabu search heuristic developed for the vehicle routing problem (VRP) by Gendreau et al. (1994). The subroutines in CARPET are however rather intricate. Recently, Muyldermans et al. (2001a) developed k-opt local search algorithms for (uncapacitated) arc and general routing problems. These procedures are very similar to the well-known k-opt procedures for the traveling salesman problem (TSP) (Croes, 1958; Lin, 1965), and are much simpler than the approaches developed by Hertz et al. (1999). In this paper, we extend the procedures of Muyldermans et al. (2001a) to deal with CARP. New moves (relocate, exchange and cross) operating between two routes, are introduced, as was done for the VRP in Savelsbergh (1988). We present a local search algorithm for the CARP embedded within a guided local search procedure (GLS), a meta-heuristic successfully applied to several combinatorial optimization problems among which: the TSP (Voudouris and Tsang, 1999), the VRP (Kilby et al., 1999) and the URPP (Muyldermans et al., 2001a). Experiments on standard benchmark problems from the literature and newly developed instances indicate that the new algorithm is capable of finding optimal or near-optimal solutions within a limited computation time.

The remainder of the paper is organized as follows. Section 2 describes the solution representation, the neighborhood moves and some implementation issues, and finally gives a high-level description of the local search algorithm for the CARP. Computational experiments are presented in Section 3 and the conclusions follow in Section 4.