Discrete OptimizationA guided local search heuristic for the capacitated arc routing problem
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 ce and a non-negative demand for service qe. The edges with positive demand make up the subset of the required edges ER. 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.
Section snippets
Solution representation
As for the URPP in Muyldermans et al. (2001a), we construct a CARP solution in a directed graph D(V,A) instead of in the original undirected graph G(V,E). The reason to do so is that the direction of travel along an arc is uniquely defined while this is not the case for edges. The graph D has the same vertex set V as G, the arc set A contains for every edge e=(vi,vj)∈E, the arcs a=(vi,vj) and aRev=(vj,vi), both with cost ce, and demand qe. The arcs a and aRev are called representatives of edge e
The test problems and calculation of the lower bounds
We tested our algorithm on CARP instances from the literature as well as on new instances. The test problems from the literature are the following:
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23 instances of Golden et al. (1983), named gdb1–gdb23 in Table 2,
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6 instances of Kiushi et al. (1995), named kshs1–kshs6 in Table 2,
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34 instances of Benavent et al. (1992), denoted 1.A–10.D in Table 2.
The data of these problems can be found in Benavent (1997). The individual instance characteristics are shown in Table 2: |V|, |ER|, the continuous
Conclusions
In this paper we present a new local search algorithm for the CARP. The procedure uses, apart from the single vehicle moves introduced in Muyldermans et al. (2001a), new moves (relocate, exchange and cross), that operate between two routes and are commonly used in a node routing context (Savelsbergh, 1988). We combine the algorithm with the meta-heuristic GLS and further use the mechanisms of neighbor lists and edge marking to save computation time. Our approach is no more complex than the
References (30)
- et al.
Multiple center capacitated arc routing problems: A Tabu search algorithm using capacitated trees
European Journal of Operational Research
(2000) - et al.
Arc routing methods and applications
Routing winter gritting vehicles
Discrete Applied Mathematics
(1994)- et al.
Computational experiments with algorithms for a class of routing problems
Computers and Operations Research
(1983) - et al.
Guided local search and its application to the traveling salesman problem
European Journal of Operational Research
(1999) - et al.
The capacitated arc routing problem: Valid inequalities and facets
Computational Optimization and Applications
(1998) - Belenguer, J.M., Benavent, E., 2000. A Cutting Plane Algorithm for the Capacitated Arc Routing Problem. Working paper,...
- Benavent, E., 1997....
- et al.
The capacitated arc routing problem. Lower bounds
Networks
(1992) - et al.
Linear programming based methods for solving arc routing problems
Fast algorithms for geometric traveling salesman problems
ORSA Journal on Computing
A method for solving traveling salesman problems
Operations Research
Arc routing problems, Part II: The rural postman problem
Operations Research
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