On matchings, T‐joins, and arc routing in road networks (original) (raw)

Fast upper and lower bounds for a large‐scale real‐world arc routing problem

Networks, 2022

Arc Routing Problems (ARPs) are a special kind of Vehicle Routing Problem (VRP), in which the demands are located on edges or arcs, instead of nodes. There is a huge literature on ARPs, and a variety of exact and heuristic algorithms are available. Recently, however, we encountered some real-life ARPs with over ten thousand roads, which is much larger than those usually considered in the literature. For these problems, we develop fast upper-and lower-bounding procedures. We also present extensive computational results.

The Capacitated Arc Routing Problem: Lower bounds

Networks, 1992

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.

Node Duplication Lower Bounds for the Capacitated Arc Routing Problem

Journal of The Operations Research Society of Japan, 1992

It is well-knawn that the tight lower bounds determine the effectiveness of the branch and bound method fbr the NP-hard problems, In this paper, we present a new lower bounding procedure fbr the capacitated arc routing problem (CARP), one of the arc routing preblems, They give the tight lower bounds aRd it is easy to develop an exact algorithm using their network structures. 1 Introduction The routing problems have been studied by many researchers for more decades. The arc routing problem (ARP) is one of the routing problems which focuses on arcs in a network. This problem includes the well-known "Chinese postman problem (CPP)", CPP is a problem of covering all arcs in a network while minimizing the total distance cost traveled, CPP was presented by Meiko-Kwan [9] and solved polynomially by Edmonds afid Johnson [5] based on the minimum-cost perfect matching problem (MCPM) in the general graph. CPP is said to be attractive, because it is an exceptionally well-solved problem in ARP alld has a number of applications like mai1 deliveiy [5]. On the other hand, since CPP is a simple structured pioblem, there are many problems in ARP to which CPP algoTithmis directly inapplicable, For example, the routing of street sweepers, snow plows [41, household refuse collection vehic}es, the spra(ying of roads with saltgrit to prevent ice formation, the inspection of electric power lines [15] gas or oil pipelines for faults and so on. In this paper, we consider one of these problems, so called the capacitated arc routing probgem (CARP). As is mentioned in [71, CARP includes such related problems as the traveling salesman problem (TSP), the Chinese postman prob}em (CPP) [2,5], the rural postman problem (RPP) [3], the capacitated Chinese postman problem (CCPP), the vehicle routiRg problem (VRP) [6,10] and the general routing problem (GRP) [11,i2]. Golden and Wong [7] showed that CARP is a NP-hard problem. Thus recent reseamchers have focused their effbrt on deve}oping and testing heuristics. Also the lower bounding procedures [1,7,13] have been developed to estimate the eficiency of the heuristics. These lower bounds can be ebtained eficiently by solving MCPM on simple structured networks. However, since they relax the capacity constraint, i.e., one of the constraints in CARP, we point out that bounds are not tight. Moreover, it is hard to construct an exact solution through a bTallch and bound method using their network structures.

Factors that impact solution run times of arc-based formulations of the Vehicle Routing Problem ∗

2005

It is well known that the Vehicle Routing Problem (VRP) becomes more difficult to solve as the problem size increases. However little is known about what makes a VRP difficult or easy to solve for problems of the same size. In this paper we investigate the effect of the formulation and data parameters on the efficiency with which we can obtain exact solutions to the VRP with a general IP solver. Our results show that solution run times for arc-based formulations with exponentially many constraints are mostly insensitive to problem parameters, whereas polynomial arc-based formulations, which can solve larger problems because of the smaller memory requirement, are sensitive to problem parameters. For instance, we observe that solution times for polynomial formulations significantly decrease for larger capacities and number of vehicles. ∗Research supported by NSF under grant CMS-0409887 †Corresponding author

Vehicle routing problems with road-network information: State of the art

Networks, 2018

Vehicle routing problems have drawn researchers' attention for more than 50 years. Most approaches found in the literature address these problems using the so-called customer-based graph, a complete graph representing the road network, where a node is introduced for every point of interest (eg, customers, depot...) and an arc represents the best path between two points. In many situations, this representation induces negative effects on the solution quality or efficiency. A growing number of works in the literature investigate these issues and propose modeling taking account of more detailed information from the road-network. In this article, we review these works and classify them with respect to the type of negative effects provoked by the customer-based graph.

Lower bounds for the mixed capacitated arc routing problem

Computers & Operations Research, 2010

Capacitated Arc Routing Problems (CARP) arise in distribution or collecting problems where activities are performed by vehicles, with limited capacity, and are continuously distributed along some pre-defined links of a network. The CARP is defined either as an undirected problem or as a directed problem depending on whether the required links are undirected or directed. The Mixed Capacitated Arc Routing Problem (MCARP) models a more realistic scenario since it considers directed as well as undirected required links in the associated network. We present a compact flow based model for the MCARP. Due to its large number of variables and constraints, we have created an aggregated version of the original model. Although this model is no longer valid, we show that it provides the same linear programming bound than the original model. Different sets of valid inequalities are also derived. The quality of the models is tested on benchmark instances with quite promising results.

Column Generation Bounds for the Capacitated Arc Routing Problem

XLII SBPO, Bento …, 2010

Arc routing problems are among the most challenging combinatorial optimization problems. We tackle the Capacitated Arc Routing Problem where demands are spread over a subset of the edges of a given graph, called the required edge set. Costs for traversing edges, demands on the required ones and the capacity of the available identical vehicles at a vertex depot are given. Routes that collect all the demands at minimum cost are sought. This work presents new lower bounds for this problem. They are based on formulations on variables representing routes. These are solved by column generation. Valid inequalities from formulations on arc variables are added to the route based formulations. Issues regarding the use of elementary and nonelementary routes are explored. Extensive experiments are presented.

Exact methods based on node routing formulations for arc routing problems

, where each required CARP arc was mapped onto a triplet of CVRP nodes. In our case, only 2 CVRP nodes are needed for every CARP required arc. The transformed instances have a structure and a dimension which make most CARP benchmarks solvable by state of the art CVRP techniques. We thus propose a general purpose transformation of arc into node routing problems and new results on lower bounds and exact methods for CARP instances.

Branch-and-bound algorithm for an arc routing problem

The Open Capacitated Arc Routing Problem (OCARP) is an NP-hard combinatorial optimization problem where, given an undirected graph, the objective is to find a minimum cost set of tours that services a subset of edges with positive demand under capacity constraints. This problem is related to the Capacitated Arc Routing Problem (CARP) but differs from it since OCARP does not consider a depot, and tours are not constrained to form cycles. Three lower bounds are proposed to the OCARP, one of them uses a subgradient method to solve a Lagrangian relaxation. These lower bounds are integrated within a branch-and-bound framework to conceive the first OCARP exact algorithm. The branch-and-bound algorithm is started with high-quality upper bounds obtained with a sucessful GRASP with evolutionary path-relinking, originally developed to solve the CARP. Computational tests compared the proposed branchand-bound with a commercial state-of-the-art ILP solver. The results show that the branch-andbound outperformed CPLEX in both overall average deviation from lower bounds and number of best lower bounds.

Arc Based Integer Programming Formulations for the Distance Constrained Vehicle Routing Problem

The literature on the Distance Constrained Vehicle Routing Problem (DVRP), which is an important extension of the classical Vehicle Routing Problem (VRP), is scarce. In this paper, two new arc based integer programming formulations, with polynomial size are presented. It is shown that, second formulation, designated as strengthened model produces stronger lower bound than the first and existing arc based formulations. This formulation is compared with the recently proposed node based formulation in terms of linear programming relaxations and CPU times computationally. It is observed that, the second model is superior with respect to both performance indicators. It is also shown that the strengthened model can be easily adapted to the cases where additional restrictions imposed on vehicle's route length.