An improved ant colony optimization based algorithm for the capacitated arc routing problem Luís Santos c , João Coutinho-Rodrigues a,c, * , John R. Current b,c a Department of Civil Engineering, Faculty of Sciences and Technology, Polo II, University of Coimbra, 3030-788 Coimbra, Portugal b Department of Management Sciences, The Fisher College of Business, The Ohio State University, 632 Fisher Hall, 2100 Neil Ave., Columbus, OH 43210-1144, USA c Researcher at INESC-Coimbra, R. Antero Quental, 199, 3000-033 Coimbra, Portugal article info Article history: Received 31 May 2008 Received in revised form 16 July 2009 Accepted 19 July 2009 Keywords: Capacitated arc routing Ant colony optimization Metaheuristics abstract The capacitated arc routing problem is a well-studied problem in the Transportation/Logis- tics/OR literature. The problem consists of identifying the minimum cost routes required to service (e.g., pickup or deliver) demand located along the edges of a network. Unfortu- nately, the problem belongs to the set of NP-Hard problems; consequently, numerous heu- ristic and metaheuristic solution approaches have been developed to solve it. In this article, an ant colony optimization based metaheuristic is presented. Modifications are introduced for various components of the ant colony metaheuristics; specifically for those associated with the ‘‘initial population”, the ‘‘ant decision rule” and the ‘‘local search procedure”. The new metaheuristic was tested on seven standard test networks for the capacitated arc routing problem. The results demonstrate that the proposed approach performs extremely well vis-à-vis the state-of-the-art metaheuristics for the problem. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Vehicle routing is a frequent and costly activity for many public and private sector enterprises. In the United States for example, freight transportation costs account for approximately 6% of the GDP (MacroSys Research and Technology, 2005). Transportation costs are a consequence of road congestion (Sankaran and Wood, 2007), accidents, travel delays (Stei- metz, 2008), and energy consumption, among others. Obviously, distance travelled impacts all of these. Therefore, small improvements in routing efficiency can result in large cost reductions. As a consequence, the design of vehicle routes continues to be an important part of the transportation literature (e.g., Brandão and Eglese, 2008; Polacek et al., 2008; Figliozzi, 2007; Imai et al., 2007; Santos et al., 2007; Ouang, 2007; Francis and Smilowitz, 2006). A common routing problem involves the pickup or delivery of items located along the streets of a road network. If there is a capacity constraint on the vehicles, such problems may be modeled as a capacitated arc routing prob- lem (CARP). The CARP was first introduced by Golden and Wong (1981) and may be stated as follows. Let G =(N, E) be an undirected graph, with a node set N, an edge set E, and a set of required edges R # E. Each edge [i, j] of E has a nonnegative cost or length c ij and nonnegative demand or weight q ij . If q ij > 0, then [i, j] is referred to as a required edge. Node s represents a depot at which k identical vehicles of capacity w ðw P max q ij Þ are based. The number of vehicles (k) is a decision variable. The CARP consists of designing a set of vehicle routes, such that: 0191-2615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.trb.2009.07.004 * Corresponding author. Address: Department of Civil Engineering, Faculty of Sciences and Technology, Polo II, University of Coimbra, 3030-788 Coimbra, Portugal. Tel.: +351 239797145; fax: +351 239797123. E-mail addresses: lsantos@dec.uc.pt (L. Santos), coutinho@dec.uc.pt (J. Coutinho-Rodrigues), current.1@osu.edu (J.R. Current). Transportation Research Part B 44 (2010) 246–266 Contents lists available at ScienceDirect Transportation Research Part B journal homepage: www.elsevier.com/locate/trb