Discrete Optimization A parallel multi-neighborhood cooperative tabu search for capacitated vehicle routing problems Jianyong Jin a , Teodor Gabriel Crainic b,⇑ , Arne Løkketangen a a Molde University College, Specialized University in Logistics, N-6411, Molde, Norway b School of Management, UQAM & Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation, Montréal, QC, Canada article info Article history: Received 7 December 2010 Accepted 2 May 2012 Available online 29 May 2012 Keywords: Routing Parallel metaheuristics Multi-neighborhood Cooperative search Solution pool abstract This paper presents a parallel tabu search algorithm that utilizes several different neighborhood struc- tures for solving the capacitated vehicle routing problem. Single neighborhood or neighborhood combi- nations are encapsulated in tabu search threads and they cooperate through a solution pool for the purpose of exploiting their joint power. The computational experiments on 32 large scale benchmark instances show that the proposed method is highly effective and competitive, providing new best solu- tions to four instances while the average deviation of all best solutions found from the collective best results reported in the literature is about 0.22%. We are also able to associate the beneficial use of special neighborhoods with some test instance characteristics and uncover some sources of the collective power of multi-neighborhood cooperation. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction The vehicle routing problem (VRP) describes the allocation of transportation tasks to a fleet of vehicles, and the simultaneous routing of each vehicle. The VRP was first described by Dantzig and Ramser (1959), and has been proved NP-hard by Lenstra and Kan (1981). Due to its high industrial applicability and complexity, the VRP has been the object of numerous studies and a great num- ber of papers have proposed solution methods. These methods comprise both exact and heuristic algorithms. Since the VRP is NP-hard, it is not always possible to solve instances to optimality within limited computing time. Exact algorithms have been used to solve the VRP instances with up to about 100 customers (La- porte, 2007). For larger problems, heuristics and metaheuristics are more appropriate, especially tabu search (TS)(Glover and Lagu- na, 1997), which has often been used successfully. For more infor- mation on the VRP, its solution methods and the recent work, we refer to the books of Toth and Vigo (2002), Golden et al. (2008) and the survey paper of Laporte (2007). Among the solution methods for solving the VRP, one trend is to adopt parallel algorithms. For instance, parallel algorithms are increasingly applied in real time vehicle routing contexts, where solutions have to be timely generated (Ghiani et al., 2003). Unlike sequential algorithms which run on a single processor and are exe- cuted sequentially, parallel algorithms run multiple processes simultaneously on available processors with the common goal of solving a given problem instance. Crainic (2008) describes the main strategies used on this group of algorithms and also provides an up-to-date survey of contributions to this rapidly evolving field. The author also points out that parallel algorithms can both speed up the search and improve the robustness and the quality of the solutions attained. Thus it would be advantageous to make use of parallelism. Another feature of the latest metaheuristics for the VRP is to use multiple neighborhoods (e.g., Li et al., 2005; Kytöjoki et al., 2007; Mester and Bräysy, 2007; Groër et al., 2011). From the outcome of these algorithms, one may perceive that it is beneficial to em- ploy multiple neighborhoods. Indeed, each neighborhood can be used to improve or modify a solution in its particular way such as reinserting a node, swapping two nodes and so forth. For a par- ticular instance, at a certain stage, a specific neighborhood may be more effective than the others by leading the search through its own distinct search trajectory and producing better solutions. We term such a capability of a neighborhood as its effectiveness. Moreover, it is also noticeable that in these methods multiple neighborhoods are used in serial manner, in that each neighbor- hood is used one after another following a fixed or randomized se- quence. One may wonder whether it can be more effective to use multiple neighborhoods in a parallel way instead. The objective of this paper is to explore the strategy of utilizing multiple neigh- borhoods in a parallel setting and analyze their effectiveness for solving the capacitated vehicle routing problem (CVRP). The CVRP, as the classical version of the VRP, is defined on a graph G ¼ðN; AÞ, where N ¼f0; ... ; ng is a vertex set and A ¼ fði; jÞ : i; j 2 Ng is an arc set. Vertex 0 is the depot, where the 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.05.025 ⇑ Corresponding author. Tel.: +1 514 343 7143; fax: +1 514 343 7121. E-mail address: TeodorGabriel.crainic@cirrelt.ca (T.G. Crainic). European Journal of Operational Research 222 (2012) 441–451 Contents lists available at SciVerse ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor