An ILP improvement procedure for the Open Vehicle Routing Problem Majid Salari, Paolo Toth à , Andrea Tramontani DEIS, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy article info Keywords: Integer Linear Programming Local search Heuristics Open Vehicle Routing Problem abstract We address the Open Vehicle Routing Problem (OVRP), a variant of the ‘‘classical’’ (capacitated and distance constrained) Vehicle Routing Problem (VRP) in which the vehicles are not required to return to the depot after completing their service. We present a heuristic improvement procedure for OVRP based on Integer Linear Programming (ILP) techniques. Given an initial feasible solution to be possibly improved, the method follows a destruct-and-repair paradigm, where the given solution is randomly destroyed (i.e., customers are removed in a random way) and repaired by solving an ILP model, in the attempt of finding a new improved feasible solution. The overall procedure can be considered as a general framework which could be extended to cover other variants of Vehicle Routing Problems. We report computational results on benchmark instances from the literature. In several cases, the proposed algorithm is able to find the new best known solution for the considered instances. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction We address the Open Vehicle Routing Problem (OVRP), a variant of the ‘‘classical’’ (capacitated and distance constrained) Vehicle Routing Problem (VRP) in which the vehicles are not required to return to the depot after completing their service. OVRP can be formally stated as follows. We are given a central depot and a set of n customers, which are associated with the nodes of a complete undirected graph G ¼(V,E) (where V ¼{0,1,y,n}, node 0 represents the depot and V \f0g is the set of customers). Each edge e A E has an associated finite cost c e Z0 and each customer v A V \f0g has a demand q v 40 (with q 0 ¼ 0). A fleet of m identical vehicles is located at the depot, each one with a fixed cost F,a capacity Q and a total distance-traveled (duration) limit D. The customers must be served by at most m Hamiltonian paths (open routes), each path associated with one vehicle, starting at the depot and ending at one of the customers. Each route must have a duration (computed as the sum of the edge costs in the route) not exceeding the given limit D of the vehicles, and can visit a subset S of customers whose total demand P v A S q v does not exceed the given capacity Q. The problem consists of finding a feasible solution covering (i.e., visiting) exactly once all the customers and having a minimum overall cost, computed as the sum of the traveled edge costs plus the fixed costs associated with the vehicles used to serve the customers. OVRP is known to be NP-hard in the strong sense, as it generalizes the Bin Packing Problem and the Hamiltonian Path Problem. In this paper we present a heuristic improvement procedure for OVRP based on Integer Linear Programming (ILP) techniques. Given an initial feasible solution to be possibly improved, the procedure iteratively performs the following steps: (a) randomly select several customers from the current solution, and build the restricted solution obtained from the current one by extracting (i.e., short-cutting) the selected customers; (b) reallocate the extracted customers to the restricted solution by solving an ILP problem, in the attempt of finding a new improved feasible solution. This method has been proposed by De Franceschi et al. [7] and deeply investigated by Toth and Tramontani [27] in the context of the classical VRP. Here, we consider a simpler version of this approach, which exploits no particular feature of the addressed problem. The method follows a destruct-and-repair paradigm, where the current solution is randomly destroyed (i.e., customers are removed in a random way) and repaired by following ILP techniques. Hence, the overall procedure can be considered as a general framework which could be extended to cover other variants of Vehicle Routing Problems. The notion of using ILP techniques to improve a feasible solution of a combinatorial optimization problem has emerged in several papers in the last few years. Addressing the split delivery VRP, Archetti et al. [2] developed a heuristic algorithm that integrates tabu search with ILP by solving integer programs to explore promising parts of the solution space identified by a tabu search heuristic. A similar approach has been presented by Archetti et al. [1] for an inventory routing problem. Hewitt et al. [15] proposed to solve the capacitated fixed charge network flow problem by combining exact and heuristic approaches. In this ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/caor Computers & Operations Research 0305-0548/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2010.02.010 à Corresponding author. Tel.: + 39 051 2093028; fax: + 39 051 2093073. E-mail addresses: majid.salari2@unibo.it (M. Salari), paolo.toth@unibo.it (P. Toth), andrea.tramontani@unibo.it (A. Tramontani). Please cite this article as: Salari M, et al. An ILP improvement procedure for the Open Vehicle Routing Problem. Computers and Operations Research (2010), doi:10.1016/j.cor.2010.02.010 Computers & Operations Research ] (]]]]) ]]]–]]]