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A Guided Tabu Search for the Vehicle Routing Problem with Two-Dimensional Loading Constraints

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A Guided Tabu Search for the Vehicle Routing Problem with Two-Dimensional Loading Constraints Available online at www.sciencedirect.com European Journal of Operational Research 195 (2009) 729–743 www.elsevier.com/locate/ejor A Guided Tabu Search for the Vehicle Routing Problem with two-dimensional loading constraints Emmanouil E. Zachariadis a, Christos D. Tarantilis b,*, Christos T. Kiranoudis a a Department of Process Analysis and Plant Design, Department of Chemical Engineering, National Technical University of Athens, Athens, Greece b Department of Management Science and Technology, Athens University of Economics and Business, Hydras Street 28, Athens 11362, Greece Received 12 January 2007; accepted 31 May 2007 Available online 21 November 2007 Abstract We present a metaheuristic methodology for the Capacitated Vehicle Routing Problem with two-dimensional loading constraints (2L- CVRP). 2L-CVRP is a generalisation of the Capacitated Vehicle Routing Problem, in which customer demand is formed by a set of two- dimensional, rectangular, weighted items. The purpose of this problem is to produce the minimum cost routes, starting and terminating at a central depot, to satisfy the customer demand. Furthermore, the transported items must be feasibly packed into the loading surfaces of the vehicles. We propose a metaheuristic algorithm which incorporates the rationale of Tabu Search and Guided Local Search. The loading aspects of the problem are tackled using a collection of packing heuristics. To accelerate the search process, we reduce the neigh- bourhoods explored, and employ a memory structure to record the loading feasibility information. Extensive experiments were con- ducted to calibrate the algorithmic parameters. The effectiveness of the proposed metaheuristic algorithm was tested on benchmark instances and led to several new best solutions. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Vehicle routing; Loading constraints; Tabu Search; Guided Local Search 1. Introduction optimisation problem which involves determining the opti- mal set of routes, for a homogenous fleet of vehicles, to ser- This paper addresses a variant of the standard version of vice a set of customers with known demands. The designed the Vehicle Routing Problem (VRP) (Toth and Vigo, routes must originate and terminate at the central depot, 2002), called the Capacitated Vehicle Routing Problem and totally satisfy customer demand. Each customer must with two-dimensional loading constraints (2L-CVRP) (Iori be visited once by one vehicle only. The total demand of et al., 2007; Iori, 2005; Gendreau et al., in press). It models the customer set, covered by a route, must not exceed the the cases when the demand of customers consists of two- maximum carrying load of the vehicles. dimensional, rectangular and weighted items. Although The problem examined in the present paper (2L-CVRP) 2L-CVRP is a practical problem, which often arises in is a generalisation of the CVRP, as it considers the demand the field of transportation logistics, only recently did of customers in the form of two-dimensional, rectangular researchers publish the first solution approaches for it. and weighted items. The 2L-CVRP can be reduced to the The standard version of the CVRP (Rochat and Tail- CVRP by setting the dimensions of the items equal to zero lard, 1995; Prins, 2004; Tarantilis and Kiranoudis, 2002; and dealing only with their weight attribute. The aim of the Tarantilis, 2005; Mester and Bra¨ysy, 2007) is an NP-hard 2L-CVRP is to generate a set of routes that respect the aforementioned CVRP constraints but also to guarantee the feasible loading/unloading of the items into/from the * Corresponding author. Fax: +30 2108816705. vehicles. We consider two versions of the 2L-CVRP: the E-mail address: [email protected] (C.D. Tarantilis). Unrestricted one that only deals with the feasible loading 0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.05.058 730 E.E. Zachariadis et al. / European Journal of Operational Research 195 (2009) 729–743 of the items into the vehicles, and the Sequential one that of the items into the vehicles, we employ a bundle of pack- considers both loading and unloading constraints, as it will ing heuristics that aim at producing diverse packing be described in the next sections. Loading the two-dimen- designs. To accelerate our methodology, the loading feasi- sional items into the vehicles is closely related to the two- bility information obtained through the progress of the dimensional bin packing problem (2BPP). 2BPP is an search is stored in a memory structure. The proposed solu- NP-hard combinatorial optimisation problem which aims tion approach was successfully tested on a 180 2L-CVRP at packing a given set of rectangular items into the mini- benchmark data set introduced by Gendreau et al. (in mum number of identical rectangular bins (Baker et al., press) and found several new best solutions. 1980; Chazelle, 1983; Martello and Vigo, 1998; Lodi The remainder of this paper is outlined as follows: in et al., 1999; Burke et al., 2004; Fekete et al., 2007). Section 2, a detailed description of the problem examined The 2L-CVRP is a particularly important problem. Its is provided. Section 3 presents the proposed algorithmic importance can be attributed to the fact that it is an inter- methodology, followed by the computational results in esting problem both from the theoretical and the practical Section 4. Finally, Section 5 concludes the paper. points of view. Regarding the theoretical viewpoint, since 2L-CVRP is, in a sense, composed of two NP-hard optimi- 2. The problem sation problems (CVRP and 2BPP), it is also a challenging NP-hard problem of high complexity. As far as its practical Following the description of Gendreau et al. (in press), importance is concerned, the 2L-CVRP has an obvious the 2L-CVRP is defined as follows: let G =(N,A)bean commercial value. A variety of real life applications in undirected graph, where N is the vertex set containing the the distribution/collection management context involves central depot and n customers, and A ={(i,j):i, j 2 N, the transportation of rectangular shaped items that cannot i – j} is the edge set. With each edge (i,j) 2 A is associated be stacked, due to their weight or fragility (household a cost cij that corresponds to the cost demanded for the appliances, delicate pieces of furniture, etc.). Pallet distri- transition from i to j. In the central depot, a fleet of vh bution is another application example of 2L-CVRP. homogenous vehicles is available. Every vehicle has a max- To the best of our knowledge, only two algorithmic imum carrying load equal to Q and a rectangular loading methodologies have been proposed for the 2L-CVRP. Iori surface of length L and width W. The demand of each cus- et al. (2007) have developed an exact methodology which tomer i consists of a set ITi of mi rectangular items of uses a branch-and-cut algorithm to deal with the routing known weight. Each item Iik in ITi(k =1,2,...,mi) has characteristics of the problem and a branch-and-bound length lik and width wik (corresponding to height and width procedure to guarantee feasible loadings of the items into in the work of Gendreau et al. (in press)). Let ai denote the the vehicles. This exact solution methodology is applied total surface area of all the items in ITi. The items must be to problem instances with no more than 30 customers placed on the loading surfaces without being rotated: their and 90 items. Since, in practical conditions, the scale of l-andw-edges must be parallel to the L- and W-edges of the problem tends to be larger, Gendreau et al. (in press) the vehicle surfaces, respectively. This constraint models focused their interest on metaheuristic approaches to solve the practical cases of automated, fixed orientation palette larger 2L-CVRP instances. In particular, their algorithmic loading. The 2L-CVRP aims at determining the minimum methodology employs Tabu Search for the routing aspects cost set of routes that satisfy the following constraints: of the problem. Customers are relocated by means of the (a) the size of the generated route set does not exceed the GENI heuristic (Gendreau et al., 1992). To guarantee the number of available vehicles vh (at most one route per vehi- loading feasibility, they solve the two-dimensional strip cle), (b) every route starts and ends at the central depot, (c) packing problem (Martello et al., 2003). In their work, they the demand of every customer is totally covered, (d) each provide 180 problem instances with diverse characteristics customer is visited once, (e) the total weight of all items as far as the size of the customer set and the number of demanded by the set of customers covered by a route must demanded items are concerned. Except for the CVRP with not exceed the capacity of the vehicle Q and (f) there must two-dimensional constraints, researchers have recently be a non-overlapping loading of all items demanded by the addressed routing problem extensions considering the set of customers covered by a route into the L Â W loading transportation of three-dimensional items (Gendreau surface of the vehicles. et al., 2006; Moura and Oliveira, 2007). As mentioned earlier, in this paper we study two ver- The purpose of this paper is to present an effective and sions of the 2L-CVRP: the Unrestricted 2L-CVRP,and efficient metaheuristic methodology designed for the 2L- the Sequential 2L-CVRP examined in the work of Iori CVRP called Guided Tabu Search (GTS). Regarding the et al. (2007). The Unrestricted version is modelled by the routing aspects of the problem examined, our algorithm aforementioned six constraints (a–f), while for the Sequen- explores the solution space by employing a search strategy tial version of the problem an additional constraint, based on Tabu Search, guided by an objective function namely Sequence Constraint is imposed: the loading of alteration mechanism.
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