Computers & Operations Research 36 (2009) 1683--1692 Contents lists available at ScienceDirect Computers & Operations Research journal homepage: www.elsevier.com/locate/cor Solving the truck and trailer routing problem based on a simulated annealing heuristic Shih-Wei Lin a , Vincent F. Yu b, ∗ , Shuo-Yan Chou b a Department of Information Management, Chang Gung University, No. 259, Wen-Hwa 1st Road, Kwei-Shan, Tao-Yuan 333, Taiwan b Department of Industrial Management, National Taiwan University of Science and Technology, No. 43, Section 4, Keelung Road, Taipei 106, Taiwan ARTICLE INFO ABSTRACT Available online 29 May 2008 Keywords: Vehicle routing problem Truck and trailer routing problem Simulated annealing Heuristic In this study, we consider the application of a simulated annealing (SA) heuristic to the truck and trailer routing problem (TTRP), a variant of the vehicle routing problem (VRP). In the TTRP, some customers can be serviced by either a complete vehicle (that is, a truck pulling a trailer) or a single truck, while others can only be serviced by a single truck for various reasons. SA has seen widespread applications to various combinatorial optimization problems, including the VRP. However, to our best knowledge, it has not been applied to the TTRP. So far, all the best known results for benchmark TTRP instances were obtained using tabu search (TS). We applied SA to the TTRP and obtained 17 best solutions to the 21 benchmark TTRP benchmark problems, including 11 new best solutions. Moreover, the computational time required by the proposed SA heuristic is less than those reported in prior studies. The results suggest that SA is competitive with TS on solving the TTRP. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction In this paper we consider the truck and trailer routing problem (TTRP), a variant of the vehicle routing problem (VRP). VRP has long been one of the most studied combinatorial optimization problems due to its complexity in nature and extensive applications in practice [1--9]. In the standard VRP, a set of customers with known demand is serviced by a fleet of homogeneous vehicles with known capacity. The goal is to design least-cost vehicle routes originating from and terminating at a central depot to fulfill individual customer demands without violating the capacity constraints of the vehicles in a way that each customer is serviced only once by exactly one vehicle. In practical applications, additional operational requirements and restrictions, such as in the case of TTRP, may be imposed on the VRP. In the TTRP, the use of trailers (a commonly neglected feature in the VRP) is considered where customers are serviced by a truck pulling a trailer. However, due to practical constraints, including government regulations, limited maneuvering space at customer site, road conditions, etc., some customers may only be serviced by a truck. These constraints exist in many practical situations. Gerdessen gave two real-world TTRP applications in [10]. The first one occurred in the distribution of dairy products by the Dutch dairy industry. In this case, many customers were located in cities with heavy traffic and limited parking spaces. Maneuvering a complete ∗ Corresponding author. Tel.: +886 2 2730 3276; fax: +886 2 2737 6344. E-mail address: vincent@mail.ntust.edu.tw (V.F. Yu). 0305-0548/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2008.04.005 vehicle (that is, a truck pulling a trailer) was very difficult. Therefore, the trailer was often parked at some point while the truck delivered the products to the customers along a certain route. Another case arose in the delivery of compound animal feed to farmers. Because there were narrow roads and/or small bridges on the delivery routes, various types of vehicles were needed to make the deliveries. One type of vehicles, called double bottoms, consisting of a truck and a trailer, was commonly used. The trailer might be parked at a parking place while the truck serviced some farmers on the delivery route. Another application related to the TTRP was given by Semet and Taillard [11]. It occurred in a major food chain store in Switzerland where 45 company owned chain stores were serviced by a fleet of 21 trucks and 7 trailers. The scheduling for the deliveries with a combination of trucks and trailers was therefore of great interest. Hoff [12] considered another real-world problem occurred at a Norwegian dairy company which collects raw milk from farmers. The company used a fleet of heterogeneous trucks with tanks for the milk. A truck could either drive the route by itself or carry a trailer with an additional tank. Since most Norwegian farms were small and inaccessible for vehicles with trailers, these vehicles needed to leave its trailer at a parking place before driving a sub-tour to the farmers to collect milk. When the truck returned to the parking place, it could fill the milk over from the truck tank to the trailer tank and go on another sub-tour from there. It could also drive the trailer to a new parking place, fill the milk over and start a new sub-tour from the new parking place. The milk could be stored up to three days at the farms so the problem could be treated as a special type of the TTRP.