A set-covering formulation for a drayage problem with single and double container loads

Journal of Industrial Engineering International https://doi.org/10.1007/s40092-018-0256-8 (0123456789().,-volV)(0123456789().,-volV) ORIGINAL RESEARCH A set-covering formulation for a drayage problem with single and double container loads A. Ghezelsoflu1 • M. Di Francesco1 • A. Frangioni2 • P. Zuddas1 Received: 1 May 2017 / Accepted: 11 January 2018  The Author(s) 2018. This article is an open access publication Abstract This paper addresses a drayage problem, which is motivated by the case study of a real carrier. Its trucks carry one or two containers from a port to importers and from exporters to the port. Since up to four customers can be served in each route, we propose a set-covering formulation for this problem where all possible routes are enumerated. This model can be efficiently solved to optimality by a commercial solver, significantly outperforming a previously proposed node-arc formulation. Moreover, the model can be effectively used to evaluate a new distribution policy, which results in an enlarged set of feasible routes and can increase savings w.r.t. the policy currently employed by the carrier. Keywords Drayage  Vehicle routing problem  Street-turns  Set covering Introduction Drayage involves the distribution of a fleet of trucks moving loaded and empty containers between intermodal facilities, importers and exporters. It typically accounts for a significant portion of total transportation costs in intermodal door-to-door container transportation. This paper investigates a drayage problem, which is motivated by the case study of a medium-size carrier providing door-to-door freight transportation services. The carrier manages a fleet of trucks and specialized 24.5-foot containers to serve two types of transportation requests: the delivery of container loads from the port to importers and the shipment of container loads from exporters to the port. & P. Zuddas A. Ghezelsoflu M. Di Francesco A. Frangioni 1 Department of Mathematics and Computer Science, University of Cagliari, Cagliari, Italy 2 Department of Computer Science, University of Pisa, Pisa, Italy Although drayage operations are frequently encountered by carriers operating around ports, this case study exhibits some special characteristics, which were seldom investigated. First, in most of the related studies containers are supposed to be left at customer locations, and drivers can move to other customers, thereby bypassing packing and unpacking operations (Wang et al. 2002; Jula et al. 2005; Chung et al. 2007; Zhang et al. 2010, 2011; Vidovic et al. 2011; Braekers et al. 2013; Nossack and Pesch 2013). Conversely, in this case study trucks move loaded containers to import customers and drivers wait for the empty containers, while unpacking operations are performed. Similarly, trucks move empty containers to export customers and drivers wait for loaded containers that will be returned after the conclusion of packing operations. Therefore, in this case study trucks carry the same containers throughout their routes. According to customers’ viewpoint, this is a high quality service, because the content of the cargo can be immediately verified by drivers: they make sure that the container loads are the correct ones, in the right quantity and without damages. Moreover, customers need to be equipped only with forklift trucks for packing and unpacking operations. From the carrier’s viewpoint, this service improves the integrity of containers, because they are never left unsupervised at customer locations. 123 Journal of Industrial Engineering International Second, in this case study routes are performed by two types of trucks carrying one or two containers, respectively, whereas most of the related literature concerns one container per truck (Wang et al. 2002; Jula et al. 2005; Zhang et al. 2011, 2010; Braekers et al. 2013; Nossack and Pesch 2013). The case more than one container has been dealt with only recently (Vidovic et al. 2017; Funke and Kopfer 2016). The assumption of one container per truck is often unnecessarily strong, as the transportation of two 20-foot containers is allowed in many countries (citeNagl2007. Although two-containers trucks have higher costs per unitary distance, they allow to serve a larger number of customers per route than one-container ones. Although in this case study the majority of routes is expected to be performed by trucks carrying two containers, it is useful to propose a model to optimally decide by which type of truck each customer is served, taking into account the number of available trucks, in order to minimize routing costs. We assume that each truck performs only one route and the cost of routing between two predefined locations depends only on the truck type. The planning of the truck routes is a complex task for the carrier of this case study, as no optimization-based tool is currently adopted to support this phase. At the moment, the carrier performs the planning of routes in accordance with this simple policy: all importers must be served before all exporters in order to exploit street-turns, i.e., the use of containers emptied at importers to collect cargoes from the exporters (Jula et al. 2006; Deidda et al. 2008). Thus, at the moment the routes consist in the shipment of container loads from the port to one or two importers, the allocation of empty containers from importers to exporters, and the final shipment of container loads from one or two exporters to the port. Since the number of container loads to be delivered to importers and collected from exporters is typically different, street-turns are typically insufficient and the carrier must also decide which customers are served by direct trips from the depot. The current carrier’s policy simplifies the planning process, but it does not necessarily yield optimal routes in terms of total cost. Indeed, streetturns can be realized by the optimization process itself, when they are useful to decrease the routing cost, instead of being forced a priori upon the routes construction. Therefore, some room for improvement exists in the current carrier policy. This paper proposes an optimization model to support the planning of routes in accordance with the current carrier policy, and evaluates the savings that could be obtained according to the different distribution policy in which customers are allowed to be visited in any order. Since up to four customers can be visited in each route, the proposed model consists in the enumeration of all feasible routes and in the definition of a set-covering problem to select the 123 subset of the generated routes that serves all customers at a minimum cost by the available vehicles. The set-covering model is compared to the node-arc model proposed in Lai et al. (2013). That model was restricted to the case where all importers are served before all exporters, in accordance with the current distribution policy. Furthermore, it could not be solved effic (...truncated)