A new bi-objective vehicle routing-scheduling problem with cross-docking: Mathematical model and algorithms
Introduction
Since the beginning of this century, Giant online retailers (e.g., Amazon and Walmart with their transportation fleet) have benefited from cross-docking as a competitive advantage in the market to decrease the cost associated with inventory holding and transportation while providing next- or same-day delivery services to customers. Moreover, third-party logistics (3PL) and carrier companies (e.g., Nexus Logistics and Menlo Logistics) also offer, not exclusively but including, cross-docking services to clients in addition to purchasing, warehousing, transportation, and delivery. In practice, while increasing the delivery cost because of the growing urban population worldwide and surging density of commodity drop in urban areas, demand for home delivery is surprisingly on the rise (Savelsbergh & Woensel, 2016). As such, the need for efficient, responsive, and sustainable delivery services has been a major challenge for not only industrial companies, but also many researchers and experts active in the transport and logistics service industry. Thus, we are motivated to address this research problem, which recognizes the challenge of offering on-time-in-full (OTIF) delivery services to customers in a retail chain sector.
From a business point of view, cross-docking, also known as an efficient strategy to regulate the flow of products in a supply chain, has received global recognition thanks to the advantage of keeping no or at least low level of temporary storage and the economies of scale in inbound and outbound transportation. The main purpose of cross-docking operations in almost all companies in retail chain and logistical services is to collect various supply products in the form of unit pallets, consolidate them into a collection of mixed pallets of outgoing goods with the same destination, and deliver them to retail stores according to the orders.
From a supply chain angle, a company can either utilize its facilities and fleets or outsource those functions to a 3PL service provider. Note that the problem dealt with in this paper can be applied to each of these cases. In the context considered here, the common sequence of events related to cross-docking is given as follows. Heavy-duty trucks collect orders from different suppliers and return to the cross-dock for consolidation. Arriving trucks are then directed to the receiving docks for the pallet unloading process. After consolidation operations, mixed pallets are moved to the staging docks and are then loaded into the outbound trucks. Once it is done, outbound trucks are then dispatched for delivering the pallets (orders) to the retailers in the same or on the next day. Note that incoming pallets are usually delivered to the cross-dock facility in full truckload (FTL), while outgoing pallets are dropped off at delivery points in FTL or less-than-truck-load (LTL) shipments. Goods consolidations in routing result in the so-called vehicle routing problem with cross-docking (VRPCD), where vehicles start predefined journeys from a cross-dock facility, visit suppliers for picking-up unit pallets in their route and return to the cross-dock for goods consolidation. Fig. 1 illustrates a typical cross-docking network considered in this paper. The solid arcs represent pickup routes of inbound fleets, while the dash-arcs show delivery routes of outbound fleets. It can be seen that all these routes start and end at the cross-dock location and that pickup node 3 and delivery node 4 are met by two vehicles (i.e., their servicing operations have been split).
In the context of the VRPCD, truck scheduling has been considered as a key issue in the pickup, cross-dock, and delivery operations. Lee, Jung, and Lee (2006) first declared that effective physical flow in a chain could be realized by taking pickup, delivery, and cross-docking operations into account. To achieve this goal, vehicle routing and scheduling have to be considered simultaneously to assure smooth physical flow in the chain. The reason behind the latter issue stems from the fact that the late arrival of the inbound fleet to any supplier facility may result in a delay in freight consolidation operations at the cross-dock facility and consequently shortage at retailers in the downstream of a retail chain. On the other hand, the early arrival of a vehicle causes long waiting times at the supplier location that affects the fleet resource utilization (Maknoon & Laporte, 2017). This matter has widely been studied by numerous researchers (see, for example, Ahkamiraad and Wang, 2018, Dondo and Cerdá, 2013, Liao et al., 2010, Musa et al., 2010, Wen et al., 2009).
Both late and early arrivals, which are known as tardiness and earliness, respectively, are due to the lack of timing in freight deployment (Dantzig & Ramser, 1959), which are used to measure customer satisfaction (Lee et al., 2006). Inspired by the concept of just-in-time (JIT) settings to minimize the total late and early arrivals (Liman and Ramaswamy, 1994, Sidney, 1977), we incorporate truck scheduling with OTIF in our study. It is important to mention that both tardiness and earliness on the outbound route give rise to product shortage and truck waiting time, respectively, at retail stores. In general, either case can jeopardize OTIF services in the retail chain sector. In cross-docking network design, vehicle routing decisions for the inbound flow should be taken along with those of the outbound flow, as coordination of inbound and outbound route settings at the cross-dock are strongly interrelated. Therefore, it is important to incorporate the truck scheduling characteristics in routing problems to assure efficient and effective delivery services, (Boysen and Fliedner, 2010, Ladier and Alpan, 2016, Wen et al., 2009).
Overall, this paper proposes an optimization model and a solution algorithm, which tackles a multi-objective VRP with cross-docking, truck scheduling, and order splitting, combined. In this study, we focus on a multi-product flow, which means that each retailer can receive more than one type of goods with different sizes in a truck. To verify our proposed algorithm, the obtained solutions are compared with those of two commonly used algorithms (i.e., the non-dominated sorting genetic algorithm (NSGA-II) and Pareto-archived evolution strategy (PAES)) over a set of instances, which confirm its efficiency.
The remainder of the paper is organized as follows. In Section 2, a brief review of the respective literature to illuminate the background of the problem is presented. The problem description and the formulation of the mathematical model are given in Section 3. The proposed solution algorithm based on a multi-objective meta-heuristic method and benchmarking algorithms is presented in Section 4. Numerical results are reported in Section 5 consisting of a set of instances borrowed from the literature and hypothetical case data. Finally, Section 6 concludes the paper with computational performance, key takeaways about the managerial aspects of this problem, and suggestions for further development.
Section snippets
Literature review
In practice, OTIF delivery plays an important role in creating value-added to the retail chain and its stockholders. Cross-docking platforms are considered as a representative of variant sharing economies, which widely has been developed in the related literature. Matzler, Veider, and Kathan (2015) underlined the impact of collaborating logistics service-providers and sharing the available resource and capacity on better consolidation, higher capacity utilization, and reduction in
Problem definition
We denote the suppliers set by (i.e., pickup nodes) and the retailers set by (i.e., demand nodes). The cross-dock node is represented by 0. Let us denote the set of all nodes in the network, including pickup nodes, delivery nodes, and the cross-dock node, by . We also assume that a node cannot be a pickup and delivery location, simultaneously. Moreover, pickup and delivery processes are not intermixed (i.e., inbound and outbound routes are separated from each
Solution methods
It should be clear that the VRPCD is classified as an NP-hard problem and thus an efficient solution algorithm to find the optimal or near-optimal solutions in reasonable computation time is required. Unfortunately, the commercial optimization solvers are unable to converge to optimality in a short time when the instance size increases. In the following, a new multi-objective evolutionary algorithm is constructed based on key features of competing methodologies and embedding local search
Numerical experiments
In this section, we present the results of two sets of instances that are designed for testing the effectiveness of the algorithms under investigation. The first set of instances is borrowed from the literature that addresses the network of a multiple-visit VRP (Dror, Laporte, & Trudeau, 1994). We compare the performances of the proposed solution method against the PAES and NSGA-II in terms of solution quality metrics. The second set of instances is taken from a hypothetical case study to
Conclusion
In this paper, we addressed a vehicle routing problem with cross-docking (VRPCD), in which splitting service in the pickup and delivery routes are allowed and every node has a time-windows to receive services. We formulated a bi-objective mixed-integer linear programming (MILP) model that minimizes (1) the transportation cost of routes and the operational cost of vehicles, and (2) the total amount of violation from the allowable interval (i.e., time windows) of each node. For solving the
CRediT authorship contribution statement
Asefeh Hasani Goodarzi: Conceptualization, Writing - original draft, Software. Reza Tavakkoli-Moghaddam: Supervision, Writing - review & editing. Mehdi Amiri-Aref: Investigation, Visualization. Alireza Amini: Resources, Project administration.
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