Exact algorithms for the multiple depot vehicle scheduling problem with heterogeneous vehicles, split loads and toll-by-weight scheme
Introduction
The multiple depot vehicle scheduling problem (MDVSP) is a well-known problem in public transport system, and it has been studied for decades in the time constrained scheduling and routing areas. It is first formulated by Carpaneto, Dell’Amico, Fischetti, and Toth (1989), and has been proven to be NP-hard in Bertossi, Carraresi, and Gallo (1987). In this problem, there is a fleet of homogeneous vehicles based at multiple depots, and a set of trips to be served by vehicles. A trip is defined by a starting location, an ending location, a service start time, and a service end time. A vehicle can only depart from its base depot, serve a sequence of trips, and return to its base depot finally. The objective of the MDVSP is to assign trips to vehicles such that all trips are fulfilled and the total transportation cost is minimized.
In this paper, we study a general version of the MDVSP, which is originated from the practice of line-haul transportation by our collaborator, a leading express delivery firm in China, whose business is typical in the industry. Line-haul transportation refers to the shipment of packages between sorting hubs, where sorting hubs are usually located in suburb areas of cities and cover the pickup/delivery outlets in nearby urban regions. At a sorting hub, packages collected from the covered pickup outlets are first sorted according to their destinations, and then are shipped to the targeted sorting hubs (that are close to destinations) by line-haul transportation. In particular, a shipment of packages from an origin sorting hub to a destination sorting hub is called a trip. Due to the time-of-delivery requirement in express delivery, each trip has to be served no later than a deadline. For example, if the packages of a trip have been sorted before 20:00 pm, then they should be shipped toward the destination sorting hub no later than 24:00 pm at the same day.
The firm has a fleet of self-owned vehicles to provide line-haul transportation services. Each vehicle is based at a unique sorting hub for regular maintenance, and if it sets off for serving trips, it has to depart from and return to its base sorting hub. The line-haul transportation cost accounts nearly one third of the total revenue to the firm. To improve profitability, the firm endeavors to reduce the line-haul transportation cost by elaborately optimizing the assignment of trips to vehicles, which is analogous to the decision-making in the MDVSP, since both are aimed at minimizing the total transportation cost while fulfilling trips.
However, different from the settings in the MDVSP, the firm has to consider the following practical features. At first, the vehicles are heterogeneous. Currently, the firm owns thousands of vehicles for line-haul transportation, and it is impossible to hold a fleet of homogeneous vehicles given this scale. Instead, there are different types of vehicles, each of which has different loading capacity and curb weight.
Second, the service start time for each trip is not fixed but flexible. Specifically, each trip has a time window that prescribes the earliest and latest time to start the service, in addition to a predefined service duration time. On the one hand, the flexibility of service start time is in favor of the firm’s management of line-haul transportation, since otherwise it may happen that no vehicle is available to serve a trip at some time points. On the other hand, as imposed by the time-of-delivery requirement, the time window of a trip cannot be very large.
Third, the transportation demand to be served on each trip depends on the package volume in business, and hence it is not restricted to vehicle’s capacity. As a result, split loads is permitted – multiple vehicles can be used to serve one trip, which is usually the case in practice. Especially with the rapid development of e-commerce, the package volume of this express delivery firm has been remarkably increased, and millions of packages are delivered on each day.
Last, the transportation cost is calculated according to the toll-by-weight scheme (Zhang et al., 2012, Luo et al., 2016, Luo et al., 2017). Precisely, there is a toll function with both the weight of vehicle and the traveling distance as the input, and the incentive is to fairly charge tolls while effectively penalizing overloading. Under this scheme, the transportation cost depends on not only the traveling distance but also the weight of vehicle, and therefore the cost is dynamic rather than fixed. In consequence, most classical vehicle routing models and arc routing models cannot be used for this problem directly, because the transportation cost in those models is fixed (e.g., the traveling distance times a constant factor).
With the above features considered, the assignment problem of trips to vehicles is called a general version of the MDVSP, and we denoted it by GMDVSP.
We first formulate the studied problem into a route based model, which is a mixed integer nonlinear program. The non-linearity stems from the concavity of the toll function, and spontaneously, it induces us to identify an optimal structure of the route based model. Taking advantage of this optimal structure and the cover concept introduced by Li, Qin, Shen, and Tsui (2019), we transform the nonlinear program into an equivalent linear program, which is called a route-cover based model.
To solve the transformed model, we design a branch-and-price algorithm and a branch-and-benders decomposition algorithm. Both are implemented in a branch-and-bound tree. The core of both algorithms is a column generation procedure to solve the involved subproblems. Besides, some acceleration techniques have been used, e.g., bounded bidirectional search, decremental search space relaxation, and Pareto-optimal cut. The effectiveness of our approach have been demonstrated in computational experiments on a set of random instances, which are generated from the benchmark procedure for the MDVSP with adaptations.
The contributions of our work are threefold.
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First, to the best of our knowledge, we are the first to define the GMDVSP. Many practical features are respected in this problem.
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Second, we formulate a tractable model for this problem, and address the non-linearity issue induced by the toll function. We also design two exact algorithms to solve the model, which overcome the difficulty arising from the exponential size of decision variables.
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Last, we generate a set of instances for this problem, and validate the effectiveness of our approach through computational experiments. The instances and the obtained solutions can serve as a reference for future research on this and relevant problems.
The rest of this paper is organized as follows. Section 2 reviews the related literature. Section 3 formally defines the problem and Section 4 presents the model formulation and transformation. In Section 5, we deliberately introduce the branch-and-price algorithm, including the column generation and label setting algorithm for solving the pricing problem. We present the branch-and-benders decomposition in Section 6. We report our computational experiments and results in Section 7, and finally conclude the paper in Section 8. Note that a glossary of notations is provided as Appendix A, and all proofs are provided as Appendix B.
Section snippets
Literature Review
In this section, we review the literature on some related problems, including the MDVSP, the vehicle routing problem with time windows (VRPTW), split delivery, and load-dependent travel cost.
The most related problem is the MDVSP. As mentioned in the introduction, the MDVSP is first formulated by Carpaneto et al. (1989), and has been proven to be NP-hard in Bertossi et al. (1987). Both exact algorithms and heuristics have been proposed for solving the MDVSP. Ribeiro and Soumis (1994) formulated
Problem Description
We first give a formal definition of this problem (GMDVSP). Given a set of trips , each trip calls for a service, i.e., requiring a demand to be transported from a starting location to an ending location . The service time (i.e., travel time) of trip is , and the service start time is subject to a time window , where and are the earliest and the latest start time, respectively. Note that this time window constraint is a general one in the sense that it
A Route Based Model
Since a vehicle starts from and ends at its base depot, a route is associated with a unique starting (ending) depot. Accordingly, we let be the set of routes that start from (end at) the depot . For each pair of route and arc , we define a binary parameter such that if arc is traversed by the route and otherwise. Similarly, for each pair of route and trip , we define a binary parameter such that if trip is visited by the route
A Branch-and-price Algorithm
In this section, we present an exact algorithm for the model RCM. The major difficulties in solving this model are twofold. First, its decision variables are integers. Second, the number of routes is exponentially increasing with the number of vertices, which leads to a huge size of decision variables .
To tackle these difficulties, we propose a branch-and-price algorithm, which has been widely used in routing problems (Costa, Contardo, & Desaulniers, 2019). This algorithm is built upon a
A Branch-and-Benders Decomposition Algorithm
In the branch-and-price algorithm, both the decisions and are LP relaxed. Recall that the decision is to select a cover for each trip, i.e., ssif cover is selected for trip and otherwise. Conceptually, relaxing breaks the “binary” nature of . As a result, the relaxation gap is not as tight as expected, which increases the number of nodes to be explored in the branch-and-bound tree consequently. We have also observed this phenomenon in the computational experiments. To
Computational Experiments
To evaluate the model and algorithms proposed in this paper, we have conducted computational experiments on a set of random instances. Our algorithm was coded in Java, using Cplex (version 12.10) for solving the master problem in Benders decomposition and the restricted master problem in column generation. All the experiments were implemented on a PC equipped with an Intel i7-8700 processor and 64 GB RAM, running 64-bit Windows 10 operating system. A time limit of 7200 s is imposed for solving
Conclusions
In this paper, we study a general version of the multiple depot vehicle scheduling problem (GMDVSP), where several practical features have been considered, including heterogeneous vehicles, service start time windows, split loads, and the toll-by-weight scheme. These features make the problem more general but also more challenging. We first formulate this problem into a route based model, which is a nonlinear program that poses computational hurdles. Taking advantage of an optimal structure of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
Hu Qin acknowledges the supports from National Key R&D Program of China (No. 2018YFB1700600) and National Natural Science Foundation of China (Grant No. 71971090). Huaxiao Shen acknowledges the support from National Natural Science Foundation of China (Grant No. 71801231). Zhou Xu acknowledges the support from the Hong Kong Polytechnic University [Grant SB84, Project ID: P0006371].
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