Model predictive control for simultaneous planning of container and vehicle routes
Graphical abstract
Introduction
The longer it takes from the moment a plan is made until it is implemented, the larger is the risk that something unexpected will happen. In container transport this unexpected event could be extreme weather delaying a barge, or an extra control check by customs delaying a container. Traditionally, such events are handled manually, hence making direct truck transport the easiest mode to use. Truck transport is however often the least environmentally friendly and the most man-hour consuming mode of transport. From an environmental, societal, and economical perspective it is therefore desirable to use other modes of transport such as rail and water instead. Multi-modal, intermodal and synchromodal transport, as well as the physical internet, supply chain logistics, etc. are all concepts that enable such a shift away from simplistic solutions and towards overall efficient solutions.
The shift towards an overall efficient approach creates new challenges on both the strategic, network design level, the tactical, flow scheduling level and on the operational, specific movements level. For synchromodal transport it can be argued that the time-horizon of decisions taken on the tactical level becomes closer to the time-horizon of decisions on the operational level [25], when the flows and services can be re-planned based on online information. A key enabler for this change is the concept of a-modal bookings where the service of transport is bought instead of a slot on a specific connection. This lets the transport supplier decide which modes and which vehicles are used to fulfil a specific transport order, and allows the supplier to change this decision during the execution of the transport.
It is however not enough to change decisions in real time, it is also necessary to take good decisions. Smart planning, disruption handling, dynamic switching, and demand aggregation are in [30] identified to be the four categories of necessary actions to obtain synchromodality. Real-time switching and integrated planning are also in the literature review [9] found to be among the 8 most important properties of synchromodality. It is thus agreed upon that the success of synchromodal transport is closely linked to the ability to switch plans when disturbances occur and the ability to plan container moves and equipment use simultaneously.
This paper presents such a framework which chooses modality and routes for containers simultaneously with routes and loading/unloading actions for trucks in real time. The framework uses model predictive control (MPC) to take decisions based on the latest available information with a conscious trade-off between the cost of transport for the containers and the utilization rates of the vehicles.
In the current literature on transport planning under uncertainty, transport suppliers create vehicle routes based on estimations of the demand. In [35], a static plan that accommodates uncertain future events is created by optimizing over different scenarios, while they in [27] are accommodated by using the probabilistic knowledge of the future events in an approximate dynamic programming method. Another approach is to plan truck flows and barge and train schedules ahead of time based on an assumed demand and handle undercapacity during implementation with expensive ad hoc alternatives (e.g., [2], [33]).
In the literature there are very few attempts that directly plan container and vehicle routes simultaneously at the operational level. The authors of [32] state that “the flexibility in transportation routes may be used in conjunction with the operational fleet deployment problem. This creates new and more complex optimisation challenges”, but the statement is not explored further. A planning model that besides container routes also decides if a specific service is operated or not is presented in [35]. The services are however not routed, which for a scenario with more import than export will lead to overcapacity of empty vehicles on the import side. In other words, the need for vehicles performing round-trips is not considered. In the container route planning model presented in [26], trucks are likewise modelled as links between locations which for a given time can be used or not. It is here taken into account that trucks may not always be available, but the model does not route the trucks. In [24], import containers, trucks, trains and barges are scheduled simultaneously by solving a mixed integer optimization problem. However, all vehicles, including trucks, have pre-determined routes and thus only the departure times are decided. In contrast, the current paper routes the trucks and handles both import and export containers.
Both container and vehicle planning problems have separately been studied extensively in the literature for several different transport systems. In [31], a comprehensive overview of the Operations Research planing models used in multimodal, intermodal, and synchromodal transport can be found. To route containers through a synchromodal network, Di Febbraro et al. [19] finds the k shortest paths through a network where barges and trains depart according to a schedule. This framework does not reconsider decisions on future actions automatically, but the ability to do so when disruptions occur is discussed. In [14], last minute decisions are used to route commodity flows online over a network with scheduled barge and train services, assuming truck capacity is infinite and instantly available. In [28], a similar problem is addressed by learning a preferred policy with Approximate Dynamic Programming. To obtain higher utilization rates of vehicles, the literature on dynamic vehicle routing problems combine pre-defined pick-up and delivery appointments in the most efficient way [23]. Most papers in this category do not relate themselves to intermodal or synchromodal transport. Some accommodate transshipments in their models (e.g., [4], [8]) and cover thereby some of the challenges of intermodal transport planning.
The ability to change decisions during transport without confirmations from shippers as well as the increasing volumes to be transported motivate the use of control methods in container transport problems. Model predictive control (MPC) has already been used to address the container routing problem, but has not yet been used to integrate the planning of container and truck routes. In aforementioned [14], receding horizon control is used to plan the container flows in a hinterland network, but in contrast to the current paper, they only consider import and assume trucks are available when needed. In [13], that model is extended to the distributed case, where the geographical network is divided into non-overlapping regions served by different cooperating stakeholders. They consider commodity flows between multiple origins and destinations, but still assume trucks to be instantly available when needed. The container routing problem is furthermore solved distributed in [7] in an MPC-like framework. Trucks are hare considered instantly available and mainly used for last-mile transport.
MPC has also been used for planning and execution of related problems. It has been used to coordinate supply to demand in different supply chains (e.g. [10], [17], [22], [34]). These models generally treat transport as a known input delay, without considering modes and timetables. Reis [1] and Wang and Rivera [21] employed MPC to improve efficiency inside container terminals. The former considers equipment as queues, and is only suitable for small geographical areas, as it does not consider the advantages of handling containers based on their geographical location. The latter considers trucks to be instantaneously available.
The current paper is an extension of the conference paper [12] with improved assumptions and additional simulated experiments that strengthen the conclusions. The MPC formulation has been modified to ensure recursive feasibility when unpredicted events change the truck travel time. In this paper, trucks can drive through nodes without unloading the container they carry and wait for vacant unloading capacity at the node. The results section has furthermore been extended to include several scenarios, two different demand profiles and both nominal cases and cases with uncertain truck travel times. The method’s sensitivity to prediction horizon length is furthermore discussed. All simulations are performed on the multi-commodity, synchromodal transport network seen in Fig. 1, considering multi-type trucks as well as scheduled trains and barges.
The paper is organized as follows. In Section 2 the transport network model is introduced. Section 3 presents the control algorithm used for the simultaneous, real-time planning. Section 4 describes the simulation scenarios used to compare the proposed benchmark method to a real-time container routing method, which is presented in the same section. The results of the comparison are presented and discussed in Section 5. Finally in Section 6 the conclusions and directions for future research are discussed.
Section snippets
Model description
The transport network is modelled as a continuous state, discrete time, state-space commodity flow model of a hinterland network. The network is described by an undirected graph, where the nodes represent locations where containers are transferred between modes, locations where containers or trucks are stored or parked for longer periods of time, or scheduled services with high capacity. The arcs represent truck routes between physical locations or (un)loading actions for scheduled services.
Proposed control method
To achieve an efficient execution of container transport and truck routing that can adapt to delays online, a convex MPC is proposed. The control variables are, for all the amount of departing trucks and the containers they bring, and the quantity to load and unload for scheduled services, and the amount of demand to satisfy udi(k) and .
The proposed control model is based on Ref. [12] and extended, such that trucks can arrive to a node
Simulation experiments
To evaluate the potential benefits of simultaneous routing of containers and trucks, simulation experiments of hinterland transport scenarios have been carried out. The experiments are performed both with the planning method presented in Section 3 that determines container and truck routes simultaneously and with a benchmark method that considers truck capacity to be infinite and instantly available. To focus on the added value of simultaneous routing, the benchmark method is an MPC-based
Results and discussion
This section presents the results of the comparison between the proposed method that considers the movements of trucks and containers simultaneously and the benchmark method that assumes infinite and instant truck capacity. Both methods compute what actions to take fast due to the optimization problems’ convex nature. The average and maximum time the MPCs needed to compute the inputs at any timestep t are shown in Table 4 for all scenarios. The benchmark method is faster than the proposed
Conclusions and future research
The often used assumption that trucks are instantly available at any location in the synchromodal network significantly changes what the optimal actions are. A plan under this assumption is thus likely to perform worse in reality where only a finite number of trucks are available. The proposed method routes trucks and containers simultaneously and successfully smooths out peaks in the needed number of trucks, even when it has a large number of trucks available. This creates better plans for
Acknowledgment
This research is supported by the project “Complexity Methods for Predictive Synchromodality” (project 439.16.120) of the Netherlands Organisation for Scientific Research (NWO).
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