Open-Source Public Transportation Mobility Simulation Engine DTALite-S: A Discretized Space–Time Network-Based Modeling Framework for Bridging Multi-agent Simulation and Optimization

1.1 Literature Review

Scheduling vehicles on congested transportation networks needs to consider both traffic flows with vehicle-to-road assignment and vehicle routing problem (VRP) with passenger-to-vehicle matching. There are numbers of studies about agent-based traffic assignment and traffic simulation. Mahmassani et al. [3] used flow-density relationships to predict time-dependent traffic flows in the Dynamic Network Assignment-Simulation Model for Advanced Roadway Telematics (DYNASMART). From a broader multi-agent optimization perspective, in the study by Nedic et al. [4], a distributed computation model is built for optimizing a sum of convex objective functions for all types of agents. For shared autonomous vehicle (SAV) operating, Fagnant et al. [5] proposed an agent-based shared autonomous vehicle relocation model in order to reduce potential users’ wait times. Following Newell’s kinematic approach [6], Zhou and Taylor [7] designed a mesoscopic traffic simulation approach and developed a time-driven open-source traffic assignment package DTALite to simulate large-scale networks with millions of vehicles. Based on the multi-source data generated from transportation network companies, Spieser et al. [8] provide an on-demand transportation service model and estimate the benefits of sharing vehicles. Focusing on modeling the microscopic behavior in virtual reality systems, Yu et al. [9] provided a hierarchical modular modeling and distributed simulation methodology. A concise overview of simulation-based transportation analysis approaches is offered by Bierlaire [10].

Transportation researchers have devoted significant attentions to both traffic and public transportation simulation models. Recently, Bradley et al. [11] conducted possible autonomous vehicle (AV) operating scenarios in a road network system, and further modeled the metro transit station as a finite capacity queuing system through a discrete-event simulation (DES) approach, which was also adopted in the study by Afaq et al. [12]. Liang et al. [13] provided a mathematical model to consider the door-to-door intermodal travel trips and found that the vehicle fleet size directly influences the performance of the taxi system. Mahmassani [14] integrated varying behavioral mechanisms for different classes of vehicles into a microsimulation framework through a series of experiments under varying market penetration rates of AVs and/or connected vehicles. Qu et al. [15] presented a computationally efficient parallel-computing framework for real-life traffic simulation for metropolitan areas. To meet simulation accuracy requirements, Martinez et al. [16] proposed an agent-based model to simulate individual daily mobility while traffic assignment conditions are updated every 5 min. Golubev et al. [17] presented an agent-based traffic modeling framework allowing users to set a specific model for each supported class. Sun et al. [18] presented an agent-based simulation for urban rail transit systems. Based on kinematic wave model, Wen et al. [19] implemented a shared autonomous mobility-on-demand (AMoD) modeling platform for simulating individual travelers and vehicles with demand–supply interaction and analyzing the system performance through various metrics of indicators.

Recently, there are many papers focusing on vehicle routing optimization models and algorithms used in large-scale optimization. Boyd et al. [20] discussed general distributed optimization and provided efficient implementation under the non-convex setting. Mahmoudi and Zhou [21] built the state-space–time network to model the vehicle routing problem with pickup and drop-off and with time windows (VRPPDTW). Based on the Lagrangian decomposition and space–time windows, Tong et al. [22] developed a joint optimization approach for customized bus services. Wei et al. [23] developed a set of integer programming and dynamic programming models to optimize simplified multi-vehicle trajectories. Zhou et al. [24] introduced a vehicle routing optimization engine VRPLite on the basis of a hyper space–time–state network representation with an embedded column generation and Lagrangian relaxation framework. Zhao et al. [25] considered an optimization framework for electric vehicles in the one-way carsharing system, and they proposed a Lagrangian relaxation-based solution approach to decompose the primal problem.

1.2 Paper Structure

The remainder of this paper is organized as follows. The next section presents a modeling framework within a space–time transportation network representation. In Sect. 3, we formulate a set of space–time network-based formulations to describe the public transportation optimization problem. Section 4 provides a simulation process of vehicular loading with pickup/drop-off services based on Lagrangian relaxation. In Sect. 5, we use ADMM algorithm to expound the linkage between multi-agent-based optimization and simulation. Numerical experiments based on Beijing subway network are conducted in Sect. 6. In conclusion, Sect. 7 provides concluding remarks and future research work.

2.1 Motivation

Among widely used traffic simulation tools, DTALite is a queue-based mesoscopic traffic simulator, documented in the paper by Zhou and Taylor [7]. It is an open-source mesoscopic DTA (dynamic traffic assignment) simulation package designed to provide transportation planners, engineers, and researchers with a theoretically rigorous and computationally efficient traffic network modeling tool.

2.2 Overall Modeling Framework

As shown in Fig. 1, there are five different elements in the overall modeling framework. The first category is (1) input data, covering a transportation network, passenger information with origin, destination, preferred departure/arrival time window and desired travel path; vehicle information with depot, time window, and capacity constraint. We consider two different types of vehicles, (I) the background vehicles, driven by their own drivers as single-occupancy vehicle (SOV) or high-occupancy vehicle (HOV), move along the given physical paths from the traffic assignment program, and (III) the MaaS vehicles operated by MaaS companies with optimized routes to meet the demand of MaaS passengers as stakeholder (II). For the benefits of stakeholder (IV), we have two coupled optimization modules, namely (2) space–time network-based traffic assignment for background vehicles to find the shortest path for each agent, (3) transportation mobility optimization for MaaS vehicles and passengers, which assigns the tasks of serving passengers to specific vehicles in a cost-optimal way, and finally, (4) traffic simulation for calculating traffic states in the transportation network. The system output (5) includes time-dependent travel times, and vehicle and passenger space–time trajectories. The traffic congestion factors, if need to reach the steady states or fixed point equilibrium, should be returned back to part of (1) input data for further consistent optimization and simulation. Please note that Roman numerals (I–IV) correspond to the stakeholders’ indices in Table 1.

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3.1 Space–Time Network Construction for a Point Queue Model with Constant Capacity

Consider a physical transportation network \(M = \left( {N,L} \right)\) with a finite set of nodes N and a finite set of links L, where nodes \(i,j \in N\) and directed link \(\left( {i,j} \right) \in L\). Set L is further divided into two subsets, i.e., passenger link set L^p and vehicle link set L^v. In this research, the physical transportation network is expanded into two coupled high-dimensional space–time networks, for passengers and vehicles. Besides typical space–time traveling/waiting arcs, additional space–time arcs, such as pickup and drop-off space–time arc, are also constructed for modeling the vehicle-passenger service process.

Let \(G = \left( {E,A} \right)\) be a space–time network, where E is the set of space–time vertexes and \(A = A^{p} \cup A^{v} \cup A^{PK}\) is the set of space–time links/arcs under the planning time horizon T. Each arc \(\left( {i,j,t,t^{\prime}} \right) \in A\) indicates a directed space–time path from node i departing at time t to node j arriving at time t′. Thanks to the discretized space–time network structure, it is easy to incorporate passengers’ travel request time windows and vehicles’ operating hour requirement, and more importantly, simplified queueing models (SQM) which can distinguish vehicles’ speed in free-flow and congested conditions, under the assumption of constant bottleneck discharge capacities. The study by Lu et al. [26] provides a more systematic comparison among modes of point queues, spatial queues, and an extended version with time-dependent capacity and spillback along the backward wave. Along this line, one can extend the method from Lawson et al. [27] to calculate the spatial and temporal extents of queue and the actual waiting time spent on upstream of a bottleneck. The schematic trajectories of vehicle v from node i to node j in space–time network are shown in Fig. 2.

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The detailed attributes of vehicles and passengers can be described in the coupled modules of simulation and optimization. Given a set of passengers P and their travel requests, pickup/delivery locations \(o_{p} /d_{p}\) for each passenger p, a set of vehicles V with vehicle capacity \(Cap_{v}\) and other routing constraints, the simulation problem aims to find a feasible set of vehicle’s space–time trajectories. Given an aggregated realization of traffic flow conditions from the simulation module, the optimization module needs to perform matching and provide system-optimal routing instructions between vehicles and passengers. In details, a space–time network can be built systematically using the following steps from Tong et al. [28]. As illustrated in Fig. 3, a vehicle could travel through a sequence of pickup and drop-off links for different passengers. For a passenger, his first pickup link along his trip is his own pickup link, while he can still travel through pickup/drop-off links of other passengers. This unique modeling feature is critically important for urban rail transit and public transportation bus systems (Table 2). A station should be represented as a pair of pickup and drop-off links, as well as a possible dedicated traveling-through link, as 3–5 and 6–8 in Fig. 3.

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Table 2

Node sequences of passenger and vehicle in their paths (the pickup links for passengers are marked in bold)

Agent	Node sequence
V 1	depot1-1-2-3-4-5-6-7-8-9-10-depot2 (1-2,4-5,7-8 are pickup links; 3-4,6-7,9-10 are drop-off links; 2-3,5-6,8-9 are moving links)
P 1	O1-1-2-3-4-5-6-7-D1 (1-2 is his own pickup link; 6-7 is his own drop-off link; 2-3 and 5-6 are moving links, 4-5 is the others’ pickup link)
P 2	O2-1-2-3-4-D2 (1-2 is pickup link; 3-4 is drop-off link; 2-3 is moving link)
P 3	O3-7-8-9-10-D3 (7-8 is pickup link; 9-10 is drop-off link; 8-9 is moving link)

3.2 Mathematic Formulations for Traffic and Public Transportation Optimization Problem on a Space–Time Network

Equations (2) to (5) impose passenger/vehicle flow balance constraints on both physical and space–time network. Equations (6) and (7) ensure that the path used in the physical network corresponds to the time-index trajectories in the space–time network for each agent. Both classes of MaaS vehicles and passengers use the offline scheduled routes from the passenger-to-vehicle assignment results in the MaaS optimization program. In the future, an on-line vehicle-to-passenger matching capability can be a nature extension; for example, following the line of research by Ma et al. [29] Alonso-Mora et al. [30], and Vazifeh et al. [31], constraint (8) can be viewed as a simplified version of point queue model with constant capacity in a space–time network with waiting arcs, and the related modeling details can be found in Lu et al. [26]. Constraint (9) aims to satisfy the vehicle carrying capacity with combined Eq. (10), which ensures that each passenger is matched to exactly one vehicle. If we need to consider intermodal transfers in the public transportation problem, then one passenger may be served by multiple vehicles in a tour sequence and Eq. (10) needs to be further extended.

4.1 Simulation Flowchart Based on Simple Data Structure

Illustrated in Algorithm 2, we need to perform two loops of time and agents across different links to check the available road and vehicle carrying capacity. As we follow a point queue-based system, without complicated data structure, we only need to concern about the key variables, namely arrival time and departure time of vehicle v on link l: \(TA\left( {v,l} \right)\), \(TD\left( {v,l} \right)\), as well as cumulative arrival/departure counts of vehicles on link l at time t, \(A(l,t)\) and \(D(l,t)\).

In step 3, illustrated in Fig. 4, the waiting time and queue length can be derived from the grant traffic state variables of \(A(l,t)\) and \(D(l,t)\). Overall, it could be memory consuming to store a full matrix of variables \(TA\left( {v,l} \right)\), \(TD\left( {v,l} \right)\), one can use dynamically allocated vectors to store and update the link sequences along their paths.

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As illustrated in Fig. 5, the software architecture designed in DTALite-S aims to bridge multi-agent simulation and optimization (integrating passenger-to-vehicle assignment and time-dependent routing) in the open-source DTALite-S modeling package. ODME (origin–destination matrix estimation) is considered throughout the dynamic assignment module adopted from Zhou et al. [32].

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4.2 Further Discussions for Multimodal Environment