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

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## Abstract

Recently, an open-source light-weight dynamic traffic assignment (DTA) package, namely DTALite, has been developed to allow a rapid utilization of advanced dynamic traffic analysis capabilities. Aiming to bridge the modeling gaps between multi-agent simulation and optimization in a multimodal environment, we further design and develop DTALite-S to simplify the traffic flow dynamic representation details in DTALite for future extensions. We hope to offer a unified modeling framework with inherently consistent space–time network representations for both optimization formulation and simulation process. This paper includes three major modeling components: (1) mathematic formulations to describe traffic and public transportation simulation problem on a space–time network; (2) transportation transition dynamics involving multiple agents in the optimization process; (3) an alternating direction method of multipliers (ADMM)-based modeling structure to link different features between multi-agent simulation and optimization used in transportation. This unified framework can be embedded in a Lagrangian relaxation method and a time-oriented sequential simulation procedure to handle many general applications. We carried out a case study by using this unified framework to simulate the passenger traveling process in Beijing subway network which contains 18 urban rail transit lines, 343 stations, and 52 transfer stations. Via the ADMM-based solution approach, queue lengths at platforms, in-vehicle congestion levels and absolute deviation of travel times are obtained within 1560 seconds.The case study indicate that the open-source DTALite-S integrates simulation and optimization procedure for complex dynamic transportation systems and can efficiently generate comprehensive space-time traveling status.

## Keywords

Space–time network Dynamic traffic assignment Multi-agent simulation Lagrangian relaxation Alternating direction method of multipliers## 1 Introduction

To understand and analyze future emerging mobility scenarios, planers and engineers need to utilize many different simulation tools to generate corresponding modeling results. The main purpose of transportation simulation is to shed more light on the underlying mechanisms or potential problems that control the behavior of a complex transportation system.

Typically, simulating a system involves a probabilistic input model, a set of dynamic equations or constraints between the inputs and outputs, and then finally produces a set of outputs under different input instances. Optimization, on the other hand, needs to search a solution in the dynamic (possibly complex) system subject to a number of constraints. There are a wide range of studies focusing on simulation-based optimization, to name a few, a leading study by Osorio et al. [1] involving stochastic urban traffic simulators, and another study by Xiong et al. [2] using the DTALite simulator. Generally, transportation planners and engineers utilize simulation tools to evaluate and further optimize a subset of system’s parameters, but there is a critical modeling gap between simulation and optimization for complex dynamic transportation systems. To bridge such a gap in a multimodal environment, this research focuses on how to offer a theoretically sound and practically useful modeling framework with a simplified traffic flow dynamic.

### 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 Problem Statement and Modeling Framework

### 2.1 Motivation

Interested stakeholders and different measures that can be optimized and simulated

Stakeholders | Stakeholder interests | System-wide metrics | Individual user-oriented performance |
---|---|---|---|

(I) (Non-MaaS) users (passenger/drivers) | • Travel driving cost • Level of service | • Total driving time • Congestion | • Waiting time at bottlenecks • Mean travel time • Travel time reliability |

(II) MaaS passengers | • MaaS service cost • Accessibility | • Total end-to-end service time • MaaS service availability | • Waiting time for MaaS vehicles • Service time window and fare • Detour factor for shared ride • Punctuality |

(III) MaaS operator | • Operating and transportation costs • Revenue and profit from fares • Fleet structure and size of fleet | • Revenue • Ridership • Punctuality • Total travel time | • Vehicle traveling distance and travel time • Distance-based average passenger load • Cost per passenger served |

(IV) Public transportation planning and management agencies | • System efficiency • Public equity • Sustainability | • Space–time time accessibility • Motorized traffic • Non-motorized trips | • Vehicle traveling distance • MaaS Service rate • Market share of green and shared transportation mode |

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.

- 1.
Provide an open-source simulation package that enables transportation researchers and students to understand the complex space–time network modeling process.

- 2.
Offer a unified framework with pickup and drop-off events that cover different traveling activities from driving-only to multiple public transportation modes, across the emerging transportation mobility spectrum, e.g., urban rail transit, synchronized bus, ride-sharing applications, as well as freight transport.

- 3.
Agent-based dynamic traffic assignment and traffic simulation are integrated and extended to tackle practical applications of vehicle routing problem (VRP), or its variants, e.g., vehicle routing problem with pickup and delivery (VRPPD), vehicle routing problem with pickup and delivery with time windows (VRPPDTW).

### 2.2 Overall Modeling Framework

**(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.

## 3 Mathematical Programing Model in a Disseized Network

### 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.

*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.

*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.

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

Agent | Node sequence |
---|---|

| depot |

| O |

| O |

| O |

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

Basic indices used to describe the space–time modeling framework

Index | Definition |
---|---|

| Physical transportation network |

| Space–time transportation network |

| Vehicle index, \(v \in V\) |

| Passenger index, \(p \in P\) |

\(EDT\left( p \right)\) | Earliest departure time of passenger |

\(LAT\left( p \right)\) | Latest arrival time of passenger |

\(EDT\left( v \right)\) | Earliest departure time of vehicle |

\(LAT\left( v \right)\) | Latest arrival time of vehicle |

| Origin node of vehicle |

| Origin node of passenger |

\(o_{p} '\) | Dummy node for passenger \(p's\) origin |

| Destination node of vehicle |

| Destination node of passenger |

\(d_{p} '\) | Dummy node for passenger \(p's\) destination |

\(\left( {i,j,t,t^{\prime}} \right)\) | Index of space–time traveling arc, \(\left( {i,j,t,t^{\prime}} \right) \in A\) |

\(TT\left( {i,j,t} \right)\) | Link travel time from node |

Basic parameters used to describe the space–time modeling framework

Notations | Definition |
---|---|

\(TT\left( {i,j,t} \right)\) | Link travel time from node |

\(Cap_{v}\) | Number of seats in vehicle |

\(CapRoad_{i,j,t}\) | Road capacity from node |

\(C\left( {v,i,j,t,t^{\prime}} \right)\) | Vehicle-specific transportation cost on arc \(\left( {i,j,t,t^{\prime}} \right)\) traveled by vehicle |

\(C\left( {p,i,j,t,t^{\prime}} \right)\) | Passenger-specific transportation cost on arc \(\left( {i,j,t,t^{\prime}} \right)\) traveled by passenger |

Key variables used to describe the space–time modeling framework

Notations | Definition |
---|---|

\(x_{i,j} \left( p \right)\) | Passenger routing variable (= 1, if physical arc ( |

\(x_{i,j} \left( v \right)\) | Vehicle routing variable (= 1, if physical arc ( |

\(y_{{i,j,t,t^{'} }} \left( p \right)\) | Passenger space–time routing variable (= 1, if space–time arc (\(i,j,t,t^{\prime}\)) is selected by passenger |

\(y_{{i,j,t,t^{'} }} \left( v \right)\) | Vehicle space–time routing variable (= 1, if space–time arc (\(i,j,t,t^{\prime}\)) is selected by vehicle |

\(z\left( {p,v} \right)\) | Passenger-to-vehicle marching variable (= 1, if passenger |

**objective function**shown in Eq. (1) is to minimize all vehicles and passengers’ total costs.

**physical path**at vehicle \(v^{\prime}s\) node

**space–time path**at vehicle \(v 's\) vertex

**physical path**at passenger \(p 's\) node

**space–time path**at passenger \(p 's\) vertex

**between physical path and space–time path**for

*p*

**between physical path and space–time path**for

*v*

**Road capacity**constraints for all the vehicles using the same link

**Passenger**carrying

**capacity**constraints for vehicles

**Matching constraints**(10) ensures that each passenger is matched to exactly one vehicle

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 Simulation Process of Vehicular Loading and Passenger Pickup and Drop-off Services

### 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 \(D(l,t)\).

### 4.2 Further Discussions for Multimodal Environment

Comparison of different applications in transportation network

Transportation mode | Agents | Focuses of modeling |
---|---|---|

Urban rail transit | • Passenger oversaturation • Passengers and vehicles have fixed routes | • Waiting time • Walking link • Transfer for \(x_{ij} \left( p \right)\) • Timetable |

Bus | • \(y_{ij} \left( v \right)\) is not stable/reliable, affect \(y_{ij} \left( p \right)\) | • Waiting time • Walking link • Transfer for \(x_{ij} \left( p \right)\) • Bus service network design for \(x_{ij} \left( v \right)\) • Timetable |

Taxi | • Real-time passenger-to-vehicle matching | • Affected by traffic congestion, capacity • Dynamically solve assignment of passengers and vehicles • Pickup and drop-off |

Driving only | • Vehicles carry passengers | • Traffic assignment problem with user equilibrium (UE) • Focus on \(y_{ij} \left( v \right)\) and \(x_{ij} \left( v \right),\) • Vehicle capacity and road capacity • Computational graph |

Bike | • Passengers carry vehicles | • Routing is decided by passenger |

Major modeling enhancements from DTALite to DTALite-S

Index | DTALite | DTALite-S |
---|---|---|

Representation scale | Mesoscopic, link-based network | Range from mesoscopic link-based and microscopic cell-based network, with additional hyper space–time–state network details for vehicle routing problem with pickup and delivery |

Traffic dynamics model | Point, spatial, and simplified kinematic wave-based queues | Point queue model |

Agent type | Vehicles | Vehicles and passengers |

Mode | Driving-only mode | Multiple modes |

Simulation functionality | Traffic assignment, queue propagation | Traffic assignment, queue propagation, vehicle routing optimization |

Algorithm | Label correcting algorithm, LR | Dynamic programming, LR, ADMM |

Time-based simulation | Discrete-time in physical network with fixed/regular time interval (6 s) | Discrete-time in space–time network with flexible simulation time interval from 0.1 s to 1 min |

## 5 An ADMM-Based Framework to Understand Linkage Between Multi-agent-Based Optimization and Simulation

ADMM is a widely used optimization algorithm, and the literature of ADMM can be traced to classical papers by Douglas and Rachford [33] and Boyd et al. [20]. Essentially, it integrates problem decomposition techniques of augmented Lagrangian and block coordination descent. In our proposed framework, ADMM is firstly used to generate vehicle routing solutions for the multi-agent optimization problem. With Lagrangian multipliers and quadratic penalty terms, the complex public transportation mobility model can be decomposed into simple subproblems that can be solved by a sequential solution scheme. The open-source vehicle routing package using the ADMM framework proposed by Yao et al. [34] can be downloaded from https://github.com/YaoYuBJTU/ADMM_Python.

*v*

_{1}and

*v*

_{2}. Given \(v_{2} 's\) trajectory up to time

*t*+

*h*, we need to schedule trajectory of

*v*

_{1}from

*t*to

*t*+

*h*, where

*h*is the optimization time horizon in the future. We can interpret the above problem from perspectives of simulation and optimization.

**Lagrangian relaxation**form for problem (11) can be rewritten as Eq. (12), where \(\lambda_{{\left( {i,j} \right)}}\) is the vector of Lagrangian multiplier for different links.

**ADMM function**can be written as Eq. (13), and the algorithm will apply a block coordinate decomposition scheme for each agent

*v*.

Through the record of \(CapRoad_{i,j,t}\) in Algorithm 2, the simulation mechanism in DTALite-S first removes the infeasible space–time regions occupied by vehicle *v*_{2}, to ensure the full feasibility trajectories of *v*_{1}. Recall that, \(CapRoad_{i,j,t} = 0\) indicates that the space–time resource capacity at link (*i*, *j*) at time *t* has been used by one previously scheduled vehicle, otherwise.

*k*, the simulation process can be viewed as a one-shot procedure from time 0 to time

*T*, with the infinite penalty parameters to ensure the full feasibility of space–time agent trajectories.

Conceptual algorithmic comparison between ADMM with simulation processes in the space–time network representation

Essentially, the ADMM-based iterative optimization procedure can optimize the system-wide costs by iteratively scheduling individual vehicle trajectories with a relatively small \(\rho\) and then increasing the value of \(\rho\) further aims to enforce the primal feasibility. In contract, the simulation-based process makes scheduling decisions only based on purely local information at current time and only schedules the trajectories at a short time horizon. Interested readers can further study the use of ADMM in a distributed multi-agent optimization environment, e.g., for distributed automated vehicles, based on a recent study by Nedic and Ozdaglar [4].

## 6 Case Study

## 7 Conclusions

Optimization tends to solve tactical/operational issues, while simulation aims to resolve more complex and realistic transportation problems. A number of researchers have devoted a great amount of efforts to integrating both optimization approaches and simulation tools to pursue future advanced mobility solutions. For example, stochastic programming and approximate dynamic programming often use simulation to estimate expected values within an optimization framework.

The proposed discrete space–time network-based modeling framework can be used to evaluate and further optimize transport strategies from new demand and supply perspectives. It intends to bridge the modeling gaps for solving complex transportation problems, such as multimodal transportation network design, VRP, urban rail transit scheduling, bus operating, etc. With the embedded Lagrangian relaxation and ADMM algorithms, the framework is able to shed more lights on many fundamental research issues in large-scale dynamic traffic assignment, mesoscopic traffic simulation, and vehicle route optimization. The proposed modeling framework is used to simulate the passenger traveling process in Beijing subway network. Via the ADMM-based solution approach, queue lengths at platforms, in-vehicle congestion levels and absolute deviation of travel times are obtained within 1560 seconds.The numerical examples indicate that the proposed simulation engine DTALite-S is able to efficiently simulate passenger traveling process.

In our future research, a more comprehensive and microscopic cell-based simulation framework will be further studied to simulate more detailed travel decisions, such as passenger route choice lane changing behavior, and car flowing behavior. In addition, we will apply and extend the proposed open-source offline modeling framework to many emerging transportation applications, in a real-time and distributed computing environment.

## Notes

### Acknowledgements

This research project, especially the large-scale Beijing Subway network and smart card data set, has been supported through Beijing Key Laboratory of Urban Traffic Operation Simulation and Decision Support and Beijing International Science and Technology Cooperation Base of Urban Transport. The last author is partially funded by National Science Foundation, USA, under NSF Grant No. CMMI 1538105 “Collaborative Research: Improving Spatial Observability of Dynamic Traffic Systems through Active Mobile Sensor Networks and Crowdsourced Data” and NSF Grant No. CMMI 1663657. “Real-time Management of Large Fleets of Self-Driving Vehicles Using Virtual Cyber Tracks”.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## References

- 1.Osorio C, Bierlaire M (2013) A simulation-based optimization framework for urban transportation problems. Oper Res 61(6):1333–1345MathSciNetCrossRefGoogle Scholar
- 2.Xiong C, Zhu Z, Chen X, Zhang L (2018) Optimal travel information provision strategies: an agent-based approach under uncertainty. Transp B Transp Dyn 6(2):129–150Google Scholar
- 3.Mahmassani HS, Fei X, Eisenman S, Zhou X, Qin X (2005) DYNASMART-X evaluation for real-time TMC application: CHART test bed. Maryland Transportation Initiative, University of Maryland, College Park, Maryland, pp 1–144Google Scholar
- 4.Nedic A, Ozdaglar A (2009) Distributed subgradient methods for multi-agent optimization. IEEE Trans Autom Control 54(1):48–61MathSciNetCrossRefGoogle Scholar
- 5.Fagnant D, Hall CJ, Kockelman KM (2014) The travel and environmental implications of shared autonomous vehicles, using agent-based model scenarios. Transp Res Part C 40(1):1–13CrossRefGoogle Scholar
- 6.Newell GF (1993) A simplified theory of kinematic waves in highway traffic, part iii: multi-destination flows. Transp Res Part B Methodol 27(4):281–287CrossRefGoogle Scholar
- 7.Zhou X, Taylor J, Pratico F (2014) DTALite: a queue-based mesoscopic traffic simulator for fast model evaluation and calibration. Cogent Eng 1(1):961345CrossRefGoogle Scholar
- 8.Spieser K, Treleaven KB, Zhang R, Frazzoli E, Morton D, Pavone M (2014) Toward a systematic approach to the design and evaluation of automated mobility-on-demand systems: a case study in Singapore. Springer International Publishing, Road Vehicle AutomationGoogle Scholar
- 9.Yu Y, El Kamel A, Gong G, Li F (2014) Multi-agent based modeling and simulation of microscopic traffic in virtual reality system. Simul Model Pract Theory 45:62–79CrossRefGoogle Scholar
- 10.Bierlaire M (2015) Simulation and optimization: a short review. Transp Res Part C Emerg Technol 55:4–13CrossRefGoogle Scholar
- 11.Kloostra B, Roorda MJ (2017) Fully autonomous vehicles: analyzing transportation network performance and operating scenarios in the Greater Toronto Area, Canada. In: TRB 2017 annual meetingGoogle Scholar
- 12.Khattak A, Yangsheng J, Lu H, Juanxiu Z (2016) Modeling and simulation of metro transit station walkway as a state-dependent queuing system based on the phase-type distribution. Int J Traffic Transp Eng 5(5):103–111Google Scholar
- 13.Liang X, de Almeida Correia GH, Van Arem B (2016) Optimizing the service area and trip selection of an electric automated taxi system used for the last mile of train trips. Transp Res Part E Logist Transp Rev 93:115–129CrossRefGoogle Scholar
- 14.Mahmassani HS (2016) 50th anniversary invited article—autonomous vehicles and connected vehicle systems: flow and operations considerations. Transp Sci 50(4):1140–1162CrossRefGoogle Scholar
- 15.Qu Y, Zhou X (2017) Large-scale dynamic transportation network simulation: a space-time-event parallel computing approach. Transp Res Part C Emerg Technol 75:1–16CrossRefGoogle Scholar
- 16.Martinez LM, Correia GH, Viegas JM (2015) An agent-based simulation model to assess the impacts of introducing a shared-taxi system: an application to Lisbon (Portugal). J Adv Transp 49(3):475–495CrossRefGoogle Scholar
- 17.Golubev K, Zagarskikh A, Karsakov A (2018) A framework for a multi-agent traffic simulation using combined behavioral models. Procedia Comput Sci 136:443–452CrossRefGoogle Scholar
- 18.Sun Y, Zhang C, Dong K, Lang M (2018) Multiagent modelling and simulation of a physical internet enabled rail-road intermodal transport system. Urban Rail Transit 4(3):141–154CrossRefGoogle Scholar
- 19.Wen J, Chen YX, Nassir N, Zhao J (2018) Transit-oriented autonomous vehicle operation with integrated demand-supply interaction. Transp Res Part C Emerg Technol 97:216–234CrossRefGoogle Scholar
- 20.Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn 3(1):1–122CrossRefGoogle Scholar
- 21.Mahmoudi M, Zhou X (2016) Finding optimal solutions for vehicle routing problem with pickup and delivery services with time windows: a dynamic programming approach based on state-space-time network representations. Transp Res Part B Methodol 89:19–42CrossRefGoogle Scholar
- 22.Tong LC, Zhou L, Liu J, Zhou X (2017) Customized bus service design for jointly optimizing passenger-to-vehicle assignment and vehicle routing. Transp Res Part C Emerg Technol 85:451–475CrossRefGoogle Scholar
- 23.Wei Y, Avcı C, Liu J, Belezamo B, Aydın N, Li PT, Zhou X (2017) Dynamic programming-based multi-vehicle longitudinal trajectory optimization with simplified car following models. Transp Res Part B Methodol 106:102–129CrossRefGoogle Scholar
- 24.Zhou X, Tong L, Mahmoudi M, Zhuge L, Yao Y, Zhang Y, Shang P, Liu J, Shi T (2018) Open-source VRPLite package for vehicle routing with pickup and delivery: a path finding engine for scheduled transportation systems. Urban Rail Transit 4(2):68–85CrossRefGoogle Scholar
- 25.Zhao M, Li X, Yin J, Cui J, Yang L, An S (2018) An integrated framework for electric vehicle rebalancing and staff relocation in one-way carsharing systems: model formulation and Lagrangian relaxation-based solution approach. Transp Res Part B Methodol 117:542–572CrossRefGoogle Scholar
- 26.Lu CC, Liu J, Qu Y, Peeta S, Rouphail NM, Zhou X (2016) Eco-system optimal time-dependent flow assignment in a congested network. Transp Res Part B Methodol 94:217–239CrossRefGoogle Scholar
- 27.Lawson TW, Lovell DJ, Daganzo CF (1997) Using input-output diagram to determine spatial and temporal extents of a queue upstream of a bottleneck. Transp Res Rec 1572(1):140–147CrossRefGoogle Scholar
- 28.Tong L, Zhou X, Miller HJ (2015) Transportation network design for maximizing space–time accessibility. Transp Res Part B Methodol 81:555–576CrossRefGoogle Scholar
- 29.Ma J, Li X, Zhou F, Hao W (2017) Designing optimal autonomous vehicle sharing and reservation systems: a linear programming approach. Transp Res Part C Emerg Technol 84:124–141CrossRefGoogle Scholar
- 30.Alonso-Mora J, Samaranayake S, Wallar A, Frazzoli E, Rus D (2017) On-demand high-capacity ride-sharing via dynamic trip-vehicle assignment. Proc Natl Acad Sci 114(3):462–467CrossRefGoogle Scholar
- 31.Vazifeh MM, Santi P, Resta G, Strogatz SH, Ratti C (2018) Addressing the minimum fleet problem in on-demand urban mobility. Nature 557(7706):534CrossRefGoogle Scholar
- 32.Lu CC, Zhou X, Zhang K (2013) Dynamic origin-destination demand flow estimation under congested traffic conditions. Transp Res Part C Emerg Technol 34:16–37CrossRefGoogle Scholar
- 33.Douglas J, Rachford HH (1956) On the numerical solution of heat conduction problems in two and three space variables. Trans Am Math Soc 82(2):421–439MathSciNetCrossRefGoogle Scholar
- 34.Yao Y, Zhu X, Dong H, Wu S, Wu H, Zhou X (2018). An alternating direction method of multiplier based problem decomposition scheme for iteratively improving primal and dual solution quality in vehicle routing problem. Working paperGoogle Scholar