Transportation Network Equilibrium Research Articles

A descriptive and prescriptive analysis of rail service subsidies in the China–Europe freight transportation market

This study delivers a descriptive and prescriptive analysis of rail service subsidies for China Railway Express (CRE) in the China-Europe freight transportation market. The analysis is conducted by advanced mathematical modeling and programming methods. Specifically, we implemented a multicommodity multimodal freight transportation network equilibrium model that can be used for predicting the commodity-specific mode-route cargo flow pattern and hence for assessing the effectiveness and limitations of the current CRE subsidy scheme. To properly quantify the impact of subsidies on individual shippers’ decision making, the model explicitly characterizes individual shippers’ mode-route choice behavior and takes into account shipping cost, transit time, capacity-induced congestion surcharge, and unobserved transportation impedances as shippers’ disutility. The solution of the network equilibrium model resorts to a disaggregate simplicial decomposition algorithm within the well-known Lagrangian relaxation framework. A bi-level network-based subsidy optimization model is constructed, in which the upper level aims at minimizing the sum of revenue loss and congestion charge, and the lower level is the aforementioned freight transportation network equilibrium model. A tabu search procedure is proposed and implemented to derive the solution of the bi-level model. The above models and algorithms are then applied to the China-Europe containerized freight transportation network, which comprises all China-Europe liner shipping lines, all CRE service lines, and the highway networks in China and Europe. The evaluation and optimization results show that the current subsidy scheme creates an imbalanced capacity utilization pattern across CRE service lines while an optimized line-specific subsidy solution can yield noteworthy improvements in the service utilization and economic efficiency of CRE.

Simplicial decomposition of variational inequalities with multiple nonlinear column generation

<abstract> <p>Simplicial decomposition (SD) of variational inequalities experiences the long-tail convergence property. That is, the equilibrium solution rapidly progresses at first but then tails off, making only a tiny amount of progress per column generation iteration, which is a drawback of SD-VI. In the context of Dantzig-Wolfe of LP, it is reported that the more proposals are used to initialize the algorithm, the faster the solution can be found by reducing the number of decomposition steps. Therefore, I proposed to solve multiple nonlinear column generation (mNCG) subproblems in each SD-VI iteration (SD-VI-mNCG) instead of solving only one subproblem as in SD-VI. Generating multiple column generation subproblem solutions in each SD-VI iteration enabled the corresponding convex hull to be rapidly enlarged. Consequently, the number of SD-VI iterations could be greatly reduced. A transportation network equilibrium problem was used to study the performance of the SD-VI-mNCG.</p> </abstract>

End-to-end learning of user equilibrium with implicit neural networks

This paper intends to transform the transportation network equilibrium modeling paradigm via an “end-to-end” framework that directly learns travel choice preferences and the equilibrium state from multi-day link flow observations. The centerpiece of the proposed framework is to use deep neural networks to represent travelers’ route choice preferences and then encapsulate the neural networks in a variational inequality that prescribes the user equilibrium flow distribution. The proposed neural network architecture ensures the existence of equilibrium and accommodates future changes in road network topology. The variational inequality is then embedded as an implicit layer in a learning framework, which takes the context features (e.g., road network and traveler characteristics) as input and outputs the user equilibrium flow distribution. By comparing computed equilibrium flows with observed flows, the neural networks can be trained. The proposed end-to-end framework is demonstrated and validated using synthesized data for the Sioux Falls network.

A multimodal multicommodity network equilibrium model with service capacity and bottleneck congestion for China-Europe containerized freight flows

As a traditional way for transporting containerized cargoes, international liner shipping has long played a dominant role in the China-Europe freight transportation market. In recent years, China Railway Express has offered a new and competitive freight transportation option, contributing to a significant increase in the trade volume by rail between China and Europe. With the goal of providing a holistic computational tool that can predict and analyze the spatial distribution of containerized freight flows and suggest strategic transportation infrastructure and service expansion options, this paper develops and implements a multimodal multicommodity freight transportation network equilibrium model, based on a newly constructed supernetwork integrating international liner shipping lines, international freight rail services, and international and domestic highway networks in relevant countries. The combined transportation mode-route choice behavior of individual shippers is characterized by the widely used multinomial logit model, where the values of time of different commodity categories are included in its utility functions. The congestion effects arising from bottleneck facilities along liner shipping and freight rail lines (i.e., seaports, water channels, and break-of-gauge stations) are first captured by queueing models and then approximated by polynomial delay functions, while the accommodation capacity of individual service lines is explicitly embedded into the model as hard constraints. This supernetwork equilibrium model is tackled by a subgradient algorithm in the Lagrangian relaxation framework embedding a disaggregate simplicial decomposition algorithm. Computational experiments are conducted with a multi-layer, hetero-structure set of supply and demand data collected from multiple data repositories, government agencies, freight carriers, and research papers and reports. The preliminary results exhibit the computational performance of the solution procedure and reveal the variation of congestion levels of bottleneck facilities, capacity usage profiles of service lines, and competitiveness conditions between liner shipping and freight rail services under different monthly demand patterns.

Optimal Deployment of Cordon Sanitaire with Available Testing Capacity

During the COVID-19 pandemic, authorities in many places have implemented various countermeasures, including setting up a cordon sanitaire to restrict population movement. This paper proposes a bi-level programming model to deploy a limited number of parallel checkpoints at each entry link around the cordon sanitaire to achieve a minimum total waiting time for all travelers. At the lower level, it is a transportation network equilibrium with queuing for a fixed travel demand and given road network. The feedback process between trip distribution and trip assignment results in the predicted waiting time and traffic flow for each entry link. For the lower-level model, the method of successive averages is used to achieve a network equilibrium with queuing for any given allocation decision from the upper level, and the reduced gradient algorithm is used for traffic assignment with queuing. At the upper level, it is a queuing network optimization model. The objective is the minimization of the system’s total waiting time, which can be derived from the predicted traffic flow and queuing delay time at each entry link from the lower-level model. Since it is a nonlinear integer programming problem that is hard to solve, a genetic algorithm with elite strategy is designed. An experimental study using the Nguyen-Dupuis road network shows that the proposed methods effectively find a good heuristic optimal solution. Together with the findings from two additional sensitivity tests, the proposed methods are beneficial for policymakers to determine the optimal deployment of cordon sanitaire given limited resources.

Emergency response facility location in transportation networks: A literature review

Emergency response activity relies on transportation networks. Emergency facility location interacts with transportation networks clearly. This review is aimed to provide a combined framework for emergency facility location in transportation networks. The article reveals emergency response activities research clusters, issues, and objectives according to keywords co-occurrence analysis. Four classes of spatial separation models in transportation networks, including distance, routing, accessibility, and travel time are introduced. The stochastic and time-dependent characteristics of travel time are described. Travel time estimation and prediction method, travel time under emergency vehicle preemption, transportation network equilibrium method, and travel time in degradable networks are demonstrated. The emergency facilities location models interact with transportation networks, involving location-routing model, location models embedded with accessibility, location models embedded with travel time, and location models employing mathematical program with equilibrium constraints are reviewed. We then point out the-state-of-art challenges: ilities-oriented, evolution landscape and sequential decision modelling, data-driven optimization approach, and machine learning-based algorithms. • Emergency response facility location problem in transportation networks is reviewed for taxonomy purpose. • Travel time estimation and forecast methods in various scenarios are summarized. • Emergency facility location models in transportation networks are summarized and categorized. • Ilities-oriented, sequential decision modeling, data-driven optimization, and machine learning algorithm are to be studied.

A Traffic Assignment Approach for Multi-Modal Transportation Networks Considering Capacity Constraints and Route Correlations

A multi-modal urban transportation network provides travelers with diversified and convenient travel options. The study of multi-modal traffic assignment encounters great challenges from the common-line problem, route correlation, and the variability of travel tools. This article starts with the construction of super-networks to solve the common-line problem of multi-modal system. From the dimensions of time, fee, comfort, and transfer penalty, a multi-modal generalized travel cost function is proposed to reflect the impact of travel mode and transfer on route choice. Based on C-logit model, considering multi-modal capacity constraint and route correlation, a nonlinear programming model equivalent to the multi-modal stochastic user equilibrium is set up. The corresponding solving algorithm is designed by combining the augmented Lagrangian multiplier method and the successive weight average algorithm. Finally, the effectiveness of the proposed model and algorithms is verified through a numerical example, and the traffic assignment approach is applied in some typical scenarios. The multi-modal transportation network equilibrium approach proposed in this article takes into account the capacity constraints of different travel modes and solves the path overlapping problem in combined modes. It provides a basis and tool to formulate the traffic management strategy for public transport and combined mode trips.

Smoothing method in multi-criteria transportation network equilibrium problem

ABSTRACTThis paper is aimed to study a single-product multi-criteria transportation network with capacity constraints. We use a vector version of the Heaviside Step function to construct an optimization problem, the solutions of which form the set of equilibria of our model. We propose two methods to solve this problem. The first one is based on a modified Frank-Wolfe gradient algorithm, and the second one is based on smoothing the objective function, the optimal solutions of which can be obtained by optimization tools. Numerical examples are also given to illustrate our approaches.

Learning Towards Transportation Network Equilibrium: A Model Comparison Study

As an interdisciplinary topic, human travel-choice behavior has attracted the interests of transportation managers, theoretical computer science researchers and economists. Recent studies on tacit coordination in iterated route choice games (i.e., a large number of subjects could achieve the transportation network equilibrium in limited rounds) have been driven by two questions. (1) Will learning behavior promote tacit coordination in route choice games? (2) Which learning model can best account for these choices/behaviors? To answer the first question, we choose a set of learning models and conduct extensive simulations to determine their success in accounting for major behavioral patterns. To answer the second question, we compare these models to one another by competitively testing their predictions on four different datasets. Although all the selected models account reasonably well for the slow convergence of the mean route choice to equilibrium, they account only moderately well for the mean frequencies of the round-to-round switches from one route to another and fail to appropriately account for substantial individual differences. The implications of these findings for model construction and testing are briefly discussed.

Evaluation of exclusive bus lanes in a tri-modal road network incorporating carpooling behavior

This paper considers the evaluation of exclusive bus lanes (EBLs) in the road network with three travel modes: bus, solo driving, and carpooling. A tri-modal transportation network equilibrium model is developed to analyze the effects of EBLs under three different policies: (i) no EBLs (called Policy 1); (ii) EBLs can only be used by bus (called Policy 2); and (iii) EBLs can be used by both bus and carpooling modes (called Policy 3). By taking into account both EBLs setting scheme and bus frequencies, a combinatorial optimization model is proposed to test the performance of the tri-modal transportation system. In a traffic corridor case with single O-D pair, numerical results show that travel demand levels will remarkably influence the total system costs under different policies. The effects of average carpooling occupancy and mode choice parameters on travelers’ choice behavior are analyzed. Finally, a tri-modal network with nineteen links is used to illustrate that the system could be more efficient when different EBLs policies are adopted on different links.

Equity based congestion pricing: considering the constraint of alternative path

Congestion pricing is considered as an effective measure to ease urban traffic congestion. This paper develops a bi-level model that designs more equitable congestion pricing with bus subsidies. In the upper-level model, the constraint of alternative path is proposed to ensure that there is at least one non-toll path for any origin–destination (OD) pair. If one link is the only way for an OD pair just like Lidingo, Stockholm, this link should not be tolled. Meanwhile, toll revenue is used to improve the service level of transit system in which transfer fare is subsidized and the frequency of the most crowded bus is improved. The lower-level model is a multimodal transportation network equilibrium model. A shuffled pattern search algorithm and method of successive averages are used to optimize congestion pricing, transfer fare and bus frequency in the bi-level model. The results of the numerical test show that the model can meet the travel demand of travelers after tolling. Moreover, traffic congestion is alleviated while service level of transit is improved.

Unified closed-form expression of logit and weibit and its extension to a transportation network equilibrium assignment

This study proposes a generalized multinomial logit model that allows heteroscedastic variance and flexible utility function shape. The novelty of our approach is that the model is theoretically derived by applying a generalized extreme-value distribution to the random component of utility, while retaining its closed-form expression. In addition, the weibit model, in which the random utility is assumed to follow the Weibull distribution, is a special case of the proposed model. This is achieved by utilizing the q-generalization method developed in Tsallis statistics. Then, our generalized logit model is incorporated into a transportation network equilibrium model. The network equilibrium model with a generalized logit route choice is formulated as an optimization problem for uncongested networks. The objective function includes Tsallis entropy, a type of generalized entropy. The generalization of the Gumbel and Weibull distributions, logit and weibit models, and network equilibrium model are formulated within a unified framework with q-generalization or Tsallis statistics.

Research on Urban Road Congestion Pricing Strategy Considering Carbon Dioxide Emissions

Congestion pricing strategy has been recognized as an effective countermeasure in the practical field of urban traffic congestion mitigation. In this paper, a bi-level programming model considering carbon dioxide emission is proposed to mitigate traffic congestion and reduce carbon dioxide emissions. The objective function of the upper level model is to minimize the sum of travel costs and the carbon dioxide emissions costs. The lower level is a multi-modal transportation network equilibrium model. To solve the model, the method of successive averages (MSA) and the shuffled frog leaping algorithm (SFLA) are introduced. The proposed method and algorithm are tested through the numerical example. The results show that the proposed congestion pricing strategy can mitigate traffic congestion and reduce carbon emissions effectively.

A Unified Closed-form Expression of Logit and Weibit and its Application to a Transportation Network Equilibrium Assignment

This study proposes a generalized multinomial logit model where heteroscedastic variance and flexible shape of utility function are allowed. The novel point of our approach is that, while the model is theoretically derived by applying a generalized extreme value distribution to the random component of utility, the model maintains its closed-form expression. Also, the weibit model, where the random utility is assumed to follow the Weibull distribution, is a special case of the proposed model. This is achieved by utilizing q-generalization method developed in Tsallis statistics. Then, the generalized logit model is incorporated into a transportation network equilibrium model. The network equilibrium model with the generalized logit route choice is formulated as an optimization problem under uncongested networks. The objective function includes Tsallis entropy, which is a type of generalized entropy. The generalization of the Gumbel and Weibull distributions, logit and weibit models, and network equilibrium model is made within a unified framework with q-analysis or Tsallis statistics.

Q-Generalized Logit Route Choice an Network Equilibrium Model

The multinomial logit model is expressed as a closed-form equation, and plays an important role in the field of tra sportation. The Gumbel-distributed utility in the multinomial logit model is restrictive in certain applications, especially in ro te choice behavior and network equilibrium analysis, although it is mathematically convenient. The range of variation of utility in the logit model is unbounded. The Gumbel distribution is left-skewed and has a very thi tail to the left. In additionFor example, the utility in the multinomial logit has a the homogeneity of homosce astic variance. In this study, the multinomial logit model is extended by generalizing the Gumbel-distributed utility to allow heteroscedastic variance and flexible shape. Then, the generalized logit model with a generalized Gumbel distribution is incorporated into the transportation network equilibrium model. The network equilibrium model with a generalized logit route choice is formulated as an optimization problem under uncongested networks. The objective function includes Tsallis entropy, which is a type of generalized entropy. In this study, the generalization of the Gumbel distribution, logit model, and network equilibrium model has a unified framework with q-analysis or Tsallis statistics.

A Link-Node Nonlinear Complementarity Model for a Multiclass Simultaneous Transportation Dynamic User Equilibria

Static transportation network equilibrium models have evolved from traditional sequential models to simultaneous (combined) models, and then to the multiclass simultaneous models to improve prediction of traffic flow. Most Dynamic Traffic Assignment (DTA) models, however, still deal only with the trip assignment step (traveler route choice) that is one of several steps in the transportation planning process. In this paper, the authors combine a dynamic link-node based discrete-time Nonlinear Complementarity Problem (NCP) DTA model with a static Multiclass Simultaneous Transportation Equilibrium Model (MSTEM) in a unified dynamic link-node based discrete-time NCP Dynamic Multiclass Simultaneous Transportation Equilibrium Model (DMSTEM) model. The new model improves the prediction process and eliminates inconsistencies that arise when the DTA or Dynamic Traffic Assignment with Departure Time (DTA-DT) is embedded in a more comprehensive transportation planning framework. An iterative solution algorithm for the proposed DMSTEM model is proposed by solving several relaxed NCPs in each iteration of the algorithm.

Beckmann's transportation network equilibrium model: Its history and relationship to the Kuhn–Tucker conditions

During 1952–54, Martin Beckmann, and his colleagues, formulated a nonlinear programming problem corresponding to behavioral assumptions from the viewpoint of an individual traveler concerning travel demand and cost-minimizing route choice over a congested road network. Their formulation was based on the conditions for a constrained maximum, recently derived by Kuhn and Tucker. This formulation was evidently the first time that economists used the Kuhn–Tucker conditions to formulate a new problem in economics, one of substantial practical importance and consequence, and quite possibly the first to use these conditions to formulate a new, large-scale problem in all fields of engineering. In this paper, an overview of the research leading to the formulation is offered. Then, the derivation presented in their monograph is described and explored in more detail. Finally, the impacts of this model on the field of transportation economics and the associated fields of transportation engineering and regional science are examined.

Combinatorial Optimization of Exclusive Bus Lanes and Bus Frequencies in Multi-Modal Transportation Network

Bus priority strategy has widely been recognized as an effective countermeasure in the practical field of urban traffic congestion mitigation. This study proposes a bi-level programming model to comprehensively analyze the combinational optimization problem of exclusive bus lanes (EBLs) with variable bus frequencies (BFs) in multi-modal transportation networks. The objective function is to minimize the sum of the road users’ travel costs and the transit operating costs. The upper level model is an optimal decision-making program for setting up EBLs and BFs, and the lower level is a multi-modal transportation network equilibrium model. An efficient genetic algorithm is designed to solve the problem. The results of the numerical analysis show that the combinatorial optimization scheme of EBLs and BFs performs well with regard to the objective function in the model. The outperformance of combinatorial optimization scheme becomes greater with increased traffic demand, whereas oversetting of EBLs can result in reduced operating efficiency of the transportation system.

Multiperiod competitive supply chain networks with inventorying and a transportation network equilibrium reformulation

In this paper, we present a multitiered dynamic supply chain network equilibrium modeling framework in which the decision-makers have sufficient information about the future and seek to determine their optimal plans that maximize their profits over the multiperiod planning horizon. We construct the finite-dimensional variational inequality governing the equilibrium of the multiperiod competitive supply chain network. The model allows us to investigate the interplay of the heterogeneous decision-makers in the supply chain in a dynamic setting, and to compute the resultant equilibrium pattern of product outputs, transactions, inventories, and product prices. We then establish the supernetwork equivalence of the multiperiod supply chain model with a properly configured transportation network, which provides a new interpretation of the equilibrium conditions of the former in terms of paths and path flows. This framework offers great modeling flexibility so that, for example, transportation delay and/or perishable products can be easily handled, as we also demonstrate. Numerical examples are provided to illustrate how such multiperiod supply chain problems can be reformulated and solved as transportation network equilibrium problems in practice.

A Framework for Intelligent Decision Support System for Traffic Congestion Management System

Traffic congestion problem is one of the major problems that face many transportation decision makers for urban areas. The problem has many impacts on social, economical and development aspects of urban areas. Hence the solution to this problem is not straight forward. It requires a lot of effort, expertise, time and cost that sometime are not available. Most of the existing transportation planning software, specially the most advanced ones, requires personnel with lots practical transportation planning experience and with high level of education and training. In this paper we propose a comprehensive framework for an Intelligent Decision Support System (IDSS) for Traffic Congestion Management System that utilizes a state of the art transportation network equilibrium modeling and providing an easy to use GIS-based interaction environment. The developed IDSS reduces the dependability on the expertise and level of education of the transportation planners, transportation engineers, or any transportation decision makers.