Equilibrium Assignment Research Articles

A Study on Two-stage Selection Model of Tourism Destination at the Scale of Urban Agglomerations

Considering that the demand of tourism destination is variable on the scale of urban agglomeration, the selection process of travel destination is divided into two stages. The traditional transportation combination model based on the multinomial Logit cannot reflect this characteristic. And it is the lack of consideration of the influence of travel distribution and the dynamic transfer of passenger flow between various transport routes. Therefore, this thesis established a combination model of travel demand distribution and transportation assignment with two-stage terminal selection characteristics based on the nested Logit. Based on the analysis of tourists' trip process on the scale of urban agglomeration, a tourist flow transport network with travel destination nest structure is constructed. The generalized cost impedance function of transportation route is constructed based on the direct cost of transportation mode and the indirect cost of travel time. Based on the characteristics of two-stage destination selection of tourists, the form of travel distribution function of tourist flow is given. Through the first-order optimization conditions, it proved that the volume of travel distribution and tourism passenger transport assignment can meet the two-stage equilibrium conditions in the equilibrium state. Based on the idea of MSA algorithm, it designed the solution algorithm of the model and verified the feasibility of the model and algorithm in a simplified example. The calculation results show that the two-stage equilibrium assignment model proposed in this paper can obtain the volume of travel distribution and transportation assignment at the same time, meanwhile compared with the multinomial logit model, the nested Logit structure fully considers the attraction measure of the city destination and the scenic spot destination, which is more in line with the choice behavior of the tourists when choosing the transportation route. Thus, it provides a new comparable method for the optimal allocation of tourism passenger flow transport network resources on the scale of urban agglomeration, and can provide data support for the transportation organization plans of government decision-making departments and tourism transport enterprises.

Transit Frequency Optimization in Bi-modal Networks Using Differential Evolution Algorithm

This study proposes a bi-level optimization model for the transit frequency setting problem in bi-modal networks. The objective of the upper-level problem is to obtain a solution set of bus line frequencies that provide the minimum total travel cost of the car and bus users. Differential Evolution (DE) algorithm is employed in the upper-level model to determine the optimal headways for a given route structure. The lower-level model is a congested multi-modal user equilibrium assignment model, which considers the interactions of car and bus flows, for determining joint mode/route preferences of the network users, which considers the interactions of car and bus flows. The developed model is tested on Mandl's benchmark network to evaluate its performance and applicability. The comparative experiments demonstrate that the proposed model leads to reductions in transportation costs. Also, the results of numerous optimization runs show that DE performs well in finding similar frequency sets in independent optimizations.

Mitigation strategy of cascading failures in urban traffic congestion based on complex networks

Urban road traffic network is becoming increasingly complex. The increasing travel demand has exceeded facility supply of transportation system. Thus, a scientific way of transportation can solve the problem of traffic congestion. If some central intersections are congested, the road network could easily lose its travel efficiency and further lead to cascading failure. In this regard, from the viewpoint of complex networks, this paper provides a cascading failure mitigation strategy for the urban road traffic network. A load redistribution strategy is represented which can redistribute congestion load reasonably. Random and intentional failures are simulated with MATLAB on Barabási and Albert (BA) scale-free network. The indices related to complex network of different methods are compared on BA network and part of the real road network in Baoding City. The load redistribution strategy in this paper is implemented combined with user equilibrium assignment, then the travel time related to urban road network is compared with another method on the Sioux Falls network. The simulation results suggest that when the capacity parameter reaches the threshold, further increase has little effect on the mitigation time. Comparisons of other mitigation approaches verify the effectiveness of our approach in lessening cascading failure caused by traffic congestion. The increase in total network travel time and additional delay by reason of load redistribution is acceptable.

Container Shipping Scheduling Method Based on the Evidence Reasoning Approach in Fluctuating CCFI and BDI Cycle

Due to the COVID-19/Omicron pandemic and the trade war and tariffs between China and America, the supply chain between Asia and America, or Asia and Europe, faced unprecedented challenges. With the outbreak of the Russian-Ukrainian war, the global supply chain has become increasingly unbalanced. In particular, the shortage of container ships and the continuous fluctuation of the BDI and CCFI make shipping scheduling increasingly important for ocean carriers. To solve this problem, in this study, we propose an analytic approach considering the fluctuation of BDI and CCFI based on interval evidence reasoning and the Hungarian algorithm. The proposed approach uses two pairs of nonlinear optimisation models to construct a Nash equilibrium assignment model to compute the shipping company’s scheduling and maximum utility in the CCFI and BDI cycles. Compared to the simulation algorithm, the analytical algorithm can take advantage of stability and efficiency. Finally, a COSCO company’s container shipping scheduling problem was examined to demonstrate the efficiency and effectiveness of the proposed approach.

Path Choice Models in Stochastic Assignment: Implementation and Comparative Analysis

In assignment models, a key role is played by the path choice simulation that evaluates the path chosen by users in relation to the perceived paths and relative costs. This study deals with the effects of the implementation of some most adopted path choice models (Logit, Weibit, Probit, and Gammit) within a Stochastic User Equilibrium assignment procedure. Some considerations on parameters needed to make results comparable and the method used to estimate them are also suggested some extensions based on Weibit model are proposed. Results obtained both on a test network and on a real one are reported.

Flow-Capture Location Model with Link Capacity Constraint Over a Mixed Traffic Network

Abstract This paper constructs and settles a charging facility location problem with the link capacity constraint over a mixed traffic network. The reason for studying this problem is that link capacity constraint is mostly insufficient or missing in the studies of traditional user equilibrium models, thereby resulting in the ambiguous of the definition of road traffic network status. Adding capacity constraints to the road network is a compromise to enhance the reality of the traditional equilibrium model. In this paper, we provide a two-layer model for evaluating the efficiency of the charging facilities under the condition of considering the link capacity constraint. The upper level model in the proposed bi-level model is a nonlinear integer programming formulation, which aims to maximize the captured link flows of the battery electric vehicles. Moreover, the lower level model is a typical traffic equilibrium assignment model except that it contains the link capacity constraint and driving distance constraint of the electric vehicles over the mixed road network. Based on the Frank-Wolfe algorithm, a modified algorithm framework is adopted for solving the constructed problem, and finally, a numerical example is presented to verify the proposed model and solution algorithm.

Risk-Based Formulation of the Transit Priority Network Design

Demand uncertainties are inevitable in transportation networks. The transit priority network design problem over more than a decade of development has been solved under deterministic conditions. This paper proposes a model to find the optimal transit priority scheme in a multimodal transportation network under uncertain demand. This model is formulated as a risk-based bi-level optimization problem. At the upper-level, a risk measure of expected social cost is minimized subject to a chance constraint on total travel time with a user-specified confidence level and a budget constraint. At the lower-level, a mode choice, a traffic user equilibrium assignment, and a transit assignment are applied. An ant colony algorithm is utilized to solve this complex design problem. Numerical results using a real world middle-size city network empirically demonstrate that the demand uncertainty has a significant impact on the solution and the proposed model is applicable to realistic networks.

A congested schedule-based dynamic transit passenger flow estimator using stop count data

A network-level dynamic transit passenger flow estimation model based on congested schedule-based transit equilibrium assignment is proposed using observations from stop count data. A solution algorithm is proposed for the mathematical program with schedule-based transit equilibrium constraints with proven quadratic space- and time-complexities. The error bound is proven to be linearly proportional to the number of measurements. Computational experiments are conducted first to verify the methodology with two synthetic data sets (one of which is Sioux Falls compared to a benchmark model), followed by a validation of the method using bus data from Qingpu District in Shanghai, China from 1 July 2016, with 4 bus lines, 120 segments, and 55 bus stops. The estimated average of segment flows is only 2.5% off from the average of the observed segment flows; relative errors among segments are 42.5%, which fares well compared with even less complex OD estimation methods in the literature.

On the Gains from Tradable Benefits‐in‐kind: Evidence for Workfare in India

Widely‐used in‐kind benefits are often rationed across potential participants in social programmes. This poses an operational challenge in implementation with limited public information. One possible solution is to randomize assignment given the available information. However, as this paper shows, neither the purposive targeting of in‐kind benefits nor randomized assignment is likely to yield a competitive equilibrium given private information on diverse personal gains. The paper characterizes equilibrium assignments and the implications for policy and evaluation. The theory is applied to an antipoverty programme providing rationed jobs on rural public works in a poor state of India. Special‐purpose survey data reveal large unexploited gains from allowing tradable assignments. The paper shows that the potential gains exceed those from poverty targeting and stylized cash transfer options. Realizing the scope for poverty reduction by such means will undoubtedly require complementary efforts in reducing the influence of other market and institutional factors creating poverty in the first place, including inequalities in access to credit and knowledge.

Sensitivity analysis for transit equilibrium assignment and applications to uncertainty analysis

Systematic uncertainty analysis can be used to quantitatively evaluate variation in model outputs and identify the critical sources of uncertainty to improve the reliability and stability of a system. To analyze the effects of uncertainties in transit networks that may be caused by probabilistic travel demand, congestion, or vehicle frequencies, this study develops a sensitivity-based uncertainty analysis approach as a post-analysis tool for equilibrium transit systems. The congestion effect is considered in the waiting time and in-vehicle travel time of a passengers’ route-choice model. The hyperpath concept is used to manage passengers’ riding strategies due to the common-line problem at transit stops. A hyperpath-based gradient projection (GP) solution algorithm is developed for the solution of the variational inequality formulation of the transit equilibrium assignment problem (TEAP). A restricted sensitivity analysis approach originally developed for road networks is re-developed for the TEAP in transit networks. An analytical sensitivity-based approach is derived to conduct uncertainty analysis for the TEAP, which enables the simultaneous propagation of uncertainties from different input sources to the model outputs. Numerical examples are provided for the following purposes. (1) To demonstrate three applications of the sensitivity analysis of the TEAP, namely the perturbed solution estimation problem, the critical parameter identification problem, and the paradox analysis problem. (2) To illustrate the use of uncertainty analysis of the TEAP, such as estimating the variance and confidence level of model outputs with respect to various model inputs/parameters as random variables, and ranking the importance of arcs using sensitivity-based uncertainty analysis. (3) To demonstrate the applicability of the proposed approach to real transit networks. The findings demonstrate not only the importance of the analytical sensitivity analysis development for the TEAP, but also for the practical applications of sensitivity and uncertainty analyses.

An electric vehicle charging station access equilibrium model with M/D/C queueing

Despite the dependency of electric vehicle (EV) fleets on charging station availability, charging infrastructure remains limited in many cities. Three contributions are made. First, we propose an EV-to-charging station user equilibrium (UE) assignment model with a M/D/C queue approximation as a nondifferentiable nonlinear program. Second, to address the non-differentiability of the queue delay function, we propose an original solution algorithm based on the derivative-free Method of Successive Averages. Computational tests with a toy network show that the model converges to a UE. A working code in Python is provided free on Github with detailed test cases. Third, the model is applied to the large-scale case study of New York City Department of Citywide Administrative Services (NYC DCAS) fleet and EV charging station configuration as of July 8, 2020, which includes unique, real data for 563 Level 2 chargers and 4 Direct Current Fast Chargers (DCFCs) and 1484 EVs distributed over 512 Traffic Analysis Zones. The arrival rates of the assignment model are calibrated in the base scenario to fit an observed average utilization ratio of 7.6% in NYC. The model is then applied to compare charging station investment policies of DCFCs to Level 2 charging stations based on two alternative criteria. Results suggest a policy based on selecting locations with high utilization ratio instead of with high queue delay.

Formulating multi-class user equilibrium using mixed-integer linear programming

We introduce an approach to formulate and solve the multi-class user equilibrium traffic assignment as a mixed-integer linear programming (MILP) problem. Compared to simulation approaches, the analytical MILP formulation makes the solution of network assignment problems more tractable. When applied in a multi-class context, it obviates the need to assume a symmetrical influence between classes and thereby allows richer traffic behavior to be taken into account. Also, it integrates naturally in optimization problems such as maintenance planning and traffic management. We develop the model and apply it for the Sioux Falls network, showing that it outperforms the traditional Beckmann-based and MSA approaches in smaller-scale problems. Further research opportunities lie in developing extensions of MILP-based assignment, with different variants of user equilibrium or dynamic assignment, and in improving the model and solution algorithms to allow large-scale application. • We formulate multi-class static traffic assignment using MILP. • The new formulation is analytical and therefore prescriptive. • We compare the performance of MILP with other existing methods in the Sioux Falls network. • This formulation opens a new axis for future works.

Prioritizing People - Mixed Equilibrium Assignment for Av Based on Occupancy

Prioritizing People - Mixed Equilibrium Assignment for Av Based on Occupancy

Multi-vehicle assignment with elastic vehicle choice behaviour: Fixed-point, deterministic process and stochastic process models

New types of vehicles, such as electricity-powered and/or autonomous cars or buses, are under rapid development, but the time needed to turn the existing stock of traditional vehicles, such as privately-owned human-driven fossil fuelled passenger cars, will likely last several years during which mixed traffic is expected. Moreover, the likely effects of the introduction of such vehicles may not be necessarily positive. Thus, a careful analysis of their various impacts through well designed tools is needed. Main tools for transportation systems analysis are based on methods for travel demand assignment to a transportation network, commonly used to support transportation project assessment and evaluation.The original contribution of this paper is twofold. First, a modelling approach to multi vehicle type assignment is proposed, in this approach the user choice proportions among vehicle types available to users are the results of an explicit choice behaviour model instead of input data as in all existing papers on multi-vehicle assignment. This approach is applied for analysing assignment to transportation networks including Fixed-Point models for equilibrium assignment and Deterministic and Stochastic Process models for day-to-to dynamic assignment.Second, statistical methods are introduced to study the bifurcation of SP models, in order to account for the stochasticity of such models. In particular, the Hartigan’s Dip statistic, which measure the degree of unimodality of a distribution, showed to be the most effective one in identifying bifurcation in SP models.Fixed-point stability and bifurcation analysis are carried out too, and some simple numerical examples are also discussed.Reported results show that innovative vehicles with less impact on congestion than traditional ones may not have a positive effect on equilibrium stability since negative effect of reducing route choice dispersion may be compensate the positive effects on congestion.

Markovian traffic equilibrium assignment based on network generalized extreme value model

This study establishes Markovian traffic equilibrium assignment based on the network generalized extreme value (NGEV) model, which we call NGEV equilibrium assignment. The use of the NGEV model for route choice modeling has recently been proposed, and it enables capturing the path correlation without explicit path enumeration. However, the theoretical properties of the model in traffic assignment have yet to be investigated in the literature, which has limited the practical applicability of the NGEV model in the traffic assignment field. This study addresses the research gap by providing the theoretical developments necessary for the NGEV equilibrium assignment. We first show that the NGEV assignment can be formulated and solved under the same path algebra as the traditional Markovian traffic assignment models. Moreover, we present the equivalent optimization formulations to the NGEV equilibrium assignment. The formulations allow us to derive both primal and dual types of efficient solution algorithms. In particular, the dual algorithm is based on the accelerated gradient method that is for the first time applied in the traffic assignment. The numerical experiments showed the excellent convergence and complementary relationship of the proposed primal and dual algorithms.

Urban Road Network Maintenance Scheduling Using Ant Colony Optimization

In this study, an optimization model for road maintenance scheduling is proposed. The proposed model determines a maintenance schedule for a given set of links in an urban road network, that minimizes the total travel time during the maintenance period under a set of assumptions and a particular crew number constraint. It is assumed that all lanes of a road to be repaired are closed to traffic during the maintenance. Ant Colony Optimization (ACO), a suitable algorithm for discrete transportation problems, is employed in the proposed model. The model generates a set of roads to be closed for each maintenance day. The total travel time during the whole maintenance period is calculated using the deterministic user equilibrium assignment model. The proposed model is applied to a test network, and more efficient schedules in terms of total travel time are obtained compared to randomly generated schedules.

Schelling games on graphs

We study strategic games inspired by Schelling's seminal model of residential segregation. These games are played on undirected graphs, with the set of agents partitioned into multiple types; each agent either aims to maximize the fraction of her neighbors who are of her own type, or occupies a node of the graph and never moves away. We consider two natural variants of this model: in jump games agents can jump to empty nodes of the graph to increase their utility, while in swap games they can swap positions with other agents. We investigate the existence, computational complexity, and quality of equilibrium assignments in these games, both from a social welfare perspective and from a diversity perspective. Some of our results extend to a more general setting where the preferences of the agents over their neighbors are defined by a social network rather than a partition into types.

Modeling the α-max capacity of transportation networks: a single-level mathematical programming formulation

Network capacity, defined as the largest sum of origin–destination (O–D) flows that can be accommodated by the network based on link performance function and traffic equilibrium assignment, is a critical indicator of network-wide performance assessment in transportation planning and management. The typical modeling rationale of estimating network capacity is to formulate it as a mathematical programming (MP), and there are two main approaches: single-level MP formulation and bi-level programming (BLP) formulation. Although single-level MP is readily solvable, it treats the transportation network as a physical network without considering level of service (LOS). Albeit BLP explicitly models the capacity and link LOS, solving BLP in large-scale networks is challenging due to its non-convexity. Moreover, the inconsideration of trip LOS makes the existing models difficult to differentiate network capacity under various traffic states and to capture the impact of emerging trip-oriented technologies. Therefore, this paper proposes the α-max capacity model to estimate the maximum network capacity under trip or O–D LOS requirement α. The proposed model improves the existing models on three aspects: (a) it considers trip LOS, which can flexibly estimate the network capacity ranging from zero to the physical capacity including reserve, practical and ultimate capacities; (b) trip LOS can intuitively reflect users’ maximum acceptable O–D travel time or planners’ requirement of O–D travel time; and (c) it is a convex and tractable single-level MP. For practical use, we develop a modified gradient projection solution algorithm with soft constraint technique, and provide methods to obtain discrete trip LOS and network capacity under representative traffic states. Numerical examples are presented to demonstrate the features of the proposed model as well as the solution algorithm.

A Bi-Level Model for Green Freight Transportation Pricing Strategy Considering Enterprise Profit and Carbon Emissions

Due to the impact of COVID-19, enterprises need effective pricing strategies to improve profits and viability. In order to fill research gaps in the literature relating to market competition among different freight modes and to adjust the freight transportation structure by optimizing transportation prices, we propose a multi-objective bi-level programming pricing model that considers market competition and the carbon emissions of the freight system in China. First, an objective function in upper-level planning was used to improve logistics enterprise profits and reduce the total carbon emissions of the freight system. Then, a generalized cost function for the freight transportation mode was designed to quantify the market competition among different transportation modes, and a user equilibrium assignment model was established to obtain the results of cargo flow assignment in lower-level planning. To solve the model, a sensitivity analysis algorithm was designed, and a logistics network example was used to verify the effectiveness of the model. The experimental results show that reasonable freight price adjustment can effectively increase enterprise profits and reduce the total carbon emissions of the freight system. In this paper, we provide a new method for freight pricing research, considering the market competition of multiple transportation modes, and provide a new idea for adjusting the transportation structure through freight price optimization, which will play a positive role in promoting the development of green freight.

Route based equilibrium assignment in congested transit networks

This work introduces a new route-based equilibrium assignment formulation and algorithm for congested transit networks. It is an extension of De Cea and Fernández (1993), which implements congestion effects in the boarding process and introduced the concept of effective frequencies. That seminal work provides a solution algorithm for the equilibrium problem that takes advantage of two main simplifications. First, it limits the number of route sections under consideration (i.e., it considers a reduced set of attractive services for users to choose from on each leg of their trip). Second, it assumes that the flow split between attractive services can be estimated using the nominal frequencies of the services instead of the effective ones.In this paper, we develop an equilibrium model and solution algorithm that extend the original formulation without relying on the two assumptions mentioned above, while still addressing three key challenges. First, user cost functions are asymmetric in nature, and therefore, an equivalent optimization problem cannot be formulated. Second, cost functions on this approach can only be expressed implicitly, which makes the implementation of a diagonalization algorithm a challenging problem. Lastly, dealing with the whole set of route sections implies working with all feasible sets of attractive services, which can make the underlying auxiliary network grow exponentially in size.We solve the equilibrium problem by using a diagonalization algorithm, obtaining approximate diagonalized functions by solving a fixed-point problem. Our methodology also includes an efficient algorithm to generate potential candidate sets of services in the network. We test our algorithm on three networks of increasing size, where we show that the approach effectively converges to a user equilibrium solution in a matter of seconds. We also show that the benchmark original algorithm converges to solutions that violate user equilibrium conditions, exhibiting unutilized routes with lower than equilibrium costs, confirming that our algorithm is the first one to find an exact solution to the route-based public transport user equilibrium problem.