Sensitivity analysis for stochastic user equilibrium traffic assignment with constraints

The stochastic user equilibrium (SUE) traffic assignment model integrates the concepts of random assignment and Wardrop equilibrium, and it makes the assignment results more reasonable. The conventional Logit-based SUE model is explicitly built upon path flow variables, which encumbers its large-scale applications and further theoretical analyses. In this paper, a link-based SUE traffic assignment model with link capacity constraints is developed by using the network equilibrium modeling approach. The solution equivalence and uniqueness of the proposed model are rigorously proved. An efficient solution algorithm is also proposed to obtain the equilibrium solution. Numerical results indicate that the proposed link-based SUE model can reflect the real route choice behavior of travelers more reasonable and also facilitate the algorithmic development and further theoretical analyses.

This paper addresses a general stochastic user equilibrium (SUE) traffic assignment problem with link capacity constraints. It first proposes a novel linearly constrained minimization model in terms of path flows and then shows that any of its local minimums satisfies the generalized SUE conditions. As the objective function of the proposed model involves path‐specific delay functions without explicit mathematical expressions, its Lagrangian dual formulation is analyzed. On the basis of the Lagrangian dual model, a convergent Lagrangian dual method with a predetermined step size sequence is developed. This solution method merely invokes a subroutine at each iteration to perform a conventional SUE traffic assignment excluding link capacity constraints. Finally, two numerical examples are used to illustrate the proposed model and solution method.

Turn‐delays in intersections contribute significantly to travel times and thus route choices in urban networks. However, turns are difficult to handle in traffic assignment models due to the asymmetric Jacobian in the cost functions. The paper describes a model where turn delays have been included in the solution algorithm of Stochastic User Equilibrium (SUE) traffic assignment. When the Jacobian is symmetric, SUE minimises the road users' ‘perceived travel resistances’. This is a probit‐model where the links cost‐functions of the links are traffic dependent. Hereby, overlapping routes are handled in a consistent way. However, no theoretical proof of convergence has been given if the Jacobian is asymmetric, although convergence can be shown probable for model data representing realistic road‐networks. However, according to the authors knowledge SUE with intersection delays have not been tested earlier on a full‐scale network. Therefore, an essential part of the paper presents practical tests of convergence. Both geometric delays and delays caused by other turns are considered for each turn. Signalised and non‐signalised intersections are handled in different ways, as are roundabouts. In signalised intersections a separate model handles queues longer than one green‐period. Green‐waves can also be taken into consideration. The model has been tested on a large‐scale network for Copenhagen with good results. To make it possible to establish the comprehensive data, a GIS‐based ‘expert system’ was implemented (see Nielsen, O.A., Frederiksen, R. D. and Simonsen, N. (1997). Using expert system rules to establish data on intersections and turns in road networks. International Transactions in Operational Research, 5, 513–529.

Public transport (PT) agencies are increasingly keen on integrating ride-hailing (RH) services with PT to improve overall mobility. Understanding the traffic flow distribution in the integrated system is vital for the policy decision-making and services design of such a system. We propose a stochastic user equilibrium (SUE) model for multimodal transport systems consisting of private car, PT and RH. The travel costs in the SUE model are investigated using a multimodal graph representation to capture the relationship of different travel modes in the integrated system. We apply the proposed model to a toy case and a real-world case. A RH subsidy strategy is compared with the benchmark to demonstrate travellers’ route and mode shifts in the integrated system. Our findings offer insights on subsidising RH services through the proposed model, and provide valuable knowledge on the planning and design of the integrated system.

When we evaluate the performance reliability of a network, it is necessary to describe user's behaviour in a partially degraded network. This paper shows that the Stochastic User Equilibrium (SUE) model assuming user's route choice behaviour under uncertain network conditions can be incorporated in the performance reliability model. The effects of providing information are analyzed using the SUE model with two different groups of route choice; informed drivers and non informed drivers. It is found that providing information generally increases network performance reliability. This depends, however, on the probability distribution of the network states.

This paper first analyzes link and route travel time distributions in a degradable road network by assuming a uniform distribution on the capacity of each link. It is postulated that travelers, by acquiring route travel time variability from past experiences, make route decisions based on a travel time budget, which is the summation of the mean route travel time and the safety margin of the travel time. This paper then formulates a novel travel time budget-based stochastic user equilibrium traffic assignment model with multiple user classes and elastic demand as a variational inequality problem. Finally, a numerical example of a small road network is presented to illustrate the properties of the proposed model.

An economic analysis of a multi-modal transportation system with ride-sourcing services and multi-class users

The work zone on the urban road network will affect the surrounding road traffic. To represent the influence area of the work zone, the concept of a subnetwork is proposed in this paper. Delineating a suitable subnetwork quantitatively is a challenging problem. To address this issue, the node synthesized indexes (NSI) are deployed as a variability measure that captures both the change of link flow and origin-destination (OD) demand. The inertia-based stochastic user equilibrium with the elastic demand (ISUEED) model is proposed to accurately provide the data of link flow and OD demand for the network with the work zone. Correspondingly, the data of the network without a work zone can be obtained by the stochastic user equilibrium (SUE) model. According to the value of NSI, the initial range of the subnetwork is determined. Finally, the connectivity and compactness can be guaranteed by the modified L-shell algorithm. To demonstrate the performance of the method, two case studies and sensitivity analyses are conducted based on the Braess network and the local road network in Changchun, China. The proposed method is beneficial to reduce the complexity of the traffic model by substituting the entire network with a subnetwork.

As one of the most promising travel modes in the modern urban transportation network, ridesharing is widely favored because of its convenience and lower travel costs. In the literature on ridesharing equilibrium modeling, the deterministic user equilibrium (DUE) and logit-based stochastic user equilibrium (SUE) are often adopted in the role choice or route choice. Although SUE models tackle the inability of DUE models that cannot consider perception errors, the logit-based SUE model assumes the perception variance is fixed for all travel modes or travel routes. In order to deal with the homogenous perception variance issue, this study proposes a weibit-based SUE model to characterize the mode choice and route choice in the stochastic ridesharing user equilibrium (SRUE) problem. In the proposed model, we focus on the travelers' conversion among three modes (i.e., solo driver, ridesharing driver, ridesharing passenger) and explicitly consider the relationship between origin-destination (O–D) supply (i.e., the number of ridesharing drivers) and O–D demand (i.e., the number of ridesharing passengers). Moreover, we consider a non-additive path cost function that consists of travel time, monetary cost, ridesharing compensation, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$a$</tex> nd/or inconvenient cost in the SRUE problem. We customize a solution framework for solving this novel model efficiently. Numerical examples are conducted to illustrate the applicability of the weibit-based SRUE model.

This study presents a comprehensive comparative analysis of the effect of route set size on stochastic user equilibrium (SUE) traffic assignment, focusing on both logit-based (Multinomial Logit (MNL) and Path Size Logit (PSL)) and weibit-based models (Multinomial Weibit (MNW) and Path Size Weibit (PSW)). The primary objective is to investigate the influence of route set size on traffic patterns and determine the minimum requisite number of routes for flow stabilization within the SUE framework. The analysis, conducted on the Winnipeg network using a customized Self-Regulated Averaging (SRA) scheme, yields three key findings. First, all models successfully converged, but the weibit-based models (MNW and PSW) converged faster than the logit-based models. Second, an analysis of perceived total travel time demonstrated that the majority of efficiency gains from route inclusion diminish after a threshold of approximately maximum 30 routes to 40 routes per O-D pair, indicating this number is sufficient for achieving stable SUE results in both model families. Third, the weibit-based model was found to be more sensitive to route overlap effects, continuing to adjust flow patterns up to maximum 45 routes per O-D pair, and exhibiting a greater tendency to allocate flow to less overlapping outer roads. This highlights the superior capability of the weibit formulation, which accounts for heterogeneous perception variance, to achieve a more behaviorally realistic equilibrium compared to the logit models.

Based on the random utility theory of passengers' boarding choices, this paper has proposed a Stochastic User Equilibrium Model with elastic demand for different trains. Taking the Stochastic User Equilibrium Model as the lower model, it has constructed a Bi-level Programming Model in which the upper model aims for maximizing the railroad department's income. It has solved the lower model with MSA algorithm to obtain the flow distribution by using Bi-level Programming with Particle Swarm Optimization. The passenger-dedicated line from Xi'an to Lanzhou is taken as an example, where rational off -line range under different moving distances on high-speed line was obtained.

This study introduces the novel concept of local detouredness, i.e. detours on subsections of a route, as a new phenomenon for understanding and modelling route choice. Traditionally, Stochastic User Equilibrium (SUE) traffic assignment models have been concerned with judging the attractiveness of a route by its total route cost. However, through empirical analysis we show that considering solely the global properties of a route is insufficient. We find that it is important to consider local detouredness both when determining realistic and tractable route choice sets and when determining route choice probabilities. For example, analysis of observed route choice data shows that route usage tends to decay with local detouredness, and that there is an apparent limit on the amount of local detouredness seen as acceptable. No existing models can account for this systematically and consistently, which is the motivation for the new route choice model proposed in this paper: the Bounded Choice Model with Local Detour Threshold (BCM-LDT). The BCM-LDT model incorporates the effect of local detouredness on route choice probability, and has an in-built mechanism that assigns zero probabilities to routes violating a bound on total route costs and/or a threshold on local detouredness. Thereby, the model consistently predicts which routes are used and unused. Moreover, the probability expression is closed-form and continuous. SUE conditions for the BCM-LDT are given, and solution existence is proven. Exploiting the special structure of the problem, a novel solution algorithm is proposed where flow averaging is integrated with a modified branch-and-bound method that iteratively column-generates all routes satisfying local and global bounds. Numerical experiments are conducted on small-scale and large-scale networks, establishing that equilibrated solutions can be found and demonstrating the influence of the BCM-LDT parameters on choice set size and flow allocation.

In order to design the traffic network more accurately, the bi-level programming model for the continuous network design problem based on the paired combinatorial Logit stochastic user equilibrium model is proposed in this study. In the model, the paired combinatorial Logit stochastic user equilibrium model which is used to characterize the route choice behaviors of the users is adopted in the lower level model, and the minimum summation of the system total costs and investment amounts is used in the upper objective function. The route-based self-regulated averaging (SRA) algorithm is designed to solve the stochastic user equilibrium model and the genetic algorithm (GA) is designed to get the optimal solution of the upper objective function. The effectiveness of the proposed combining algorithm which contains GA and SRA is verified by using a simple numerical example. The solutions of the bi-level models which use the paired combinatorial Logit stochastic user equilibrium model in the lower level model with different demand levels are compared. Finally, the impact of the dispersion coefficient parameter which influences the decision results of the network design problem is analyzed.

Differential sensitivity analysis is indispensable in fitting parameters, understanding uncertainty, and forecasting the results of both thought and lab experiments. Although there are many methods currently available for performing differential sensitivity analysis of biological models, it can be difficult to determine which method is best suited for a particular model. In this paper, we explain a variety of differential sensitivity methods and assess their value in some typical biological models. First, we explain the mathematical basis for three numerical methods: adjoint sensitivity analysis, complex perturbation sensitivity analysis, and forward mode sensitivity analysis. We then carry out four instructive case studies. (a) The CARRGO model for tumor-immune interaction highlights the additional information that differential sensitivity analysis provides beyond traditional naive sensitivity methods, (b) the deterministic SIR model demonstrates the value of using second-order sensitivity in refining model predictions, (c) the stochastic SIR model shows how differential sensitivity can be attacked in stochastic modeling, and (d) a discrete birth-death-migration model illustrates how the complex perturbation method of differential sensitivity can be generalized to a broader range of biological models. Finally, we compare the speed, accuracy, and ease of use of these methods. We find that forward mode automatic differentiation has the quickest computational time, while the complex perturbation method is the simplest to implement and the most generalizable.

Sensitivity analysis characterises input–output relationships for mathematical models, and has been widely applied to deterministic models across many applications in the life sciences. In contrast, sensitivity analysis for stochastic models has received less attention, with most previous work focusing on well-mixed, non-spatial problems. For explicit spatio-temporal stochastic models, such as random walk models (RWMs), sensitivity analysis has received far less attention. Here we present a new type of sensitivity analysis, called parameter-wise prediction, for two types of biologically-motivated and computationally expensive RWMs. To overcome the limitations of directly analysing stochastic simulations, we employ continuum-limit partial differential equation (PDE) descriptions as surrogate models, and we link these efficient surrogate descriptions to the RWMs using a range of biophysically-motivated measurement error models. Our approach is likelihood-based, which means that we also consider likelihood-based parameter estimation and identifiability analysis along with parameter sensitivity. The new approach is presented for two important classes of lattice-based RWM including a classical model where crowding effects are neglected, and an exclusion process model that explicitly incorporates crowding. Our workflow illustrates how different process models can be combined with different measurement error models to reveal how each parameter impacts the outcome of the expensive stochastic simulation. Open-access software to replicate all results is available on GitHub.