Fuzzy preference based traffic assignment problem

This paper introduces a fuzzy preference based model of route choice. The core of the model is FiPV (Fuzzy individuelle Präferenzen von Verkehrsteilnehmern or fuzzy traveler preferences), that is a choice function based on fuzzy preference relations for travel decisions. The proposed model may be the first application of fuzzy individual choice in traffic assignment and probably also the first in this class to consider the spatial knowledge of individual travelers. It is argued that travelers do not or cannot always follow the maximization principle. Therefore we formulate a model that also takes into account the travelers with non-maximizing behavior. The model is based on fuzzy preference relations, of which elements are fuzzy pairwise comparisons between the available alternatives.     

On the dual approach to the traffic assignment problem

This paper attempts to explore the possibility of solving the traffic assignment problem with elastic demands by way of its dual problem. It is shown that the dual problem can be formulated as a nonsmooth convex optimization problem of which the objective function values and subgradients are conveniently calculated by solving shortest path problems associated with the transportation network. A subgradient algorithm to solve the dual problem is presented and limited computational experience is reported. The computational results are encouraging enough to demonstrate the effectiveness of the proposed approach.     

The problem of cyclic flows in traffic assignment

A fundamental property of traffic assignment is that cyclic flows from a common origin or to a common destination cannot exist in an equilibrium solution. However, cyclic flows can easily be created by the Frank-Wolfe (F-W) assignment procedure, especially during its first several iterations. The PARTAN technique—a more rapidly converging derivative of the F-W method—can also create cyclic flows during its procedure. We show in this paper that once cyclic flows become part of a combined assignment, they are difficult to correct, thus presenting one impediment to convergence. We then present modifications to the F-W and PARTAN procedures that prevent cyclic flows from being created between adjacent pairs of nodes. The avoidance of cyclic flows in test problems is shown to accelerate the convergence of both the F-W and PARTAN techniques, particularly in the first several iterations. While the impossibility of cyclic flows in a true equilibrium solution is an important property of traffic assignment, this paper shows that (1) the F-W and PARTAN procedures eventually reduce cyclic flows to zero if they occur, (2) avoiding cyclic flows can be most helpful in the early iterations of these procedures, and (3) avoiding cyclic flows in large networks is very difficult because of large computational requirements.     

A path-based flow formulation for the traffic assignment problem

In order to improve cooperation between traffic management and travelers, traffic assignment is the key component to achieve the objectives of both traffic management and route choice decisions for travelers. Traffic assignment can be classified into two models based on the behavioral assumptions governing route choices: User Equilibrium (UE) and System Optimum (SO) traffic assignment. According to UE and SO traffic assignment, travelers usually compete to choose the least cost routes to minimize their own travel costs, while SO traffic assignment requires travelers to work cooperatively to minimize overall cost in the road network. Thus, the paradox of benefits between UE and SO indicates that both are not practical. Thus, a solution technique needs to be proposed to balance UE and SO models, which can compromise both sides and give more feasible traffic assignments. In this paper, Stackelberg game theory is introduced to the traffic assignment problem, which can achieve the trade-off process between traffic management and travelers. Since traditional traffic assignments have low convergence rates, the gradient projection algorithm is proposed to improve efficiency.     

The dual of the traffic assignment problem with elastic demands

This article is concerned with the dual of the traffic assignment problem, and of the combined generation, distribution, and assignment problem. The duals, and duality relations, for the arc-chain and node-arc formulations of the problem are derived using only the Kuhn-Tucker conditions for convex programs. This has the advantage of being more familiar to most readers than the conjugate function presentation which has been used elsewhere.     

A modified Frank-Wolfe algorithm for solving the traffic assignment problem

This paper presents a very simple modification of the Frank-Wolfe algorithm for the solution of the traffic assignment problem. It is shown that the modified algorithm can be implemented without much increase in computational effort over the original one. Convergence of the algorithm is proved and computational results are reported to demonstrate the validity of the modification.     

An equilibrium algorithm for the spatial aggregation problem of traffic assignment

Present traffic assignment methods require that all possible origins and destinations of trips taking place within a study area be represented as if they were taking place to and from a small set of points or centroids. Each centroid is supposed to represent the location of all trip-ends within a given zone, and this necessarily misrepresents points located at the edges of the zone.In order to alleviate this problem (which we refer to as the spatial aggregation problem) one could use smaller zones and more centroids, but existing traffic assignment algorithms cannot efficiently handle many centroids.This paper introduces an algorithm procedure which is designed to handle a substantially larger number of centroids. In the paper that follows, the technique is further developed to take into account a continuous distribution of population.     

An equilibration scheme for the traffic assignment problem with elastic demands

This note presents an algorithm for the solution of the traffic assignment problem with elastic demands. The algorithm is based on the concept of “equilibration operator” introduced by Dafermos and Sparrow (1969) for the solution of the traffic assignment problem with fixed demands. Computational experience is provided for linear and nonlinear problems for both the algorithm proposed here and the Dafermos-Sparrow algorithm applied to the “excess-demand” reformulations of the problems.     

Cost scaling based successive approximation algorithm for the traffic assignment problem

This paper presents a cost scaling based successive approximation algorithm, called ε-BA (ε-optimal bush algorithm), to solve the user equilibrium traffic assignment problem by successively refining ε-optimal flows. As ε reduces to zero, the user equilibrium solution is reached. The proposed method is a variant of bush-based algorithms, and also a variant of the min-mean cycle algorithm to solve the min-cost flow by successive approximation. In ε-BA, the restricted master problem, implying traffic equilibration restricted on a bush, is solved to ε-optimality by cost scaling before bush reconstruction. We show that ε-BA can reduce the number of flow operations substantially in contrast to Dial’s Algorithm B, as the former operates flows on a set of deliberately selected cycles whose mean values are sufficiently small. Further, the bushes can be constructed effectively even if the restricted master problem is not solved to a high level of convergence, by leveraging the ε-optimality condition. As a result, the algorithm can solve a highly precise solution with faster convergence on large-scale networks compared to our implementation of Dial’s Algorithm B.     

A cell-based Merchant-Nemhauser model for the system optimum dynamic traffic assignment problem

A cell-based variant of the Merchant-Nemhauser (M-N) model is proposed for the system optimum (SO) dynamic traffic assignment (DTA) problem. Once linearized and augmented with additional constraints to capture cross-cell interactions, the model becomes a linear program that embeds a relaxed cell transmission model (CTM) to propagate traffic. As a result, we show that CTM-type traffic dynamics can be derived from the original M-N model, when the exit-flow function is properly selected and discretized. The proposed cell-based M-N model has a simple constraint structure and cell network representation because all intersections and cells are treated uniformly. Path marginal costs are defined using a recursive formula that involves a subset of multipliers from the linear program. This definition is then employed to interpret the necessary condition, which is a dynamic extension of the Wardrop’s second principle. An algorithm is presented to solve the flow holding back problem that is known to exist in many discrete SO-DTA models. A numerical experiment is conducted to verify the proposed model and algorithm.     

Formulating the within-day dynamic stochastic traffic assignment problem from a Bayesian perspective

This study proposes a formulation of the within-day dynamic stochastic traffic assignment problem. Considering the stochastic nature of route choice behavior, we treat the solution to the assignment problem as the conditional joint distribution of route traffic, given that the network is in dynamic stochastic user equilibrium. We acquire the conditional joint probability distribution using Bayes’ theorem. A Metropolis–Hastings sampling scheme is developed to estimate the characteristics (e.g., mean and variance) of the route traffic. The proposed formulation has no special requirements for the traffic flow models and user behavior models, and so is easily implemented.     

Quadratic approximation and convergence of some bush-based algorithms for the traffic assignment problem

This paper first shows that LUCE (Gentile, 2012), a recent addition to the family of bush-based algorithms, is closely related to OBA (Bar-Gera, 2002). LUCE’s promise comes mainly from its use of the greedy method for solving the quadratic approximation of node-based subproblems, which determines the search direction. While the greedy algorithm accelerates the solution of the subproblems and reduces the cost of line search, it unexpectedly disrupts the overall convergence performance in our experiments, which consistently show that LUCE failed to converge beyond certain threshold of relative gap. Our analysis suggests that the root cause to this interesting behavior is the inaccurate quadratic approximation constructed on faulty information of second-order derivatives. Because the quadratic approximations themselves are inaccurate, the search directions generated from them are sub-optimal. Unlike OBA, however, LUCE does not have a mechanism to correct these search directions through line search, which explains why its convergence performance suffers the observed breakdowns. We also attempt to improve LUCE using the ideas that have been experimented for the improvement of OBA. While these improvements do work, their effects are not enough to counteract the inability to adjust sub-optimal search directions. Importantly, the fact that the search direction has to be corrected in line search to ensure smooth convergence attests to the limitation of origin-based flow aggregation shared by both OBA and LUCE. These findings offer guidelines for the design of high performance traffic assignment algorithms.     

General stochastic user equilibrium traffic assignment problem with link capacity constraints

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.     

Direct formulation and algorithms for the probit-based stochastic user equilibrium traffic assignment problem

This paper proposes simple and direct formulation and algorithms for the probit-based stochastic user equilibrium traffic assignment problem. It is only necessary to account for random variables independent of link flows by performing a simple transformation of the perceived link travel time with a normal distribution. At every iteration of a Monte-Carlo simulation procedure, the values of the random variables are sampled based on their probability distributions, and then a regular deterministic user equilibrium assignment is carried out to produce link flows. The link flows produced at each iteration of the Monte-Carlo simulation are averaged to yield the final flow pattern. Two test networks demonstrate that the proposed algorithms and the traditional algorithm (the Method of Successive Averages) produce similar results and that the proposed algorithms can be extended to the computation of the case in which the random error term depends on measured travel time.     

A microeconomic interpretation for the system optimal traffic assignment problem with nonadditive path cost

Using a Bergson–Samuelson welfare function, we outline a microeconomic interpretation of the effects of the non-linearity in the time/cost relationship for travellers in a congested transport network. It is demonstrated that a marginal cost traffic flow assignment following Wardrop's second principle, although it minimizes the total cost of a transport network, may reduce social welfare compared to the market equilibrium assignment based on Wardrop's first principle. A welfare-maximizing assignment model is presented and used to show that if the travellers' utility functions are linear, the assignment that maximizes social welfare will be the same as the assignment that minimizes total network cost, but if users' utility functions are non-linear (reflecting the traditional non-satiation and diminishing marginal utility axioms), the two assignments will be different. It is further shown that the effects of this non-linearity are such that a welfare-maximizing assignment will meet with less user resistance than a minimum total network cost assignment.     

Rail freight traffic assignment

This paper describes an algorithm for routing freight over a rail network whose tracks are controlled by several carriers in a manner that follows current industrial practices. In particular, the freight is routed to minimize the number of interline transfers and to maximize the revenue division for the originating carrier. The algorithm is based upon generalizing a shortest path algorithm to allow vector valued link impedances.     

On the variance of recurrent traffic flow for statistical traffic assignment

This paper generalizes and extends classical traffic assignment models to characterize the statistical features of Origin-Destination (O-D) demands, link/path flow and link/path costs, all of which vary from day to day. The generalized statistical traffic assignment (GESTA) model has a clear multi-level variance structure. Flow variance is analytically decomposed into three sources, O-D demands, route choices and measurement errors. Consequently, optimal decisions on roadway design, maintenance, operations and planning can be made using estimated probability distributions of link/path flow and system performance. The statistical equilibrium in GESTA is mathematically defined. Its multi-level statistical structure well fits large-scale data mining techniques. The embedded route choice model is consistent with the settings of O-D demands considering link costs that vary from day to day. We propose a Method of Successive Averages (MSA) based solution algorithm to solve for GESTA. Its convergence and computational complexity are analyzed. Three example networks including a large-scale network are solved to provide insights for decision making and to demonstrate computational efficiency.     

Network representation,continuum approximations and a solution to the spatial aggregation problem of traffic assignment

This paper complements the preceding one, which showed how one could modify equilibrium traffic assignment algorithms for networks with many centroids. In this one it is shown how one can substitute centroids by zones with continuous population densities. The technique, which is mathematically guaranteed to approximate a flow pattern in accordance with Wardrop's user equilibrium criterion, requires some geometrical calculations which can be handled off-line. With the suggested approach, it is possible to represent the spatial distribution of trip ends more realistically and this allows traffic assignment models to capture the effects of short and intrazonal trips in a way previously impossible. The method is shown to be computationally feasible.     

The existence of an equilibrium solution to the traffic assignment problem when there are junction interactions

We consider a network with interactions and capacity constraints at each junction. We give conditions on the interactions and constraints which, if satisfied at each separate junction, ensure that any feasible assignment problem has an equilibrium solution. Two illustrative examples are provided; the first arises naturally and does not satisfy our conditions, while the second does satisfy our conditions but is somewhat unnatural.     

A computational model for the probit‐based dynamic stochastic user optimal traffic assignment problem

This paper focuses on computational model development for the probit‐based dynamic stochastic user optimal (P‐DSUO) traffic assignment problem. We first examine a general fixed‐point formulation for the P‐DSUO traffic assignment problem, and subsequently propose a computational model that can find an approximated solution of the interest problem. The computational model includes four components: a strategy to determine a set of the prevailing routes between each origin–destination pair, a method to estimate the covariance of perceived travel time for any two prevailing routes, a cell transmission model‐based traffic performance model to calculate the actual route travel time used by the probit‐based dynamic stochastic network loading procedure, and an iterative solution algorithm solving the customized fixed‐point model. The Ishikawa algorithm is proposed to solve the computational model. A comparison study is carried out to investigate the efficiency and accuracy of the proposed algorithm with the method of successive averages. Two numerical examples are used to assess the computational model and the algorithm proposed. Results show that Ishikawa algorithm has better accuracy for smaller network despite requiring longer computational time. Nevertheless, it could not converge for larger network. Copyright © 2010 John Wiley & Sons, Ltd.