Perceived Travel Cost Research Articles

Boundedly rational user equilibria (BRUE): Mathematical formulation and solution sets

Boundedly rational user equilibria (BRUE) represent traffic flow distribution patterns where travellers can take any route whose travel cost is within an ‘indifference band’ of the shortest path cost. Those traffic flow patterns satisfying the above condition constitute a set, named the BRUE solution set. It is important to obtain all the BRUE flow patterns, because it can help predict the variation of the link flow pattern in a traffic network under the boundedly rational behavior assumption. However, the methodology of constructing the BRUE set has been lacking in the established literature. This paper fills the gap by constructing the BRUE solution set on traffic networks with fixed demands. After defining ε-BRUE, where ε is the indifference band for the perceived travel cost, we formulate the ε-BRUE problem as a nonlinear complementarity problem (NCP), so that a BRUE solution can be obtained by solving a BRUE–NCP formulation. To obtain the BRUE solution set encompassing all BRUE flow patterns, we propose a methodology of generating acceptable path set which may be utilized under the boundedly rational behavior assumption. We show that with the increase of the indifference band, the acceptable path set that contains boundedly rational equilibrium flows will be augmented, and the critical values of indifference band to augment these path sets can be identified by solving a family of mathematical programs with equilibrium constraints (MPEC) sequentially. The BRUE solution set can then be obtained by assigning all traffic demands to the acceptable path set. Various numerical examples are given to illustrate our findings.

Optimal transit fare in a bimodal network under demand uncertainty and bounded rationality

SUMMARYThis paper investigates the optimal transit fare in a simple bimodal transportation system that comprises public transport and private car. We consider two new factors: demand uncertainty and bounded rationality. With demand uncertainty, travelers are assumed to consider both the mean travel cost and travel cost variability in their mode choice decision. Under bounded rationality, travelers do not necessarily choose the travel mode of which perceived travel cost is absolutely lower than the one of the other mode. To determine the optimal transit fare, a bi‐level programming is proposed. The upper‐level objective function is to minimize the mean of total travel cost, whereas the lower‐level programming adopts the logit‐based model to describe users' mode choice behaviors. Then a heuristic algorithm based on a sensitivity analysis approach is designed to solve the bi‐level programming. Numerical examples are presented to illustrate the effect of demand uncertainty and bounded rationality on the modal share, optimal transit fare and system performance. Copyright © 2013 John Wiley & Sons, Ltd.

Boundedly Rational User Equilibria (BRUE): Mathematical Formulation and Solution Sets

Boundedly rational user equilibria (BRUE) represent traffic flow distribution patterns where travellers can take any route whose travel cost is within an ‘indifference band’ of the shortest path cost. Those traffic flow patterns satisfying the above condition constitute a set, named the BRUE solution set. It is important to obtain all the BRUE flow patterns, because it can help predict the variation of the link flow pattern in a traffic network under the boundedly rational behavior assumption. However, the methodology of constructing the BRUE set has been lacking in the established literature. This paper fills the gap by constructing the BRUE solution set on traffic networks with fixed demands connecting multiple OD pairs. According to the definition of the ɛ-BRUE, where ɛ is the indifference band for the perceived travel cost, we formulate the ɛ-BRUE problem as a nonlinear complementarity problem (NCP), so that a BRUE solution can be obtained by solving a BRUE-NCP formulation. To obtain the whole BRUE solution set encompassing all BRUE flow patterns, we firstly propose a methodology of generating various path combinations which may be utilized under the boundedly rational behavior assumption. We find out that with the increase of the indifference band, the path set that contains boundedly rational equilibrium flows will be augmented, and the critical values of indifference bands to augment these path sets can be identified by solving a family of mathematical programs with equilibrium constraints (MPEC) sequentially. After these utilized path sets are attained, the BRUE solution set can be obtained when we assign all traffic demands to these utilized paths. Various numerical examples are given to illustrate our findings.

A reliability-based land use and transportation optimization model

We consider a transportation network with a set of origins and a set of destinations. Given a set of budgets for the residential and employment developments, and network enhancement, the problem is one of allocating the resources within the system, so that the probability of overloading the links in the network is minimized. For the improved transportation system with new household and employment distribution patterns, a combined distribution and assignment model is used to map the land-use pattern to the link-loading pattern in the network. Assuming that the actual demand of each origin–destination (O–D) pair follows a certain distribution, the road users choose their destination and route in accordance with the user equilibrium principle that is based on the long-run perceived travel cost. Moreover, it is assumed that despite the short-term demand fluctuation, the O–D-link choice proportion remains unchanged for the choices of destination and route, and the stochastic demand of all O–D pairs are independent. Explicit formulae for the mean and variance of the traffic volume on each link are derived. Using the central limit theorem, the probability that the traffic volume does not exceed the link capacity can be estimated, from which we can calculate a network reliability index. This index represents the probability that all links in the network are within the respective capacities. The problem can be formulated as a bi-level program, in which the upper-level sub-program maximizes the network reliability index with respect to the residential and employment allocations and network enhancements, whereas the lower-level subprogram is the combined distribution and assignment model with long-run travel cost functions. The problem is solved by a genetic algorithm. A numerical example is used to demonstrate the effectiveness of the methodology.

Comparison of Different Toll Policies in the Dynamic Second-best Optimal Toll Design Problem: Case study on a Three-link network

In this paper, the dynamic optimal toll design problem is considered as a one leader-many followers hierarchical non-cooperative game. On a given network the road authority as the leader tolls some links in order to reach its objective, while travelers as followers minimize their perceived travel costs. So far toll has always been considered either as constant or as time-varying. Inspired by the San Diego's Interstate 15 congestion pricing project, in which heuristics with toll proportional to traffic flow are applied on a real two-link highway network, we consider toll as proportional to traffic flows in the network. On a three-link network we investigate various toll schemes and their influence on the outcome of the game for the road authority. We show that the use of alternative toll schemes may improve system performance remarkably.

User-Oriented Model to Support Funding Decisions in Pavement Management

A new choice model for corrective road maintenance work based on an economic evaluation of users’ expectations and perceptions about road quality is proposed. Because uncertainty affects human subjective perception processes, the model uses fuzzy sets to deal with this kind of uncertainty. Accordingly, the international roughness index was first fuzzified by combining users’ perceptions about pavement conditions with ranges of speeds for four different categories of rural roads. These fuzzy values were then used to calculate vehicle operating costs and freeflow speed. The latter can be considered a function of the international roughness index and the law enforcement factor. To calculate this factor, a fuzzy inference system was set up. Vehicle operating costs and free-flow speed, as well as the value of time, were then used to calculate the travel cost perceived by users. The model finds, through fuzzy maximization of the difference between perceived travel costs before and after interventions, the best allocation of the available budget for a rural road network. In other words, the results suggest the extent of interventions on specific road sections. An application to a real network shows how the proposed model may be used.

Comparison of Different Toll Policies in the Dynamic Second-best Optimal Toll Design Problem: Case study on a Three-link network

In this paper, the dynamic optimal toll design problem is considered as a one leader-many followers hierarchical non-cooperative game. On a given network the road authority as the leader tolls some links in order to reach its objective, while travelers as followers minimize their perceived travel costs. So far toll has always been considered either as constant or as time-varying. Inspired by the San Diego's Interstate 15 congestion pricing project, in which heuristics with toll proportional to traffic flow are applied on a real two-link highway network, we consider toll as proportional to traffic flows in the network. On a three-link network we investigate various toll schemes and their influence on the outcome of the game for the road authority. We show that the use of alternative toll schemes may improve system performance remarkably.

Sensitivity Analysis Based Method for Optimal Road Network Pricing

Road pricing is an important economic measure for optimal management of transportation networks. The optimization objectives can be the total travel time or total cost incurred by all the travelers, or some other environmental objective such as minimum emission of dioxide, an so on. Suppose a certain toll is posed on some link on the network, this will give an impact on flows over the whole network and brings about a new equilibrium state. An equilibrium state is a state of traffic network at which no traveler could decrease the perceived travel cost by unilaterally changing the route. The aim of the toll setting is to achieve such an equilibrium state that a certain objective function is optimized. The problem can be formulated as a mathematical program with equilibrium constraints (MPEC). A key step for solving such a MPEC problem is the sensitivity analysis of traffic flows with respect to the change of link characteristics such as the toll prices. In this paper a sensitivity analysis based method is proposed for solving optimal road pricing problems.

Temporal-Spatial Microassignment and Sequencing of Travel Demand with Activity-Trip Chains

A stochastic temporal–spatial microassignment and activity sequencing model for activity–trip chains is presented. In this model, trip-chain patterns are defined by the respective locations of destinations in the chain, preferred arrival times at these destinations, and the activity durations at the intermediate destinations; they are given as input to the model. A stochastic dynamic user equilibrium problem is formulated and solved for this purpose. In this problem, drivers simultaneously seek to determine their departure time, route choice, and sequence of their intermediate activities at the origin to minimize their perceived travel cost. This perceived cost is typically a function of the travel time and the schedule delay at the intermediate and final destinations. The model is presented through a study of the relative efficiency of carpooling and trip-chaining travel behavior in a network context. In that example, the performance of travelers who have the option to carpool and chain trips is compared with that of households with single-occupant and individual trip-based travel. Several measures of travel performance, including travel distance, travel time, and schedule delay, are considered for that comparison.

A Second Order Stochastic Network Equilibrium Model, I: Theoretical Foundation

Existing models of stochastic network equilibrium route choice in transport networks are able to represent exogenously specified variations in drivers' actual or perceived travel costs, but assume throughout that flows are deterministic. In this paper, a new notion of equilibrium is presented based on stochastic flow variables, in which the impact of variable flows on the variability in travel costs is endogenously handled. Firstly, a very general notion of equilibrium is deduced as a fixed point condition on the joint probability distribution of network flows. Then, an approximation to this condition is derived, which operates by equilibrating moments of order n and below of the joint flow probability distribution, and is termed a Generalized Stochastic User Equilibrium of order n, being denoted GSUE(n). The GSUE(1) model is seen to be a conventional Stochastic User Equilibrium. The paper goes on to focus on the second order model, GSUE(2). Conditions are presented to guarantee the existence of GSUE(2) solutions. Conditions are deduced to guarantee (a) uniqueness of solutions in networks with a single interzonal movement, and (b) proximity of solutions in networks with multiple interzonal movements. Finally, a simple example is presented.

Using fuzzy integral for evaluating subjectively perceived travel costs in a traffic assignment model

A good traffic assignment model can be a powerful tool to describe the characteristics of traffic behavior in a road network. The traffic assignment results often play an important role in transportation planning, e.g., an optimal and economical network design. Many traditional traffic assignment models rely heavily on the travel cost function established by Wardrop’s principles; however, the Wardrop’s travel cost function has been proven to be weak for explaining the uncertainty and interactivity of traffic among links. This study tries to construct a traffic assignment model that is different from Wardrop’s in many aspects. First, it considers the cross-effect among the links. Second, a fuzzy travel cost function is established based on the possibility concept instead of precise calculation of traffic volumes. Third, the techniques of fuzzy measure and fuzzy integral are applied to calculate the subjectively perceived travel costs during traffic assignment. Furthermore, in order to validate our model, a detailed network with 22 nodes and 36 links is used to illustrate it. Study results show that our model explains more interactivity and uncertainty of traffic among links when compared with the traditional model of Wardrop’s.

Algorithms for logit-based stochastic user equilibrium assignment

The paper proposes an efficient algorithm for determining the stochastic user equilibrium solution for logit-based loading. The commonly used Method of Successive Averages typically has a very slow convergence rate. The new algorithm described here uses Williams’ result [ Williams, (1977)On the formation of travel demand models and economic evaluation measures of user benefit. Environment and Planning 9A(3), 285–344] which enables the expected value of the perceived travel costs S rs to be readily calculated for any flow vector x. This enables the value of the Sheffi and Powell, 1982objective function [Sheffi, Y. and Powell, W. B. (1982) An algorithm for the equilibrium assignment problem with random link times. Networks 12(2), 191–207], and its gradient in any specified search direction, to be calculated. It is then shown how, at each iteration, an optimal step length along the search direction can be easily estimated, rather than using the pre-set step lengths, thus giving much faster convergence. The basic algorithm uses the standard search direction (towards the auxiliary solution). In addition the performance of two further versions of the algorithm are investigated, both of which use an optimal step length but alternative search directions, based on the Davidon–Fletcher–Powell function minimisation method. The first is an unconstrained and the second a constrained version. Comparisons are made of all three versions of the algorithm, using a number of test networks ranging from a simple three-link network to one with almost 3000 links. It is found that for all but the smallest network the version using the standard search direction gives the fastest rate of convergence. Extensions to allow for multiple user classes and elastic demand are also possible.

Some Remarks on Stochastic User Equilibrium

The behavioural foundation of Stochastic User Equilibrium is that each traveller attempts to minimize his or her perceived travel costs, where these costs are composed of a deterministic measured cost and a random term which can be interpreted as perceptual error. In principle such a definition, which is in terms of random errors, should imply an equilibrium probability distribution over feasible flow patterns on the transport network. Such a probability distribution could potentially allow between day variability in network flows to be represented. However, traditionally a deterministic, large sample approximation has been used as the ‘solution’ of Stochastic User Equilibrium. In this paper a representation of Stochastic User Equilibrium as a probability distribution is developed. This distribution is defined by the conditional route selection of each individual given the choices of all other travellers. An interpretation of the resulting assignment model as the limit (in the infinite future) of a continuous time assignment process is discussed. The limiting behaviour as the travel demand becomes large is also investigated, and convergence to the traditional deterministic form of Stochastic User Equilibrium is proved.

Probability-Based Models of Spatial Search

The author develops and explores probability-based decision strategies for the consumer spatial search problem. These strategies extend experimental and analytical work in psychology on repeated choice among alternatives with varying reinforcement rates. Normative models postulate that the searcher attempts to maximize the frequency with which the lowest acquisition cost will be achieved over a multiple-trip horizon. A behavioral model postulates that the searcher maximizes the frequency with which the maximum utility will be achieved over a multiple-trip horizon. These models encompass a wide range of behavioral factors, including the perceived travel cost, decision foresight, learning, price motivation, and risk attitudes. Sensitivity experiments conducted with the models establish their basic characteristics and assess the influence of these behavioral factors on the spatial search process.

Changes in perceived travel cost and time for the work trip during a period of increasing gasoline costs

This study was designed to examine the relationship between actual and perceived values of cost and time for the work trip and to examine how perceptions have changed over a period of dramatically increased travel costs. Variations in the relationship between perceived and actual values were examined as a function of situational and attitudinal variables. Two telephone surveys were conducted one year apart (Fall 1978 and Fall 1979). On the next working day following a survey, a research assistant recreated the respondent's work trip, recorded time values and used distance measures, car type information and parking costs to compute travel cost. The first survey revealed that most auto users were unable to articulate dollars-and-cents driving costs for the work trip, but auto users in the second survey were able to provide fairly accurate cost estimates. Dramatic changes in fuel prices between surveys is probably the main reason for the change in driving cost awareness. Auto users were also asked to rate relative costs of driving a car compared to using the bus for the work trip. These ratings showed that auto users tended to underestimate driving costs relative to bus costs, but this tendency decreased from the first to the second time period. Commuters in all modal groups at both time periods tended to overestimate travel times. Perception of travel time varied as a function of mode, perceived comfort (for car users), and perceived convenience and number of transfers (for bus users).