Travel Cost Functions Research Articles

A nonlinear equation system approach to the dynamic stochastic user equilibrium simultaneous route and departure time choice problem

In dynamic stochastic user equilibrium simultaneous route and departure time choice (DSUE-SRDTC) problems, route travel costs can be non-monotone even if route travel times are monotone with respect to route flows. As a result, the mapping function of the variational inequality (VI) problems for the DSUE-SRDTC problems can be non-monotone, and many existing solution algorithms developed for the DSUE-SRDTC problems do not guarantee convergence under this non-monotone condition. This paper formulates the DSUE-SRDTC problem with fixed demand as a system of nonlinear equations. The mapping function of the proposed system of nonlinear equations is defined by a dynamic route choice problem, which can also be formulated as a VI problem with a strictly monotone mapping function under some assumptions. This property enables that the solution algorithm for the DSUE-SRDTC problem can avoid the requirement of the monotonicity of the route travel cost functions for the convergence of the solution procedure. A backtracking inexact Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is adopted to solve the system of nonlinear equations, and iterative methods are developed to generate an initial solution for the BFGS method and solve the dynamic route choice problem. Finally, numerical examples are set up to show that the proposed method outperforms many existing algorithms for solving the DSUE-SRDTC problem in terms of guaranteeing solution convergence.

Mapping the Cost of a Balanced Diet, as a Function of Travel Time and Food Price

We present a new method for analyzing spatial variation in the cost of a balanced diet, as an alternative to food desert classification. Our specific hypothesis is that the cost of a balanced diet varies according to where one lives, as a function of travel and food item costs. We collected price data for the USDA Thrifty Food Plan from approximately 30 percent of food retail outlets of various kinds in the three Gulf Coast counties of Mississippi, and these prices were extrapolated to the remaining stores. Transportation costs were calculated for both driving by automobile and the combination of walking and public transportation by bus, accounting for both the shoppers' time and the cost of automobile mileage. We developed a traveling purchaser problem algorithm to estimate the lowest-cost combination of travel and food costs for purchasing all items in the Thrifty Food Plan for each residential parcel in the study area, and mapped the resulting costs and examined their variation. Estimated costs varied more because of transportation costs than food prices, and ranged from US$109 to US$215 for automobile travel and from US$111 to US$439 for a combination of walking and public transportation. In general, costs were lowest in the more populated areas near the coast and higher in more rural areas further inland. Results of this analysis demonstrate that the cost of acquiring a balanced diet varies considerably and more or less continuously. Food is not equally cheap for all; it depends on where one lives. For any given location, an estimate of the cost of a balanced diet, including both food price and transportation, is more useful than a classification as food desert or not in understanding access issues and needs. Furthermore, policy alternatives that are intended to influence access should be evaluated based on how much they influence costs, and for whom, depending on where people live.

Pre-trip information and route-choice decisions with stochastic travel conditions: Experiment

This paper studies the effects of pre-trip information on route-choice decisions when travel conditions on two alternative congestible routes vary unpredictably. It presents and discusses an experiment designed to test a model recently proposed in a companion paper by Lindsey et al. (2013). That model predicts that if free-flow costs on the two routes are unequal, travel cost functions are convex, and capacities are positively and perfectly correlated, then in equilibrium, paradoxically, total expected travel costs increase with the provision of pre-trip information about travel conditions on each route. By contrast, when capacities vary independently, total expected travel costs are predicted to decrease with pre-trip information. We reformulate the model for finite populations, and then test and find support for its predictions in an experiment where under different capacity scenarios, and with and without pre-trip information, subjects are asked to choose routes with payoff contingent on their performance.

Pre-trip information and route-choice decisions with stochastic travel conditions: Theory

This paper studies the effects of pre-trip information on route-choice decisions when travel conditions are congested and stochastic. We adopt a model based on the classical two-route network in which free-flow travel times and/or capacities on each route vary unpredictably due to such shocks as bad weather, accidents, and special events. We show that the benefits of information depend on differences between routes in free-flow costs, the shape of the travel cost functions, the severity of congestion and capacity shocks, and the degree of correlation between routes in travel conditions. Information is more likely to be welfare-reducing when free-flow travel costs differ appreciably, travel cost functions are convex, shocks are similar in size on the routes, and route conditions are strongly and positively correlated.

Analysis of interaction among land use, transportation network and air pollution using stochastic nonlinear programming

This paper presents two novel models for land use and transportation to address the development of different functional zones in urban areas by considering the design of an efficient transportation network and reducing air pollution. Objective functions of the first model are maximizing utility function and maximizing reliability index. The utility is formulated as a function of travel cost and zonal attractiveness. Reliability index is defined as the probability that flow in each link of the network is less than the design capacity. Maximizing this probability is equivalent to minimizing congestion in the network. In addition, maximizing utility and minimizing carbon monoxide emission in the network are considered as objective functions in the second model. The formulated models are nonlinear and stochastic. We implement the e-constraint method for solving these bi-objective optimization problems. We analyze the models and solution characteristics of some examples. In addition, we evaluate the relation between computing time and complexity of the model. In this study, for the first time in the open literature, stochastic bi-objective optimization models are formulated to analyze interaction among land use, transportation network and air pollution. We also extract and summarize some useful insights on the relationship among land use, transportation network and environmental impact associated with them.

Bounding the Efficiency Loss of Mixed Equilibrium in the Transportation Network with Selfish and Altruistic Users

Existing research mainly focuses on the efficiency loss of homogeneous users in the transportation network while little effort has been made to explore the efficiency loss of heterogeneous users. The aim of this article is to investigate the efficiency loss of mixed traffic assignment in the transportation network with selfish and altruistic users. The selfish user chooses a travel path based on the classical user equilibrium (UE) principle and the altruistic user aims to minimize their perceived travel cost (here, the perceived travel cost of each altruistic user is a linear combination of a selfish and altruistic component). Firstly, this article establishes a Variational Inequality (VI) model to depict this mixed traffic assignment. Secondly, the upper bound of this mixed equilibrium traffic assignment is derived by analytic derivation and the relation between the upper bound and the network parameters is obtained. Finally, a numerical example is carried out to validate the analytical result. The analytical and numerical results show that the upper bound of efficiency loss is related to the maximal altruism coefficient, the minimal altruism coefficient and the link travel cost functions.

The Stability of Multi-modal Traffic Network

There is an explicit and implicit assumption in multimodal traffic equilibrium models, that is, if the equilibrium exists, then it will also occur. The assumption is very idealized; in fact, it may be shown that the quite contrary could happen, because in multimodal traffic network, especially in mixed traffic conditions the interaction among traffic modes is asymmetric and the asymmetric interaction may result in the instability of traffic system. In this paper, to study the stability of multimodal traffic system, we respectively present the travel cost function in mixed traffic conditions and in traffic network with dedicated bus lanes. Based on a day-to-day dynamical model, we study the evolution of daily route choice of travelers in multimodal traffic network using 10000 random initial values for different cases. From the results of simulation, it can be concluded that the asymmetric interaction between the cars and buses in mixed traffic conditions can lead the traffic system to instability when traffic demand is larger. We also study the effect of travelers' perception error on the stability of multimodal traffic network. Although the larger perception error can alleviate the effect of interaction between cars and buses and improve the stability of traffic system in mixed traffic conditions, the traffic system also become instable when the traffic demand is larger than a number. For all cases simulated in this study, with the same parameters, traffic system with dedicated bus lane has better stability for traffic demand than that in mixed traffic conditions. We also find that the network with dedicated bus lane has higher portion of travelers by bus than it of mixed traffic network. So it can be concluded that building dedicated bus lane can improve the stability of traffic system and attract more travelers to choose bus reducing the traffic congestion.

Estimating the demand for drop-off recycling sites: A random utility travel cost approach

Drop-off recycling is one of the most widely adopted recycling programs in the United States. Despite its wide implementation, relatively little literature addresses the demand for drop-off recycling. This study examines the demand for drop-off recycling sites as a function of travel costs and various site characteristics using the random utility model (RUM). The findings of this study indicate that increased travel costs significantly reduce the frequency of visits to drop-off sites implying that the usage pattern of a site is influenced by its location relative to where people live. This study also demonstrates that site specific characteristics such as hours of operation, the number of recyclables accepted, acceptance of commingled recyclables, and acceptance of yard-waste affect the frequency of visits to drop-off sites.

A Departure Time Choice for Morning Commute Considering Train Capacity of a Rail Transit Line

Every commuter utilizing urban rail transit decides the departure time from home to a station according to individual judgment for the biggest possibility to board a train as soon as possible after the arrival. Therefore, the departure time choice behavior of the commuters is complicated especially when the transport capacity of this transit line cannot meet the travel demands of its users in rush hour. This research first develops a travel cost function mainly considering the travel time to rationally describe the volume distribution of the commuters arriving at a station at different time periods. Furthermore, an optimization model is accordingly proposed to rationalize the arrival distribution of the commuters on the basis of the amount of the arrived commuters who are able to board a train from the perspective of the systematic operation of a rail transit line for the minimal total travel cost of all the commuters along this line. It is found that the total travel cost can be reduced by 19.0% at most through the arrival distribution rationalization with the optimization model.

Model of Urban Road Congestion Pricing Based on Game Theory

This paper proposes a game-based model to conduct the issue of road congestion pricing. The ride comfort of travel modes, e.g., cars or buses, is introduced into the travel cost function of the traditional bottleneck model. Furthermore, based on different travel cost functions of various travel modes, the Nash equilibriums are achieved among the government and various travelers. The results can be employed to describe internality and externality of traffic system respectively. Finally, numerical examples are presented. The findings of our work indicate a relationship between the government’s goal and the charge rate and that the emergence of ride comfort obviously is a key determinant of travelers’ behavior.

Hierarchical network-based equilibrium model and algorithm for a mixed-traffic urban transport system

Many equilibrium models and algorithms based on homogeneous motorized traffic have been devised to model urban transport systems in developed countries, but they are inadequate when it comes to represent mixed-traffic urban transport systems, including automobiles, transit, bicycles, and pedestrians, in developing countries such as China or India. In these cases, traffic flow on a road segment is an aggregated result of travellers' combined mode/route choices and corresponding interactions. Therefore, a special assignment model and algorithm are needed for modeling these distinct behaviors. In this article, the structure of a mixed-traffic urban transport system is analyzed and then expanded and represented using a hierarchical network model based on graph theory. Based on the analysis of travelers' combined mode/route choices, generalized travel cost functions and link impedance functions for different modes are formulated, where the interferences between different modes on the same road segments are taken into account. Due to the ‘asymmetric’ nature of these functions, a variational inequality model is proposed to represent the equilibrium assignment problem in a mixed-traffic urban transport system. The corresponding solution algorithm is also presented. Finally, a numerical example is provided to illustrate the practicality of the proposed model and algorithm.

Efficiency Loss of Mixed Equilibrium Behaviors with Polynomial Cost Functions

This paper investigates the efficiency loss of selfish routing simultaneously with user equilibrium (UE) player and Cournot-Nash (CN) players. The upper bound of the efficiency loss of UE-CN mixed equilibrium with polynomial cost functions is obtained by the scaling method and the nonlinear programming method, respectively. It is shown that the upper bound of efficiency loss obtained by the scaling method only depends on the polynomial degree of the link travel cost function and the upper bound of efficiency loss obtained by the nonlinear programming method depends on the number of the CN players besides the aforementioned factors. The numerical tests validate our analytical results. KEY WORDS: efficiency loss, UE-CN mixed equilibrium, variational inequality, selfish routing game

Parcel distribution network design problem

This paper presents and discusses the parcel distribution network design problem for ground shipments. An optimization model is developed to formulate this problem. The model is distinguished by its ability to: (i) incorporate actual economies of scale rather than relying on the theoretical discount factor commonly used in the literature, (ii) deal with more than one level of facilities and locate them simultaneously, (iii) find the optimal hub-to-hub connections rather than assuming a complete hub-to-hub network. More capabilities of the formulation include size-reduction features and piecewise travel cost functions. The model is illustrated with an application to a real-world problem; the generated design decreases the annual operating cost by around 9%.

Model and Algorithm of Competition between High-speed Railway and Air Transport

Transport planning is usually based on models’ forecasts, but the reliability of their outputs depends much on the analysis of relationship between transportation modes. The paper considers the access and egress cost of passengers, and put forward a passenger’s total travel cost function. Firstly, we employ the concept of “linear city” to analyze the market share between high-speed rail (HSR) and airline (AIR), with the analyzing the relationship between the two competitors with the non-cooperative game theory. In addition, we take the fixed fare and variable fare rate as decision variables, and establish an optimizing model to calculate the best decision of each mode; then, a heuristic algorithm is developed to solve the model. At last, taking the Wuhan-Guangzhou transportation corridor as example, the model and algorithm are used to calculate the optimizing fares, and analyze change of solutions with different value of time. The result demonstrates that HSR can attract more passengers than AIR in the distance around 1,000km, and the best strategy of AIR is reducing price.

Location of Multiple-Server Common Service Centers or Facilities, for Minimizing General Congestion and Travel Cost Functions

The authors propose a model for locating a fixed number of multiple-server service centers or facilities that may become congested. Customers arriving at these centers must wait in line until served. The locations of the facilities and the allocations of customers to them are chosen by the planner, so to minimize both travel costs and system-wide congestion (or queuing) at centers. All the customers arriving at the facilities must be served, up to a certain maximum line length. The travel cost in which customers incur is a function of the length of the trip to the facility, while the congestion cost at a facility is a general function of the number of customers waiting on line or being served at the facility. The resulting model is a nonlinear p-median formulation. A solution method for this nonlinear model is proposed, and computational experience is presented.

Location of Multiple Server Common Service Centers or Public Facilities for Minimizing General Congestion and Travel Cost Functions

We propose a model and solution methods, for locating a fixed number of multiple-server, congestible common service centers or congestible public facilities. Locations are chosen so to minimize consumers' congestion (or queuing) and travel costs, considering that all the demand must be served. Customers choose the facilities to which they travel in order to receive service at minimum travel and congestion cost. As a proxy for this criterion, total travel and waiting costs are minimized. The travel cost is a general function of the origin and destination of the demand, while the congestion cost is a general function of the number of customers in queue at the facilities.

Properties of Expected Travel Cost Function with Uncertain Travel Time

The paper presents a theoretical analysis of travelers’ scheduling preferences and the resulting form of the expected utility that includes travel time reliability measures. A series of research papers and reports used the mean–standard deviation approach to evaluate policies that improve travel time reliability. Recently, this approach was theoretically substantiated under the conventional assumptions of constant marginal utility of time (MUT) at the origin, two discrete MUT values at the destination, and constant standardized travel time distribution. In this paper, properties of the minimal expected travel cost are investigated with smooth MUTs at the origin and destination of the trip. The influence of small variations in travel time on travel cost is well approximated by a term proportional to the travel time variance and independent of the distribution form of travel time. Two examples of MUT functions are provided: the minimal expected travel cost can be analytically expressed through moments or through a moment generating function of travel time, and conditions are stated guaranteeing that the expected travel cost is exactly additive by independent parts of the trip. These results provide justification in particular for the mean–variance approach to modeling drivers’ decisions under uncertain travel times. This formulation is convenient especially for scheme evaluation in large road networks because it allows the use of conventional network assignment routines by just modifying the volume delay functions to include the travel time variability term.

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.

Trip distribution for limited destinations: a case study for grocery shopping trips in the Netherlands

In this paper, we introduce a new trip distribution model for destinations that are not homogeneously distributed. The model is a gravity model in which the spatial configuration of destinations is incorporated in the modeling process. The performance was tested on a survey with reported grocery shopping trips in the Dutch city of Almelo. The results show that the new model outperforms the traditional gravity model. It is also superior to the intervening opportunities model, because the distribution can be described as a function of travel costs, without increasing the computational time. In this study, the distribution was described by a simple function of Euclidean distance, which provides a good fit to the survey data. The slope of the distribution is quite steep. This shows that most trips are made to nearby supermarkets. However, a significant fraction of trips, mainly made by car, still goes to supermarkets further away. We argue that modeling of these trips by the new method will improve traffic flow predictions.

Novel travel cost functions based on morning peak commuting equilibrium

This paper derives two novel travel cost functions by formulating a morning commuting equilibrium model that incorporates traffic congestion based on fundamental traffic flow diagram. The travel cost functions have components to represent traversal cost, waiting queuing costs and early arrival penalty. It is found that the equilibrium travel cost is a concave function of the total demand in the uncongested regime but an increasing linear function of the total demand in the congested regime.