Mathematical Problems in Engineering

Volume 2015, Article ID 313249, 10 pages

http://dx.doi.org/10.1155/2015/313249

## Modeling Stochastic Route Choice Behaviors with Equivalent Impedance

Research Centre of Intelligent Transportation Systems, School of Engineering, Sun Yat-sen University, Guangzhou 510275, China

Received 14 December 2014; Accepted 7 June 2015

Academic Editor: Wei (David) Fan

Copyright © 2015 Jun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A Logit-based route choice model is proposed to address the overlapping and scaling problems in the traditional multinomial Logit model. The nonoverlapping links are defined as a subnetwork, and its equivalent impedance is explicitly calculated in order to simply network analyzing. The overlapping links are repeatedly merged into subnetworks with Logit-based equivalent travel costs. The choice set at each intersection comprises only the virtual equivalent route without overlapping. In order to capture heterogeneity in perception errors of different sizes of networks, different scale parameters are assigned to subnetworks and they are linked to the topological relationships to avoid estimation burden. The proposed model provides an alternative method to model the stochastic route choice behaviors without the overlapping and scaling problems, and it still maintains the simple and closed-form expression from the MNL model. A link-based loading algorithm based on Dial’s algorithm is proposed to obviate route enumeration and it is suitable to be applied on large-scale networks. Finally a comparison between the proposed model and other route choice models is given by numerical examples.

#### 1. Introduction

Stochastic route choice models capture the perception errors in drivers where they are assumed as rational choosers to minimize their perceived costs rather than actual least costs. It provides the probability that each path is chosen, and thus the future traffic demand can be calculated for transportation planning and management. Multinomial Logit model (MNL) is a basic type of Logit model and used to model relationships between a polytomous response variable and a set of regressor variables. Under the assumption, the probability of Multinomial Logit (MNL) route choice model really depends on the route’s costs or what is called impedance. Alternatively, the Logit model can be formulated as a mathematical programming problem in equilibrium traffic assignment, which includes an extremely large number of available routes [1]. Dial [2] proposed an efficient algorithm for fixed cost network, which can be used to solve Logit route choice problem by method of successive averages [3]. Recently, path-based solution algorithms have been proposed for Fisk model [4]. Dial’s algorithm is very efficient in solving Logit network loading problem by considering only subset of alternative routes, namely, “reasonable” routes. However it was found that some routes with smaller cost are excluded while the larger ones are included in the reasonable paths and Dial’s algorithm sometimes fails in the real world application due to the deficiency of Dial’s “reasonable” route [5]. All route Logit models have been proposed to address this problem, which consider all possible routes including those routes with finite and infinite loops so that the number of routes is infinite [5–7]. Including loops in all route models makes the IIA problem even worse. Since too many links are defined as “unreasonable” in Dial’s algorithm, Li et al. [8] proposed a topological scan based algorithm to find more “reasonable” routes by removing “unreasonable” link only when cycle exists. Other efforts to solve “reasonable” issue include algorithm excluding all cyclic flows [9], but it is difficult to be applied to the large-scale networks because the algorithm requires route enumeration.

There are two main deficiencies of the Logit route choice model due to its Independently and Identically extreme value Distributed (IID) assumption; consequently, the MNL model cannot interpret the correlated degree of paths because of its IIA (independence of irrelevant alternatives) property, which leads to enlarged probabilities for correlated routes; secondly, it cannot represent the heterogeneity in perception errors which would produce unreasonable results, namely, the scaling problem. The improvements for the first issue, the correlation of paths, fall into three categories.

*(1) MNL Model with Utility Correction*. One correction term is added to the utility function to account for the correlated degree of paths. The motivation is that the unnecessarily higher choice probabilities are given to the correlated routes, so additional costs for the correlated degree can reduce their attractiveness. The C-Logit model was proposed by introducing an attribute called commonality factor (CF) to interpret the correlated degree of routes [10]. The value of the CF term is proportional to the correlated parts. The Path Size Logit (PSL) model was proposed with a similar idea [11, 12]. The correction term, called the Path Size (PS), is derived from the property of the Gumbel distribution to aggregate alternatives. A Path Size Correction (PSC) term is also proposed similarly to the PSL model [13]. However the PSL and the PSCL (Path Size Correction Logit) models might be sensitive to the composition of the route choice set [13–15].

*(2) Nested Structure*. The motivation is to categorize the routes into a nest if they share the same links. The link-based Cross Nested Logit (CNL) is the most wildly used CNL route choice model [16–18]. It treats each link as a nest, and the routes sharing the same link belong to the same nest. Some researches provide approximated formulas for the CNL model [16, 18]. Besides, some researchers suggest that the parameters can be achieved by solving a system of equations of the correlation and constraints [19, 20]. The Paired Combinatorial Logit (PCL) model also has a nest structure and it processes routes in pairs [21–23]. The correlated degrees of paths are captured by the similarity index and the specifications are provided by Gliebe [24], Prashker, and Bekhor [25].

*(3) Other Distributions*. The mixed Logit [26–28] and the probit are most commonly used [29–31]. The mixed Logit model incorporates other distributions, mainly, the normal, into the logit model to interpret correlated degree of paths. The probit model is assumed to be normal distribution but it does not have the closed-form expression when there are more than three alternatives. Their estimation and prediction all require the simulation-based methods. Researches [18, 28] show that the simulation-based method requires a large number of draws to achieve stable predictions. Besides, currently there is no efficient path-based SUE traffic assignment for solving the route choice model [32].

Regarding the scaling problem, Pravinvongvuth and Chen [32] proposed an origin-destination specific scaling (dispersion) factor to represent the different scale of diverse networks. Chen et al. [23] examine the scaling effect when applying route choice model in stochastic equilibrium models. Miwa et al. [33] examine how to set the scale parameter (dispersion parameter) and apply a multiclass stochastic user equilibrium (SUE) assignment model to consider differences in drivers’ perception errors. The relative impedance was proposed based on the same motivation and improved models are derived from the properties of extreme value distribution [34]. The CNL model demonstrates that each link has its own scale parameter to interpret the perception error. In application it is usually fixed to the relationships among link-path topology [16–18].

Despite weakness of Logit route choice models, Logit route choice is widely used because it has its closed-form probability expression and an equivalent mathematical programming formulation and can be solved by efficient algorithm. For example, Logit route choice model has been applied to dynamic stochastic route choice [35, 36].

In the following sections, a Logit-based route choice model is proposed to reduce the effects of overlapping problem by using a new Logit dispersion parameter setting and a new definition of Logit equivalent link, followed by an efficient network loading algorithm based on Dial’s algorithm to obviate route enumeration. Numerical examples are followed to compare the proposed model with some previous models. The final section concludes.

#### 2. Methodology

##### 2.1. Logit Parameters

The probability distribution of Logit route choice model depends heavily on the Logit dispersion parameter; moreover, the probability distribution only depends on the difference between alternatives and is irrelevant to mean travel costs of routes. Consider a road network comprising a set of nodes and a set of links . Let denote the travel time on link , and then the route travel time on route connecting origin and destination can be calculated as follows:where if link is a part of route ; , otherwise. The probability that drivers choose route iswhere is the set of all routes connecting origin and destination ; is Logit dispersion parameter. Consider three networks in Figure 1 with travel time 100 and 105. The probabilities of route choice by different methods are given.