Mathematical Problems in Engineering

Volume 2015, Article ID 729089, 9 pages

http://dx.doi.org/10.1155/2015/729089

## Modelling Stochastic Route Choice Behaviours with a Closed-Form Mixed Logit Model

Guangdong Provincial Key Laboratory of Intelligent Transportation System, School of Engineering, Sun Yat-sen University, Guangzhou 510270, China

Received 5 June 2014; Revised 16 September 2014; Accepted 25 September 2014

Academic Editor: Kang Li

Copyright © 2015 Xinjun Lai and Jun Li. 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 closed-form mixed Logit approach is proposed to model the stochastic route choice behaviours. It combines both the advantages of Probit and Logit to provide a flexible form in alternatives correlation and a tractable form in expression; besides, the heterogeneity in alternative variance can also be addressed. Paths are compared by pairs where the superiority of the binary Probit can be fully used. The Probit-based aggregation is also used for a nested Logit structure. Case studies on both numerical and empirical examples demonstrate that the new method is valid and practical. This paper thus provides an operational solution to incorporate the normal distribution in route choice with an analytical expression.

#### 1. Introduction

Route choice is one of the crucial issues in transportation analysis because it models the travelling behaviours so as to provide predictions for the future demand. Drivers always try to maximize their travelling welfare when choosing a path from a given origin-destination (OD) pair. However not all of them choose the best alternative because of the imperfect knowledge of network. To model this perception error and the stochastic route choice behaviour, Probit and Logit models are two of the most wildly used methods. The utility of each alternative is decomposed into a deterministic and a random portion. Assume that there are paths between an OD pair and the route choice set is ; the utility (welfare) of an alternative path can be represented aswhere is the utility of path , is the deterministic part which is composed by attributes such as length and cost that can be explicitly captured, and is the random term that captures the perception error. A rational traveller would select a path with the maximum utility among the alternatives in .

Probit assumes that the random portion is normally distributed; besides, it provides a highly flexible structure for correlation. However, it is limited due to the computation burden. It does not have a closed-form formula when there are more than two alternatives. Generally, the computation of multinomial Probit requires either Clark’s approximation [1, 2], Monte Carlo simulation [3], or numerical integration [4]. Yai et al. [5] used the multinomial Probit model in the context of route choice in the Tokyo rail network, but the maximum number of alternatives is limited to four. On the other hand, Logit is more popular for its analytical tractability. Logit assumes that the error term is type I extreme value (EV) distributed. Moreover, it assumes each of the error terms is independently identically distributed (IID), which leads to a closed-form mathematical structure to simplify the computation in estimation and prediction. As a consequence, Logit has two main disadvantages because of the IID assumption: (1) it cannot represent the path correlation which leads to enlarged probabilities of the overlapped paths, namely, the overlapping problem and (2) it cannot represent the heterogeneity in perception errors which would produce unreasonable results, namely, the scaling problem.

There are several modified methods to address the two drawbacks of Logit in the context of route choice. Regarding the first disadvantage, the overlapping problem, the improved models are classified into two types.(i)Modifications of multinomial Logit (MNL), such as path size Logit (PSL) [6–9], C-Logit [10], and implicit availability/perception (IAP) model [11]: in these models, an additional term is introduced in the utility function to capture the correlations of paths, so as to decrease the attraction of the overlapped path. This method maintains the simple form of MNL. Besides, the log-likelihood function of this method is globally concave, so it guarantees a global optimum for parameters estimations. However, the additional terms are convenient approximations. Previous researches show that they might be too sensitive to the composition of the choice set [9, 12].(ii)Generalized extreme value (GEV) proposed by McFadden [13]: the most widely used methods of this type in route choice are the link-based crossed nested Logit (CNL) [14, 15] and the paired combinatorial Logit (PCL) [16–19] model. These two models all have a tree structure to represent the link-path relation, where alternatives with shared attributes are classified into the same nest so the correlation can be explicitly captured. The link-based CNL model treats each link as a nest, and each path uses several links which are classified into the corresponding nests. The PCL model compares paths by paired combinations, and each path pair is a nest. The CNL model has a large set of parameters that need to be estimated, so some researches provide approximated formulas [14, 20]. Besides, some researchers suggest that the parameters can be achieved by solving a system of equations of the correlation and constraints [21, 22]. Likewise, the PCL model also requires a parameter to represent the correlation, and the specifications are provided by Gliebe [23] and Prashker and Bekhor [24].

As for the second drawback of Logit, the scaling problem, Pravinvongvuth and Chen [18] propose origin-destination specific scaling factor to represent the different scale of diverse networks. Chen et al. [25] examine the scaling effect when applying route choice model in stochastic equilibrium models. Miwa et al. [26] 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.

Some researches combine both advantages of Probit and Logit, and the most representative model is the mixed Logit [27], also named as Logit kernel, error component, or hybrid Logit. It incorporates other distributions other than type I EV to provide a flexible and tractable form to represent the correlation across alternatives, the alternative specific variances, and also taste heterogeneity. Frejinger and Bierlaire [8] use the error component to model the subnetwork so as to represent the path overlap in an abstract network. Bekhor et al. [28] estimate an error component model based on the Boston route choice data. However, the mixed Logit does not have a closed-form expression; consequently the estimation and prediction all require the simulation-based method. Researches [20, 29] 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 with the mixed Logit model [18].

To fully use the advantages of Probit and Logit, this paper proposes a mixed Logit method with a closed-form to model the stochastic route choice behaviours. With a closed-form expression, the computation burden in estimation and prediction would be relieved. Moreover, the closed-form formula alleviates the difficulties in the path-based SUE assignment with a mixed Logit model. The paper is organized as follows. Section 2 describes the methodology, including the nested model structure and the Probit-based aggregation. The validation from a numerical example is presented in Section 3. The new method is applied to real data in Section 4. Finally conclusions and discussions for future study are given in Section 5.

#### 2. Methodology

##### 2.1. Model Structure

The proposed model has a similar structure as the PCL model. Paths are compared by paired combination. Consider alternatives in the choice set between an OD pair; by paired combination there are totally paths pairs. The new model has a two-level nest structure. Each path pair is a nest; within the nest there is actually a binary choice case. The expected maximum utility [7] of each path pair is used as the utility of the nest. In the upper level, it is a multinomial choice model with nests. Consider a three-alternative case, as shown in Figure 1, paths , , and , and the path pairs are , , and . The probability that path is chosen among three paths is a combination of the marginal probability of the nest and the conditional probability within the nest, which is