Modelling and Simulation in Engineering

Volume 2018, Article ID 8169036, 38 pages

https://doi.org/10.1155/2018/8169036

## Performance Comparison of Mode Choice Optimization Algorithm with Simulated Discrete Choice Modeling

^{1}Faculty of Engineering and Applied Science, Environmental Systems Engineering, University of Regina, Regina, SK, Canada S4S 0A2^{2}City of Regina, Regina, SK, Canada S4P 3C8^{3}Department of Civil Engineering, Birla Institute of Technology and Science Pilani, Rajasthan 333031, India

Correspondence should be addressed to Hyuk-Jae Roh; moc.liamg@hor.eajkuyh

Received 8 December 2017; Accepted 30 January 2018; Published 9 May 2018

Academic Editor: Luis Carlos Rabelo

Copyright © 2018 Hyuk-Jae Roh 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

Until recently, a majority of modeling tasks of transportation planning, especially in discrete choice modeling, is conducted with the help of commercial software and only concerned about the result of parameter estimates to get a policy-sensitive interpretation. This common practice prevents researchers from gaining a systematic knowledge involved in estimation mechanism. In this research, to shed a light on these limited modeling practices, a standard discrete choice model’s parameter is estimated using Quasi-Newton method, DFP, and BFGS. Two extended algorithms, called DFP-GSM and BFGS-GSM, are proposed for the first time to overcome the weakness of the Quasi-Newton method. The golden section method (GSM) incorporates a nonlinear programming technique to choose an optimal step size automatically. Partial derivatives of log-likelihood function are derived and coded using Visual Basic Application (VBA). Through extensive numerical evaluation, estimation capability of each proposed estimation algorithms is compared in terms of performance measures. The proposed algorithms show a stable estimation performance and the reasons were studied and discussed. Furthermore, useful insights educated in custom-built modeling are present.

#### 1. Introduction

Discrete choice modeling is widely used across disciplines to predict a certain choice situation through a mathematical inferring of a choice model’s parameters [1]. In transportation demand forecasting, this choice analysis method is generally used to describe a variety of choices. Applications of discrete choice theory in various fields are very useful tool for policy analysis and planning. Applications in the transportation sector are very active and diverse and apply to almost every choice situation where alternatives exist, for example, the choice of travel mode [2–7], mode and departure time selection [8, 9], route selection [10–13], and airport selection [14–16]. It is common practice to estimate model parameters by relying on commercially available software.

Until recently, a concern of discrete choice modelers is mainly about to find an interpretable parameter estimates sound in their statistical accuracy level and in their reflected meaning in a policy-sensitive way. Estimation algorithm and calculation mechanism are actually not concern of researchers. However, as a specification of model becomes more complicated to model a real situation in a persuasive way, and as a more realistic assumption is frequently adopted in a model developing process, researchers (or modelers) might meet difficulties in dealing with modeling tasks. Employing a new probability distribution function (PDF) for describing an unobserved part of a choice model can be prohibited by a limitation of software packages. Choosing and applying a specific algorithm for satisfying a number of occasions in modeling tasks are not easy to practice with a normal software package. In particular, opportunity to have access to all the details of calculation that enable researchers to have a systematic understanding of modeling mechanism is hardly to get unless modeling is in custom-built procedure. A few literature items on this topic can be found in the econometrics field [17–23], but these studies have addressed a different application environment and, therefore, it is difficult to apply these to modeling transportation related decisions [24, 25]. To tackle this kind of challenges that can occur in model developing, researchers must have a capability of doing modeling with custom-built computer codes that are tailored for each unique modeling case. To be able to reach this level of stage, researchers should be more familiar with a calculation mechanism and a flow to be undertaken before final parameter estimates are taken.

In this research, a standard discrete choice model is estimated using self-made computer code that is developed by using Visual Basic Application (VBA) in EXCEL [26–28]. In order to guarantee convergence of a test model, excellent and robust in convergence properties even for the problems that cannot be solved in satisfactory manner [29, 30], two practically and pedagogically important estimation algorithms are employed to calculate parameter estimates in every iteration stage. These are DFP and BFGS algorithms. More importantly, for the first time, two new estimation algorithms are proposed to improve a way of finding an optimal step size by incorporating a golden section method (GSM) into general routine of both DFP and BFGS. These are DFP-GSM and BFGS-GSM. These four algorithms are compared in terms of performance measures to show a different operational characteristic according to application of several critical factors that are identified and experimented through extensive numerical trials.

The next parts of this document consist of the following. Section 2 is about a model specification assumed within a concept of multinomial logit choice model and data utilized in this research. Section 3 is dedicated to the issues that researchers should have knowledge on in making custom-built modeling procedures. Section 4 deals with the results obtained from extensive numerical experiments that show a relative operational performance among algorithms, responding to changing of criteria. The last section summarizes the results obtained from this research and presents conclusions.

#### 2. Model and Data

As a standard discrete choice model, multinomial logit (MNL) model has been widely applied to describe a choice behavior of decision maker due to its simplicity of its probabilistic choice function, which is the results derived based on two big assumptions, first one is about an utility maximization which explains a choice mechanism based on an utility concept and second one is about an probability distribution function applied to the distribution of unobserved part of total utility. The simple choice function is shown below: where is a choice probability of mode and and are a representative utility function of mode , respectively. The final form of the multinomial logit model is composed of two parts, a denominator which is the sum of the systematic utility of all the alternatives in the choice set and a numerator which is the systematic utility of the alternative chosen by decision makers.

As a revealed preference (RP) type data, 540 travels interviewed on site for passengers who travel to airport using possible modes are used to estimate a mode choice model. Five alternatives are considered and the representative utility function of each alternative has a specification as follows:

#### 3. Components in Custom-Built Discrete Choice Modeling

##### 3.1. Formulating Log-Likelihood Function

To describe estimation process with computer codes using maximum likelihood estimator (MLE), a high-order nonlinear likelihood function containing whole information of the surveyed data is to be built. Likelihood function specific this research is below:By a log transformation, the above equation (3) changed into a tractable form from the mathematical standpoint as shown in (4). By differentiating the equation in first and second order with respect to each parameter, all elements contained in a vector of gradients and the Hessian matrix can be expressed with mathematical expressions that provide a numeric value in iterative estimation process. This estimation process will be discussed in detail in the following section:

##### 3.2. Iteration Rule

As the most frequently adopted in commercial statistical package, Quasi-Newton method shows a better performance in running time and is considered to be robust in convergence properties compared to any other algorithms [30]. More importantly, this method might be excellent in convergence properties even for problems that cannot be solved by other algorithms in a satisfactory manner [29]. As noted in the text of Train [30], this unique excellence of the algorithm is the results of selecting a different approach in updating the Hessian matrix, compared to other estimation algorithms such as Newton Raphson, BHHH, BHHH-2, and Steepest Ascent.

A general iteration rule applied in this research for estimating parameters of discrete choice model is shown below: where is the parameter estimates after iterations, is the parameter estimates after iterations, is a step size (can be assumed by a researcher at the initial time of model parameter estimation), is the Hessian matrix in iteration in each algorithm, and is a vector of gradient in iteration . This rule is commonly applied to DFP and BFGS.

As compared in Table 1, each algorithm might be generally tried in custom-built modeling and it is important to know how these algorithms work in estimation process from the pedagogical aspect [30]. In fact, they have been widely adopted as an estimation routine in commercial software packages. The first four algorithms (i.e., Newton Raphson, BHHH, BHHH-2, and Steepest Ascent) are referred from the earlier research of Roh and Khan [25] and Train [30]. For more details refer to the above two researches. In this research, Quasi-Newton method (the last two algorithms of Table 1) is mainly considered in finding parameters of the test model with developed computer codes for this specific purpose. Furthermore, an extension of the method is proposed to present a new technique of searching a step size automatically.