Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 821574, 12 pages

http://dx.doi.org/10.1155/2015/821574

## A Capacity-Restraint Transit Assignment Model When a Predetermination Method Indicates the Invalidity of Time Independence

^{1}Jiangsu Key Laboratory of Urban ITS, Southeast University, Nanjing 210096, China^{2}Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies, Nanjing 210096, China

Received 8 May 2015; Revised 25 July 2015; Accepted 28 July 2015

Academic Editor: Yuanchang Xie

Copyright © 2015 Haoyang Ding 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

The statistical independence of time of every two adjacent bus links plays a crucial role in deciding the feasibility of using many mathematical models to analyze urban transit networks. Traditional research generally ignores the time independence that acts as the ground of their models. Assumption is usually made that time independence of every two adjacent links is sound. This is, however, actually groundless and probably causes problematic conclusions reached by corresponding models. Many transit assignment models such as multinomial probit-based models lose their effects when the time independence is not valid. In this paper, a simple method to predetermine the time independence is proposed. Based on the predetermination method, a modified capacity-restraint transit assignment method aimed at engineering practice is put forward and tested through a small contrived network and a case study in Nanjing city, China, respectively. It is found that the slope of regression equation between the mean and standard deviation of normal distribution acts as the indicator of time independence at the same time. Besides, our modified assignment method performs better than the traditional one with more reasonable results while keeping the property of simplicity well.

#### 1. Introduction

Urban transit network is becoming a hot issue especially in developing countries like China where transit priority has risen to become a national policy. Developed countries in Europe witness this trend as well. Even in the US where car traffic dominates, as a major focus of transit network, transit accessibility is widely researched (e.g., [1–3]). The essential part of transit network analysis, transit assignment, is paid attention to concerning its framework, algorithm, equilibrium solution, and so on. For example, the solution algorithms for the multicriteria multimodal shortest path problem (M-SPP), which is known as NP-hard, in urban transit network were proposed [4]. This was followed by a label-setting algorithm for finding optimal hyperpaths in large transit networks with realistic headway distributions [5]. Among those models aimed at transit network, many of them have something to do with statistical independence, especially stochastic transit assignment models. A conditional logit model applied to traffic assignment and modal split was established which was able to cope with the independence of irrelevant alternatives (IIA) phenomenon in a very natural way [6]. Nested logit model that is widely believed to overcome the IIA characteristic was put forward as an extension of conditional logit model [7]. Cascetta et al. [8] introduced a commonality factor for overcoming the IIA problem in route choice, which was made good use of to estimate link flow variance and the variance of the path choice proportion in a proposed stochastic assignment model [9, 10]. Independence of route choice probability has been a focus since discrete choice model was brought in the area of traffic assignment.

The issue “independence” acts as one of the prerequisites that make a great difference to the validity of many models targeted at analyzing transit network. Traditional works, however, hardly paid enough attention to time independence of every two adjacent bus links. Substantially, this independence can never be neglected. MNP model, which is short for multinomial probit model, was first used in stochastic traffic assignment [11]. The model has no requirement on independence of irrelevant route choice but requires time independence of every two adjacent bus links. The requirement was applied to various logit models as well [12]. When probit-based stochastic user equilibrium (SUE) models were researched, most of them were implicitly based on the assumption of the correctness of time independence. Those researches varied from algorithms and dynamic pricing to sensitivity analysis [13–15]. Yet none of them concerned the validity of independence assumption, which might lead to inaccurate, even useless mathematical models when it came to their applications to real life. Meanwhile, plenty of research had something to do with independence test method in the statistics field. Entropy Theory is frequently made use of to test independence and the entropy was used as a measure of dependence, like serial dependence and spatial dependence, in these studies [16–18]. In addition, many other methods were proposed to conduct independence test. Broock et al. [19] presented an independence test method on the basis of an in-depth analysis on correlation dimension. Sugiyama and Suzuki [20] introduced least squares theory to nonparametric independence test. Those methods of test are subtle and theoretic. It is, however, rather hard to put them into practice for their complexity, especially to introduce them to time independence test in large urban transit networks.

This work is targeted at engineering practice of transit systems planning and management while two simple methods are proposed. To start with, we put forward a method to predetermine time independence of every two adjacent bus links. A characteristic quantity, , the indicator of time independence, is derived to conduct independence predetermination, laying a foundation of the usage of many models. It is found that time independence is invalid when , and data collected in metropolitan area of Nanjing city, China, is used to make a preliminary test on the practicability of our method. In the case , transit network models on the basis of MNP lose their effects, especially those probit-based stochastic transit assignment models, so the number of alternative assignment models decreases. Meanwhile, capacity-restraint assignment method lacks the multipath characteristic [21] though widely used in engineering practice out of its simplicity and not bad precision. Therefore, a modified capacity-restraint transit assignment method is proposed following the predetermination method while taking good advantage of some predetermination results. The method maintains its effectiveness under the circumstance of invalidity of time independence. After introducing 95% quantile of normal distribution to the traditional capacity-restraint method to realize multipath transit assignment, it turns out that the results of the modified method are more sensible than the traditional one through a small contrived transit network.

#### 2. Methodology

##### 2.1. Time Independence Predetermination

###### 2.1.1. Basic Assumptions

(1)Most bus drivers are technical in the network.(2)The (the ratio of road traffic volume to road capacity) can be somewhat large but no congestions occur when there exists a transit exclusive lane. Otherwise, is medium or low.(3)There is only a little interference to transit operations caused by pedestrians and bicycles.(4)Transit priority control is advanced, so intersections have limited effect on transit operations.

These four basic assumptions are not impractical. Assumptions , , and are valid in most cities of developed countries, while assumption may be a bit questionable in some developing countries like China. These countries are known for mixed traffic flow that does make a big difference to transit operations in their big cities like Shanghai. Anyway, bicycles and pedestrians have little effect on public transit on the road with separation infrastructures. The effects are also quite limited in most of their medium and small cities. The most demanding assumption is probably, which is easy to be satisfied in many European countries. It is, however, very difficult to guarantee an advanced transit priority control system. Anyhow, the priority control does exist in most cities of developed countries and many big cities of developing countries. In a word, it is likely that assumption is the most accessible while assumption is to the contrary.

As for time independence predetermination, the “time” here is to be defined strictly. The two sides of a bus line have a bus stop. When a bus begins to pull over at the upstream stop, the time is . Similarly, is defined. We call “stop-stop time” that is simply denoted by SST, which is composed of the parking time at the upstream bus stop and travel time between two adjacent stops. If all basic assumptions are satisfied, we could infer that SST at the bus link conforms to normal distribution according to the data in Nanjing. This can be seen from the section of distribution test below. To analyze the whole transit network basically satisfying all the assumptions above, we assume that all the links in the network conform to normal distribution, which is the foundation of the proposed methodology. More insight concerning the normal distribution can be seen in Discussion.

###### 2.1.2. Mean and Standard Deviation Fitting

We fit the mean and std, which is short for standard deviation in this paper, of the normal distribution mentioned above. Bus links that satisfy all the basic assumptions to the greatest degree can be selected in a transit network. Accordingly, we could get several normal distributions: . and are estimated using the samples collected in transit survey. Note that and have the same dimension, so regression analysis is decided to be aimed at the std rather than variance. Furthermore, and , so instead of doing linear regression directly, we fit the and in the Double Logarithmic Coordinate System and get the regression equation as follows:

The regression model is the fundamental of time independence predetermination and the modified capacity-restraint model. According to the data in Nanjing, when all chosen links meet the assumptions well, the goodness of fitting is perfect with high -square which can be seen in Section 4. To analyze the whole transit network, we assume the regression equation has high goodness of fitting. If the regression model is not sufficiently significant, we may appropriately select other or more bus links that satisfy all basic assumptions well to avoid poor goodness of fitting. More analysis about the regression model can be seen in Section 5.

###### 2.1.3. Indicator of Time Independence

After the regression analysis, we could predetermine time independence of every two adjacent bus links in a network. Time independence predetermination is to predetermine the independence of SST at every two adjacent links. Note that all bus links in the network conform to normal distribution and the goodness of fitting of regression equation is high on the basis of our analysis above. In this part, we put forward an indicator of independence whose value can simply predetermine time independence with practicability.

Considering normal distribution with the mean and std , according to the previous section of fitting,

and are regression coefficients of the regression equation. The slope is supposed to be greater than zero () because the std will increase when the mean increases. Another form of equation derived by deforming the regression equation above is

Then the variance is given by the following expression:

A transit route is given in Figure 1. Here A represents the origin and B represents the destination. The mean of a random variable , the difference between the moment when a bus stops at station A and that at station B, is set at fixed value . () represents the intermediate bus stop, and () is the mean of SST at bus link. Obviously, can be expressed as follows: