Journal of Advanced Transportation

Volume 2017 (2017), Article ID 9216864, 16 pages

https://doi.org/10.1155/2017/9216864

## Three Extensions of Tong and Richardson’s Algorithm for Finding the Optimal Path in Schedule-Based Railway Networks

^{1}Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong^{2}Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong

Correspondence should be addressed to J. Xie

Received 2 July 2016; Accepted 26 October 2016; Published 12 January 2017

Academic Editor: Richard S. Tay

Copyright © 2017 J. Xie 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

High-speed railways have been developing quickly in recent years and have become a main travel mode between cities in many countries, especially China. Studying passengers’ travel choices on high-speed railway networks can aid the design of efficient operations and schedule plans. The Tong and Richardson algorithm that is used in this model offers a promising method for finding the optimal path in a schedule-based transit network. However, three aspects of this algorithm limit its application to high-speed railway networks. First, these networks have more complicated common line problems than other transit networks. Without a proper treatment, the optimal paths cannot be found. Second, nonadditive fares are important factors in considering travel choices. Incorporating these factors increases the searching time; improvement in this area is desirable. Third, as high-speed railways have low-frequency running patterns, their passengers may prefer to wait at home or at the office instead of at the station. Thus, consideration of a waiting penalty is needed. This paper suggests three extensions to improve the treatments of these three aspects, and three examples are presented to illustrate the applications of these extensions. The improved algorithm can also be used for other transit systems.

#### 1. Introduction

Traffic assignment in a transit network is unlike that in a road network, because transit vehicles run on fixed routes and predetermined timetables. Vehicles on roads have more freedom in route selection, and they do not need to depart at specific times [1]. There are two approaches to consider the time dimension and the fixed routes in a transit network, that is, the frequency-based model and the schedule-based model. The frequency-based model [2–4] simplifies the transit network by regarding each line as having many runs and the headway of runs as having a mean. The schedule-based model [1] is based on the real timetable. Thus, in evaluating the characteristics of these two approaches, it seems that the scheduled-based model is more promising for dynamic transit networks. Generally, there are three methodologies for path generation in a schedule-based network.

##### 1.1. Branch and Bound Method

Tong and Richardson [1] suggested an optimal path algorithm which combined Dijkstra’s algorithm [5] with the branch and bound method. Later, Tong and Wong [6] extended this algorithm by considering nonadditive fares. Khani et al. [7] improved the searching speed of this algorithm by setting a lower bound to reduce the size of searches.

Other researchers have introduced several important variables to limit path searching. These variables include the latest departure time [8], the preset time window between the planned departure time and the arrival time [9–14], the maximum number of transfers [8, 9, 11, 13], the maximum waiting time for transfers [9, 11], and the walking distance [13]. Applying these variables (or bounds) can improve the branch and bound method, but the effectiveness of the bounds depends on their preset values. Compared with Tong and Richardson’s algorithm, these bounds are not tight enough to effectively reduce the searching size.

##### 1.2. Event Dominance Method

Florian [15] suggested an event dominance method, which can be used to find optimal paths to one or multiple destinations. However, the optimality of the path found by this method is questionable, because the searching stops after one event associated with the destination is found. In addition, the real optimal path might be ruled out at an earlier stage. This problem was investigated by Nielsen and Frederiksen [16] using a counterexample. They reported using hidden waiting time at the origin point to solve it and explained the method with a two-line example. However, the search size then becomes larger, so they needed to also use the graph cut approach, a modification that may fail to help if the optimal path has more than one transfer, due to the event dominance principle.

##### 1.3. Other Methods

Nielsen et al. [17] suggested a specified design method for the Copenhagen suburban rail network. They divided the path choices into two kinds, that is, paths with no transfers and paths with transfers. The optimal path with no transfers was selected as the first train departing at the origin to the destination after the planned departure time. For the path with transfers, the whole path was divided into several segments, according to the transfer points that were planned in advance, based on the network. Each of the segments is treated like the path with no transfer. This method greatly reduces the computation time, but it is only suitable for simple networks with several transfer stations and with full information about passenger choices.

Chen and Yang [18] modified the traditional graphic network by adding departure time windows on nodes. That is, if the arrival time at node is not within the preset departure time window, the passengers will need to wait and decide whether to depart at the next time window. This kind of algorithm provided a way to consider the time dimension, but it is not a real schedule-base algorithm, and it simply treats waiting time in the same way as the in-vehicle time.

Compared with the other approaches, Tong and Richardson’s algorithm has an improved capacity for effectively and flexibly handling complex transit networks in real time. However, in its present form it still has three limitations. This paper aims to extend Tong and Richardson’s algorithm to improve its results in dynamic transit networks. This improved algorithm can be used as a tool for information provision. Passengers can plan the best path according to their own preference, such as one without transfer or with less waiting time. In the future, we will develop this algorithm for network assignment and the planning of schedules, but at this stage the algorithm is mainly for transit trip navigation.

#### 2. Tong and Richardson’s Algorithm

##### 2.1. Description of Tong and Richardson’s Algorithm

The Tong and Richardson algorithm can be described as follows.

*Step 1. *The time-dependent quickest path algorithm is developed from Dijkstra’s algorithm [5], and it uses travel time between nodes as the path cost (as based on the timetable) to search for the time-dependent quickest paths from the origin to all other nodes, if departing at . The variable is the actual clock time for departure at the origin , unit: mins. After the quickest path search, the earliest arrival time at node , namely, , can be found.

*Step 2. *Based on the quickest path from the origin to the destination , the total weighted cost can be calculated. The weighted cost function for path from node to node iswhere is the journey time (including waiting time, walking time, and transit time) for path from node to , unit: mins.

is the weighted function for journey time in using path , unit: mins:wherewhere is the walking time for path , unit: mins; is the waiting time at stations for path , unit: mins; is the in-vehicle time for path , unit: mins; is the penalty for walking time; is the penalty for waiting time; is the penalty for in-vehicle time; is the penalty for transfer, unit: mins; is the number of transfers for path from node to .

*Step 3. *Set the upper boundary of the latest arrival time at the destination :

*Step 4. *Reverse the timetable, and set the reversed upper boundary as the departure time for searching the time-dependent quickest paths from the destination to all other nodes, so that we have the earliest arrival time at node from the destination in the reversed network . Then, reverse to get the latest departure time to the destination at node , .

*Step 5. *Compare and to check the accessibility of node , :Exclude inaccessible nodes, so that inaccessible nodes are not included in the optimal path search (i.e., in Step 6).

*Step 6. *The optimal path search finds all of the possible paths by searching all accessible nodes. The weighted time function is calculated for node along path . If , the search for path ends.

*Step 7. *The smallest is found, and the path is identified as the optimal path from the origin to destination .

##### 2.2. Limitations of the Existing Tong and Richardson Algorithm

This algorithm can be used for most transit networks. However, it cannot deal with the following three common situations in railway networks.

###### 2.2.1. Nonadditive Fare

Tong and Richardson’s algorithm does not consider travel fares. If the fare is link-based and additive (i.e., the fare proportionally increases with the distance or travel time), Tong and Richardson’s algorithm still works when adding an additional linear extension for the weighted time function (such as that provided by Friedrich et al. [9]) along the optimal path search. However, the fare system for the railway is usually nonadditive, so the additive-fare function [9] cannot be used. A later improvement by Tong and Wong [6] allowed for the consideration of the nonadditive fare. However, introducing the fare into the weighted cost function enlarged the bounding (i.e., the searching size), so Tong and Wong used the cheapest fare to tighten the bound. Nevertheless, due to the gap between the fare for the quickest path and the cheapest fare, the searching size can still be large. Therefore, treatments are needed to accelerate the searching process. This paper suggests an effective way to include nonadditive fares.

###### 2.2.2. Common Lines Problem

A line has a sequence of stations and can be separated into sections. Each line section consists of two successive stations with an arc connecting these two nodes. According to Tong and Richardson’s algorithm, arcs with the same starting stations, end stations, and running times can be aggregated into a link with a unique transit time. The arcs of a link are listed in the time sequence. During the path search, for each link only the first arc having a departure time later than the train’s arrival time at node is used. However, this found path may not be the optimal path. An example of this problem is shown in Figure 1.