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Optimizing train stopping patterns for congestion management

Abstract

In this paper, we optimize train stopping patterns during the morning rush hour in Japan. Since trains are extremely crowded, we need to determine stopping patterns based not only on travel time but also on congestion rates of trains. We exploit a Wardrop equilibrium model to compute passenger flows subject to congestion phenomena and present an efficient local search algorithm to optimize stopping patterns which iteratively computes a Wardrop equilibrium. The framework of the proposed algorithm is extended to solve the problem of optimizing the number of services for each train type. We apply our algorithms to railway lines in Tokyo including the Keio Line with six types of trains and demonstrate that we succeed in relaxing congestion.

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Notes

  1. The set \(\mathcal {A}_\mathrm{change}\) models a common situation in Japan, where a local train is connected to a superior train at a station. In such a situation, the superior train catches and overtakes the local train. Thus, the order of the two trains is changed at the station.

  2. This is a common assumption in Japan.

  3. The first four lines are operated by the private railway operator Keio Corporation, while the last line is operated by Bureau of Transportation, Tokyo Metropolitan Government.

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Acknowledgements

The authors would like to thank the reviewers for the careful reading and helpful comments.

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Correspondence to Mizuyo Takamatsu.

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This work was supported in part by JST CREST, Grant Number JPMJCR14D2, Japan. The research of the second author was supported in part by JSPS KAKENHI Grant Number 16K16356. The research of the third author was supported in part by JSPS KAKENHI Grant Number 15H02969. A preliminary version appeared in Proceedings of the 17th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS 2017) (Yamauchi et al. 2017).

Appendices

A Computation of passenger flows

We explain the details of source vertices and a sink vertex described in the last paragraph of Sect. 3.2. Recall that \(\mathcal {C}\) denotes the set of OD pairs. For an OD pair \(k\in \mathcal {C}\), we know the first station \(v_k^\mathrm{dep}\), the last station \(v_k^\mathrm{arr}\), the number of passengers \(n_k\), and rough departure time \(t_k^\mathrm{dep}\).

We define \(\mathcal {E}_\mathrm{in}=\{k\mid k\in \mathcal {C} \}\) and a new vertex s. Let \(g^\circ \) denote the first train which departs from \(v_k^\mathrm{dep}\) after time \(t_k^\mathrm{dep}\). We define

$$\begin{aligned} \mathcal {A}_\mathrm{in}&=\{(k,(g,v_k^\mathrm{dep},\mathrm{dep}))\mid k\in \mathcal {E}_\mathrm{in}, t_k^\mathrm{dep}-T_0 \le \varPi _{(g,v_k^\mathrm{dep},\mathrm{dep})} \le t_k^\mathrm{dep} \} \\& \cup \{(k,(g^\circ ,v_k^\mathrm{dep},\mathrm{dep})) \}, \\ \mathcal {A}_\mathrm{out}&=\{((g,v,\mathrm{arr}),s)\mid (g,v,\mathrm{arr})\in \mathcal {E}_\mathrm{arr} \}. \end{aligned}$$

We set \(T_0=120\) [min] in the case study in Sect. 6.2. The definition of \(\mathcal {A}_\mathrm{in}\) allows passengers to board a train after time \(t_k^\mathrm{dep}-T_0\). Passengers can also board a train after time \(t_k^\mathrm{dep}\) by using \((k,(g^\circ ,v_k^\mathrm{dep},\mathrm{dep}))\) and arcs in \(\mathcal {A}_\mathrm{next}\).

Passengers who depart before \(t_k^\mathrm{dep}\) have to get up early, while those who depart after \(t_k^\mathrm{dep}\) might be late for work. Depending on departure time \(\varPi _{(g,v_k^\mathrm{dep},\mathrm{dep})}\) of train g, we define arc length of \(\mathcal {A}_\mathrm{in}\) by

$$\begin{aligned} L_{(k,(g,v_k^\mathrm{dep},\mathrm{dep}))}= {\left\{ \begin{array}{ll} \varPi _{(g,v_k^\mathrm{dep},\mathrm{dep})}-t_k^\mathrm{dep} &{} g=g^\circ , \\ 0 &{} t_k^\mathrm{dep}-\dfrac{T_0}{2}\le \varPi _{(g,v_k^\mathrm{dep},\mathrm{dep})}\le t_k^\mathrm{dep}, \\ t_k^\mathrm{dep}-\dfrac{T_0}{2}-\varPi _{(g,v_k^\mathrm{dep},\mathrm{dep})} &{} t_k^\mathrm{dep}-T_0\le \varPi _{(g,v_k^\mathrm{dep},\mathrm{dep})}\le t_k^\mathrm{dep}-\dfrac{T_0}{2}. \end{array}\right. } \end{aligned}$$

This means that passengers who depart \(\dfrac{T_0}{2}\) [min] earlier have no cost and others have costs according to their extra time. Moreover, we set \(L_{((g,v,\mathrm{arr}),s)}=0\) for arcs of \(\mathcal {A}_\mathrm{out}\).

In the event-activity network, each vertex \(i\in \mathcal {E}_\mathrm{arr}\cup \mathcal {E}_\mathrm{dep}\) has time \(\varPi _i\) determined from the timetable \(\varPi \). The length of each arc a is defined by \(L_a=\varPi _j-\varPi _i\) for \(a=(i,j)\). In order to assign passengers in the event-activity network, we add \(\mathcal {E}_\mathrm{in}\cup \{s\}\) and \(\mathcal {A}_\mathrm{in}\cup \mathcal {A}_\mathrm{out}\) to the vertex set and the arc set, respectively.

In computation of a Wardrop equilibrium in the resulting event-activity network \((\mathcal {E},\mathcal {A})\), we use the BPR function (Bureau of Public Roads 1964) as a cost function:

$$\begin{aligned} t_a(f_a)&=L_a \left( 1+\alpha \left( \dfrac{f_a}{C_a}\right) ^\beta \right) \quad (a\in \mathcal {A}_\mathrm{drive}\cup \mathcal {A}_\mathrm{wait}), \end{aligned}$$
(7)
$$\begin{aligned} t_a(f_a)&=L_a \quad (a\in \mathcal {A}\setminus (\mathcal {A}_\mathrm{drive}\cup \mathcal {A}_\mathrm{wait})), \end{aligned}$$
(8)

where \(f_a\) denotes a flow on arc \(a\in \mathcal {A}\) and \(C_a\) is the capacity of a train. The term \(f_a/C_a\) represents the congestion rate. We set parameters \(\alpha \) and \(\beta \) by \(\alpha =0.15\) and \(\beta =4\).

B Estimation of OD demand

We estimate OD pairs who use Keio Railway Lines from available data. We make use of commuter passengers’ data in the report (Ministry of Land, Infrastructure, Transport and Tourism of Japan 2010). This report lists 83,838 OD pairs, and each OD pair has the following information: origin station, destination station, and the number of passengers.

We need to extract passengers who get on Keio Railway Lines from 83,838 OD pairs. We first construct a railway network in the Tokyo metropolitan area given in Fig. 18. Next, we compute an optimal route for each OD pair with respect to distance and the number of transfers, and then extract OD pairs using Keio Railway Lines. Their routes are divided into four types:

  • Use only Keio Railway Lines.

  • First use another line and then transfer to Keio Railway Lines.

  • First use Keio Railway Lines and then transfer to another line.

  • Use another line, transfer to Keio Railway Lines, and transfer to another line again.

As a result, we obtain 9717 OD pairs and 805,344 passengers who get on Keio Railway Lines.

Fig. 18
figure 18

Tokyo Railway network with 2128 vertices and 3041 edges, where red edges represent Keio Railway Lines

Since information about the departure time is not available, we estimate it in the following way. Let \(\mathcal {C}\) be the set of 9717 OD pairs. For an OD pair \(k\in \mathcal {C}\), we know the origin and destination station (not necessarily in Keio Railway Lines), the first station \(v_k^\mathrm{dep}\) and last station \(v_k^\mathrm{arr}\) in Keio Railway Lines, and the number of passengers \(n_k\). We denote by \(\tau _k\) [min] travel time from \(v_k^\mathrm{dep}\) to the destination station, which can be computed by finding an optimal route in the railway network.

We assume that passengers are required to arrive at the destination station before 08:30, because most office workers start to work from 9 o’clock in Japan. Under this assumption, OD pair k departs from station \(v_k^\mathrm{dep}\) around time \(t_k^\mathrm{dep}:=08{:}30-\tau _k\). We use \(t_k^\mathrm{dep}\) as the rough departure time needed for construction of the event-activity network explained in Appendix A. The accurate departure time is computed by finding a Wardrop equilibrium in the event-activity network.

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Yamauchi, T., Takamatsu, M. & Imahori, S. Optimizing train stopping patterns for congestion management. Public Transp (2021). https://doi.org/10.1007/s12469-021-00286-w

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Keywords

  • Train stopping pattern
  • Wardrop equilibrium
  • Local search algorithm
  • Event-activity network