Skip to main content

Optimizing train stopping patterns for congestion management


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17


  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.


  • Beckmann M, McGuire CB, Winsten CB (1956) Studies in the economics of transportation. Yale University Press, New Haven

    Google Scholar 

  • Borndörfer R, Karbstein M (2012) A direct connection approach to integrated line planning and passenger routing. In: Delling D, Liberti L (eds) 12th Workshop on algorithmic approaches for transportation modelling, optimization, and systems, vol 25. OpenAccess Series in Informatics (OASIcs), pp 47–57, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany.

  • Borndörfer R, Grötschel M, Pfetsch ME (2007) A column-generation approach to line planning in public transport. Transp Sci 41:123–132

    Article  Google Scholar 

  • Borndörfer R, Hoppmann H, Karbstein M (2017) Passenger routing for periodic timetable optimization. Public Transp 9:115–135.

    Article  Google Scholar 

  • Bureau of Public Roads (1964) Bureau of Public Roads: traffic assignment manual. U.S. Dept. of Commerce, Urban Planning Division, Washington, DC

  • Burggraeve S, Bull SH, Vansteenwegen P, Lusby RM (2017) Integrating robust timetabling in line plan optimization for railway. Transp Res Part C Emerg Technol 77:134–160

    Article  Google Scholar 

  • Bussieck MR, Kreuzer P, Zimmermann UT (1997) Optimal lines for railway systems. Eur J Oper Res 96:54–63

    Article  Google Scholar 

  • Chang YH, Yeh CH, Shen CC (2000) A multiobjective model for passenger train services planning: application to Taiwan’s high-speed rail line. Transp Res Part B Methodol 34:91–106

    Article  Google Scholar 

  • Chankong V, Haimes YY (2008) Multiobjective decision making: theory and methodology. Dover Publications, New York

    Google Scholar 

  • Cochran JJ, Cox LAJ, Keskinocak P, Kharoufeh JP, Smith JC (eds) (2011) Wiley encyclopedia of operations research and management science. John Wiley & Sons Inc, Hoboken

    Google Scholar 

  • Florian M (1999) Untangling traffic congestion: application of network equilibrium models in transportation planning. OR MS Today 26(2):52–57

    Google Scholar 

  • Frank M, Wolfe P (1956) An algorithm for quadratic programming. Naval Res Logist 3:95–110

    Article  Google Scholar 

  • Fu Q, Liu R, Hess S (2012) A review on transit assignment modelling approaches to congested networks: a new perspective. Proc Soc Behav Sci 54:1145–1155

    Article  Google Scholar 

  • Fu H, Nie L, Meng L, Sperry BR, He Z (2015) A hierarchical line planning approach for a large-scale high speed rail network: the China case. Transp Res Part A Policy Pract 75:61–83

    Article  Google Scholar 

  • Goerigk M, Schmidt M (2017) Line planning with user-optimal route choice. Eur J Oper Res 259:424–436

    Article  Google Scholar 

  • Goossens JW, van Hoesel S, Kroon L (2004) A branch-and-cut approach for solving railway line-planning problems. Transp Sci 38:379–393

    Article  Google Scholar 

  • Ieda H (1995) Commuter railways–can congestion be relieved? Jpn Railw Transp Rev 4:8–15

    Google Scholar 

  • Jamili A, Aghaee MP (2015) Robust stop-skipping patterns in urban railway operations under traffic alteration situation. Transp Res Part C Emerg Technol 61:63–74

    Article  Google Scholar 

  • Jiang F, Cacchiani V, Toth P (2017) Train timetabling by skip-stop planning in highly congested lines. Transp Res Part B Methodol 104:149–174

    Article  Google Scholar 

  • Kaspi M, Raviv T (2013) Service-oriented line planning and timetabling for passenger trains. Transp Sci 47:295–311

    Article  Google Scholar 

  • Kroon L, Maróti G, Nielsen L (2015) Rescheduling of railway rolling stock with dynamic passenger flows. Transp Sci 49:165–184

    Article  Google Scholar 

  • Lien JW, Mazalov VV, Melnik AV, Zheng J (2016) Wardrop equilibrium for networks with the BPR latency function. In: Kochetov Y, Khachay M, Beresnev V, Nurminski E, Pardalos P (eds) Discrete optimization and operations research, vol 9869. Lecture Notes in Computer Science, Springer, pp 37–49

  • Ministry of Land, Infrastructure, Transport and Tourism of Japan (2010) Ministry of Land, Infrastructure, Transport and Tourism: the eleventh census data. (in Japanese)

  • Niu H, Zhou X, Gao R (2015) Train scheduling for minimizing passenger waiting time with time-dependent demand and skip-stop patterns: nonlinear integer programming models with linear constraints. Transp Res Part B Methodol 76:117–135

    Article  Google Scholar 

  • Patriksson M (1994) The traffic assignment problem–models and methods. VSP, Utrecht

    Google Scholar 

  • Schmidt ME (2014) Integrating routing decisions in public transportation problems. Springer, New York

    Book  Google Scholar 

  • Schöbel A (2006) Optimization in public transportation. Springer, New York

    Google Scholar 

  • Schöbel A (2012) Line planning in public transportation: models and methods. OR Spectr 34:491–510

    Article  Google Scholar 

  • Schöbel A, Scholl S (2006) Line planning with minimal traveling time. In: Kroon LG, Möhring RH (eds) 5th Workshop on algorithmic methods and models for optimization of railways (ATMOS’05), OpenAccess Series in Informatics (OASIcs), vol 2. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany.

  • Serafini P, Ukovich W (1989) A mathematical model for periodic event scheduling problems. SIAM J Discret Math 2:550–581

    Article  Google Scholar 

  • Shang P, Li R, Liu Z, Yang L, Wang Y (2018) Equity-oriented skip-stopping schedule optimization in an oversaturated urban rail transit network. Transp Res Part C Emerg Technol 89:321–343

    Article  Google Scholar 

  • Taguchi A (2005a) A paradox of staggered commuting. Commun Oper Res Soc Jpn 50:555–559 (in Japanese)

    Google Scholar 

  • Taguchi A (2005b) Time dependent traffic assignment model for commuter traffic in Tokyo metropolitan railway network. Trans Oper Res Soc Jpn 48:85–108 (in Japanese)

    Article  Google Scholar 

  • Taguchi A (2017) Can Tokyo’s efficient rail system handle Olympic strain? Trans Oper Res Soc Jpn 62:5–14 (in Japanese)

    Google Scholar 

  • Taguchi A, Kashima S, Toriumi S, Sato M (2005) Real-time forecasting system of commuters in the public railway network: use of a area type time-space network model. Transp Policy Stud Rev 8:31–35 (in Japanese)

    Google Scholar 

  • Takamatsu M, Taguchi A (2020) Bus timetable design to ensure smooth transfers in areas with low-frequency public transportation services. Transp Sci 54:1238–1250

    Article  Google Scholar 

  • Wang Y, Schutter BD, van den Boom TJJ, Ning B, Tang T (2014) Efficient bilevel approach for urban rail transit operation with stop-skipping. IEEE Trans Intell Transp Syst 15:2658–2670

    Article  Google Scholar 

  • Yamauchi T, Takamatsu M, Imahori S (2017) Optimizing train stopping patterns for congestion management. In: Proceedings of the 17th workshop on algorithmic approaches for transportation modelling, optimization, and systems (ATMOS 2017), vol 59, no 13.

  • Yan F, Goverde RM (2019) Combined line planning and train timetabling for strongly heterogeneous railway lines with direct connections. Transp Res Part B Methodol 127:20–46

    Article  Google Scholar 

  • Yang L, Qi J, Li S, Gao Y (2016) Collaborative optimization for train scheduling and train stop planning on high-speed railways. Omega 64:57–76

    Article  Google Scholar 

  • Yue Y, Wang S, Zhou L, Tong L, Saat MR (2016) Optimizing train stopping patterns and schedules for high-speed passenger rail corridors. Transp Res Part C Emerg Technol 63:126–146

    Article  Google Scholar 

Download references


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

Author information

Authors and Affiliations


Corresponding author

Correspondence to Mizuyo Takamatsu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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).


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:

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

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yamauchi, T., Takamatsu, M. & Imahori, S. Optimizing train stopping patterns for congestion management. Public Transp (2021).

Download citation

  • Accepted:

  • Published:

  • DOI:


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