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:
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Use only Keio Railway Lines.
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First use another line and then transfer to Keio Railway Lines.
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First use Keio Railway Lines and then transfer to another line.
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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.
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.