3.1 Problem statement
Metro system generally includes lines of two directions, namely up and down directions. Each train of certain line follows a back and forth route from origin to destination then turn back, and different lines are actually operated separately to a certain extent, which can be analyzed individually. Passenger demand is generally different for the stations along the line. As demonstrated in Fig. 2, passenger demand between Stations 6 and 18 (the shadow section) is significantly larger than the other stations (with the correspondence of station name and number listed in Appendix 1) during AM peak hour (07:30–08:30) and the demands of up and down directions in each station are largely different. As a result, overcrowding occurring among these stations will affect passengers’ travel experience. In contrast, no large difference on traffic demand and directional non-uniformity for different stations was found during PM off peak hour (14:30–15:30). Therefore, these overcrowding sections are the investigating focus of this study, in order to provide rational and efficient short turning operating policies. As short turning pattern need to be coordinated with present operation strategies, the question turns to determine the specific short turning zone and the coordinated schedule.
A regular symmetric full-length service and a short turning service between Stations 2 and n-2 are illustrated in Fig. 3. The vertical axis represents travel time from origin to destination and the horizontal axis represents travel distance. Overlapping area of full length and short turning service have larger transport capacity than those outside of the short turning zone. As the cumulative demand in short turning zone is larger, the short turning strategy could relieve overcrowding within the zone. With an objective to reduce overcrowding in a high traffic demand area, the following steps were conducted. First, selecting short turning zone with comparable higher traffic demand, and which is congested in peak hours. Then, defining the operating timetable, such as offset of inserting new ones to existing ones.
Although applying short turning pattern may need additional trains and larger frequency, which is generally considered as part of public competencies, the cost constraint was relaxed from passengers’ welfare perspective. For simplicity, other assumptions were made as follows.
-
a)
For both full length and short turning services, all trains have to stop at every station in between and operate at the same speed with fixed headway.
-
b)
No turn around constraints exist along a line since turn around facilities can be built without many particular difficulties, although only a subset of stations containing short-turning facilities may be established.
-
c)
All passengers waiting on the platforms take the recent arrival train and are able to aboard.
-
d)
Passengers arrive uniformly during the peak period regardless of stochastic demand.
3.2 Model formulation
Without loss of generality, a metro line can be modeled as a directed graph, G = (V, L), in which V denotes station (vertex) set, L denotes link set. Station is denoted as i ∈ V = {1, 2, …, n}, and link, between any two consecutive stations, is denoted as l ∈ L = {1, 2, …, n − 1}. Route denotes a train runs back and forth, indicated by r ∈ R. Two routes are presented in Fig. 3, one (Route 1) covers Station 1 to Station n, and the other (Route 2) covers Station 2 to Station n-2 overlapping with Route 1. Passenger demand between different stations is denoted by OD (Origin-Destination) pairs, an n by n matrix.
3.2.1 Parameters
M-a sufficiently large constant (chosen as 9999 during the numerical experiment);
h-full length headway;
h
min
-the minimum headway;
h‘-the offset between short turning and full-length services;
q
ij
-hourly demands from Station i to Station j;
u-upstream direction;
d-downstream direction;
\( {Q}_k^u \)-hourly demand of Station k in the upstream direction (u-up direction);
\( {Q}_k^d \)-hourly demand of Station k in the downstream direction (d-down direction);
Cap-capacity of each train (310 passengers per carriage unit, 6 passengers/m2);
α
ijr
- binary variable, whether trips from Station i to Station j are covered by Route r, 1-yes, 0-no;
β
ij
- binary variable, whether passengers can reach destination j directly from Station i, 1-yes, 0-no;
γ
yz
, binary variable, whether the zone from Station y to Station z is selected for short turning services, 1-yes, 0-no;
η-load factor;
δ
rk
-binary variable, whether Route r covers Station k, 1-yes, 0-no;
\( {\pi}_{r{k}^{\prime }} \)-binary variable, whether turn back Station k′ is part of Route r, 1-yes, 0-no;
\( {\uptau}_{k^{\prime }} \)-headway of Station k′ for turning back;
S
r
-binary variable, whether Route r is selected, 1-yes, 0-no;
h‘-the offset between short-turning and full-length services, continuous.
f
r
-frequency of Route r, continuous;
θ
ij
-whether OD pair i to j can be served directly without interchange.
3.2.2 Decision variables
Short turning zone: r
yz
, y, z ∈ V′ ∈ V, route is denoted as r ∈ R;
Frequency of route r: f
r
.
3.3 Constraints
Train capacity constraint
Different station-based passenger demands exist in upstream and downstream direction because of directional non-equilibrium of passenger flow during AM peak period. Demand of up and down directions can be represented as follows, respectively:
$$ {Q}_k^u=\sum \limits_{i,j\in V:i\le k<j}{q}_{ij},\forall k\in V $$
(1)
$$ {Q}_k^d=\sum \limits_{i,j\in V:j\le k<i}{q}_{ij},\forall k\in V $$
(2)
where, \( {Q}_k^u \) and \( {Q}_k^d \) denote the cumulatively demand of Station k in up and down direction, respectively.
If multiple operating patterns exist, such as a full-length service together with a short-turning service, stations are actually served with different train capacity. The capacity constraints of outside and inside short turning zone are defined as follows.
V′ ∈ V- a subset of stations containing short-turning facilities.
h‘- the offset between short-turning and full-length services.
γ
yz
(y, z ∈ V′ : y < z), binary variable, 1 if the zone is selected for short turning services.
Then, the constraint is expressed by Eq. (3) as below:
$$ \sum \limits_{y,z\in {V}^{\prime }:y<z}{\gamma}_{yz}=1 $$
(3)
As stations are served by different routes (full-length route and short turning route), capacity constraints of different zones are expressed as follows:
-
1)
Full-length service outside the short-turning zone:
$$ {Q}_k^{u(d)}\le \eta Cap\sum \limits_r{f}_r{\delta}_{rk}+M\left(1-{\gamma}_{yz}\right),y,z\in {V}^{\prime }:y<z;k<y\ or\ k>z $$
(4)
-
2)
Full-length service inside the short-turning zone:
$$ {Q}_k^{u(d)}\frac{h^{`}}{h}\le \eta Cap\sum \limits_r{f}_r{\delta}_{rk}+M\left(1-{\gamma}_{yz}\right),y,z\in {V}^{\prime }:y<z;y\le k\le z $$
(5)
-
3)
Short-turning service:
$$ {Q}_k^{u(d)}\left(1-\frac{h^{`}}{h}\right)\le \eta Cap\sum \limits_r{f}_r{\delta}_{rk}+M\left(1-{\gamma}_{yz}\right),y,z\in {V}^{\prime }:y<z;y\le k\le z $$
(6)
where,\( {Q}_k^{u(d)},{Q}_k^{u(d)}\frac{h^{`}}{h},\kern0.5em {Q}_k^{u(d)}\left(1-\frac{h^{`}}{h}\right) \) denote accumulative passengers for Station k in both up and down directions; Cap indicates the capacity of a train (number of passengers). f
r
denotes frequency of Route r, and δ
rk
is a binary variable, indicating whether Station k is covered by Route r. Load factor η
k
indicates utilization of the capacity at Station k. As a result, \( {\eta}_k\ast Cap\ast \sum \limits_r{f}_r{\delta}_{rk} \) indicates the practical capacity of Route r during the peak hours at Station k.
To avoid heavy overcrowded and prevent resources wasting simultaneously, the loading factor η
k
of each train must be within a certain rational range, which is general low at the initial departure station and increases gradually, and then decreases. Therefore, load factor of a train indicates the maximum short-term load factor during one-way operating period. Capacity of each train is calculated based on full loaded with criterion of 6 passengers/m2. To avoid overcrowded, the maximum load factor η
k
is limited to no more than 1.2. Meanwhile, to avoid resources wasting, short turning zone has to be selected from stations with an original load factor η
k
≥ 0.8.
Line capacity constraint
Each metro line within the subway system generally has a theoretically minimal headway h
min
from technical perspective (1.5 min for subway system using A-type vehicle). As presented in Eq. (7), the cumulatively frequency \( \sum \limits_r{f}_r{\delta}_{rk} \) for any station k can’t exceed theoretically maximum frequency \( \frac{1}{h_{\mathrm{min}}} \).
$$ {h}_{min}<\frac{1}{\sum \limits_r{f}_r{\delta}_{rk}},\forall k\in V $$
(7)
Turn around capacity constraint
The minimum time for turning around is possibly ranging from 1 to 5 min (Canca et al., 2014b). As presented in Eq. (8), the combined frequency of any route should not exceed the maximal allowable turn-back frequency.
$$ {\uptau}_{k^{\prime }}\le \frac{1}{\sum \limits_r{f}_r{\pi}_{r{k}^{\prime }}} $$
(8)
3.3.1 Objective function
As the overcrowding status of train can be controlled by a predefined load factor η, travel time is used for evaluating travel experience depending on the departure frequency, short-turning zone selection etc. As all trains are supposed to run at the same speed, and the on-train time is identical, the only difference lies in the platform waiting time. Therefore, the passengers’ waiting time was chosen as the objective of the model.
Other parameters, α
ijr
- indicates whether trips from Station i to Station j are covered by Route r, binary variable. S
r
-whether Route r is selected, binary variable. δ
rk
-whether Route r covers station k, binary variable. \( {\beta}_{ij}=\underset{r}{\mathit{\max}}\left\{{\alpha}_{ij r}{S}_r\right\},\forall i,j\in V \)-whether passengers can reach destination j directly from Station i, binary variable.
If α
ijr
=1, passengers from Station i can travel to destination j directly without transferring. If β
ij
= 1, the average waiting time is calculated as below:
$$ {t}_{ij}=\frac{1}{2\sum \limits_r{\alpha}_{ij r}{f}_r},\forall i,j\in V $$
(9)
If α
ijr
· S
r
=0, based on the assumption that passengers on the platform are all able to aboard, these passengers at most have one transferring and consequently, the average waiting time is calculated as:
$$ {t}_{ij}=\frac{1}{2}\left(\frac{1}{\sum \limits_r{\delta}_{ri}{f}_r}+\frac{1}{\sum \limits_r{\delta}_{rj}{f}_r}\right),i,j\in V $$
(10)
As a result, the objective function of model is formulated as follows:
Objective function
$$ \mathit{\min}\sum \limits_i\sum \limits_j{q}_{ij}\left({\beta}_{ij}\cdotp \frac{1}{2\sum \limits_r{\alpha}_{ij r}{f}_r}+\left(1-{\beta}_{ij}\right)\cdotp \frac{1}{2}\left(\frac{1}{\sum \limits_r{\delta}_{ri}{f}_r}+\frac{1}{\sum \limits_r{\delta}_{rj}{f}_r}\right)\right) $$
(11)
subject to:
Train capacity constraints: (3), (4), (5), (6).
Load factor: 0.8 ≤ max {η
k
, k ∈ V′} ≤ 1.2.
Line capacity constraint: (7)
Turn-back capacity constraint: (8)
The objective is to minimize passengers’ average waiting time under necessary constraints. The train capacity constraint ensures that the supply could meet passenger demand at certain level of service. Line capacity constraint guarantees the operating frequency does not exceed the permissible value, while the turn back constraint makes trains turn around without exceeding the turn-back capacity. Load factor was highlighted here for ensuring the degree of occupancy.
