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025-1174
A Synergistic Time-Dependent Timetable for the Kaohsiung Mass Rapid Transit
System
Shou-Ren Hu1, Chih-Peng Chu
2, Chao-Tang Liu
3
POMS 23rd
Annual Conference
Chicago, Illinois, U.S.A.
April 20 to April 23, 2012
1
Associate Professor, Department of Transportation and Communication
Management Science, National Cheng Kung University, Tainan 70101, Taiwan,
e-mail: [email protected].
2 Professor, Department of Business of Administration and Logistics Management,
National Dong-Hwa University, Shou-Feng, Hualien County 97401, Taiwan, e-mail:
3 Graduate Student, Department of Transportation and Communication Management
Science, National Cheng Kung University, Tainan 70101, Taiwan, e-mail:
ABSTRACT
The aim of operating a mass rapid transit (MRT) system is to minimize total
system costs while maintaining a satisfactory level of service. For a MRT system, the
main service is to provide passengers with a cost-affordable mobility, while meets the
minimum standards regulated by the government to ensure a certain level of MTR
service. Therefore, how to establish an optimal timetable for train and crew
scheduling purposes is one of the key operational issues for a MRT system. Therefore,
a synergistic time-dependent timetable for a MRT system is crucial to minimize total
system cost while maintaining a certain level of train service.
The purpose of this research is to establish a time-dependent timetable model
which is able to dynamically adjust train schedule depending on the passenger
spatio-temporal distribution demands under a daily operation basis. The developed
model is developed by considering both the operating cost and passenger’s waiting
cost. Finally, numerical case study and sensitivity analysis will be conducted to
demonstrate the feasibility and effectiveness of the proposed models and solution
algorithms.
Key Words: Mass Rapid Transit, Train Service Plan, Timetable, Mathematical
Programming
INTRUCTION
Under the regulation of MRT service indicators, currently the headways of the
Kaohsiung mass rapid transit (KMRT) system are 6 minutes in the peak period and 10
minutes in the off-peak period. The KMRT system has already been operating for four
years, but the ridership is still low (especially in off-peak hours). The KMRT system
is designed to provide approximately 450,000 passenger trips per day, but currently it
serves averagely 130,000 trips per day (only one-tenth of the ridership of the Taipei
MRT system). In order to reduce the KMRT operating cost and improve the
operational efficiency of the system, this research focuses on a flexible MRT
timetable design and adjusts train schedule depending on the historical ridership data.
Traveler’s auto-prone travel behaviors in Kaohsiung City and accessibility of the
KMRT system are the two most important factors which affect the ridership of the
KMRT system. In Kaohsiung City, most people commute by private modes such as
passenger cars and motor scooters, because it is much cheaper and/or convenient than
the MRT system. Under this situation, the ridership of the KMRT system in the past
four years grows slowly and does not expect to increase in the near future. Therefore,
this research aims to analyze the distribution of passenger’s trip O-D demands in the
past years and accordingly adjust the current MTR timetable to establish a new
timetable (called a time-dependent timetable). Basically, the time-dependent timetable
needs to abide by the government regulations of MRT service indicators. Thereby, the
main purpose of this research is to design a flexible timetable that achieves the
following goals:
1. reduce the total system cost (including manpower cost, energy cost, maintenance
cost and cost associated with passenger’s waiting time); and
2. maintain a certain level of train service (by reducing passenger’s waiting time).
Currently, there are three types of timetable which are operating in the KMRT
system (one for Monday to Thursday, another for Friday, and the other for Saturday to
Sunday). The target of this research is to establish the new MRT system timetable
types by considering passengers’ trip O-D patterns and incorporating the factors
which will influence the operating costs or system performance (such as passenger
waiting cost, transferring cost and energy used). The ultimate goal is to increase the
efficiency of the system while maintaining the originally designed service quality for
the passenger. Some operational information and system characteristics about the
KMRT system are described below.
1. The KMRT system operates two lines which are red line and orange line. There
are 37 stations and 3 maintenance depots on the system located at R4a, R23 and
OT1 stations, respectively). Figure 1 shows the system map of the KMRT system.
Many stations were constructed together with other public transportation systems,
department stores, schools and other community centers (such as R8, R11, R14
and R16 stations).
Figure 1. System route map of the KMRT system (37 stations and 3 deports) [11]
2. Regulated by the MRT service indicators, the headway during the peak period
must be less than 6 minutes and less than 10 minutes in the off-peak period.
Since the KMRT system started to operate on Sept. 16, 2008, the timetable has
been adjusted several times. The newly one is 4 to 6 minutes for the peak period
and 7 to 8 minutes for the off-peak period.
In this study, a time-dependent timetable model is formulated by balancing the
costs associated with the KMRT’s operating cost and passenger waiting cost, and
mathematical programming methods and solution algorithms are adopted to solve this
problem. Specifically, the multi-objective programming (MOP) method is applied to
solve the problem in view of the two competing goals in the objective function.
BACKGROUND INFORMATION
Under the government laws and regulations, the operator of the MRT system
needs to plan a Train Service Plan (TSP) for daily operation. Before a new line starts
to provide service, the operator needs to design a TSP submitted to the government
for approval. After being approved by the government, the operator starts to design
the timetable and crew scheduling. To ensure the TSP satisfying the passenger
demand, a TSP consists of the following major components:
1. Characteristics of system operating: including operation line, running distance,
running trip time (running time, dwelling time and turnover time), train operating
model, number of trains and capacity of system. Figures 2 and 3 are the train
operating model of the red and orange lines.
Figure 2. The train operating model of the red line [9]
Figure 3. The train operating model of the orange line [9]
2. System headway design: design the headway in peak and off-peak periods by
considering the following factors:
(1) System characteristics: including line length, gradient, route design, track
alignment, signal equipment, train, station and round trip time.
(2) Service standard regulated by the government service indicators (less than 6
minutes in peak period and 10 minutes in off-peak period).
(3) System capacity and transportation demand.
3. Operating type: including operating daily type, peak/off-peak period, departure
time of the first and last trains.
4. Reschedule the TSP: in case of the following situations, the TSP needs to be
rescheduled:
(1) when the system breakdown by accident or signal error;
(2) on holiday or special activity; and
(3) natural disaster.
Figure 4 shows the framework of the operating cycle of the KMRT’s TSP.
Currently, based on regular daily operational data, we could observe that the
passenger trip O-D distribution is centralized in specific time period and spatial line
segment. Beyond those time period and spatial locations, the system was running
without cost-effectiveness. Basically, we need to regularly analyze passenger’s
volume in order to feed back into the TSP design.
Figure 4. Framework of the operating cycle of the KMRT’s TSP [9]
PROBLEM STSTEMENT
Ridership of the KMRT system is still low even the system has already operated
for four years. Because the KMRT system has only two lines and the commuters of
Kaohsiung City are used to take motorcycles and cars. The train ridership distribution
in a daily operation is not averagely spread in the peak and off-peak periods (also the
difference between weekdays and holidays). The timetable of the KMRT is relatively
inefficient. In order to reduce total system cost and improve performance, we aim to
redesign a timetable which accommodates the passenger’s trip spatio-temporal
distribution. Therefore, this research aims to establish a time-dependent timetable
which depends on the current ridership distribution of different daily types.
The capacity provided based on the timetable in 2008~2011 is providing
excessive volume. If we could design a time-dependent timetable, we may save trains
and drivers to achieve the final goal. At the same time, the energy consumption is the
key factor which could influence the total system cost, and will be incorporated into
the proposed model.
LITERATURE REVIEW
MRT system performance
Guihaire and Hao (2008) [8] proposed the crucial strategy and technical steps in
a transit network. Overall, a transit network design consists of routing design,
frequencies setting, timetabling, vehicle scheduling and crew scheduling and/or roster.
According to those issues, they defined the transit system construction is a Transit
Network Design and Scheduling Problem (TNDSP). On the timetabling issue, one
always needs to concern about frequencies setting and achieve the target that
minimizes passenger waiting time and transfer waiting time.
MRT headway design
Lo (1998) [14] used the Visual SLAM to construct a model which decides
immediately the trains dispatch timing by the ridership of the Muh-Cha line in Taipei
City. The model checks the passenger volume of the Muh-Cha line every 5 seconds,
when it reaches the capacity of a train (336 persons per train), the system will dispatch
the next train. If the volume doesn’t reach the train capacity in 75 seconds, the system
also dispatches the next train, since the Muh-Cha line’s headway is 75 seconds.
Chen (2002) [3] developed multiple headways in the single line of a bus system.
Under the stochastic situation of passenger behaviors, the model considered about
passenger demand, running time and off-line situation (i.e., if the passenger waits too
long and he or she leaves the bus station) to construct the model formulation which
consists of ticket revenue, operating cost and passenger waiting time.
Shiu (2005) [16] developed a Multiple-Objective Programming (MOP) model
which incorporates operating cost and passenger waiting cost into the objective
function. In in the case study of the Bannan line of the Taipei rapid transit system,
when the passenger volume is over the capacity of the platform, the model will
dispatch a shuttle train to alleviate the congestion level.
Lin (2009) [13] used the Genetic Algorithm method to find out the best trains
operating model and the headways in the Taipei Rapid Transit Corporation (TRTC)
system. To design a desirable operating model one has to consider lots of constraints,
including track alignment, passenger demand, number of trains and drivers, train
speed, etc.
Caprara et al. [2] focused on a single one-way track lining two major stations.
They used a graph theoretic formulation for the timetable problem, and found out the
every train’s maximum profit (corresponding to the departure time and running to the
original timetable).
Other factors
Chang (2002) [4], Shie (2003) [15] and Chen (2008) [5] used different
algorithms to search the bets MRT operating strategy and headways when the system
encounters an accident causing a major delay.
Albrecht et al. (2010) [1] used the Problem Space Search to create the timetable
by considering both train operating and track maintenance assignment. The
maintenance task would interrupt the daily operating schedule, and cause traffic delay,
especially on a single track network system.
Kim and Chien (2010) [10] considered train energy consumption as one of the
key factors which influences operating cost. They adjusted the train speed by
considering track algorithm, speed limitation and timetable adherence. Adopting the
Simulating Annealing algorithm the research aimed to find the best timing of
accelerating, cruising, coasting, breaking and standing.
Regarding the past research of the operations of a MTR, most of them is to find
the optimal solution between operator and passenger costs. In the operator side, they
aim to minimize the total operational cost (e.g., costs of power and water, human
resource, maintenance, security, etc.). In the passenger side, they expect high quality
of service (e.g., less waiting time, pleasant riding environment, more safety, etc.).
Nevertheless, these two groups of factors are conflicting to each other; in this research
we will use a Mathematical Programming method aiming to seek the balance between
both stakeholders.
MODEL CONSTRUCTION
We model the problem stated before from two sides: operator and passenger. The
developed model is aimed to minimize the total system cost. But these two objectives
are usually conflicted with each other, we could use the Multi-Objectives
Programming (MOP) [12] method to construct the basic model.
Model structure
The model is constructed by the following steps.
1. Collect passenger’s 15-minute O-D data: We collect the passenger trip O-D data
in March 2011, which is collected from the KMRT RA system (one kind of
analysis system used in the KMRT system). Through this system, we could
analyze the trip trend at every fifteen minutes of an operational day in Kaohsiung
City and establish the operating management strategy accordingly.
2. Establish a new timetable: According to the result of passenger’s trip O-D data
distribution, we could re-design the headways in each hour.
3. Timetable performance calculation: We use the MOP method to establish the
model formulation and calculate the system performance which includes
operating cost, passenger waiting time and ticket revenue.
4. Compare the original timetable: In this step, if the new timetable has better
performance, managers will accept the new one. Otherwise, we need to re-design
and adjust the timetable again.
5. Sensitivity analysis: In this part, we will observe what kind of factors (weight of
the passenger waiting cost, accident, cost setting, ticket fare and runtime reserve)
that will influence the model outputs. In this research, we evaluate the effect of
the weight of passenger waiting time cost related to operational cost on the
adjusted timetable.
6. Mission arrangement and scheduling: When the new timetable is accepted, we
will design the mission arrangement and driver scheduling. However, this
component is out of the scope of this research, and will not be discussed later.
Figure 5 shows the flowchart of the model structure.
Figure 5. Model structure
Model formulation
1. Objective function
To evaluate the relative importance of different cost items, we give different
weights to the operating cost and passenger cost in the objective function, shown in
Eq. (1):
(M1) minimize (1)
where,
TC: system total cost, including operating cost and passenger waiting time cost;
OC: operating cost, including driver cost, energy cost and maintenance cost;
W: relative weight;
PWC: passenger waiting time cost.
Details of each component are described below.
A. Operating cost
When a MRT system changes the headway of the timetable at different time
periods, the operating cost will be affected. There are three major items, human cost,
energy cost and maintenance cost, the equation is determined as follows:
(2)
The formulation for HC is defined as the following:
[ ( ) ( ) ( )] [∑ (
) ∑ (
)
( ) ] (3)
where,
DC: unit driver cost per minute, NTD;
DF: human factor;
Ht: headway of each hour in operating time, minute;
O(Ht): time of drivers operating the train on the main line, minute;
RD(Ht): time of retrieve and depart the train from the depot, minute;
P(Ht): time of prepare trains for operating, minute;
TO: the run trips time of red line/orange line, minute;
TR: time of retrieve or depart one train, minute;
TP: time of prepare one train, minute.
The formulation for EC is described as the following:
( ) ( ) ∑ (
)
∑ (
)
(4)
where,
RT(Ht): run trip, times;
Kw: kilowatt per run trip of red line/orange line, kWh;
Ck: cost of per kilowatt hour, NTD;
RDM(Ht): kilometer of retrieve and depart the train, km;
Dm: kilometer between main line and depot, km;
Km: kilowatt per car kilometer, kWh.
The formulation for MC is defined as the following:
( ) = ∑ (
)
(5)
where,
RT(Ht): run trip, times;
L: length of the red/orange lines, kilometer;
Cm: cost of maintenance per kilometer per hour, NTD.
B. Passenger waiting cost
The formulation for PWC is defined as the following:
PWC =TP * Cp (6)
where,
TP: total waiting time of passengers, minute;
Cp: cost of passenger waiting time per minute, NTD.
The headway of the KMRT system is relatively stable, so we assume that the
waiting time of every passenger is one-half of the headway at each period. Because
there is a transferring situation between two lines, we also consider passenger
transferring cost in this formulation. The TP formulation is defined as the following:
where,
Pij: trips of station i to j;
Pro: trips of red line to orange line;
Por: trips of orange line to red line;
n: number of station;
k: direction of trips, 1 means the north direction, 2 indicates the south direction.
The optimization problem that minimizes both the operating cost and passenger
waiting cost can be rewritten as the following equation,
(M1) minimize
[∑ (
) ∑ (
)
( )
] ∑ (
) ∑ (
)
∑ (
)
W* ∑ ∑ [
]
∑ ∑ [
]
∑ ∑ [
]
(8)
2. Model constraints
The model constraints are depicted as follows.
A. Headway
The headway must satisfy the request of government and under the system
∑ ∑ [
]
∑ ∑ [
]
∑ ∑ [
]
(7)
operating constraint, the formulation is as the following:
(9)
where,
HS: the minimal headway that the KMRT system could provide;
HG: the maximal headway of the service indicators regulated by the government;
B. System operating time
The first trains are dispatched at 05:50 AM and the last trains at 11:55 PM from
R3 and R23 stations.
Model variant
We further incorporate the ticket revenue into the model, and investigate the net
profit of the KMRT system. The formulation is as the following:
(M2) maximize NI=TR –TC (10)
where,
NI: net income, NTD;
TR: ticket revenue, NTD.
The formulation for TR is as the following:
∑ ∑ [ ]
(11)
where,
Fij: ticket fare of station i to station j.
( )
∑ ∑ [ ]
(12)
EMPIRICAL STUDY
O-D data analysis
We collect the O-D data of March 2011, and calculate the ridership in each hour
of every daily type, figure 6 illustrates the O-D demand trend and table 1 shows the
total ridership for each daily type.
Figure 6. O-D distribution of daily type
Table 1. Total ridership for each daily type
Daily types Total ridership
Monday 104,773
Tuesday 104,152
Wednesday 104,771
Thursday 108,039
Friday 134,422
Saturday 159,328
Sunday 136,527
According to the O-D trend, we design two new types of time-dependent
timetable, one for Monday to Friday operation and the other for Saturday to Sunday.
The KRTC has three types of timetable for operation before the re-design, one for
Monday to Thursday, another for Friday and the other for Saturday to Sunday.
Assumption on the designed hourly headway
We set up the headway abided by the government service indicator limitation for
the two lines in each hour (must less than 6 minute in peak period and 10 minute in
off-peak period). Then we define the headway to be the decision variable. Accordingly,
we use the model to find out the optimal headway setting in each time interval. Table
2 shows the headway design to be solved in the model.
Table 2. Headway setting for each hour on daily type
Headway for Monday to Friday time-depend timetable
Line Hour Lower acceptance
limit/upper acceptance limit
Decision
variable
Red line 0600-0700 6/10 H1
0700-0800 2/6 H2
0800-0900 2/6 H3
0900-1600 6/10 H4
1600-1700 6/10 H5
1700-1800 2/6 H6
1800-1900 2/6 H7
1900-2200 6/10 H8
2200-0100 6/10 H9
Orange
line
0600-0700
0900-1600
6/10 H10
1800-0100
0700-0900
1600-1800
2/6 H11
Headway for Saturday to Sunday time-depend timetable
Line Hour Lower acceptance
limit/upper acceptance limit
Decision
variable
Red line 0600-1100 6/10 H12
1100-1600 6/10 H13
1600-1700 6/10 H14
1700-1800 6/10 H15
1800-1900 6/10 H16
1900-2200 6/10 H17
2200-0100 6/10 H18
Orange
line
0600-0100 6/10 H19
RESULTS
Scenario 1: Minimize the total system cost
Depending on the previous settings, and assign the model parameters (see table
3) into Model 1, the result is showing in table 4.
Table 3. Parameter settings of the model
Parameter Value
DC 7.317 (NTD, per minutes)
DF 1.511
TO 50 (minute, run trip time of red line)
30 (minute, run trip time of the orange line)
TR 20 (minute, time retrieve the train from the depot)
TP 25 (minute, time depart the train from the depot)
Kw 243.3 (kilowatt per round trip of the red line)
131.8 (kilowatt per round trip of orange line)
Ck 2.8 (NTD, per kilowatt)
2.27(NTD, per kilowatt)
Dm
4.8 (kilometer between the main line and South depot)
4.1(kilometer between the main line and North depot)
1.1(kilometer between the main line and Daliao depot)
Km 9.1(kilowatt per car kilometer of the red line)
10.7(kilowatt per car kilometer of the red line)
L 28.242 (kilometer of the red line)
13.143 (kilometer of the orange line)
Cm 6.455(NTD, per car kilometer)
Cp 1.58(NTD, per car kilometer)
n 37 (number of stations)
HS 2(minute)
HG 6/10(minute, in peak period/ in off-peak period)
Table 4. Total system cost and new headways of the time-dependent timetable under
different weights
Line HourLower acceptance limit/upper
acceptance limitDecision variable W=0.25 W=0.5 W=0.75 W=1 W=1.5 W=2 W=2.5
0600-0700 6/10 H1 8 7 6 6 6 6 6
0700-0800 2/6 H2 6 5 4 4 3 3 2
0800-0900 2/6 H3 6 6 5 5 4 3 3
0900-1600 6/10 H4 10 9 8 7 6 6 6
1600-1700 6/10 H5 7 6 6 6 6 6 6
1700-1800 2/6 H6 6 5 4 3 3 2 2
1800-1900 2/6 H7 6 5 4 4 3 3 2
1900-2200 6/10 H8 9 7 6 6 6 6 6
2200-0100 6/10 H9 10 8 7 6 6 6 6
0600-0700
0900-1600
1800-0100
6/10 H10 10 9 8 8 7 6 6
0700-0900
1600-18002/6 H11 6 6 6 6 5 4 4
0600-1100 6/10 H12 10 8 6 6 6 6 6
1100-1600 6/10 H13 8 6 6 6 6 6 6
1600-1700 6/10 H14 6 6 6 6 6 6 6
1700-1800 6/10 H15 6 6 6 6 6 6 6
1800-1900 6/10 H16 6 6 6 6 6 6 6
1900-2200 6/10 H17 7 6 6 6 6 6 6
2200-0100 6/10 H18 8 7 6 6 6 6 6
Orange line 0600-0100 6/10 H19 9 8 7 7 6 6 6
6,898,459 8,207,563 9,364,824 10,445,160 12,451,270 14,408,380 16,288,580
Red line
Model 1- Total cost(TC)
Headway for Monday to Friday time-depend timetable
Red line
Orange line
Headway for Saturday to Sunday time-depend timetable
Scenario 2: Maximize the net profit
Incorporating the relative parameters (see table 3) into Model 2, the result is
showing in table 5.
Table 5. Net profit and new headways of the time-dependent timetable under different
weights
Comparison of the performance of the current and time-dependent timetables of
the KRTC system
1. Total cost (operating cost, passenger waiting cost) and net profit
We use model 1 and 2 to calculate the new timetables that the KRTC operates
now, table 6 shows the total cost and net profit among different weights.
Line HourLower acceptance
limit/upper acceptance limitDecision variable W=0.25 W=0.5 W=0.75 W=1 W=1.5 W=2 W=2.5
0600-0700 6/10 H1 8 7 6 6 6 6 6
0700-0800 2/6 H2 6 5 4 4 3 3 2
0800-0900 2/6 H3 6 6 5 5 4 3 3
0900-1600 6/10 H4 10 9 8 7 6 6 6
1600-1700 6/10 H5 7 6 6 6 6 6 6
1700-1800 2/6 H6 6 5 4 3 3 2 2
1800-1900 2/6 H7 6 5 4 4 3 3 2
1900-2200 6/10 H8 9 7 6 6 6 6 6
2200-0100 6/10 H9 10 8 7 6 6 6 6
0600-0700
0900-1600
1800-0100
6/10 H10 10 9 8 8 7 6 6
0700-0900
1600-18002/6 H11 6 6 6 6 5 4 4
0600-1100 6/10 H12 10 8 6 6 6 6 6
1100-1600 6/10 H13 8 6 6 6 6 6 6
1600-1700 6/10 H14 6 6 6 6 6 6 6
1700-1800 6/10 H15 6 6 6 6 6 6 6
1800-1900 6/10 H16 6 6 6 6 6 6 6
1900-2200 6/10 H17 7 6 6 6 6 6 6
2200-0100 6/10 H18 8 7 6 6 6 6 6
Orange line 0600-0100 6/10 H19 9 8 7 7 6 6 6
14,405,720 13,096,620 11,939,360 10,859,020 8,852,911 6,895,802 5,015,606
Headway for Monday to Friday time-depend timetable
Red line
Orange line
Headway for Saturday to Sunday time-depend timetable
Red line
Model 2- Net income(NI)
Table 6. Total cost and net profit associated with the timetables of the KRTC
In table 6, some important results are summarized:
A. W=1: If we give the same weights between operating cost and passenger waiting
time, and using the developed model to calculate the time-dependent headway, the
KRTC can save 0.86% of the total cost and increases 7.88% net benefit.
B. W=1.5~3: If we increase the relative weight on the passenger waiting time, which
means that we provide shorter headway in each hour, the new timetable provided
by the developed model gives better performance in total cost and net profit,
especially in the net profit.
C. W=0.25~0.75: If we place more importance on the operating cost, which means
that we would like to reduce daily operating cost, and provide longer headway in
each hour, the system performance result is still better than the KRTC’s current
timetable’s performance.
2. Headway in each hour
Timetable type\ Weight W=0.25 W=0.5 W=0.75 W=1 W=1.5 W=2 W=2.5
KRTC timetable operating cost(A) 3,348,088 3,348,088 3,348,088 3,348,088 3,348,088 3,348,088 3,348,088
KRTC timetable passenger waiting cost(B) 3,674,702 4,845,705 6,016,708 7,187,712 9,529,712 11,871,722 14,213,732
KRTC timetable total cost(C=A+B) 7,022,790 8,193,793 9,364,796 10,535,800 12,877,800 15,219,810 17,561,820
New time-depend timetable operating cost(D) 3,156,237 3,569,363 3,997,756 4,139,064 4,582,288 4,938,098 5,122,127
New time-depend timetable passenger waiting cost(E) 3,742,222 4,638,200 5,367,068 6,306,096 7,868,982 9,470,282 11,166,453
New time-depend timetable total cost(F=D+E) 6,898,459 8,207,563 9,364,824 10,445,160 12,451,270 14,408,380 16,288,580
Operating cost saving rate ((D-A)/A,%) -5.73% 6.61% 19.40% 23.62% 36.86% 47.49% 52.99%
Paseenger waiting cost saving rate ((E-B)/B,%) 1.84% -4.28% -10.80% -12.27% -17.43% -20.23% -21.44%
Total cost saving rate ((F-C)/C,%) -1.77% 0.17% 0.00% -0.86% -3.31% -5.33% -7.25%
KRTC timetable net income(G) 13,579,070 12,408,070 11,237,060 10,066,060 7,724,055 5,382,049 3,040,043
New time-depend timetable net income(H) 14,405,720 13,096,620 11,939,360 10,859,020 8,852,911 6,895,802 5,015,606
Income increasing rate((H-G)/G),% 6.09% 5.55% 6.25% 7.88% 14.61% 28.13% 64.98%
Model 1 result
Model 2 result
As shown in table 7, the KRTC’s current timetable provides three types (one for
Monday to Thursday, another for Friday and the other for Saturday and Sunday).
Comparing the contents in table 4 to those in table 7, we can observe, when W=1, in
most operating hour, Model 1 and 2 can produce better headway and maintain a
certain service level for passengers. Especially, when we place more priority on
passenger waiting time (meaning that we increase the value of W in the model), we
can obtain better system performance.
Table 7. Headways of the KRTC timetable
Scenario 3: Adopting the extra trains
To analyze the O-D distribution, we can find out several key characteristic as
following:
1. There are 75% O-D ridership centralized in red line and 25% in orange line.
HourHeadway for Monday to
ThurdayHour Headway for Friday Hour
Headway for Saturday to
Sunday
0600-0630 8 0600-0630 8 0600-1030 8
0630-0830 4-6 0630-0830 4-6 1030-1500 7-8
0830-1630 8 0830-1630 8 1500-1630 5-7
1630-1830 4-6 1630-1830 4-6 1630-1830 4-6
1830-2300 8 1830-2130 6-8 1830-2100 6
2130-2300 7-8 2100-2300 6-8
0600-0630 8
0630-0830 4-6 0600-0630 8 0600-1630 8
0830-1630 8 0630-0830 4-6 1630-1830 6-8
1630-1830 4-6 0830-1630 8
1630-1830 4-6
1830-2300 81830-2300 8
1830-2300 7-8
Red line
Orange line
Red line
Orange line
Red line
Orange line
2. 41% O-D ridership centralizes in peak-period (07:00 AM~09:00AM and 17:00
PM~19:00 PM).
Thereby, we evaluate the extra train operation in peak period, and take W=1 for
example, the result is shown in table 8.
Table 8. Total system cost of the time-dependent timetable by adopting extra train in
peak period
We use model 1 and model 2 to calculate the cost of the new timetables that the
KRTC operates now, table 8 and table 9 shows the total system cost and net income
by adopting extra train operation. When we adopt extra train operation in peak period,
the more extra trains we used (meaning that the headway becomes shorter), the less
total cost we saved. Based on the level of service, the KRTC can spend little money
and provide better service quality for passenger.
Line HourLower acceptance limit/upper
acceptance limitDecision variable W=1 W=1 W=1 W=1 W=1 W=1 W=1 W=1 W=1 W=1 W=1
0600-0700 6/10 H1 6 6 6 6 6 6 6 6 6 6 6
0700-0800 2/6 H2 4 60/16 60/17 60/18 60/19 60/19 60/19 60/19 60/19 60/19 3
0800-0900 2/6 H3 5 60/13 60/14 60/14 60/14 60/14 60/14 60/14 60/14 60/14 4
0900-1600 6/10 H4 7 7 7 7 7 7 7 7 7 7 7
1600-1700 6/10 H5 6 6 6 6 6 6 6 6 6 6 6
1700-1800 2/6 H6 3 60/21 60/22 60/23 60/24 60/25 60/26 60/27 60/28 60/29 2
1800-1900 2/6 H7 4 60/16 60/17 60/18 60/19 60/19 60/19 60/19 60/19 60/19 3
1900-2200 6/10 H8 6 6 6 6 6 6 6 6 6 6 6
2200-0100 6/10 H9 6 6 6 6 6 6 6 6 6 6 6
0600-0700
0900-1600
1800-0100
6/10 H10 8 8 8 8 8 8 8 8 8 8 8
0700-0900
1600-18002/6 H11 6 6 6 6 6 6 6 6 6 6 6
0600-1100 6/10 H12 6 6 6 6 6 6 6 6 6 6 6
1100-1600 6/10 H13 6 6 6 6 6 6 6 6 6 6 6
1600-1700 6/10 H14 6 6 6 6 6 6 6 6 6 6 6
1700-1800 6/10 H15 6 6 6 6 6 6 6 6 6 6 6
1800-1900 6/10 H16 6 6 6 6 6 6 6 6 6 6 6
1900-2200 6/10 H17 6 6 6 6 6 6 6 6 6 6 6
2200-0100 6/10 H18 6 6 6 6 6 6 6 6 6 6 6
Orange line 0600-0100 6/10 H19 7 7 7 7 7 7 7 7 7 7 7
10,445,160 10,443,560 10,446,530 10,451,410 10,458,550 10,462,850 10,467,530 10,472,540 10,477,850 10,483,430 10,495,920
Red line
Orange line
Red line
Model 1- Total cost(TC)
Headway for Monday to Friday time-depend timetable-Extra train
Headway for Saturday to Sunday time-depend timetable
Table 9. Comparison of total system cost and net profit of the timetable of the
KRTC by adopting extra train operation
Scenario 4: Adopting the shuttle trains
To analyze the O-D ridership of every daily type, we observe a significant spatial
distribution, there are almost 50% ridership centralized between R3 station to R16
station in daily operation. So we segment the red line into two individual sections and
use different headway settings to operating. Table 10 and table 11 show the results of
total system cost, net profit and headway settings for different line segments.
Line Hour W=1 W=1 W=1 W=1 W=1 W=1 W=1 W=1 W=1 W=1 W=1
0700-0800 4 60/16 60/17 60/18 60/19 60/19 60/19 60/19 60/19 60/19 3
0800-0900 5 60/13 60/14 60/14 60/14 60/14 60/14 60/14 60/14 60/14 4
1700-1800 3 60/21 60/22 60/23 60/24 60/25 60/26 60/27 60/28 60/29 2
1800-1900 4 60/16 60/17 60/18 60/19 60/19 60/19 60/19 60/19 60/19 3
10,445,160 10,443,560 10,446,530 10,451,410 10,458,550 10,462,850 10,467,530 10,472,540 10,477,850 10,483,430 10,495,920
4,172,235 4,209,041 4,245,846 4,273,451 4,301,055 4,310,257 4,319,458 4,328,660 4,337,861 4,347,063 4,383,868
6,272,925 6,234,519 6,200,684 6,177,959 6,157,495 6,152,593 6,148,072 6,143,880 6,139,989 6,136,367 6,112,052
-0.86% -0.88% -0.85% -0.80% -0.73% -0.69% -0.65% -0.60% -0.55% -0.50% -0.38%
24.62% 25.71% 26.81% 27.64% 28.46% 28.74% 29.01% 29.29% 29.56% 29.84% 30.94%
-12.73% -13.26% -13.73% -14.05% -14.33% -14.40% -14.46% -14.52% -14.58% -14.63% -14.97%
10,848,350 10,849,960 10,846,990 10,842,110 10,834,960 10,830,660 10,825,990 10,820,980 10,815,670 10,810,090 10,797,600
10,066,060 10,066,060 10,066,060 10,066,060 10,066,060 10,066,060 10,066,060 10,066,060 10,066,060 10,066,060 10,066,060
7.77% 7.79% 7.76% 7.71% 7.64% 7.60% 7.55% 7.50% 7.45% 7.39% 7.27%
M2-Net income (G)
KRTC timetable net income (H)
Income increasing rate((G-H)/H),%
Model 1 result-Adopting extra train
Red line
Operating cost saving rate ((B-D)/D,%)
Paseenger waiting cost saving rate ((C-E)/E,%)
M1-Total cost (A=B+C)
Operating cost (B)
Passenger waiting cost (C)
KRTC timetable operating cost (D)
KRTC timetable passenger waiting cost (E)
Total cost saving rate ((A-F)/F,%)
KRTC timetable total cost (F=D+E) 10,535,800
3,348,088
7,187,712
Table 10. Total system cost and new headways of the time-dependent timetable in the
two individual sections under different weights
Line HourLower acceptance
limit/upper acceptanceDecision variable W=0.25 W=0.5 W=0.75 W=1 W=1.5 W=2 W=2.5
0600-0700 6/10 H1 6 6 6 6 6 6 6
0700-0800 2/6 H2 4 4 3 3 2 2 2
0800-0900 2/6 H3 5 4 4 4 3 3 3
0900-1600 6/10 H4 9 7 6 6 6 6 6
1600-1700 6/10 H5 6 6 6 6 6 6 6
1700-1800 2/6 H6 5 4 4 3 3 2 2
1800-1900 2/6 H7 4 4 3 3 2 2 2
1900-2200 6/10 H8 7 6 6 6 6 6 6
2200-0100 6/10 H9 7 6 6 6 6 6 6
0600-0700 6/10 H20 10 10 9 8 7 6 6
0700-0800 2/6 H21 6 6 5 5 4 3 3
0800-0900 2/6 H22 6 6 6 6 5 4 4
0900-1600 6/10 H23 10 10 9 8 6 6 6
1600-1700 6/10 H24 10 8 7 6 6 6 6
1700-1800 2/6 H25 6 5 4 4 3 3 2
1800-1900 2/6 H26 6 6 5 4 4 3 3
1900-2200 6/10 H27 10 8 7 6 6 6 6
2200-0100 6/10 H28 10 10 9 8 7 6 6
0600-0700
0900-1600
1800-0100
6/10 H10 10 9 8 8 7 6 6
0700-0900
1600-18002/6 H11 6 6 6 6 5 4 4
0600-1100 6/10 H12 7 6 6 6 6 6 6
1100-1600 6/10 H13 6 6 6 6 6 6 6
1600-1700 6/10 H14 6 6 6 6 6 6 6
1700-1800 6/10 H15 6 6 6 6 6 6 6
1800-1900 6/10 H16 6 6 6 6 6 6 6
1900-2200 6/10 H17 6 6 6 6 6 6 6
2200-0100 6/10 H18 6 6 6 6 6 6 6
0600-1100 6/10 H29 10 9 8 7 6 6 6
1100-1600 6/10 H30 9 7 6 6 6 6 6
1600-1700 6/10 H31 6 6 6 6 6 6 6
1700-1800 6/10 H32 8 6 6 6 6 6 6
1800-1900 6/10 H33 8 6 6 6 6 6 6
1900-2200 6/10 H34 10 7 6 6 6 6 6
2200-0100 6/10 H35 10 10 8 7 6 6 6
Orange line 0600-0100 6/10 H19 9 8 7 7 6 6 6
6,854,499 8,151,512 9,327,100 10,425,090 12,472,600 14,413,830 16,294,700
Red line
Headway for Saturday to Sunday time-depend timetable-R3~R16
Headway for Saturday to Sunday time-depend timetable-R16~R23
Red line
Model 1- Total cost(TC)
Headway for Monday to Friday time-depend timetable-R16~R23
Red line
Orange line
Red line
Headway for Monday to Friday time-depend timetable-R3~R16
Table 11. Net profit and new headways of the time-dependent timetable in the two
individual sections under different weights
Table 12 show the result by comparing the KRTC current timetable on total cost,
operating cost, passenger waiting cost and net profit to those of the new timetable.
Line HourLower acceptance
limit/upper acceptanceDecision variable W=0.25 W=0.5 W=0.75 W=1 W=1.5 W=2 W=2.5
0600-0700 6/10 H1 6 6 6 6 6 6 6
0700-0800 2/6 H2 4 4 3 3 2 2 2
0800-0900 2/6 H3 5 4 4 4 3 3 3
0900-1600 6/10 H4 9 7 6 6 6 6 6
1600-1700 6/10 H5 6 6 6 6 6 6 6
1700-1800 2/6 H6 5 4 4 3 3 2 2
1800-1900 2/6 H7 4 4 3 3 2 2 2
1900-2200 6/10 H8 7 6 6 6 6 6 6
2200-0100 6/10 H9 7 9 6 6 6 6 6
0600-0700 6/10 H20 10 10 9 8 7 6 6
0700-0800 2/6 H21 6 6 5 5 4 3 3
0800-0900 2/6 H22 6 6 6 6 5 4 4
0900-1600 6/10 H23 10 10 9 8 6 6 6
1600-1700 6/10 H24 10 8 7 6 6 6 6
1700-1800 2/6 H25 6 5 4 4 3 3 2
1800-1900 2/6 H26 6 6 5 4 4 3 3
1900-2200 6/10 H27 10 8 7 6 6 6 6
2200-0100 6/10 H28 10 10 9 8 7 6 6
0600-0700
0900-1600
1800-0100
6/10 H10 10 9 8 8 7 6 6
0700-0900
1600-18002/6 H11 6 6 6 6 5 4 4
0600-1100 6/10 H12 7 6 6 6 6 6 6
1100-1600 6/10 H13 6 6 6 6 6 6 6
1600-1700 6/10 H14 6 6 6 6 6 6 6
1700-1800 6/10 H15 6 6 6 6 6 6 6
1800-1900 6/10 H16 6 6 6 6 6 6 6
1900-2200 6/10 H17 6 6 6 6 6 6 6
2200-0100 6/10 H18 6 6 6 6 6 6 6
0600-1100 6/10 H29 10 9 8 7 6 6 6
1100-1600 6/10 H30 9 7 6 6 6 6 6
1600-1700 6/10 H31 6 6 6 6 6 6 6
1700-1800 6/10 H32 8 6 6 6 6 6 6
1800-1900 6/10 H33 8 6 6 6 6 6 6
1900-2200 6/10 H34 10 7 6 6 6 6 6
2200-0100 6/10 H35 10 10 8 7 6 6 6
Orange line 0600-0100 6/10 H19 9 8 6 6 6 6 6
14,449,580 13,152,560 11,976,980 10,878,980 8,831,478 6,890,246 5,009,375
Headway for Monday to Friday time-depend timetable-R3~R16
Red line
Headway for Saturday to Sunday time-depend timetable-R3~R16
Headway for Saturday to Sunday time-depend timetable-R16~R23
Red line
Model 2- Net income(NI)
Headway for Monday to Friday time-depend timetable-R16~R23
Red line
Orange line
Red line
Table 12. Compare total cost and net income to the timetable of KRTC by adopting
shuttle train
As shown in table 12, when we increase the relative weight on the passenger
waiting time, the KRTC can save 1.05% total system cost and increase 8.08% net
profit. Focusing on the headway, as demonstrated in tables 10 and 11, in most hours
we also can provide better headways in the two individual sections than the original
timetable. Particularly, even we place more priority on the passenger waiting time, we
still obtain a certain performance in terms of total system cost saving and/or net profit
gaining.
Timetable type\ Weight W=0.25 W=0.5 W=0.75 W=1 W=1.5 W=2 W=2.5
KRTC timetable operating cost(A) 3,348,088 3,348,088 3,348,088 3,348,088 3,348,088 3,348,088 3,348,088
KRTC timetable passenger waiting cost(B) 3,674,702 4,845,705 6,016,708 7,187,712 9,529,712 11,871,722 14,213,732
KRTC timetable total cost(C=A+B) 7,022,790 8,193,793 9,364,796 10,535,800 12,877,800 15,219,810 17,561,820
New time-depend timetable operating cost(D) 3,274,236 3,595,035 3,950,187 4,093,645 4,603,643 4,974,024 5,015,240
New time-depend timetable passenger waiting cost(E) 3,580,263 4,556,477 5,376,913 6,331,445 7,868,957 9,439,806 11,279,460
New time-depend timetable total cost(F=D+E) 6,854,499 8,151,512 9,327,100 10,425,090 12,472,600 14,413,830 16,294,700
Operating cost saving rate ((D-A)/A,%) -2.21% 7.38% 17.98% 22.27% 37.50% 48.56% 49.79%
Paseenger waiting cost saving rate ((E-B)/B,%) -2.57% -5.97% -10.63% -11.91% -17.43% -20.48% -20.64%
Total cost saving rate ((F-C)/C,%) -2.40% -0.52% -0.40% -1.05% -3.15% -5.30% -7.22%
KRTC timetable net income(G) 13,579,070 12,408,070 11,237,060 10,066,060 7,724,055 5,382,049 3,040,043
New time-depend timetable net income(H) 14,449,580 13,152,560 11,976,980 10,878,980 8,831,478 6,890,246 5,009,375
Income increasing rate((H-G)/G),% 6.41% 6.00% 6.58% 8.08% 14.34% 28.02% 64.78%
Model 2 result
Model 1 result
CONCLUSIONS
In this research, first we establish models 1 and 2 to find out the system’s best
performance under the government service indicators by considering the operating
cost and passenger waiting time, than we adopt different operating models (i.e. extra
train and shuttle train) to provide the decision maker with a beneficial advice on train
timetable design. Following are some results found in different test scenarios.
1. Using the models 1 and 2 under the government service indicator limitation, we
can get the better cost-effective and net profit, and also obtain a certain level
service in the daily operating under different relative weights.
2. If the KRTC wants to provide better service in the peak hours, they can adopt the
extra train operation. Use this strategy, the KRTC just increases 5% of the
operating cost (approximately spending 30,000 NTD per day and adding 19
trains in peak hour), the system can provide shorter headways in peak periods.
3. The shuttle train mode is one of the operating strategies that the KRTC could
consider to adopt. But there are several constraint factors needed to be
considered, including track infrastructure, train retrieving and departing, etc.
FUTURE WORK
In this paper, we didn’t consider some constraints of the real operating situations.
In the near future, the following factors can be studied in later research.
1. Runtime reserve: Runtime reserve is one of the factors which can influence the
energy consumption. In the KRTC system, the runtime reserve in the red line is
13% and 10% for the orange line. The more degree of runtime reserve is adopted,
and the trains can accelerate and decelerate smoothly, and the system will save
more energy. The issue of runtime reserve on total system cost saving could be
further incorporated into the developed model.
2. Mission arrangement and driving scheduling: For a MRT system, it is a
continuous and integrated event in timetable, mission arrangement and driving
scheduling. In this research, we just focus on the timetable optimization. Later
research could incorporate the above task items into an integrated and
comprehensive time-dependent timetable design model.
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