ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΙΑΣ
Minimizing Waiting Time at Intermediate Nodes
for Intercity Public Bus Transportation
Saharidis G.K.D.
Dimitropoulos Ch.
Skordilis E.
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Introduction
The purpose of this research is to create a
suitable timetable for intercity buses,
departing from various nodes of the network,
such that the total waiting time of passengers
at intermediate nodes is minimized.
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Description of the problem
• By using the current infrastructure of the
public bus companies in Greece (dubbed KTEL),
we tried to reform the bus schedule, for better
service of passengers who use intermediate
nodes to reach their destination.
• In many cases, the waiting time at these nodes
is very long. The reason for this is that there
are interconnections of various itineraries,
which increments the difficulty of a suitable
time schedule.
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Formulation of the problem
• The problem is formulated as a mixed integer
linear program.
• The objective function is the minimization of
the sum of waiting times for every intermediate
node of the network
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Formulation of the problem
• Assumptions:
1. Steady travel time between nodes
2. Steady number of itineraries (routes) between connected
nodes
3. Sufficient parking space in bus terminals (no bottleneck)
4. Generally, parameters describing the problem are
considered steady
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Decision variables
• The departure time of every bus (𝐿𝑇𝑖,𝑗,𝑚 )
• The waiting time at an intermediate node
𝑚,𝑛
(𝑊𝑇𝑖,𝑘,𝑗 )
• A binary variable that defines the combination
of routes to and from an intermediate node as
𝑚,𝑛
active or inactive (𝑌𝑖,𝑘,𝑗 )
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Decision variables
• The time difference between the bus
departure and a high priority time (will be
used as penalty factor) (𝐺𝑖,𝑗,𝑚,𝑠 )
• A binary variable that defines the number of
buses affected by the high priority time
(𝐵𝑖,𝑗,𝑚,𝑠 )
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Illustration of the problem
Το σχήμα
περιλαμβάνει
3 παράλληλες
γραμμές
που αναπαριστούν
τους
κόμβους
αναχώρησης,
άφιξης
και τον ενδιάμεσο.
Το χρησιμοποιούμε
για να δείξουμε
πως υπολογίζουμε
τον χρόνο
αναμονής στον
ενδιάμεσο.
Saharidis G.K.D., Dimitropoulos Ch.,
Skordilis E.
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Sets, Parameters
Saharidis G.K.D., Dimitropoulos Ch.,
Skordilis E.
9
Sets, Parameters
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Constraints
• Constraint (1) defines the combination of
routes to and from an intermediate node as
active or inactive, based on whether the bus
from the intermediate node has already left or
not
sm  ( LTk , j ,n  LTi ,k ,m  TT i ,k  WT _ MIN k )  BigM  (2  D i ,k  D k , j  D i , j )  Yi ,mk ,,nj
Yi ,mk ,,nj  1  sm  ( LTk , j ,n  LTi ,k ,m  TT i ,k  WT _ MIN k )  BigM  (2  D i ,k  D k , j  D i , j )
i, k , j , m  Ri ,k , n  Rk , j
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Constraints
• Constraint (2) calculates the waiting time between the arrival
on each intermediate node and the first two available
departing routes from that node towards the destination
node.
LTk , j ,n  LTi ,k ,m  TT i ,k  WT _ MIN k  BigM  (2  D i ,k  D k , j  D i , j )  WTi ,mk ,,nj
i, k , j , m  Ri ,k
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Constraints
• Constraint (3) assures that a bus departing
from any node will return to this node before
and after a certain time
 BigM  (1  H i , j  H j ,i )  LTi , j ,m  TT i , j  BR _ MIN i , j  LT j ,i ,m i, j , m  Ri , j
LT j ,i ,m  LTi , j ,m  TT i , j  BR _ MAX i , j  BigM  (1  H i , j  H j ,i ) i, j , m  Ri , j
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Constraints
• Constraint (4) assures that each bus departs
after another.
 BigM  (1  D i , j )  LTi , j , m 1  NR _ MIN i , j  LTi , j ,m i, j , m  (1,..., Ri , j )
LTi , j ,m  LTi , j ,m1  NR _ MAX i , j  BigM  (1  D i , j ) i, j , m  (1,..., Ri , j )
• Constraint (5) defines the time window in
which a bus must depart
ST i , j ,m  LTi , j ,m  ET i , j ,m
i, j , m  Ri , j
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Extensions
Furthermore, an extension regarding high
priority times is introduced. By using this
extension bus departures are gravitated towards
certain times.
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Extensions
• Constraints (7) and (8) calculates the time
difference between the departing time of a
route and the time where a high passenger
demand occurs.
Gi , j ,m ,s  BT i , j ,s  LTi , j ,m  BigM  (2  D i , j  Bi , j ,m ,s )
i, j , s, m  Ri , j
Gi , j ,m ,s   ( BT i , j ,s  LTi , j ,m )  BigM  (2  D i , j  Bi , j ,m ,s ) i, j , s, m  Ri , j
• Constraint (9) defines the number of buses
affected by these high priority times.
M
B
m 0
i , j ,m , s
 C i , j ,s
i, j , s, m  Ri , j
C i , j ,s
 L i , j ,s
, if Li , j ,s mod MC  0

MC


  Li , j ,s   1, if Li , j ,s mod MC  0
  MC 

Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Objective function
• Minimization of the waiting time for all
available buses departing from the
intermediate node
I
K
I
Ri , k Rk , j
min  ( Rk , j  n  1)WTi ,mk ,,nj
k  i, k  j , i  j
i 0 k 0 j 0 m 0 n 0
• Using the penalty factor :
I
K
I
Ri , k Rk , j
min  ( Rk , j  n  1)WT
i 0 k 0 j 0 m 0 n 0
m ,n
i ,k , j
I
I
Ri , j
S
  Ai , j ,s  Gi , j ,m ,s
i 0 j 0 m 0 s 0
k  i, k  j , i  j
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Case study
• The formulation was applied to the intercity
bus network of the island of Crete.
• Numerical data for Crete network:
122 nodes total.
13 of these nodes considered intermediate.
Number of routes between nodes varies
from 2 to 25.
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Case study
We will examine three different cases:
Case 1 : Existing bus timetable (benchmark)
Case 2 : Minimizing waiting time without
penalty factor
Case 3 : Minimizing waiting time with penalty
factor to better approach the existing bus
timetables
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Case study
• For the penalty factor we introduced five high
priority times, each corresponding to five time
zones
High Priority Time:
Minute of the day:
330
Zone 1
Zone 2
435
645
540
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
Zone 3
Zone 4
855
750
Zone 5
1065
960
1275
1170
1380
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Case study
• For each time zone, the number of buses affected by the
high priority time was set by the number of buses departing,
based on the existing bus timetable
• This also applies to the weight factor of the penalty cost in
the objective function
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Results
Total
Waiting
Time
(%)*
Number of (%)*
Active
Routes
Average
Waiting
Time
(%)*
Case 1
1,994,385
-
33051
-
60
-
Case 2
493,924
75.2
19361
41.4
25
58.3
Case 3
925,444
53.5
25565
22.6
36
40.0
*Percentage of decrease over existing timetable (Case 1).
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Results
• Distribution of waiting times :
• Distribution of itineraries :
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Conclusions
• For all the examined cases there is a clear
reduction of the waiting times
• The use of a penalty factor (Case 3)results in an
increased waiting time compared to not using
it (Case 2)
• However, it is greatly reduced compare to the
existing bus timetable
• Using of a penalty factor (Case 3) leads to
timetable more suitable for realistic cases,
giving passengers more choices considering
their departure from intermediate nodes
Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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Saharidis G.K.D., Dimitropoulos Ch., Skordilis E.
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