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webinar of the ALC BRT - COE
july 2014
Integrating timetabling and vehicle scheduling to analyze
the trade-off between transfers and the fleet size
Omar Jorge Ibarra Rojas
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Outline
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• Context
• Transit network characteristics
• Timetabling problem
• Vehicle scheduling problem
• Integrated approach
• Conclusions and future research
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4.
Transit network planning
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context
Frequency setting
Timetabling
Vehicle scheduling
Crew assignment
tactical decisions
operational decisions
level of service
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Transit network planning
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context
Frequency setting
Timetabling
Vehicle scheduling
Crew assignment
tactical decisions
operational decisions
costs $$$
level of service
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6.
How to solve the planning problem?
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context
Frequency setting
Timetabling
Vehicle scheduling
Crew assignment
solution
feedback
solution
feedback
solution
feedback
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7.
Drawbacks of sequential approaches
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context
• Suboptimal solutions, even for subproblems.
• Restrictive for the last subproblems solved due to solution of previous
subproblems.
• Deﬁning feedback.
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8.
Drawbacks of sequential approaches
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context
• Suboptimal solutions, even for subproblems.
• Restrictive for the last subproblems solved due to solution of previous
subproblems.
• Deﬁning feedback.
Alternative: Integrate subproblems to
jointly determine their decisions
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9.
Motivation
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Our goal: help to decision makers of
transport system management by
integrating subproblems of the planning
problem through operations research
techniques
context
Frequency
setting
Integrated
Timetabling
and
Vehicle scheduling
Crew
assignment
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10.
Integrated approach
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context
• Advantage: possible to ﬁnd optimal solution for each subproblem
considering the degrees of freedom of the integrated subproblems.
• Handicaps: Exploring a large solution space and to deﬁning a proper
objective function.
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12.
Passengers demand
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Transit Network
• Each day can be divided into different planning periods such as morning peak-
hour, morning non peak-hour, afternoon peak hour, and so on.
• Constant passenger demand in each period => regular service is desired.
• The number of passengers transferring from one line to another is
proportional to the bus load of the feeding line.
• Frequency setting previously solved => the number of trips is given for each
line and planning period (no capacity issues).
• Small delays (up to 10% of the even headway) do not affect the passengers
demand.
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13.
Bus lines
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• There are planning periods with mid/low frequencies where well-timed
transfers are needed.
• Passengers may transfer from a line A to a line B and not necessarily vice
versa.
• Buses can not be held at stops.
• Lines start and end at the same point.
• Accurate estimation of the travel times from depot to each transfer node, for
all lines and periods.
Transit Network
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15.
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Timetabling problem
Problem deﬁnition
Determine the departure times for all trips that maximizes the
number of passengers beneﬁt from well-timed passenger transfers.
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16.
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Timetabling problem
Input
Set of lines
Set of planning periods for each
Frequency of line i for period s
Stops where passengers transfer from i to j
Number of passengers that need to transfer from line i to line j at stop
b in planning period s considering a regular service.
S
I
fi
s
Bij
[as, bs] s 2 S
paxijb
s
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17.
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Timetabling problem
Input
as = 8 : 00 bs = 8 : 40
a) Even headway hi
s =
bs as
fi
s
8 : 05 8 : 15 8 : 25 8 : 35
as = 8 : 00 bs = 8 : 40
b) Almost even headway times. Flexibility parameter
[ ]
i
s = 1 min
[ ][ ] [ ]
Di
2 = [8 : 14, 8 : 16]
i
s
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18.
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Timetabling problem
Input
b
line i
line j
timeib
p
timejb
q
⇥
MinWijb
pq , MaxWijb
pq
⇤
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19.
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Timetabling problem
Decisions
• : Departure time for each trip p of line i
• :Auxiliary variable to identify if separation time between trip q of line j
and trip p of line i at node b are within
• : Number of passengers transferring from trip p of line i to line j at
node b considering the departure time
Xi
p
Y ijb
pq ⇥
MinWijb
pq , MaxWijb
pq
⇤
PAXijb
p
PAXijb
p := paxijb
s 1 +
Xi
p Xi
p 1 hi
s
hi
s
!
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20.
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Timetabling problem
Mathematical formulation
max FTT(X) =
X
i2I
X
j2J(i)
X
b2Bij
fi
X
p=1
PSijb
p
Xi
p 2 Di
p
(Xj
q + tjb
q ) (Xi
p + tib
p ) 2
⇥
MinWijb
pq , MaxWijb
pq
⇤
! Y ijb
pq = 1
PSijb
p = PAXijb
p
fj
X
q=1
Y ijb
pq
(1)
(2)
(3)
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22.
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Problem deﬁnition
Determine the trip-vehicle assignment to minimize the ﬂeet size
Vehicle scheduling problem
Vehicle Scheduling I:
Fixed Schedules
It is better to
doubt what is
true than accept
what isn’t
No. of
vehicles
Scheduler
Gantt chart
Time
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Input
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Vehicle scheduling problem
ri
p
• A timetable.
• F: Set of ﬂeets where each ﬂeet f cover a set of lines L(f)
• :Turnaround time for trip p of line i
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Decisions
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Vehicle scheduling problem
o o’i(1) i(2) i(fi
) j(1) j(fj
). . . . . .
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25.
Decisions
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Vehicle scheduling problem
o o’i(1) i(2) i(fi
) j(1) j(fj
). . . . . .
V ijf
pq =
⇢
1 if a vehicle of ﬂeet f makes trip j(q) after ﬁnishing trip i(p),
0 otherwise,
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Mathematical formulation
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Vehicle scheduling problem
X
j2I(f)
fj
X
q=1
V ijf
pq =
X
j2I(f)
fj
X
q=1
V jif
qp = 1 8f, p, i (4)
min FV S(V ) =
X
f2F
X
i2I(f)
fi
X
p=1
V if
op
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Common approaches
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Integrated Approach
Sequential or
min w1FT T (X) + w2FV S(V )
X 2 X
V 2 V
Guihaire and Hao (2010)
Fleurent et al. (2009)
Guihaire and Hao (2008)
van den Heuvel et al. (2008)
Liu and Shen (2007)
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29.
Objectives conﬂict nature
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Integrated Approach
Users costs Agency costs
100 60
70 100
Which is the best solution?
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Pareto front
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Integrated Approach
Analyze the trade-off between criteria by ﬁnding efﬁcient solutions
Feasible solution
space
Non-convex
Pareto curve
FT T
FV S
F✏
T T (x)
✏
Efﬁcient solutions
Dominated solutions
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Common approach drawbacks
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Integrated Approach
• It misses solution points on the non-convex part of the Pareto surface.
• Even distribution of weights does not translate to uniform distribution of
the solution points.
• The distribution of solution points is highly dependent on the relative
scaling of the objective.
• Misinterpretation of the theoretical and practical meaning of the weights
can make the process of intuitively selecting non-arbitrary weights an
inefﬁcient chore.
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Our integrated formulation
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Integrated Approach
Timetabling constraints
Vehicle scheduling constraints
(1)-(3)
Xj
q Xi
p + ri
p M 1 V ijf
pq (5)8 f, i, j, p, q
(4)
[max FT T (X), min FV S(V )]
Text
+ epsilon constraint
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Solution approach: epsilon-constraint
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Integrated Approach
Feasible solution
space
Non-convex
Pareto curve
FT T
FV S
F✏
T T (x)
✏
Algorithm 1 : ✏-constraint for TT-VS
Input: TT-VS instance
Output: ListPareto: Pareto optimal points
1: ListPareto = ;
2: Find V S⇤
= {min FVS(V ) : (1)-(5)}
3: Find TT⇤
= {max FTT(X) : (1)-(5)}
4: Find P⇤
1 = {max FTT(X) : (1)-(5), FVS(V ) V S⇤
}
5: Find P⇤
2 = {min FVS(V ) : (1)-(5), FTT(X) TT⇤
}
6: ListPareto = ListPareto [ {(TT⇤
, P⇤
2 ) , (P⇤
1 , V S⇤
)}
7: Let ✏ = P⇤
2 1
8: while ✏ > V S⇤
do
9: Find P⇤
✏ = {max FTT(X) : (1)-(5), FVS(V ) ✏}
10: Update ListPareto considering (P⇤
✏ , ✏)
11: ✏ = ✏ 1
12: end while
4. Experimental Study94
ﬁndextremepointsﬁllParetofront
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Test instances
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Integrated Approach
Instances T1 T2 T3 T4 T5 T6
|I| 10 50 10 50 10 50
|B| 1 5 1 5 1 5
100
i
s
hi
s
2 [7.5,12.5] [7.5,12.5] [11.25,18.75] [11.25,18.75] [15,25] [15,25]
Table 1: Instance types and parameter values.
4.2. Analysis of Results328
Our ✏-constraint algorithm described by Algorithm 1 was implemented on a Macbook air329
1.3 GHz Intel Core i5 processor with 4 GB 1600 MHz of RAM. We used the integer linear330
programming solver CPLEX 12.6. Table 2 shows the computational time in seconds (Time)331
Instances based on a transit network in Mexico (Ibarra-Rojas et al., 2014)
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Conclusions
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• It is possible to identify instances where the conﬂict of objectives is present.
• It is possible to measure the “cost” of a vehicle in terms of well-timed
passenger transfers.
• Computational times are acceptable since the input (lines and frequency) are
modiﬁed in long periods, e.g., once every six months.
Conclusions
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42.
Future research
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• Heterogeneous ﬂeets.
• Multiple-depots.
• Other criteria such as total waiting time for larger ﬂexibility parameters and
deadhead costs for vehicles.
Conclusions
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43.
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References
Ibarra-Rojas, O., Giesen, R., Ríos-Solis,Y.A. An integrated approach for timetabling and vehicle scheduling problems to analyze the trade-off
between level of service and operating costs of transit networks. under revision in Transportation Research B.
Ibarra-Rojas, O., López-Irarragorri, F., Rios-Solis,Y.A., (2014). Multiperiod synchronization bus timetabling.Transportation Science (in press).
Ibarra-Rojas, O., Rios-Solis,Y.A., (2012). Synchronization of bus timetabling.Transportation Research B: Methodological 46, 599-614.
Guihaire, V., Hao, J.K., (2010). Transit network timetabling and vehicle assignment for regulating authorities. Computers and Industrial
Engineering 59, 16-23.
Fleurent, C., Lessard, R., (2009). Integrated Timetabling andVehicle Scheduling in Practice.Technical Report. GIRO Inc. Montreal, Canada.
van den Heuvel, A., van den Akker, J., van Kooten, M., (2008). Integrating timetabling and vehicle scheduling in public bus transportation.
Technical Report UUCS-2008-003. Department of Information and Computing Sciences, Utrecht University, Utrecht,The Netherlands.
Guihaire, V., Hao, J.K., (2008). Transit network re-timetabling and vehicle scheduling, in: Le Thi, H.A., Bouvry, P., Pham Dinh, T. (Eds.),
Modelling, Computation and Optimization in Information Systems and Management Sciences. Springer Berlin Heidelberg. volume 14 of
Communications in Computer and Information Science, pp. 135-144.
Liu, Z., Shen, J., (2007). Regional bus operation bi-level programming model integrating timetabling and vehicle scheduling. Systems
Engineering-Theory & Practice 27, 135-141.
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