webinar: integrating timetabling and vehicle scheduling to analyze the trade-off between transfers...

webinar of the ALC BRT - COEjuly 2014

Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and the fleet size

Omar Jorge Ibarra Rojas

Outline

2/

• Context

• Transit network characteristics

• Timetabling problem

• Vehicle scheduling problem

• Integrated approach

• Conclusions and future research

40

Transit network planning

3/

context

Frequency setting

Timetabling

Vehicle scheduling

Crew assignment

tactical decisions

operational decisions

40


3/

context

Frequency setting

Timetabling

Vehicle scheduling

Crew assignment

tactical decisions


level of service

40


3/

context

Frequency setting

Timetabling

Vehicle scheduling

Crew assignment

tactical decisions


costs $$$

level of service

40

How to solve the planning problem?

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context

Frequency setting

Timetabling

Vehicle scheduling

Crew assignment

solution

feedback

solution

feedback

solution

feedback

40

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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Drawbacks of sequential approaches

5/

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

40

Motivation

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Our goal: help to decision makers of t ranspor t sys tem management by integrating subproblems of the planning problem through operations research techniques

context

Frequency setting

Integrated Timetabling

andVehicle scheduling

Crew assignment

40

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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Transit network characteristics

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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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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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11/

Timetabling problem

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12/

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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13/

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

f is

Bij

[as, bs] s 2 S

pax

ijbs

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14/

Timetabling problem

Input

as = 8 : 00 bs = 8 : 40

a) Even headway his =

bs � asf is

8 : 05 8 : 15 8 : 25 8 : 35

as = 8 : 00 bs = 8 : 40

b) Almost even headway times. Flexibility parameter

[ ]

�is = 1 min

[ ][ ] [ ]Di

2 = [8 : 14, 8 : 16]

�is

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15/

Timetabling problem

Input

bline i

line j

timeibp

timejbq

⇥MinW

ijbpq ,MaxW

ijbpq

⇤

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16/

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

Xip

Y ijbpq ⇥

MinW

ijbpq ,MaxW

ijbpq

⇤

PAXijbp

PAX

ijbp := pax

ijbs

1 +

X

ip �X

ip�1 � h

is

h

is

!

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17/

Timetabling problem

Mathematical formulation

maxFTT(X) =

X

i2I

X

j2J(i)

X

b2Bij

fiX

p=1

PSijbp

Xip 2 Di

p

(Xjq + t

jbq )� (Xi

p + t

ibp ) 2

⇥MinW

ijbpq ,MaxW

ijbpq

⇤! Y

ijbpq = 1

PSijbp = PAXijb

p

fjX

q=1

Y ijbpq

(1)

(2)

(3)

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18/

Vehicle scheduling problem

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19/

Problem definition

Determine the trip-vehicle assignment to minimize the fleet size


Vehicle Scheduling I:Fixed Schedules

It is better todoubt what is

true than acceptwhat isn’t

No. ofvehicles

Sched

uler

Gantt chart

Time

7

Ch07-H6166 2/23/07 2:56 PM Page 163

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Input

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rip

• 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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o o’i(1) i(2) i(f i) j(1) j(f j). . . . . .

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Decisions

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o o’i(1) i(2) i(f i) j(1) j(f j). . . . . .

V ijfpq =

⇢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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X

j2I(f)

fjX

q=1

V ijfpq =

X

j2I(f)

fjX

q=1

V jifqp = 1 8f, p, i (4)

minFV S

(V ) =X

f2F

X

i2I(f)

f

iX

p=1

V if

op

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23/

Integrated Approach

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Common approaches

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Integrated Approach

Sequential or

minw1FTT (X) + w2FV S(V )

X 2 XV 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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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

FTT

FV S

F

✏TT (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)

Xjq �

�Xi

p + rip�� M

�1� V ijf

pq

�(5)8 f, i, j, p, q

(4)

[maxFTT (X),minFV S(V )]

Text

+ epsilon constraint40

Solution approach: epsilon-constraint

29/

Integrated Approach

Feasible solution space

Non-convex Pareto curve

FTT

FV S

F

✏TT (x)

✏

Algorithm 1 : ✏-constraint for TT-VSInput: TT-VS instanceOutput: ListPareto: Pareto optimal points

1: ListPareto = ;2: Find V S

⇤ = {minFVS(V ) : (1)-(5)}3: Find TT

⇤ = {maxFTT(X) : (1)-(5)}4: Find P

⇤1 = {maxFTT(X) : (1)-(5), FVS(V ) V S

⇤}5: Find P

⇤2 = {minFVS(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

⇤ do9: Find P

⇤✏

= {maxFTT(X) : (1)-(5), FVS(V ) ✏}10: Update ListPareto considering (P ⇤

✏

, ✏)11: ✏ = ✏ � 112: end while

4. Experimental Study294

We base our experimental study on cases inspired by the transit network of Monterrey,295

Mexico, where the bus transit system is private and di↵erent bus agencies share passenger296

demand. Competition between them creates particular characteristics: such as many bus297

lines with a mid/low frequency of service; a high concentration of bus lines in specific298

zones, such as universities, the central business district, and the main avenues; and finally,299

passengers only have an estimate of their waiting time to take the next bus. In general,300

passenger transfers are needed at specific stops of the transit network and the operation of301

the system allows flexible departure times but service regularity must be guaranteed. In this302

paper, this flexibility is used to improve the level of service and reduce operating costs.303

4.1. Scenarios Studied304

Based on information from the transit networks planners, the following assumptions are305

made to define our scenarios: one synchronization node per ten bus lines; the number of306

pairs of lines to be synchronized at each node is between one and seven; bus lines start and307

end at the same depot, which avoids deadhead implementation; each day can be divided into308

planning periods with specific demand and travel times; finally, the agencies are capable of309

defining values for the flexibility parameters to satisfy their operating policies. Since the310

operating costs are strongly related to bus purchases, bus maintenance, and the drivers’311

wages, the minimization of the fleet size is a justifiable objective to benefit the agency of312

our case study. Thus, cv� = 1 and cdh

ij� = 0 for each � 2 �, i, j 2 I(�). To perform our313

experiments, we introduce several scenarios with di↵erent sizes and flexibility characteristics.314

The network size is determined by the number of lines |I| and the number of transfer315

nodes |B|. All instance types have six planning periods of T = 240 minutes each. The316

frequency for each line i is randomly generated between [13,18]; turnaround times r

i

p

are317

randomly generated between 80 and 150; setup parameters dhij are zero; travel times from318

13

find extreme points

fill Pareto front

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Test instances

30/

Integrated Approach

Instances T1 T2 T3 T4 T5 T6

|I| 10 50 10 50 10 50

|B| 1 5 1 5 1 5

100 �

is

h

is

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

and the number of solutions in the Pareto Frontier (PF) for each one of the proposed332

instances. Note that our ✏-constraint algorithm is capable of finding the Pareto optimal333

solutions for all instances of our case study.

T1 T2 T3 T4 T5 T6

time PF time PF time PF time PF time PF time PF

1 26.25 1 569.66 2 15.66 1 586.85 1 123.96 2 73093.8 5*

2 42.06 1 246.51 1 31.06 1 237.18 1 375.062 2 21313.9 2

3 28.52 1 384.69 2 28.71 1 1030.64 2 152 2 25493.9 3

4 49.65 1 381.59 1 88.93 2 508.43 2 304.99 3* 45338.4 3

5 319.30 2* 265.82 1 266.81 2* 990.71 2 139.33 1 25408.7 3

6 34.04 1 3957.69 3 36.74 1 1175.55 2 7484.51 2 33401.3 3

7 44.84 2 305.49 2 49.02 2 2307.14 3 186.30 1 25420 3

8 42.49 1 1120.39 1 41.63 1 571.80 2 19035.7 2 23678.8 4

9 226.29 1 1851.18 3* 161.96 1 3848.58 5* 4080.64 2 45109 3

10 14.60 1 1093.22 3 16.59 1 843.19 2 192.68 1 39750.3 4

Table 2: Computational results using our ✏-constraint algorithm for instances T1–T6.

334

From Table 2, it is remarkable to observe that 40% of the instances have only one335

optimal solution. Therefore, with a fixed number of vehicles, the timetable is able to o↵er336

19

Instances based on a transit network in Mexico (Ibarra-Rojas et al., 2014)

40

Numerical results

31/

Integrated Approach

Instances T1 T2 T3 T4 T5 T6

|I| 10 50 10 50 10 50

|B| 1 5 1 5 1 5

100 �

is

⌘

is

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

and the number of solutions in the Pareto Frontier (PF) for each one of the proposed332

instances. Note that our ✏-constraint algorithm is capable of finding the Pareto optimal333

solutions for all instances of our case study.

T1 T2 T3 T4 T5 T6

time PF time PF time PF time PF time PF time PF

1 26.25 1 569.66 2 15.66 1 586.85 1 123.96 2 73093.8 5*

2 42.06 1 246.51 1 31.06 1 237.18 1 375.062 2 21313.9 2

3 28.52 1 384.69 2 28.71 1 1030.64 2 152 2 25493.9 3

4 49.65 1 381.59 1 88.93 2 508.43 2 304.99 3* 45338.4 3

5 319.30 2* 265.82 1 266.81 2* 990.71 2 139.33 1 25408.7 3

6 34.04 1 3957.69 3 36.74 1 1175.55 2 7484.51 2 33401.3 3

7 44.84 2 305.49 2 49.02 2 2307.14 3 186.30 1 25420 3

8 42.49 1 1120.39 1 41.63 1 571.80 2 19035.7 2 23678.8 4

9 226.29 1 1851.18 3* 161.96 1 3848.58 5* 4080.64 2 45109 3

10 14.60 1 1093.22 3 16.59 1 843.19 2 192.68 1 39750.3 4

Table 2: Computational results using our ✏-constraint algorithm for instances T1–T6.

334

From Table 2, it is remarkable to observe that 40% of the instances have only one335

optimal solution. Therefore, with a fixed number of vehicles, the timetable is able to o↵er336

19

40

Some Pareto fronts

32/

Integrated Approach

T1_5T1_5 T2_9T2_9 T3_5T3_5 T4_9T4_9 T5_4T5_4 T6_?T6_?Trans Veh Trans Veh Trans Veh Trans Veh Trans Veh Trans Veh1004 71 6727 372 1426 71 8319 367 2753 79922 70 6524 371 1313 70 8310 366 2750 78

5397 370 8151 365 2624 777826 3647150 363

69

70

71

72

915 939 963 986 1010

Pareto front of T1_5

Num

ber o

f bus

es

Passenger Transfers

369

370

371

372

5300 5675 6050 6425 6800


Num

ber o

f bus

es

Passenger Transfers

69

70

71

72

1300 1333 1365 1398 1430


Num

ber o

f bus

es

Passenger Transfers

362

363

364

365

366

367

368

7140 7435 7730 8025 8320


Num

ber o

f bus

es

Passenger Transfers

76

77

78

79

80

2620 2660 2700 2740 2780


Num

ber o

f bus

es

Passenger Transfers

40

Some Pareto fronts

33/

Integrated Approach

T1_5T1_5 T2_9T2_9 T3_5T3_5 T4_9T4_9 T5_4T5_4 T6_?T6_?Trans Veh Trans Veh Trans Veh Trans Veh Trans Veh Trans Veh1004 71 6727 372 1426 71 8319 367 2753 79922 70 6524 371 1313 70 8310 366 2750 78

5397 370 8151 365 2624 777826 3647150 363

69

70

71

72

915 939 963 986 1010


Num

ber o

f bus

es

Passenger Transfers

369

370

371

372

5300 5675 6050 6425 6800


Num

ber o

f bus

es

Passenger Transfers

69

70

71

72

1300 1333 1365 1398 1430


Num

ber o

f bus

es

Passenger Transfers

362

363

364

365

366

367

368

7140 7435 7730 8025 8320


Num

ber o

f bus

es

Passenger Transfers

76

77

78

79

80

2620 2660 2700 2740 2780


Num

ber o

f bus

es

Passenger Transfers

40

Some Pareto fronts

34/

Integrated Approach

T1_5T1_5 T2_9T2_9 T3_5T3_5 T4_9T4_9 T5_4T5_4 T6_1T6_1Trans Veh Trans Veh Trans Veh Trans Veh Trans Veh Trans Veh1004 71 6727 372 1426 71 8319 367 2753 79 10647 363922 70 6524 371 1313 70 8310 366 2750 78 10622 362

5397 370 8151 365 2624 77 10520 3617826 364 10495 3607150 363 10423 359

69

70

71

72

915 939 963 986 1010


Num

ber o

f bus

esPassenger Transfers

369

370

371

372

5300 5675 6050 6425 6800


Num

ber o

f bus

es

Passenger Transfers

69

70

71

72

1300 1333 1365 1398 1430


Num

ber o

f bus

es

Passenger Transfers

362

363

364

365

366

367

368

7140 7435 7730 8025 8320


Num

ber o

f bus

es

Passenger Transfers

76

77

78

79

80

2620 2660 2700 2740 2780


Num

ber o

f bus

es

Passenger Transfers

358

360

361

363

10400 10463 10525 10588 10650


Num

ber o

f bus

es

Passenger Transfers

40

Using one more vehicle yields . . .

35/

Integrated Approach

0257

10121417192224

[0,50] [51,100] [101,150] [151,200] [201,250] [300, 350] [600,700] [1100,1200]

Passengers benefited by using one more vehicle

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36/

Conclusions

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Conclusions

37/

• 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

40

Future research

38/

• Heterogeneous fleets.

• Multiple-depots.

• Other criteria such as total waiting time for larger flexibility parameters and deadhead costs for vehicles.

Conclusions

40

39/

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 and Vehicle 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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FIN

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webinar: integrating timetabling and vehicle scheduling to analyze the trade-off between transfers...

Engineering

planning problem

transit network planning

integrated timetabling

integrated subproblems

context advantage

timetabling problem

context suboptimal solutions

different planning periods