TemporalSpreadingOfAlternativeTrainsPresentation

Temporal Spreading of Alternative Trains in order to Minimise Passenger Travel Time in Practice
Temporal Spreading of Alternative Trains
in order to Minimise
Passenger Travel Time in Practice
Peter Sels1,2,3
, Thijs Dewilde3
, Dirk Cattrysse3
, Pieter
Vansteenwegen3
1
Infrabel, Traﬃc Management & Services,
Fonsnylaan 13, 1060 Brussels, Belgium
2
Logically Yours BVBA,
Plankenbergstraat 112 bus L7, 2100 Antwerp, Belgium
e-mail: sels.peter@gmail.com, corresponding author
3
Katholieke Universiteit Leuven,
Leuven Mobility Research Centre, CIB,
Celestijnenlaan 300, 3001 Leuven, Belgium
March 24, 2015

Table of Contents
1 Timetabling Context
Business Problem
Solution Process Flow
Reﬂowing & Retiming
Results without Temporal Spreading
2 Temporal Spreading of Alternative Trains
For Whom & How Much?
ILP Model: Constraints
ILP Model: Objective Function
ILP Model: Piecewise Linearisation
ILP Model: Variable Linearisation
3 Results
4 Conclusions & Future Work
5 Questions / Next Steps

Timetabling Context
Business Problem
Business Problem
Belgian Infrastructure Management Company: Infrabel:
Find Timetable that Minimises Expected Passenger Travel Time
(includes: ride, dwell, transfer time and primary & secondary delays)
Note:
Reduce Expected Passenger Time ⇒ Optimises Robustness
Fixed:
Infrastructure, Train Lines, Halting Pattern, Primary Delay Distributions
Variable:
Timing: Supplement Times at every Ride, Dwell, Transfer Action,
⇒ variable inter-Train Heading Times ⇒ variable Train Orders
Speciﬁcs:
One Busy Day, Morning Peak Hour

Timetabling Context
Solution Process Flow
Context: FAPESP: Two Phased
FAPESP
zodzo,zd

Figure: Two Phased implies Iterations

Timetabling Context
Reﬂowing Retiming
Graph for Reﬂowing: add Source Sink Edges
transferdwellridesource sink
(a)
(a)
(b)
(b)
space increase
time increase

Timetabling Context
Reflowing Retiming
Result of Reflowing: Disc Area = Daily Flow
WllUInfoGraphicsUhereUisUautomaticallyUwrittenUoutUbyURhinoêros9
RhinoêrosUInputDULINEUPLWNNINGURESULTSD
nameUKURouter_5_5VUnetwerkUcodeUKU+EVUscaleUKUg5Upixels8km9VUnVerticesUKUg3z64VUnEdgesUKU7)gz4
correspondsUtoUaUstation
TrainsUinUnetworkUd+EdUrideUonUtheUright9
nParentVertices_UKU6)gVUnParentEdges_UKUgzg5VUnRideEdges_UKU4zg'
correspondsUtoDUBdistanceUKU3UkmVUdistanceUKU35Upixels19
dreddUmeansUtrainsUcontainingUdDgDdUareUpresent9
correspondsUtoDUBdistanceUKU4UkmVUdistanceUKU45Upixels19
dgreendUmeansUtrainsUcontainingUD3DUareUpresent9
correspondsUtoDUBdistanceUKUg5UkmVUdistanceUKUg55Upixels19
dorangedUmeansUtrainsUcontainingUDgDUandUtrainsUcontaintingUD3DUareUpresent9
firstUhourUKU(VULastUhourUKU5VUdInitSizeUKU35)6zVUD_MIN_UKU5VUD_MWX_UKU5VUinitSizeUKU73('zVUtotalômmutersPerDayUKU))5(7)
totalRideTrackKmForParentEdgesUKU44'(9(3VUaverageRideTrackKmPerParentEdgeUKU39z57zz
RhinoêrosUReflowUPhaseUOutputDUPWSSENGERUFLOWURESULTSD
totalDayKmForPassengersUKUg9767(ze/55)VUaverageDayKmPerPassengerUKUg)9z44g
correspondsUtoDUBdayUflowUKU3555UpassengersVUareaUKU(79((3pixels^3VUradiusUKU'945g46Upixels1
correspondsUtoDUBdayUflowUKU4555UpassengersVUareaUKUg4z9g44pixels^3VUradiusUKU)9gg)(7Upixels1
correspondsUtoDUBdayUflowUKUg5555UpassengersVUareaUKU7g697gpixels^3VUradiusUKUg595(46Upixels1

Timetabling Context
Reﬂowing Retiming
Graph for Retiming: add Knock-On Edges Cycles
transferdwellride knock-on
(and headway)cycle lin. comb. of cycles
space increase
time increase
train 1
train 2

Timetabling Context
Reﬂowing Retiming
Graph for Retiming: All Constraints
dwell
ride
turn around
symmetry
transfer
primary edges secondary edges
knock on
b + m + s = e b + m + s + (d * T) = e
b(egin), m(inimum), s(upplement), e(nd), d(integer), T(period)
constants: m, T
variables: b, s, e, d
symmetry
e = T - b or
e = T/2 - b or
...
cycles
sum_e_in_cycle :
sign_e_in_cycle *
(m_e + s_e + (d_e * T)) = 0

Timetabling Context
Reﬂowing Retiming
Reﬂowing decides on Rectangle Heights
Retiming (Timetabling) decides on Rectangle Widths
(a) Original Schedule
(b) Optimized Version

Timetabling Context
Results without Temporal Spreading
Expected (Non-)Linear Time, as used in Evaluation
orig.m original orig.s Exp_Train_Lin_Time opt.m optimal opt.s
020000400006000080000120000
86.79%
13.21%
improvement:0.93%
87.60%
12.40%
Ride(s) Dwell(s)
orig.m original orig.s opt.m optimal opt.s
0.0e+005.0e+071.0e+081.5e+08
63.01%
9.88%
0.00% 0.07%3.99%
10.43%
5.75%
0.94%0.00%
5.93%
improvement:6.03%
67.06%
3.77%0.00% 0.11%4.24%
16.80%
6.12%
0.26%0.00%
1.63%
Source(s) Transfer(s) Sink(s) KnockOn(s)
020000400006000080000120000
86.76%
13.24%
improvement:0.84%
87.50%
12.50%
Ride(m) Dwell(m)
0.0e+005.0e+071.0e+081.5e+08
63.70%
10.00%
0.00% 0.24%4.03%
10.18%
5.81%
0.98%0.00%
5.06%
improvement:3.81%
66.23%
3.80%0.00% 0.45%
4.19%
17.10%
6.04%
0.33%0.00%
1.86%
Source(m) Transfer(m) Sink(m) KnockOn(m)

For Whom How Much?
Cui Bono? To Whose Beneﬁt?
”It is a Latin adage that is used either to suggest a
hidden motive or to indicate that the party responsible for something may
not be who it appears at ﬁrst to be.”

For Whom How Much?
Cui Bono? To Whose Benefit?
Passengers arriving at station randomly minimise their waiting time
before departure
”inter-departure waiting time”
”inter-arrival waiting time”
do benefit:
random arrival passengers (fraction r)
do NOT/barely benefit:
passengers adapting to train departure schedule (fraction 1 − r)

For Whom How Much?
Categories Determining r
informed
caring
adaptable adapting
in for dep.
(1 − r)
passenger
time
choice
for trsfr. model
over for dep.
r
time for trsfr. random
unadaptable
not
arrival
not-caring
adapting
model
uninformed
Table: Sub-categories of passengers: fraction (1-r) showing passenger choice
model behaviour and fraction r showing random arrival behaviour. (dep. =
departure, trsf. = transfer)

For Whom How Much?
How Much do they Beneﬁt/Suﬀer?
[Welding(1957)], [Holroyd and Scraggs(1966)],
[Osuna and Newell(1972)]:
E(w) = E(h)/2 · 1 + Cv (h)2
(1)
Cv (h) = σ(h)/µ(h) (2)
E(h) =
N−1
i=0
pi · Hi =
N−1
i=0
(Hi /T) · Hi =
N−1
i=0
H2
i /T (3)
E(f · w) =
f
2T
N−1
i=0
H2
i · 1 + Cv (h)2
(4)

∀(O, D) :
∀i ∈ IN {N − 1} : bi ≤ bi+1
bN−1 ≥ b0
(5)
∀(O, D) : ∀i ∈ IN :



bi = j∈IN
pi,j · bj
∀j ∈ IN : pi,j ∈ {0, 1}
j∈IN
pi,j = 1 = j∈IN
pj,i
(6)
∀(O, D) :



∀i ∈ IN {N − 1} : si = bi+1 − bi
sN−1 = (b0 + T) − bN−1
∀i ∈ IN : 0 ≤ si ≤ T − δ
(7)
∀(i, j) ∈ In : 0 ≤ bi ≤ T − δ and so for the non-decreasingly ordered bi it
holds that ∀i ∈ IN {N − 1} : 0 ≤ bi+1 − bi ≤ T − δ.
∀(O, D) :
i∈IN
si = T (8)

ILP Model: Objective Function
ILP Model: Objective Function = How Much Penalty?
Passengers arriving at O and wanting to go to D, before having taken
ﬁrst train:
their expected waiting time for randomly arriving passengers
between train i and train i+1 is
ui =
si
0
(si − t) · f dt = si f
si
0
dt − f
si
0
t dt = f
s2
i
2 . (9)
their total expected waiting time for randomly arriving passengers is
∀(O, D) : U =
i∈IN
ui = f /2 ·
i∈IN
s2
i . (10)

∀(O, D) : ∀i ∈ IN :



(si,0, ui,0) = (0, 0)
(si,1, ui,1) = T
N , f
2
T
N
2
(si,2, ui,2) = T, f
2 T2
(11)
∀(O, D) : ∀i ∈ IN :



ui ≥ ui,0 +
ui,1−ui,0
si,1−si,0
· (si − si,0)
= 0 +
fT2
2N2
T
N
(si − 0) = fT
2N si
ui ≥ ui,1 +
ui,2−ui,1
si,2−si,1
· (si − si,1)
= fT2
2N2 +
fT2
2 − fT2
2N2
T− T
N
(si − T
N )
= fT2
2N2 + N
(N−1)·T
fT2
N2
−fT2
2N2 (si − T
N )
= fT2
2N2 1 + N(N+1)
T (si − T
N )
(12)

Introduce helper variables hi,j , such that
∀(O, D) : ∀i, j ∈ IN : hi,j = pi,j · bj , (13)
which is in linearised form:
∀(O, D) : ∀i, j ∈ IN :
(bl − bu)(1 − pi,j ) ≤ hi,j − bj ≤ (bu − bl)(1 − pi,j )
bl · pi,j ≤ hi,j ≤ bu · pi,j .
(14)
So, now
∀(O, D) : ∀i ∈ IN : bi =
j∈IN
pi,j · bj (15)
can be replaced with:
∀(O, D) : ∀i ∈ IN : bi =
j∈IN
hi,j . (16)

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