scheduling problems in traffic and transport · scheduling problems in traffic and transport 19...
TRANSCRIPT
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DFG Research Center MATHEON
Mathematics for Key Technologies
LBW
Scheduling Problems and Algorithms in Traffic and Transport
MAPSP 2011 Nymburk, 24.06.11
Ralf Borndörfer Zuse-Institute Berlin
Joint work with Ivan Dovica, Martin Grötschel, Olga Heismann, Andreas Löbel, Markus Reuther, Elmar Swarat, Thomas Schlechte, Steffen Weider
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Optimization in Public Transit
Scheduling Problems in Traffic and Transport 2
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Trip 1
Trip 2
Trip 4
Trip 3
Trip 5
Trip 6
Trip 7
Railway Challenges
Basic Rolling Stock Rostering Problem = Multicommodity Flow Problem
Can be solved efficiently for networks with 109 arcs
Constraints complicating rolling stock rostering
Discretization: Space/Time ("Multiscale Problems")
Robustness: Delay Propagation
Path Constraints: Maintenance, Parking
Configuration Constraints: Track Usage, Train Composition, Uniformity
Scheduling Problems in Traffic and Transport 3
Photo courtesy of DB Mobility Logistics AG
We want to avoid this! Simplon Tunnel
Visualization based on JavaView
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LBW
Scheduling Problems in Traffic and Transport 4
Integrated Routing and Scheduling
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Integrated Routing and Scheduling
Routing Scheduling
Scheduling Problems in Traffic and Transport 5
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Timetable
Scheduling Problems in Traffic and Transport 6
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Train Routes are Flexible in Space and Time
Scheduling Problems in Traffic and Transport 7
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Conflict
Scheduling Problems in Traffic and Transport 8
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Track Allocation Graph
Scheduling Problems in Traffic and Transport 9
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Track Allocation/Train Timetabling Problem
Combinatorial Optimization Problem
Path Packing Problem
Scheduling Problems in Traffic and Transport 10
…
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Literature
Charnes and Miller (1956), Szpigel (1973), Jovanovic and Harker (1991),
Cai and Goh (1994), Schrijver and Steenbeck (1994), Carey and Lockwood (1995)
Nachtigall and Voget (1996), Odijk (1996) Higgings, Kozan and Ferreira (1997)
Brannlund, Lindberg, Nou, Nilsson (1998), Lindner (2000), Oliveira and Smith (2000)
Caprara, Fischetti and Toth (2002), Peeters (2003)
Kroon and Peeters (2003), Mistry and Kwan (2004)
Barber, Salido, Ingolotti, Abril, Lova, Tormas (2004)
Semet and Schoenauer (2005),
Caprara, Monaci, Toth and Guida (2005)
Kroon, Dekker and Vromans (2005),
Vansteenwegen and Van Oudheusden (2006), Liebchen (2006)
Cacchiani, Caprara, T. (2006), Cachhiani (2007)
Caprara, Kroon, Monaci, Peeters, Toth (2006)
Borndoerfer, Schlechte (2005, 2007), Caimi G., Fuchsberger M., Laumanns M., Schüpbach K. (2007)
Fischer, Helmberg, Janßen, Krostitz (2008)
Lusby, Larsen, Ehrgott, Ryan (2009)
Caimi (2009), Klabes (2010)
...
Scheduling Problems in Traffic and Transport 11
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Path/Arc Packing Model
Scheduling Problems in Traffic and Transport 12
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Path Packing Model
Scheduling Problems in Traffic and Transport 13
Integ.,}1,0{(iii)
Conflicts1(ii)
Flow,)((i)
max(APP)
),(
)( )(
IiAax
Kkx
IiVvvxx
xc
i
a
kia
i
a
i
va va
i
a
i
a
Ii Aa
i
a
i
a
i i
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Configuration Model
Scheduling Problems in Traffic and Transport 14
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Configuration Model
Scheduling Problems in Traffic and Transport 15
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Packing- and Configuration Model
Scheduling Problems in Traffic and Transport 16
Integ.,}1,0{(iii)
Conflicts1(ii)
Flow,)((i)
max(APP)
),(
)( )(
IiAax
Kkx
IiVvvxx
xc
i
a
kia
i
a
i
va va
i
a
i
a
Ii Aa
i
a
i
a
i i
Integ.}1,0{(v)
Integ.}1,0{(iv)
Coupling0(iii)
Configs1(ii)
Trains1(i)
max(PCP)
Qqy
Ppx
Aayx
Jjy
Iix
xc
q
p
Qqa
q
Ppa
p
q
Pp
p
Ii Pp pa
p
i
a
j
i
i
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Track Allocation Models
Theorem (B., Schlechte
[2007]):
= vLP(PCP) = vLP(ACP)
= vLP (APP) = vLP(PPP)
≤ vLP(APP').
All LP-relaxations can be
solved in polynomial time.
= vIP(PCP) = vIP(ACP)
= vIP (APP) = vIP(PPP)
= vIP(APP').
Scheduling Problems in Traffic and Transport 17
APP
ACP PCP
PPP
APP'
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Packing- and Configuration Model
Scheduling Problems in Traffic and Transport 18
Integ.,}1,0{(iii)
Conflicts1(ii)
Flow,)((i)
max(APP)
),(
)( )(
IiAax
Kkx
IiVvvxx
xc
i
a
kia
i
a
i
va va
i
a
i
a
Ii Aa
i
a
i
a
i i
Integ.}1,0{(v)
Integ.}1,0{(iv)
Coupling0(iii)
Configs1(ii)
Trains1(i)
max(PCP)
Qqy
Ppx
Aayx
Jjy
Iix
xc
q
p
Qqa
q
Ppa
p
q
Pp
p
Ii Pp pa
p
i
a
j
i
i
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Configuration Model
Scheduling Problems in Traffic and Transport 19
0,,(iii)
Configs,0(ii)
Paths,(i)
min(DUA)
JjQq
IiPpc
j
qa
aj
i
pa
i
a
pa
ai
Ii Jj
ji
Integ.0(v)
Integ.0(iv)
Coupling0(iii)
Configs1(ii)
Trains1(i)
max(PLP)
Qqy
Ppx
Aayx
Jjy
Iix
xc
q
p
Qqa
q
Ppa
p
q
Pp
p
Ii Pp pa
p
i
a
j
i
i
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Configuration Model
Proposition:
Route pricing = acyclic shortest
path problem with arc weights
ca = ca+a.
Scheduling Problems in Traffic and Transport 20
0,(iii)
Configs,0(ii)
Paths,(i)
min(DUA)
JjQq
IiPpc
j
qa
aj
i
pa
i
a
pa
ai
Ii Jj
ji
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Configuration Model
Proposition:
Config pricing = acyclic shortest
path problem with arc weights
ca = a.
Scheduling Problems in Traffic and Transport 21
0,(iii)
Configs,0(ii)
Paths,(i)
min(DUA)
JjQq
IiPpc
j
qa
aj
i
pa
i
a
pa
ai
Ii Jj
ji
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Configuration Model
Scheduling Problems in Traffic and Transport 22
Integ.0(v)
Integ.0(iv)
Coupling0(iii)
Configs1(ii)
Trains1(i)
max(PLP)
Qqy
Ppx
Aayx
Jjy
Iix
xc
q
p
Qqa
q
Ppa
p
q
Pp
p
Ii Pp pa
p
i
a
j
i
i
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Mathematische Optimierung 23
Lagrange Funktion des PCP
(PCP) (LD)
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Mathematical Optimization and Public Transportation
Bundle Method (Kiwiel [1990], Helmberg [2000])
24
Problem
Algorithm
Subgradient
Cutting Plane Model
Update
Quadratic Subproblem
Primal Approximation
Inexact Bundle Method
f
T T( ) : min ( )
x Xf c x b Ax
T T( ) ( )f c x b Ax
2
1ˆ ˆargmax ( )
2k
k k k
uf
ˆ ( ) : min ( )k
kJ
f f
2ˆ ˆmax ( )
2k
k k
uf
2ˆmax
2
s.t. ( ), for all
kk
k
uv
v f J
2
1ˆmax ( ) ( )2
s.t. 1
0 1, for all
k k
k
kJ J
J
k
f b Axu
J
1
k
kJ
x x
0 ( )kb Ax k
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Mathematical Optimization and Public Transportation
Bundle Method (Kiwiel [1990], Helmberg [2000])
25
Problem
Algorithm
Subgradient
Cutting Plane Model
Update
Quadratic Subproblem
Primal Approximation
Inexact Bundle Method
f
1
1f
T T( ) : min ( )
x Xf c x b Ax
T T( ) ( )f c x b Ax
2
1ˆ ˆargmax ( )
2k
k k k
uf
ˆ ( ) : min ( )k
kJ
f f
2ˆ ˆmax ( )
2k
k k
uf
2ˆmax
2
s.t. ( ), for all
kk
k
uv
v f J
2
1ˆmax ( ) ( )2
s.t. 1
0 1, for all
k k
k
kJ J
J
k
f b Axu
J
1
k
kJ
x x
0 ( )kb Ax k
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Mathematical Optimization and Public Transportation
Bundle Method (Kiwiel [1990], Helmberg [2000])
26
f
1
1f
2
f̂
Problem
Algorithm
Subgradient
Cutting Plane Model
Update
Quadratic Subproblem
Primal Approximation
Inexact Bundle Method
T T( ) : min ( )
x Xf c x b Ax
T T( ) ( )f c x b Ax
2
1ˆ ˆargmax ( )
2k
k k k
uf
ˆ ( ) : min ( )k
kJ
f f
2ˆ ˆmax ( )
2k
k k
uf
2ˆmax
2
s.t. ( ), for all
kk
k
uv
v f J
2
1ˆmax ( ) ( )2
s.t. 1
0 1, for all
k k
k
kJ J
J
k
f b Axu
J
1
k
kJ
x x
0 ( )kb Ax k
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Mathematical Optimization and Public Transportation
Bundle Method (Kiwiel [1990], Helmberg [2000])
27
f
1
1f
2
f̂
3
Problem
Algorithm
Subgradient
Cutting Plane Model
Update
Quadratic Subproblem
Primal Approximation
Inexact Bundle Method
T T( ) : min ( )
x Xf c x b Ax
T T( ) ( )f c x b Ax
2
1ˆ ˆargmax ( )
2k
k k k
uf
ˆ ( ) : min ( )k
kJ
f f
2ˆ ˆmax ( )
2k
k k
uf
2ˆmax
2
s.t. ( ), for all
kk
k
uv
v f J
2
1ˆmax ( ) ( )2
s.t. 1
0 1, for all
k k
k
kJ J
J
k
f b Axu
J
1
k
kJ
x x
0 ( )kb Ax k
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Mathematical Optimization and Public Transportation
Bundle Method (Kiwiel [1990], Helmberg [2000])
28
1
1f
2
f̂
3
Problem
Algorithm
Subgradient
Cutting Plane Model
Update
Quadratic Subproblem
Primal Approximation
Inexact Bundle Method
T T( ) : min ( )
x Xf c x b Ax
T T( ) ( )f c x b Ax
2
1ˆ ˆargmax ( )
2k
k k k
uf
ˆ ( ) : min ( )k
kJ
f f
2ˆ ˆmax ( )
2k
k k
uf
2ˆmax
2
s.t. ( ), for all
kk
k
uv
v f J
2
1ˆmax ( ) ( )2
s.t. 1
0 1, for all
k k
k
kJ J
J
k
f b Axu
J
1
k
kJ
x x
0 ( )kb Ax k
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Mathematical Optimization and Public Transportation 29
Rapid Branching
Perturbation Branching
Sequence of perturbed IP objectives cji+1 := cj
i – (xji)2, j, i=1,2,…
Fixing candidates in iteration i Bi := { j : xji 1 – }
Potential function in iteration i vi := cTxi – w|Bi |
Go on while not integer and potential decreases, else
Perturb for kmax additional iterations, if still not successful
Fix a single variable and reset objective every ks iterations
Set of fixed variables (many) B* := Bargmin vi
Binary Search Branching
Set of fixed variables (many) B* := {j1, ... , jm}, cj1 ... cjm
Sets Qjk at pertubation branch j Qj
k := { x : xj1=...=xjk
=1 },
k=0,...,m
Branch on Qjm
Repeat perturbation branching to plunge
Backtrack to Qjm/2 and set m := m/2 to prune
29
Qj2
Qjm/4
Qjm/2
Qjm
Qj-1p/q
Qj1
Qj4
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Ralf Borndörfer 30
A Simple LP-Bound
Lemma (BS [2007]):
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Solving the LP-Relaxation
Scheduling Problems in Traffic and Transport 31
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Mathematische Optimierung
Solving the IP
HaKaFu, req32, 1140 requests, 30 mins time windows
Scheduling Problems in Traffic and Transport 32
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Mathematische Optimierung
Track Allocation and Train Timetabling
BAB: Branch-and-Bound
PAB: Price-and-Branch
BAP: Branch-and-Price
Scheduling Problems in Traffic and Transport 33
Article Stations Tracks Trains Modell/Approach
Szpigel [1973] 6 5 10 Packing/Enumeration
Brännlund et al. [1998] 17 16 26 Packing/ Lagrange, BAB
Caprara et al. [2002] 74 (17) 73 (16) 54 (221) Packing/ Lagrange, BAB
B. & Schlechte [2007] 37 120 570 Config/PAB
Caprara et al. [2007] 102 (16) 103 (17) 16 (221) Packing/PAB
Fischer et al. [2008] 656 (104) 1210 (193) 117 (251) Packing/Bundle, IP Rounding
Lusby et al. [2008] ??? 524 66 (31) Packing/BAP
B. & Schlechte [2010] 37 120 >1.000 Config/Rapid Branching
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LBW
Scheduling Problems in Traffic and Transport 34
Discretization and Scheduling
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Railway Infrastructure Modeling
Detailed railway infrastucture data given by simulation programs
(Open Track)
Switches
Signals
Tracks (with max. speed, acceleration, gradient)
Stations and Platforms
Mathematische Optimierung Scheduling Problems in Traffic and Transport 35
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Microscopic Model
Simplon micrograph: 1154 nodes and 1831 arcs, 223 signals etc.
Mathematische Optimierung Scheduling Problems in Traffic and Transport 36
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Headways
Simulation tools provide exact running and blocking times
Basis for calculation of minimal headway times
Mathematische Optimierung Scheduling Problems in Traffic and Transport 37
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Macroscopic Network Generation
38
Simulation of all possible routes with appropiate train types
EC R
GV Auto Brig-Iselle GV ROLA
GV SIM
GV MTO
Chosen TrainTypes
BRTU SGAA IS_A IS
BRRB
BR VAR MOGN PRE
DOFM
DO DOBI_A
Mathematische Optimierung Scheduling Problems in Traffic and Transport 38
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Interaction of Train Routes
Generation of artifical nodes – „pseudo“ stations
No interactions between train routes
IS
Macro network definition is based on set of train routes
Mathematische Optimierung Scheduling Problems in Traffic and Transport 39
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Interaction of Train Routes
Generation of artifical nodes – „pseudo“ stations
Diverging of train routes
IS_P IS
The same holds for converging routes
Mathematische Optimierung Scheduling Problems in Traffic and Transport 40
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Interaction of Train Routes
Generation of artifical nodes – pseudo stations
crossing of train routes
IS_P1 IS IS_P2
Two pseudo stations were generated
Mathematische Optimierung Scheduling Problems in Traffic and Transport 41
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Reduced Macrograph (53 nodes and 87 track arcs for 28 train routes)
Mathematische Optimierung Scheduling Problems in Traffic and Transport 42
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Station Aggregation
Frequently many macroscopic station nodes are in the area of big stations
Further aggregation is needed
k k
k =
EC 2
R 4
GV Auto 2
GV Rola 2
GV SIM 4
GV MTO 6
Mathematische Optimierung Scheduling Problems in Traffic and Transport 43
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Micro-Macro Transformation
Planned times in macro network are possible in micro network
Valid headways lead to valid block occupations (no conflicts)
feasible macro timetable can be transformed to feasible micro timetable
Mathematische Optimierung Scheduling Problems in Traffic and Transport 44
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Micro-Macro-Transformation: Simplon Case
Micro
12 stations
1154 OpenTrack nodes
1831 OpenTrack edges
223 signals
8 track junctions
100 switches
6 train types
28 “routes“
230 ”block segments“
Macro
18 macro nodes
40 tracks
6 Train types
Mathematische Optimierung Scheduling Problems in Traffic and Transport 45
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Time Discretization
Cumulative Rounding Procedure
Compute macroscopic running time with specific rounding procedure
Consider again routes of trains (represented by standard trains)
Example with
Station Dep/Pass Rounded Buffer
A 0 0 0
B 11 12 (2) 1
C 20 24 (4) 4
D 29 30 (5) 1
6
Theorem: If micro-running time d for all tracks of the current train
route, the cumulative rounding error (buffer) is always in . ),0[
Mathematische Optimierung Scheduling Problems in Traffic and Transport 46
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Complex Traffic at the Simplon
Slalom route
ROLA trains traverse the tunnel on the “wrong“
side
Crossing of trains
complex crossings of AUTO trains in Iselle
Conflicting routes
complex routings in station area Domodossola
and Brig
Source: Wikipedia
Mathematische Optimierung Scheduling Problems in Traffic and Transport 47
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Dense Traffic at the Simplon
Scheduling Problems in Traffic and Transport 48
0
5
10
15
20
25
30
00-04 04-08 08-12 12-16 16-20 20-24
Sum
PV
EC
GV Auto
R
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Estimation of the maximum theoretical corridor capacity
Network accuracy of 6s
Consider complete routing through stations
Saturate by additional cargo trains
Conflict free train schedules in simulation software (1s accuracy)
Saturation
Scheduling Problems in Traffic and Transport 49
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Manual Reference Plan
Aggregation-Test (Micro->Macro->Micro)
Microscopic feasible 4h (8:00-12:00) reference plan in Open Track
Reproducing this plan by an Optimization run
Reimport to Open Track
Scheduling Problems in Traffic and Transport 50
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Theoretical Capacities
180 trains for network
small (without station
routing and buffer times)
196 trains for network big
with precise routing
through stations (without
buffer times)
175 trains for network big
with precise routing
through stations and
buffer times
Scheduling Problems in Traffic and Transport 51
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Retransformation to Microscopic Level (Network big)
No delays, no early coming
Feasible train routing and block occupation
Timetable is valid in micro-simulation
52 Scheduling Problems in Traffic and Transport 52
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Valid blocking time stairs
53
Network big with buffer times
Scheduling Problems in Traffic and Transport 53
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Network big with buffer times
54
Time Discretization Analysis
Time discretization dt/s 6 10 30 60
Number of trains 196 187 166 146
Cols in IP 504314 318303 114934 61966
Rows in IP 222096 142723 53311 29523
Solution time in secs 72774.55 12409.19 110.34 10.30
Scheduling Problems in Traffic and Transport 54
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Scheduling Problems in Traffic and Transport 55
Hypergraph
Scheduling
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Trip Network
Scheduling Problems in Traffic and Transport 56
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Cyclic Timetable for Standard Week
Scheduling Problems in Traffic and Transport 57 (Visualization based on JavaView) 57
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Rotation
Scheduling Problems in Traffic and Transport 58
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Rotation
Scheduling Problems in Traffic and Transport 59
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Rotation Schedule (Blue: Timetable, Red: Deadheads)
Scheduling Problems in Traffic and Transport 60 (Visualization based on JavaView)
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(Operational) Uniformity
Scheduling Problems in Traffic and Transport 61
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Uniformity (Blue: Uniform, …, Red: Irregular)
Scheduling Problems in Traffic and Transport 62 (Visualization based on JavaView) 62
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Uniformity (Blue/Yellow: Uniform, …, Red: Irregular, Fat: Maintenance)
Scheduling Problems in Traffic and Transport 63 (Visualization based on JavaView)
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Rotation Schedule
Scheduling Problems in Traffic and Transport 64
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Uniformity
Scheduling Problems in Traffic and Transport 65
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Uniformity
Scheduling Problems in Traffic and Transport 66
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Modelling Uniformity Using Hyperarcs
Scheduling Problems in Traffic and Transport 67
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Hyperassignment
Scheduling Problems in Traffic and Transport 68
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Hyperassignment Problem
Definition: Let D=(V,A) be a directed hypergraph w. arc costs ca
H⊆A hyperassigment : +(v)H = -(v)H = 1
Hyperassignment Problem : argmin c(H), H hyperassignment
Literature
Cambini, Gallo, Scutellà (1992): Minimum cost flows on hypergraphs;
solves only the LP relaxation
Jeroslow, Martin, Rarding, Wang (1992): Gainfree Leontief substitution
flow problems; does not hold for the hyperassignment problem
Theorem: The HAP is NP-hard (even for simple cases).
Scheduling Problems in Traffic and Transport 69
A
T
x
Vvvx
Vvvx
xc
}1,0{
1))((
1))((
min
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Further Complexity Results
Theorem: The LP/IP gap of HAP can be arbitrarity large.
Scheduling Problems in Traffic and Transport 70
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Further Complexity Results
Theorem: The LP/IP gap of HAP can be arbitrarity large.
Scheduling Problems in Traffic and Transport 71
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Further Complexity Results
Theorem: The LP/IP gap of HAP can be arbitrarity large.
Scheduling Problems in Traffic and Transport 72
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Further Complexity Results
Theorem: The LP/IP gap of HAP can be arbitrarity large.
Proposition: The determinants of basis matrices of HAP can be
arbitrarily large, even if all hyperarcs have head and tail size 2.
Proposition: HAP is APX-complete for hyperarc head and tail size
2 in general and for hyperarc head and tail cardinality 3 in the
revelant cases.
Scheduling Problems in Traffic and Transport 73
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Computational Results (CPLEX 12.1.0)
Scheduling Problems in Traffic and Transport 74
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Partitioned Hypergraph and Configurations
Scheduling Problems in Traffic and Transport 75
ICE 4711 (Mo)
ICE 4711 (Tu)
ICE 4711 (Su)
ICE 4711 (Mo)
ICE 4711 (Tu)
ICE 4711 (Su)
ICE 4711 (Mo)
ICE 4711 (Tu)
ICE 4711 (Su)
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Extended Configuration Formulation
Theorem: There is an extended formulation of HAP with O(V8)
variables that implies all clique constraints.
Scheduling Problems in Traffic and Transport 76
Ca
a
A
T
y
AaxaCy
AaxaCy
x
Vvvx
Vvvx
xc
}1,0{
))((
))((
}1,0{
1))((
1))((
min
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LBW
Scheduling Problems in Traffic and Transport 77
Stochastic Scheduling
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LBW
Delays
Cost of delays
72 €/minute average cost of gate delay over 15 minutes, cf. EUROCONTROL [2004]
840 – 1200 millions € annual costs caused by gate delays in Europe
Benefits of robust planning
Cost savings
Reputation
Less operational changes
The Tail Assignment Problem – assign legs to aircraft in order to fulfill operational constraints such as preassignments, maintenance rules, airport curfews, and minimum connection times between legs, cf. Grönkvist [2005]
We consider the tail assignment problem in a research project based on real-world data from a European carrier using the NetLine system
Scheduling Problems in Traffic and Transport 78
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LBW
Delay Propagation
Scheduling Problems in Traffic and Transport 79
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LBW
Delay Propagation Along Rotations
EDP (bad) EDP (good)
Scheduling Problems in Traffic and Transport 80
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LBW
Delay Propagation
Goal: Decrease impact of delays
Primary delays: genuine disruptions, unavoidable
Propagated delays: consequences of aircraft routing, can be minimized
Rule-oriented planning
Ad-hoc formulas for buffers
These rules are costly and it is uncertain how efficient they are
Calibrating these rules is a balancing act: supporting operational stability, while staying cost efficient
Goal-oriented planning
Minimize occurrence of delay propagation on average
Scheduling Problems in Traffic and Transport 81
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LBW
Stochastic Model (similar to Rosenberger et. al. [2002])
Delay distribution
Delays are not homogeneously spread in the network
Stochastic model must captures properties of individual airports and legs
Structure of the stochastic model
Gate phase, representing time spent on the ground
Flight phase, representing time spent en-route
Phase durations are modelled by probability distribution
Gj is random variable for delay of gate phase of leg j
Fj is random variable for duration of flight phase of leg j
Scheduling Problems in Traffic and Transport 82
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LBW
k
k
r
Rj
k
j
b
k
p
k Rp
bp
k Rrrlr
r
k Rr
k
rr
Rrkx
kx
Bbrxa
Llx
xd
k
k
k
k
,1,0
1
1
min
,:
Robust Tail Assignment Problem
Mathematical model:
Minimize non-robustness
Cover all legs
Fulfill side constraints
One rotation for each aircraft
Integrality
Scheduling Problems in Traffic and Transport 83
Set partitioning problem with side constraints
Problem has to be resolved daily for period of a few days
Solved by Netline/Ops Tail xOPT (state-of-the-art column generation solver by Lufthansa Systems)
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Column Generation
Scheduling Problems in Traffic and Transport 84
Start
Solve Tail Assignment Problem (IP)
Solve Tail
Assignment
Problem (LP)
Stop?
All fixed? Stop
Yes No
Yes No
Compute
rotations
Compute
prices
Fix
rotations
Conflict?
Backtrack?
Yes
No
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LBW
Column Generation
Scheduling Problems in Traffic and Transport 85
Start
Solve Tail Assignment Problem (IP)
Solve Tail
Assignment
Problem (LP)
Stop?
All fixed? Stop
Yes No
Yes No
Compute
robust
rotations
Compute
prices
Fix
rotations
Conflict?
Backtrack?
Yes
No
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LBW
Pricing Robust Rotations
Robustness measure: total probability of delay propagation (PDP)
Resource constraint shortest path problem
where is random variable of delay propagated to leg i in rotation r and are dual variables corresponding to cover, aircraft, and side constraints
To solve this problem one must compute along rotations
Scheduling Problems in Traffic and Transport 86
Bb
kbbr
ri
irRr
adk
min
0P
r
i
ri
r PDd
r
iPD
Bb
kbbr
ri
i
r
i
riRr
aPDk
0Pmin
bki ,,
r
iPD
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LBW
Computing PDi Along a Rotation
Delay distribution Hj of leg j
Hj = Gj + Fj
Delay propagation from leg j to leg k via buffer bjk
PDk = max( Hj - bjk , 0)
Delay distribution Hk of next leg k
Hk = PDk + Gk + Fk
and so on…
Scheduling Problems in Traffic and Transport 87
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Convolution
Convolution
H = F + G and f, g and h are their probability density functions
Numerical convolution based on discretization
where are stepwise constant approximations
of functions f, g
Alternative approaches
Analytical convolution, cf. Fuhr [2007]
Scheduling Problems in Traffic and Transport 88
t
dxxtgxfth0
)()()(
t
i
ititit ggfh1
1 2/)(
gf ,
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Path Search
Scheduling Problems in Traffic and Transport 89
Flight 1
Flight 2
Flight 4
Flight 3
Flight 5
Flight 6
Flight 7
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Path Search
Scheduling Problems in Traffic and Transport 90
Flight 1
Flight 2
Flight 4
Flight 3
Flight 5
Flight 6
Flight 7
1
2c
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LBW
Path Search
Scheduling Problems in Traffic and Transport 91
1
2c
1
5cFlight 1
Flight 2
Flight 4
Flight 3
Flight 5
Flight 6
Flight 7
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LBW
Path Search
Scheduling Problems in Traffic and Transport 92
1
2c
1
5c 1
7c
Flight 1
Flight 2
Flight 4
Flight 3
Flight 5
Flight 6
Flight 7
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LBW
Path Search
Scheduling Problems in Traffic and Transport 93
1
2c
1
5c 1
7c
2
3c
Flight 1
Flight 2
Flight 4
Flight 3
Flight 5
Flight 6
Flight 7
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LBW
Path Search
Scheduling Problems in Traffic and Transport 94
1
2c
2
5c 1
7c
2
3c
Flight 1
Flight 2
Flight 4
Flight 3
Flight 5
Flight 6
Flight 7
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LBW
Path Search
Scheduling Problems in Traffic and Transport 95
1
2c
2
5c 2
7c
2
3c
Flight 1
Flight 2
Flight 4
Flight 3
Flight 5
Flight 6
Flight 7
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Accuracy vs. Speed
Instance SC1: reference solution
100 legs, 16 aircraft, no preassignments, no maintenace
Optimizer produces the same solution for each step size
CPU time differs only in computation of the convolutions
PDP values differ because of approximation error
* Comparison with average of 100 000 iterations
Scheduling Problems in Traffic and Transport 96
step size [min]
CPU [s]
PDP error [%]
SC1 0.1 15.4 25.0586 0.11
SC1 0.5 1.0 25.0672 0.15
SC1 1 0.5 25.0917 0.25
SC1 2 0.4 25.2227 0.77
SC1 3 0.4 25.4775 1.79
SC1 4 0.3 25.7667 2.94
Simulation* 25.0303
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Accuracy vs. Speed
Instance SC1: optimized solution
Different discretization step sizes may produce different solutions
CPU time and PDP are not straightforward to compare
*Value is average of 10 000 iterations
Scheduling Problems in Traffic and Transport 97
step size [min]
PDP optimized
CPU [s]
PDP simulated*
SC1 0.1 19.7268 4450 19.7469
SC1 0.5 19.7362 231 19.7382
SC1 1 19.7450 70 19.7239
SC1 2 19.8693 45 19.7313
SC1 3 20.0651 29 19.7239
SC1 4 20.3353 31 19.7562
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LBW
Test Instances
Analyzed data
approx. 350000 flights / 300 – 650 flights per day
28 months, 4 subfleets
European airline with hub-and-spoke network
Test instances
We optimize single day instances of one subfleet
Data for 4 months, no maintenance rules and preassignments
Scheduling Problems in Traffic and Transport 98
min max avg
#days
Legs aircraft flight time [min]
legs aircraft flight time [min]
legs aircraft flight time [min]
January 26 44 12 3840 105 17 8830 88 15 7447
February 22 94 15 8295 118 17 10065 109 16 9339
March 21 94 15 7900 121 17 10390 110 16,3 9483
April 27 93 15 7080 118 18 9750 103 16 8648
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LBW
Gate Phase
Probability of delay Depends on day time and departure airport
Distribution of delay Independent of daytime and departure airport
Scheduling Problems in Traffic and Transport 99
0
0,05
0,1
0,15
0,2
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58
pro
babili
ty
probability of departure delay during the day
on various airports
distribution of the length of gate primary delays
on various airports
Gate phase
gate delay distribution Gj of flight j
where Ln() is probability density function of Log-normal distribution with Power-law distributed tail and , t(j) is departure time of flight j and a(j) is departure airport of flight j
0),,Ln(
01]Pr[
xxp
xpxG
j
j
j
))(),(( jajtcp j
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Flight Phase
Distribution of deviation from scheduled duration
Depends on scheduled leg duration
Flight phase
flight delay distribution Fj of flight j
where Llg() is probability density function
of Log-logistic distribution and lj is scheduled flight duration of leg j
Scheduling Problems in Traffic and Transport 100
Histogram of the flight duration and its representation by random variable. left: scheduled flight
duration 80 minutes, right: scheduled flight duration 45 minutes
RxlxxFjj lljj ),,Llg(]Pr[
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LBW
Model Verification
Parameters of the model:
for every airport and day hour
for every flight length
Parameters are estimated by automatic scripts in R and quality is proofed by Chi-Square test.
Model applied to South American airline data
Validation of various assumptions of the model
Stability of parameters over time, …
Scheduling Problems in Traffic and Transport 101
,
p
,
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LBW
Gain of the Method
ORC
Standard KPI method
Bonus for ground buffer minutes
Threshold value for maximal ground buffer time (15 minutes)
PDP
Total probability of delay propagation
EAD – expected arrival delay
Robust Tail Assignment 102
ORC PDP Savings
#days PDP EAD [min]
CPU [s] PDP EAD [min]
CPU [s] PDP EAD [min]
January 26 414,51 28488 28 395,46 28085 66 19,05 403
February 22 540,48 31870 31 530,42 31652 89 10,06 218
March 21 516,69 30363 31 507,91 30174 75 8,78 189
April 27 465,48 34453 42 449,16 34159 71 16,51 294
102 Scheduling Problems in Traffic and Transport
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Gain in Detail
Estimation of monetary savings by the cost model developed based on EUROCONTROL [2004]
Lufthansa Systems estimates annual saving of the method in the tail assignment to 300,000 € for short haul carrier with 30 aircraft
Application in other planning stages may increase the benefit
Scheduling Problems in Traffic and Transport 103
ORC vs. PDP on a single disruption scenario
ORC outperforms PDP only in 21% of cases
PDP saves on average 29 minutes of arrival delay
For more disrupted days, PDP saves on average 62 minutes of arrival delay
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Planning in Public Transport (Product, Project, Planned)
Scheduling Problems in Traffic and Transport 104
B3
Cost R
eco
very
Fa
res
Constru
ction C
osts
Netw
ork
Topolo
gy
Velo
cities
Lin
es
Serv
ice L
eve
l Fre
quencie
s Connectio
ns
Tim
eta
ble
Sensitiv
ity
Rota
tions
Relie
f Poin
ts D
utie
s D
uty
Mix
Roste
ring
Fairn
ess
Cre
w A
ssignm
ent
Disru
ptio
ns
Opera
tions C
ontro
l
multidepartmental Departments multidepotwise Depots multiple line groups Line Groups multiple lines Lines multiple rotations Rotations
B1 AN-OPT/B5 BS-OPT
IS-OPT
VS-OPT DS-OPT APD B1
VS-OPT2 B15
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Visit ISMP 2012!
Scheduling Problems in Traffic and Transport 105
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Thank your for your attention
PD Dr. habil. Ralf Borndörfer Zuse-Institute Berlin Takustr. 7 14195 Berlin-Dahlem Fon (+49 30) 84185-243 Fax (+49 30) 84185-269 [email protected] www.zib.de/borndoerfer
Scheduling Problems in Traffic and Transport 106