track allocation and path configurations · (iii) { 0 ,1 } , integ. (ii) 1 conflicts (i) ( ) ,...
TRANSCRIPT
Freie Universität Berlin
Track Allocation and Path Configurations
Ralf Borndörfer
2015 Workshop on Combinatorial Optimization with Applications in
Transportation and Logistics
Beijing, 29.07.2015
1 Track Allocation and Path Configurations | CO∈TL 2015
Freie Universität Berlin
Track Allocation and Path Configurations | CO∈TL 2015 4
Track Allocation:
Block Occupation
Freie Universität Berlin Track Allocation (Train Timetabling) Problem
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, 2009, 2010, 2011, 2012), Caimi G., Fuchsberger M., Laumanns M., Schüpbach K. (2007)
Fischer, Helmberg, Janßen, Krostitz (2008, 2011)
Lusby, Larsen, Ehrgott, Ryan (2009)
Caimi (2009), Klabes (2010)
...
Track Allocation and Path Configurations | CO∈TL 2015 5
Freie Universität Berlin Track Allocation (= Routes in Space-Time)
Railway Network + Trains Train Routes + Timetable
Track Allocation and Path Configurations | CO∈TL 2015 6
Freie Universität Berlin Integrated Train Routing and Scheduling
Routing: Flow Scheduling: Headway constraints
Track Allocation and Path Configurations | CO∈TL 2015 7
Freie Universität Berlin Block Occupation Conflict
Track Allocation and Path Configurations | CO∈TL 2015 8
Freie Universität Berlin Track Allocation Problem
Train Path Packing Problem Track Allocation and Path Configurations | CO∈TL 2015 9
…
Freie Universität Berlin Multicommodity Flow + Arc Packing Model
Block occupation conflict graph (here: interval graph, perfect)
Conflict cliques
Local for each block
Packing constraints = local constraints
Track Allocation and Path Configurations | CO∈TL 2015 10
Freie Universität Berlin
Integ.,}1,0{(iii)
Conflicts1(ii)
Trains,)((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
Arc Packing Model
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Theorem (Lukac [2004]):
Let H be a quadrangle-linear headway matrix for train types Y, i.e.,
𝐻 𝑖, 𝑗 + 𝐻 𝑗, 𝑘 ≥ 𝐻 𝑖, 𝑘 + 𝐻(𝑗, 𝑗) ∀𝑖, 𝑗, 𝑘.
Then all maximal cliques in 𝐺𝐻 are of the form
𝑖 × [𝑡𝑖 , 𝑡𝑖 + 𝐻 𝑖, 𝑖 − 1]
𝑖∈𝑌
,
if after setting 𝑡𝑖0 = 0 for some 𝑖0 ∈ 𝑌 the following holds for any different 𝑖, 𝑗:
𝐻 𝑖, 𝑖 − 𝐻 𝑗, 𝑖 ≤ 𝑡𝑗 − 𝑡𝑖 ≤ 𝐻 𝑖, 𝑗 − 𝐻(𝑗, 𝑗).
Freie Universität Berlin
Integ.,}1,0{(iii)
Conflicts1(ii)
Trains,)((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
Arc Packing Model
Track Allocation and Path Configurations | CO∈TL 2015 12
Freie Universität Berlin
Block Occupation Configurations (B., Schlechte [2007])
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Freie Universität Berlin
Integ.}1,0{(v)
Integ.,}1,0{(iv)
Coupling,(iii)
Configs1(ii)
Trains,)((i)
max(ACP)
)( )(
Qqy
IiAax
IiAaxy
Jjy
IiVvvxx
xc
q
ia
ia
Qqaq
Qqq
iva va
ia
ia
Ii Aa
ia
ia
j
i i
Integ.,}1,0{(iii)
Conflicts1(ii)
Trains,)((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
Arc Packing and Arc Configuration Model
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Freie Universität Berlin
Integ.}1,0{(v)
Integ.,}1,0{(iv)
Coupling(iii)
Configs1(ii)
Trains,)((i)
max(ACP)
)( )(
Qqy
IiAax
Aaxy
Jjy
IiVvvxx
xc
q
ia
Ii
ia
Qqaq
Qqq
iva va
ia
ia
Ii Aa
ia
ia
j
i i
Integ.,}1,0{(iii)
Conflicts1(ii)
Trains,)((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
Arc Packing and Configuration Model
Track Allocation and Path Configurations | CO∈TL 2015 15
Freie Universität Berlin
Nonneg.}1,0{(v)
Nonneg.}1,0{(iv)
Coupling0(iii)
Configs1(ii)
Trains1(i)
max(PCP)
Qqy
Ppx
Aayx
Jjy
Iix
xc
q
p
Qqaq
Ppap
Qqq
Ppp
Ii Pp pap
ia
j
i
i
Integ.,}1,0{(iii)
Conflicts1(ii)
Trains,)((i)
max(APP)
),(
)( )(
IiAax
Kkx
IiVvvxx
xc
i
a
kia
i
a
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va va
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a
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Ii Aa
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Arc Packing and Path Configuration Model
Track Allocation and Path Configurations | CO∈TL 2015 16
Freie Universität Berlin Model Comparison
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APP
APP'
PCP
PPP
ACP
Theorem (B., Schlechte [2007]):
vIP(PCP) = vIP(ACP)
= vIP (APP) = vIP(PPP)
= vIP(APP')
and
vLP(PCP) = vLP(ACP)
vLP (APP) = vLP(PPP)
vLP(APP').
Freie Universität Berlin Column Generation Algorithm
Track Allocation and Path Configurations | CO∈TL 2015 18
Solve Integer Program
Solve Linear
Program
All fixed?
begin
Fix variables
stop?
end
Yes
No
Yes
No
Compute duals
Generate configs
Generate paths
Freie Universität Berlin
Configuration Pricing = Shortest Path Problem (B., Schlechte [2007])
Track Allocation and Path Configurations | CO∈TL 2015 19
Freie Universität Berlin Configuration Pricing = Shortest Path Problem
Track Allocation and Path Configurations | CO∈TL 2015 20
Freie Universität Berlin
Nonneg.0(v)
Nonneg.0(iv)
)(Coupling0(iii)
)(Configs1(ii)
)(Trains1(i)
max(PLP)
Qqy
Ppx
Aayx
πJjy
γIix
xc
q
p
a
Qqa
q
Ppa
p
j
q
i
Pp
p
Ii Pp pa
p
i
a
j
i
i
Configuration LP
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0,,(iii)
Configs,0(ii)
Paths,(i)
min(DUA)
JjQq
IiPpc
j
qa
aj
i
pa
i
a
pa
ai
Ii Jj
ji
Freie Universität Berlin
Prop. (B., Schlechte [2007]):
Config pricing is an acyclic shortest path problem with arc weights ca = a, that can be solved in polynomial time.
Configuration Pricing = Shortest Path Problem
Track Allocation and Path Configurations | CO∈TL 2015 22
0,,(iii)
Configs,0(ii)
Paths,(i)
min(DUA)
JjQq
IiPpc
j
qa
aj
i
pa
i
a
pa
ai
Ii Jj
ji
Freie Universität Berlin
Prop. (B., Schlechte [2007]):
Route pricing is an acyclic shortest path problem with arc weights 𝑐 𝑎 = −𝑐𝑎 + 𝜆𝑎, that can be solved in polynomial time.
Path Pricing = Shortest Path Problem
Track Allocation and Path Configurations | CO∈TL 2015 23
0,,(iii)
Configs,0(ii)
Paths,(i)
min(DUA)
JjQq
IiPpc
j
qa
aj
i
pa
i
a
pa
ai
Ii Jj
ji
Freie Universität Berlin Model Comparison
Track Allocation and Path Configurations | CO∈TL 2015 24
APP
APP'
PCP
PPP
ACP
Theorem (B., Schlechte [2007]):
vIP(PCP) = vIP(ACP)
= vIP (APP) = vIP(PPP)
= vIP(APP')
and
vLP(PCP) = vLP(ACP)
vLP (APP) = vLP(PPP)
vLP(APP').
vLP(PCP) and vLP(ACP) can be
computed in polynomial time.
Freie Universität Berlin A Dual Bound
Prop. (B., Schlechte [2007]): ([+]+,[+]+,) is dual feasible for PLP and
(, , )= [+]+i+ [+]+
j
is an upper bound on v(PLP).
Track Allocation and Path Configurations | CO∈TL 2015 25
0,,(iii)
Configs,0(ii)
Paths,(i)
min(DUA)
JjQq
IiPpc
j
qa
aj
i
pa
i
a
pa
ai
Ii Jj
ji
)( iγ
j
pa
ajjj
qa
aQq
j
i
pa
a
i
aiii
pa
a
i
aPp
i
QqJj
PpIicc
j
i
,max
,)()(max
Freie Universität Berlin Solving the LP-Relaxation
Track Allocation and Path Configurations | CO∈TL 2015 26
Freie Universität Berlin
Rapid Branching & Track Allocation (B., Schlechte, Weider, Reuther [2013])
Track Allocation and Path Configurations | CO∈TL 2015 27
HaKaFu, req32, 1140 requests, 30 mins time windows
Freie Universität Berlin
Packing vs. Configuration Model ( Schlechte [2011])
Scenario REQ_36 from ttplib.zib.de (Packing , Config )
Track Allocation and Path Configurations | CO∈TL 2015 28
Freie Universität Berlin Track Allocation and Train Timetabling
BAB: Branch-and-Bound
PAB: Price-and-Branch
BAP: Branch-and-Price
Track Allocation and Path Configurations | CO∈TL 2015 29
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
Freie Universität Berlin Thank you for your attention
Ralf Borndörfer Freie Universität Berlin Zuse-Institute Berlin Takustr. 7 14195 Berlin-Dahlem Fon (+49 30) 84185-243 Fax (+49 30) 84185-269
[email protected] www.zib.de/borndoerfer
Track Allocation and Path Configurations | CO∈TL 2015 51 (Photo courtesy of DB Mobility Logistics AG)