The Ideal Train Timetabling Problem
Tomáš Robenek
Jianghang Chen Michel Bierlaire
July 17, 2014
1 / 28
Liberalisation – 01.01.2010
2 / 28
Liberalisation – Overview
3 / 28
Public Sector – Accessibility/Mobility
1
Figure : Mobility evolution in Switzerland1
– source: Entwicklung der MIV und OV Erreichbarkeit in der Schweiz:
1950-2000; Ph. Frohlich, M. Tschopp and K.W. Axhausen
Private Sector
Market Settings
Travel Time is the same
Serve Different Destinations
Better Quality
Better Price
Better Departure Times
Goal: Better Timetables!
• How to measure goodness of a
timetable?
• Timetable design in the literature
– non-cyclic: using so called "ideal
timetables"
– cyclic: does not take into
account anything
• In the industry – historical
7 / 28
Update of Planning
STRATEGIC - several years
Demand
Line
Planning
TACTICAL - >= 1 year
Lines
Ideal Train
Timetabling
Ideal
Timetables
OPERATIONAL - < 1 year
Train
Timetabling
Actual
Timetables
Train
Platforming
Platform
Assignment
s
Actual
Timetables
Rolling
Stock
Planning
Train
Assignment
s
Actual
Timetables
Crew
Planning
Crew
Assignment
s
TOC
IM
8 / 28
How to Measure Quality of a Timetable?
Speed (Time)
Being On Time
Timetable
Transfers
Waiting Time
Being on Time
Scheduled Delay
f_2
f_1
Time
Ideal Time
• Scheduled delay times value of time (Arnott et al. (1990))
10 / 28
The Rest
Speed (Time)
• Running time multiplied by the value of time (Axhausen et al.
(2008))
Waiting Time
• Waiting time multiplied by the value of waiting time
(Wardman (2004))
Transfers
• Minimum transfer time multiplied by the number of transfers
and the value of waiting time (Wardman (2004))
11 / 28
References
Arnott, R., de Palma, A. and Lindsey, R.
(1990). Economics of a bottleneck, Journal
of Urban Economics 27(1): 111 – 130.
Axhausen, K. W., Hess, S., König, A., Abay,
G., Bates, J. J. and Bierlaire, M. (2008).
Income and distance elasticities of values of
travel time savings: New swiss results,
Transport Policy 15(3): 173 – 185.
Wardman, M. (2004). Public transport values
of time, Transport Policy 11(4): 363 – 377.
12 / 28
Ideal Timetable
The ideal timetable consists of such train departures that the
passengers’ global costs are minimized, i.e. the fastest most
convenient path to get from the origin to the destination
traded-off by a timely arrival to the destination for every
passenger.
Inputs
i ∈I
t∈T
t0 ∈ T i
l ∈L
v ∈ Vl
p ∈ Pi
l ∈ Lp
ripl
hipl
Dit
m
c
q1
q2
f1
f2
0
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
set of origin-destination pairs
set of time steps t in the planning horizon
set of ideal times for OD pair i
set of operated lines
set of available vehicles on line l
set of possible paths between OD pair i
set of lines in the path p
running time between OD pair i on path p using line l
time to arrive from the starting station of the line l to
the origin of the pair i
demand between OD i with ideal time t 0
minimum transfer time
cycle
value of the waiting time
value of the in vehicle time
coefficient of being early
coefficient of being late
Decisions
Cit
0
–
wit
0
wit
0
p
–
wit
0
pl
–
xit
0
sit
0
sit
0
–
p
–
–
p
–
dvl
0
yit plv
–
–
zvl
–
the total cost of the passengers with ideal time t 0 between
OD pair i
the total waiting time of the passengers with ideal time
t 0 between OD pair i
the total waiting time of the passengers with ideal time
t 0 between OD pair i using path p
the waiting time of the passengers with ideal time t 0
between OD pair i on the line l that is part of the path
p
1 – if the passengers with ideal time t 0 between OD pair
i choose path p; 0 – otherwise
the final scheduled of the passengers with ideal time t 0
between OD pair i
scheduled delay of the passengers with ideal time t 0 between OD pair i traveling on the path p
the departure time of a train v on the line l
1 – if the passengers with ideal time t 0 between OD pair i
on the path p take the train v on the line l; 0 – otherwise
frequency within cyclicity
Objective
Speed (Time)
Being On Time
min
Timetable
X X
0
Dit · Cit
0
i∈I t 0 ∈T i
Transfers
Waiting Time
16 / 28
Pricing Constraints
0
0
Cit = q1 · wit + q1 · m ·
X
xit
0
p
· (|Lp | − 1)
p∈P
+ q2 ·
XX
p∈P
ripl · xit
0
p
+ q2 ·
l∈Lp
X
0
sit ,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P
0
wit ≥ wit
0
0
− M · 1 − xit p ,
X t 0 pl
0
wit p ≥
wi ,
p
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp \1
wit
0
pl
0
0
0
≥ dvl + hipl − dvl 0 + hipl + ripl + m
0
0 0 0
− M · 1 − yit pl v − M · 1 − yit plv ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
≤
0
0
≥M·
X
0
dvl + hipl − dvl 0 + hipl + ripl + m
0 0 0
0
+ M · 1 − yit pl v + M · 1 − yit plv ,∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
0
l
t pl (V +1)
yi
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp
0
sit p
sit
0
p
0
0
0
sit ≥ sit p − M · 1 − xit p ,
|L|
p|L|
t 0 p|L|v
≥ f2 · dv|L| + hi + ri
− t 0 − M · 1 − yi
,
|L|
p|L|
t 0 p|L|v
0
|L|
≥ f1 · t − dv + hi + ri
− M · 1 − yi
,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
Pricing Constraints
0
0
Cit = q1 · wit + q1 · m ·
X
xit
0
p
· (|Lp | − 1)
p∈P
+ q2 ·
XX
p∈P
ripl · xit
0
p
+ q2 ·
l∈Lp
X
0
sit ,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P
0
wit ≥ wit
0
0
− M · 1 − xit p ,
X t 0 pl
0
wit p ≥
wi ,
p
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp \1
wit
0
pl
0
0
0
≥ dvl + hipl − dvl 0 + hipl + ripl + m
0
0 0 0
− M · 1 − yit pl v − M · 1 − yit plv ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
≤
0
0
≥M·
X
0
dvl + hipl − dvl 0 + hipl + ripl + m
0 0 0
0
+ M · 1 − yit pl v + M · 1 − yit plv ,∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
0
l
t pl (V +1)
yi
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp
0
sit p
sit
0
p
0
0
0
sit ≥ sit p − M · 1 − xit p ,
|L|
p|L|
t 0 p|L|v
≥ f2 · dv|L| + hi + ri
− t 0 − M · 1 − yi
,
|L|
p|L|
t 0 p|L|v
0
|L|
≥ f1 · t − dv + hi + ri
− M · 1 − yi
,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
Pricing Constraints
0
0
Cit = q1 · wit + q1 · m ·
X
xit
0
p
· (|Lp | − 1)
p∈P
+ q2 ·
XX
p∈P
ripl · xit
0
p
+ q2 ·
l∈Lp
X
0
sit ,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P
0
wit ≥ wit
0
0
− M · 1 − xit p ,
X t 0 pl
0
wit p ≥
wi ,
p
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp \1
wit
0
pl
0
0
0
≥ dvl + hipl − dvl 0 + hipl + ripl + m
0
0 0 0
− M · 1 − yit pl v − M · 1 − yit plv ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
≤
0
0
≥M·
X
0
dvl + hipl − dvl 0 + hipl + ripl + m
0 0 0
0
+ M · 1 − yit pl v + M · 1 − yit plv ,∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
0
l
t pl (V +1)
yi
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp
0
sit p
sit
0
p
0
0
0
sit ≥ sit p − M · 1 − xit p ,
|L|
p|L|
t 0 p|L|v
≥ f2 · dv|L| + hi + ri
− t 0 − M · 1 − yi
,
|L|
p|L|
t 0 p|L|v
0
|L|
≥ f1 · t − dv + hi + ri
− M · 1 − yi
,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
Pricing Constraints
0
0
Cit = q1 · wit + q1 · m ·
X
xit
0
p
· (|Lp | − 1)
p∈P
+ q2 ·
XX
p∈P
ripl · xit
0
p
+ q2 ·
l∈Lp
X
0
sit ,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P
0
wit ≥ wit
0
0
− M · 1 − xit p ,
X t 0 pl
0
wit p ≥
wi ,
p
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp \1
wit
0
pl
0
0
0
≥ dvl + hipl − dvl 0 + hipl + ripl + m
0
0 0 0
− M · 1 − yit pl v − M · 1 − yit plv ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
≤
0
0
≥M·
X
0
dvl + hipl − dvl 0 + hipl + ripl + m
0 0 0
0
+ M · 1 − yit pl v + M · 1 − yit plv ,∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
0
l
t pl (V +1)
yi
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp
0
sit p
sit
0
p
0
0
0
sit ≥ sit p − M · 1 − xit p ,
|L|
p|L|
t 0 p|L|v
≥ f2 · dv|L| + hi + ri
− t 0 − M · 1 − yi
,
|L|
p|L|
t 0 p|L|v
0
|L|
≥ f1 · t − dv + hi + ri
− M · 1 − yi
,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
Pricing Constraints
0
0
Cit = q1 · wit + q1 · m ·
X
xit
0
p
· (|Lp | − 1)
p∈P
+ q2 ·
XX
p∈P
ripl · xit
0
p
+ q2 ·
l∈Lp
X
0
sit ,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P
0
wit ≥ wit
0
0
− M · 1 − xit p ,
X t 0 pl
0
wit p ≥
wi ,
p
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp \1
wit
0
pl
0
0
0
≥ dvl + hipl − dvl 0 + hipl + ripl + m
0
0 0 0
− M · 1 − yit pl v − M · 1 − yit plv ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
≤
0
0
≥M·
X
0
dvl + hipl − dvl 0 + hipl + ripl + m
0 0 0
0
+ M · 1 − yit pl v + M · 1 − yit plv ,∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
0
l
t pl (V +1)
yi
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp
0
sit p
sit
0
p
0
0
0
sit ≥ sit p − M · 1 − xit p ,
|L|
p|L|
t 0 p|L|v
≥ f2 · dv|L| + hi + ri
− t 0 − M · 1 − yi
,
|L|
p|L|
t 0 p|L|v
0
|L|
≥ f1 · t − dv + hi + ri
− M · 1 − yi
,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
Pricing Constraints
0
0
Cit = q1 · wit + q1 · m ·
X
xit
0
p
· (|Lp | − 1)
p∈P
+ q2 ·
XX
p∈P
ripl · xit
0
p
+ q2 ·
l∈Lp
X
0
sit ,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P
0
wit ≥ wit
0
0
− M · 1 − xit p ,
X t 0 pl
0
wit p ≥
wi ,
p
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp \1
wit
0
pl
0
0
0
≥ dvl + hipl − dvl 0 + hipl + ripl + m
0
0 0 0
− M · 1 − yit pl v − M · 1 − yit plv ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
≤
0
0
≥M·
X
0
dvl + hipl − dvl 0 + hipl + ripl + m
0 0 0
0
+ M · 1 − yit pl v + M · 1 − yit plv ,∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
0
l
t pl (V +1)
yi
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp
0
sit p
sit
0
p
0
0
0
sit ≥ sit p − M · 1 − xit p ,
|L|
p|L|
t 0 p|L|v
≥ f2 · dv|L| + hi + ri
− t 0 − M · 1 − yi
,
|L|
p|L|
t 0 p|L|v
0
|L|
≥ f1 · t − dv + hi + ri
− M · 1 − yi
,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
Pricing Constraints
0
0
Cit = q1 · wit + q1 · m ·
X
xit
0
p
· (|Lp | − 1)
p∈P
+ q2 ·
XX
p∈P
ripl · xit
0
p
+ q2 ·
l∈Lp
X
0
sit ,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P
0
wit ≥ wit
0
0
− M · 1 − xit p ,
X t 0 pl
0
wit p ≥
wi ,
p
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp \1
wit
0
pl
0
0
0
≥ dvl + hipl − dvl 0 + hipl + ripl + m
0
0 0 0
− M · 1 − yit pl v − M · 1 − yit plv ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
≤
0
0
≥M·
X
0
dvl + hipl − dvl 0 + hipl + ripl + m
0 0 0
0
+ M · 1 − yit pl v + M · 1 − yit plv ,∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
0
l
t pl (V +1)
yi
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp
0
sit p
sit
0
p
0
0
0
sit ≥ sit p − M · 1 − xit p ,
|L|
p|L|
t 0 p|L|v
≥ f2 · dv|L| + hi + ri
− t 0 − M · 1 − yi
,
|L|
p|L|
t 0 p|L|v
0
|L|
≥ f1 · t − dv + hi + ri
− M · 1 − yi
,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
Pricing Constraints
0
0
Cit = q1 · wit + q1 · m ·
X
xit
0
p
· (|Lp | − 1)
p∈P
+ q2 ·
XX
p∈P
ripl · xit
0
p
+ q2 ·
l∈Lp
X
0
sit ,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P
0
wit ≥ wit
0
0
− M · 1 − xit p ,
X t 0 pl
0
wit p ≥
wi ,
p
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp \1
wit
0
pl
0
0
0
≥ dvl + hipl − dvl 0 + hipl + ripl + m
0
0 0 0
− M · 1 − yit pl v − M · 1 − yit plv ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
≤
0
0
≥M·
X
0
dvl + hipl − dvl 0 + hipl + ripl + m
0 0 0
0
+ M · 1 − yit pl v + M · 1 − yit plv ,∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
0
l
t pl (V +1)
yi
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp
0
sit p
sit
0
p
0
0
0
sit ≥ sit p − M · 1 − xit p ,
|L|
p|L|
t 0 p|L|v
≥ f2 · dv|L| + hi + ri
− t 0 − M · 1 − yi
,
|L|
p|L|
t 0 p|L|v
0
|L|
≥ f1 · t − dv + hi + ri
− M · 1 − yi
,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
Pricing Constraints
0
0
Cit = q1 · wit + q1 · m ·
X
xit
0
p
· (|Lp | − 1)
p∈P
+ q2 ·
XX
p∈P
ripl · xit
0
p
+ q2 ·
l∈Lp
X
0
sit ,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P
0
wit ≥ wit
0
0
− M · 1 − xit p ,
X t 0 pl
0
wit p ≥
wi ,
p
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp \1
wit
0
pl
0
0
0
≥ dvl + hipl − dvl 0 + hipl + ripl + m
0
0 0 0
− M · 1 − yit pl v − M · 1 − yit plv ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
≤
0
0
≥M·
X
0
dvl + hipl − dvl 0 + hipl + ripl + m
0 0 0
0
+ M · 1 − yit pl v + M · 1 − yit plv ,∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp :
0
l > 1, l 0 = l − 1, ∀v ∈ V l , ∀v 0 ∈ V l ,
wit
0
pl
0
l
t pl (V +1)
yi
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
l∈Lp
0
sit p
sit
0
p
0
0
0
sit ≥ sit p − M · 1 − xit p ,
|L|
p|L|
t 0 p|L|v
≥ f2 · dv|L| + hi + ri
− t 0 − M · 1 − yi
,
|L|
p|L|
t 0 p|L|v
0
|L|
≥ f1 · t − dv + hi + ri
− M · 1 − yi
,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀v ∈ V |L| ,
Feasibility Constraints
X
0
xit p = 1,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P i
X
0
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp ,
= c · zvl ,
∀l ∈ L, ∀v ∈ V : v > 1,
yit plv = 1,
v ∈V l +1
l
dv − dvl −1
domain
constraints
18 / 28
Feasibility Constraints
X
0
xit p = 1,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P i
X
0
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp ,
= c · zvl ,
∀l ∈ L, ∀v ∈ V : v > 1,
yit plv = 1,
v ∈V l +1
l
dv − dvl −1
domain
constraints
18 / 28
Feasibility Constraints
X
0
xit p = 1,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P i
X
0
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp ,
= c · zvl ,
∀l ∈ L, ∀v ∈ V : v > 1,
yit plv = 1,
v ∈V l +1
l
dv − dvl −1
domain
constraints
18 / 28
Feasibility Constraints
X
0
xit p = 1,
∀i ∈ I, ∀t 0 ∈ T i ,
p∈P i
X
0
∀i ∈ I, ∀t 0 ∈ T i , ∀p ∈ P i , ∀l ∈ Lp ,
= c · zvl ,
∀l ∈ L, ∀v ∈ V : v > 1,
yit plv = 1,
v ∈V l +1
l
dv − dvl −1
domain
constraints
18 / 28
Case Study – Israel
• OD Matrix for an
average working day
(Sunday to Thursday)
in Israel during 2008
• 47 Stations
• 2162 ODs
• 36 (unidirectional) lines
• 389 trains
• Min. transfer – 4 mins
• VOT – 21.12 NIS per
hour
Too Heavy – Branch-and-Price Framework
Start
Initial Columns
Solve Master
Problem
Dual Variables
New Columns
with Negative
Reduced Cost
• Initial Solution
Pricing Solver
• Column Generation – Lower Bound
No Columns
with Negative
Reduced Cost
Integer
Solution?
• Branch and Bound – Optimal Integer
No
Branch & Bound
Solution
yes
Output Solution
End
20 / 28
Initial Solution
Start
Initial Columns
Solve Master
Problem
Dual Variables
New Columns
with Negative
Reduced Cost
Pricing Solver
No Columns
with Negative
Reduced Cost
Integer
Solution?
No
Branch & Bound
yes
Output Solution
End
21 / 28
Master Problem (MP)
Start
Initial Columns
Solve Master
Problem
Dual Variables
Idea
New Columns
with Negative
Reduced Cost
• Formulated as a Set Partitioning Problem
Pricing Solver
• Decision variables are relaxed, solution
No Columns
with Negative
Reduced Cost
Integer
Solution?
yes
space restricted
No
Branch & Bound
• Consists of the feasibility constraints
Go to model
Output Solution
End
22 / 28
MP – Column
Start
Time 1
Initial Columns
Line 1
Solve Master
Problem
Dual Variables
Time 1
New Columns
with Negative
Reduced Cost
Pricing Solver
OD Pair i at time t’
No Columns
with Negative
Reduced Cost
Integer
Solution?
No
Time 2
Line 2
Time 2
Line 3
Time 1
Branch & Bound
Time 2
yes
Output Solution
Line 4
Time 1
Time 2
End
23 / 28
Sub-Problem (SP)
Start
Initial Columns
Idea
Solve Master
Problem
Dual Variables
• run for each OD pair i, each ideal time
New Columns
with Negative
Reduced Cost
t 0 ∈ T i and each path p ∈ P i separately
Pricing Solver
• Consists of the pricing constraints
No Columns
with Negative
Reduced Cost
Integer
Solution?
yes
Output Solution
Go to model
No
Branch & Bound
Dual Variables
0
• αit , βlt
End
24 / 28
Results – 6 a.m. to 8 a.m.
Current
[NIS]
220
218
219
217
213
279.32
564.49
271.56
644.50
957.08
Column Generation
# iter.
2
2
2
2
2
time
48m
48m
46m
48m
45m
19s
15s
12s
01s
00s
Ω1
228
230
233
228
226
859
941
627
849
772
lb [NIS]
132
131
131
130
127
371.26
582.85
420.41
494.59
586.58
Cyclic
decrease [%]
40
40
40
40
40
ub [NIS]
209
207
207
208
205
791.62
717.86
886.98
275.88
328.12
Non-Cyclic
ub [NIS]
204
202
201
203
199
770.06
209.67
949.28
133.08
417.99
25 / 28
Conclusions
• New planning phase based on the demand
• In line with the new market structure
• Can handle both non- and cyclic timetables
• Takes care of the connections, in the current practice:
– non-cyclic – does not exist
– cyclic – always imposed
• Returns ideal timetables, its cost and the routings of the
passengers
26 / 28
Future Work
• Unsolvable using exact method
• Brute force algorithm to evaluate current timetable
• Heuristic to solve the whole day
27 / 28
Thank you for your attention.
MP – Inputs
a∈Ω
i ∈I
t∈T
t0 ∈ T i
–
–
–
–
l ∈L
c
Ca
Da
nl
–
–
–
–
–
Bait
Ealt
0
=
=
Go back
set of all possible assignments
set of origin-destination pairs
set of all time steps
set of times that there is a demand between OD
i
set of operated lines
cycle
cost of the assignment a
demand using assignment a
number of available train units on line l
(
1 if OD pair i at time t 0 is assigned in assignment a,
0 otherwise.
(
1 if the assignment a is using line l at time t,
0 otherwise.
MP – Decisions
Start
Initial Columns
Solve Master
Problem
Dual Variables
New Columns
with Negative
Reduced Cost
λa
Pricing Solver
No Columns
with Negative
Reduced Cost
Integer
Solution?
yes
Output Solution
End
xlt
No
Branch & Bound
–
–
n
1
0
if assignment a is a part of the solution,
otherwise.
n
1
0
if there is a train scheduled on line l at time t,
otherwise.
MP
min
X
Da · Ca · λa
a∈Ω
0
Bait · λa = 1,
∀i ∈ I, ∀t 0 ∈ T i ,
Ealt · λa ≤ xlt ,
∀l ∈ L, ∀t ∈ T ,
X
a∈Ω
X
a∈Ω
X
xlt ≤ nl ,
∀l ∈ L,
t∈T
min(t+c,T )
X
00
xlt ≤ 1,
∀l ∈ L, ∀t ∈ T ,
t 00 =t
λa ∈ {0, 1} ,
xlt
∈ {0, 1} ,
∀a ∈ Ω,
∀l ∈ L, t ∈ T .
MP
min
X
Da · Ca · λa
a∈Ω
0
Bait · λa = 1,
∀i ∈ I, ∀t 0 ∈ T i ,
Ealt · λa ≤ xlt ,
∀l ∈ L, ∀t ∈ T ,
X
a∈Ω
X
a∈Ω
X
xlt ≤ nl ,
∀l ∈ L,
t∈T
min(t+c,T )
X
00
xlt ≤ 1,
∀l ∈ L, ∀t ∈ T ,
t 00 =t
λa ∈ {0, 1} ,
xlt
∈ {0, 1} ,
∀a ∈ Ω,
∀l ∈ L, t ∈ T .
MP
min
X
Da · Ca · λa
a∈Ω
0
Bait · λa = 1,
∀i ∈ I, ∀t 0 ∈ T i ,
Ealt · λa ≤ xlt ,
∀l ∈ L, ∀t ∈ T ,
X
a∈Ω
X
a∈Ω
X
xlt ≤ nl ,
∀l ∈ L,
t∈T
min(t+c,T )
X
00
xlt ≤ 1,
∀l ∈ L, ∀t ∈ T ,
t 00 =t
λa ∈ {0, 1} ,
xlt
∈ {0, 1} ,
∀a ∈ Ω,
∀l ∈ L, t ∈ T .
MP
min
X
Da · Ca · λa
a∈Ω
0
Bait · λa = 1,
∀i ∈ I, ∀t 0 ∈ T i ,
Ealt · λa ≤ xlt ,
∀l ∈ L, ∀t ∈ T ,
X
a∈Ω
X
a∈Ω
X
xlt ≤ nl ,
∀l ∈ L,
t∈T
min(t+c,T )
X
00
xlt ≤ 1,
∀l ∈ L, ∀t ∈ T ,
t 00 =t
λa ∈ {0, 1} ,
xlt
∈ {0, 1} ,
∀a ∈ Ω,
∀l ∈ L, t ∈ T .
MP
min
X
Da · Ca · λa
a∈Ω
0
Bait · λa = 1,
∀i ∈ I, ∀t 0 ∈ T i ,
Ealt · λa ≤ xlt ,
∀l ∈ L, ∀t ∈ T ,
X
a∈Ω
X
a∈Ω
X
xlt ≤ nl ,
∀l ∈ L,
t∈T
min(t+c,T )
X
00
xlt ≤ 1,
∀l ∈ L, ∀t ∈ T ,
t 00 =t
λa ∈ {0, 1} ,
xlt
∈ {0, 1} ,
∀a ∈ Ω,
∀l ∈ L, t ∈ T .
MP
min
X
Da · Ca · λa
a∈Ω
0
Bait · λa = 1,
∀i ∈ I, ∀t 0 ∈ T i ,
Ealt · λa ≤ xlt ,
∀l ∈ L, ∀t ∈ T ,
X
a∈Ω
X
a∈Ω
X
xlt ≤ nl ,
∀l ∈ L,
t∈T
min(t+c,T )
X
00
xlt ≤ 1,
∀l ∈ L, ∀t ∈ T ,
t 00 =t
λa ∈ {0, 1} ,
xlt
∈ {0, 1} ,
∀a ∈ Ω,
∀l ∈ L, t ∈ T .
MP
min
X
Da · Ca · λa
a∈Ω
0
Bait · λa = 1,
∀i ∈ I, ∀t 0 ∈ T i ,
Ealt · λa ≤ xlt ,
∀l ∈ L, ∀t ∈ T ,
X
a∈Ω
X
a∈Ω
X
xlt ≤ nl ,
∀l ∈ L,
t∈T
min(t+c,T )
X
00
xlt ≤ 1,
∀l ∈ L, ∀t ∈ T ,
t 00 =t
λa ∈ {0, 1} ,
xlt
∈ {0, 1} ,
∀a ∈ Ω,
∀l ∈ L, t ∈ T .
SP – Inputs
Start
Initial Columns
Solve Master
Problem
Dual Variables
New Columns
with Negative
Reduced Cost
Pricing Solver
Go back
i
t0
p
t∈T
l ∈ Lp
–
–
–
–
–
rl
hl
–
–
m
q1
q2
f1
f2
–
–
–
–
–
No Columns
with Negative
Reduced Cost
Integer
Solution?
yes
Output Solution
End
No
Branch & Bound
the origin destination pair
ideal travel time for OD pair i
path of the sub-problem
set of all time steps
the sequence of lines used to get
from the origin to the destination
running time of line l
running time to get from the starting station of the line l to the first
station on the same line included in
the current path
the minimum transfer time
the value of time spent waiting
the value of time spent in vehicle
coefficient of being early
coefficient of being late
SP – Decision Variables
Start
(
Initial Columns
Solve Master
Problem
Dual Variables
New Columns
with Negative
Reduced Cost
betalt
–
w
–
wl
–
s
–
C
–
Pricing Solver
No Columns
with Negative
Reduced Cost
Integer
Solution?
No
Branch & Bound
yes
Output Solution
End
1 if line l is used at time t,
0 otherwise.
the total waiting time of the
passengers
the waiting time of the passengers when transferring to line l
scheduled delay of the passengers
the cost of the passengers
SP – Model
!
min C −
α+
XX
l∈L
t
βl
·
t
betal
t∈T
!
C =
q1 · w + q1 · m · (|L| − 1) + q2 ·
X
l
r + q2 · s
l∈L
X
t
betal = 1,
p
l ∈L ,
t∈T
w =
X
wl ,
l∈L\1
wl ≥
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
wl ≤
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
s ≥ f1 ·
t
t · beta|L| + h|L|
0
t −
t
,
,
−t
t · beta|L| + h|L|
0
domain
00
∈T :
+ hl−1 + rl−1
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
s ≥ f2 ·
00
00
00
+ hl−1 + rl−1
∀t ∈ T ,
∀t ∈ T ,
constraints
∈T :
SP – Model
!
min C −
α+
XX
l∈L
t
βl
·
t
betal
t∈T
!
C =
q1 · w + q1 · m · (|L| − 1) + q2 ·
X
l
r + q2 · s
l∈L
X
t
betal = 1,
p
l ∈L ,
t∈T
w =
X
wl ,
l∈L\1
wl ≥
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
wl ≤
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
s ≥ f1 ·
t
t · beta|L| + h|L|
0
t −
t
,
,
−t
t · beta|L| + h|L|
0
domain
00
∈T :
+ hl−1 + rl−1
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
s ≥ f2 ·
00
00
00
+ hl−1 + rl−1
∀t ∈ T ,
∀t ∈ T ,
constraints
∈T :
SP – Model
!
min C −
α+
XX
l∈L
t
βl
·
t
betal
t∈T
!
C =
q1 · w + q1 · m · (|L| − 1) + q2 ·
X
l
r + q2 · s
l∈L
X
t
betal = 1,
p
l ∈L ,
t∈T
w =
X
wl ,
l∈L\1
wl ≥
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
wl ≤
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
s ≥ f1 ·
t
t · beta|L| + h|L|
0
t −
t
,
,
−t
t · beta|L| + h|L|
0
domain
00
∈T :
+ hl−1 + rl−1
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
s ≥ f2 ·
00
00
00
+ hl−1 + rl−1
∀t ∈ T ,
∀t ∈ T ,
constraints
∈T :
SP – Model
!
min C −
α+
XX
l∈L
t
βl
·
t
betal
t∈T
!
C =
q1 · w + q1 · m · (|L| − 1) + q2 ·
X
l
r + q2 · s
l∈L
X
t
betal = 1,
p
l ∈L ,
t∈T
w =
X
wl ,
l∈L\1
wl ≥
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
wl ≤
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
s ≥ f1 ·
t
t · beta|L| + h|L|
0
t −
t
,
,
−t
t · beta|L| + h|L|
0
domain
00
∈T :
+ hl−1 + rl−1
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
s ≥ f2 ·
00
00
00
+ hl−1 + rl−1
∀t ∈ T ,
∀t ∈ T ,
constraints
∈T :
SP – Model
!
min C −
α+
XX
l∈L
t
βl
·
t
betal
t∈T
!
C =
q1 · w + q1 · m · (|L| − 1) + q2 ·
X
l
r + q2 · s
l∈L
X
t
betal = 1,
p
l ∈L ,
t∈T
w =
X
wl ,
l∈L\1
wl ≥
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
wl ≤
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
s ≥ f1 ·
t
t · beta|L| + h|L|
0
t −
t
,
,
−t
t · beta|L| + h|L|
0
domain
00
∈T :
+ hl−1 + rl−1
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
s ≥ f2 ·
00
00
00
+ hl−1 + rl−1
∀t ∈ T ,
∀t ∈ T ,
constraints
∈T :
SP – Model
!
min C −
α+
XX
l∈L
t
βl
·
t
betal
t∈T
!
C =
q1 · w + q1 · m · (|L| − 1) + q2 ·
X
l
r + q2 · s
l∈L
X
t
betal = 1,
p
l ∈L ,
t∈T
w =
X
wl ,
l∈L\1
wl ≥
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
wl ≤
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
s ≥ f1 ·
t
t · beta|L| + h|L|
0
t −
t
,
,
−t
t · beta|L| + h|L|
0
domain
00
∈T :
+ hl−1 + rl−1
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
s ≥ f2 ·
00
00
00
+ hl−1 + rl−1
∀t ∈ T ,
∀t ∈ T ,
constraints
∈T :
SP – Model
!
min C −
α+
XX
l∈L
t
βl
·
t
betal
t∈T
!
C =
q1 · w + q1 · m · (|L| − 1) + q2 ·
X
l
r + q2 · s
l∈L
X
t
betal = 1,
p
l ∈L ,
t∈T
w =
X
wl ,
l∈L\1
wl ≥
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
wl ≤
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
s ≥ f1 ·
t
t · beta|L| + h|L|
0
t −
t
,
,
−t
t · beta|L| + h|L|
0
domain
00
∈T :
+ hl−1 + rl−1
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
s ≥ f2 ·
00
00
00
+ hl−1 + rl−1
∀t ∈ T ,
∀t ∈ T ,
constraints
∈T :
SP – Model
!
min C −
α+
XX
l∈L
t
βl
·
t
betal
t∈T
!
C =
q1 · w + q1 · m · (|L| − 1) + q2 ·
X
l
r + q2 · s
l∈L
X
t
betal = 1,
p
l ∈L ,
t∈T
w =
X
wl ,
l∈L\1
wl ≥
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
wl ≤
t
t · betal + hl
−
t
00
t 00
· betal−1 + hl−1 + rl−1 + m ,
s ≥ f1 ·
t
t · beta|L| + h|L|
0
t −
t
,
,
−t
t · beta|L| + h|L|
0
domain
00
∈T :
+ hl−1 + rl−1
p
∀l ∈ L : l > 1, ∀t, t
t ≥t
s ≥ f2 ·
00
00
00
+ hl−1 + rl−1
∀t ∈ T ,
∀t ∈ T ,
constraints
∈T :