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Line Planning in Public Transportation€¦ · September 2006 2 / 78. Line planning in public...
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Line Planning in Public Transportation Anita Schöbel Institut für Numerische und Angewandte Mathematik Georg-August Universität Göttingen 27. September 2006 Anita Schöbel (NAM) 27. September 2006 1 / 78
Transcript of Line Planning in Public Transportation€¦ · September 2006 2 / 78. Line planning in public...
Line Planning in Public Transportation
Anita Schöbel
Institut für Numerische und Angewandte MathematikGeorg-August Universität Göttingen
27. September 2006
Anita Schöbel (NAM) 27. September 2006 1 / 78
Line planning in public transportation
Planning in public transportation
companie’sperspective
VehicleScheduling/rolling
stock planning|
crew scheduling|
rostering
Stations|
Lines|
— Timetable —
Customers’ pointof view
Tariffs|
Disposition
Anita Schöbel (NAM) 27. September 2006 2 / 78
Line planning in public transportation
Planning in public transportation
companie’sperspective
VehicleScheduling/rolling
stock planning|
crew scheduling|
rostering
Stations|
Lines|
— Timetable —
Customers’ pointof view
Tariffs|
Disposition
Anita Schöbel (NAM) 27. September 2006 2 / 78
Line planning in public transportation
Planning in public transportation
companie’sperspective
VehicleScheduling/rolling
stock planning|
crew scheduling|
rostering
Stations|
Lines|
— Timetable —
Customers’ pointof view
Tariffs|
Disposition
Anita Schöbel (NAM) 27. September 2006 2 / 78
Line planning in public transportation
Planning in public transportation
companie’sperspective
VehicleScheduling/rolling
stock planning|
crew scheduling|
rostering
Stations|
Lines|
— Timetable —
Customers’ pointof view
Tariffs|
Disposition
Anita Schöbel (NAM) 27. September 2006 2 / 78
Line planning in public transportation
Planning in public transportation
companie’sperspective
VehicleScheduling/rolling
stock planning|
crew scheduling|
rostering
Stations|
Lines|
— Timetable —
Customers’ pointof view
Tariffs|
Disposition
Anita Schöbel (NAM) 27. September 2006 2 / 78
Line planning in public transportation
Planning in public transportation
companie’sperspective
VehicleScheduling/rolling
stock planning|
crew scheduling|
rostering
Stations|
Lines|
— Timetable —
Customers’ pointof view
Tariffs|
Disposition
Anita Schöbel (NAM) 27. September 2006 2 / 78
Line planning in public transportation
Line planning problem
Givena public transportation network PTN=(V , E)
I with its stops VI and its direct connections E
Find a set of lines, i.e.,determine the number of lines,the route of the lines,and their frequencies
Anita Schöbel (NAM) 27. September 2006 3 / 78
Line planning in public transportation
Line planning problem
Givena public transportation network PTN=(V , E)
I with its stops VI and its direct connections E
Find a set of lines, i.e.,determine the number of lines,the route of the lines,and their frequencies
Anita Schöbel (NAM) 27. September 2006 3 / 78
Line planning in public transportation
Example
Anita Schöbel (NAM) 27. September 2006 4 / 78
Line planning in public transportation
Example
Anita Schöbel (NAM) 27. September 2006 4 / 78
Line planning in public transportation
Example
Anita Schöbel (NAM) 27. September 2006 4 / 78
Line planning in public transportation
Example
Anita Schöbel (NAM) 27. September 2006 4 / 78
Line planning in public transportation
Example
Anita Schöbel (NAM) 27. September 2006 4 / 78
Line planning in public transportation
Example
lf =1
lf =4
lf =1
f =2l
Anita Schöbel (NAM) 27. September 2006 4 / 78
Line planning in public transportation
Literature
Patz (1925), Lampkin and Saalmans (1967), Wegel (1974), Dienst (1970), Sonntag (1977 and
1979), Newell (1979) Simonis (1980 and 1981), Reinecke (1992), Israeli and Ceder (1993),
Claessens (1994), Carey (1994), Oltrogge (1994), Reinecke (1995), Bussieck, Kreuzer and
Zimmermann (1996) Zwaneveld, Claessens, van Dijk (1996), Claessens, van Dijk, Zwaneveld
(1996), Bussieck and Zimmermann (1997), Zimmermann, Bussieck, Krista and Wiegand (1997),
Bussieck (1998), Claessens, van Dijk and Zwaneveld (1998), Klingele (2000), Völker (2001),
Goessens, Hoesel, and Kroon (2001), Schmidt (2001), Goessens, Hoesel, and Kroon (2002),
Bussieck, Lindner, and Lübbecke (2004), Quack (2003), Liebchen and Möhring (2004), Laporte,
Marin, Mesa, Ortega (2004), Schöbel and Scholl (2004) Borndörfer, Grötschel and Pfetsch
(2005), Scholl (2005), Schneider (2005), Borndörfer and Pfetsch (2006), Schöbel and Scholl
(2006), Schöbel and Schwarze (2006) . . .
Anita Schöbel (NAM) 27. September 2006 5 / 78
Line planning in public transportation
Basic notation
NotationA line P is a path in the public transportation network
The frequency fl of a line l says how often service is offered along linel within a (given) time period I.
A line concept (L, f ) is a set of lines L together with their frequenciesfl for all l ∈ L.
Anita Schöbel (NAM) 27. September 2006 6 / 78
Line planning in public transportation
Line planning problemGiven
a public transportation network PTNI with its stops VI and its direct connections E
Find a set of lines, i.e.,determine the number of lines,the route of the lines,and their frequencies
such that
?
All customers should be transportedPublic transport should be convenient for the customersCosts should be small
We are now formalizing the problem and its objective functions.
Anita Schöbel (NAM) 27. September 2006 7 / 78
Line planning in public transportation
Line planning problemGiven
a public transportation network PTNI with its stops VI and its direct connections E
Find a set of lines, i.e.,determine the number of lines,the route of the lines,and their frequencies
such that ?
All customers should be transportedPublic transport should be convenient for the customersCosts should be small
We are now formalizing the problem and its objective functions.
Anita Schöbel (NAM) 27. September 2006 7 / 78
Line planning in public transportation
Line planning problemGiven
a public transportation network PTNI with its stops VI and its direct connections E
Find a set of lines, i.e.,determine the number of lines,the route of the lines,and their frequencies
such that
?
All customers should be transportedPublic transport should be convenient for the customersCosts should be small
We are now formalizing the problem and its objective functions.
Anita Schöbel (NAM) 27. September 2006 7 / 78
Line planning in public transportation
Line planning problemGiven
a public transportation network PTNI with its stops VI and its direct connections E
Find a set of lines, i.e.,determine the number of lines,the route of the lines,and their frequencies
such that
?
All customers should be transportedPublic transport should be convenient for the customersCosts should be small
We are now formalizing the problem and its objective functions.Anita Schöbel (NAM) 27. September 2006 7 / 78
Line planning in public transportation
Minimal and maximal frequencies
Notation
Let A denote the (fixed) capacity of a vehicle, and f mine , f max
e denote theminimal and maximal allowed frequency on edge e ∈ E.
Example:f mine is the minimal number of vehicles needed to transport allcustomers.
f mine = dwe
Ae.
f maxe : due to security reasons or noise avoidance
NotationThe edge-frequency of e w.r.t. a line concept (L, f ) is given as
f (e) =∑l:e∈l
fl .
Anita Schöbel (NAM) 27. September 2006 8 / 78
Line planning in public transportation
Minimal and maximal frequencies
Notation
Let A denote the (fixed) capacity of a vehicle, and f mine , f max
e denote theminimal and maximal allowed frequency on edge e ∈ E.
Example:f mine is the minimal number of vehicles needed to transport allcustomers.
f mine = dwe
Ae.
f maxe : due to security reasons or noise avoidance
NotationThe edge-frequency of e w.r.t. a line concept (L, f ) is given as
f (e) =∑l:e∈l
fl .
Anita Schöbel (NAM) 27. September 2006 8 / 78
Line planning in public transportation
Minimal and maximal frequencies
Notation
Let A denote the (fixed) capacity of a vehicle, and f mine , f max
e denote theminimal and maximal allowed frequency on edge e ∈ E.
Example:f mine is the minimal number of vehicles needed to transport allcustomers.
f mine = dwe
Ae.
f maxe : due to security reasons or noise avoidance
NotationThe edge-frequency of e w.r.t. a line concept (L, f ) is given as
f (e) =∑l:e∈l
fl .
Anita Schöbel (NAM) 27. September 2006 8 / 78
Line planning in public transportation
The line pool
Two different possibilities:from pool: Choose the lines for the line concept (L, fl) from agiven line pool L0, i.e. L ⊆ L0
from scratch: Construct the lines L from scratch
Remark: Nearly all publications use a line pool.
From now on: Let L0 be a given line pool.
Anita Schöbel (NAM) 27. September 2006 9 / 78
Line planning in public transportation
The line pool
Two different possibilities:from pool: Choose the lines for the line concept (L, fl) from agiven line pool L0, i.e. L ⊆ L0
from scratch: Construct the lines L from scratch
Remark: Nearly all publications use a line pool.
From now on: Let L0 be a given line pool.
Anita Schöbel (NAM) 27. September 2006 9 / 78
Line planning in public transportation
The line pool
Two different possibilities:from pool: Choose the lines for the line concept (L, fl) from agiven line pool L0, i.e. L ⊆ L0
from scratch: Construct the lines L from scratch
Remark: Nearly all publications use a line pool.
From now on: Let L0 be a given line pool.
Anita Schöbel (NAM) 27. September 2006 9 / 78
Basic model
Basic line planning problem
(LP0) (Finding a feasible line concept)
Given a PTN, a set L0 of potential lines, and lower and upper frequen-cies f min
e ≤ f maxe for all e ∈ E , find a feasible line concept (L, f ) with
L ⊆ L0 and fl ∈ IN0 ∀ l ∈ L.
Anita Schöbel (NAM) 27. September 2006 10 / 78
Basic model
Complexity of (LP0)
Theorem
(LP0) is NP-complete.
Anita Schöbel (NAM) 27. September 2006 11 / 78
Basic model
Algorithm for (LP0)
Special case: no upper frequencies
Algorithm: Finding a feasible line concept without upper fre-quencies
Input: PTN, set of potential lines L0, lower frequencies f mine for
all e ∈ E .Output: A feasible line concept (L, f ), if one exists.Step 1. Set L = ∅, fl := 0 for all l ∈ L0.Step 2. If for all e ∈ E :
∑l∈L:e∈ fl ≥ f min
e stop. Output: (L0, f ) isa feasible line concept.Otherwise take some e ∈ E with
∑l∈L:e∈ fl < f min
e
Step 3. If there is a line l ∈ L0 with e ∈ l define L := L ∪ {l},fl := f min
e and goto Step 2.Otherwise stop. No feasible line concept exists.
Anita Schöbel (NAM) 27. September 2006 12 / 78
Cost-oriented models
Cost-oriented models
Anita Schöbel (NAM) 27. September 2006 13 / 78
Cost-oriented models
Cost-oriented models
Anita Schöbel (NAM) 27. September 2006 13 / 78
Cost-oriented models Simple cost model
Contents
1 Line planning in public transportation
2 Basic model
3 Cost-oriented modelsSimple cost modelAlgorithmsExtended cost model
4 Customer-oriented modelsDirect travelers approachMinimizing traveling time
5 Other (more recent) models
Anita Schöbel (NAM) 27. September 2006 14 / 78
Cost-oriented models Simple cost model
The costs of a line concept
costs of a line concept (L, f ): ∑l∈L
cost l
where the costs cost l of a line l ∈ L0 depend onthe frequency of lthe length of l ,the time needed for a complete trip along line l ,and the costs per kilometer and per minute driving
(Fixed costs are neglected.)
→ How can cost l be estimated ?
Anita Schöbel (NAM) 27. September 2006 15 / 78
Cost-oriented models Simple cost model
The costs of a line concept
costs of a line concept (L, f ): ∑l∈L
cost l
where the costs cost l of a line l ∈ L0 depend onthe frequency of lthe length of l ,the time needed for a complete trip along line l ,and the costs per kilometer and per minute driving
(Fixed costs are neglected.)
→ How can cost l be estimated ?
Anita Schöbel (NAM) 27. September 2006 15 / 78
Cost-oriented models Simple cost model
Estimating the costs of a line
Notation
costtime = time-dependent costs for running a vehicle (per minute)costkm = distance-dependent costs for running a vehicle (per km)Lengthl = the length of line l (in kilometers)Timel = time needed for driving a complete run of line l.
The costs are dependent on the following questions:1 How many vehicles are necessary to run line l?2 How much time is spent by all vehicles serving line l within I?3 How many kilometers are driven by all vehicles on line l within I?
Then: cost l = Time on line l ∗costtime + Kilometers on line l ∗costkm.
Anita Schöbel (NAM) 27. September 2006 16 / 78
Cost-oriented models Simple cost model
Estimating the costs of a line
Notation
costtime = time-dependent costs for running a vehicle (per minute)costkm = distance-dependent costs for running a vehicle (per km)Lengthl = the length of line l (in kilometers)Timel = time needed for driving a complete run of line l.
The costs are dependent on the following questions:1 How many vehicles are necessary to run line l?2 How much time is spent by all vehicles serving line l within I?3 How many kilometers are driven by all vehicles on line l within I?
Then: cost l = Time on line l ∗costtime + Kilometers on line l ∗costkm.
Anita Schöbel (NAM) 27. September 2006 16 / 78
Cost-oriented models Simple cost model
Estimating the costs of a line
Notation
costtime = time-dependent costs for running a vehicle (per minute)costkm = distance-dependent costs for running a vehicle (per km)Lengthl = the length of line l (in kilometers)Timel = time needed for driving a complete run of line l.
The costs are dependent on the following questions:1 How many vehicles are necessary to run line l?2 How much time is spent by all vehicles serving line l within I?3 How many kilometers are driven by all vehicles on line l within I?
Then: cost l = Time on line l ∗costtime + Kilometers on line l ∗costkm.
Anita Schöbel (NAM) 27. September 2006 16 / 78
Cost-oriented models Simple cost model
How many vehicles are needed?
Special case: Timel = I. Then the number of vehicles needed for l is fl .
For arbitrary Timel :
number of vehicles on line l = dTimel · flI
e
due to rule of proportion.
Example: I = 60 min, Timel = 120 min, fl = 4 =⇒ 8 vehicles needed.
Anita Schöbel (NAM) 27. September 2006 17 / 78
Cost-oriented models Simple cost model
How many vehicles are needed?
Special case: Timel = I. Then the number of vehicles needed for l is fl .
For arbitrary Timel :
number of vehicles on line l = dTimel · flI
e
due to rule of proportion.
Example: I = 60 min, Timel = 120 min, fl = 4 =⇒ 8 vehicles needed.
Anita Schöbel (NAM) 27. September 2006 17 / 78
Cost-oriented models Simple cost model
How many vehicles are needed?
Special case: Timel = I. Then the number of vehicles needed for l is fl .
For arbitrary Timel :
number of vehicles on line l = dTimel · flI
e
due to rule of proportion.
Example: I = 60 min, Timel = 120 min, fl = 4
=⇒ 8 vehicles needed.
Anita Schöbel (NAM) 27. September 2006 17 / 78
Cost-oriented models Simple cost model
How many vehicles are needed?
Special case: Timel = I. Then the number of vehicles needed for l is fl .
For arbitrary Timel :
number of vehicles on line l = dTimel · flI
e
due to rule of proportion.
Example: I = 60 min, Timel = 120 min, fl = 4 =⇒ 8 vehicles needed.
Anita Schöbel (NAM) 27. September 2006 17 / 78
Cost-oriented models Simple cost model
How much time is spent on line l?
total time spent on l within I = number of vehicles on line l · I
=Timel · fl
I∗ I
= fl Timel
(neglecting the rounding in the number of vehicles)
Anita Schöbel (NAM) 27. September 2006 18 / 78
Cost-oriented models Simple cost model
How many kilometers are spent on line l?
kilometers on line l within I
= number of vehicles on line l · Lengthl ·I
Timel
=Timel · fl
I· Lengthl ·
ITimel
= fl Lengthl
Anita Schöbel (NAM) 27. September 2006 19 / 78
Cost-oriented models Simple cost model
Finally: The costs of line l
cost l = Time on line l · costtime + Kilometers on line l · costkm
= Timel fl costtime + Lengthl fl costkm
= fl (Timel costtime + Lengthl costkm︸ ︷︷ ︸=:costl
)
= fl costl .
Note: costl , defined by the equation above, does not depend on thefrequency fl .
Anita Schöbel (NAM) 27. September 2006 20 / 78
Cost-oriented models Simple cost model
Finally: The costs of line l
cost l = Time on line l · costtime + Kilometers on line l · costkm
= Timel fl costtime + Lengthl fl costkm
= fl (Timel costtime + Lengthl costkm︸ ︷︷ ︸=:costl
)
= fl costl .
Note: costl , defined by the equation above, does not depend on thefrequency fl .
Anita Schöbel (NAM) 27. September 2006 20 / 78
Cost-oriented models Simple cost model
Cost-oriented model
(LP1) (Cost-oriented line concept)
Given a PTN, a set L0 of potential lines, lower and upper frequenciesf mine ≤ f max
e for all e ∈ E , and parameters costl for all l ∈ L0, find afeasible line concept (L, f ) with minimal overall costs
cost(L, f ) =∑l∈L
fl costl .
Anita Schöbel (NAM) 27. September 2006 21 / 78
Cost-oriented models Simple cost model
Complexity of (LP1)
Theorem(LP1) is NP-hard.
Proof: Special case of (LP0)
Note that (LP0) is trivially solvable without upper frequencies.What about this case in the cost model?
Theorem
(LP1) is NP-hard, even without considering upper frequencies (i.e.without constraints (2)) and with costl = 1 for all l ∈ L0 and f min
e = 1 forall e ∈ E.
Anita Schöbel (NAM) 27. September 2006 22 / 78
Cost-oriented models Simple cost model
Complexity of (LP1)
Theorem(LP1) is NP-hard.
Proof: Special case of (LP0)
Note that (LP0) is trivially solvable without upper frequencies.What about this case in the cost model?
Theorem
(LP1) is NP-hard, even without considering upper frequencies (i.e.without constraints (2)) and with costl = 1 for all l ∈ L0 and f min
e = 1 forall e ∈ E.
Anita Schöbel (NAM) 27. September 2006 22 / 78
Cost-oriented models Simple cost model
Complexity of (LP1)
Theorem(LP1) is NP-hard.
Proof: Special case of (LP0)
Note that (LP0) is trivially solvable without upper frequencies.What about this case in the cost model?
Theorem
(LP1) is NP-hard, even without considering upper frequencies (i.e.without constraints (2)) and with costl = 1 for all l ∈ L0 and f min
e = 1 forall e ∈ E.
Anita Schöbel (NAM) 27. September 2006 22 / 78
Cost-oriented models Simple cost model
Integer programming formulation
variables: fl
min∑l∈L0
fl costl
s.t.∑
l∈L0,e∈l
fl ≥ f mine ∀ e ∈ E (1)
∑l∈L0,e∈l
fl ≤ f maxe ∀ e ∈ E (2)
fl ∈ IN0.
Note: solution fl for all l ∈ L0, then line concept (L, f ) is given through
L = {l ∈ L0 : fl > 0}.
Anita Schöbel (NAM) 27. September 2006 23 / 78
Cost-oriented models Simple cost model
Cost model without upper frequencies
Observations: (LP1) without upper frequencies is a multi coveringproblem.
(MC) (Multi covering problem):
Given an M × N-matrix A = (aij) with elements aij ∈ {0, 1}, a vectorb ∈ IN0
M and some integer K , does there exist x ∈ IN0N with Ax ≥ b
and∑N
n=1 xn ≤ K ?
Anita Schöbel (NAM) 27. September 2006 24 / 78
Cost-oriented models Algorithms
Contents
1 Line planning in public transportation
2 Basic model
3 Cost-oriented modelsSimple cost modelAlgorithmsExtended cost model
4 Customer-oriented modelsDirect travelers approachMinimizing traveling time
5 Other (more recent) models
Anita Schöbel (NAM) 27. September 2006 25 / 78
Cost-oriented models Algorithms
Algorithms
General idea:for (LP1-Pool) without upper frequencies
Basic idea: Let f mine be the required frequency for edge e.
1 While there is an uncovered edge e (i.e. with f mine > 0)
2 Choose a line l , such that g(l) minimal3 Choose frequency of line as fl := f (l)4 Update f min
e accordingly
Anita Schöbel (NAM) 27. September 2006 26 / 78
Cost-oriented models Algorithms
Applying Dobson, 1982
1 While there is an uncovered edge e (i.e. with f mine > 0)
2 Choose a line l , such that g(l) minimal
3 Choose frequency of line as fl := f (l)
4 Update f mine accordingly
withg(l) = cl
|{e∈l:f mine >0}|
f (l) = fl + 1
Theorem
HEU ≤ OPT H(maxl |l |), where H(d) =∑d
i=11i
Anita Schöbel (NAM) 27. September 2006 27 / 78
Cost-oriented models Algorithms
Applying Dobson, 1982
1 While there is an uncovered edge e (i.e. with f mine > 0)
2 Choose a line l , such that g(l) minimal
3 Choose frequency of line as fl := f (l)
4 Update f mine accordingly
withg(l) = cl
|{e∈l:f mine >0}|
f (l) = fl + 1
Theorem
HEU ≤ OPT H(maxl |l |), where H(d) =∑d
i=11i
Anita Schöbel (NAM) 27. September 2006 27 / 78
Cost-oriented models Algorithms
Applying van Slyke/Xiaoming, 1984
1 While there is an uncovered edge e (i.e. with f mine > 0)
2 Choose edge e with maximal f mine
3 Choose a line l , covering e such that g(l) minimal
4 Choose frequency of line fl = f (l)
5 Update f mine accordingly
withg(l) = cl
f (l) = fl + f mine
TheoremHEU ≤ OPT maxe∈E |L(e)|
Anita Schöbel (NAM) 27. September 2006 28 / 78
Cost-oriented models Algorithms
Applying van Slyke/Xiaoming, 1984
1 While there is an uncovered edge e (i.e. with f mine > 0)
2 Choose edge e with maximal f mine
3 Choose a line l , covering e such that g(l) minimal
4 Choose frequency of line fl = f (l)
5 Update f mine accordingly
withg(l) = cl
f (l) = fl + f mine
TheoremHEU ≤ OPT maxe∈E |L(e)|
Anita Schöbel (NAM) 27. September 2006 28 / 78
Cost-oriented models Algorithms
Line planning heuristic
1 While there is an uncovered edge e (i.e. with f mine > 0)
2 Choose edge e with maximal f mine
3 Choose a line l , covering e such that g(l) minimal
4 Choose frequency of line fl = f (l)
5 Update f mine accordingly
withg(l) = cl
|{e∈l:f mine >0}|mine∈l:fmin
e >0 f mine
f (l) = fl + mine∈l:f mine >0 f min
e
Anita Schöbel (NAM) 27. September 2006 29 / 78
Cost-oriented models Algorithms
Two combinations
1 While there is an uncovered edge e (i.e. with f mine > 0)
2 Choose edge e with maximal f mine
3 Choose a line l , covering e such that g(l) minimal
4 Choose frequency of line fl = f (l)
5 Update f mine accordingly
withg(l) = cl
|{e∈l:f mine >0}|mine∈l:fmin
e >0 f mine
f (l) = fl + 1or
g(l) = cl|{e∈l:f min
e >0}|
f (l) = fl + mine∈l f mine
Anita Schöbel (NAM) 27. September 2006 30 / 78
Cost-oriented models Algorithms
Numerical results
Data:Simulated data with
1000 stations,4000 edges and30, 60, 90, 120, . . . , 1350, 1500, 3000 possible lines
Algorithms tested on 500 instances.
Anita Schöbel (NAM) 27. September 2006 31 / 78
Cost-oriented models Algorithms
Numerical results
sizeHeuristic 30 60 90 120 150 180 210 240 270 300
Dobson 620 1470 2060 2720 3340 4090 4730 5640 6100 6880
Sly/Xia 880 2640 4290 6210 8500 10680 13440 16330 17330 19260
LiPla 650 1520 2150 2730 3490 4220 4980 5800 6430 7130
Comb 1 610 1390 1950 2620 3190 3940 4610 5500 6020 6700
Comb 2 580 1380 1910 2640 3220 3910 4690 5450 6240 6830
Anita Schöbel (NAM) 27. September 2006 31 / 78
Cost-oriented models Algorithms
Numerical results
540 600 750 900 1200 1350 1500 3000
12810 14470 18520 22560 30760 35000 39960 85100
40980 47050 61240 75490 111030 125130 143430 318920
12990 14650 18430 22470 30800 35020 39890 85310
11130 12580 14080 18030 22020 30360 34680 84950
12730 14300 18330 22180 30380 34710 39180 84890
Anita Schöbel (NAM) 27. September 2006 31 / 78
Cost-oriented models Algorithms
Results of heuristics
Winner: The two combinations (where the first combination isslightly better)then Dobson and line planing heuristicFar away: van Slyke and Xiaoming
Anita Schöbel (NAM) 27. September 2006 32 / 78
Cost-oriented models Extended cost model
Contents
1 Line planning in public transportation
2 Basic model
3 Cost-oriented modelsSimple cost modelAlgorithmsExtended cost model
4 Customer-oriented modelsDirect travelers approachMinimizing traveling time
5 Other (more recent) models
Anita Schöbel (NAM) 27. September 2006 33 / 78
Cost-oriented models Extended cost model
Extended cost model
Evaluate costs of line conceptDetermine the lines and the frequenciesDetermine also the type of vehicle t = 1, . . . , T which is used torun a lineIn rail transportation: determine also the number of cars of thetrains for each line
Anita Schöbel (NAM) 27. September 2006 34 / 78
Cost-oriented models Extended cost model
Additional parameters
ccost tfix = fixed cost per car of type t
ccost tkm = cost per kilometer with a car of type t
vcost tkm = cost per kilometer with a vehicle of type tAt = capacity of a car of type t
cmint = min number of cars allowed for a train of type t
cmaxt = max number of cars allowed for a train of type t
Anita Schöbel (NAM) 27. September 2006 35 / 78
Cost-oriented models Extended cost model
Recall . . .
Number of vehicles needed to run line l is
d fl Timel
Ie.
Moreover, we definef max = max
e∈Ef maxe
as the upper bound over all upper frequencies.
Anita Schöbel (NAM) 27. September 2006 36 / 78
Cost-oriented models Extended cost model
Objective function
variables:
X tcl =
{1 if line l is served by vehicles of type t with c cars0 otherwise
.
min∑l∈L0
T∑t=1
cmaxt∑
c=cmint
(ccost tfix
Timel
Ic + Lengthl ccost t
km c
+Lengthl vcost tkm)fl X tc
l
Anita Schöbel (NAM) 27. September 2006 37 / 78
Cost-oriented models Extended cost model
But:
A quadratic term fl X tcl in the objective!
Solution: Substitute fl X tcl by a new variable X tfc
l with
X tfcl =
1 if line l ∈ L0 is served by vehicles of type t
with c cars and frequency f0 otherwise
Anita Schöbel (NAM) 27. September 2006 38 / 78
Cost-oriented models Extended cost model
Extended cost model (LP2)
min∑l∈L0
T∑t=1
f max∑f=1
cmaxt∑
c=cmint
(ccost tfix · d
Timel
I· f e · c + Lengthl · ccost t
km · f · c
+ Lengthl · vcost tkm · f ) · X tfc
l
s.t.∑
l∈L0:e∈l
T∑t=1
f max∑f=1
cmaxt∑
c=cmint
At · f · c · X tfcl ≥ we ∀ e ∈ E (3)
f mine ≤
∑l∈L0:e∈l
T∑t=1
f max∑f=1
cmaxt∑
c=cmint
f · X tfcl ≤ f max
e ∀ e ∈ E (4)
f max∑f=1
cmaxt∑
c=cmint
X tfcl ≤ 1 ∀l ∈ L0,∀t = 1, . . . , T (5)
X tfcl ∈ {0, 1} ∀l , f , t , c (6)
Anita Schöbel (NAM) 27. September 2006 39 / 78
Cost-oriented models Extended cost model
Compare (LP1) with (LP2)
The main changes of (LP2) compared to (LP1) are the following:The type of vehicles and the number of cars of the vehicles isdetermined within (LP2).The approximation of the costs is more detailed by having the newparameters.Constraints (3) are needed to make sure that all passengers canbe transported.(LP2) is NP hard as special case of (LP1).
Anita Schöbel (NAM) 27. September 2006 40 / 78
Cost-oriented models Extended cost model
Solving (LP2)
see Claessens, M.T. and van Dijk, N.M. and Zwaneveld, P.J., EJOR,1996
get rid of many of the X tfcl variables
use commercial IP-solver
Anita Schöbel (NAM) 27. September 2006 41 / 78
Cost-oriented models Extended cost model
Literature
Claessens (1994), Zwaneveld, Claessens, van Dijk (1996), Claessens,van Dijk, Zwaneveld (1996), Bussieck and Zimmermann (1997),Claessens, van Dijk and Zwaneveld (1998), Goessens, Hoesel, andKroon (2001, 2002), Bussieck, Lindner, and Lübbecke (2003)
Anita Schöbel (NAM) 27. September 2006 42 / 78
Customer-oriented models
Customer-oriented models
Anita Schöbel (NAM) 27. September 2006 43 / 78
Customer-oriented models
Customer-oriented models
Anita Schöbel (NAM) 27. September 2006 43 / 78
Customer-oriented models Direct travelers approach
Contents
1 Line planning in public transportation
2 Basic model
3 Cost-oriented modelsSimple cost modelAlgorithmsExtended cost model
4 Customer-oriented modelsDirect travelers approachMinimizing traveling time
5 Other (more recent) models
Anita Schöbel (NAM) 27. September 2006 44 / 78
Customer-oriented models Direct travelers approach
Goal
NotationGiven a line concept (L, fl). A customer does not change the line onhis/her journey is called a direct traveler
Goal of direct travelers approach:design the lines in such a way that as many customers as possiblehave a direct connection, i.e. maximize the number of direct travelers.
Anita Schöbel (NAM) 27. September 2006 45 / 78
Customer-oriented models Direct travelers approach
Preferable paths
Assumption: Customers use preferable paths (e.g. shortest paths)which are known beforehand for each OD-pair i , j ∈ V × V .
Notation
Pij denotes a shortest (or preferable) path between station i andstation j.Pij ⊆ l if there exists a shortest (or preferable) path between i andj which is completely contained in line l.
Anita Schöbel (NAM) 27. September 2006 46 / 78
Customer-oriented models Direct travelers approach
ExampleGraph with 5 nodes, all edge weights are 1.
2
34 5
1
l
line l=(1,2,3,4,5)
Anita Schöbel (NAM) 27. September 2006 47 / 78
Customer-oriented models Direct travelers approach
ExampleGraph with 5 nodes, all edge weights are 1.
2
34 5
1 P contained in line l23
l
line l=(1,2,3,4,5)
Anita Schöbel (NAM) 27. September 2006 47 / 78
Customer-oriented models Direct travelers approach
ExampleGraph with 5 nodes, all edge weights are 1.
2
34 5
1 P contained in line l
l
14
line l=(1,2,3,4,5)
Anita Schöbel (NAM) 27. September 2006 47 / 78
Customer-oriented models Direct travelers approach
ExampleGraph with 5 nodes, all edge weights are 1.
2
34 5
1
l
15P NOT contained in line l
line l=(1,2,3,4,5)
Anita Schöbel (NAM) 27. September 2006 47 / 78
Customer-oriented models Direct travelers approach
Formulation of objective
Variables:dijl for all i , j ∈ V and all lines l ∈ L0,
denotes the number of direct travelers between i and j that use linel ∈ L0.
Objective:max
∑i,j∈V ,l∈L0:Pij⊆l
dijl
Anita Schöbel (NAM) 27. September 2006 48 / 78
Customer-oriented models Direct travelers approach
Formulation of objective
Variables:dijl for all i , j ∈ V and all lines l ∈ L0,
denotes the number of direct travelers between i and j that use linel ∈ L0.
Objective:max
∑i,j∈V ,l∈L0:Pij⊆l
dijl
Anita Schöbel (NAM) 27. September 2006 48 / 78
Customer-oriented models Direct travelers approach
Recall . . .
A denotes the (fixed) capacity of a vehicle.Wij denotes the number of travelers between station i and stationj .
Anita Schöbel (NAM) 27. September 2006 49 / 78
Customer-oriented models Direct travelers approach
Model(LP3)
max∑l∈L0
∑i,j∈V :
Pij⊆l
dijl
s.t.∑l∈L0:Pij⊆l
dijl ≤ Wij for all i , j ∈ V (7)
∑i,j∈V :
e∈Pij⊆l
dijl ≤ A · fl for all e ∈ E , l ∈ L0 (8)
f mine ≤
∑l∈L0:e∈l
fl ≤ f maxe for all e ∈ E (9)
dijl , fl ∈ IN0 for all i , j ∈ V , l ∈ L0
Anita Schöbel (NAM) 27. September 2006 50 / 78
Customer-oriented models Direct travelers approach
Complexity of (LP3)
Theorem(LP3) is NP-hard.
Proof: For A and Wij sufficiently large, the feasible set of (LP3)coincides with (LP0), hence (LP3) is NP-hard.
Anita Schöbel (NAM) 27. September 2006 51 / 78
Customer-oriented models Direct travelers approach
Simplifying (LP3)
Problem: O(|V |2 · |L0|) variables.Idea: aggregate the dijl variables to Dij =
∑l∈L0:Pij⊆l dijl .
(LP3’)
max∑i,j∈V
Dij
s.t. Dij ≤ Wij for all i , j ∈ V
Dij ≤ A∑
l∈L0:Pij⊆l
fl for all i , j ∈ V
f mine ≤
∑l∈L0:e∈l
fl ≤ f maxe for all e ∈ E
Dij , fe ∈ IN0 for all i , j ∈ V , e ∈ E
Anita Schöbel (NAM) 27. September 2006 52 / 78
Customer-oriented models Direct travelers approach
Simplifying (LP3)
Problem: O(|V |2 · |L0|) variables.Idea: aggregate the dijl variables to Dij =
∑l∈L0:Pij⊆l dijl .
(LP3’)
max∑i,j∈V
Dij
s.t. Dij ≤ Wij for all i , j ∈ V
Dij ≤ A∑
l∈L0:Pij⊆l
fl for all i , j ∈ V
f mine ≤
∑l∈L0:e∈l
fl ≤ f maxe for all e ∈ E
Dij , fe ∈ IN0 for all i , j ∈ V , e ∈ E
Anita Schöbel (NAM) 27. September 2006 52 / 78
Customer-oriented models Direct travelers approach
Relation between (LP3) and (LP3’)
LemmaThe optimal objective value of (LP3’) is an upper bound for the optimalobjective value of (LP3).
feas. sol.
feas. sol.of (LP3)
of (LP3’)
I.e. (LP3’) is a relaxation of (LP3).
Anita Schöbel (NAM) 27. September 2006 53 / 78
Customer-oriented models Direct travelers approach
Relation between (LP3) and (LP3’)
LemmaThe optimal objective value of (LP3’) is an upper bound for the optimalobjective value of (LP3).
feas. sol.
feas. sol.of (LP3)
of (LP3’)
I.e. (LP3’) is a relaxation of (LP3).
Anita Schöbel (NAM) 27. September 2006 53 / 78
Customer-oriented models Direct travelers approach
Solving (LP3’)
see M.R. Bussieck and P. Kreuzer and U.T. Zimmermann, EJOR, 1996further relax integrality constraints of the Dij variables of (LP3’)use a cutting-plane approach
Anita Schöbel (NAM) 27. September 2006 54 / 78
Customer-oriented models Minimizing traveling time
Contents
1 Line planning in public transportation
2 Basic model
3 Cost-oriented modelsSimple cost modelAlgorithmsExtended cost model
4 Customer-oriented modelsDirect travelers approachMinimizing traveling time
5 Other (more recent) models
Anita Schöbel (NAM) 27. September 2006 55 / 78
Customer-oriented models Minimizing traveling time
Goal
Observation:Customers choose a path P with minimal “traveling time” I(P),including
riding time on trains (proportional to length of trip)and time for transfers (dependent on number of transfers).
Goal:Design the lines in such a way that the sum of all traveling times overall customers is minimal.
Anita Schöbel (NAM) 27. September 2006 56 / 78
Customer-oriented models Minimizing traveling time
Goal
Observation:Customers choose a path P with minimal “traveling time” I(P),including
riding time on trains (proportional to length of trip)and time for transfers (dependent on number of transfers).
Goal:Design the lines in such a way that the sum of all traveling times overall customers is minimal.
Anita Schöbel (NAM) 27. September 2006 56 / 78
Customer-oriented models Minimizing traveling time
Illustration
Examle:
Anita Schöbel (NAM) 27. September 2006 57 / 78
Customer-oriented models Minimizing traveling time
IllustrationMinimizing riding times . . .
Anita Schöbel (NAM) 27. September 2006 57 / 78
Customer-oriented models Minimizing traveling time
Illustration
Minimize number of transfers . . .
Anita Schöbel (NAM) 27. September 2006 57 / 78
Customer-oriented models Minimizing traveling time
Objective function
Idea: Take both efects into account!
Inconvenience=k1· Riding Time + k2· number of transfers
K2 is an estimate for the average waiting time when changing (sincetimetable is not known in the phase of line planning).
Anita Schöbel (NAM) 27. September 2006 58 / 78
Customer-oriented models Minimizing traveling time
Travel-time model
(LP4) (travel-time line concept)
Given a PTN, a set L0 of potential lines, costl for all l ∈ L0, budget Band an OD-matrix Wstfind a line concept (L, f ) with
∑l∈L flcostl ≤ B and paths Pst for all OD-
pairs (s, t) minimizing the sum of all inconveniences,
min∑
s,t∈V
Wst I(Pst).
Anita Schöbel (NAM) 27. September 2006 59 / 78
Customer-oriented models Minimizing traveling time
Relation to direct travelers approach
Note: The minimal number of transfers needs not be the same as themaximal number of direct passengers!
Example:
Solution 1: 4 direct passengers, 6 transfersSolution 2: 3 direct passengers, 4 transfers
Anita Schöbel (NAM) 27. September 2006 60 / 78
Customer-oriented models Minimizing traveling time
Relation to direct travelers approachNote: The minimal number of transfers needs not be the same as themaximal number of direct passengers!
Example:
Solution 1: 4 direct passengers, 6 transfersSolution 2: 3 direct passengers, 4 transfers
Anita Schöbel (NAM) 27. September 2006 60 / 78
Customer-oriented models Minimizing traveling time
Relation to direct travelers approachNote: The minimal number of transfers needs not be the same as themaximal number of direct passengers!
Example:
2
3
2
Solution 1: 4 direct passengers, 6 transfers
Solution 2: 3 direct passengers, 4 transfers
Anita Schöbel (NAM) 27. September 2006 60 / 78
Customer-oriented models Minimizing traveling time
Relation to direct travelers approachNote: The minimal number of transfers needs not be the same as themaximal number of direct passengers!
Example:
2
3
2
Solution 1: 4 direct passengers, 6 transfersSolution 2: 3 direct passengers, 4 transfers
Anita Schöbel (NAM) 27. September 2006 60 / 78
Customer-oriented models Minimizing traveling time
Complexity of (LP4)
Theorem(LP4) is NP complete, even
if only the number of transfers is counted in the objectivethe network is a linear graphand all costs are equal to one.
Proof: Reduction to set covering
s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s6 t6
l1 l2
l3
l4
Anita Schöbel (NAM) 27. September 2006 61 / 78
Customer-oriented models Minimizing traveling time
Complexity of (LP4)
Theorem(LP4) is NP complete, even
if only the number of transfers is counted in the objectivethe network is a linear graphand all costs are equal to one.
Proof: Reduction to set covering
s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s6 t6
l1 l2
l3
l4
Anita Schöbel (NAM) 27. September 2006 61 / 78
Customer-oriented models Minimizing traveling time
The change & go graph
s1 s2s3
s4
s5s6
s7
s8
l2
l3
l1
Anita Schöbel (NAM) 27. September 2006 62 / 78
Customer-oriented models Minimizing traveling time
The change & go graph
l1
s2,l1 s3,l1 s4,l1s1,l1
Anita Schöbel (NAM) 27. September 2006 62 / 78
Customer-oriented models Minimizing traveling time
The change & go graph
l2
l3
l1
s1,l2
s5,l2
s4,l2
s8,l3
s6,l2
s6,l3s7,l3
s2,l1 s3,l1 s4,l1s1,l1
s3,l3
Anita Schöbel (NAM) 27. September 2006 62 / 78
Customer-oriented models Minimizing traveling time
The change & go graph
s1 s2s3
s4
s5s6
s7
s8
l2
l3
l1
Anita Schöbel (NAM) 27. September 2006 62 / 78
Customer-oriented models Minimizing traveling time
The change & go graph
l2
l3
l1
s1,l2
s5,l2
s4,l2
s8,l3
s6,l2
s6,l3s7,l3
s2,l1 s3,l1 s4,l1s1,l1
s3,l3
Anita Schöbel (NAM) 27. September 2006 62 / 78
Customer-oriented models Minimizing traveling time
The change & go graph
l2
l3
l1
s1,l2
s5,l2
s4,l2
s8,l3
s1, 0
s6,l2
s6,l3s7,l3
s2,l1 s3,l1 s4,l1s1,l1
s3,l3
Anita Schöbel (NAM) 27. September 2006 62 / 78
Customer-oriented models Minimizing traveling time
The change & go graph
l2
l3
l1
s1,l2
s5,l2
s4,l2
s8,l3
s7,0
s1, 0
s6,0 s8,0
s4,0
s6,l2
s6,l3s7,l3
s2,l1 s3,l1 s4,l1s1,l1
s3,l3
s5,0
s3,0s2,0
Result: Change & Go Graph N = (E ,A)
Anita Schöbel (NAM) 27. September 2006 62 / 78
Customer-oriented models Minimizing traveling time
Advantages of the model
paths with minimal traveling time can be calculated as shortestpathsvery flexible due to different possibilities for the weights ca for allactivities a ∈ A:
I weight all arcs by their (estimated) durationsminimize traveling time
I weight changing arcs by 1, all others by zerominimize number of changes
I weight changing arcs by zero, others by their lengthsminimize length of trip (and hence costs)
Anita Schöbel (NAM) 27. September 2006 63 / 78
Customer-oriented models Minimizing traveling time
IP-FormulationVariables:
xast =
{1 if activity a ∈ A is used on a shortest path from s to t in N0 otherwise
yl =
{1 if line l is established0 otherwise
Parameters: Θ as node-arc-incidence matrix of N ,
bist =
1 if i = (s, 0)
−1 if i = (t , 0)0 otherwise
LemmaAny solution xst ∈ {0, 1}|A| of Θ xst = bst is a path from s to t in N
Anita Schöbel (NAM) 27. September 2006 64 / 78
Customer-oriented models Minimizing traveling time
IP-Formulation
min∑
s,t∈V∑
a∈A Wstcaxast
s.t. xast ≤ yl for all s, t ∈ V , l ∈ L, a ∈ l
Θxst = bst for all s, t ∈ V with Wst > 0∑l∈L ylcostl ≤ B
xast , yl ∈ {0, 1}
this model assumes unlimited capacity of the vehicleswith limited capacity A of the trains:
relax xast and fl = yl to integers and replace
xast ≤ yl by
∑s,t∈V
xast ≤ flA for all l ∈ L, a ∈ l
and Θxst = bst is a network flow problem.
Anita Schöbel (NAM) 27. September 2006 65 / 78
Customer-oriented models Minimizing traveling time
IP-Formulation
min∑
s,t∈V∑
a∈A Wstcaxast
s.t. xast ≤ yl for all s, t ∈ V , l ∈ L, a ∈ l
Θxst = bst for all s, t ∈ V with Wst > 0∑l∈L ylcostl ≤ B
xast , yl ∈ {0, 1}
this model assumes unlimited capacity of the vehicleswith limited capacity A of the trains:
relax xast and fl = yl to integers and replace
xast ≤ yl by
∑s,t∈V
xast ≤ flA for all l ∈ L, a ∈ l
and Θxst = bst is a network flow problem.
Anita Schöbel (NAM) 27. September 2006 65 / 78
Customer-oriented models Minimizing traveling time
IP-Formulation
min∑
s,t∈V∑
a∈A Wstcaxast
s.t. xast ≤ yl for all s, t ∈ V , l ∈ L, a ∈ l
Θxst = bst for all s, t ∈ V with Wst > 0∑l∈L ylcostl ≤ B
xast , yl ∈ {0, 1}
this model assumes unlimited capacity of the vehicleswith limited capacity A of the trains:
relax xast and fl = yl to integers and replace
xast ≤ yl by
∑s,t∈V
xast ≤ flA for all l ∈ L, a ∈ l
and Θxst = bst is a network flow problem.
Anita Schöbel (NAM) 27. September 2006 65 / 78
Customer-oriented models Minimizing traveling time
Block structure of (LP4)
min∑
s,t∈V∑
a∈A Wstcaxast
s.t. xast −yl ≤ 0 for all s, t ∈ V , l ∈ L, a ∈ l
Θxst = bst for all s, t ∈ V with Wst > 0∑l∈L ylcostl ≤ B
xast , yl ∈ {0, 1}
Consequence: one block for each OD-pair s, t and a y -variable block.
Anita Schöbel (NAM) 27. September 2006 66 / 78
Customer-oriented models Minimizing traveling time
Block structure of (LP4)
min∑
s,t∈V∑
a∈A Wstcaxast
s.t. xast −yl ≤ 0 for all s, t ∈ V , l ∈ L, a ∈ l
Θxs1t1. . .
Θxsr tr
=bs1t1. . .bsr tr∑
l∈L ylcostl ≤ Bxa
st , yl ∈ {0, 1}
Consequence: one block for each OD-pair s, t and a y -variable block.
Anita Schöbel (NAM) 27. September 2006 66 / 78
Customer-oriented models Minimizing traveling time
Variants of the coupling constraints
(VAR1)
min∑
s,t∈V∑
a∈A Wstcaxast
s.t. xast ≤ yl for all s, t ∈ V , l ∈ L, a ∈ l
Θxst = bst for all s, t ∈ V with Wst > 0∑l∈L ylcostl ≤ B
xast , yl ∈ {0, 1}
The coupling constraints can equivalently replaced by:
(VAR2)∑
a∈Al xast ≤ |Al |yl ∀ l ∈ L, (s, t) ∈ R
(VAR3)∑
(s,t)∈R xast ≤ |R|yl ∀ l ∈ L, a ∈ Al
(VAR4)∑
(s,t)∈R∑
a∈Al xast ≤ |R||Al |yl ∀ l ∈ L
Anita Schöbel (NAM) 27. September 2006 67 / 78
Customer-oriented models Minimizing traveling time
How strong are the relaxations?
(VAR1) xast ≤ yl ∀ (s, t) ∈ R, a ∈ Al : l ∈ L
(VAR2)∑
a∈Al xast ≤ |Al |yl ∀ l ∈ L, (s, t) ∈ R
(VAR3)∑
(s,t)∈R xast ≤ |R|yl ∀ l ∈ L, a ∈ Al
(VAR4)∑
(s,t)∈R∑
a∈Al xast ≤ |R||Al |yl ∀ l ∈ L
LPP2 LPP3LPP1
LPP4
Anita Schöbel (NAM) 27. September 2006 68 / 78
Customer-oriented models Minimizing traveling time
Solving (LP4)
see Schöbel and Scholl, DROPS, 2006excluding trivial solutionsDantzig Wolfe decomposition in different variantsBranch and Price
Anita Schöbel (NAM) 27. September 2006 69 / 78
Customer-oriented models Minimizing traveling time
LiteratureMaximize number of direct travelers with respect to upperline frequency requirements:Patz (1925), Wegel (1974), Dienst (1970) Reinecke (1992) andReinecke (1995), Bussieck, Kreuzer and Zimmermann (1996)Bussieck and Zimmermann (1997), Zimmermann, Bussieck,Krista and Wiegand (1997), Bussieck (1998)Maximize number of direct travelers w.r.t. budget constraint:Simonis (1980 and 1981)Maximize number of travelers within a reasonable amount oftraveling time with respect to budget constraint: Laporte,Marin, Mesa, Ortega (2004)Minimize traveling time with respect to budget constraint:Schöbel and Scholl (2004), Borndörfer, Grötschel and Pfetsch(2005), Schneider (2005), Scholl (2006), Schöbel andScholl(2006)
Anita Schöbel (NAM) 27. September 2006 70 / 78
Other (more recent) models
Other recent approaches
Black-Box-Model of Deutsche BahnIntegrating the vehicle schedules in cooperation with GÖVBGame-theoretic approachPath-based models
Anita Schöbel (NAM) 27. September 2006 71 / 78
Other (more recent) models
Black-Box-Model
Line Concept
Profit
Anita Schöbel (NAM) 27. September 2006 72 / 78
Other (more recent) models
Black-Box-Model
Line Concept
Profit
Anita Schöbel (NAM) 27. September 2006 72 / 78
Other (more recent) models
Black-Box-Model
Line Concept
Costs
cost parameters
Profit
Anita Schöbel (NAM) 27. September 2006 72 / 78
Other (more recent) models
Black-Box-Model
potentialcustomers Line Concept
Costs
cost parameters
Profit
Anita Schöbel (NAM) 27. September 2006 72 / 78
Other (more recent) models
Black-Box-Model
potentialcustomers Line Concept
Costs
Modal Split
cost parametersreal number of
customers
Profit
Anita Schöbel (NAM) 27. September 2006 72 / 78
Other (more recent) models
Black-Box-Model
potentialcustomers Line Concept
CostsIncome
Modal Split
cost parameters
tariff system
real number ofcustomers
Profit
Anita Schöbel (NAM) 27. September 2006 72 / 78
Other (more recent) models
Black-Box-Model
potentialcustomers Line Concept
CostsIncome
Modal Split
cost parameters
tariff system
real number ofcustomers
Profit
Anita Schöbel (NAM) 27. September 2006 72 / 78
Other (more recent) models
Black-Box-Model
potentialcustomers Line Concept
CostsIncome
Modal Split
cost parameters
tariff system
real number ofcustomers
Profit
Anita Schöbel (NAM) 27. September 2006 72 / 78
Other (more recent) models
Black-Box-Model
Profit =∑
s,t∈V
customerss,t · Prices,t −∑l∈L
costl
Costs: similar as in cost modelsIncome: Determine number of customers and evaluate using theticket prices
Anita Schöbel (NAM) 27. September 2006 73 / 78
Other (more recent) models
Black-Box-ModelLet Wst be the number of potential travelers from s to t
W (m)st := number of travelers who will use the system,
if the number of transfers is less or equal to m.
number oftravelers
number of transfers
Anita Schöbel (NAM) 27. September 2006 74 / 78
Other (more recent) models
Integrating vehicle schedules
Project work of Michael Schachtebeck
in cooperation with GÖVBDer Stadtbus
Info-Telefon: 38 444 444www.goevb.de
Esebeck
HoltensenElliehausen
Grone
Rosdorf
Weende-Nord
Bovenden
Geismar
Weende
Zietenterrassen
Treuenhagen
Roringen
Herberhausen
Nikolausberg
Uni-Nord
Ostviertel
Klausberg
Papenberg
Knutbühren
Hetjershausen
Groß Ellershausen Grone-Süd
Leineberg
Industriegebiet
Holtenser Berg
Charlottenburger Straße
PlauenerStraße
Rosmarinweg
Theaterstraße
Friedrichstraße
Jüdenstraße
Geismartor
Kornmarkt
MarktGronerStraße
Weender Straße-WestWeender Straße-Ost
Nikolaistraße
Gotth
elf-Le
imba
ch-S
tr.Ro
bert-
Bosc
h-Brei
te
Geismar-Süd
Am Kalten BornAm Rischen
Kampstraße
Auf dem Paul
Geismar Landstraße
David-Hilbert-Straße
Rudo
lf-Wink
el-St
r.
GronerTor
Bahnhof
Landgericht
Grete-Henry-Straße
Hannah-Vogt-Straße
Ehrengard-Schramm-Weg
Alva-Myrdal-Weg
Alfred-Delp-Weg
Gulden-hagen
LönswegAm Steinsgraben
Keplerstraße
Bürgerstraße
Nikolausberger Weg
Am Papenberg
Faßberg
Burckhardtweg
Kellnerweg
Tammannstraße
GoldschmidtstraßePetrikirchstraße
Blauer Turm
PosthofStadt-
friedhofGottes-lager
Tulpenweg
Asternweg
Schützen-platz
NeuesRathaus
Schillerstraße
Reinholdstraße
Fritz-Reuter-Straße
Treuenhagen
Baumschulenweg Stadtstieg
Teichstraße
Vor dem Walde
Schöneberger Straße
Gehrenring
Adolf-Sievert-Straße
Stumpfe Eiche
Im Hassel
Helle-weg
Valentinsbreite
Humboldtallee
Bühlstraße
Theaterplatz
Lichtenwalder Straße Greitweg Auf dem Greite Saline Königsstieg
Söseweg
Allerstraße
Fuldaweg
Rosdorfer Weg
WiesenstraßeCramer-straße
Am Hasengraben
Leibniz-straße
Mitteldorfstraße
Kurmainzer Weg
Eislebener Weg
Pommerneck Stettiner Straße
Ortelsburger StraßeBreslauer
Straße Flüthenweg Görlitzer Straße
An der Tillyschanze
Himmelsruh
Weidenbreite MittelbergSpandauer
WegSanders-
beekTegeler
Weg
Pfalz-Grona-Breite Im Rinschenrott
Maschmühlenweg
Hildebrandstraße
Bahnhofsallee
Campus
Hennebergstraße
Adolf
-Hoy
er-St
r.
Wilh
elm-La
mbrec
ht-St
r.Flo
renz-S
artor
ius-S
tr.
Marien-straße
Otto-Frey-Brücke
Levin
park
Bornbreite
Twechte
Holtenser Landstraße
Lindenweg
Elliehäuser Weg
Domäne
Auf dem Hagen
Nußanger
Esebeck
Kleehöfen
Harrenacker
Talgraben
Lehnshof
Gerhard-Zillich-Straße
Am Burggraben
Gesundbrunnen
Am Eikborn
Am Anger
Hermann-Kolbe-Straße
Willi-Eichler-Straße
Otto-
Bren
ner-S
tr.
Gustav-Bielefeld-Straße
Freibad Weende
Friedhof Junkerberg
Heinrich-A.-Zachariä-Bogen
Karl-Schwarzschild-Weg
Edward-Schröder-Bogen
James-Franck-Ring
FesthalleWeende
Ostlandweg
Rudolf-Diesel-Straße
Christophorusweg
Goßlerstraße
Theodor-Heuss-Straße
Waldweg
Beyerstraße
Robert-Koch-Straße
Klinikum
Hermann-Rein-Straße
Sprangerweg
Vor der Laakenbreite
Hohler Graben
Krankenhaus-Weende Hoffmannshof Luttertal
Knochenmühle
Eulenloch
Herberhausen
Roringen
Am MenzelbergLange Straße
Rottenanger
Auf der Lieth
Eschen-breite
AmSchlehdorn
Augustiner-straße
UlrideshuserStraße
Albaniplatz
Hermann-Föge-Weg
Nonnenstieg
Eichendorffplatz
Dahlmannstraße
Wagnerstraße
Corvinuskirche
Hainbundstraße
Jugendherberge
Ewaldstraße
Rohns
KlausbergStauffenbergring
Thomas-Dehler-Weg
Bramwaldstraße
Zollstock
Deisterstraße
Elmweg
Sollingstraße
Süntel-weg
Harzstr.
Backhausstraße
KrugstraßeSt.-Heinrich-Straße
St.-Martini-Straße
Siek
höhe
Herb
ert-Q
uand
t-Str.
DransfelderStraße
GroßEllershausen
Mittelbergschule
In der Wehm
HasenwinkelAm Winterberg
Knutbühren
Ithweg
Am Alten Krug
Olenhuser Weg
Auf der Schanze
Hauptstraße
Kiefernweg
Magdeburger Weg
Gothaer Platz
Kiessee-straße
Merkelstraße
Friedländer Weg
Werner-Heisenberg-Platz
Lenglerner Straße
Europaallee
Straßburgstraße
Eschenweg
Londonstraße
Grünberger Straße
Plesseweg
Rathaus Bovenden
Liegnitzer StraßeEibenweg
St.-Godehard-Kirche
AmKirschberge
Grätz
elstr.
Walkemühlenweg
Lotzestraße
Zeppelinstraße
Eiswiese
Sandweg
Jahnstadion
Ascherberg
RischenwegHambergstraßeFreibadRosdorfObere StraßeHagenbreiteFriedensstraße
Leinestraße
Spickenweg
Klosterweg
Göttinger Straße
Roter Berg
Zimmermann-straße
Angerstraße
Auditorium
Karl-Grüneklee-Straße
Friedrich-Ebert-Straße
Lutteranger
An der Lutter
Grüner Weg
Liebrechtstraße
Kreuzbergring
Diedershäuser Str.
Kauf Park
LegendeHaltestelle
Haltestelle wird nur in Pfeilrichtung angefahren
Endhaltestelle13VSN-Umsteigehaltestelle
gültig ab 12. Dezember 2004
Linien 1, 2
Linien 6, 7, 8, 13
Linien 3, 5, 9, 10Linien 4, 14
Holtenser Berg <> ZietenterrassenLinie 1
Geismar-Charlottenburger Straße <> Weende-NordLinie 2
Weende-Nord <> Grone-SüdLinie 3
Kauf Park <> Geismar-Schöneberger StraßeLinie 4
Knutbühren/Hetjershausen/Groß Ellershausen/Kauf Park/Grone <> Nikolausberg
Linie 5
Klausberg <> BahnhofLinie 6
Holtensen <> ZietenterrassenLinie 7
Weende-Ost/Papenberg <> Grone-NordLinie 8
Ostviertel <> LeinebergLinie 9
Bahnhof <> Herberhausen/RoringenLinie 10
Geismar-Schöneberger Straße <> Holtenser BergLinie 12
Geismar-Süd <> Elliehausen/Esebeck/Kauf ParkLinie 13
Linie 14 Rosdorf <> Bovenden
24-Std. Hotline: (05 51) 99 80 99
5
3
4
6
9
10
13
8
7
12 1
143 2
8
5
10
10
6
9
132
4
71
5
5
14
8
12
13
andere Tarifzone
andere Tarifzone
Taxizwischen Knutbühren
und In der Wehm
P+R
Anita Schöbel (NAM) 27. September 2006 75 / 78
Other (more recent) models
Integrating vehicle schedules
Project work of Michael Schachtebeck
in cooperation with GÖVBDer Stadtbus
Info-Telefon: 38 444 444www.goevb.de
Esebeck
HoltensenElliehausen
Grone
Rosdorf
Weende-Nord
Bovenden
Geismar
Weende
Zietenterrassen
Treuenhagen
Roringen
Herberhausen
Nikolausberg
Uni-Nord
Ostviertel
Klausberg
Papenberg
Knutbühren
Hetjershausen
Groß Ellershausen Grone-Süd
Leineberg
Industriegebiet
Holtenser Berg
Charlottenburger Straße
PlauenerStraße
Rosmarinweg
Theaterstraße
Friedrichstraße
Jüdenstraße
Geismartor
Kornmarkt
MarktGronerStraße
Weender Straße-WestWeender Straße-Ost
Nikolaistraße
Gotth
elf-Le
imba
ch-S
tr.Ro
bert-
Bosc
h-Brei
te
Geismar-Süd
Am Kalten BornAm Rischen
Kampstraße
Auf dem Paul
Geismar Landstraße
David-Hilbert-Straße
Rudo
lf-Wink
el-St
r.
GronerTor
Bahnhof
Landgericht
Grete-Henry-Straße
Hannah-Vogt-Straße
Ehrengard-Schramm-Weg
Alva-Myrdal-Weg
Alfred-Delp-Weg
Gulden-hagen
LönswegAm Steinsgraben
Keplerstraße
Bürgerstraße
Nikolausberger Weg
Am Papenberg
Faßberg
Burckhardtweg
Kellnerweg
Tammannstraße
GoldschmidtstraßePetrikirchstraße
Blauer Turm
PosthofStadt-
friedhofGottes-lager
Tulpenweg
Asternweg
Schützen-platz
NeuesRathaus
Schillerstraße
Reinholdstraße
Fritz-Reuter-Straße
Treuenhagen
Baumschulenweg Stadtstieg
Teichstraße
Vor dem Walde
Schöneberger Straße
Gehrenring
Adolf-Sievert-Straße
Stumpfe Eiche
Im Hassel
Helle-weg
Valentinsbreite
Humboldtallee
Bühlstraße
Theaterplatz
Lichtenwalder Straße Greitweg Auf dem Greite Saline Königsstieg
Söseweg
Allerstraße
Fuldaweg
Rosdorfer Weg
WiesenstraßeCramer-straße
Am Hasengraben
Leibniz-straße
Mitteldorfstraße
Kurmainzer Weg
Eislebener Weg
Pommerneck Stettiner Straße
Ortelsburger StraßeBreslauer
Straße Flüthenweg Görlitzer Straße
An der Tillyschanze
Himmelsruh
Weidenbreite MittelbergSpandauer
WegSanders-
beekTegeler
Weg
Pfalz-Grona-Breite Im Rinschenrott
Maschmühlenweg
Hildebrandstraße
Bahnhofsallee
Campus
Hennebergstraße
Adolf
-Hoy
er-St
r.
Wilh
elm-La
mbrec
ht-St
r.Flo
renz-S
artor
ius-S
tr.
Marien-straße
Otto-Frey-Brücke
Levin
park
Bornbreite
Twechte
Holtenser Landstraße
Lindenweg
Elliehäuser Weg
Domäne
Auf dem Hagen
Nußanger
Esebeck
Kleehöfen
Harrenacker
Talgraben
Lehnshof
Gerhard-Zillich-Straße
Am Burggraben
Gesundbrunnen
Am Eikborn
Am Anger
Hermann-Kolbe-Straße
Willi-Eichler-Straße
Otto-
Bren
ner-S
tr.
Gustav-Bielefeld-Straße
Freibad Weende
Friedhof Junkerberg
Heinrich-A.-Zachariä-Bogen
Karl-Schwarzschild-Weg
Edward-Schröder-Bogen
James-Franck-Ring
FesthalleWeende
Ostlandweg
Rudolf-Diesel-Straße
Christophorusweg
Goßlerstraße
Theodor-Heuss-Straße
Waldweg
Beyerstraße
Robert-Koch-Straße
Klinikum
Hermann-Rein-Straße
Sprangerweg
Vor der Laakenbreite
Hohler Graben
Krankenhaus-Weende Hoffmannshof Luttertal
Knochenmühle
Eulenloch
Herberhausen
Roringen
Am MenzelbergLange Straße
Rottenanger
Auf der Lieth
Eschen-breite
AmSchlehdorn
Augustiner-straße
UlrideshuserStraße
Albaniplatz
Hermann-Föge-Weg
Nonnenstieg
Eichendorffplatz
Dahlmannstraße
Wagnerstraße
Corvinuskirche
Hainbundstraße
Jugendherberge
Ewaldstraße
Rohns
KlausbergStauffenbergring
Thomas-Dehler-Weg
Bramwaldstraße
Zollstock
Deisterstraße
Elmweg
Sollingstraße
Süntel-weg
Harzstr.
Backhausstraße
KrugstraßeSt.-Heinrich-Straße
St.-Martini-Straße
Siek
höhe
Herb
ert-Q
uand
t-Str.
DransfelderStraße
GroßEllershausen
Mittelbergschule
In der Wehm
HasenwinkelAm Winterberg
Knutbühren
Ithweg
Am Alten Krug
Olenhuser Weg
Auf der Schanze
Hauptstraße
Kiefernweg
Magdeburger Weg
Gothaer Platz
Kiessee-straße
Merkelstraße
Friedländer Weg
Werner-Heisenberg-Platz
Lenglerner Straße
Europaallee
Straßburgstraße
Eschenweg
Londonstraße
Grünberger Straße
Plesseweg
Rathaus Bovenden
Liegnitzer StraßeEibenweg
St.-Godehard-Kirche
AmKirschberge
Grätz
elstr.
Walkemühlenweg
Lotzestraße
Zeppelinstraße
Eiswiese
Sandweg
Jahnstadion
Ascherberg
RischenwegHambergstraßeFreibadRosdorfObere StraßeHagenbreiteFriedensstraße
Leinestraße
Spickenweg
Klosterweg
Göttinger Straße
Roter Berg
Zimmermann-straße
Angerstraße
Auditorium
Karl-Grüneklee-Straße
Friedrich-Ebert-Straße
Lutteranger
An der Lutter
Grüner Weg
Liebrechtstraße
Kreuzbergring
Diedershäuser Str.
Kauf Park
LegendeHaltestelle
Haltestelle wird nur in Pfeilrichtung angefahren
Endhaltestelle13VSN-Umsteigehaltestelle
gültig ab 12. Dezember 2004
Linien 1, 2
Linien 6, 7, 8, 13
Linien 3, 5, 9, 10Linien 4, 14
Holtenser Berg <> ZietenterrassenLinie 1
Geismar-Charlottenburger Straße <> Weende-NordLinie 2
Weende-Nord <> Grone-SüdLinie 3
Kauf Park <> Geismar-Schöneberger StraßeLinie 4
Knutbühren/Hetjershausen/Groß Ellershausen/Kauf Park/Grone <> Nikolausberg
Linie 5
Klausberg <> BahnhofLinie 6
Holtensen <> ZietenterrassenLinie 7
Weende-Ost/Papenberg <> Grone-NordLinie 8
Ostviertel <> LeinebergLinie 9
Bahnhof <> Herberhausen/RoringenLinie 10
Geismar-Schöneberger Straße <> Holtenser BergLinie 12
Geismar-Süd <> Elliehausen/Esebeck/Kauf ParkLinie 13
Linie 14 Rosdorf <> Bovenden
24-Std. Hotline: (05 51) 99 80 99
5
3
4
6
9
10
13
8
7
12 1
143 2
8
5
10
10
6
9
132
4
71
5
5
14
8
12
13
andere Tarifzone
andere Tarifzone
Taxizwischen Knutbühren
und In der Wehm
P+R
Anita Schöbel (NAM) 27. September 2006 75 / 78
Other (more recent) models
Integrating vehicle schedules
Project work of Michael Schachtebeck
in cooperation with GÖVBDer Stadtbus
Info-Telefon: 38 444 444www.goevb.de
Esebeck
HoltensenElliehausen
Grone
Rosdorf
Weende-Nord
Bovenden
Geismar
Weende
Zietenterrassen
Treuenhagen
Roringen
Herberhausen
Nikolausberg
Uni-Nord
Ostviertel
Klausberg
Papenberg
Knutbühren
Hetjershausen
Groß Ellershausen Grone-Süd
Leineberg
Industriegebiet
Holtenser Berg
Charlottenburger Straße
PlauenerStraße
Rosmarinweg
Theaterstraße
Friedrichstraße
Jüdenstraße
Geismartor
Kornmarkt
MarktGronerStraße
Weender Straße-WestWeender Straße-Ost
Nikolaistraße
Gotth
elf-Le
imba
ch-S
tr.Ro
bert-
Bosc
h-Brei
te
Geismar-Süd
Am Kalten BornAm Rischen
Kampstraße
Auf dem Paul
Geismar Landstraße
David-Hilbert-Straße
Rudo
lf-Wink
el-St
r.
GronerTor
Bahnhof
Landgericht
Grete-Henry-Straße
Hannah-Vogt-Straße
Ehrengard-Schramm-Weg
Alva-Myrdal-Weg
Alfred-Delp-Weg
Gulden-hagen
LönswegAm Steinsgraben
Keplerstraße
Bürgerstraße
Nikolausberger Weg
Am Papenberg
Faßberg
Burckhardtweg
Kellnerweg
Tammannstraße
GoldschmidtstraßePetrikirchstraße
Blauer Turm
PosthofStadt-
friedhofGottes-lager
Tulpenweg
Asternweg
Schützen-platz
NeuesRathaus
Schillerstraße
Reinholdstraße
Fritz-Reuter-Straße
Treuenhagen
Baumschulenweg Stadtstieg
Teichstraße
Vor dem Walde
Schöneberger Straße
Gehrenring
Adolf-Sievert-Straße
Stumpfe Eiche
Im Hassel
Helle-weg
Valentinsbreite
Humboldtallee
Bühlstraße
Theaterplatz
Lichtenwalder Straße Greitweg Auf dem Greite Saline Königsstieg
Söseweg
Allerstraße
Fuldaweg
Rosdorfer Weg
WiesenstraßeCramer-straße
Am Hasengraben
Leibniz-straße
Mitteldorfstraße
Kurmainzer Weg
Eislebener Weg
Pommerneck Stettiner Straße
Ortelsburger StraßeBreslauer
Straße Flüthenweg Görlitzer Straße
An der Tillyschanze
Himmelsruh
Weidenbreite MittelbergSpandauer
WegSanders-
beekTegeler
Weg
Pfalz-Grona-Breite Im Rinschenrott
Maschmühlenweg
Hildebrandstraße
Bahnhofsallee
Campus
Hennebergstraße
Adolf
-Hoy
er-St
r.
Wilh
elm-La
mbrec
ht-St
r.Flo
renz-S
artor
ius-S
tr.
Marien-straße
Otto-Frey-Brücke
Levin
park
Bornbreite
Twechte
Holtenser Landstraße
Lindenweg
Elliehäuser Weg
Domäne
Auf dem Hagen
Nußanger
Esebeck
Kleehöfen
Harrenacker
Talgraben
Lehnshof
Gerhard-Zillich-Straße
Am Burggraben
Gesundbrunnen
Am Eikborn
Am Anger
Hermann-Kolbe-Straße
Willi-Eichler-Straße
Otto-
Bren
ner-S
tr.
Gustav-Bielefeld-Straße
Freibad Weende
Friedhof Junkerberg
Heinrich-A.-Zachariä-Bogen
Karl-Schwarzschild-Weg
Edward-Schröder-Bogen
James-Franck-Ring
FesthalleWeende
Ostlandweg
Rudolf-Diesel-Straße
Christophorusweg
Goßlerstraße
Theodor-Heuss-Straße
Waldweg
Beyerstraße
Robert-Koch-Straße
Klinikum
Hermann-Rein-Straße
Sprangerweg
Vor der Laakenbreite
Hohler Graben
Krankenhaus-Weende Hoffmannshof Luttertal
Knochenmühle
Eulenloch
Herberhausen
Roringen
Am MenzelbergLange Straße
Rottenanger
Auf der Lieth
Eschen-breite
AmSchlehdorn
Augustiner-straße
UlrideshuserStraße
Albaniplatz
Hermann-Föge-Weg
Nonnenstieg
Eichendorffplatz
Dahlmannstraße
Wagnerstraße
Corvinuskirche
Hainbundstraße
Jugendherberge
Ewaldstraße
Rohns
KlausbergStauffenbergring
Thomas-Dehler-Weg
Bramwaldstraße
Zollstock
Deisterstraße
Elmweg
Sollingstraße
Süntel-weg
Harzstr.
Backhausstraße
KrugstraßeSt.-Heinrich-Straße
St.-Martini-Straße
Siek
höhe
Herb
ert-Q
uand
t-Str.
DransfelderStraße
GroßEllershausen
Mittelbergschule
In der Wehm
HasenwinkelAm Winterberg
Knutbühren
Ithweg
Am Alten Krug
Olenhuser Weg
Auf der Schanze
Hauptstraße
Kiefernweg
Magdeburger Weg
Gothaer Platz
Kiessee-straße
Merkelstraße
Friedländer Weg
Werner-Heisenberg-Platz
Lenglerner Straße
Europaallee
Straßburgstraße
Eschenweg
Londonstraße
Grünberger Straße
Plesseweg
Rathaus Bovenden
Liegnitzer StraßeEibenweg
St.-Godehard-Kirche
AmKirschberge
Grätz
elstr.
Walkemühlenweg
Lotzestraße
Zeppelinstraße
Eiswiese
Sandweg
Jahnstadion
Ascherberg
RischenwegHambergstraßeFreibadRosdorfObere StraßeHagenbreiteFriedensstraße
Leinestraße
Spickenweg
Klosterweg
Göttinger Straße
Roter Berg
Zimmermann-straße
Angerstraße
Auditorium
Karl-Grüneklee-Straße
Friedrich-Ebert-Straße
Lutteranger
An der Lutter
Grüner Weg
Liebrechtstraße
Kreuzbergring
Diedershäuser Str.
Kauf Park
LegendeHaltestelle
Haltestelle wird nur in Pfeilrichtung angefahren
Endhaltestelle13VSN-Umsteigehaltestelle
gültig ab 12. Dezember 2004
Linien 1, 2
Linien 6, 7, 8, 13
Linien 3, 5, 9, 10Linien 4, 14
Holtenser Berg <> ZietenterrassenLinie 1
Geismar-Charlottenburger Straße <> Weende-NordLinie 2
Weende-Nord <> Grone-SüdLinie 3
Kauf Park <> Geismar-Schöneberger StraßeLinie 4
Knutbühren/Hetjershausen/Groß Ellershausen/Kauf Park/Grone <> Nikolausberg
Linie 5
Klausberg <> BahnhofLinie 6
Holtensen <> ZietenterrassenLinie 7
Weende-Ost/Papenberg <> Grone-NordLinie 8
Ostviertel <> LeinebergLinie 9
Bahnhof <> Herberhausen/RoringenLinie 10
Geismar-Schöneberger Straße <> Holtenser BergLinie 12
Geismar-Süd <> Elliehausen/Esebeck/Kauf ParkLinie 13
Linie 14 Rosdorf <> Bovenden
24-Std. Hotline: (05 51) 99 80 99
5
3
4
6
9
10
13
8
7
12 1
143 2
8
5
10
10
6
9
132
4
71
5
5
14
8
12
13
andere Tarifzone
andere Tarifzone
Taxizwischen Knutbühren
und In der Wehm
P+R
Anita Schöbel (NAM) 27. September 2006 75 / 78
Other (more recent) models
A game-theoretic approach
Idea: Distribute freuqencies of lines equally over the network to avoiddelays due to capacity constraints.
The line planning game:Players: linesStrategies: frequenciesCost function: Due to delays
see: Dissertation of Silvia Schwarze
Anita Schöbel (NAM) 27. September 2006 76 / 78
Other (more recent) models
A game-theoretic approach
Idea: Distribute freuqencies of lines equally over the network to avoiddelays due to capacity constraints.
The line planning game:Players: linesStrategies: frequenciesCost function: Due to delays
see: Dissertation of Silvia Schwarze
Anita Schöbel (NAM) 27. September 2006 76 / 78
Other (more recent) models
A game-theoretic approach
Idea: Distribute freuqencies of lines equally over the network to avoiddelays due to capacity constraints.
The line planning game:Players: linesStrategies: frequenciesCost function: Due to delays
see: Dissertation of Silvia Schwarze
Anita Schöbel (NAM) 27. September 2006 76 / 78
Other (more recent) models
A game-theoretic approach
Idea: Distribute freuqencies of lines equally over the network to avoiddelays due to capacity constraints.
The line planning game:Players: linesStrategies: frequenciesCost function: Due to delays
see: Dissertation of Silvia Schwarze
Anita Schöbel (NAM) 27. September 2006 76 / 78
Other (more recent) models
Other approaches
BlackBox Model of Deutsche BahnIntegrating the vehicle schedules in cooperation with GÖVBGame-theoretic approachPath-based models
I Laporte, Mesa and Ortega (optimize modal split)I Borndörfer and Pfetsch (generate lines dynamically)
→ next Lecture!
Anita Schöbel (NAM) 27. September 2006 77 / 78
Other (more recent) models
Other approaches
BlackBox Model of Deutsche BahnIntegrating the vehicle schedules in cooperation with GÖVBGame-theoretic approachPath-based models
I Laporte, Mesa and Ortega (optimize modal split)I Borndörfer and Pfetsch (generate lines dynamically)
→ next Lecture!
Anita Schöbel (NAM) 27. September 2006 77 / 78
Other (more recent) models
Other approaches
BlackBox Model of Deutsche BahnIntegrating the vehicle schedules in cooperation with GÖVBGame-theoretic approachPath-based models
I Laporte, Mesa and Ortega (optimize modal split)I Borndörfer and Pfetsch (generate lines dynamically)
→ next Lecture!
Anita Schöbel (NAM) 27. September 2006 77 / 78
Other (more recent) models
The end . . .
THANK YOU!
Anita Schöbel (NAM) 27. September 2006 78 / 78
