[PPT] - Line Planning in Public Transportation Anita Schbel Institut fr PowerPoint Presentation

SLIDE 1

Line Planning in Public Transportation

Anita Schöbel

Institut für Numerische und Angewandte Mathematik Georg-August Universität Göttingen

27. September 2006

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Line planning in public transportation

Planning in public transportation

Stations | Lines | — Timetable —

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Line planning in public transportation

Planning in public transportation

Vehicle Scheduling/rolling stock planning | crew scheduling | rostering Stations | Lines | — Timetable —

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Line planning in public transportation

Planning in public transportation

Vehicle Scheduling/rolling stock planning | crew scheduling | rostering Stations | Lines | — Timetable — Tariffs | Disposition

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Line planning in public transportation

Planning in public transportation

companie’s perspective Vehicle Scheduling/rolling stock planning | crew scheduling | rostering Stations | Lines | — Timetable — Tariffs | Disposition

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Line planning in public transportation

Planning in public transportation

companie’s perspective Vehicle Scheduling/rolling stock planning | crew scheduling | rostering Stations | Lines | — Timetable — Customers’ point

f view

Tariffs | Disposition

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SLIDE 7

Line planning in public transportation

Planning in public transportation

companie’s perspective Vehicle Scheduling/rolling stock planning | crew scheduling | rostering Stations | Lines | — Timetable — Customers’ point

f view

Tariffs | Disposition

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SLIDE 8

Line planning in public transportation

Line planning problem

Given a public transportation network PTN=(V, E)

◮ with its stops V ◮ and its direct connections E Anita Schöbel (NAM)

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SLIDE 9

Line planning in public transportation

Line planning problem

Given a public transportation network PTN=(V, E)

◮ with its stops V ◮ 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

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SLIDE 10

Line planning in public transportation

Example

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Line planning in public transportation

Example

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Line planning in public transportation

Example

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SLIDE 13

Line planning in public transportation

Example

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Line planning in public transportation

Example

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SLIDE 15

Line planning in public transportation

Example

l

f =1

l

f =4

l

f =1 f =2

l Anita Schöbel (NAM)

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SLIDE 16

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) . . .

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SLIDE 17

Line planning in public transportation

Basic notation

Notation

A line P is a path in the public transportation network The frequency fl of a line l says how often service is offered along line l within a (given) time period I. A line concept (L, f) is a set of lines L together with their frequencies fl for all l ∈ L.

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SLIDE 18

Line planning in public transportation

Line planning problem

Given a public transportation network PTN

◮ with its stops V ◮ 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

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SLIDE 19

Line planning in public transportation

Line planning problem

Given a public transportation network PTN

◮ with its stops V ◮ 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 ?

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SLIDE 20

Line planning in public transportation

Line planning problem

Given a public transportation network PTN

◮ with its stops V ◮ 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 transported Public transport should be convenient for the customers Costs should be small

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SLIDE 21

Line planning in public transportation

Line planning problem

Given a public transportation network PTN

◮ with its stops V ◮ 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 transported Public transport should be convenient for the customers Costs should be small We are now formalizing the problem and its objective functions.

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SLIDE 22

Line planning in public transportation

Minimal and maximal frequencies

Notation

Let A denote the (fixed) capacity of a vehicle, and f min

e

, f max

e

denote the minimal and maximal allowed frequency on edge e ∈ E.

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SLIDE 23

Line planning in public transportation

Minimal and maximal frequencies

Notation

Let A denote the (fixed) capacity of a vehicle, and f min

e

, f max

e

denote the minimal and maximal allowed frequency on edge e ∈ E. Example: f min

e

is the minimal number of vehicles needed to transport all customers. f min

e

= ⌈we A ⌉. f max

e

: due to security reasons or noise avoidance

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SLIDE 24

Line planning in public transportation

Minimal and maximal frequencies

Notation

Let A denote the (fixed) capacity of a vehicle, and f min

e

, f max

e

denote the minimal and maximal allowed frequency on edge e ∈ E. Example: f min

e

is the minimal number of vehicles needed to transport all customers. f min

e

= ⌈we A ⌉. f max

e

: due to security reasons or noise avoidance

Notation

The edge-frequency of e w.r.t. a line concept (L, f) is given as f(e) =

l:e∈l

fl.

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SLIDE 25

Line planning in public transportation

The line pool

Two different possibilities: from pool: Choose the lines for the line concept (L, fl) from a given line pool L0, i.e. L ⊆ L0 from scratch: Construct the lines L from scratch

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SLIDE 26

Line planning in public transportation

The line pool

Two different possibilities: from pool: Choose the lines for the line concept (L, fl) from a given line pool L0, i.e. L ⊆ L0 from scratch: Construct the lines L from scratch Remark: Nearly all publications use a line pool.

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SLIDE 27

Line planning in public transportation

The line pool

Two different possibilities: from pool: Choose the lines for the line concept (L, fl) from a given 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.

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SLIDE 28

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 max

e

for all e ∈ E, find a feasible line concept (L, f) with L ⊆ L0 and fl ∈ IN0 ∀ l ∈ L.

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Basic model

Complexity of (LP0)

Theorem

(LP0) is NP-complete.

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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 min

e

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) is

a 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.

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Cost-oriented models

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Cost-oriented models

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SLIDE 33

Cost-oriented models Simple cost model

The costs of a line concept

costs of a line concept (L, f):

l∈L

costl where the costs costl of a line l ∈ L0 depend on the frequency of l the 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.)

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SLIDE 35

Cost-oriented models Simple cost model

The costs of a line concept

costs of a line concept (L, f):

l∈L

costl where the costs costl of a line l ∈ L0 depend on the frequency of l the 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 costl be estimated ?

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SLIDE 36

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.

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SLIDE 37

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?

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SLIDE 38

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: costl = Time on line l ∗costtime + Kilometers on line l ∗costkm.

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SLIDE 39

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.

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SLIDE 40

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 = ⌈Timel · fl I ⌉ due to rule of proportion.

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SLIDE 41

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 = ⌈Timel · fl I ⌉ due to rule of proportion. Example: I = 60 min, Timel = 120 min, fl = 4

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SLIDE 42

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 = ⌈Timel · fl I ⌉ due to rule of proportion. Example: I = 60 min, Timel = 120 min, fl = 4 = ⇒ 8 vehicles needed.

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SLIDE 43

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)

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SLIDE 44

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 · I Timel = fl Lengthl

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SLIDE 45

Cost-oriented models Simple cost model

Finally: The costs of line l

costl = Time on line l · costtime + Kilometers on line l · costkm = Timel fl costtime + Lengthl fl costkm = fl (Timel costtime + Lengthl costkm

=:costl

) = fl costl.

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SLIDE 46

Cost-oriented models Simple cost model

Finally: The costs of line l

costl = 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 the frequency fl.

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SLIDE 47

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 frequencies f min

e

≤ f max

e

for all e ∈ E, and parameters costl for all l ∈ L0, find a feasible line concept (L, f) with minimal overall costs cost(L, f) =

l∈L

fl costl.

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SLIDE 48

Cost-oriented models Simple cost model

Complexity of (LP1)

Theorem

(LP1) is NP-hard. Proof: Special case of (LP0)

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SLIDE 49

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?

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SLIDE 50

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 for all e ∈ E.

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SLIDE 51

Cost-oriented models Simple cost model

Integer programming formulation

variables: fl min

l∈L0

fl costl s.t.

l∈L0,e∈l

fl ≥ f min

e

∀ e ∈ E (1)

l∈L0,e∈l

fl ≤ f max

e

∀ 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}.

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SLIDE 52

Cost-oriented models Simple cost model

Cost model without upper frequencies

Observations: (LP1) without upper frequencies is a multi covering problem. (MC) (Multi covering problem): Given an M × N-matrix A = (aij) with elements aij ∈ {0, 1}, a vector b ∈ IN0M and some integer K, does there exist x ∈ IN0N with Ax ≥ b and N

n=1 xn ≤ K?

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SLIDE 53

Cost-oriented models Algorithms

Algorithms

General idea: for (LP1-Pool) without upper frequencies Basic idea: Let f min

e

be the required frequency for edge e.

1

While there is an uncovered edge e (i.e. with f min

e

> 0)

2

Choose a line l, such that g(l) minimal

3

Choose frequency of line as fl := f(l)

4

Update f min

e

accordingly

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SLIDE 55

Cost-oriented models Algorithms

Applying Dobson, 1982

1

While there is an uncovered edge e (i.e. with f min

e

> 0)

2

Choose a line l, such that g(l) minimal

3

Choose frequency of line as fl := f(l)

4

Update f min

e

accordingly

with g(l) =

cl |{e∈l:f min

e

>0}|

f(l) = fl + 1

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SLIDE 56

Cost-oriented models Algorithms

Applying Dobson, 1982

1

While there is an uncovered edge e (i.e. with f min

e

> 0)

2

Choose a line l, such that g(l) minimal

3

Choose frequency of line as fl := f(l)

4

Update f min

e

accordingly

with g(l) =

cl |{e∈l:f min

e

>0}|

f(l) = fl + 1

Theorem

HEU ≤ OPT H(maxl |l|), where H(d) = d

i=1 1 i

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SLIDE 57

Cost-oriented models Algorithms

Applying van Slyke/Xiaoming, 1984

1

While there is an uncovered edge e (i.e. with f min

e

> 0)

2

Choose edge e with maximal f min

e

3

Choose a line l, covering e such that g(l) minimal

4

Choose frequency of line fl = f(l)

5

Update f min

e

accordingly

with g(l) = cl f(l) = fl + f min

e

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SLIDE 58

Cost-oriented models Algorithms

Applying van Slyke/Xiaoming, 1984

1

While there is an uncovered edge e (i.e. with f min

e

> 0)

2

Choose edge e with maximal f min

e

3

Choose a line l, covering e such that g(l) minimal

4

Choose frequency of line fl = f(l)

5

Update f min

e

accordingly

with g(l) = cl f(l) = fl + f min

e

Theorem

HEU ≤ OPT maxe∈E |L(e)|

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SLIDE 59

Cost-oriented models Algorithms

Line planning heuristic

1

While there is an uncovered edge e (i.e. with f min

e

> 0)

2

Choose edge e with maximal f min

e

3

Choose a line l, covering e such that g(l) minimal

4

Choose frequency of line fl = f(l)

5

Update f min

e

accordingly

with g(l) =

cl |{e∈l:f min

e

>0}| mine∈l:fmin

e >0 f min e

f(l) = fl + mine∈l:f min

e

>0 f min e

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SLIDE 60

Cost-oriented models Algorithms

Two combinations

1

While there is an uncovered edge e (i.e. with f min

e

> 0)

2

Choose edge e with maximal f min

e

3

Choose a line l, covering e such that g(l) minimal

4

Choose frequency of line fl = f(l)

5

Update f min

e

accordingly

with g(l) =

cl |{e∈l:f min

e

>0}| mine∈l:fmin

e >0 f min e

f(l) = fl + 1

r

g(l) =

cl |{e∈l:f min

e

>0}|

f(l) = fl + mine∈l f min

e

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SLIDE 61

Cost-oriented models Algorithms

Numerical results

Data: Simulated data with 1000 stations, 4000 edges and 30, 60, 90, 120, . . . , 1350, 1500, 3000 possible lines Algorithms tested on 500 instances.

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SLIDE 62

Cost-oriented models Algorithms

Numerical results

size Heuristic

30 60 90 120 150 180 210 240 Dobson 620 1470 2060 2720 3340 4090 4730 5640 6100 Sly/Xia 880 2640 4290 6210 8500 10680 13440 16330 17330 LiPla 650 1520 2150 2730 3490 4220 4980 5800 6430 Comb 1 610 1390 1950 2620 3190 3940 4610 5500 Comb 2 580 1380 1910 2640 3220 3910 4690 5450 6240

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SLIDE 63

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

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SLIDE 64

Cost-oriented models Algorithms

Results of heuristics

Winner: The two combinations (where the first combination is slightly better) then Dobson and line planing heuristic Far away: van Slyke and Xiaoming

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SLIDE 65

Cost-oriented models Extended cost model

Extended cost model

Evaluate costs of line concept Determine the lines and the frequencies Determine also the type of vehicle t = 1, . . . , T which is used to run a line In rail transportation: determine also the number of cars of the trains for each line

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SLIDE 67

Cost-oriented models Extended cost model

Additional parameters

ccostt

fix

= fixed cost per car of type t ccostt

km

= cost per kilometer with a car of type t vcostt

km

= cost per kilometer with a vehicle of type t At = capacity of a car of type t cmin

t

= min number of cars allowed for a train of type t cmax

t

= max number of cars allowed for a train of type t

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SLIDE 68

Cost-oriented models Extended cost model

Recall . . .

Number of vehicles needed to run line l is ⌈fl Timel I ⌉. Moreover, we define f max = max

e∈E f max e

as the upper bound over all upper frequencies.

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SLIDE 69

Cost-oriented models Extended cost model

Objective function

variables: X tc

l

= 1 if line l is served by vehicles of type t with c cars

therwise

. min

l∈L0

T

t=1

cmax

t

c=cmin

t

(ccostt

fix

Timel I c + Lengthl ccostt

km c

+Lengthl vcostt

km)fl X tc l

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SLIDE 70

Cost-oriented models Extended cost model

But:

A quadratic term fl X tc

l

in the objective! Solution: Substitute fl X tc

l

by a new variable X tfc

l

with X tfc

l

=    1 if line l ∈ L0 is served by vehicles of type t with c cars and frequency f

therwise

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SLIDE 71

Cost-oriented models Extended cost model

Extended cost model (LP2)

min

l∈L0

T

t=1

f max

f=1

cmax

t

c=cmin

t

(ccostt

fix · ⌈Timel

I · f⌉ · c + Lengthl · ccostt

km · f · c

+ Lengthl · vcostt

km · f) · X tfc l

s.t.

l∈L0:e∈l

T

t=1

f max

f=1

cmax

t

c=cmin

t

At · f · c · X tfc

l

≥ we ∀ e ∈ E (3) f min

e

≤

l∈L0:e∈l

T

t=1

f max

f=1

cmax

t

c=cmin

t

f · X tfc

l

≤ f max

e

∀ e ∈ E (4)

f max

f=1

cmax

t

c=cmin

t

X tfc

l

≤ 1 ∀l ∈ L0, ∀t = 1, . . . , T (5) X tfc

l

∈ {0, 1} ∀l, f, t, c (6)

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SLIDE 72

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 is determined within (LP2). The approximation of the costs is more detailed by having the new parameters. Constraints (3) are needed to make sure that all passengers can be transported. (LP2) is NP hard as special case of (LP1).

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SLIDE 73

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 tfc

l

variables use commercial IP-solver

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SLIDE 74

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, and Kroon (2001, 2002), Bussieck, Lindner, and Lübbecke (2003)

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SLIDE 75

Customer-oriented models

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SLIDE 76

Customer-oriented models

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SLIDE 77

Customer-oriented models Direct travelers approach

Goal

Notation

Given a line concept (L, fl). A customer does not change the line on his/her journey is called a direct traveler Goal of direct travelers approach: design the lines in such a way that as many customers as possible have a direct connection, i.e. maximize the number of direct travelers.

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SLIDE 79

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 and station j. Pij ⊆ l if there exists a shortest (or preferable) path between i and j which is completely contained in line l.

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SLIDE 80

Customer-oriented models Direct travelers approach

Example

Graph with 5 nodes, all edge weights are 1.

2 3 4 5 1

l

line l=(1,2,3,4,5)

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SLIDE 81

Customer-oriented models Direct travelers approach

Example

Graph with 5 nodes, all edge weights are 1.

2 3 4 5 1

P contained in line l

23

l

line l=(1,2,3,4,5)

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SLIDE 82

Customer-oriented models Direct travelers approach

Example

Graph with 5 nodes, all edge weights are 1.

2 3 4 5 1

P contained in line l l

14 line l=(1,2,3,4,5)

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SLIDE 83

Customer-oriented models Direct travelers approach

Example

Graph with 5 nodes, all edge weights are 1.

2 3 4 5 1

l

15

P NOT contained in line l

line l=(1,2,3,4,5)

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SLIDE 84

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 line l ∈ L0.

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SLIDE 85

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 line l ∈ L0. Objective: max

i,j∈V,l∈L0:Pij⊆l

dijl

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SLIDE 86

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 station j.

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SLIDE 87

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 min

e

≤

l∈L0:

e∈l

fl ≤ f max

e

for all e ∈ E (9) dijl, fl ∈ IN0 for all i, j ∈ V, l ∈ L0

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SLIDE 88

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.

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SLIDE 89

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.

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SLIDE 90

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 min

e

≤

l∈L0:e∈l

fl ≤ f max

e

for all e ∈ E Dij, fe ∈ IN0 for all i, j ∈ V, e ∈ E

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SLIDE 91

Customer-oriented models Direct travelers approach

Relation between (LP3) and (LP3’)

Lemma

The optimal objective value of (LP3’) is an upper bound for the optimal

bjective value of (LP3).

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SLIDE 92

Customer-oriented models Direct travelers approach

Relation between (LP3) and (LP3’)

Lemma

The optimal objective value of (LP3’) is an upper bound for the optimal

bjective value of (LP3).
feas. sol.
feas. sol.
f (LP3)
f (LP3’)

I.e. (LP3’) is a relaxation of (LP3).

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SLIDE 93

Customer-oriented models Direct travelers approach

Solving (LP3’)

see M.R. Bussieck and P . Kreuzer and U.T. Zimmermann, EJOR, 1996 further relax integrality constraints of the Dij variables of (LP3’) use a cutting-plane approach

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SLIDE 94

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).

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SLIDE 96

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 over all customers is minimal.

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SLIDE 97

Customer-oriented models Minimizing traveling time

Illustration

Examle:

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Customer-oriented models Minimizing traveling time

Illustration

Minimizing riding times . . .

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SLIDE 99

Customer-oriented models Minimizing traveling time

Illustration

Minimize number of transfers . . .

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SLIDE 100

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 (since timetable is not known in the phase of line planning).

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SLIDE 101

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 B and an OD-matrix Wst find 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).

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SLIDE 102

Customer-oriented models Minimizing traveling time

Relation to direct travelers approach

Note: The minimal number of transfers needs not be the same as the maximal number of direct passengers! Example:

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SLIDE 103

Customer-oriented models Minimizing traveling time

Relation to direct travelers approach

Note: The minimal number of transfers needs not be the same as the maximal number of direct passengers! Example:

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SLIDE 104

Customer-oriented models Minimizing traveling time

Relation to direct travelers approach

Note: The minimal number of transfers needs not be the same as the maximal number of direct passengers! Example:

2 3 2

Solution 1: 4 direct passengers, 6 transfers

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SLIDE 105

Customer-oriented models Minimizing traveling time

Relation to direct travelers approach

Note: The minimal number of transfers needs not be the same as the maximal number of direct passengers! Example:

2 3 2

Solution 1: 4 direct passengers, 6 transfers Solution 2: 3 direct passengers, 4 transfers

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SLIDE 106

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 objective the network is a linear graph and all costs are equal to one.

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SLIDE 107

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 objective the network is a linear graph and all costs are equal to one. Proof: Reduction to set covering

s1 t1 s2 t2 s3 t3 s4 t4 s5 t5 s6 t6

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SLIDE 108

Customer-oriented models Minimizing traveling time

The change & go graph

s1 s2 s3 s4 s5 s6 s7 s8 l2 l3

l1

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SLIDE 109

Customer-oriented models Minimizing traveling time

The change & go graph

l1

s2,l1 s3,l1 s4,l1 s1,l1

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SLIDE 110

Customer-oriented models Minimizing traveling time

The change & go graph

l2 l3

l1

s1,l2 s5,l2 s4,l2 s8,l3 s6,l2 s6,l3 s7,l3 s2,l1 s3,l1 s4,l1 s1,l1 s3,l3

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SLIDE 111

Customer-oriented models Minimizing traveling time

The change & go graph

s1 s2 s3 s4 s5 s6 s7 s8 l2 l3

l1

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SLIDE 112

Customer-oriented models Minimizing traveling time

The change & go graph

l2 l3

l1

s1,l2 s5,l2 s4,l2 s8,l3 s6,l2 s6,l3 s7,l3 s2,l1 s3,l1 s4,l1 s1,l1 s3,l3

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SLIDE 113

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,l3 s7,l3 s2,l1 s3,l1 s4,l1 s1,l1 s3,l3

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SLIDE 114

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,l3 s7,l3 s2,l1 s3,l1 s4,l1 s1,l1 s3,l3 s5,0 s3,0 s2,0

Result: Change & Go Graph N = (E, A)

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SLIDE 115

Customer-oriented models Minimizing traveling time

Advantages of the model

paths with minimal traveling time can be calculated as shortest paths very flexible due to different possibilities for the weights ca for all activities a ∈ A:

◮ weight all arcs by their (estimated) durations

minimize traveling time

◮ weight changing arcs by 1, all others by zero

minimize number of changes

◮ weight changing arcs by zero, others by their lengths

minimize length of trip (and hence costs)

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SLIDE 116

Customer-oriented models Minimizing traveling time

IP-Formulation

Variables: xa

st =

1 if activity a ∈ A is used on a shortest path from s to t in N

therwise

yl = 1 if line l is established

therwise

Parameters: Θ as node-arc-incidence matrix of N, bi

st =

   1 if i = (s, 0) −1 if i = (t, 0)

therwise

Lemma

Any solution xst ∈ {0, 1}|A| of Θ xst = bst is a path from s to t in N

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SLIDE 117

Customer-oriented models Minimizing traveling time

IP-Formulation

min

s,t∈V
a∈A Wstcaxa

st

s.t. xa

st

≤ yl for all s, t ∈ V, l ∈ L, a ∈ l Θxst = bst for all s, t ∈ V with Wst > 0

l∈L ylcostl

≤ B xa

st, yl ∈ {0, 1}

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SLIDE 118

Customer-oriented models Minimizing traveling time

IP-Formulation

min

s,t∈V
a∈A Wstcaxa

st

s.t. xa

st

≤ yl for all s, t ∈ V, l ∈ L, a ∈ l Θxst = bst for all s, t ∈ V with Wst > 0

l∈L ylcostl

≤ B xa

st, yl ∈ {0, 1}

this model assumes unlimited capacity of the vehicles with limited capacity A of the trains:

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SLIDE 119

Customer-oriented models Minimizing traveling time

IP-Formulation

min

s,t∈V
a∈A Wstcaxa

st

s.t. xa

st

≤ yl for all s, t ∈ V, l ∈ L, a ∈ l Θxst = bst for all s, t ∈ V with Wst > 0

l∈L ylcostl

≤ B xa

st, yl ∈ {0, 1}

this model assumes unlimited capacity of the vehicles with limited capacity A of the trains: relax xa

st and fl = yl to integers and replace

xa

st ≤ yl by

s,t∈V

xa

st ≤ flA for all l ∈ L, a ∈ l

and Θxst = bst is a network flow problem.

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SLIDE 120

Customer-oriented models Minimizing traveling time

Block structure of (LP4)

min

s,t∈V
a∈A Wstcaxa

st

s.t. xa

st

−yl ≤

for all s, t ∈ V, l ∈ L, a

Θxst = bst

for all s, t ∈ V with Wst

l∈L ylcostl

≤ B xa

st, yl ∈ {0, 1}

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SLIDE 121

Customer-oriented models Minimizing traveling time

Block structure of (LP4)

min

s,t∈V
a∈A Wstcaxa

st

s.t. xa

st

−yl ≤

for all s, t ∈ V, l ∈

Θxs1t1 . . . Θxsrtr = bs1t1 . . . bsrtr

l∈L ylcostl

≤ B xa

st, yl ∈ {0, 1}

Consequence: one block for each OD-pair s, t and a y-variable block.

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SLIDE 122

Customer-oriented models Minimizing traveling time

Variants of the coupling constraints

(VAR1) min

s,t∈V
a∈A Wstcaxa

st

s.t. xa

st

≤ yl for all s, t ∈ V, l ∈ L, a ∈ l Θxst = bst for all s, t ∈ V with Wst > 0

l∈L ylcostl

≤ B xa

st, yl ∈ {0, 1}

The coupling constraints can equivalently replaced by: (VAR2)

a∈Al xa

st

≤ |Al|yl ∀ l ∈ L, (s, t) ∈ R (VAR3)

(s,t)∈R xa

st

≤ |R|yl ∀ l ∈ L, a ∈ Al (VAR4)

(s,t)∈R
a∈Al xa

st

≤ |R||Al|yl ∀ l ∈ L

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SLIDE 123

Customer-oriented models Minimizing traveling time

How strong are the relaxations?

(VAR1) xa

st

≤ yl ∀ (s, t) ∈ R, a ∈ Al : l ∈ L (VAR2)

a∈Al xa

st

≤ |Al|yl ∀ l ∈ L, (s, t) ∈ R (VAR3)

(s,t)∈R xa

st

≤ |R|yl ∀ l ∈ L, a ∈ Al (VAR4)

(s,t)∈R
a∈Al xa

st

≤ |R||Al|yl ∀ l ∈ L

LPP2 LPP3 LPP1 LPP4

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SLIDE 124

Customer-oriented models Minimizing traveling time

Solving (LP4)

see Schöbel and Scholl, DROPS, 2006 excluding trivial solutions Dantzig Wolfe decomposition in different variants Branch and Price

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SLIDE 125

Customer-oriented models Minimizing traveling time

Literature

Maximize number of direct travelers with respect to upper line frequency requirements: Patz (1925), Wegel (1974), Dienst (1970) Reinecke (1992) and Reinecke (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 of traveling 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 and Scholl(2006)

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SLIDE 126

Other (more recent) models

Other recent approaches

Black-Box-Model of Deutsche Bahn Integrating the vehicle schedules in cooperation with GÖVB Game-theoretic approach Path-based models

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SLIDE 127

Other (more recent) models

Black-Box-Model

Line Concept Profit

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SLIDE 128

Other (more recent) models

Black-Box-Model

Line Concept Profit

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SLIDE 129

Other (more recent) models

Black-Box-Model

Line Concept Costs cost parameters Profit

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SLIDE 130

Other (more recent) models

Black-Box-Model

potential customers Line Concept Costs cost parameters Profit

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SLIDE 131

Other (more recent) models

Black-Box-Model

potential customers Line Concept Costs Modal Split cost parameters real number of customers Profit

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SLIDE 132

Other (more recent) models

Black-Box-Model

potential customers Line Concept Costs Income Modal Split cost parameters tariff system real number of customers Profit

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SLIDE 133

Other (more recent) models

Black-Box-Model

potential customers Line Concept Costs Income Modal Split cost parameters tariff system real number of customers Profit

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SLIDE 134

Other (more recent) models

Black-Box-Model

potential customers Line Concept Costs Income Modal Split cost parameters tariff system real number of customers Profit

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SLIDE 135

Other (more recent) models

Black-Box-Model

Profit =

s,t∈V

customerss,t · Prices,t −

l∈L

costl Costs: similar as in cost models Income: Determine number of customers and evaluate using the ticket prices

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SLIDE 136

Other (more recent) models

Black-Box-Model

Let 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 of travelers number of transfers Anita Schöbel (NAM)

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SLIDE 137

Other (more recent) models

Integrating vehicle schedules

Project work of Michael Schachtebeck

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SLIDE 138

Other (more recent) models

Integrating vehicle schedules

Project work of Michael Schachtebeck in cooperation with GÖVB

Der Stadtbus

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Legende

Haltestelle Haltestelle wird nur in Pfeilrichtung angefahren Endhaltestelle

13

VSN-Umsteigehaltestelle

gültig ab 12. Dezember 2004

Linien 1, 2 Linien 6, 7, 8, 13 Linien 3, 5, 9, 10 Linien 4, 14 Holtenser Berg <> Zietenterrassen Linie 1 Geismar-Charlottenburger Straße <> Weende-Nord Linie 2 Weende-Nord <> Grone-Süd Linie 3 Kauf Park <> Geismar-Schöneberger Straße Linie 4 Knutbühren/Hetjershausen/ Groß Ellershausen/Kauf Park/Grone <> Nikolausberg Linie 5 Klausberg <> Bahnhof Linie 6 Holtensen <> Zietenterrassen Linie 7 Weende-Ost/Papenberg <> Grone-Nord Linie 8 Ostviertel <> Leineberg Linie 9 Bahnhof <> Herberhausen/Roringen Linie 10 Geismar-Schöneberger Straße <> Holtenser Berg Linie 12 Geismar-Süd <> Elliehausen/Esebeck/Kauf Park Linie 13 Linie 14 Rosdorf <> Bovenden

24-Std. Hotline: (05 51) 99 80 99

5 3 4 6 9 10 13 8 7 12 1 14 3 2 8 5 10 10 6 9 13 2 4 7 1 5 5 14 8 12 13

andere Tarifzone andere Tarifzone Taxi zwischen Knutbühren und In der Wehm P+R

Anita Schöbel (NAM)

27. September 2006

75 / 78

SLIDE 139

Other (more recent) models

Integrating vehicle schedules

Project work of Michael Schachtebeck in cooperation with GÖVB

Der Stadtbus

Info-Telefon: 38 444 444 www.goevb.de Esebeck Holtensen Elliehausen 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 Plauener Straße Rosmarinweg Theaterstraße Friedrichstraße Jüdenstraße Geismartor Kornmarkt Markt Groner Straße Weender Straße-West Weender Straße-Ost Nikolaistraße Gotthelf-Leimbach-Str. Robert-Bosch-Breite Geismar-Süd Am Kalten Born Am Rischen Kampstraße Auf dem Paul Geismar Landstraße David- Hilbert- Straße Rudolf-Winkel-Str. Groner Tor Bahnhof Landgericht Grete- Henry-Straße Hannah-Vogt- Straße Ehrengard- Schramm-Weg Alva-Myrdal- Weg Alfred- Delp-Weg Gulden- hagen Lönsweg Am Steinsgraben Keplerstraße Bürgerstraße Nikolausberger Weg Am Papenberg Faßberg Burckhardtweg Kellnerweg Tammannstraße Goldschmidtstraße Petrikirchstraße Blauer Turm Posthof Stadt- friedhof Gottes- lager Tulpenweg Asternweg Schützen- platz Neues Rathaus 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 Wiesenstraße Cramer- straße Am Hasengraben Leibniz- straße Mitteldorfstraße Kurmainzer Weg Eislebener Weg Pommerneck Stettiner Straße Ortelsburger Straße Breslauer Straße Flüthenweg Görlitzer Straße An der Tillyschanze Himmelsruh Weidenbreite Mittelberg Spandauer Weg Sanders- beek Tegeler Weg Pfalz-Grona- Breite Im Rinschenrott Maschmühlenweg Hildebrandstraße Bahnhofsallee Campus Hennebergstraße Adolf-Hoyer-Str. Wilhelm-Lambrecht-Str. Florenz-Sartorius-Str. Marien- straße Otto-Frey- Brücke Levinpark 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-Brenner-Str. Gustav-Bielefeld-Straße Freibad Weende Friedhof Junkerberg Heinrich-A.-Zachariä-Bogen Karl-Schwarzschild-Weg Edward-Schröder-Bogen James-Franck-Ring Festhalle Weende 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 Menzelberg Lange Straße Rottenanger Auf der Lieth Eschen- breite Am Schlehdorn Augustiner- straße Ulrideshuser Straße Albaniplatz Hermann- Föge-Weg Nonnenstieg Eichendorffplatz Dahlmannstraße Wagnerstraße Corvinuskirche Hainbundstraße Jugendherberge Ewaldstraße Rohns Klausberg Stauffenbergring Thomas-Dehler-Weg Bramwaldstraße Zollstock Deisterstraße Elmweg Sollingstraße Süntel- weg Harzstr. Backhausstraße Krugstraße St.- Heinrich- Straße St.-Martini-Straße Siekhöhe Herbert-Quandt-Str. Dransfelder Straße Groß Ellershausen Mittelbergschule In der Wehm Hasenwinkel Am 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 Straße Eibenweg St.-Godehard- Kirche Am Kirschberge Grätzelstr. Walkemühlenweg Lotzestraße Zeppelinstraße Eiswiese Sandweg Jahnstadion Ascherberg Rischenweg Hambergstraße Freibad Rosdorf Obere Straße Hagenbreite Friedensstraße 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

Legende

Haltestelle Haltestelle wird nur in Pfeilrichtung angefahren Endhaltestelle

13

VSN-Umsteigehaltestelle

gültig ab 12. Dezember 2004

Linien 1, 2 Linien 6, 7, 8, 13 Linien 3, 5, 9, 10 Linien 4, 14 Holtenser Berg <> Zietenterrassen Linie 1 Geismar-Charlottenburger Straße <> Weende-Nord Linie 2 Weende-Nord <> Grone-Süd Linie 3 Kauf Park <> Geismar-Schöneberger Straße Linie 4 Knutbühren/Hetjershausen/ Groß Ellershausen/Kauf Park/Grone <> Nikolausberg Linie 5 Klausberg <> Bahnhof Linie 6 Holtensen <> Zietenterrassen Linie 7 Weende-Ost/Papenberg <> Grone-Nord Linie 8 Ostviertel <> Leineberg Linie 9 Bahnhof <> Herberhausen/Roringen Linie 10 Geismar-Schöneberger Straße <> Holtenser Berg Linie 12 Geismar-Süd <> Elliehausen/Esebeck/Kauf Park Linie 13 Linie 14 Rosdorf <> Bovenden

24-Std. Hotline: (05 51) 99 80 99

5 3 4 6 9 10 13 8 7 12 1 14 3 2 8 5 10 10 6 9 13 2 4 7 1 5 5 14 8 12 13

andere Tarifzone andere Tarifzone Taxi zwischen Knutbühren und In der Wehm P+R

Anita Schöbel (NAM)

27. September 2006

75 / 78

SLIDE 140

Other (more recent) models

A game-theoretic approach

Idea: Distribute freuqencies of lines equally over the network to avoid delays due to capacity constraints.

Anita Schöbel (NAM)

27. September 2006

76 / 78

SLIDE 141

Other (more recent) models

A game-theoretic approach

Idea: Distribute freuqencies of lines equally over the network to avoid delays due to capacity constraints. The line planning game: Players: lines Strategies: frequencies Cost function: Due to delays

Anita Schöbel (NAM)

27. September 2006

76 / 78

SLIDE 142

Other (more recent) models

A game-theoretic approach

Idea: Distribute freuqencies of lines equally over the network to avoid delays due to capacity constraints. The line planning game: Players: lines Strategies: frequencies Cost function: Due to delays see: Dissertation of Silvia Schwarze

Anita Schöbel (NAM)

27. September 2006

76 / 78

SLIDE 143

Other (more recent) models

A game-theoretic approach

Idea: Distribute freuqencies of lines equally over the network to avoid delays due to capacity constraints. The line planning game: Players: lines Strategies: frequencies Cost function: Due to delays see: Dissertation of Silvia Schwarze

Anita Schöbel (NAM)

27. September 2006

76 / 78

SLIDE 144

Other (more recent) models

Other approaches

BlackBox Model of Deutsche Bahn Integrating the vehicle schedules in cooperation with GÖVB Game-theoretic approach Path-based models

Anita Schöbel (NAM)

27. September 2006

77 / 78

SLIDE 145

Other (more recent) models

Other approaches

BlackBox Model of Deutsche Bahn Integrating the vehicle schedules in cooperation with GÖVB Game-theoretic approach Path-based models

◮ Laporte, Mesa and Ortega (optimize modal split) ◮ Borndörfer and Pfetsch (generate lines dynamically) Anita Schöbel (NAM)

27. September 2006

77 / 78

SLIDE 146

Other (more recent) models

Other approaches

BlackBox Model of Deutsche Bahn Integrating the vehicle schedules in cooperation with GÖVB Game-theoretic approach Path-based models

◮ Laporte, Mesa and Ortega (optimize modal split) ◮ Borndörfer and Pfetsch (generate lines dynamically)

→ next Lecture!

Anita Schöbel (NAM)

27. September 2006

77 / 78

SLIDE 147

Other (more recent) models

The end . . .

THANK YOU!

Anita Schöbel (NAM)

27. September 2006

78 / 78