Mode l s f o r r o l l i ng s t o ck p l ann ing - - PowerPoint PPT Presentation

θ / . ρ

NSRZKLA4-P1

Mode l s f

• r

r

• l

l i ng s t

• ck

p l ann ing

Leo Kroon , AM O RE mee t i ng Pa t ra s , Oc t /Nov 2001

θ / . ρ

NSRZKLA4-P2

Ro l l i ng s t

• ck

p l anni ng

• I

n r u sh hour s : Al l

• c

a t i

• n
• f

s c a r c e c apa c i t y

• Ou

t s i d e r u s h hou r s : Ef f i c i e n cy

θ / . ρ

NSRZKLA4-P3

Kop loper wi t h 3

• r

4 c a r r i ag e s Doub l e Decker w i th 3

• r

4 c a r r i ag e s

θ / . ρ

NSRZKLA4-P3

θ / . ρ

NSRZKLA4-P4

1731

Rtd Ut Dv A mf Es

21731 1 6 3 6 1 6 4 1 7 3 8 1 7 3 4 1 6 3 2 1 7 3 20533 21735 20537 21739 20541 529 533 1735 537 1739 1 7 3 1 1 6 2 9 1 7 2 7 1 6 3 3 1 7 3 5 1 6 3 7 524 1726 528 1730 532 1734

Gvc Ut

1731 1 7 3 5 537 1 7 3 9 5 3 3 541

9 :00 10 :00 11 :00 12 :00 9 :30 10 :30 11 :30

1 6 3 6

θ / . ρ

NSRZKLA4-P5

Line 3000

From: Den Helder (Hdr) To: Nijmegen (Nm) Freq: 2 trains per hour (both directions)

Asd Ut Ah N m Hdr A mr

θ / . ρ

NSRZKLA4-P6

Hdr Amr Ah Nm

Details per station

Hdr: Incoming compositions of the trains are unchanged Amr: Going north: units may be uncoupled from the rear Going south: units may be coupled to the front Ah: Front and rear of the train are interchanged Nm: Units may be coupled or uncoupled (not both) at the (incoming) front

Details per track

Hdr-Amr: Max. length of the trains is 9 carriages Amr-Nm: Max. length of the trains is 12 carriages

Line 3000: 12 compositions (5 Koploper, 7 Double Decker)

θ / . ρ

NSRZKLA4-P7

Questions:

• What is the minimum capacity of the rolling stock that is necessary for

providing a certain service level?

• What is the minimum number of carriage kilometers that is necessary

for providing a certain service level?

• What is the maximum service level that can be provided with a given

capacity of rolling stock?

• What is the maximum service level that can be provided with a given

capacity of rolling stock and within a given number of carriage kilometers?

• …

θ / . ρ

NSRZKLA4-P8

Minimum required capacity per trip:

• Based on counting figures by conductors

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

µ+σ µ

15%

θ / . ρ

NSRZKLA4-P9

1 type of train units: min cost flow problem

with additional constraints

Additional constraints

• Service constraints
• Going north in Amr: units must not be be coupled
• Going south in Amr: units must not be be uncoupled
• Max. parking space at/near stations
• Circulation constraints

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NSRZKLA4-P10

2 types of train units: multi-commodity flow problem

However: not only the # of train units per train, but also their order in the train is relevant, because of the shunting possibilities at the stations

Example

?

• nly the red unit can be uncoupled

Hdr Nm Amr

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NSRZKLA4-P11

The approach of Schrijver & Groot:

• Composition graph per composition: look for paths in these graphs
• Link this graph to the multi-commodity flow problem

Hdr Amr Nm Amr Amr

……

3 33 333 3333 4 44 444 34 43 334 343 433 344 434 443

θ / . ρ

NSRZKLA4-P12

The approach of Schrijver & Groot:

Algorithm:

• Solve the integer multi-commodity flow problem

without taking into account the composition graphs

• Fix the total number of units of the two types
• For each of the trips do

For each of the possible capacities do

• solve the continuous multi-commodity flow problem
• if the capacity does not fit for the trip, then delete

the corresponding nodes from the composition graph

• If one of the composition graphs becomes disconnected,

then increase the total number of train units, and restart

• Otherwise solve the integer multi-commodity flow problem,

thereby taking into account the reduced composition graphs

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NSRZKLA4-P13

Nm Nm Nm Amr Hdr Amr Amr Amr Amr Amr Hdr

An alternative approach (Set Covering ++):

Example for 1 composition:

• Generate all potential duties
• Select an appropriate subset of the potential duties

that fit together and are optimal in some sense

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NSRZKLA4-P14

Relevant constraints:

• If a unit is coupled underway in X and

uncoupled underway in Y, then X=Y

A A

• LIFO coupling and uncoupling for each side of the train

A

• Coupling and uncoupling cannot take place at the same time and station

θ / . ρ

NSRZKLA4-P15

Objective:

A combination of shortages of seats (in rush hours) and efficiency (outside rush hours)

Decision Variables:

Xd

= potential duty d is/is not used

Td

= # train units of length 3 in duty d

Fd

= # train units of length 4 in duty d Decision variables for shortages on trips (1st and 2nd class) Decision variables for stock keeping of train units in stations (Hdr, Amr, Nm)

θ / . ρ

NSRZKLA4-P16

Constraints:

d d d d

X M F T X ⋅ ≤ + ≤

∑

∈

+ − ≥

d t d d d t t

F c T c P S

: 24 23 2 2

) ( 1 1

} ' : ' { ' ≤

+

∑

d d d d d

X M X

!"

Maximum/minimum train lengths on tracks Stock keeping of both types of train units in stations

∑

∈

+ − ≥

d t d d d t t

F c T c P S

: 14 13 1 1

) (

for all duties d for all trips t for all trips t for all duties d

θ / . ρ

NSRZKLA4-P17

2 ) (

:

≥ +

∑

∈d t d d d

F T

Valid inequalities:

3 ) 2 (

:

≥ +

∑

∈d t d d d

F T Valid inequalities for trips outside rush hours:

2 : 24 23

) (

t d t d d d

P F c T c ≥ +

∑

∈

Example: suppose

24 23 2 23

2 c c P c

t

+ ≤ ≤

Original constraint:

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NSRZKLA4-P18

Single Deck and Double Deck combined:

d d d d

X M F T X ⋅ ≤ + ≤

d d d d

XX M FF TT XX ⋅ ≤ + ≤ 1 ≤ +

d d

XX X

c d

Double X − ≤1

c d

Double XX ≤

for all duties d for all duties d for all duties d for all compositions c and duties d involving composition c

}

Extension of the valid inequalities for trips outside rush hours

θ / . ρ

NSRZKLA4-P19

Computa t i

• na

l r e su l t s

Imp l emen t ed i n OPL S t ud i

• (

ILOG) , b a s ed

• n

2001 t ime t a b l e So l v ed by CPLEX 7 .

• n

PC wi t h 900 MHz , 2 56 M memor y Only Doub l e Deck ( 3 & 4 ) # v a r i a b l e s / # c

• n

s t r a i n t s :

• a

bou t 1100 / 2000 compu t a t i

• n

t i me :

• m

i nu t e s

• h
• ur

s

• v

a l i d i n equa l i t i e s s

• me

t ime s u s e f u l Doub l e Deck and S ing l e Deck comb ined ( 3 & 4 ) # v a r i a b l e s / # con s t r a i n t s :

• a

bou t 2000 / 3200 compu t a t i

• n

t i me :

• h

i gh l y d ependen t

• n

a v a i l a b l e c apa c i t y : m inu t e s

• d

ay s

• v

a l i d i n equa l i t i e s c r u c i a l Genera l r emark s

• c

l

• s

i ng t h e g a p t a k e s t ime

• m

i no r imp rovemen t s by a l l

• wing

s ho r t a g e s

• u

t s i d e r u s h hou r s

θ / . ρ

NSRZKLA4-P20

Computa t i

• na

l r e su l t s Computa t i

• na

l r e su l t s ( Doub l e Decker )

W1 = 2 W2 = 1 WS = 4 W C K = 1 No sho r t a g e s

• u

t s i d e r u sh hou r s : ( 9 : 30

• 15

: 30 ) + ( 18 : 30 ,

• >

) t

• t

s hor t ckm t (VI ) t ( no VI ) i n f 8 361 1533 2229 1 5 1 5 18 / 14 12546 2600 2146 2 18 5 24 18 / 12 18193 4028 2081 2651 1310 18 / 10 24878 5726 1974 5 51 1449 16 / 10 30322 7104 1906 2 50 1 32 15 / 10 33621 7943 1849 1 94 1 38 16 / 9 34811 8246 1827 9 9 9 9 15 / 9 38897 9280 1777 8 9 6 3

θ / . ρ

NSRZKLA4-P21

Computa t i

• na

l r e su l t s ( Doub l e Decker )

W1 = 2 W2 = 1 WS = 4 W C K = 1 Fu l l

• p

t im i s a t i

• n w

i t hou t Va l i d I n equa l i t i e s t

• t

shor t ckm t ∆ ∆ ∆ ∆ t

• ∆

∆ ∆ ∆ t

• t

i n f 8 361 1533 2229 2 5 18 / 14 12546 2600 2146 4 48 230 18 / 12 18193 4028 2081 2048 738 18 / 10 24806 5702 1998 1190 6 39 7 2 16 / 10 30118 7055 1898 1 97 6 5 2 04 15 / 10 33126 7819 1850 5 97 359 4 95 16 / 9 34083 8059 1847 1 83 8 4 7 28 15 / 9 37423 8912 1775 1 15 5 2 1474

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NSRZKLA4-P22

Asd Sh l Amf Zl Lw Gn Ut Gvc Rtd Es Dv

Combin ing and sp l i t t i n g

Noord

• Oos

t Nor th

• Eas

t

500 = R td /Gvc- Gn /Lw 700 = Asd /Sh l- Gn /Lw 1600 = Asd /Sh l-E s 1700 = R td /Gvc- E s Al l s e r i e s : 1 x p e r hou r Al l t r a ck s : 2 x p e r hou r

θ / . ρ

NSRZKLA4-P23

1731

Rtd Ut Dv A mf Es

21731 1 6 3 6 1 6 4 1 7 3 8 1 7 3 4 1 6 3 2 1 7 3 20533 21735 20537 21739 20541 529 533 1735 537 1739 1 7 3 1 1 6 2 9 1 7 2 7 1 6 3 3 1 7 3 5 1 6 3 7 524 1726 528 1730 532 1734

Gvc Ut

1731 1 7 3 5 537 1 7 3 9 5 3 3 541

9 :00 10 :00 11 :00 12 :00 9 :30 10 :30 11 :30

1 6 3 6

θ / . ρ

NSRZKLA4-P24

Objective:

• Minimize shortages of seats during the rush hours

Decision Variables:

• Tfrom_to_t = # units of length 3 on train t on the route from from to to
• Ffrom_to_t = # units of length 4 on train t the route from from to to
• Decision variables for shortages on trips (1st and 2nd class)
• Decision variables for stock keeping of train units in stations

θ / . ρ

NSRZKLA4-P25

Constraints:

• Link between the allocated capacities per train and the shortages per trip
• Stock keeping of both types of train units in stations
• Maximum/minimum train lengths on tracks
• Combining and splitting in Utrecht, Amersfoort, Zwolle
• Return trips on end points
• Shunting in Deventer

θ / . ρ

NSRZKLA4-P26

Compl i c a t i ng f a c t

• r

: c

• mb

i n ing i n Z land s p l i t t i ng i n Amf Compos i t i

• n

b e tween Z la ndAmf : f r

• m Lw

f r

• m Lw

f r

• m Gn

f r

• m Gn

Gn-Asd d i r e c t

• r Lw-Sh

ld i r e c t , b u t no t b

• t

h

) 1 ( 5

t Lw_Shl_t Lw_Shl_t

Y F T − ≤ + } 1 , { ∈

t

Y ) 1 ( 5

t Gn_Asd_t Gn_Asd_t

Y F T − ≤ +

θ / . ρ

NSRZKLA4-P27

} 1 , { ∈

t

Z

Compl i c a t i ng f a c t

• r

: r e turn i n A sd Compos i t i

• n

u pon a r r i v a l i nAsd : Capac i t y c an b e r e a l l

• c

a t e d i n t

• max

ima l l y

• ne

d i r e c t i

• n

f r

• m Lw

f r

• m Lw

f r

• m Gn

f r

• m Gn

t Gn_Asd_t Asd_Gn_r

Z T T 5 + ≤ ) 1 ( 5

t Lw_Asd_t Asd_Lw_r

Z T T − + ≤

t Gn_Asd_t Asd_Gn_r

Z F F 5 + ≤

) 1 ( 5

t Lw_Asd_t Asd_Lw_r

Z F F − + ≤

r = r(t)

θ / . ρ

NSRZKLA4-P28

Mode l f

• r

t h eNoord Oos t Imp l emen t ed i n OPL S t ud i

• (

ILOG) So l v ed by CPLEX 7 .

• n

PC wi t h 900 MHz , 2 56 M Tes t ed

• n

p r e l i mina ry NSR 2001 t ime t ab l e ( Monday ) 4 l i n e s , 5 t r i p s i n e a ch d i r e c t i

• n

Compu t a t i

• n

t i me : < 3 m inu t e s # v a r i a b l e s : abou t 1000 # con s t r a i n t s : abou t 1000 Manua l s

• l

u t i

• n

: 4528 s e a t s s ho r t ( 2

nd c

l a s s ) 1 88 s e a t s s ho r t ( 1

s tc

l a s s ) Opt ima l s

• lu

t i

• n

: 3385 s e a t s s ho r t ( 2

nd c

l a s s ) 1 05 s e a t s s ho r t ( 1

s tc

l a s s ) Unl im i t ed c apac i t y : 1874 s e a t s s ho r t ( 2

nd c

l a s s ) 3 4 s e a t s s ho r t ( 1

s tc

l a s s )

θ / . ρ

NSRZKLA4-P29

200 400 600 800 1000 1200 1400 1600 1800

shortage

500 700 1600 1700

train series INF OPT NU

Shor tage s i n t h e Nor th

• Eas

t