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

Mode l s f o r r o l l i ng s t o ck p l ann ing Leo Kroon , AM O RE mee t i ng Pa t ra s , Oc t /Nov 2001 NSRZKLA4-P1 θ / . ρ

Ro l l i ng s t ock p l anni ng •I n r u sh hour s : Al l o c a t i on o 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 NSRZKLA4-P2 θ / . ρ

Kop loper wi t h 3 o r 4 c a r r i ag e s Doub l e Decker w i th 3 o r 4 c a r r i ag e s NSRZKLA4-P3 NSRZKLA4-P3 θ / θ / . ρ . ρ

9 :00 9 :30 10 :00 10 :30 11 :00 11 :30 12 :00 Gvc 537 1 1731 5 1 541 7 7 3 3 3 3 9 5 Ut Rtd 21731 20533 21735 20537 21739 20541 Ut 529 533 1734 1731 528 1730 1735 532 537 1739 524 1726 A mf 1 1 1 1 1 1 6 6 7 6 7 7 2 3 3 3 3 2 9 7 7 1 3 5 6 3 6 1 Dv 2 6 0 4 8 0 3 3 3 3 3 4 6 7 7 6 7 6 1 1 1 1 1 1 Es NSRZKLA4-P4 θ / . ρ

Line 3000 Hdr From: Den Helder (Hdr) A mr To: Nijmegen (Nm) Freq: 2 trains per hour Asd (both directions) Ah Ut N m NSRZKLA4-P5 θ / . ρ

Line 3000: 12 compositions (5 Koploper, 7 Double Decker) 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 NSRZKLA4-P6 θ / . ρ

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? • … NSRZKLA4-P7 θ / . ρ

Minimum required capacity per trip: • Based on counting figures by conductors 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 15% 0 µ µ+σ NSRZKLA4-P8 θ / . ρ

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 NSRZKLA4-P9 θ / . ρ

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 Hdr ? Amr Nm only the red unit can be uncoupled NSRZKLA4-P10 θ / . ρ

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 NSRZKLA4-P11 θ / . ρ

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 NSRZKLA4-P12 θ / . ρ

An alternative approach (Set Covering ++): Example for 1 composition: Hdr Amr Amr Amr Nm Amr Amr Nm Amr Hdr Nm • Generate all potential duties • Select an appropriate subset of the potential duties that fit together and are optimal in some sense NSRZKLA4-P13 θ / . ρ

Relevant constraints: • LIFO coupling and uncoupling for each side of the train A A • Coupling and uncoupling cannot take place at the same time and station A • If a unit is coupled underway in X and uncoupled underway in Y , then X=Y NSRZKLA4-P14 θ / . ρ

Objective: A combination of shortages of seats (in rush hours) and efficiency (outside rush hours) Decision Variables: X d = potential duty d is/is not used T d = # train units of length 3 in duty d F d = # 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) NSRZKLA4-P15 θ / . ρ

Constraints: ≤ + ≤ ⋅ X T F M X for all duties d d d d d ≥ − + ∑ 2 2 S P ( c T c F ) for all trips t t t 23 d 24 d ∈ : d t d ≥ − + ∑ 1 1 for all trips t S P ( c T c F ) t t 13 d 14 d ∈ : d t d 1 + ' ≤ ∑ X X 1 for all duties d d d M !" { ' : ' } d d d Maximum/minimum train lengths on tracks Stock keeping of both types of train units in stations NSRZKLA4-P16 θ / . ρ

Valid inequalities for trips outside rush hours: ≤ ≤ + 2 2 c P c c Example: suppose 23 t 23 24 Original constraint: + ≥ ∑ 2 ( c T c F ) P 23 d 24 d t ∈ : d t d Valid inequalities: ∑ + ≥ ( T F ) 2 d d ∈ d : d t ∑ + ≥ ( T 2 F ) 3 d d ∈ d : d t NSRZKLA4-P17 θ / . ρ

Single Deck and Double Deck combined: ≤ + ≤ ⋅ X T F M X for all duties d d d d d ≤ + ≤ ⋅ XX TT FF M XX for all duties d d d d d + ≤ 1 X XX for all duties d d d ≤ 1 − X Double } d c for all compositions c and duties d involving composition c XX ≤ Double d c Extension of the valid inequalities for trips outside rush hours NSRZKLA4-P18 θ / . ρ

Computa t i o na l r e su l t s Imp l emen t ed i n OPL S t ud i o ( ILOG) , b a s ed on 2001 t ime t a b l e So l v ed by CPLEX 7 . 0 on 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 on s t r a i n t s : -a bou t 1100 / 2000 compu t a t i o n t i me : - m i nu t e s -h our s -v a l i d i n equa l i t i e s s ome 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 o n t i me : -h i gh l y d ependen t on 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 o 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 o wing NSRZKLA4-P19 s ho r t a g e s ou t s i d e r u s h hou r s θ / . ρ

Computa Computa t t i i o o na na l l r r e e su su l l t t s s ( Doub l e Decker ) W1 = 2 W2 = 1 WS = 4 W C K = 1 No sho r t a g e s ou t s i d e r u sh hou r s : ( 9 : 30 - 15 : 30 ) + ( 18 : 30 , -> ) t o 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 NSRZKLA4-P20 θ / . ρ

Computa t i o 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 o p t im i s a t i o n w i t hou t Va l i d I n equa l i t i e s ∆ ∆ ∆ ∆ - ∆ ∆ ∆ ∆ t o t shor t ckm t t t o t i n f 8 361 1533 2229 2 0 5 0 18 / 14 12546 2600 2146 4 48 230 0 18 / 12 18193 4028 2081 2048 738 0 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 NSRZKLA4-P21 θ / . ρ

Gn Combin ing and sp l i t t i n g Lw Noord -Oos t Nor th -Eas t Zl 500 = R td /Gvc- Gn /Lw 700 = Asd /Sh l- Gn /Lw Asd 1600 = Asd /Sh l-E s Dv Es 1700 = R td /Gvc- E s Sh l Amf Gvc Ut 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 Rtd NSRZKLA4-P22 θ / . ρ

9 :00 9 :30 10 :00 10 :30 11 :00 11 :30 12 :00 Gvc 537 1 1731 5 1 541 7 7 3 3 3 3 9 5 Ut Rtd 21731 20533 21735 20537 21739 20541 Ut 529 533 1734 1731 528 1730 1735 532 537 1739 524 1726 A mf 1 1 1 1 1 1 6 6 7 6 7 7 2 3 3 3 3 2 9 7 7 1 3 5 6 3 6 1 Dv 2 6 0 4 8 0 3 3 3 3 3 4 6 7 7 6 7 6 1 1 1 1 1 1 Es NSRZKLA4-P23 θ / . ρ