Transit Network Design under Stochastic Demand

Transcription

1 Transit Network Design uner Stochastic Deman Hong K. LO Civil an Environmental Engineering Hong Kong University of Science an Technology 1

2 Public transport Backgroun Fixe route transit (FRT) : fixe route an fixe scheule, large capacity, exclusive right of way, such as metro, bus, or regular ferry Deman responsive transit (DRT): flexible route, small or meium vehicle, complementary to FRT, such as riesharing services, taxi, etc. Transit Network Design Problem (TNDP) Fin a most cost efficient operating plan for FRT while satisfying passenger eman Challenges in TNDP Jointly optimize the FRT an DRT as an integrate system uner stochastic eman, har capacity constraints, an user equilibrium flows

3 Scheule-base TNDP Backgroun Ferry service network (Lai an Lo, 004, Wan an Lo, 009) Time-space network Decision variables: route, scheule, fleet size Integer (a lot) an real variables Frequency-base TNDP Rapi transit network (Bruno et al. 1998, Laporte et al. 005, Samanta et al. 011, Marin, 007, Wan an Lo 003, 009) Objective: minimize construction an passenger cost Decision variables: route alignment, service frequency Integer an real variables 3

4 Backgroun Deterministic TNDP Given OD matrix First formulate by Magnanti (1984) as a mixe integer linear program (MILP) Stochastic TNDP Stochastic programming approach (Ruszczynski, 008) Stochastic eman follows a known probability istribution Monte-Carlo simulation to approximate the cost expectation by sample average, two-stage stochastic problem Robust optimization approach Stochastic eman capture by an uncertainty set Min-max problem, worst case scenario (Ben-tal et al., 004) 4

5 Backgroun Solution algorithm for stochastic approach Exact metho: multi-imensional integral evaluation, formiable task Heuristic approach: search the neighborhoo of the initial solution, efficient but solution quality not guarantee (Hoff et al. 011) Approximation metho: L-shape/ Multi-cut metho, long computation time, global optimal solution is not guarantee Solution algorithm for robust optimization approach Polyheral uncertainty set, linearization Same imension as its eterministic counterpart Conservative, epenent on the size of the uncertainty set 5

6 Objectives To evelop a moeling framework for combining FRT an DRT network esign uner stochastic eman Investigate the benefits of the integrate services uner stochastic eman Develop a service reliability (SR) base formulation an solution algorithm to aress eman uncertainty To assess the performance of SR-base formulation Two application contexts: ferry network an rapi transit network Two passenger flow istribution pattern: system optimal (SO) an user equilibrium (UE) 6

7 Transit Network Design with Stochastic Deman uner System Optimal Flows Lo, H., K. An an W. Lin Ferry Network Design uner Deman Uncertainty. Transportation Research Part E, 59,

8 Time-space network escription Ferry time-space network Passenger time-space network Time Origin Pier Intermeiate Piers Destination Pier Pier I Pier II Pier N 07:00 Legen 07:00 07:15 Wait arc Time 07:15 07:30 3:30 4:00.. Legen 07:30 Wait arc 3:30 Service arc 4:00 Notations M.. Service arc Origin arc Destination arc Deman arrivals f S Y = { Yij}, ij f W Ferry service arc Ferry waiting arc S X = { X ij }, ij W Passenger service arc Passenger waiting arc 8

9 Research objective Regular services A-hoc services Regular ferry services (FRT) Fixe scheule Large capacity Low unit cost + A-hoc ferry services (DRT) Flexible scheule Small capacity High unit cost Objectives Capture eman uncertainty on ferry service eployment Develop a moeling framework for combining FRT an DRT 9

10 Problem challenge Objective: Regular Services cost + A-hoc services cost + Passenger cost Decisions: Regular service Scheule (Y) A-hoc services eployment (Z) Y Y X 1 Y Y X X 1 X X 3 X 3 Z 1 Z The original problem Z 3 The eployment of these two services are relate Two-stage stochastic program Represent stochastic eman by a large number of iscrete scenarios A large size MILP 10

11 Service reliability ρρ The probability of passengers carrie by regular ferry services A vector, one for an OD pair Service reliability Regular service A-hoc service Traeoff high more less low Less more 7:00 B 1 X Y X 14, Y=1 14, Z 1 B 1 B 7:30 B X 3 X, Y, Z 5 5 Y = 4 8: ρ 1 =0.3 ρ =0.7 11

12 SR-base stochastic formulation (Phase-1) Phase-1 regular service eployment Y (1) min Y j N i N j N \ N Yc ij F ij ij ij S f f f f b b 1 Yc Fixe cost B if i is the O of OD X ij X ki = B if i is the D of OD i N k N 0 otherwise Operating cost Passenger flow conservation () B = Ψ 1 ( ρ ) 0 ρ 1 Stochastic eman Service reliability constraint Deman realize within the service reliability bounary can be carrie by regular ferry services BB (ρ) 1

13 SR-base stochastic formulation (Phase-1) Y Decision variable min Y i N j N \ N Yc ij F ij ij ij S f f f f b b Yc Fixe cost Operating cost (3) 3 j N Y Y = 0 i N ij k N 0 Yij Uij Y integer ij A ij ki Regular services connection constraints (4) f f f b b i N j N \ N Y ij V Fleet size constraint (5) R X Y ζ ij S ij ij f Capacity constraints 13

14 SR-base stochastic formulation (Phase-) Phase- A-hoc services eployment Ze Z e (1) () min Z e j N R 3 Q( ρ) = pe cz e + cx ij ij, e e E R R ij A A-hoc cost Passenger cost = Be Ze,if kisthe O of OD Xkj, e Xik, e = Ze Be,if kisthe D of OD, k N, R i N = 0 otherwise X Y ζ ij S ij, e ij 1 f Passenger flow conservation Capacity constraints Stochastic eman Deman beyon the service reliability is to be carrie by a-hoc services 14

15 SR-base graient approach 15 Phase-1 Phase- ρ Y 1.. ρ ρ ρ ρ ρ ρ ρ + Total cost g Q ( ) g Q φ = + ρ 1 1 ( ) ( ) φ φ ρ ρ ρ ρ Disavantage: computation time highly epens on the number of OD pairs ' ρ Fin the optimal service reliability ρ ρ Fin the graient of total cost w.r.t ρ increase ρ by a small number each time until Y changes FRT DRT ' Y

16 Avantage of the SR-base moel Y Y Y X X Z X Z X Z The original problem L-shape metho SR-base graient metho The constraint structures Take avantage of the special structure of the problem Separate the large size MILP into one smaller size MILP an one LP Can be extene to inclue the user equilibrium assignment principle 16

17 Transit Network Design with Stochastic Deman uner User Equilibrium Flows An, K. an H. Lo Ferry Service Network Design with Stochastic Deman uner User Equilibrium Flows. Transportation Research Part B, 66,

18 Scheme A (with passenger reservation) Allow avance reservation for passengers Passenger eman for the next ay is obtaine one-ay in avance Provie sufficient amount of a-hoc services consiering user equilibrium 7:00 Pier A B 1 1 X 1, Y, Z B Y =1 1 B B 1 X 1 7:30 8:00 B X 3 3 Y 5, = 5, X Y Z Realize eman Realize eman Deman realization b1=300, b=300, regular service capacity= 50, 500 no nee to provie a-hoc services, reuce operating cost 18

19 Scheme A (with passenger reservation) User equilibrium with stochastic eman Not sensible to fin the long-term UE as eman varies from ay to ay Fin the short-term UE : for a eman realization, same minimum traveling cost for passengers on the same origin-estination (OD) Passenger options: take the congeste irect service, wait for the next irect service or take a etour 7:00 7:30 Pier A B C B 1 1 X 1 X, Y, Z X, Y, Z 1 B 4 X 3 6 8: Travelling cost= waiting + on vehicle + overflow elay 19

20 Stochastic Formulation uner UE (Phase-1) Phase-1 Regular ferry services routes an scheule Y The same as SO min Y j N i N j N \ N Yc ij F ij ij ij S f f f f b b input Yc B if i is the O of OD X ij X ki = B if i is the D of OD i N k N 0 otherwise Regular services cost 1)Regular services connectivity constraints )Passenger flow conservation constraints 3)Capacity constraints 0

21 Stochastic Formulation uner UE (Phase-) Phase- Dynamic a-hoc services eployment Ze e for ay scenario min Z e 3 Q( ρ) = pe cz e + cij Xij, e + βij, e Xij, e e E ij A ij A A-hoc cost Travel time Overflow elay (P4.) For a certain a-hoc services cost min f X,Z = cij ij A X ij, e Passenger cost θ θ θ Travel time cost e,min e e,max j N D Be Ze,if kisthe origin of OD Xkj, e Xik, e =, k N i N 0 otherwise X ζ Y ij A ij, e ij, cz = θ 4 e e input β ij, e ij A A-hoc cost βij, e X ij, e Overflow elay Proposition 4.1: P4. yiels a UE flow pattern uner capacity constraints, with the negative Lagrange multiplier associate with the link capacity constraint representing the corresponing passenger overflow elay. 1

22 SR-base graient solution proceure Phase-1:Regular ferry services routes an scheule Phase-:Dynamic a-hoc eployment Total cost= Phase-1 + Phase- cost * ρ B 1 Fin the optimal service reliability * θ Fin the optimal a-hoc services provision level Increase θ by κ each time an the one with the lowest cost is kept as the optimal solution cost x Phase- cost a-hoc cost travel time cost overflow elay cost

23 Scheme B (no avance reservation) Passenger eman is reveale only when they arrive at the piers Provie a-hoc services whenever there is eman overflow 7:00 Pier A B 1 1 B Y =1 X 141, Y 14, Z 1 B B 1 X 1 7:30 8:00 B X 3 3 Y 5, = 5, X Y Z Realize eman Realize eman Deman realization b1=300, b=300, provie a-hoc services 50, spaces waste for B 3

24 Scheme B (no avance reservation) No waiting or etour, only irect service, best for passengers Most costly plan for the company Pier A B C 7:00 7:30 B 1 B X 1 X 3 1 X, Y, Z X, Y, Z : Travelling time = on vehicle travel time 4

25 SR-base stochastic formulation (single phase) Phase-1: Regular ferry services routes an scheule Y Up to eman B = Ψ 1 ( ρ ) Phase-: Dynamic a-hoc eployment Expecte a-hoc services cost can be calculate by Single phase problem Z = max{0, B B } e, e, θ ( ρ) = Ψ B B, e 3 c Ze, e 1 4 min φ( ) = ( ) T T ρ c Y + θ + (c ) t B h,y, ρ Regular service cost A-hoc service cost Passenger cost s.t. Regular services connection constraints Fleet size constraint Capacity constraints Passenger flow conservation Value of reservation: cost ifference between scheme A an B 5

26 Numerical stuy Ferry service network in Hong Kong 15 min, $ min, $ min, $ 5000 Stuying horizon: 7am-9 am Time interval: 15 min Time Slice CBD-MW MW-CBD CBD-PC PC-CBD MW-PC PC-MW

27 Solution proceure illustration Different starting points comparison Computation time: SR-base metho : 17 secons versus L-shape metho: 6 hours 7 7

28 Result analysis Group: ifferent unit a-hoc cost an COV (coefficient of variation) Total company cost uner SO is lower than that uner UE The value of reservation between scheme A an B increase with COV.5 x 105 cost(hk$) Regular service cost (SO) A-hoc service cost (SO) Regular service cost (Scheme A) A-hoc service cost (Scheme A) Regular service cost (Scheme B) A-hoc service cost (Scheme B) group inex Service eployment comparison between UE an SO solutions 8 8

29 Robust Rapi Transit Network Design An, K. an H. Lo Two-Phase Stochastic Program for Transit Network Design uner Deman Uncertainty. Transportation Research Part B, 84, An, K. an H. Lo Robust Transit Network Design with Stochastic Deman Consiering Development Density. 1th International Symposium on Traffic an Transportation Theory (ISTTT) an Transportation Research Part B, 81,

30 Research objectives Rapi TNDP: location of stations, route alignment an frequency + Rapi transit services (FRT) Dial-a-rie services (DRT) Objectives Investigate the TNDP by robust optimization The optimal level of robustness to minimize the system cost Methoology Apply the SR-base two-phase formulation Improve the efficiency of the graient solution proceure 30

31 Robust optimization to aress eman Robust optimization uncertainty Day to ay variation in eman, min-max problem, worst case scenario (Ben-tal et al., 004) Conservative solutions Evaluate the outcome of uncertainty set Search for the optimal robustness level to hege against uncertainty Deman fluctuation, uneconomical to rely on transit lines alone Stochastic eman Uncertainty set system cost vs. robustness level 31

32 Service reliability to aress eman uncertainty Service reliability ρρ: the probability that passengers can be carrie by rapi transit services Conveys the level of robustness, size of the uncertainty set Dial-a-rie services cost evaluate the outcomes beyon the uncertainty set B 1 ρ 1 =0.3 Rapi transit services + Dial-a-rie services 3

33 Network representation Multi-line esign: ummy origin an estination Passenger transfer an waiting: line sub noe an station sub noe Dummy O an D: ark gray noe Dummy arcs: black ashe line Transfer time: blue line Get on/off time: green line On vehicle travel time: orange line Deman: impose on the station sub noe Dummy O an D for line 1 Line 1 sub noe Station sub noe Line sub noe Dummy O an D for line S 1 T 1 A 1 C 1 B 1 eman eman eman A 0 B 0 C 0 A C B S T 33

34 Two-phase Robust Formulation uner System Optimal Flows 34

35 Two phase robust formulation Phase-1 Rapi Transit line (RTL) alignment an frequencies WYf,, min r r W,f,Y c fy + cy + cw 1 3 ij r ij ij ij i r R ij A r R ij A i N Transit operating Line construction Station construction Phase- Dynamic ial-a-rie services eployment Z e 3 4 ( ρ) = + Passenger cost min Q pe cz e cij Xij, e Z e h H ij A ij A D Dial-a-rie cost Total cost= Phase-1 + Phase- cost Decision variables: W: Station location Y: Route alignment f: Frequency Z: Dial-a-rie services 35

36 5.1 Solution algorithm : fin the ecent irection (revise) 1 ρ ρ.. ρ ρ Phase-1 r r ( ) min g= c fy + cy + cw ' ρ f,y 1 1 ρ + ρ ρ.. ρ 1 3 ij r ij ij ij i r R ij A i N Y f Q= e E pq e Phase- e 3 4 min Qe = cz e + cij Xij, e, e E XZ, D ij A D r r s.. t Xij, h C ( Yij + Yji ) fr, ual βij, h r 1 r r g Q= β, * CY ( Y) f h + h ij r φ ρ Total cost ( ρ) g Q( ρ) φ = + ij h ij ji r φ ( ρ ) φ ρ Avantages: computation time oes not epen on the number of OD pairs Assumption: A small perturbation in ρ changes frequency f only while maintaining the line alignment Y 36

37 Two Phase Robust Formulation uner User Equilibrium Flows 37

38 . 5. Two-phase robust formulation Short term UE with stochastic eman Passenger options: ifferent routes choices on transit line Over flow elay: transit services operate at capacity Passenger travelling cost= on-vehicle travel time+ overflow elay Phase-1: Feasible region for ρ [0,1] Phase-: Dial-a-rie provision θ [ θ, θ ], e for one eman realization e e,min e,max External variables (P6.4): Total require ial-a-rie services Lower boun c 3 T min θ e,min = ( ) Z Z e θ e,min just enough to carry eman overflow (P6.): Maximum ial-a-rie services cost θ e,max Upper boun When the overflow elay is zero, every passenger can take the shortest path 38

39 5. Solution proceure (P6.3): Fin the optimal ial-a-rie services provision level θe,min θe θe,max Feasibility range 000 Cost phase- cost overflow elay cost flexible services cost passenger travel time cost Flexible services cost Detect the feasibility range: θe, θe Avantage: (1) be able to fin the exact optimal solution () reuce the computation time to 1/10 39

40 Case stuy Three lines were generate Frequency SO: f = 16, f = 13, f = 7 vehicle / h 1 3 Frequency UE: f = 19, f = 16, f = 7 vehicle / h

41 Result analysis RTL carries most of the passengers while ial-a-rie serves as a supplementary role UE requires more ial-a-rie services to bring own the overflow elay cost SO UE OD pair OD eman Path by RTL Expecte RTL Patronage Transfer Distance by RTL Path by RTL Expecte RTL Patronage Transfer Distance by RTL Distance % 87 % 41

42 5.3 Result analysis The SR-base approach obtains the optimal level of robustness with lowest total cost Higher level of robustness inicates higher transit cost an lower Dial-a-rie cost Level of Company cost Passenger cost Total robustness RTL Dial-a-rie RTL Dial-a-rie cost SR-base metho SO SR-base metho UE

43 43

44 Conclusions Scheule base ferry service network esign problem uner eman uncertainty Combination of regular ferry services an a-hoc services Service reliability to separate the two service types eployment into two phases Two passenger flow patterns SO an UE are investigate Value of reservation to the company uner stochastic eman Frequency base rapi transit network esign problem Multi-line esign without preetermine origin an estination Transfer costs are accounte for Consier the problem from the perspective of robust optimization Service reliability conveys the level of robustness Improve the current SR base graient solution algorithm 44

45 Key Takeways Jointly optimize FRT an DRT as an integrate system can achieve substantial cost efficiency in aressing stochastic eman. The notion of Service Reliability (SR) offers an efficient way to reformulate an solve stochastic programs, which also allows the incorporation of three ifficult extensions in TND: aressing stochastic eman, har capacity constraints, an user equilibrium conitions. 45