Scheduled Service Network Design for Rail Carriers

Transcription

1 Scheduled Service Network Design for Rail Carriers E.Zhu 1 T.G.Crainic 2 M.Gendreau 3 1 Département d informatique et de recherche opérationnelle Université de Montréal 2 Département de management et technologie Université du Québec à Montréal 3 Département de mathématiques et de génie industriel École Polytechnique de Montréal Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique 09 1 / 21

2 Outline 1 Rail Freight Transportation 2 Service Network Design Problem 3 3-Layer Time-Space Network and Model 4 Slope Scaling Algorithm 5 Computational Experiments 6 Observations and Perspectives Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique 09 2 / 21

3 Rail Freight Transportation Rail freight transportation is one important transportation way for modern society; carried 357 million tons of freight in Canada (2007); achieved $9.44 billion revenue in Canada (2007). 0 A number of articles exist for solving problems in various aspects of rail transportation. Most of the models address only a part of the rail freight transportation. 0 Statistics Canada Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique 09 3 / 21

4 Rail Freight Transportation A block is a group of cars which are transported as a unit from the block origin to the block destination. Cars are classified and formed into blocks in classification yards. Classification: cars are moved into and accumulated in a classification track. A service is a train following a physical route, and stops at some intermediate yards. Blocks are transported by services. At an intermediate stop, some blocks are loaded or unloaded. Transfer: blocks are unloaded from one service and later loaded onto another service. A block takes only service sections. From its origin to its destination, each car may go through several blocks. Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique 09 4 / 21

5 Service Network Design Problem The operating plan conducts the operating efficiency, balances customer satisfaction and rail profits, and is vital to rail carriers. The service network design problem is studied to provide a good operating plan. scheduled (time-dependent) service plan blocking policy train make-up policy traffic (loaded and empty cars) distribution Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique 09 5 / 21

6 3-Layer Time-Space Network B D A C Cyclic time dimension. 1 2 T 1 T t= T 2 T 1 Two nodes: and, to represent a yard at a time point. Three layers: Service-Layer, Block-Layer, Car-Layer. Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique 09 6 / 21

7 3-Layer Time-Space Network Moving Link Serv Stop Link Serv Load Block Unload Block Block Transfer Link Block Transfer Delay Link Block Form Up Block Broken Down Car Holding Link Car Classification Link Car Car Waiting Link Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique 09 7 / 21

8 Services Each service is represented by a path in the service layer. For each service, specific departure time, and fixed service time; fixed cost representing the locomotive and crew cost. flow capacity in number of cars; a service section is a sub-service between two (not necessary consecutive) stops; flow cost one each section: sum of the flow cost on moving links; Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique 09 8 / 21

9 Service Sections B D A C Yard D Physical Network Service Layer Block Layer Car Layer Yard C Yard B 3 Layer Network Yard A Moving Link Stop Link Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique 09 9 / 21

10 Service Sections B D A C Yard D Physical Network Service Layer Block Layer Car Layer Yard C Yard B 3 Layer Network Yard A Moving Link Stop Link Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique 09 9 / 21

11 Blocks Each block can be represented by a path, the path is formed by a series of service sections, which are connected by transfer-delay links, transfer links and vertical links; flow cost is the sum of the flow costs on its service sections; an approximated classification track occupancy time at the origin yard; fixed cost representing the classification track occupancy cost. Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

12 Blocks B A C D Yard D Physical Network Yard C Service Layer Block Layer Yard B Car Layer 3 Layer Network Yard A Service Section (Projection) Transfer Delay Link Transfer Link Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

13 Blocks B A C D Yard D Physical Network Yard C Service Layer Block Layer Yard B Car Layer 3 Layer Network Yard A Service Section (Projection) Transfer Delay Link Transfer Link Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

14 Blocks B A C D Yard D Physical Network Yard C Service Layer Block Layer Yard B Car Layer 3 Layer Network Yard A Service Section (Projection) Transfer Delay Link Transfer Link Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

15 Blocks B A C D Yard D Physical Network Yard C Service Layer Block Layer Yard B Car Layer 3 Layer Network Yard A Service Section (Projection) Transfer Delay Link Transfer Link Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

16 Traffic Flow Distribution (Vertical Projection) A B C D Yard D D 3 D 4 b 6 D 1 D 2 Physical Network b 2 b 1 Yard C Service Layer b 4 Block Layer Yard B Car Layer O 4 b 5 3 Layer Network Yard A O 1 O 2 O 3 Block (Projection) Car Holding Link Classification Link Car Waiting Link Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

17 Formulation min Φ = p P s.t. + c(p, a) x p a c f (b) y b + a A b B l L x p a a A + (n) a A (n) p P b B f F(b) p P b B(a) l L(e,t) p P f F(l) a A v (f ) c f (l) z l (1) x p a = wp n n N, p P; (2) x p a ua a Ac ; (3) x p b z lu l a A v, l L; (4) y b u v(a) a A h ; (5) z l u e e E, t {1,, T}; (6) x p b y bu b b B; (7) y b z l u l A v (l) l L; (8) f F(l) b B f F(b) x p a 0 a A, p P; (9) y b {0, 1} b B; (10) z l {0, 1} l L. (11) Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

18 Formulation min Φ = p P s.t. + c(p, a) x p a c f (b) y b + a A b B l L x p a a A + (n) a A (n) p P b B f F(b) p P b B(a) l L(e,t) p P f F(l) a A v (f ) c f (l) z l (1) x p a = wp n n N, p P; (2) x p a ua a Ac ; (3) x p b z lu l a A v, l L; (4) y b u v(a) a A h ; (5) z l u e e E, t {1,, T}; (6) x p b y bu b b B; (7) y b z l u l A v (l) l L; (8) f F(l) b B f F(b) x p a 0 a A, p P; (9) y b {0, 1} b B; (10) z l {0, 1} l L. (11) Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

19 Formulation min Φ = p P s.t. + c(p, a) x p a c f (b) y b + a A b B l L x p a a A + (n) a A (n) p P b B f F(b) p P b B(a) l L(e,t) p P f F(l) a A v (f ) c f (l) z l (1) x p a = wp n n N, p P; (2) x p a ua a Ac ; (3) x p b z lu l a A v, l L; (4) y b u v(a) a A h ; (5) z l u e e E, t {1,, T}; (6) x p b y bu b b B; (7) y b z l u l A v (l) l L; (8) f F(l) b B f F(b) x p a 0 a A, p P; (9) y b {0, 1} b B; (10) z l {0, 1} l L. (11) Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

20 Formulation min Φ = p P s.t. + c(p, a) x p a c f (b) y b + a A b B l L x p a a A + (n) a A (n) p P b B f F(b) p P b B(a) l L(e,t) p P f F(l) a A v (f ) c f (l) z l (1) x p a = wp n n N, p P; (2) x p a ua a Ac ; (3) x p b z lu l a A v, l L; (4) y b u v(a) a A h ; (5) z l u e e E, t {1,, T}; (6) x p b y bu b b B; (7) y b z l u l A v (l) l L; (8) f F(l) b B f F(b) x p a 0 a A, p P; (9) y b {0, 1} b B; (10) z l {0, 1} l L. (11) Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

21 Observation The Service Network Design Problem (SNDP) is a mixed-integer programming problem with a large number of variables and constraints. much more complicated than the model we studied before; cycle-based neighborhood fails to provide good approximation of reduced cost; development of new heuristic. Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

22 Slope Scaling Given a flow distribution x, a design ỹ( x) can be obtained by opening only the blocks/services bearing flow. On design ỹ( x), the total fixed cost is b B ỹ b =1 cf (b) + l L z l =1 cf (l). Linear factors are adjusted in order to keep the equation, total fixed cost = (α b + β l(f ) ) x p b p P b B f F(b) Surrogate block flow cost : c (p, b) = c(p, b) + α b + l F(b) β l. A multi-commodity network flow problem. augmenting path heuristic solves the large-size network flow problem. Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

23 Slope Scaling Algorithm General Procedure Initialization {initialize the linear factors α and β} while stop criteria do not meet do repeat Heuristically solved the LNFP α b = block fixed cost/total car flow on the block β l = service fixed cost/total car flow on the service until no improvement is achieved on LNFP perturbation {adjust the linear factors} end while Stop Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

24 Perturbation with Long-Term Memory History information on each block and each service. average flow. maximal flow opening frequency Perturbation based on long-term memory. Intensification - favor the blocks/services with high opening frequency and stable flow. Diversification - favor the blocks/services rarely used. Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

25 Random Instances Inst Block Service Yard Track Period Demand p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

26 Numerical Results Inst CplexSol time(s) OptGap S.S.Sol CplexGap time(s) TimeGap S.S.(24h) CplexGap p t 9.49% % % % p % % % % p % % % % p % % % % p % % % % p t 9.46% % % % p t 18.46% % % % p t 12.42% % % % p t 38.23% % % % p t 31.56% % % % p t 25.37% % % % p t 5.53% % % % p t p t p t p % % % % p t 8.50% % % % p t 5.09% % % % p t 3.17% % % % p t 4.28% % % % p t 3.03% % % % p t 14.12% % % % p t 39.35% % % % p t 50.30% % % % p t 33.02% % % % p t 28.42% % % % p t 42.46% % % % p t p t p t t = 36,000 sec Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

27 Observations and Perspectives Observations Complicated MIP service network design formulation. Relation between services and blocks makes the problem even more complicated. Good results can be obtained with slope scaling algorithm. Effective heuristic for the linear flow problem. Long-term memories on both blocks and services. Perspectives Apply the model in rail. Decompositions. Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21

28 Thanks for your attention. Questions? Zhu, Crainic, Gendreau (CIRRELT) Scheduled Rail Service Network Design Journée Scientifique / 21