A New Solution for Rail Service Network Design Problem

Transcription

1 A New Solution for Rail Service Network Design Problem E.Zhu 1 T.G.Crainic 2 M.Gendreau 3 1 Département d informatique et de recherche opérationnelle Université de Montréal 2 École des sciences de la gestion Université du Québec à Montréal 3 Département de mathématiques et de génie industriel École Polytechnique de Montréal Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 1 / 26

2 Outline 1 Rail Freight Transportation 2 Scheduled Service Network Design 3 Multi-layer Time-Space Model 4 A New Meta-heuristic 5 Conclusions Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 2 / 26

3 Double-Consolidation Operations Rail transportation = Trains Train = More than meets the eye Freight is loaded into cars (by customers) Cars are classified (sorted) and consolidated (put together) into blocks at classification yards A block is a group of cars moved together as a unit from the block origin to the block destination Blocks are grouped together to make up trains at classification yards Trains move blocks-cars through the rail network Blocks may be transferred from one train to another at an intermediate yard Cars are moved from particular origin to particular destination on sequences of trains and blocks = itineraries Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 3 / 26

4 Services and Schedules Service: What the carrier operates to address demand Train route (physical) and intermediate stops (eventually) Type: main/feeder/..., speed/priority, power requirements, etc. Capacity (weight, length,...) Schedule: frequency, times of departure (origin yard), arrival & departure (intermediary stops), arrival (destination yard) Schedule: How the carrier operates Fixed length (e.g., week) Repetitive over the planning horizon (e.g., season) Departure & arrival times of trains at stations (& meeting points) To simplify: service = particular train & departure time (from its origin) Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 4 / 26

5 The Scheduled Service Network Design Problem Build the operating plan and schedule for next season = Tactical planning activity Select train services to operate with departure times Select blocks to build: blocking policy Determine block routing = train make up Determine cargo routing = demand itineraries Do not forget empty car distribution, yard capacities, etc. To minimize costs, minimize delays, maximize on-time delivery and service quality... We proposed new modeling approach and a meta-heuristic We present an enhancement based on dynamic block generation to address larger instances Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 5 / 26

6 3-Layer Time-Space Network Three layers - Service, Block, & Car - for selection of movements of each type of flow and consolidation level Service Layer Block Layer Car Layer Inter-cycle arcs Cars to form/break blocks Blocks to make up/break down trains or to transfer (attach/detach) Cyclic time dimension for repetitive schedule 1 2 T 1 T t= T 2 T 1 Two nodes, and, to represent each yard at each time point Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 6 / 26

7 A 3-Layer Time-Space Network Illustration Service Layer Block Layer Car Layer Moving Link Stop Link Moving Link Attach Block to Service Detach Block from Service Transfer Link Transfer Delay Link Cars Grouped, Block Ready to Go Block Break Down, Return Cars Car Holding Link Classification Link Car Waiting Link Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 7 / 26

8 Services Each service s S is represented by a path in the service layer Specific departure time and duration time Route = sequence of moving (set of tracks & travel time) and stop links Track capacity in terms of number of services Fixed cost (power, crew, etc.) Unit flow (demand) cost on each car-moving link Capacity in terms of number of cars Service section: Service provided between two (not necessary consecutive) stops Service selection decision variables z s {1, 0} Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 8 / 26

9 Service Sections & Service layer B A C D Yard D Physical Network Service Layer Block Layer Yard C Yard B Car Layer 3 Layer Network Yard A Moving Link Stop Link Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 9 / 26

10 Service Sections & Service layer B A C D Yard D Physical Network Service Layer Block Layer Yard C Yard B Car Layer 3 Layer Network Yard A Moving Link Stop Link Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 9 / 26

11 Blocks Each block b B is represented by a path in the block layer Series of block-moving links = service sections (projected) Connected by within-layer transfer (inter yards) links, transfer-delay (holding/waiting at yard) links, and Inter-layer vertical links to attach/detach the block to train services Fixed cost to build and transfer (yard crew & equipment, classification track occupancy) Approximated classification-track occupancy time at the origin yard Yard capacity in terms of number of blocks = classification tracks Classification track capacity in terms of number of cars Capacity in terms of number of cars Unit flow cost on each block-moving link Block selection decision variables y b {1, 0} Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 10 / 26

12 Block Layer 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) Rail Scheduled Service Design FORMS 2010, Austin, TX 11 / 26

13 Block Layer 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) Rail Scheduled Service Design FORMS 2010, Austin, TX 11 / 26

14 Block Layer 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) Rail Scheduled Service Design FORMS 2010, Austin, TX 11 / 26

15 Block Layer 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) Rail Scheduled Service Design FORMS 2010, Austin, TX 11 / 26

16 Cars Enter & exit the system, are classified (sorted and grouped), and wait in the car layer Moved as blocks on train services The itineraries: Multi-layer series of Classification links Move (hump, haul) from receiving to classification tracks Car-waiting links (on receiving tracks): Classification delay Holding links: Waiting on classification tracks for the block to be full Car-moving links = blocks (projected) Connected by inter-layer vertical links to indicate blocks are ready to go or arrived and are broken down Yard classification capacity in terms of number of cars handled per period Unit flow cost on each link Car flow-on-link decision variables x ap 0 Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 12 / 26

17 Car Layer B A 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 Car Moving Link = Block Projection Classification Link Car Waiting Link Car Holding Link Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 13 / 26

18 Formulation SNDP = s.t. min c ap x ap + c F b y b+ c F s zs (1) p P a A b B s S x ap x ap = w p n n N C, p P; (2) a A + (n) a A (n) x ap u a a A CC ; (3) p P z s u e e E, t {0,, T 1}; (4) s S(e,t) y b u v v V, t {0,, T 1}; (5) b B(v,t) x bp z su s a A SM, s S; (6) l L(s) a A SM (l) b B l L(b) p P x bp y b u b b B; (7) p P y b z s(l) l L(b), b B; (8) x ap 0 a A, p P; (9) y b {0, 1} b B; (10) z s {0, 1} s S. (11) Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 14 / 26

19 Algorithmic Observations A large-scale Service Network Design Problem (SNDP) A combinatorial MIP formulation Difficult formally and computationally A meta-heuristic combining a number of interesting concepts Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 15 / 26

20 The Original Meta-heuristic Slope scaling (SS) for rapid identification of fairly good solutions (Kim and Pardalos 1999) Address a SNDP relaxation by iteratively solving a linear approximation Relaxation of service & block capacity constraints Approximating the service & block fixed-cost A minimum cost multi-commodity network flow problem Long-term memory perturbation (LMP) for guiding SS out of local optima (Crainic et al. 2004, Kim et al. 2006) Ellipsoidal Search (ES) to improve a set of elite solutions built by SS Define a restricted SNDP from the elite set (Path relinking & local branching ideas to restrict the space) Solve MIP subproblems exactly for feasibility & exploration Long-term memories for guidance Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 16 / 26

21 Algorithmic Observations 2 Two interlaced layers of design decisions Very large number of design variables Blocks = combinations of service sections Consider blocks implicitly and generate them dynamically Define a block-design relaxation for the Slope Scaling A MIP SNDP with block-selection decisions and implicitly considered blocks Generate new blocks at each move of the cycle-based Tabu Search addressing the SNDP Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 17 / 26

22 The New Meta-heuristic Slope Scaling Service Layer Block Generation Block Layer Approximation Problem Car Layer Service Layer more blocks residual flow Block Layer Car Layer Service Layer Perturbation Ellipsoidal Search Long term Memories Tabu Search Block Layer Car Layer Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 18 / 26

23 Slope Scaling Address a SNDP relaxation by iteratively solving a block-design MIP approximation problem AP(β) Remove the service (4) capacity constraints Relaxed SNDP Approximate the service fixed-cost part with a linear function β s x sp p P s S Solve the resulting MIP SNDP AP(β) Given a solution, c F s is paid for service s with positive flow β s coefficients are adjusted to satisfy fixed service cost c F s = total flow on service s β s for s open Next iteration is initiated Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 19 / 26

24 Approximation Problem AP(β) = min c ap x ap + c F b y b + p P a A b B s S x sp β s p P Subject to flow conservation and capacity constraints, and y b {0, 1}. b B A cycle-based Tabu Search addresses the block-network design problem The cycle-based neighborhood is evaluated through residual networks Add/remove γ flow on the block forward/backward arc (projection of services on block layer) with positive/negative cost Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 20 / 26

25 Block Generation A block = a projected-service path in β, γ-block layer A parallel block set for each - node pair (different yards and time periods) Parallel forward arcs in residual network Shortest path (linearization of transfer costs) Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 21 / 26

26 Experimental Setting Two random problem sets = 30 instances 5-10 yards tracks 7 and 10 time periods 100 to 900 demands C++; CPLEX 10.1; 10 hours 2.4Hz CPU, 16 GB RAM, Linux Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 22 / 26

27 Only few times on small instances with 5 yards (p01-p06, p16-p21), CPLEX finds optima. Results With 7-yard 7 Time instances, Periods CPLEX returns a significant optimal gap. For larger instances with 10 yards (p13-p15, p28-p30) CPLEX fails to identify any feasible solution in the limited computing time. The next two columns display the previous SS+ES+LMP results with the same CPU time and their gap with the CPLEX solution. Inst CplSol OptGap SS+ES+LMP CplGap BEST CplGap SS+ES+LMP Gap p % % % 0.00% p % % % 0.13% p % % % 0.02% p % % % 0.00% p % % % 0.06% p % % % 0.19% p % % % 1.55% p % % % 1.10% p % % % 0.07% p % % % 0.15% p % % % 1.46% p % % % 1.66% p % p % p % Avg -0.04% Table 2: BEST Results on Instance Set S Column BEST displays the solution from the heuristic proposed in this paper. For the purpose of comparison, same computing time as CPLEX and SS+ES+LMP is imposed. The initialization terminates with maximal 300 iterations, or maximal 5 hours Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 23 / 26

28 Results 10 Time Periods Inst CplSol OptGap SS+ES+LMP CplGap BEST CplGap SS+ES+LMP Gap p % % % 0.27% p % % % 0.64% p % % % 0.32% p % % % 0.18% p % % % 0.00% p % % % 1.51% p % % % 1.26% p % % % 0.36% p % % % 1.23% p % % % 0.20% p % % % 1.52% p % % % 1.35% p % p % p % Avg 0.02% Table 3: BEST Results on Instance Set L Compared with SS+ES+LMP, BEST finds very close solutions with the same computing effort. Results of the BEST algorithm is a bit weak than the previous heuristic on small and Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 24 / 26

29 Conclusions An integrated scheduled service network design model An algorithmic framework able to address interesting-size instances Largest instance tested: 30 yards, 135 tracks, 7 time periods, 1000 demands Outperforms state-of-the-art solver Interesting heuristic dynamic generation of path variables within tabu search Perspectives for multi-layer network design Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 25 / 26

30 Thank you for your attention Zhu, Crainic, Gendreau (CIRRELT) Rail Scheduled Service Design FORMS 2010, Austin, TX 26 / 26