Rail Car Allocation Problems

Transcription

1 Rail Car Allocation Problems Marco E. Lübbecke and Uwe T. Zimmermann Mathematical Optimization Braunschweig Germany Rail Car Allocation Problems p.1

2 Freight Cars... Rail Car Allocation Problems p.2

3 Freight Cars sit and wait Rail Car Allocation Problems p.2

4 Freight Cars... O? D Rail Car Allocation Problems p.3

5 Empty Car Distribution Cars are repositioned due to imbalances in demands Delayed empty cars Car shortage Inaccurate data Safety inventories Low utilization Large fleet sizes Rail Car Allocation Problems p.4

6 Empty Car Distribution Supply Demand Rail Car Allocation Problems p.5

7 Empty Car Distribution Time Rail Car Allocation Problems p.5

8 Empty Car Distribution Problem:? Time expanded integer multi commodity flow problem Rail Car Allocation Problems p.5

9 Empty Car Distribution Glickman, Sherali (1985): Pooling of cars Holmberg, Joborn, and Lundgren (1996, 1998): Explicit train schedules, train capacities Many more papers 5,000 cars, 18 (aggregated) types, 50 terminals Future: Simultaneously plan empty and loaded cars Rail Car Allocation Problems p.6

10 Railroad Blocking A block is a group of cars with common OD pair O D O Ideally: Only direct blocks Source of delay and unreliable service D Rail Car Allocation Problems p.7

11 Railroad Blocking Assign each car a sequence of blocks, observing the maximal tractable volume of cars per station minimizing the total number of reclassifications or delay or mileage Considerable body of literature Barnhart, Jin, and Vance (1997): 1080 stations, 12,000 shipments Rail Car Allocation Problems p.8

12 Yard Operations Shunting with capacity constraints Dahlhaus et al. (2000): Regrouping of cars older papers: e.g., queueing models Rail Car Allocation Problems p.9

13 Yard Operations Shunting with capacity constraints Dahlhaus et al. (2000): Regrouping of cars older papers: e.g., queueing models Also from passenger transportation: Routing trough stations (Kroon, Zwaneveld 1996) Shunting trams (Winter, Zimmermann 2000) This afternoon session... Rail Car Allocation Problems p.9

14 Further Issues Assign blocks to trains What is a good fleet size? Where and when to clean, maintain, and repair?... Rail Car Allocation Problems p.10

15 Integrated Planning Why not propose one big model for all stages? Rail Car Allocation Problems p.11

16 Integrated Planning Why not propose one big model for all stages? Of course it has been proposed! Rail Car Allocation Problems p.11

17 Integrated Planning Why not propose one big model for all stages? Of course it has been proposed! But (today) it is illusive to solve it optimally! Rail Car Allocation Problems p.11

18 Integrated Planning Why not propose one big model for all stages? Of course it has been proposed! But (today) it is illusive to solve it optimally! But... Rail Car Allocation Problems p.11

19 In-Plant Railroads Rail Car Allocation Problems p.12

20 In-Plant Railroads Customer oriented! Rail Car Allocation Problems p.12

21 Rail Car Management ➀ Transportation request specified by Terminal/track Quantity Goods type/car type Deadline Substitution car types ➁ As long as available, assign cars dedicated, pooled, incoming, in repair... and build blocks ➂ Otherwise: Rent additional rail cars Rail Car Allocation Problems p.13

22 Blocks Rail Car Allocation Problems p.14

23 Blocks Rail Car Allocation Problems p.14

24 Blocks Rail Car Allocation Problems p.14

25 Blocks Rail Car Allocation Problems p.14

26 Split into Regions Rail Car Allocation Problems p.15

27 Split into Regions Rail Car Allocation Problems p.15

28 Split into Regions Rail Car Allocation Problems p.15

29 Split into Regions Rail Car Allocation Problems p.15

30 Split into Regions Rail Car Allocation Problems p.15

31 Transportation Problem x τ i j : Cars of type τ from region i for request j min s.t. c i,r xi,r τ + M τ xs τ τ,r i,r,τ T r r,τ T r xi,r τ a τ i i,τ r:τ T r xi,r τ + xs τ τ,r b r r i,τ T r τ T r x τ i,r 0 i,r,τ T r Rail Car Allocation Problems p.16

32 Unsplit Supply Fulfill a demand from a single origin? rent cars a 1 = a 2 = 1 2 i b i b 1 b 2 b n Solution without car rental Partition of {b 1,b 2,...,b n } Rail Car Allocation Problems p.17

33 Shunting Minimization Demand: D = (1,2,2) c 1 c 2 c 3 c 4 c 5 Each moved car on track i costs c i Q +, No space limitations, no ordering/sequence Rail Car Allocation Problems p.18

34 Shunting Minimization Demand: D = (1,2,2) c 1 c 2 c 3 c 4 c 5 Each moved car on track i costs c i Q +, e.g. (1 + 1) c c c 5 No space limitations, no ordering/sequence Rail Car Allocation Problems p.18

35 Greedy Take cheapest car(s) for each color, respectively Demand: D = (1,1,1,1,1,1), D = n Cost: Greedy O(n 2 ), optimal O(n) 1 2 ε 3 ε 4 ε 5 ε. n ε Rail Car Allocation Problems p.19

36 Complexity Demand: D = (K,1,1,1,1,1) 1/3 1/4 1/4 1/ feasible shunting plan at cost K vertex cover of cardinality K 1 Rail Car Allocation Problems p.20

37 Integer Program z t,g : Access group g on track t? y t,g : Number of chosen cars from group g on track t, at most Q t,g min c t [(Q t,g y t,g ) z t,g+1 + y t,g ] t,g s.t. z t,g z t,g 1 t,1 < g y t,g Q t,g z t,g t,g y t,g D τ types τ t,g: color(t,g)=τ y t,g 0 t,g z t,g {0,1} t,g Rail Car Allocation Problems p.21

38 Integer Program z t,g : Access group g on track t? y t,g : Number of chosen cars from group g on track t, at most Q t,g min min c t [(Q t,g y t,g ) z t,g+1 + y t,g ] t,g c t Q t,g z t,g t,g s.t. z t,g z t,g 1 t,1 < b y t,g Q t,g z t,g t,g y t,g D τ types τ t,g: color(t,g)=τ y t,g 0 t,g z t,g {0,1} t,g Rail Car Allocation Problems p.21

39 Integrated Planning min s.t. c i,r xi,r τ + M τ xs τ τ,r i,r,τ T r r,τ T r r:τ T r x τ i,r a τ r i,τ xi,r τ + xs τ τ,r b r r i,τ T r τ T r x τ i,r Z + i,r,τ T r Rail Car Allocation Problems p.22

40 Integrated Planning min s.t. c i,r xi,r τ + M τ xs τ τ,r i,r,τ T r r,τ T r xi,r τ y i,t,g r:τ T r i,t,g: color(i,t,g)=τ xi,r τ + xs τ τ,r b r r i,τ T r τ T r i,τ x τ i,r Z + i,r,τ T r Rail Car Allocation Problems p.22

41 Integrated Planning min s.t. c i,r xi,r τ + M τ xs τ τ,r + i,r,τ T r r,τ T r n i=1 c t Q i,t,g z i,t,g t,g xi,r τ y i,t,g r:τ T r i,t,g: color(i,t,g)=τ xi,r τ + xs τ τ,r b r r i,τ T r τ T r i,τ z i,t,g z i,t,g 1 i,t,1 < b y i,t,g Q i,t,g z i,t,g i,t,g x τ i,r Z + i,r,τ T r y i,t,g 0 i,t,g z i,t,g {0,1} i,t,g Rail Car Allocation Problems p.22

42 Computational Results 683 tracks, 168 terminals, 42 regions; 1575 cars, 123 car types; 18 requests Rail Car Allocation Problems p.23

43 Conclusion Almost generic approach yields relevant results Experiments feed back into practice Test runs planned for September 2002 Results encourage extensions... Rail Car Allocation Problems p.24

44 Conclusion Almost generic approach yields relevant results Experiments feed back into practice Test runs were planned for September 2002 Results encourage extensions... Rail Car Allocation Problems p.24