1. Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

2. Maintenance Routing Gábor Maróti Leo Kroon Astrid Roelofs CWI, Amsterdam NS Reizigers, Utrecht Erasmus University, Rotterdam NS Reizigers, Utrecht Free University, Amsterdam NS Reizigers, Utrecht Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

3. Maintenance Routing  Problem formulation  successive shortest paths  Computational results Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001  What do the planners now do?  A graph representation  Models multicommodity flow  network flow and node potential  Future

4. Maintenance Routing Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001   Problem formulation  successive shortest paths  Computational results  What do the planners now do?  A graph representation  Models multicommodity flow  network flow and node potential  Future

5. Problem formulation Train units After reaching a kilometer limit, they have to be checked. In practice: the most urgent units go for maintenance. The operational plan must be changed. Bottleneck: shunting Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

6. Problem formulation Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 (Very) naive idea: solve the shunting problem at each station Natural decomposition: solve the problem separately for the rolling stock types (and try to estimate the shunting difficulty) Solution: new rolling stock schedule in the planning horizon

7. Problem formulation Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Input:  duties: sequences of tasks on each day  list of urgent units  deadlines  the actual operational plan Output:  new operational plan, such that  the urgent units can reach the maintenance station  “the cost is minimal”

8. Problem formulation Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 The planning horizon is short (e.g. 3 days).  delays  shortage of crew  shortage of rolling stock   necessary changes in the plan  cancelled trains

9. Maintenance Routing Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001   Problem formulation  successive shortest paths  Computational results  What do the planners now do?  A graph representation  Models multicommodity flow  network flow and node potential  Future

10. What do they now do? Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 1. Assign a most urgent unit to a first available maintenance job If no solution, change a bit the deadlines (  1 day). 2. Try to route it there 3. Call the local sunting crew: “Is the route feasible?” 4. Iterate this process

11. What do they now do? Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 1 3 2 Deadlines for urgent units Days Nights

12. Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 What do they now do? Urgent unit Assigned maintenance job Night change

13. Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 What do they now do? Urgent unit Assigned maintenance job Night change

14. Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 What do they now do? Urgent unit Assigned maintenance job Daily change

15. Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 What do they now do? Urgent unit Assigned maintenance job Daily change: maybe possible

16. Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 What do they now do? Urgent unit Assigned maintenance job Daily change: maybe possible

17. Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 What do they now do? Urgent unit Assigned maintenance job Daily change

18. Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 What do they now do? Urgent unit Assigned maintenance job Daily change (and a night change)

19. Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 What do they now do? Urgent unit Assigned maintenance job Empty train movement

20. Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 What do they now do? Urgent unit Assigned maintenance job Empty train movement (taking care of the balance)

21. Maintenance Routing Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001   Problem formulation  successive shortest paths  Computational results  What do the planners now do?  A graph representation  Models multicommodity flow  network flow and node potential  Future

22. A graph representation Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Nodes: arrival and departure events Arcs: operational plan + extra possibilities

23. A graph representation Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 “Grey box”: permitted or forbidden arcs A perfect matching is required Night arcs:

24. A graph representation Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Night arcs: Assumption: a small number of changes can be carried out

25. A graph representation Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Day arcs: Simple daily change possibility

26. A graph representation Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Day arcs: If we allow only one change for each train unit   it is enough to insert all these arcs

27. A graph representation Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Day arcs: In case we allow also more complex changes   the graph becomes more complicated.

28. A graph representation Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Day arcs: However, we did not implement multiple changes because they did not give any extra possibility (in the test data)

29. A graph representation Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Empty train arcs: extra arcs between the boxes: all or some of them (a small number is enough)

30. Maintenance Routing Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001   Problem formulation  successive shortest paths  Computational results  What do the planners now do?  A graph representation  Models multicommodity flow  network flow and node potential  Future

31. Models Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Solution: a new operational plan, i.e.  perfect matching on the Night Arcs  perfect matching on the Day Arcs such that each urgent unit gets to the maintenance facility.

32. Models Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Quality of a solution: the extra shunting cost Linear cost function: cost on the arcs c (a) = 0 if a is in the original plan c (a)  0 otherwise Minimize the total sum of arc costs. Idea: “the closer to the original plan the better”

33. Models Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Example: Station Utrecht expensive cheapexpensive

34. Models Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Night arcs: cheap, not too expensive or almost impossible Day arcs: typically more expensive (more risky) Empty train arcs: very expensive  carriage kilometer  crew schedule

35. Models Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Test data: rolling stock type “Sprinter”  52 units (duties)  1 maintenance job on each workday  1 maintenance station  10 terminal stations  2 further possible (daily) shunting stations

36. Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

38. Maintenance Routing Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001   Problem formulation  successive shortest paths  Computational results  What do the planners now do?  A graph representation  Models multicommodity flow  network flow and node potential  Future

39. Successive shortest paths Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 1. Match the urgent train units to the maintenance jobs 2. For each urgent unit: determine a shortest path in the graph delete this path from the graph take the next urgent unit Algorithm

40. Successive shortest paths Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Easy, simple, very fast Takes no care of matching conditions (day, night) Ad hoc ideas are necessary

41. Maintenance Routing Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001   Problem formulation  successive shortest paths  Computational results  What do the planners now do?  A graph representation  Models multicommodity flow  network flow and node potential  Future

42. Solution = Perfect matching in each box 1 3 2 s.t. the deadline conditions are fullfilled Multicommodity flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

43. Matching variables m on the Night Arcs and Day Arcs (0-1 valued). Still needed: linear inequalities expressing that 1. each urgent unit reaches the maintenance facility 2. in the time limit Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Multicommodity flow

44. A 1-flow for each urgent unit 1 3 2 Multicommodity flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

45. 2 Possible terminal nodes Deadline Multicommodity flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

46. Variables:  matching variables m  flows x 1, x 2, x 3, … Constraints:  matching constraints  conservation rule for each flow  starting and terminal constraints for the flows   x i  m ( e ) Multicommodity flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

47. Multicommodity flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Objective function: minimize  ( c ( a ) m(a) : a  Night or Day Arcs )

48. If m is integral, the values x may be chosen float (read-valued). If x and m are integral on the Day Arcs, the other variables may be chosen float. Multicommodity flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

49. Maintenance Routing Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001   Problem formulation  successive shortest paths  Computational results  What do the planners now do?  A graph representation  Models multicommodity flow  network flow and node potential  Future

50. Having fixed a matching m, s t set two new nodes s and t, set all arc capacities 1. Does there exist an s - t network flow of value 3? Network flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

51.  x : Arcs  [0 ; 1]  conservation rule for every nodes  s, t  the flow value is 3 (# of urgent units)  x(e)  m(e) for Night Arcs Network flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

52. Network flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Given a digraph G and a function C : Arcs  R   is a node potential (for the longest path) if C(uv)   (u)   (v) uv for every arc uv.

53. Network flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Having fixed a matchig m :  longest path = the only path    is an upper bound on the distance from the maintenance nodes (with appropriate initial values) ( C  1) 0 0 0 Big

54. Network flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 1 3 2 Instead of the deadlines: 1 5 3 distances. d(u) := 2 deadline( u )  1

55. Network flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 The important inequalities:  (u)  (v)  Big    m(uv)) for Day and Night Arcs  (u)  d(u) for urgent unit starting nodes LB ( v )   (v)  UB (v) for each node The bounds LB and UB from the graph structure Then Big := UB (v)  LB (u)  1

56. Network flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001  matching variables m  flow variables x  potential variables  integral may be chosen float Variables:

57. Network flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Constraints  matching variables m  flow variables x  potential variables  Variables:  matching constraints  flow constraints  potential constraints

58. Network flow Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Objective function: minimize  ( c ( a) m(a) : a  Night or Day Arcs )

59. Maintenance Routing Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001   Problem formulation  successive shortest paths  Computational results  What do the planners now do?  A graph representation  Models multicommodity flow  network flow and node potential  Future

60. Computational results Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001  Test data: rolling stock type “Sprinter”  3 - 5 days planning horizon  3 - 5 urgent units  IBM PC, Pentium III 900 MHz, 256 MB RAM  Software: ILOG OPL Studio 3.0, CPLEX 7.0

61. Computational results Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 Only night connections (5 nights):

62. Computational results All possibilities: Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

63. Maintenance Routing Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001   Problem formulation  successive shortest paths  Computational results  What do the planners now do?  A graph representation  Models multicommodity flow  network flow and node potential  Future

64. Future Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001 A lot cooperation with planners and shunting crew in  modelling the night shunting possibilities (costs)  determining the practical relevance of the solutions  finding the set of day connections New criteria for the rolling stock scheduling

65. Thank you.