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Vehicle Circulation and the Hungarian Method

Vehicle Circulation and the Hungarian Method. Martin Grötschel joint work with Ralf Borndörfer Andreas Löbel Celebration Day of the 50th Anniversary of the Hungarian Method Budapest, October 31, 2005. About the assignment problem.

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Vehicle Circulation and the Hungarian Method

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  1. Vehicle Circulation andthe Hungarian Method Martin Grötschel joint work with Ralf Borndörfer Andreas Löbel Celebration Day of the 50th Anniversary of the Hungarian MethodBudapest, October 31, 2005

  2. About the assignment problem • The assignment problem is a mathematical problem. Mathematicians have spent an awful lot of time to create “real-life interpretations” that look like applications to “prove” that it is useful. And hence, the Hungarian Method is of no practical value. • The “truth”, in fact is the other way around. Practitioners have “tuned” their applied problems in order to be able to employ the Hungarian Method. Martin Grötschel

  3. Contents • What is vehicle circulation/scheduling? • Single depot vehicle scheduling • Multiple depot vehicle scheduling • Extensions Martin Grötschel

  4. Contents • What is vehicle circulation/scheduling? • Single depot vehicle scheduling • Multiple depot vehicle scheduling • Extensions Martin Grötschel

  5. Planning Public Transportation Martin Grötschel

  6. The ZIB Transportation Team,including former members: Public Transport: Ralf BorndörferFridolin KlostermeierChristian KüttnerAndreas LöbelSascha LukacMarc PfetschThomas SchlechteSteffen Weider Online Transportation: Norbert AscheuerPhilipp FrieseSven O. KrumkeDiana PoensgenJörg RambauLuis Miguel TorresAndreas TuchschererTjark Vredeveld plus several master students Martin Grötschel

  7. Cost Recovery Fares Construction Costs Network Topology Velocities Lines Service Level Frequencies Connections Timetable Sensitivity Rotations Relief Points Duties Duty Mix Rostering Fairness Crew Assignment Disruptions Operations Control Planning in Public Transport(Product, Project, Planned) multidepartmental Departments multidepotwise Depots multiple line groups Line Groups multiple lines Lines multiple rotations Rotations VS-OPT2 B15 IS-OPT Martin Grötschel APD DS-OPT VS-OPT BS-OPT AN-OPT B1 B3 B1

  8. The ZIB Transportation Teamspin-off companies Intranetz: Fridolin Klostermeier Christian Küttner Norbert Ascheuer LBW: Ralf Borndörfer Andreas Löbel Steffen Weider Martin Grötschel

  9. What is vehicle circulation/scheduling? • We are given a transportation system in a region. • It is subdivided by carrier/vehicle types (busses, trams, subways, planes, ships…). • For each carrier type, a (daily, weekly, or monthly,..) timetable (the scheduled/timetabled trips) is given. • Task: Assign the available vehicles to the scheduled trips of the timetable such that some objective function is optimized and a (usually large) system of side constraints is satisfied. Martin Grötschel

  10. What is vehicle circulation/scheduling?Somewhat more precise: • Each vehicle (usually) has a home base. In colloquial language this is called its depot. Transportation professionals have to be more precise. A depot consists of all indistinguishable vehicles that have their home base in the same physical location. • In most cases, a vehicle leaves its depot in the “morning” and returns to its depot in the “evening” of the planning period. Thus, every vehicle “circulates” along a tour of the region. • The vehicle circulation problem is hence the task to find, for each available vehicle and for the given planning horizon, a tour such that all scheduled/timetabled trips are covered by exactly one tour and some objective is optimized and certain side constraints respected. Martin Grötschel

  11. What is vehicle circulation/scheduling?The objective function • Minimize the number of vehicles that are necessary to cover all scheduled trips. • Minimize the cost of the deadhead trips.(Deadhead trips are moves of a vehicle without passengers; a move can be just a break where the vehicle keeps waiting in a parking lot.) • A combination of these two. • Interlining • Turns • Pull-in pull-out trips Martin Grötschel

  12. Leuthardt Survey(Leuthardt 1998, Kostenstrukturen von Stadt-, Überland- und Reisebussen, DER NAHVERKEHR 6/98, pp. 19-23.) annual cost: 150 – 250 thousand US dollars per bus Martin Grötschel

  13. Vehicle Scheduling in Berlin The transportation research group at ZIB has produced software with which the • busses • street cars, and • subways in Berlin have been scheduled. A film shows some of the problems of bus scheduling: Martin Grötschel

  14. Vihicle Circulation Film Martin Grötschel

  15. Some Users Martin Grötschel

  16. Contents • What is vehicle circulation/scheduling? • Single depot vehicle scheduling • Multiple depot vehicle scheduling • Extensions Martin Grötschel

  17. Single Depot Vehicle Scheduling(Assignment Model) depot (in the morning) D D 1 2 1 2 with starting time and location timetabled trip 1 1 2 2 with ending time and location 4 timetabled trips 4 4 3 3 • A single depot: • one location • one bus type 3 3 4 4 depot (in the evening) D D Martin Grötschel

  18. Single Depot Vehicle Scheduling(Assignment Model) D 1 D D 2 3 D 4 1 2 1 2 3 D D 1 2 4 1 1 2 2 D 1 D 2 3 4 D 1 D 2 3 4 4 4 3 3 3 3 3 D D 1 2 4 4 4 3 D D 1 2 4 D D 4 timetabled trips plus 12 deadhead trips 1 blue and 1 red bus circulation 2 depot nodes foreach available bus The assignment model of thesingle depot vehicle circulation problem Martin Grötschel

  19. But • In the seventies the available computers were not able to solve large size assignment problems due to time and space problems. • The Hungarian method was the algorithm of choice. There was nothing better. Martin Grötschel

  20. Problem Specific Size Reduction:HOT = Hamburger OptimierungsTechnik HOT only looked atpeak times (about 7 a.m) and made heuristic (manual = interactive) choices toreduce the problem size. ~ 1975 beginning of code development ~ 2003 last installations replaced Martin Grötschel

  21. All other companies did basically the same;but it is hard to find out what they really did. Martin Grötschel

  22. Surprise • Due to expertise and practical experience, the HOT specialists were able to come up with very good (and often almost optimal) solutions when “number of busses” was the major objective. Martin Grötschel

  23. Single Depot Vehicle Scheduling 1 2 3 4 3 6 7 3 3 7 8 10 1 2 3 • The Assignment Problem • Input: 3 Buses, 3 trips, costs • Output: cost minimal assignment Buses Solution Cost = 20 Trips Martin Grötschel

  24. Single Depot Vehicle Scheduling 1 2 3 4 3 6 7 3 3 7 8 10 1 2 3 • The Greedy-Heuristik • heuretikos (gr.): inventiveheuriskein (gr.): to find Buses Solution Cost = 17 Trips Martin Grötschel

  25. Single Depot Vehicle Scheduling 1 2 3 4 3 6 7 3 3 7 8 10 1 2 3 • The Greedy-Heuristik • heuretikos (gr.): inventiveheuriskein (gr.): to find Busses Solution Cost = 16 Trips Martin Grötschel

  26. The "Primal Problem" Minimum Cost Assignment The "Dual Problem" Maximum Sales Revenues "Shadow Prices" Single Depot Vehicle Scheduling 5 4 0 4 3 6 7 3 3 7 8 10 7 8 9 Buses Optimum Cost = 15 Trips Martin Grötschel

  27. Graph Theoretic Model Integer Programming Model Linear Programming Relaxation Mathematical Models(Assignment Problem) 2 1 3 4 7 3 6 3 3 10 8 7 3 2 1 Martin Grötschel

  28. Single Depot Vehicle Scheduling 0 0 0 4 4 3 3 6 6 7 7 3 3 3 3 7 7 8 8 10 10 0 0 0 • The „Successive Shortest Path“ Algorithm Martin Grötschel

  29. Single Depot Vehicle Scheduling 0 0 0 0 0 0 0 0 0 0 4 4 3 3 6 6 7 7 3 3 3 3 7 7 8 8 10 10 0 0 0 • The „Successive Shortest Path“-Algorithm Buses Bound cost = 15 Partial sol. cost = 0 Trips Martin Grötschel

  30. Single Depot Vehicle Scheduling 0 0 0 0 0 0 0 0 0 0 4 0 3 0 6 2 7 4 3 0 3 0 7 4 8 5 10 6 0 0 0 • The „Successive Shortest Path“ Algorithm Buses +0 +0 +0 Bound cost = 10 Partial sol. cost = 3 Trips +3 +3 +4 Martin Grötschel

  31. Single Depot Vehicle Scheduling • The „Successive Shortest Path“ Algorithm 0 0 0 0 0 0 0 Buses 0 0 0 Bound cost = 10 Partial sol. cost = 3 4 0 3 0 6 2 7 4 3 0 -3 0 7 4 8 5 10 6 Trips 3 3 4 Martin Grötschel

  32. Single Depot Vehicle Scheduling • The „Successive Shortest Path“ Algorithm 0 0 0 0 0 0 0 Buses 0 +0 +0 0 0 +0 Bound cost = 10 Partial sol. cost = 6 4 0 3 0 6 2 7 4 3 0 -3 0 7 4 8 5 10 6 Trips 3 +0 3 +0 4 +0 Martin Grötschel

  33. Single Depot Vehicle Scheduling • The „Successive Shortest Path“ Algorithm 0 0 0 0 0 0 0 Buses 0 0 0 Bound cost = 10 Partial sol. cost = 6 4 0 -3 0 6 2 7 4 -3 0 3 0 7 4 8 5 10 6 Trips 3 3 4 Martin Grötschel

  34. Single Depot Vehicle Scheduling • The „Successive Shortest Path“ Algorithm 0 0 0 0 0 0 0 Buses 0 +4 +5 0 0 +0 Bound cost = 15 Partial sol. cost = 15 4 0 -3 0 6 2 7 4 -3 0 3 0 7 4 8 5 10 6 Trips 3 +4 3 +5 4 +5 Martin Grötschel

  35. Single Depot Vehicle Scheduling • The „Successive Shortest Path“ Algorithm 0 0 0 0 0 0 0 Buses 5 4 0 Bound cost = 15 Partial sol. cost = 15 -4 0 -3 0 6 1 7 0 3 0 3 1 7 3 -8 0 10 1 Trips 7 8 9 Martin Grötschel

  36. Single Depot Vehicle Scheduling 5 4 0 4 0 3 0 6 1 7 0 3 0 3 1 7 3 8 0 10 1 7 8 9 • The „Successive Shortest Path“ Algorithm • Path Search • Solution + Proof • Efficient Buses Bound cost = 15 Solution cost = 15 Trips Martin Grötschel

  37. SPEC Andreas Löbel Martin Grötschel

  38. Contents • What is vehicle circulation/scheduling? • Single depot vehicle scheduling • Multiple depot vehicle scheduling • Extensions Martin Grötschel

  39. Martin Grötschel

  40. Martin Grötschel

  41. Vehicle Scheduling • InputTimetabled and deadhead tripsVehicle types and depot capacitiesVehicle costs (fixed and variable) • OutputVehicle rotations • ProblemCompute rotations to cover all timetabled trips • GoalsMinimize number of vehiclesMinimize operation costsMinimize line hopping etc. vehiclecirculationsrotationsblocksschedules Martin Grötschel

  42. Graph Theoretic Model deadhead trips pull-out trips pull-in trips timetabled trips Martin Grötschel

  43. Example: Regensburg Map deleted Martin Grötschel

  44. Vehicle Scheduling Depot capacities: soft upper limits Fleet minimum: pull-in trips No line changes: interlining trips Peaks: pull-in/pull-out trips Turning: turns • Definition + cost of deadhead trips • Precise control at point, time, or trip • Changes of vehicles, lines, modes, turning, etc. • Automatic generation of pull-in/pull-out trips • Maintencance of all possible deadhead trips • Depot capacities (soft) Martin Grötschel

  45. Timelines d d Needs Fow model assignment model will be too large Martin Grötschel

  46. Integer Programming Model(Multicommodity Flow Problem) Martin Grötschel

  47. Theoretical Results • Observation: The LP relaxation of the Multicommodity Flow Problem does in general not produce integeral solutions. • Theorem: The Multicommodity Flow Problem is NP-hard. • Theorem (Tardos et. al.): There are pseudo-polynomial time approximation algorithms to solve the LP-relaxation of Multicommodity Flow Problems which are faster than general LP methods. Martin Grötschel

  48. Lagrangean Relaxation f3 x2 x1 f2 f4 P(A,b)P(B,d) f P(A,b) f1 x3 x4 Martin Grötschel

  49. Bundle Method(Kiwiel [1990], Helmberg [2000]) 2 1 3 • Max X polyhedral (piecewise linear) f  Martin Grötschel

  50. Primal Approximation • Theorem converges to a point Martin Grötschel

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