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Seminar for verkehr

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  • 1. Algorithms for the Urban Transit Routing Problem Exact and Metaheuristic Bruno Coswig Fiss 1 Institut fur Technische Informatik und Mikroelektronik ¨ ¨ Technische Universitat Berlin VSP Internal SeminarBruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 1 / 35
  • 2. Outline1 Introduction Motivation to the UTRP Existing Solutions Problem Statement2 Our Algorithms Exact Genetic3 Current State Results Work in Progress Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 2 / 35
  • 3. IntroductionShort Intro to Myself My university in Brazil: Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 3 / 35
  • 4. Introduction Motivation to the UTRPOutline1 Introduction Motivation to the UTRP Existing Solutions Problem Statement2 Our Algorithms Exact Genetic3 Current State Results Work in Progress Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 4 / 35
  • 5. Introduction Motivation to the UTRPPublic Transportation Bus in Porto Alegre. Crowded, late. Low resources? Are they being well employed? Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 5 / 35
  • 6. Introduction Motivation to the UTRPPublic Transportation Route network in Porto Alegre with about 230 routes. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 6 / 35
  • 7. Introduction Motivation to the UTRPAutomatization and Computer Assistance Complexity of network design is enormous. Human planners take decisions. Is that enough? ”I think there is a world market for maybe five computers.” – allegedly Thomas Watson, chairman of IBM, 1943 Computers can help in the process of planning. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 7 / 35
  • 8. Introduction Motivation to the UTRPUTNDP: UTRP and UTSP. This problem has been studied, and is know as the Urban Transit Network Design Problem (UTNDP). Commonly divided: Urban Transit Routing Problem and Urban Transit Scheduling Problem. Scheduling depends on previous step. New schedules are easier to test. Focus here: UTRP Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 8 / 35
  • 9. Introduction Existing SolutionsOutline1 Introduction Motivation to the UTRP Existing Solutions Problem Statement2 Our Algorithms Exact Genetic3 Current State Results Work in Progress Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 9 / 35
  • 10. Introduction Existing SolutionsExisting SolutionsThe list of existing solutions is long, including: Multiple step solutions. Metaheuristics. Mixed non-linear mathematical models. Ad-hoc solutions. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 10 / 35
  • 11. Introduction Existing SolutionsTool Example ´ Computational tool by Alvarez et al. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 11 / 35
  • 12. Introduction Existing SolutionsRoom for ImprovementCurrent issues with the existing solutions: Many different problem definitions. The quality of these solutions depends fundamentally on the chosen algorithms. Large search space (there is no free lunch). Comparison is necessary! Our Goal: develop and test appropriate algorithms and methods for the UTRP using a well-known problem definition (and with common benchmarks). Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 12 / 35
  • 13. Introduction Problem StatementOutline1 Introduction Motivation to the UTRP Existing Solutions Problem Statement2 Our Algorithms Exact Genetic3 Current State Results Work in Progress Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 13 / 35
  • 14. Introduction Problem StatementInput and Route Sets Two inputs: graph and demand matrix. Transport, route and transit networks, respectively [1]. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 14 / 35
  • 15. Introduction Problem StatementAssociated costs Operator cost: sum of weight of edges used. Passenger cost: total travel time. Multi-objective. Conditions that can be considered: Number of routes. Lenght of routes. Cycles and backtracks. Penalty for making transfers. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 15 / 35
  • 16. Introduction Problem StatementOutput Approximation for Pareto-optimal curves 240 GA Solutions with up to 8 routes GA Solutions with up to 6 routes 220 GA Solutions with up to 4 routes Fitting curve (58.22/(x-9.86) + 44.36) Total route set length in minutes 200 180 160 140 120 100 80 60 10 10.5 11 11.5 12 12.5 13 13.5 Average travel time in minutes Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 16 / 35
  • 17. Our Algorithms ExactOutline1 Introduction Motivation to the UTRP Existing Solutions Problem Statement2 Our Algorithms Exact Genetic3 Current State Results Work in Progress Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 17 / 35
  • 18. Our Algorithms ExactExact Solution Summary ”The problem of designing a good or efficient route set (or route network) for a transit system is a difficult optimization problem which does not lend itself readily to mathematical programming formulations and solutions using traditional techniques” – Dr. Partha Chakroborty, Transportation Engineer Mathematical solution has been created to test feasibility and correctness. Uses a Mixed Integer Programming formulation. Achieved global optimal solutions for Mandl’s Swiss road network (to be shown) with 2 and 3 routes. Very slow, but useful linear relaxation and for divide and conquer. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 18 / 35
  • 19. Our Algorithms GeneticOutline1 Introduction Motivation to the UTRP Existing Solutions Problem Statement2 Our Algorithms Exact Genetic3 Current State Results Work in Progress Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 19 / 35
  • 20. Our Algorithms GeneticGenetic Algorithm Overview Maintain a population of potential solutions, ie. route sets. Create or modify routes in a route set and, if dominating another route set, take its place. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 20 / 35
  • 21. Our Algorithms GeneticCreating New Routes Take it from a pool of base routes. Apply operators to existing solutions (route sets). Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 21 / 35
  • 22. Our Algorithms GeneticBase routesBase routes are intrinsic to a graph: Shortest path for every pair of nodes (in original network). Minimum Spanning Tree (!). Paths with highest covered demand. Routes with high percentages in the linear relaxation of the MIP solution. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 22 / 35
  • 23. Our Algorithms GeneticMinimum Spanning Tree Demo a 7 8 5 b c 7 9 5 15 d e 6 9 8 11 f g Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 23 / 35
  • 24. Our Algorithms GeneticMinimum Spanning Tree Demo a 7 8 5 b c 7 9 5 15 d e 6 9 8 11 f g Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 23 / 35
  • 25. Our Algorithms GeneticMinimum Spanning Tree Demo a 7 8 5 b c 7 9 5 15 d e 6 9 8 11 f g Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 23 / 35
  • 26. Our Algorithms GeneticMinimum Spanning Tree Demo a 7 8 5 b c 7 9 5 15 d e 6 9 8 11 f g Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 23 / 35
  • 27. Our Algorithms GeneticMinimum Spanning Tree Demo a 7 8 5 b c 7 9 5 15 d e 6 9 8 11 f g Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 23 / 35
  • 28. Our Algorithms GeneticMinimum Spanning Tree Demo a 7 8 5 b c 7 9 5 15 d e 6 9 8 11 f g Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 23 / 35
  • 29. Our Algorithms GeneticMinimum Spanning Tree Demo a 7 8 5 b c 7 9 5 15 d e 6 9 8 11 f g Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 23 / 35
  • 30. Our Algorithms GeneticOperators Mutate route Add a node to or remove a node from the extremity of a route. Simplify route set If the route set contains 9-3-4-5-6 and 4-5-6-12-2, we replace with 9-3-4-5-6-12-2 Cross-over two routes Join two routes at a certain intersection (cut cycles if necessary). Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 24 / 35
  • 31. Current State ResultsOutline1 Introduction Motivation to the UTRP Existing Solutions Problem Statement2 Our Algorithms Exact Genetic3 Current State Results Work in Progress Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 25 / 35
  • 32. Current State ResultsTested NetworksTwo networks were used: Mandl’s and artificial British(based) city with110 nodes and 275 links. Mandl’s Swiss road network [1]. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 26 / 35
  • 33. Current State ResultsSolution Quality QuantitiesAll results are evaluated using the following quantities, as in previousworks: di is the percentage of the demand satisfied with i transfers. ATT is the average travel time (in minutes per passenger), including transfer penalties. CO is the cost for the operator, i.e., the total route length (in minutes, considering constant transport speed). ATTwop = ATT − i≤TMAX tpen di i. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 27 / 35
  • 34. Current State ResultsMandl’s Network Exact Solutions Best possible route sets found using the Mixed Integer formulation Number of routes 2 3 d0 84.90 % 93.67 % d1 14.00 % 5.43 % d2 1.10 % 0.90 % ATT 11.33 min. 10.50 min. CO 98 min. 150 min. Processing time (s) 1065 78992 Two Routes 6-14-7-5-2-1-4-3-11-10-9-13-12 0-1-3-5-7-9-6-14-8 Three Routes 4-3-11-10-12-13-9-7-5-2-1-0 4-3-1-2-5-14-6-9-10-11 0-1-4-3-5-7-9-6-14-8 Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 28 / 35
  • 35. Current State ResultsGenetic Algorithm on Mandl’s Network Comparison between best UTRP multi-objective solutions on Mandl’s Network Scenario Qp Best previous results Our metaheuristic ([1]) approach results Best for Passenger d0 94.54 % 98.84 % d1 5.46 % 1.16 % d2 0.00 % 0.00 % ATT 10.36 min. 10.10 min. CO 283 min. 259 min. Compromise Solution d0 93.19 % 93.61 % (CO ≤ 148) d1 6.23 % 6.20 % d2 0.58 % 0.19 % ATT 10.46 min. 10.43 min. CO 148 min. 147 min. Compromise Solution d0 90.88 % 91.23 % (CO ≤ 126) d1 8.35 % 7.84 % d2 0.77 % 0.93 % ATT 10.65 min. 10.59 min. CO 126 min. 126 min. Best for Operator d0 66.09 % 77.78 % d1 30.38 % 21.32 % d2 3.53 % 0.90 % ATT 13.34 min. 12.97 min. CO 63 min. 63 min. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 29 / 35
  • 36. Current State ResultsGenetic Algorithm on Artificial British Network Comparison between best UTRP multi-objective solutions on artificial British city Scenario Qp Best previous results Our metaheuristic ([1]) approach results I-Passenger d0 72.91 % 55.80 % ATT 36.28 min. 36.35 min. ATTwop 34.60 min. 34.12 min. CO 2986 min. 8406 min. II-Passenger d0 71.21 % 46.25 % ATT 37.52 min. 36.61 min. ATTwop 35.68 min. 33.77 min. CO 2378 min. 5181 min. I-Operator d0 48.62 % 9.48 % ATT 40.88 min. 55.08 min. ATTwop 37.36 min. 45.66 min. CO 1077 min. 319 min. II-Operator d0 46.97 % 8.47 % ATT 41.26 min. 55.48 min. ATTwop 37.655 min. 47.90 min. CO 1265 min. 319 min. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 30 / 35
  • 37. Current State Work in ProgressOutline1 Introduction Motivation to the UTRP Existing Solutions Problem Statement2 Our Algorithms Exact Genetic3 Current State Results Work in Progress Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 31 / 35
  • 38. Current State Work in ProgressPerformance Time used in each function. Dijkstra takes 90% of processing time.To explore the search space faster: use GPU. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 32 / 35
  • 39. Current State Work in ProgressSimulationsTwo scenarios are being simulated: Porto Alegre: test effectiveness in comparison to existing network. Demands are artificial. Berlin: use MATSim as the objective function. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 33 / 35
  • 40. Thank you!Conclusion New methods and algorithms for the UTRP can make public transport better! Suggestions or questions? Thank you! Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 34 / 35
  • 41. Thank you!References Lang Fan, Christine L. Mumford, and Dafydd Evans. A simple multi-objective optimization algorithm for the urban transit routing problem. In Proceedings of the Eleventh conference on Congress on Evolutionary Computation, CEC’09, pages 1–7, Piscataway, NJ, USA, 2009. IEEE Press. Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 35 / 35