Seminar for verkehr

Algorithms for the Urban Transit Routing ProblemExact and Metaheuristic

Bruno Coswig Fiss

1Institut fur Technische Informatik und MikroelektronikTechnische Universitat Berlin

VSP Internal Seminar

Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 1 / 35

Outline

1 IntroductionMotivation to the UTRPExisting SolutionsProblem Statement

2 Our AlgorithmsExactGenetic

3 Current StateResultsWork in Progress


Introduction

Short Intro to Myself

My university in Brazil:


Introduction Motivation to the UTRP

Outline






Public Transportation

Bus in Porto Alegre.

Crowded, late.Low resources? Are they being well employed?



Public Transportation

Route network in Porto Alegre with about 230 routes.



Automatization 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, 1943Computers can help in the process of planning.



UTNDP: UTRP and UTSP.

This problem has been studied, and is know as the Urban TransitNetwork Design Problem (UTNDP).Commonly divided: Urban Transit Routing Problem and UrbanTransit Scheduling Problem.Scheduling depends on previous step.New schedules are easier to test.Focus here: UTRP


Introduction Existing Solutions

Outline






Existing Solutions

The list of existing solutions is long, including:Multiple step solutions.Metaheuristics.Mixed non-linear mathematical models.Ad-hoc solutions.



Tool Example

Computational tool by Alvarez et al.



Room for Improvement

Current issues with the existing solutions:Many different problem definitions.The quality of these solutions depends fundamentally on thechosen algorithms.Large search space (there is no free lunch).Comparison is necessary!Our Goal: develop and test appropriate algorithms and methodsfor the UTRP using a well-known problem definition (and withcommon benchmarks).


Introduction Problem Statement

Outline






Input and Route Sets

Two inputs: graph and demand matrix.

Transport, route and transit networks, respectively [1].



Associated 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.



Output

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Average travel time in minutes

Approximation for Pareto-optimal curves

GA Solutions with up to 8 routesGA Solutions with up to 6 routesGA Solutions with up to 4 routes

Fitting curve (58.22/(x-9.86) + 44.36)

Pareto-optimal set approximation.Bruno Coswig Fiss (TU Berlin) Algorithms for the UTRP June 13, 2012 16 / 35

Our Algorithms Exact

Outline





Our Algorithms Exact

Exact Solution Summary

”The problem of designing a good or efficient route set (or routenetwork) for a transit system is a difficult optimization problemwhich does not lend itself readily to mathematical programmingformulations and solutions using traditional techniques” – Dr.Partha Chakroborty, Transportation EngineerMathematical solution has been created to test feasibility andcorrectness.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.


Our Algorithms Genetic

Outline






Genetic Algorithm Overview

Maintain a population of potential solutions, ie. route sets.Create or modify routes in a route set and, if dominating anotherroute set, take its place.



Creating New Routes

Take it from a pool of base routes.Apply operators to existing solutions (route sets).



Base routes

Base 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 MIPsolution.



Minimum Spanning Tree Demo

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Operators

Mutate routeAdd a node to or remove a node from the extremity of a route.

Simplify route setIf the route set contains 9-3-4-5-6 and 4-5-6-12-2, we replace with9-3-4-5-6-12-2

Cross-over two routesJoin two routes at a certain intersection (cut cycles if necessary).


Current State Results

Outline






Tested Networks

Two networks were used: Mandl’s and artificial British(based) city with110 nodes and 275 links.

Mandl’s Swiss road network [1].



Solution Quality Quantities

All 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 (inminutes, considering constant transport speed).ATTwop = ATT −

∑i≤TMAX

tpendi i .



Mandl’s Network Exact Solutions

Best possible route sets found using the Mixed Integer formulation

Number of routes 2 3d0 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 78992Two Routes 6-14-7-5-2-1-4-3-11-10-9-13-12

0-1-3-5-7-9-6-14-8Three Routes 4-3-11-10-12-13-9-7-5-2-1-0

4-3-1-2-5-14-6-9-10-110-1-4-3-5-7-9-6-14-8



Genetic 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 %


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 %




Genetic 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.


I-Operator d0 48.62 % 9.48 %ATT 40.88 min. 55.08 min.


II-Operator d0 46.97 % 8.47 %ATT 41.26 min. 55.48 min.



Current State Work in Progress

Outline






Performance

Time used in each function. Dijkstra takes 90% of processing time.

To explore the search space faster: use GPU.



Simulations

Two scenarios are being simulated:Porto Alegre: test effectiveness in comparison to existing network.Demands are artificial.Berlin: use MATSim as the objective function.


Thank you!

Conclusion

New methods and algorithms for the UTRP can make publictransport better!Suggestions or questions?Thank you!


Thank you!

References

Lang Fan, Christine L. Mumford, and Dafydd Evans.A simple multi-objective optimization algorithm for the urban transitrouting problem.In Proceedings of the Eleventh conference on Congress onEvolutionary Computation, CEC’09, pages 1–7, Piscataway, NJ, USA,2009. IEEE Press.


Seminar for verkehr

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