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

2. Outline
1 Introduction
Motivation to the UTRP
Existing Solutions
Problem Statement
2 Our Algorithms
Exact
Genetic
3 Current State
Results
Work in Progress
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3. Introduction
Short 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 UTRP
Outline
1 Introduction
Motivation to the UTRP
Existing Solutions
Problem Statement
2 Our Algorithms
Exact
Genetic
3 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 UTRP
Public Transportation
Bus in Porto Alegre.
Crowded, late.
Low resources? Are they being well employed?
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6. Introduction Motivation to the UTRP
Public Transportation
Route network in Porto Alegre with about 230 routes.
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7. Introduction Motivation to the UTRP
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, 1943
Computers can help in the process of planning.
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8. Introduction Motivation to the UTRP
UTNDP: 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
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9. Introduction Existing Solutions
Outline
1 Introduction
Motivation to the UTRP
Existing Solutions
Problem Statement
2 Our Algorithms
Exact
Genetic
3 Current State
Results
Work in Progress
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10. Introduction Existing Solutions
Existing Solutions
The list of existing solutions is long, including:
Multiple step solutions.
Metaheuristics.
Mixed non-linear mathematical models.
Ad-hoc solutions.
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11. Introduction Existing Solutions
Tool Example
´
Computational tool by Alvarez et al.
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12. Introduction Existing Solutions
Room for Improvement
Current 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).
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13. Introduction Problem Statement
Outline
1 Introduction
Motivation to the UTRP
Existing Solutions
Problem Statement
2 Our Algorithms
Exact
Genetic
3 Current State
Results
Work in Progress
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14. Introduction Problem Statement
Input and Route Sets
Two inputs: graph and demand matrix.
Transport, route and transit networks, respectively [1].
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15. Introduction Problem Statement
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.
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16. Introduction Problem Statement
Output
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
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17. Our Algorithms Exact
Outline
1 Introduction
Motivation to the UTRP
Existing Solutions
Problem Statement
2 Our Algorithms
Exact
Genetic
3 Current State
Results
Work in Progress
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18. Our Algorithms Exact
Exact 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.
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19. Our Algorithms Genetic
Outline
1 Introduction
Motivation to the UTRP
Existing Solutions
Problem Statement
2 Our Algorithms
Exact
Genetic
3 Current State
Results
Work in Progress
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20. Our Algorithms Genetic
Genetic 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.
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21. Our Algorithms Genetic
Creating New Routes
Take it from a pool of base routes.
Apply operators to existing solutions (route sets).
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22. Our Algorithms Genetic
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 MIP
solution.
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23. Our Algorithms Genetic
Minimum Spanning Tree Demo
a
7
8
5 b c
7
9 5
15
d e
6 9
8
11
f g
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24. Our Algorithms Genetic
Minimum Spanning Tree Demo
a
7
8
5 b c
7
9 5
15
d e
6 9
8
11
f g
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25. Our Algorithms Genetic
Minimum 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 Genetic
Minimum 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 Genetic
Minimum 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 Genetic
Minimum 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 Genetic
Minimum 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 Genetic
Operators
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).
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31. Current State Results
Outline
1 Introduction
Motivation to the UTRP
Existing Solutions
Problem Statement
2 Our Algorithms
Exact
Genetic
3 Current State
Results
Work in Progress
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32. Current State Results
Tested Networks
Two networks were used: Mandl’s and artificial British(based) city with
110 nodes and 275 links.
Mandl’s Swiss road network [1].
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33. Current State Results
Solution Quality Quantities
All results are evaluated using the following quantities, as in previous
works:
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.
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34. Current State Results
Mandl’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
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35. Current State Results
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 %
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.
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36. Current State Results
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.
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.
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37. Current State Work in Progress
Outline
1 Introduction
Motivation to the UTRP
Existing Solutions
Problem Statement
2 Our Algorithms
Exact
Genetic
3 Current State
Results
Work in Progress
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38. Current State Work in Progress
Performance
Time used in each function. Dijkstra takes 90% of processing time.
To explore the search space faster: use GPU.
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39. Current State Work in Progress
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.
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40. Thank you!
Conclusion
New methods and algorithms for the UTRP can make public
transport better!
Suggestions or questions?
Thank you!
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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.
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