Seminar for verkehr

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

Outline

1 Introduction
Motivation to the UTRP
Existing Solutions
Problem Statement

2 Our Algorithms
Exact
Genetic

3 Current State
Results
Work in Progress


Introduction

Short Intro to Myself

My university in Brazil:


Introduction Motivation to the UTRP

Outline

1 Introduction
Existing Solutions
Problem Statement

2 Our Algorithms
Exact
Genetic

3 Current State
Results
Work in Progress



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, 1943
Computers can help in the process of planning.



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


Introduction Existing Solutions

Outline

1 Introduction
Existing Solutions
Problem Statement

2 Our Algorithms
Exact
Genetic

3 Current State
Results
Work in Progress



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 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).


Introduction Problem Statement

Outline

1 Introduction
Existing Solutions
Problem Statement

2 Our Algorithms
Exact
Genetic

3 Current State
Results
Work in Progress



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

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

Our Algorithms Exact

Outline

1 Introduction
Existing Solutions
Problem Statement

2 Our Algorithms
Exact
Genetic

3 Current State
Results
Work in Progress


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.


Our Algorithms Genetic

Outline

1 Introduction
Existing Solutions
Problem Statement

2 Our Algorithms
Exact
Genetic

3 Current State
Results
Work in Progress



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.



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 MIP
solution.



Minimum Spanning Tree Demo

a
7

8
5 b c
7
9 5
15
d e
6 9
8
11
f g



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).


Current State Results

Outline

1 Introduction
Existing Solutions
Problem Statement

2 Our Algorithms
Exact
Genetic

3 Current State
Results
Work in Progress



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].



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.



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



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.
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.
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.



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.
CO 2378 min. 5181 min.
I-Operator d0 48.62 % 9.48 %
ATT 40.88 min. 55.08 min.
CO 1077 min. 319 min.
II-Operator d0 46.97 % 8.47 %
ATT 41.26 min. 55.48 min.
CO 1265 min. 319 min.


Current State Work in Progress

Outline

1 Introduction
Existing Solutions
Problem Statement

2 Our Algorithms
Exact
Genetic

3 Current State
Results
Work in Progress



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 public
transport 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 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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