ant colony optimization and local search for the probabilistic traveling salesman problem

200 Pages·2006·1.51 MB·English

UNIVERSITE LIBRE DE BRUXELLES, UNIVERSITE D’EUROPE
Faculte´ des Sciences applique´es Anne´e acade´mique 2005-2006
ANT COLONY OPTIMIZATION AND LOCAL SEARCH
FOR THE
PROBABILISTIC TRAVELING SALESMAN PROBLEM:
A CASE STUDY IN
STOCHASTIC COMBINATORIAL OPTIMIZATION
Directeur de Me´moire: Me´moire de fin d’e´tudes pre´sente´ par
Prof. Marco Dorigo LeonoraBianchienvuedel’obtentiondu
titre de Docteur en Sciences Applique´es
Codirecteur de Me´moire:
Prof. Luca Maria Gambardella

Ant Colony Optimization and Local Search
for the
Probabilistic Traveling Salesman Problem:
A Case Study in
Stochastic Combinatorial Optimization
Leonora Bianchi
Universite´ Libre de Bruxelles

Summary
In this thesis we focus on Stochastic Combinatorial Optimization Problems (SCOPs),
a wide class of combinatorial optimization problems under uncertainty, where part of
the information about the problem data is unknown at the planning stage, but some
knowledge about its probability distribution is assumed.
Optimizationproblemsunderuncertaintyarecomplexanddifficult,andoftenclassi-
cal algorithmic approaches based on mathematical and dynamic programming are able
to solve only very small problem instances. For this reason, in recent years metaheuris-
ticalgorithmssuchasAntColonyOptimization,EvolutionaryComputation,Simulated
Annealing, Tabu Search and others, are emerging as successful alternatives to classical
approaches.
Inthisthesis, metaheuristicsthathavebeenappliedsofartoSCOPsareintroduced
and the related literature is thoroughly reviewed. In particular, two properties of
metaheuristics emerge from the survey: they are a valid alternative to exact classical
methods for addressing real-sized SCOPs, and they are flexible, since they can be quite
easily adapted to solve different SCOPs formulations, both static and dynamic. On
the base of the current literature, we identify the following as the key open issues in
solving SCOPs via metaheuristics: (1) the design and integration of ad hoc, fast and
effective objective function approximations inside the optimization algorithm; (2) the
estimationoftheobjectivefunctionbysamplingwhennoclosed-formexpressionforthe
objective function is available, and the study of methods to reduce the time complexity
and noise inherent to this type of estimation; (3) the characterization of the efficiency
of metaheuristic variants with respect to different levels of stochasticity in the problem
instances.
We investigate the above issues by focusing in particular on a SCOP belonging to
the class of vehicle routing problems: the Probabilistic Traveling Salesman Problem
(PTSP). For the PTSP, we consider the Ant Colony Optimization metaheuristic and
wedesignefficientlocalsearchalgorithmsthatcanenhanceitsperformance. Weobtain
state-of-the-artalgorithms, butweshowthattheyareeffectiveonlyforinstancesabove
a certain level of stochasticity, otherwise it is more convenient to solve the problem
as if it were deterministic. The algorithmic variants based on an estimation of the
objective function by sampling obtain worse results, but qualitatively have the same
behavior of the algorithms based on the exact objective function, with respect to the
level of stochasticity. Moreover, we show that the performance of algorithmic variants
based on ad hoc approximations is strongly correlated with the absolute error of the
approximation, and that the effect on local search of ad hoc approximations can be
very degrading.
Finally, we briefly address another SCOP belonging to the class of vehicle routing
problems: the Vehicle Routing Problem with Stochastic Demands (VRPSD). For this
problem, we have implemented and tested several metaheuristics, and we have studied
the impact of integrating in them different ad hoc approximations.
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Acknowledgments
The research work of this thesis has been mainly done at IDSIA, the Dalle Molle
Institute for Artificial Intelligence in Lugano, Switzerland. I express my sincere thanks
to all people that have been at IDSIA since I arrived there in the year 2000, because
of the friendly and special working environment they created.
AnimportantpartofthisthesisisrootedinonemonthspentatIRIDIA,Universit´e
Libre de Bruxelles, Brussels, Belgium. I must thank all the people working there, and
particularly Joshua Knowles, because, without already being involved in my subject
of research, he was very open and we could have a profitable exchange of ideas, that
resulted in my first journal paper. From IRIDIA I also thank the secretary, Muriel
Decreton, for her help in the bureaucratic formalities that she carried out for me while
I was in Switzerland.
I also thank all the people involved in the “Metaheuristics Network”, particularly
thosewithwhomIworkedfortheresearchaboutthestochasticvehicleroutingproblem:
Mauro Birattari, Marco Chiarandini, Max Manfrin, Monaldo Mastrolilli, my husband
Fabrizio Oliverio, Luis Paquete, Olivia Rossi-Doria, and Tommaso Schiavinotto. From
each of them I learnt something very useful for my research. I especially thank Mauro
Birattari and Marco Chiarandini for their support with statistical analysis of results.
Iacknowledgefinancialsupportbytwosources: theSwissNationalScienceFounda-
tion project titled “On-line fleet management”, grant 16R10FM; and the “Metaheuris-
tics Network”, a Research and Training Network founded by the Improving Human
Potential Programme of the Commission of the European Communities, grant HPRN-
CT-1999-00106.
I am particularly grateful to Luca Maria Gambardella for having supported and
guidedmyresearchsincethebeginning, andtoMarcoDorigoforhiscarefulsupervision
of my work.
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Contents
Summary iv
Acknowledgments vi
Contents x
List of Algorithms xi
Original contributions and outline xiii
I Metaheuristics for Stochastic Combinatorial Optimization 1
1 Introduction 3
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Modeling approaches to uncertainty . . . . . . . . . . . . . . . . . . . . 4
2 Formal descriptions of SCOPs 9
2.1 General but tentative SCOP definition . . . . . . . . . . . . . . . . . . . 9
2.2 Static SCOPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Dynamic SCOPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Objective function computation . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Ad hoc approximations . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.2 Sampling approximation . . . . . . . . . . . . . . . . . . . . . . . 15
3 Metaheuristics for SCOPs 17
3.1 Ant Colony Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Introduction to ACO . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 ACO for SCOPs . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.2.1 Exact objective and ad hoc approximation . . . . . . . 20
3.1.2.2 Sampling approximation . . . . . . . . . . . . . . . . . 21
3.1.2.3 Markov Decision Processes . . . . . . . . . . . . . . . . 23
3.2 Evolutionary Computation . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.1 Introduction to EC . . . . . . . . . . . . . . . . . . . . . . . . . . 24
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3.2.2 EC for SCOPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2.1 Exact objective and ad hoc approximation . . . . . . . 25
3.2.2.2 Sampling approximation . . . . . . . . . . . . . . . . . 27
3.2.2.3 Markov Decision Processes . . . . . . . . . . . . . . . . 29
3.3 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Introduction to SA . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2 SA for SCOPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.2.1 Exact objective and ad hoc approximation . . . . . . . 31
3.3.2.2 Sampling approximation . . . . . . . . . . . . . . . . . 31
3.4 Tabu Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4.1 Introduction to TS . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4.2 TS for SCOPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.2.1 Exact objective and ad hoc approximation . . . . . . . 37
3.4.2.2 Sampling approximation . . . . . . . . . . . . . . . . . 37
3.5 Stochastic Partitioning Methods . . . . . . . . . . . . . . . . . . . . . . 39
3.5.1 Stochastic Partitioning Methods for SCOP’s. . . . . . . . . . . . 39
3.5.1.1 Exact objective and ad hoc approximation . . . . . . . 40
3.5.1.2 Sampling approximation . . . . . . . . . . . . . . . . . 40
3.6 Other algorithmic approaches to SCOPs . . . . . . . . . . . . . . . . . . 43
3.7 Discussion and open issues . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.7.1 Using the Sampling approximation . . . . . . . . . . . . . . . . . 44
3.7.2 Experimental comparisons among different metaheuristics . . . . 45
3.7.3 Theoretical convergence properties . . . . . . . . . . . . . . . . . 47
II ACO and local search for the PTSP 49
4 The Probabilistic Traveling Salesman Problem 51
4.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Benchmark of PTSP instances . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Lower bound of the optimal solution value . . . . . . . . . . . . . . . . . 57
4.5 Simple constructive heuristics . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5.1 Experimental analysis . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Ant Colony Optimization 65
5.1 A straightforward implementation . . . . . . . . . . . . . . . . . . . . . 65
5.1.1 The pACS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.2 Experimental analysis . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1.2.1 Computational environment . . . . . . . . . . . . . . . 68
5.1.2.2 Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 The use of objective function approximations in ACO . . . . . . . . . . 75
5.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
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