Table Of ContentUNIVERSITE LIBRE DE BRUXELLES
Année Académique 2006-2007
Faculté des Sciences Appliquées
ANT COLONY OPTIMIZATION ALGORITHM
FOR BIOBJECTIVE PERMUTATION FLOWSHOP
PROBLEMS
Directeur de Mémoire: Mémoire de fin d’études
Prof. Ph. VINCKE
présenté par Trung Truc HUYNH
Cosuperviseurs:
en vue de l’obtention du grade
M. BIRATTARI
d’Ingénieur civil en
Y. DE SMET
Electromécanique
T. STÜTZLE
Abstract
Ant Colony Optimization (ACO) is a metaheuristic for solving combinatorial
optimization problems which belongs to swarm intelligence approaches. It is the most
successful and the most studied technique among these methods which are inspired by
social behaviour of insects and other animals.
In this master thesis, we proposed two ACO algorithms using respectively one and
two pheromone matrices to tackle a biobjective permutation flowshop scheduling problem
where the makespan and the total tardiness are the objectives to minimize. The two
algorithms are aggregation methods and their underlying idea is to force each colony to
search for solutions in different directions of the space by changing the importance of each
objective all along the procedure.
These two algorithms and their different variants have been studied and tested on
four instances. They have been compared through the analysis of more than one hundred
pairwise comparisons of the different configurations. The results obtained have helped us
to have a better understanding of the problem and provide indications and suggestions for
further research.
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Résumé
Ant Colony Optimization (ACO) ou l’Optimisation par Colonies de Fourmis (OCF)
est une métaheuristique destinée à la résolution de problème d’optimisation combinatoire.
Cette technique est la plus efficace et la plus étudiée parmi les méthodes dérivées de
l’intelligence en essaim qui s’inspirent du comportement social de certains insectes et de
certains animaux.
Dans ce Mémoire de fin d’études, nous proposons deux algorithmes basés sur
l’OCF utilisant respectivement une et deux matrices de phéromone, pour la résolution d’un
problème biobjectif d’ordonnancement flowshop où le makespan et le total tardiness sont
les deux critères à optimiser.
Ces deux algorithmes sont des approches agrégatives dont l’idée est de forcer
chaque colonie à chercher les solutions dans différentes directions de l’espace en faisant
varier l’importance de chaque objectif tout au long de la procédure.
Ces deux algorithmes et leurs différentes variantes ont été étudiés et testés sur
quatre instances et comparées par l’analyse de plus de 100 comparaisons par paires de ces
différentes configurations. Les résultats obtenus nous ont aidés à avoir une meilleure
compréhension du problème et fournissent des indications et des suggestions pour de
futures recherches plus approfondies.
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Acknowledgements
I wish to thank Prof. Philippe Vincke the research Director of the operational
research laboratory of the Engineering Faculty of the Université Libre de Bruxelles (SMG)
for the opportunity to work in combinatorial optimization problems.
I wish to thank Yves Desmet researcher of the operational research laboratory of the
Engineering Faculty of the Université Libre de Bruxelles (SMG) for his advices in writing
and his support. I’m grateful to him for the opportunity to work in such an interesting field
and for having been comprehensible with my special situation these last two years.
I am deeply grateful to Mauro Birattari and Thomas Stützle, researchers of the
Institut de Recherches Interdisciplinaires et de Développements en Intelligence Artificelle
(IRIDIA), for their helpful advices and direction throughout the entire work. I want to
express my gratitude to them for having explained things clearly, helping me in the
programmation and for having been available for me and all my questions.
I also wish to thank Manuel López Ibáñez for having kindly provided his script to
plot the difference of two attainment functions and for his help to understand it.
I wish to express my gratitude to my parents and family, which have been
understanding and supportive, contributing through their encouragement to the completion
of the work.
I also wish to express my gratitude to my friends and the different physiotherapists
and neuropsychologists with whom I have worked these last two years and half for their
support and their encouragement.
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Table of contents
Abstract................................................................................................................. - 1 -
Acknowledgements............................................................................................... - 3 -
Table of contents................................................................................................... - 4 -
List of Figures....................................................................................................... - 7 -
List of tables........................................................................................................ - 10 -
1 Introduction..................................................................................................... - 12 -
2 Definition of the problem................................................................................ - 14 -
2.1 Classification.............................................................................................. - 14 -
2.2 Nomenclature of parameters....................................................................... - 16 -
2.3 Open shop................................................................................................... - 19 -
2.4 Job shop...................................................................................................... - 20 -
2. 5 Flow shop .................................................................................................. - 20 -
2.6 Objective functions..................................................................................... - 22 -
2.6.1 Time orientated functions...........................................................................- 22 -
2.6.2 Cost orientated functions............................................................................- 26 -
2.7 Diversity of flowshop problems................................................................. - 26 -
2.8 Biobjective permutation flowshop problem............................................... - 28 -
2.8.1 Makespan....................................................................................................- 29 -
2.8.2 Total tardiness.............................................................................................- 30 -
3 Optimization algorithms for Permutation Flowshop Scheduling Problem- 31 -
3.1 Heuristics.................................................................................................... - 32 -
3.1.1 Constructive heuristics................................................................................- 32 -
3.1.2 Improvement heuristics...............................................................................- 36 -
3.2 Metaheuristics............................................................................................. - 36 -
3.2.1 Hybrid metaheuristics.................................................................................- 37 -
3.2.2 Local search................................................................................................- 37 -
3.2.3 Genetic algorithms......................................................................................- 40 -
- 4 -
3.2.4 Tabu search.................................................................................................- 42 -
3.2.5 Simulated annealing....................................................................................- 44 -
3.2.6 Iterated local search....................................................................................- 46 -
3.3 Ant Colony Optimization........................................................................... - 47 -
3.3.1 The origin....................................................................................................- 47 -
3.3.2 Characteristics of artificial ants..................................................................- 50 -
3.3.3 Description of the ACO metaheuristic........................................................- 52 -
3.3.4 Important Choices in the application of an ACO algorithm.......................- 56 -
3.3.5 Development of different ACO algorithms................................................- 59 -
3.3.6 Applications of ACO algorithms................................................................- 65 -
3.3.7 Conclusion..................................................................................................- 71 -
4 Multiobjective optimization ........................................................................... - 73 -
4.1 Definition.................................................................................................... - 74 -
4.2 Techniques of optimization for Multiobjective combinatorial problem.... - 76 -
4.2.1 Scalar approaches........................................................................................- 76 -
4.2.2 Non-Pareto/non-scalar approaches.............................................................- 77 -
4.2.3 Pareto approaches.......................................................................................- 78 -
4.3 Metaheuristics for multiobjective optimization ......................................... - 78 -
4.3.1 Tabu search.................................................................................................- 78 -
4.3.2 Genetic programming.................................................................................- 79 -
4.3.3 Simulated annealing....................................................................................- 79 -
4.3.4 Ant colony optimisation..............................................................................- 79 -
4.3.5 Hybrid metaheuristics.................................................................................- 83 -
4.3.6 Parallel algorithms......................................................................................- 83 -
4.4 Two different ACO approaches for a biobjective flowshop problem........ - 85 -
4.4.1 Non dominated local search........................................................................- 87 -
4.4.2 ACO algorithm for biobjective problems...................................................- 87 -
4.4.3 ACO algorithm using one pheromone matrix (1phero)..............................- 88 -
5.4.4 ACO algorithm using two pheromone matrices (2phero)...........................- 89 -
4.5 Performance measures................................................................................ - 92 -
4.5.1 Quality indicator.........................................................................................- 94 -
4.5.2 Attainment functions...................................................................................- 97 -
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5 Experiments................................................................................................... - 101 -
5.1 Description of the ACO algorithms.......................................................... - 102 -
5.1.1 Max Min Ant System................................................................................- 102 -
5.1.2 Max Min Ant System incorporating the summation rule.........................- 103 -
5.2 Local search.............................................................................................. - 103 -
5.3 Single objective approach......................................................................... - 104 -
5.3.1 Instances and parameters..........................................................................- 104 -
5.3.2 Results.......................................................................................................- 105 -
5.4 Biobjective approach................................................................................ - 106 -
5.4.1 ACO algorithm using one pheromone matrix (1phero)............................- 107 -
5.4.2 ACO algorithm using two pheromone matrices(2phero)..........................- 107 -
5.4.3 Instances and parameters..........................................................................- 108 -
5.4.4 Comparison procedure..............................................................................- 108 -
5.4.5 Aggregation strategy.................................................................................- 111 -
5.4. 6 Influence of the non dominated local search...........................................- 117 -
5.4. 7 Comparisons between the different configurations.................................- 118 -
6 Conclusion...................................................................................................... - 126 -
Bibliography...................................................................................................... - 129 -
Appendices........................................................................................................ - 144 -
1 Single objective approach: makespan ......................................................... - 144 -
2 Single objective approach: Total tardiness.................................................. - 147 -
3 Outcomes of multiobjective optimizer for different local search................ - 150 -
4 Influence of the number of aggregations weights....................................... - 153 -
5 Results of the optimizers for different direction changes............................ - 162 -
6 Influence of the non dominated local search............................................... - 173 -
7 Comparison 1phero – 2phero approach...................................................... - 175 -
8 Comparison scratch – 2phase approach...................................................... - 188 -
9 Comparison global – local strategy............................................................. - 192 -
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List of Figures
Figure 1: Illustration of a flowshop production line with buffer.....................................- 16 -
Figure 2: Illustration of some element of the nomenclature............................................- 17 -
Figure 3: Scheme of an open shop production line..........................................................- 19 -
Figure 4: Scheme of a job shop production line..............................................................- 20 -
Figure 5: Scheme of a flow shop production line...........................................................- 20 -
Figure 6: Illustration of some objectives.........................................................................- 25 -
Figure 7: Illustration of identical parallel stations flowshop..........................................- 27 -
Figure 8: Illustration of non-identical parallel stations flowshop...................................- 28 -
Figure 9: Illustration of the makespan.............................................................................- 29 -
Figure 10: Illustration of the total tardiness.....................................................................- 30 -
Figure 11 : Transpose neighbourhood.............................................................................- 38 -
Figure 12: Exchange neighbourhood...............................................................................- 39 -
Figure 13: Insert neighbourhood......................................................................................- 39 -
Figure 14: Cycle of reproduction in a genetic algorithm.................................................- 40 -
Figure15: Pictorial summary of ILS................................................................................- 46 -
Figure 16: Illustration of the symmetrical binary bridge................................................- 47 -
Figure 17: Illustration of the asymmetrical binary bridge...............................................- 49 -
Figure 18: Global procedure of an Ant colony metaheuristic..........................................- 55 -
Figure 19: Illustration of Pareto dominance....................................................................- 75 -
Figure 20: Procedure of 1phero algorithm.......................................................................- 86 -
Figure 21: Procedure of 2phero algorithm.......................................................................- 86 -
Figure 22: Procedure of 2pheroG algorithm....................................................................- 91 -
Figure 23: Procedure of 2pjeroL algorithm.....................................................................- 92 -
Figure 24: Limitations of a comparison based only on the dominance...........................- 93 -
Figure 25: Illustration of an unary indicator: hypervolume.............................................- 95 -
Figure 26: Illustration of a binary indicator: coverage....................................................- 96 -
Figure 27: Plot of the attainment surface.........................................................................- 97 -
Figure 28: Superposition of 5 sets of non dominated points...........................................- 98 -
Figure 29: Superposition of the 5 corresponding attainment function............................- 98 -
Figure 30: Plot of the difference of two empirical attainment functions.........................- 99 -
Figure 31: Procedure of the single objective ACO algorithm.......................................- 101 -
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Figure 32: Differences of EAFs, global scratch/local scratch / 2pheroG 2phase (50x30-1) -
110 -
Figure 33: Grey scale encoding of the difference EAFs................................................- 110 -
Figure 34 : Differences of EAFs, 1phero 2phase |W|=11-|W|=41 (50x10-2)................- 112 -
Figure 35 : Differences of EAFs, 1phero 2phase |W|=11-|W|=81 (50x30-2)................- 113 -
Figure 36 : Differences of EAFs, 2pheroL 2phase for direction changes (50x10-2)....- 116 -
Figure 37: Differences of EAFs, 1phero 2phase with/without ND_LS (50x10-2).......- 117 -
Figure 38 : Differences of EAFs, 1phero 2phase/2pheroL 2phase (50x10-1)..............- 120 -
Figure 39: Differences of EAFs, 1phero scratch/2pheroG scratch (50x30-1)..............- 121 -
Figure 40: Differences of EAFs, 2pheroG scratch /2pheroG 2phase (50x30-2)..........- 122 -
Figure 41: Differences of EAFs, 2pheroG scratch/2pheroL scratch (50x10-1)............- 124 -
Figure 42: Differences of EAFs, 2pheroG scratch/2pheroL scratch (50x30-1)...........- 125 -
Figure 43: Differences of EAFs, 1phero 2phase |W|=11-|W|=41 (50x10-2).................- 154 -
Figure 44: Differences of EAFs, 1phero 2phase |W|=11-|W|=81 (50x10-2).................- 154 -
Figure 45: Differences of EAFs, 1phero 2phase |W|=11-|W|=41 (50x30-2).................- 156 -
Figure 46: Differences of EAFs, 1phero 2phase |W|=11-|W|=81 (50x30-2).................- 156 -
Figure 47: Differences of EAFs, 2pheroG 2phase |W|=11-|W|=41 (50x30-2)..............- 157 -
Figure 48: Differences of EAFs, 2pheroG 2phase |W|=11-|W|=81 (50x30-2)..............- 158 -
Figure 49: Differences of EAFs, 2pheroG 2phase |W|=41-|W=81| (50x30-2)..............- 158 -
Figure 50: Differences of EAFs, 2pheroL 2phase |W|=11-|W|=41 (50x30-2)..............- 159 -
Figure 51: Differences of EAFs, 2pheroL 2phase |W|=11-|W|=81| (50x30-2)..............- 160 -
Figure 52: Differences of EAFs, 2pheroL 2phase |W|=41-|W|=81 (50x30-2)...............- 160 -
Figure 53: Differences of EAFs, 1phero 2phase for direction changes,(50x10-2).......- 163 -
Figure 54: Differences of EAFs, 1phero 2phase for direction changes ,(50x10-2)......- 163 -
Figure 55: Differences of EAFs, 1phero 2phase for direction changes ((50x10-2)......- 164 -
Figure 56: Differences of EAFs, 1phero 2phase for direction changes ((50x10-2)......- 164 -
Figure 57: Differences of EAFs, 2pheroG 2phase for direction changes (50x10-2).....- 165 -
Figure 58: Differences of EAFs, 2pheroG 2phase for direction changes (50x10-2).....- 166 -
Figure 59 : Differences of EAFs, 1phero 2phase for direction changes 1,50x30-2).....- 167 -
Figure 60: Differences of EAFs, 1phero 2phase for direction changes (2,50x30-2)....- 168 -
Figure 61: Differences of EAFs, 1phero 2phase for direction changes (3,50x30-2)....- 168 -
Figure 62: 2pheroG 2phase for direction changes (50x30-2)........................................- 169 -
Figure 63: 2pheroL 2phase for direction changes (1,50x30-2).....................................- 170 -
Figure 64: 2pheroL 2phase for direction changes (2, 50x30-2)....................................- 171 -
- 8 -
Figure 65: Differences of EAFs,1phero 2phase with/without ND_LS (50x10-2)........- 173 -
Figure 66: Differences of EAFs,2pheroG 2phase with/without ND_LS (50x10-2)......- 174 -
Figure 67: Differences of EAFs, 1phero 2phase/2pheroG 2phase (50x10-1)...............- 175 -
Figure 68: Differences of EAFs, 1phero 2phase/2pheroL 2phase (50x10-1)..............- 176 -
Figure 69: Differences of EAFs, 1phero 2phase/2pheroL 2phase (50x10-2)...............- 177 -
Figure 70: Differences of EAFs, 1phero scratch/2pheroG scratch (50x30-1)..............- 178 -
Figure 71: Differences of EAFs, 1phero 2phase/2pheroG 2phase (50x30-1)...............- 178 -
Figure 72: Differences of EAFs, 1phero scratch/2pheroL scratch (50x30-1)...............- 179 -
Figure 73: Differences of EAFs, 1phero scratch/2pheroG scratch (50x30-2)..............- 180 -
Figure 74: Differences of EAFs, 1phero 2phase/2pheroG 2phase (50x30-2)...............- 180 -
Figure 75: Differences of EAFs, 1phero 2phase/2pheroL 2phase (50x30-2)...............- 181 -
Figure 76: Differences of EAFs,1phero 2phase/2pheroG 2phase for λ=0-1(50x10-2)- 182 -
Figure 77: Differences of EAFs, 1phero 2phase/2pheroL 2phase for λ=0-1(50x10-2)- 183 -
Figure 78: Differences of EAFs, 1phero 2phase/2pheroG 2phase for λ=0-1-0 (50x10-2)....-
184 -
Figure 79: Differences of EAFs, 1phero 2phase/2pheroL 2phase for λ=0-1-0 (50x10-2).....-
184 -
Figure 80: Differences of EAFs, 1phero 2phase/2pheroL 2phase for λ=1-0-1 (50x10-2).....-
185 -
Figure 81: Differences of EAFs, 1phero 2phase/2pheroL 2phase for λ=0-1(50x30-2)- 186 -
Figure 82: Differences of EAFs, 1phero 2phase /2pheroG 2phase for λ=1-0-1 (50x30-2)...-
187 -
Figure 83: Differences of EAFs, 2pheroL scratch/2pheroL 2phase (50x10-1).............- 188 -
Figure 84: Differences of EAFs, 1phero scratch/1phero 2phase (50x30-1).................- 189 -
Figure 85: Differences of EAFs, 2pheroG scratch/2pheroG 2phase (50x30-1)...........- 190 -
Figure 86: Differences of EAFs, 2pheroG scratch /2pheroG 2phase (50x30-2)..........- 191 -
Figure 87: Differences of EAFs, 2pheroG scratch/2pheroL scratch (50x10-1)............- 192 -
Figure 88: Differences of EAFs, 2pheroG scratch/2pheroL scratch (50x30-1)...........- 193 -
Figure 89: Differences of EAFs, 2pheroG 2phase/2pheroL 2phase (50x30-1)............- 194 -
Figure 90: Differences of EAFs, 2pheroG scratch/2pheroL scratch (50x30-2)...........- 194 -
Figure 91: Differences of EAFs, 2pheroG 2phase/2pheroL 2phase (50x30-2)............- 195 -
Figure 92: Differences of EAFs, 2pheroG 2phase/2pheroL 2phase for λ=1-0-1 (50x10-2)..-
196 -
- 9 -
Description:social behaviour of insects and other animals. l'OCF utilisant respectivement une et deux matrices de phéromone, pour la résolution d'un problème
