Venables, Harry (2011) Ant Colony Optimisation

222 Pages·2015·5.99 MB·English

Ant Colony Optimisation – A
Proposed Solution Framework for
the Capacitated Facility Location
Problem
Harry Venables
A thesis submitted in partial fulfilment of
the requirements of the
University of Sunderland
for the degree of
Doctor of Philosophy
June 2011
Abstract
This thesis is a critical investigation into the development, application and evalu-
ation of ant colony optimisation metaheuristics, with a view to solving a class of
capacitated facility location problems. The study is comprised of three phases.
The first sets the scene and motivation for research, which includes; key con-
cepts of ant colony optimisation, a review of published academic materials and a
research philosophy which provides a justification for a deductive empirical mode
of study. This phase reveals that published results for existing facility location
metaheuristics are often ambiguous or incomplete and there is no clear evidence
of a dominant method. This clearly represents a gap in the current knowledge
base and provides a rationale for a study that will contribute to existing knowl-
edge, by determining if ant colony optimisation is a suitable solution technique for
solving capacitated facility location problems.
The second phase is concerned with the research, development and appli-
cation of a variety of ant colony optimisation algorithms. Solution methods pre-
sented include combinations of approximate and exact techniques. The study
identifies a previously untried ant hybrid scheme, which incorporates an exact
method within it, as the most promising of techniques that were tested. Also a
novel local search initialisation which relies on memory is presented. These hy-
bridisations successfully solve all of the capacitated facility location test problems
available in the OR-Library.
The third phase of this study conducts an extensive series of run-time analy-
ses, to determine the prowess of the derived ant colony optimisation algorithms
against a contemporary cross-entropy technique. This type of analysis for mea-
suring metaheuristic performance for the capacitated facility location problem is
not evident within published materials. Analyses of empirical run-time distribu-
tionsrevealthatant colonyoptimisation issuperiorto itscontemporary opponent.
All three phases of this thesis provide their own individual contributions to ex-
istingknowledgebases: theproductionofaseriesofrun-timedistributionswillbe
a valuable resource for future researchers; results demonstrate that hybridisation
of metaheuristics with exact solution methods is an area not to be ignored; the
hybridmethodsemployedinthisstudytenyearsagowouldhavebeenimpractical
or infeasible; ant colony optimisation is shown to be a very flexible metaheuristic
that can easily be adapted to solving mixed integer problems using hybridisation
techniques.
Acknowledgements
Initially, my sincere thanks must go to Dr. Mitchell Andrews, a former colleague
fromtheUniversityofSunderlandBusinessSchool, whocoercedmeintostarting
a Ph.D. study. Next, I would like to thank my original supervisor Prof. Alfredo
Moscardini. He gave me that initial spark about agent based modelling, that
made me think of ants and how they solved everyday problems by their social
interactions; I owe him a great deal. His encouragement for letting me just get
on with it, allowing me to develop the confidence to present my work to academic
peersandusetheexperienceasavitalpartoftheresearchapprenticeship. Upon
Alfredo’s retirement from the university, Dr. Valentina Plekhanova, had the task of
takingover. Thankyouforyourthoughtsandguidancethroughthoseinitialwriting
stages at a vulnerable time of life.
However, my biggest thanks must go to my wife Laura for all of the emotional
support through the highs and lows of this study. This work would not have been
worth it or even completed without her being there for me when I needed more
than just kind words. Finally, my two children Louis and Rachel their infectious
giggles and laughs make everything worthwhile.
Contents
1 Introduction 1
1.1 Research Study Outline . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Ant Colony Optimisation . . . . . . . . . . . . . . . . . . . . . . . . 3
2 A Review of Facility Location 9
2.1 Facility Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 P-Median Facility Location Problems . . . . . . . . . . . . . . . . . 12
2.3 Uncapacitated Facility Location Problems . . . . . . . . . . . . . . 14
2.4 Capacitated Facility Location Problems . . . . . . . . . . . . . . . 15
2.4.1 Lagrangean Relaxation . . . . . . . . . . . . . . . . . . . . 17
2.4.2 Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.1 Research Question . . . . . . . . . . . . . . . . . . . . . . . 25
3 Research Methodology 26
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Research Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Research Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1 Key Features for Empirical Analysis of the CFLP . . . . . . 30
3.3.2 Research Hypothesis . . . . . . . . . . . . . . . . . . . . . 31
3.4 Research Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 OR-Library Test Problems . . . . . . . . . . . . . . . . . . . . . . . 36
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Preliminary Development and Experimentation 40
4.1 Formal Specification of the CFLP . . . . . . . . . . . . . . . . . . . 40
4.1.1 Complexity of the CFLP . . . . . . . . . . . . . . . . . . . . 42
4.2 ACO Framework: Modelling Criteria . . . . . . . . . . . . . . . . . 45
iii
Contents iv
4.2.1 Characteristics of an ACO Construction Graph for the CFLP 45
4.3 Design of an ACO Algorithm . . . . . . . . . . . . . . . . . . . . . 47
4.3.1 Solution Construction Phase . . . . . . . . . . . . . . . . . 48
4.3.2 Pheromone Update Phase . . . . . . . . . . . . . . . . . . 51
4.3.3 Ant System and Max-Min Ant System for the CFLP . . . . . 52
4.4 Computational Design and Experimentation . . . . . . . . . . . . . 55
4.5 Initial Conclusions and Recommendations . . . . . . . . . . . . . . 57
5 Hybrid-ACO Development for the CFLP 63
5.1 ACO Hybridisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Hybrid Construction Phase . . . . . . . . . . . . . . . . . . . . . . 66
5.3 Transportation Problem Approximation . . . . . . . . . . . . . . . . 71
5.4 Derivation of Ant Visibility . . . . . . . . . . . . . . . . . . . . . . . 72
5.5 Hybridisation of MMAS . . . . . . . . . . . . . . . . . . . . . . . . 75
5.6 Local Search Methods for Solution Improvement . . . . . . . . . . 77
5.6.1 Drop Facilities . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6.2 Swap Facilities . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.7 Hybrid MMAS: Initial Experimentation . . . . . . . . . . . . . . . 81
5.7.1 MMAS and Basic DROP . . . . . . . . . . . . . . . . . . . 81
5.7.2 Pheromone Based DROP and DROP-SWAP . . . . . . . . 82
5.8 Hybrid MMAS: Initial Evaluation . . . . . . . . . . . . . . . . . . . 84
5.9 Hybrid MMAS: An Alternative Approach . . . . . . . . . . . . . . 88
5.9.1 Larger OR-Library Instances . . . . . . . . . . . . . . . . . 93
5.10 Hyper-cube Framework for the CFLP . . . . . . . . . . . . . . . . . 95
5.11 HCF: Restricted Pheromone Interval . . . . . . . . . . . . . . . . . 97
5.11.1 HCF: Pheromone Update . . . . . . . . . . . . . . . . . . . 98
5.12 HCF: Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.13 Hybrid-ACO: Conclusions and Recommendations . . . . . . . . . 101
6 ACO: Run-Time Analysis and Evaluation 106
6.1 Classification of Stochastic Local Search Algorithms . . . . . . . . 108
6.2 Empirical Run-Time Analysis for Stochastic Local Search . . . . . 110
6.3 Deriving ACO and CE Empirical RTDs for the CFLP . . . . . . . . 111
6.3.1 Measuring RTDs . . . . . . . . . . . . . . . . . . . . . . . . 113
6.4 Qualitative Analysis of Empirical RTDs . . . . . . . . . . . . . . . . 114
6.5 Comparative Qualitative Analysis . . . . . . . . . . . . . . . . . . . 117
6.6 Comparative Quantitative Analysis . . . . . . . . . . . . . . . . . . 121
Contents v
6.6.1 RTD Median Run-Times . . . . . . . . . . . . . . . . . . . . 122
6.6.2 Investigation of the Run-Time Median . . . . . . . . . . . . 124
7 Conclusions 131
7.1 Conclusions: Study Rationale . . . . . . . . . . . . . . . . . . . . . 132
7.2 Conclusions: ACO for the CFLP . . . . . . . . . . . . . . . . . . . 134
7.3 Conclusions: Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 136
7.4 Testing the Research Hypothesis . . . . . . . . . . . . . . . . . . . 138
7.5 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.6 Contribution to Knowledge . . . . . . . . . . . . . . . . . . . . . . . 145
Appendix 166
A Research Output 167
B Algorithmic Solution Quality: Empirical Run-Time Distributions 169
C Algorithmic Solution Quality: Graphical Descriptive Summaries of
Run-Time Distributions 180
D Algorithmic Solution Quality: Graphical Descriptive Summaries of
Bootstrapped Median Distributions 191
E Algorithmic Solution Quality: Bootstrapped 95% Confidence Inter-
vals for Median Run-Times 202
List of Tables
3.1 OR-Library test problems . . . . . . . . . . . . . . . . . . . . . . . 38
4.1 Parameter setting for AS and MMAS . . . . . . . . . . . . . . . . 57
4.2 Experimental results using a bipartite graphical representation for
the CFLP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.1 Non-local search (Z ) and facility “DROP” local search (Z ) re-
NLS D
sults for problem instances using a best-improvement technique
and approximate transportation solutions; where Z∗ are the known
optimum solutions (Beasley, 1993) . . . . . . . . . . . . . . . . . . 83
5.2 Resultsforaselectionofprobleminstancesusinglocalsearch: Z
D
method of Venables and Moscardini (2006) and pheromone based
DROP and DROP-SWAP heuristics Z and Z . . . . . . . . . . 85
Dτ Sτ
5.3 Available literature results of B93 - Beasley (1993), BA981 and
BA982 -BornsteinandAzlan(1998),BC04-BornsteinandCampelo
(2004)andtheapproximatehybridMMASACOalgorithm-MMAS
Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Sτ
5.4 OR-Library test problems using various MMAS parameter set-
tings: A – α = 2.5, β = 0.8 and ρ = 0.06; B – α = 2.5, β = 0.8 and
ρ = 0.9; C – α = 1.0,β = 0.0 and ρ = 0.9; secs – seconds; its –
iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5 Computational results of the OR-Library large test problems (100
× 1000): MMAS Single Ant A - α = 2.5, β = 0.8 and ρ = 0.9;
MMAS Colony (5 ants) B - α = 2.5, β = 0.8 and ρ = 0.9; secs –
seconds; its – iterations . . . . . . . . . . . . . . . . . . . . . . . . 94
5.6 OR-Librarytestproblemsusingparametersettings: C-α = 1.0,β =
0.0 and ρ = 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.7 Computational results of the OR-Library large test problems (100
× 1000): HCF (5 ants) C - α = 1.0, β = 0.0 and ρ = 0.9 . . . . . . 101
vi
List of Tables vii
6.1 Qualitative Summary Table . . . . . . . . . . . . . . . . . . . . . . 120
6.2 RTDs: Variable Width Notched Box Plots Rankings . . . . . . . . . 125
6.3 Boostrapped Median Sampling Distribution Rankings . . . . . . . . 129
6.4 Bootstrapped 95% Confidence Intervals . . . . . . . . . . . . . . . 130
List of Figures
1.1 Random Search and Convergence . . . . . . . . . . . . . . . . . . 5
1.2 A foraging strategy by African army ants . . . . . . . . . . . . . . . 6
1.3 A mature ant trail . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.1 Bipartite graphical representation of potential assignments of cus-
tomers to facilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Example of an ant’s tour to assign customers to facilities . . . . . . 51
4.3 Run-time experimentation for two OR-Library Instances . . . . . . 59
5.1 Schematic for an ACO hybrid algorithm for the CFLP . . . . . . . . 66
5.2 Construction graph for a hybrid CFLP ACO algorithm consisting of
facilities I ∈ {1,2,...,i}. The plain links between facility nodes
represent possible pathways that an ant could take during the con-
struction phase, whereas the arrowed links represent an example
of a pathway taken by an ant. . . . . . . . . . . . . . . . . . . . . . 68
5.3 Schematic for DROP and SWAP local search mechanisms . . . . 79
6.1 Justification and Verification of Sample Size . . . . . . . . . . . . . 116
6.2 Comparative RTDs for two OR-Library Instances . . . . . . . . . . 119
6.3 ACO and CE RTDs: Variable Width Notched Box Plots for Cap63
and Cap 113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 Boostrapped Median Run-Time Sampling Distributions: Box Plots
and 95% Confidence Intervals . . . . . . . . . . . . . . . . . . . . 127
viii
Chapter 1
Introduction
Many behavioural characteristics of ants have been studied by scientists includ-
ing their social interactions, brood sorting, colony welfare, hierarchical systems,
division of labour, cooperative transport and adaptive foraging strategies. These
same topics have provided substance for many humorous children’s stories and
even several animated big-screen adventures. A fascinating statistic is that:
“Ants are everywhere on earth. When combined, all ants in the world
weigh about as much as all humans ...” (Ho¨lldobler and Wilson, 1994)
A simple fact is that ants are very successful which is primarily due to their
adaptive nature. There are literally tens of thousands of different species of ants
that have evolved throughout the natural world. Their intrinsic behaviour to work
andsearchforfoodallforthegoodofthecolonytowhichtheybelongisincredible.
Their peculiar foraging behaviour often incites laughter from both adult and child
observers. If we can use what is known about their incessant quest for food and
their success at delivering it safely back to the colony nest, then we ought to
able to solve many logistical and transportation problems. In essence, it is the
ants desire for food that motivates and provides a rationale for using Ant Colony
Optimisation on problems that can be modelled as a network of pathways, such
1