IMPROVED VERSIONS OF THE BEES ALGORITHM FOR GLOBAL OPTIMISATION By SHAFIE KAMARUDDIN A thesis submitted to the University of Birmingham for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering School of Engineering College of Engineering and Physical Sciences University of Birmingham June 2017 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. ABSTRACT This research focuses on swarm-based optimisation algorithms, specifically the Bees Algorithm. The basic version of the algorithm was introduced in 2005 and was inspired by the foraging behaviour of honey bees in nature. The Bees Algorithm employs a combination of exploration and exploitation to find the solutions of optimisation problems. This thesis presents three improved versions of the Bees Algorithm aimed at speeding up its operation and facilitating the location of the global optimum. For the first improvement, an algorithm referred to as the Nelder and Mead Bees Algorithm (NMBA) was developed to provide a guiding direction during the neighbourhood search stage. The second improved algorithm, named the recombination-based Bees Algorithm (rBA), is a variant of the Bees Algorithm that utilises a recombination operator between the exploited and abandoned sites to produce new candidates closer to optimal solutions. The third improved Bees Algorithm, called the guided global best Bees Algorithm (gBA), introduces a new neighbourhood shrinking strategy based on the best solution so far for a more effective exploitation search and develops a new bee recruitment mechanism to reduce the number of parameters. The proposed algorithms were tested on a set of unconstrained numerical functions and constrained mechanical engineering design problems. The performance of the algorithms on numerical functions was compared with the standard Bees Algorithm and other swarm based algorithms in terms of the solutions found and the convergence speed. In terms of the application on constraint mechanical engineering design problems, the performance was compared with the results of the standard Bees Algorithm and the results of other algorithms in the literature. In addition, a paired test statistical analysis was also carried out on those results. i The results showed that the improved Bees Algorithms performed better than the standard Bees Algorithm and other algorithms on most of the problems tested. Furthermore, the algorithms also involve no additional parameters and a reduction on the number of parameters as well. ii ACKNOWLEDGEMENTS First and foremost, gratitude and Praise is to Allah, the Most Merciful and Most Compassionate for His Blessings, that I am to complete this research. I would also like to record my appreciation to these outstanding individuals for their great contributions: I am deeply indebted to my supervisor, Prof. D T Pham for all his guidance, advices, knowledge and supervision that guided me throughout my academic program. His immeasurable supports and confidence in me inspired me to complete this thesis. In addition, I also would like to thanks my wife, Dahlia Nursyarmimi and all my family members for always supporting me during my PhD journey. Furthermore, I would like also to express my outermost appreciation to all my colleagues, Muhammad Syahril, Nik Mohd Farid, Dr. Silah Hayati, Al Antoni Akhmad and other closet friends for their encouragements and advices during my research period Also, special thanks to my sponsor, the Ministry of Higher Education (MOHE) of Malaysia and International Islamic University Malaysia (IIUM) for giving me the opportunity to further my study. Lastly, I offer my regards and blessing to all of those supported me in any respect during completion of the research. This thesis was copy edited for conventions of language, spelling, and grammar by Szakif Enterprise Academic Services. iii TABLE OF CONTENTS ABSTRACT ............................................................................................................................... i ACKNOWLEDGEMENTS .................................................................................................. iii TABLE OF CONTENTS ....................................................................................................... iv LIST OF FIGURES ............................................................................................................... vii LIST OF TABLES .................................................................................................................. ix LIST OF SYMBOLS AND ABBREVIATIONS .................................................................. xi Chapter 1: Introduction ......................................................................................................... 1 1.1 Background ...................................................................................................................... 1 1.2 Motivation ........................................................................................................................ 1 1.3 Aim and Objectives.......................................................................................................... 3 1.4 Research Methodology .................................................................................................... 4 1.5 Outline of the Thesis ........................................................................................................ 4 Chapter 2: Literature Review ................................................................................................. 6 2.1 Preliminaries .................................................................................................................... 6 2.2 Optimisation ..................................................................................................................... 6 2.2 Direct Search Methods ..................................................................................................... 8 2.2.1 Nelder and Mead Method ......................................................................................... 8 2.2.2 Hooke and Jeeves Method ........................................................................................ 9 2.3 Metaheuristic.................................................................................................................. 10 2.4 Single-solution based metaheuristics ............................................................................. 11 2.4.1 Simulated Annealing (SA) ...................................................................................... 12 2.4.2 Tabu Search (TS) .................................................................................................... 12 2.4.3 Iterated Local Search (ILS) ..................................................................................... 13 2.5 Population-based metaheuristics .................................................................................... 13 2.5.1 Evolutionary Computation (EC) ............................................................................. 14 2.5.2 Nature inspired Swarm Intelligence (SI) ................................................................ 17 2.5.2.1 Firefly Algorithm (FA) .................................................................................... 18 iv 2.5.2.2 Ant Colony Optimisation (ACO) ..................................................................... 19 2.5.2.3 Particle Swarm Optimisation (PSO) ................................................................ 19 2.5.2.4 Honey Bees Inspired Algorithm ...................................................................... 20 2.6 The Bees Algorithm ....................................................................................................... 28 2.7 Improvements ................................................................................................................ 30 2.8 Applications ................................................................................................................... 35 2.9 Summary ........................................................................................................................ 39 Chapter 3: The Bees Algorithm with Nelder and Mead Method ...................................... 40 3.1 Preliminaries .................................................................................................................. 40 3.2 The Bees Algorithm with Nelder and Mead (NM) Method .......................................... 41 3.2.1 Experimental Setup ................................................................................................. 45 3.2.2 Experimental Results .............................................................................................. 48 3.2.3 Discussion ............................................................................................................... 55 3.3 Mechanical Design Applications ................................................................................... 58 3.4 Summary .......................................................................................................................