programme through the Baroness Cox Scholarship and also Lady Caroline Cox for her motherly care and warmth during the course of the programme. Also.
EFFECT OF GENDER RESTRICTED SELECTION ON GENETIC ALGORITHMS FOR SOLVING NUMERICAL OPTIMISATION PROBLEMS
By
Ezeani, Ignatius Majesty
A thesis submitted in partial fulfilment of the requirements for the degree of
Master of Science, Advanced Software Engineering
Bournemouth University, UK
September 2006
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ABSTRACT
THE EFFECT OF GENDER RESTRICTED SELECTION ON GENETIC ALGORITHMS FOR SOLVING NUMERICAL OPTIMISATION PROBLEMS
By: Ezeani, Ignatius Majesty Project Supervisor: Dr. Jonathan Vincent School of Design Engineering and Computing Different models of gender-restricted selection techniques are being recently proposed by some researchers in the Evolutionary Computing community. The possible impact of such strategy on the performance of a standard GA has not been clearly established. The effects of such models as compared to some common standard techniques on some benchmark numerical optimisation problems are investigated under fairly similar conditions and the analysis of the result of the comparisons is also presented.
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TABLE OF CONTENTS
ABSTRACT ............................................................................................................................... 2 TABLE OF CONTENTS ......................................................................................................... 3 LIST OF FIGURES .................................................................................................................. 6 DEDICATION ........................................................................................................................... 8 ACKNOWLEDGMENT .......................................................................................................... 9 1: INTRODUCTION .............................................................................................................. 11 1.1 1.2 1.3
CONTEXT OF THE STUDY ........................................................................................ 11 AIMS OF THE STUDY ............................................................................................... 13 ORGANISATION OF THE RESEARCH WORK ............................................................. 14
2: BACKGROUND OF THE STUDY .................................................................................. 15 2.1 GENETIC ALGORITHMS: HISTORY AND DEVELOPMENT......................................... 15 2.2 GA CONCEPTS AND STRUCTURE ........................................................................... 17 2.3 GA OPERATORS ..................................................................................................... 18 2.3.1 Selection operation in GA ................................................................................ 19 2.3.2 Crossover operator in GA ................................................................................ 20 2.3.3 Mutation operator in GA........................................................................................ 21 2.3.4 Replacement in GA ................................................................................................. 22 2.4 EXPLORATION, EXPLOITATION AND DIVERSITY MANAGEMENT .......................... 23 2.5 OVERVIEW OF SOME COMMON SELECTION STRATEGIES ...................................... 24 2.5.1 Roulette Wheel Selection .................................................................................. 24 2.5.2 Stochastic Universal Sampling (SUS).............................................................. 25 2.5.3 Rank Based Selection ....................................................................................... 26 2.5.4 Tournament Selection ....................................................................................... 27 2.6 APPROACHES TO GENDER RESTRICTED SELECTION IN GAS ................................. 27 2.6.1 The GenderedGA .............................................................................................. 28 2.6.2 SexualGA .......................................................................................................... 31 2.6.3 GA with Gender ................................................................................................ 33 2.6.4 MSGA................................................................................................................ 34 2.6.5 MSGA II ............................................................................................................ 35 2.6.6 Simulated Gender Separation and Mating Constraints for GAs .................... 36 2.6.7 Sexual Selection for Genetic Algorithms ......................................................... 37 2.7 NUMERICAL OPTIMISATION ................................................................................... 38 3: METHODOLOGY ............................................................................................................. 40 3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.4
THE TEST PROBLEMS ............................................................................................. 40 The Rastrigin Function..................................................................................... 40 The Schwefel function....................................................................................... 41 Whitley’s F8F2 Function ................................................................................. 41 THE ALGORITHMIC MODELS ................................................................................. 43 The general GA features .................................................................................. 43 The “StandardGA” .......................................................................................... 43 The GenderedGA(1), GGA1 ............................................................................. 44 The GenderedGA(2), GGA2 ............................................................................. 45 THE META GA AND PARAMETER OPTIMISATION.................................................. 46 METRICS AND STATISTICS ..................................................................................... 46
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3.4.1 3.4.2 3.4.3 3.4.4
Best fitness ........................................................................................................ 46 Average fitness.................................................................................................. 46 Genotypic Diversity .......................................................................................... 46 Standard Deviation........................................................................................... 47
4: RESULTS ............................................................................................................................ 48 4.1 RASTRIGIN FUNCTION ............................................................................................ 48 4.1.1 Rastrigin with 10 Parameters .......................................................................... 48 4.1.1(a) 5000 Maximum Evaluation ......................................................................... 49 4.1.1(b) 10000 Maximum Evaluation ....................................................................... 51 4.1.1(c) 15000 Maximum Evaluation ....................................................................... 53 4.1.2 Rastrigin with 20 Parameters .......................................................................... 55 4.1.2(a) 5000 Maximum Evaluation ......................................................................... 55 4.1.2(b) 10000 Maximum Evaluation ....................................................................... 58 4.1.2(c) 15000 Maximum Evaluation ....................................................................... 60 4.1.3 Rastrigin with 30 Parameters .......................................................................... 61 4.1.3(a) 5000 Maximum Evaluation ......................................................................... 61 4.1.3(b) 10000 Maximum Evaluation ....................................................................... 62 4.1.3(c) 15000 Maximum Evaluation ....................................................................... 64 4.2 SCHWEFEL FUNCTION ............................................................................................ 66 4.2.1 Schwefel with 10 Parameters ........................................................................... 66 4.2.1(a) 5000 Maximum Evaluation ......................................................................... 66 4.2.1(b) 10000 Maximum Evaluation ....................................................................... 68 4.2.1(c) 15000 Maximum Evaluation ....................................................................... 70 4.2.2 Schwefel with 20 Parameters ........................................................................... 72 4.2.2(a) 5000 Maximum Evaluation ......................................................................... 72 4.2.2(b) 10000 Maximum Evaluation ....................................................................... 73 4.2.2(c) 15000 Maximum Evaluation ....................................................................... 73 4.2.3 Schwefel with 30 Parameters ........................................................................... 75 4.2.3(a) 5000 Maximum Evaluation ......................................................................... 75 4.2.3(b) 10000 Maximum Evaluation ....................................................................... 76 4.2.3(c) 15000 Maximum Evaluation ....................................................................... 77 4.3 F8F2 FUNCTION ..................................................................................................... 78 4.3.1 F8F2 with 10 Parameters ................................................................................ 78 4.3.1(a) 5000 Maximum Evaluation ......................................................................... 78 4.3.1(b) 10000 Maximum Evaluation ....................................................................... 79 4.3.1(c) 15000 Maximum Evaluation ....................................................................... 80 4.3.2 F8F2 with 20 Parameters ................................................................................ 81 4.3.2(a) 5000 Maximum Evaluations........................................................................ 81 4.3.2(b) 10000 Maximum Evaluations ..................................................................... 82 4.3.2(c) 15000 Maximum Evaluations ..................................................................... 83 4.3.3 F8F2 with 30 Parameters ................................................................................ 84 4.3.3(a) 5000 Maximum Evaluations........................................................................ 84 4.3.3(b) 10000 Maximum Evaluations ..................................................................... 85 4.3.3(c) 15000 Maximum Evaluations ..................................................................... 86 4.4 ROBUSTNESS .......................................................................................................... 87 4.4.1 Rastrigin ........................................................................................................... 87 4.4.2 Schwefel ............................................................................................................ 87 4.4.3 F8F2 ................................................................................................................. 87 5: CONCLUSION AND FUTURE RESEARCH DIRECTIONS ..................................... 88 5.1 OVERVIEW OF THE RESEARCH AIMS ................................................................................ 88 5.2 THE EXPERIMENTS CONDUCTED...................................................................................... 89 5.3 RESEARCH FINDINGS ....................................................................................................... 89
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5.4 LIMITATIONS OF THIS STUDY .......................................................................................... 90 5.5 FUTURE RESEARCH DIRECTION ....................................................................................... 91
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LIST OF FIGURES
Figure
Page number
FIG 1: The Basic Structure of a Genetic Algorithm ···························································· 9 FIG 2: Point Crossovers in GAs ················································································· 11 FIG 3: Representation of Solutions by Liz and Eiben [Lis96] ················································ 25 FIG 4: Gender based selection model by Goh et al. [Goh03] ················································· 29 FIG 5: Rastrigin Function Graph for n=2 [Hed06] ···························································· 31 FIG 6: Schwefel Function Graph for n=2 [Hed06]···························································· 32 FIG 7: Rosenbrock Function Graph for n=2 [Hed06] ························································ 32 FIG 8: Griewangk Function Graph for n=2 [Hed06] ························································· 33 FIG 9: The Model of GenderedGA1, GGA1 ·································································· 35 FIG 10: The Model of GenderedGA2, GGA2 ································································· 36 FIG 11: Best Fitness Graph Rastrigin_10_5000 ································································ 40 FIG 12: Last 2000 evals Best Fitness Graph Rastrigin_10_5000 ·············································· 41 FIG 13: First 1000 evals Average Fitness Graph Rastrigin_10_10000 ········································ 42 FIG 14: Best Fitness Graph Rastrigin_10_10000 ······························································ 43 FIG 15: Last 3000 evals Best Fitness Graph Rastrigin_10_5000 ·············································· 43 FIG 16: Diversity Graph Rastrigin_10_10000 ·································································· 44 FIG 17: First 1000 evals Average Fitness Graph Rastrigin_10_15000 ······································· 45 FIG 18: Diversity Graph Rastrigin_10_15000 ·································································· 44 FIG 19: Last 2500 evals Best fitness Graph Rastrigin_10_15000 ············································ 46 FIG 20: Average Fitness Graph for Rastrigin_20_5000 ······················································· 47 FIG 21: Diversity Graph Rastrigin_20_5000 ··································································· 47 FIG 22: Last 1500 evals Best Fitness Graph Rastrigin_20_5000 ············································· 48 FIG 23: Best Fitness Graph Rastrigin_10_10000 ······························································ 49 FIG 24: Best Fitness Graph Rastrigin_20_10000 ······························································ 50 FIG 25: Diversity Graph Rastrigin_10_10000 ·································································· 50 FIG 26: Last 3000 evals for Best Fitness Graph Rastrigin_10_15000 ········································ 51 FIG 27: Best Fitness Graph Rastrigin_30_5000································································ 52 FIG 28: Last 2000 evals for Best Fitness Graph Rastrigin_30_5000 ·········································· 53 FIG 29: First 2000 evals for Best Fitness Graph Rastrigin_30_10000 ········································ 53 FIG 30: Last 6000 evals for Best Fitness Graph Rastrigin_30_10000 ········································ 54 FIG 31: Diversity Graph Rastrigin_30_10000 ·································································· 54 FIG 32: Best Fitness Graph Rastrigin_30_15000 ······························································ 55 FIG 33: Last 9000 evals Best Fitness Graph Rastrigin_30_15000············································· 55 FIG 34: Diversity Graph Rastrigin_30_15000 ·································································· 56 FIG 35: Best Fitness Graph Schwefel_10_5000 ······························································· 57 FIG 36: Diversity Graph Schwefel _10_5000 ·································································· 58 FIG 37: Best Fitness Graph Schwefel _10_10000 ····························································· 59
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FIG 38: Last 5000 evals Best Fitness Graph Schwefel _10_10000 ············································ 60 FIG 39: Diversity Graph Schwefel _10_10000 ································································· 60 FIG 40: Best Fitness Graph Schwefel _10_15000 ····························································· 61 FIG 41: Diversity Graph Schwefel _10_15000 ································································· 62 FIG 42: Best Fitness Graph Schwefel _20_5000 ······························································· 63 FIG 43: Best Fitness Graph Schwefel _20_15000 ····························································· 65 FIG 44: Best Fitness Graph Schwefel _30_5000 ······························································· 66 FIG 45: Best Fitness Graph Schwefel _30_10000 ····························································· 67 FIG 46: Best Fitness Graph Schwefel _30_15000 ····························································· 68 FIG 47: Best Fitness Graph F8F2 _10_5000 ··································································· 69 FIG 48: Best Fitness Graph/zoom on fitness < 1 F8F2 _10_10000 ········································ 70 FIG 49: Best Fitness Graph and zoom on fitness < 1 F8F2 _10_15000 ···································· 71 FIG 50: Best Fitness Graph; fitness < 0.2; Last 1000 evals F8F2 _20_5000 ································· 72 FIG 51: Best Fitness Graph; fitness < 0.5; Last 6000 evals F8F2 _20_10000 ······························· 73 FIG 52: Average Fitness and Diversity Graph F8F2 _20_15000 ············································ 74 FIG 53: Best Fitness and Diversity Graphs (Inset: last 3000 evals) F8F2 _30_5000 ························ 75 FIG 54: Best Fitness and Diversity Graph (Inset: last 5000 evals) F8F2 _30_10000 ······················· 76 FIG 55: Average Fitness / Diversity Graph (Inset: last 2000 evals for best fitness) F8F2 _30_15000···· 76
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DEDICATION
To God Almighty for His inspiration and guidance; and to my family for the prayers, love and support.
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ACKNOWLEDGMENT "Not everything that counts can be counted and not everything that can be counted counts." - Sign hanging in Einstein's office at Princeton
Works of this nature usually involves direct and indirect contributions from many quarters. This acknowledgement seeks understanding from two categories of readers: those who really ought to be mentioned but were not, due to lack of space and those who might find the long list rather boring. I did my best to reconcile both. I thank my dissertation adviser, Dr. Jonathan Vincent, who introduced me to the world of Evolutionary Computing and Genetic Algorithms. He sat on his scepticisms and patiently led me through the necessary research and presentational skills required to complete this work. I also thank all my tutors: Prof Bogdan Gabrys, Dr. Frank Milsom, Prof Richard Wynne, Dr. Keith Phalp, Ian Bray, John Kanyaru for contributing the bits that made up the big picture. Many thanks to my programme administrator, Lindsay Main, for dousing the fears, worries and confusions that were necessarily part of this experience and Dr Marek Machura, my programme leader, for being so generous with his time. I thank Bournemouth University, UK in a special way for sponsoring this programme through the Baroness Cox Scholarship and also Lady Caroline Cox for her motherly care and warmth during the course of the programme. Also special thanks to Nnamdi Azikiwe University, Nigeria for granting me the leave and to all my friends and colleagues who kept in touch. I thank all my friends in BU who were very helpful in many ways especially Pascal, without whom it may have been difficult and Bolanle (Aunty Bola) for her care and concern. Syndicate (Okey Nnoli) was particularly emotionally and psychologically supportive. Sam Dansobe’s contributions were of immense help.
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Chuzzy Anyakaoha is a giant physically as well as academically and so standing on is shoulder just made a lot of sense. Also my profound gratitude goes to all the staff of the Additional Learning Needs Department of Bournemouth University, especially Ruthi Chesney, Alex Balerdi. My dad, mum, siblings and other family members, will take the honour of ending this acknowledgement, their support was simply overwhelming. And to He who sits on the Throne be ever lasting glory. I feel so humbled and grateful.
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1: INTRODUCTION Selection is a key concept in evolutionary algorithms of which genetic algorithms is one of the major “dialects” [Mit96] [Eib02] [Ken97]. These algorithms are generally population based probabilistic search heuristics modelled after the concept of the Darwinian Theory of Natural Evolution [Gol89] [Hol92] [Mit96]. With such algorithms, selection plays an important role in the “birth, life and death” of the individuals that make up the population across generations [Gol89]. The balance between exploration and exploitation of the search space by evolutionary algorithms can be influenced by the selection technique used [Col91], [Aff04], [Eib98], [Bli95], [Bli95], [Leg05]. This has led to the adoption of different selection techniques with varying abilities to reasonably explore the solution space as well as guide the search to useful direction. Some have argued that considering the origin of search algorithms, naturally inspired selection techniques are likely to improve the adaptive characteristics of such algorithms [Aff04],[San031]. Several attempts have been made at designing selection models that have the potential to enhance the performance of a Genetic Algorithm (GA) in terms of effective exploration of the search space and as well as guiding the GA towards the optimal solution [Gol91],[Bli95]. Recently, different gender restricted selection models have also been proposed and reported by researchers in this area as having been successfully applied in solving classical GA problems [San031], [San032], [Vra02], [Wag04], [Lis96], [All92]. In this section, context and aims of the study as well as the structure and organisation of the dissertation would be presented. 1.1
Context of the study
Evolutionary algorithms are very commonly applied in solving complex optimisation problems. Optimisation is a recurrent concept in many real world applications. Many systems battle with the need to maximise or minimise a
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certain objective function subject to certain constraints. This may be in the form of increasing benefits (profit, gains, productivity) of some sort or reducing cost (losses, pains, risks). The fundamental principle guiding the optimisation process is to achieve the best under the given circumstances [Pin05]. Typical optimisation problems often define, in addition to the objective function, a set of constraint functions that should not be violated. Simple linear models or unimodal and continuous functions are fairly easier to solve with standard mathematical approaches. So, traditionally, optimisation problems were tackled with mostly calculus-based techniques e.g. hill-climbing1 [Gol89]. With such techniques, an iterative process would be followed to improve a one-point solution to a problem by moving about the search space in the direction of the steepest ascent which was calculated using derivatives [All92]. With such methods, certain assumptions that require the domain-specific knowledge about the problem, e.g. the continuity of the function, need to be made for such methods to work [All92]. Therefore the specificity of such techniques makes it difficult to generally apply them to a wide range of problems. Besides, one of its major problems is the tendency to get trapped in local optima. Random approach is another method which is more generalised and therefore could be used with less assumptions regarding the search domain [Gol89][Vin03]. This took care of the specificity problem of the calculus-based approach. But it is then less directed and so renders the search process somewhat inefficient [All92]. Another traditional method was the enumeration method. This approach, which is also referred to as exhaustive search, was simply too slow and impracticable for a complex multi-dimensional search space [All03].
1 Hill-climbing may be defined as any local search method that defines a neighbourhood, then moves to the first position found in that neighbourhood that offers improvement (e.g., next ascent) or to the position offering the best improvement in the neighbourhood (e.g., steepest ascent)[Whi96].
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Real life optimisation problems comprise of complex highly constrained multiobjective optimisation issues (see section 2.7). Such often require searching through a huge number of possible solutions and cannot be solved using linear approaches. Typical examples range from searching through a vast number of possible combinations of amino acids for a protein with specified properties, to predicting the ups and downs in the financial market as with foreign currency or stock exchange market [Mit96]. With such problems, not much useful assumptions could be made about the nature of the solution space because no domain-specific knowledge is available a priori. Evolutionary algorithms such as GAs are usually better options in such cases than the traditional methods mentioned above. The basic concepts involves searching through the solution domain with a population of solutions and using the genetic operators, crossover and mutation to produce new and (hopefully) better individuals (solutions) that will replace old ones. The selection of individuals that are recombined plays a fundamental role in driving the algorithm to good performance [Muh95], [Sar98]. Many works in this area have introduced and reviewed some of the classical selection strategies – which are discussed in the review – and have established their weaknesses as well as strengths. However, some recent works have claimed that GAs can be more naturally adaptive should they borrow more from nature [San032], [Wag04], [All92]. They argue that selection strategies with gender restrictions would improve the performance of a GA. This work therefore thoroughly examines the effect of some gender based models on the overall performance of typical GAs. 1.2
Aims of the study
The selection stage is very crucial in all evolutionary algorithms. This could be traced to its direct impact on the exploration and exploitation activities during the search process [Bey98], [Bli95], [Eib96], [Eib98]. Therefore, other performance criteria and characteristics are affected as well such as the selection
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pressure, loss/preservation of diversity, speed of the search and the quality of the solution. The basic aim of this research work is to study, analyse and report the behaviours of two gender restricted selection schemes based on the works of [Goh03], [All92], [Vra02] as compared to that of a standard selection scheme. The performance metrics used for the analysis include among others: fitness evaluations and diversity measures. Some classic numeric optimisation problems – Rastrigin, Schwefel, and F8F2 [Whi96] (described in section 3) – are used as test problems. The description of these test functions, the GA models used and the metrics are presented in section 3: Methodology. 1.3
Organisation of the research work
The rest of this research work is organised as follows. Section 2 – Background of the Study – presents the background of the issues and concepts raised in this work from related literature in the areas of Genetic Algorithms, Evolutionary Computation and Biogenetics. Section 3 – Methodology – presents and describes the test functions, their nature and characteristics, the GA models used, the design and approach to the experiments at different stages were presented. Also, this section has the justification of methods, strategies and metrics drawn from relevant literatures. In Section 4 – Research Findings and Analysis – presents a clear analysis of the observed behaviour of the models based on the experimental data. And finally, Section 5 – Conclusion and future research directions – the conclusion from this work, the limitations of the study as well as possible future directions in this area are presented.
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2: BACKGROUND OF THE STUDY
The use of evolutionary algorithms including Genetic Algorithms in black-box2 optimisation has engaged the attention of both users and designers of such algorithms. This is because they have been found to be computationally cost effective with complex problems. These algorithms use stochastic means to sample only a subset of the search space that are likely to give good results. This section presents an overview of the concepts and developments based on some works done in this area. It concludes with an overview of numerical optimisation. 2.1
Genetic Algorithms: History and development
Genetic Algorithms (GAs) are well known function optimisers which belong to the group of computational models inspired by the Darwinian Theory of Natural Evolution [Gol89], [Mit96], [For96], [Bac97]. The study of these models led to the development of the branch of Computer Science conveniently classified as Evolutionary Computation (EC) [Bac97], [Eib02]. The origin of EC can be traced to the 1960s when computer scientists from different parts of the globe were independently working on different but related evolution-inspired mechanism [Eib02], [Ken97], [Fog06]. These evolution-inspired mechanisms were to become relevant in the area of optimisation and the development of adaptive systems [Mit96], [Gol89]. GA, developed by Holland (1962, 1975) is one of the major dialects of EC [Eib02]. However, Evolution Strategies (ES) developed in Germany by Rechenberg and Schwefel (1965, 1973) and Genetic Programming (GP) by Fogel, Owens and Walsh (1966) in the US, are also prominent in the EC community [Mit96], [Eib02]. Eiben and Schoenauer [Eib02], observed that these techniques have a “common underlying idea”. They work by exerting the artificial “environmental pressure” on a 2
In black box optimisation, no domain knowledge of the characteristics of the function is assumed.
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population of individuals (possible solutions to a problem) and as such fitter individuals are more likely to be given the opportunity to reproduce thereby improving the average fitness of the entire population over time. Better approaches to designing these algorithms have generated massive research interest in the areas of optimisation and machine learning. Holland was originally working on the development of adaptive systems and therefore was researching on ways of implementing the mechanisms of natural adaptation on computer systems [Hol92]. So GA was initially presented as an abstraction of biological evolution [Mit96]. The concept was simple: individual data structures often encoded in a chromosome-like form (i.e. bit strings – 0s and 1s) make up the population; genetic-inspired operators (selection, crossover, mutation, inversion) were used to produce a new generation of solutions which replaced the old. The choice of averagely fitter individuals ensures the population improves in overall fitness. The process was then repeated over a given number of generations or until a certain acceptable level of solution quality is reached. Other techniques – ES, GP, EP – share the same basic underlying concept but differ mainly in the arity as well as the operator used [Bac97], [Eib02], [Whi94], [Whi01]. However, in recent times, the interactions between researchers in EC have continued to weaken the differences among the EC techniques and the GAs of today are quite far from what Holland originally described. The GA researchers have also, over these years, suggested different variations of the original canonical model proposed by Holland. Whitley [Whi94] noted that the term GA in a broader sense actually refers to “any population-based model that uses the selection and recombination3 [crossover] to generate new sample points in the search space”.
3
Most authors refer to crossover and recombination as the same process, so they mean the same in this work.
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2.2
GA Concepts and Structure
The concept of GA mimics the basic “rules” of natural evolution which are remarkably simple. In natural evolution species evolve by means of random variation (i.e. mutation, recombination) followed by natural selection in which the fittest survive and reproduce, thus propagating the genetic materials through future generations [Mit96]. A GA typically begins by randomly generating the individuals4 that make up the initial population. It then iteratively goes through the selection process to pick some individuals for the intermediate population (often referred to as the “mating pool” [Mit95],[Whi94]). Selection is usually biased toward the fitter chromosomes and so on the average, their genetic properties are propagated to future generations more than those of the weaker chromosomes [Est96], [Gol91], [Bli95]. The next stage involves the probabilistic operations of crossover and mutation which, in each case, produce at least one offspring5 (new individual) to make up a new generation of individuals. The last stage is the replacement of either the entire current population with the newly created population (generational replacement) or the worst individuals with the new offspring (steady state). The processes – selection, crossover+mutation, replacement – are repeated until a stopping condition is reached e.g. at the end of a given number of generations or when an acceptable solution quality is reached. The overall (or so far) best solution is retained and the average quality of the entire population of solutions improves through the evolving generations of individuals and this eventually finds the last set of solutions clustering around the optimum solution. Below is a description of the GA basic structure: 1. Generate (P(0)) 2. t := 0 3. while not Termination Criterion(P(t)) 4. do 5. Evaluate(P(t)) 4
As mentioned above, these individuals are simply data structures holding the encoded chromosomes.
5
Actually more than one offspring can be produced depending on the GA variation implemented.
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6. P’(t) := Selection(P(t)) 7. P’(t) := Recombination(P’(t)) 8. P’(t) := Mutation(P’(t)) 9. P(t+1) := Replace(P(t), P’(t)) 10. t := t+1 11. return Best_Solution_Found
Fig 1: The Basic Structure of a Genetic Algorithm [Vin03]
The encoding of candidates solutions into a chromosomal representation is equally a very fundamental aspect of the design of the GA. Early GAs used low cardinality alphabets (e.g. binary, Gray code) encoding as proposed by Holland [Hol92], [Gol89]. The motivation for this encoding stems mainly from the schema theorem. High cardinality encoding (e.g. real, integer) were introduced by researchers [Ball04], [Deb02] who have argued that in addition to its better performance, they are more convenient to use than the binary encoding. 2.3 GA Operators From the above basic structure of a GA, there are basic operations that can be easily seen such as: selection, crossover (recombination), mutations, replacement. A brief overview of each is presented in the following sections
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2.3.1 Selection operation in GA Selection is a very crucial operation in GAs. Blickle and Thiele [Bli95] define it as “a function that transforms a fitness distribution into another fitness distribution”. It involves choosing individual parents from the current population for the procreation of new offspring. Although a stochastic approach, selection usually gives more opportunities to fitter individuals based on some fitness function in order to “exploit their good properties” [Hol92]. The fitness function in GAs is used to derive the fitness score of a sample which determines its likelihood of being selected [Bli95]. This can be directly mapped to the objective function. However, when individuals seem to converge and there is reduced selection pressure6, a proper fitness transformation ensures that above-average individuals have better chance of being selected for reproduction and thereby driving the search process further to improved solutions [Vin03]. Selection, therefore, acts as the “driving force” which directs the GA to better solutions [Amo05], [Bli95]. Estivill-Castro [Est96] asserts that “selection implies convergence”. The selection mechanism should be designed to have adequate selection pressure to ensure that the above-average individual has high chance of being selected. It is the selection pressure that controls the speed of convergence, and hence the balance between exploration and exploitation in spite of the effect of mutation [Bac94]. Wagner and Affenzella [Wag04] assert that selection is the “most relevant diversity reducing force in GAs” which is very necessary for the “goal-directedness” of the evolutionary search process. Some of the many selection schemes introduced by researchers to make up for the weaknesses in the original mechanism proposed by Holland – the Roulette Wheel Selection (RWS) will be discussed in section 2.5.
6
Selection pressure: the ratio of the probability of selecting the best individual to the probability of selecting an average individual [Vin03].
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2.3.2 Crossover operator in GA Crossover (or recombination) operator is a very essential part of the GA. It was the only operator in the canonical GA and is the key operator in the reproduction of new offspring [Bea931], [Str05]. The earliest version of crossover – one-point – works by swapping the fragments between the selected parents at a randomly selected point [Eib97]. At the early stage of the genetic search, i.e. before the individuals become similar, crossover offers a source of exploration by recombining differing structures [Vin03].
Fig 2: Point Crossovers in GAs Eshelman [Esh89] observed that one-point crossover was not a very effective approach when combining separate schema because it has strong positional bias and low distribution bias. Multi-point (n-point) crossover and indeed other variants such as Uniform crossover, multi-parent have been explored in the bid to solve this problem. Some researchers suggest that crossover should be applied to every reproduction event and then the disruptiveness is tuned instead of crossover probability [Vin03]. The above mentioned methods were very commonly applied to binary encoded GAs and can be applied to numerical (integer or real) encodings. However, they do not introduce new information to the search as they do in binary encoding and therefore not very useful for an explorative search. With numerical representation, “more intuitive and meaningful” crossover operators could be designed [Vin03]. Some kind of blending of the parents can be performed such
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as the one described by Haupt [Hau98] where the offspring z1 and z2 can be defined from the parents x and y as:
z1 = x – (x – y) and z2 = y + (x – y) ·························· (1) where is a uniform random number in the range of 0 to 1. This approach also termed flat crossover ensures that out-of-bounds offspring are not produced by valid parents because offspring parameters are determined by uniform sampling between the two parent values inclusively. However, Eshelman and Schaffer [Esh91], later generalised this crossover to BLX- which extends sampling range on both sides of the parent. BLX-0.5 was found to perform quite well across a range of test problems whilst flat crossover appears to be converging to the population mean. Wright [Wri91] equally proposed other alternative crossover operators for real parameter GAs: real crossover and linear crossover. He observed that a combination of real and linear crossover outperforms the real crossover across a number of test functions. With this range of viable alternatives that could be defined for real parameter optimisation, it offers the best opportunity for exploiting the full potential of crossover in GAs.
2.3.3 Mutation operator in GA Mutation is the probabilistic replacement of an allele value of a chromosome. This helps to maintain diversity within the population which is necessary for an explorative search [Eib98]. The canonical GAs did not originally work with this operator and so depended on the good design of the selection and crossover strategies to manage diversity. However, the interactions by research in the EC community over the years led to the integration of the mutation operator which was a key operator in other evolutionary techniques (e.g. ES), but was at that time seen as “background operator” [Hin95], [Eib98].
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In binary encodings, mutation is achieved by either bit-flipping (i.e. 01, 10) or by randomly7 assigning a new bit value to the allele. Mutation is also typically applied with low probability for it not to be too destructive. However, with floating point (real) parameter encoding, mutation is achieved by simply replacing the allele value with a new random value. However, some works have been done in achieving less destructive mutations. Wright [Wri91] proposed a real-valued mutation operator known as creep mutations which probabilistically alters the allele value slightly. There is also an equal chance between increasing and decreasing the allele value.
2.3.4 Replacement in GA Newly produced offspring solutions replace the old parent solutions during the evolutionary process. Holland [Gol89] originally proposed the generational replacement in GAs. This means that new generation of individuals produced through crossover and mutation simply replaces the old generation. This is also referred to as block replacement. The approach does not guarantee the survival of the best individual from the old generation. Elitism was then designed to check this. Elitism ensures that the best solution found during the search is preserved during the replacement. Another form of elitism – population elitism – is achieved by ranking together all the individuals in the parent and the offspring populations and then selecting the best ones for the new generation. Another common approach to replacement is called the steady state (or incremental) replacement in which one or a few individual solutions are replaced [Whi94]. The steady state strategy often involves replacing the worst individual(s) with better offspring. So by nature, this approach is elitist [Vin03]. A comparison of generational and steady state replacement show relative weaknesses and strengths of both strategies [Bea931]. Steady state is found to be 7
The random approach halves the effective mutation rate i.e. an allele value has a 50-50 chance of either retaining its original value or loosing it [Vin03].
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more susceptible to genetic drift. But besides being inherently elitist, another strong point for the steady state is that it provides the opportunity for the algorithm to use the genetic materials introduced into the population immediately [Bea931]. 2.4 Exploration, Exploitation and Diversity Management The effective exploration and exploitation8 of the solution space is one of the main issues usually considered in the designing (and parameter tuning) of evolutionary algorithms [Eib96], [Eib98]. “The key for an efficient search is the balance between exploration and exploitation”, [Amo05]. The exploration of the search space in the context of GAs emphasises the ability of the search algorithm to discover new regions within the domain of possible solutions and as such “retain diversity” [Mau84]. On the other hand, exploitation describes the algorithm’s tendency to move, as Beyer [Bey98] puts it, towards “the direction of desired improvement”. If the algorithm is highly exploitative, then there will be the danger of premature convergence9 especially in multimodal landscapes [Eib98], [Eib96]. If, on the other hand, the explorative effect is high, then it might be too disruptive for useful search. It is therefore not surprising that these two concepts, though very useful on their own, are seen as opposing forces. Eshelman et al [Esh89] asserts that “increased exploration by selection leads to decreased exploration by crossover”. Several discussions in this area have focused on how to “balance” their individual impact on the performance of the algorithm [Bey98], [Esh91], [Eib96], [Eib98]. GA basic operators (crossover and mutation) and selection have also been conveniently classified as contributing to either the explorative or exploitative aspects of the search. So while mutation and crossover are seen as explorative, selection (as already stated above) is said to be exploitative [Eib98]. Echelman et al [Esh91] did emphasise the fact that “Selection according to fitness is the source of exploitation; while mutation and crossover are sources of exploration”. There are 8
9
In some heuristic approaches like Tabu search, these concepts can be likened to intensification and diversification. [Eze06]. Individual solutions becoming too similar that productive search can not continue.
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still concerns about the appropriate level of balance that is right for both concepts but many seem to believe that it does depend on the nature of the problem, the encoding and the type of operators used. This work is mainly concerned with the fact that selection plays a fundamental role in the process and therefore its design ought to be motivated by the need to make it more purposeful to the search; hence the investigation of the impact some gendered-selection models have on the performance. 2.5 Overview of some Common Selection Strategies Good selection strategy, as shown above, is indeed required to effectively implement a GA. Mitchell [Mit96] noted that the purpose of selection is to highlight highly fit individuals with the hope that their offspring will even be fitter. In so doing, the exploration – exploitation balance should be kept because “too-strong selection means the sub-optimal highly fit individuals will take over the population; [where as] too weak selection slows down the evolution” [Mit96]. Many selection schemes have been proposed and applied in GA literatures. This section will present an overview of some common ones. 2.5.1 Roulette Wheel Selection The Roulette Wheel Selection (aka Spinning Wheel Selection [Goh03]) is a fitness–proportionate selection scheme introduced with Holland’s original GA. It works by allocating slots on an imaginary roulette wheel to individual chromosomes according to their fitness values. The sum of all the fitness values represents the area covered by the roulette wheel. To select individuals for mating, the weighted roulette wheel is then spun for ntimes (n being the number of parents required for reproduction). At each spin, the individual with the slot under the wheels marker is selected for mating. So with this method, higher chances for mating are given to individuals with higher fitness values.
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Although this approach was classified as preservative [Bac91] (i.e. every individual has a non-zero chance of being selected)it however does not guarantee that an individual’s expected value (i.e. the number of offspring suggested by its fitness) is produced. Sampling errors were seen to reduce as population sizes increased but for practical sizes, it was difficult to produce an even sampling. This therefore does not guarantee stable selection pressure required for the evolution to proceed successfully. De Jong [DeJ75] proposed up with an approach which assigns an expected value to each individual chromosome determined by the ratio of its fitness to the mean fitness of all individuals. According to him, the selection of an individual chromosome decrements its expected value by the value of 0.5 (if used for crossover) or the value of 1 (if otherwise). When the chromosomes expected value falls below 0, it would no longer be available for selection. This approach showed improved performance over the standard roulette wheel selection as well as reduced allele loss. 2.5.2 Stochastic Universal Sampling (SUS) Baker [Bak87] proposed a different approach to sampling – Stochastic Universal Sampling – with a view to reducing bias since accurate sampling is crucial. This was aimed at minimising the spread, the range of possible actual values, given an expected value [Mit96]. SUS works by spinning the wheel (as in RWS) only once to select n–parents but with n equally spaced pointers. Mitchell [Mit96] however noted – “SUS does not solve the major problem with fitness – proportionate selection. Typically, early in the search, the fitness variance in the population is high and a small number of individuals are much fitter than others…they and their descendants multiply quickly in the population in effect preventing the GA from further exploration. This is known as ‘premature convergence’”. Scaling is commonly used with these fitness proportionate functions to keep the selection pressure constant. This is usually done by removing the offset in the objective function evaluations often by subtracting the minimum value that the function can yield for valid input [Vin03], [Mit96]. A typical example is the Sigma
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scaling which produces a scaled fitness fscaled from the actual fitness, factual using the formular:
fscaled = S + factual – fmean ····················································· (2) 2σ where: S is the scaling factor, 1 ≤ S ≤ 5; fmean is the mean fitness of all individuals in the population; σ is the standard deviation [Mit96], [Goh03] 2.5.3 Rank Based Selection Ranking is also proposed [Bak85] to reduce the risk of high convergence. It considers the individuals relative position in the population and not their absolute fitness values. Although ranking tends to discard some of the information content10 in the population, it still maintains constant selection pressure during the search process. It achieves this by avoiding giving the far largest share of offspring to a small group of highly fit individuals. This way, it reduces the selection pressure when the fitness variance is high and also keeps it up when the fitness variance is low. So the ratio of the expected values of individuals remains constant irrespective of their absolute fitness differences. In Baker’s linear ranking approach, each individual is ranked in increasing order of fitness, from 1 to n (n being the population size). An expected value max is then defined for the rank N individual (where max ≥ 0). The expected values of each individual, i, in the population at the time, t, is given by:
ExpVal(i, t) = min + (max – min) rank (i, t) – 1 ···················· (3)
n –1
where: min is the expected value of individual with rank 1. Given the constraint max ≥ 0 and ∑i ExpVal (i,t) = n (population size is constant). Basic practice 10
It might be important to know in some cases that one individual is far fitter than its nearest neighbour.
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recommends that 1 ≤ max ≤ 2 and min = 2 – max (see [Mit96] for the derivations). 2.5.4 Tournament Selection This is one of the most commonly used selection techniques. It is also said to be simple, computationally more efficient than the above and more amenable to parallel implementation [Bli952], [Mil95], [Goh03]. It loops through the population n – times (n is the number of parents required for recombination) and selects t individuals each of the n - times (t is the tournament size and t = 2 is usually recommended [Bli952], [Goh03]). It then picks the best of the t individuals and replaces them for possible re–selection. This naturally keeps the selection pressure under control, although increased tournament size increases the selection pressure. Also a variety of this scheme exists where individual are chosen from the tournament probabilistically rather than deterministically. A possible way to achieve this is by defining a threshold, t (usually 0.75 ≤ t ≤ 1) [Mit96], [Goh03] and a random number, r, is generated to decide which to from the tournament; r ≥ t selects the fitter while r < t selects the weaker. From the few common techniques discussed above, it is obvious that there is a need to be in control of the selection pressure [Bäc94]. It is important that in the design of a GA, a selection scheme that guarantees stable selection pressure is adopted. Although, in some cases it might be desirable to have varying selection at different stages of the search, it is necessary that it is controlled possible through adaptive mechanisms [Dav91]. 2.6 Approaches to Gender Restricted Selection in GAs Although GAs work well on a wide variety of complex optimisation problems, some researchers still view it as a “drastic simplification of [natural] evolution” [Wag04]. As Mitchell and Forrest [Mit95] suggested, “perhaps the most obvious area for extending the GA is to the study of evolution itself”. This, they argue, hinders the effective simulation of the organisms’ response to the complex task of survival through adaptation. Page 27 of 98
So, as the need for a robust, self adaptive approach to controlling evolutionary forces persists, some researchers began to consider “more natural” approaches to GA design [All92], [San031], [San032], [Lis96]. The concept of gender restricted selection, as a way of controlling the selection pressure and therefore maintain a reasonable balance between the exploration and exploitation in GAs, has being examined and certain approaches presented by some researchers [Goh03], [Wag04]. The central idea in such designs is simple: the population is usually (but not always) split into two genders – male and female – and individuals from the same gender are not allowed to mate. Also different selection strategies, often drawn from the common existing ones, are designed for each gender and so their combined effect on the evolutionary process becomes different from their individual effects and as such can be adjusted to achieve more desirable results. Sanchez-Velazco and Bullinaria [San031] noted that, “gender is not only responsible for preserving diversity in genes and maintaining a successful genetic pool by means of selection, crossover and mutation, but it is also responsible for the optimisation of the different tasks vital to survival. There is much evidence that the specialisation of individuals is mainly by gender groups”. Some of the works done in the area of gender based GAs including the inspirations, view points, and approaches adopted by the authors are presented below. 2.6.1 The GenderedGA In their works, Sanchez–Velazco and Bullinaria [San031], [San032] observed that through gender came the development of sexual selection, a component of natural selection where “reproductive success depends on the interaction with a partner of the opposite sex to produce offspring”. Their model – The GenderedGA – integrated some of the principle strategies of the gendered societies which includes patterns of competitive and cooperative behaviours derived from sexual selection as presented by [Gad97]. They also emphasised the fact that mutation
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rates differ in both sexes as presented by Dennis and Gallagher [Den01]. The core components of their works are based on the following factors:
a clear distinction between males and females as well as different selection strategies
inclusion of the principles of cooperation between elements of different gender to optimise the chances for survival.
inclusion of the principle of competition in each gender group to ensure that cooperative entities will be fit individuals
simulation of gender based mutation rates
simulation of a relation between age, fertility and fitness as in biological systems affecting the selection process
So with proposed strategy, P is the population and X, Y are two proper subsets of P representing each gender (females and males respectively) such that:
P = XY ··········································································· (4) and XY = ··········································································· (5) X, Y P ············································································· (6) aP, aX aY ······························································· (7)
Y is the parameter they used to denote the fraction of the individuals in the set Y, so at any given time, the probability of an individual a being in set Y or X is p(a, Y) = Y; p(a, X) = 1 – Y ···················································· (8) Selection was generally by Roulette Wheel Selection (see section 2.5.1) for both genders. However, fitness evaluation for males was based on the direct outcomes of quality of the solution (competitive fitness) while that of the
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females was based on the competitive fitness as well as on other attributes such as age, fertility and the differential between the male parent and the male offspring (cooperative fitness). So the selection of the female parent based on the cooperative fitness, could be defined by the function: xSel = Sel w1f(x) + w2f(y) + w3g(Age(y)) ·········································· (9) w1 + w2 + w3 where: i.
wi are a set of fixed weighting parameters that are used to specify the appropriate relative importance of the three components: fitness, male fitness differential, age and fertility.
ii.
f(y) = f(yson) – f(YSel)---------------------------------------------(10) is the difference between the fitness of the male parent and that of the male offspring11.
iii.
g(Age(y)) is the scaling function that determines the effect of age on the individuals chances of being selected.
Another crucial aspect of this model is the fact that different mutation rates, mY, aY and mX, aX are used for the male and female gender groups. The males were given higher mutation rates than the females12, mY > mX, based on the works of [Den01], [Fre01]. Hence, the effective mutation rate becomes: mEff = Y mY + (1 – Y) mX ·······················································(11) This new scheme was actually used on a Travelling Salesman Problem and was reported to have outperformed a standard GA with similar configuration
11
Females that produce highly fit male offspring are more likely to be allowed more chance to reproduce based on the indirect fitness scheme [Fre01], [Den01].
12
This is the simulation of male vigor [Fre01], [Den01] confirming that males have more dynamic or ‘active’ genetic pool than females.
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parameters. However, the rest of the implementation details were based on the specific characteristics of the TSP problem. 2.6.2 SexualGA Wagner and Affenzeller [Wag04] worked on a “gender-specific” selection scheme based male vigour and female choice in population genetics [Den01], [Fre01]. They presented an alternative way to control the selection pressure in GAs by using different schemes on the two genders. They argued that since the processes of selecting mating partners in nature are different for male and female individuals, it will only be more ‘natural’ and appropriate to use different selection schemes for the two genders. Inspired by this view, Wagner and Affenzeller [Wag04] examined the combined effects of the various combinations of different selection schemes on the selection pressure and thus showed that the selection pressure, and indeed genetic diversity, can be scaled as desired by implementing the gender specific structure. They asserted that selection plays a key role in nature and even considered to be “selection is considered to be the most important (and only) evolutionary mechanism for adaptation to the environment”, [Wag04, p2]. Reviewing a typical RWS strategy, they [Wag04] identified its two major deficiencies: i.
super individuals have very high probability of being selected for reproduction and so spread their genetic materials across the population (i.e. very high selection pressure). Such strong domination causes loss of diversity and eventually premature convergence.
ii.
RWS becomes completely undirected as soon as the individuals become too similar and so drifts towards some kind of random selection structure.
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Selection pressure therefore varies along evolutionary process without control and this often has undesirable effects. To check this, several “non-proportional” selection strategies have been developed. Wagner and Affenzeller considered two basic categories: ranking and tournament strategies (see sections 2.5.3 and 2.5.4). They observed that even with these strategies it was still not easy to measure the selection pressure empirically as with ES-like13 algorithms. To make selection pressure measurable, they “abstracted” it as a measurement of how hard it is for an individual to pass its genetic characteristics from one generation to the next and considered it reasonable to define it in terms of the ratio of the population size to the number of individuals selected as parents in the next generation.
SP
= 1 – |PAR|------------------------------------------------------(12) |POP|
The equation above defines a 0 – selection pressure if all the individuals are used and 1 –
1 |POP|
if clones of only one individual are made to form offspring.
In the bid to make a GA user have more control over the selection pressure, SexualGA was introduced. It does no more than split the population into two genders – male and female – and use different selection schemes for the genders. The algorithm below describes the SexualGA strategy: Initialize total number of iterations nrOfIterations N Initialize population size |POP| Initialize mutation probability mutProb Randomly produce an initial population POP0of size |POP| Calculate fitness of each individual of POP0 for i = 1 to nrOfIterations do Initialize next population of POPi+1 while |POPi+1| |POP| do Select father par1 from POPi by first selection scheme Select mother par2 from POPi by second selection scheme
13
In ES like algorithms (eg. BreederGA[Muh93]) selection pressure is defined as / based on the concept of birth surplus by Darwin. - number of offspring; - number of parents.
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Generate a new child c from par1 and par2 by crossover Mutate c with probability mutProb Calculate fitness of c Insert c into POPi+1 end while end for
So besides simulating sexual selection in natural populations “more precisely” i.e. bringing the concept of GA more towards their biological archetype; SexualGA “offers far more flexibility concerning the adaptability of the selection pressure” [Wag04]. 2.6.3 GA with Gender Allenson [All92], whose main inspiration was drawn from the fact that “if it works with nature…it is worth trying”, was one of the earliest researchers that applied gender genetics in solving multi-objective optimisation problem. In his work, he had to minimise both the length of a pipeline (and hence its cost) and also the damage its construction would cause to the environment. A large array of biodiversity values for specific locations with the start and end points for the construction of the pipelines were provided. A gendered approach was one of the options tried and it worked well. A GA was originally developed to minimise the “economic” function (i.e. the length/cost) of the pipeline which was measured using the Euclidean metric. A binary tournament selection was used for the selection and the replace-theworst14
approach was used with the one-point crossover (which modified to
take into consideration the varying chromosome length). Also a different but similar GA was developed to minimise the biodiversity (environmental) destruction. This also worked well. However, it became obvious that some members of the Pareto-optimal15 set of solutions were of quite a cost (economic fitness function) but of low biodiversity destruction (environmental damage function) while some have the 14
This is just replacing the worst individual in the population with the new offspring produced.
15
See section 2.7
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reverse case. So to find members that take both functions into account, the concept of gender was introduced. Each of the function was assigned a gender: male for the environment and females for the economic functions and only male female pairings were allowed for the recombination. The initial population contains equal number of individuals with both genders to keep the gender ration under control. The gender of the offspring was randomly allocated at birth and it replaces the worst member of its gender. This, they reported, worked well. 2.6.4 MSGA Lis and Eiben [Lis96] implemented a Multi-Sexual Genetic Algorithm (MSGA) for solving multi-objective optimisation problem. The MSGA has a similar structure to the algorithm proposed by Allenson [All92]. There are often more than just two optimisation criteria in a multi objective function (see section 2.7). The MSGA defines as many genders as the number of optimisation criteria in the problem – not just males and females. The basic structure of the MSGA describes a normal binary encoding with a “sex marker16” as one of the genes. Optimisation criteria are then mapped to the sexes on a one to one basis and individuals are assigned different evaluation functions based on their criteria. A probabilistic recombination operator is used such that if recombination will happen, exactly one parent will be selected from each gender and then multi-parent recombination will be used (see [Eib94]); otherwise, one parent from the current population is copied into the next generation based on the gender randomly selected and the rank of the individual. Simple bit flip mutation with low mutation probability was used. A new child’s gender was inherited from the parent with the highest contribution to the genetic code; and if there is a tie between parents with high genetic contributions, the child’s gender was randomly drawn from those of such parents. 16
An integer number from the interval [1..N] where N is the number of the optimisation criteria (see figure 4)
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Sex Marker Binary chromosome Fig 3: Representation of Solutions by Lis and Eiben [Lis96]
On two different test problems, the performance of the MSGA was reported as “promising” however, Lis and Eiben [Lis96] were considering more thorough assessment on the effects of different crossovers, combining fewer individuals than the number of sexes, allowing mating of individuals from the same gender. 2.6.5 MSGA II Bonnisone and Subbu [Bon03] and Richter-von Hagen et al.[Ric05] adapted the MSGA structure based on the work of Lis and Eiben [Lis96] each with slight modifications. Bonnisone and Subbu [Bon03] worked on a multiobjective flexible manufacturing optimisation problem. The adopted the Gray encoding to avoid the Hamming cliff17 problem often encountered with the normal binary encoding [Vin03]. The gender tagging approach and the selection were retained but other approaches to gender assignment for new offspring were proposed. Genotypic gender assignment is the one described by Lis and Eiben [Lis96] (reported above). Phenotypic gender assignment is based on the vector-evaluated fitness function i.e. each offspring is evaluated on all fitness components and the offspring with the strongest normalised score for each criterion takes its corresponding gender. The replacement strategy was generational with the elitism operator that guarantees that the best individual is copied into the next generation. This modified version of MSGA caused, as reported, over 250% improvement on the number of non-dominated solutions found on the two test functions used by Lis and Eiben [Lis96]. On a 17
Hamming cliffs in binary codes makes two very similar binary numbers, when combined to give a very different result (e.g. a one-point crossover of decimal values 127 and 128 can lead to offspring of 0 and 255). [Vin03]
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third function, their algorithm eliminated all the dominated solutions against what was observed in the report of the work by other researchers on the same function. Richter-von Hagen et al. [Ric05] successfully ran a customised version of MSGA in their use of a GA approach to organizing knowledge intensive processes. They adopted ternary encoding, {-1, 0, 1} and selection was by either of the two ways: the usual RWS and best-half18 operator. Their algorithm found “acceptable Pareto solutions”. 2.6.6 Simulated Gender Separation and Mating Constraints for GAs Vrajitoru, [Vra02], introduced a simulation of the phenomena of gender separation and mating restrictions in GAs. He studied the occurrence of gendered species in reproductive models and their impact on GA performance. Several reproductive models were explored on different gendered species. The gender models used in their work were males, females, hermaphrodites and self– fertilizing organisms and these were evenly spread across the initial population of the GA. The offspring are made to inherit the gender types of their parents and the size of each gender type is not fixed. Also the influence of social factors such as the population size and mating restrictions on the adaptation of the GA were examined. Mating restrictions were implemented in different ways. For instance individuals close to each other were not always allowed to mate in order to encourage exploration since they are likely to share similar genetic materials. Also the female parents could only be picked from within a certain radius around the male. The inspiration was drawn from the fact the in natural selection, “sexual reproduction is the most important mechanism for ensuring the genetic diversity of a population
18
The best-half operator sorts the entire population according to their fitness values. Then all members of the best half are selected with some randomly chosen member.
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and increasing the potentials for adaptation of species to hostile environments”; thus emphasizing the need to preserve diversity in GAs [Vra02 p2], [Aff04]. 2.6.7 Sexual Selection for Genetic Algorithms Goh et al. [Goh03] presented a review of some of the commonly applied selection schemes and the difficulties encountered while using them especially on the basis of exploration and exploitation. They then proposed the gender separation techniques as a way to clearly separate the explorative and exploitative activities “without the inconvenience of parameter tuning” [Goh03 p.3]. They separated the population into two sexes – males and females. The selection of females handled the exploration in the search while the selection of males handled the exploitation. Ultimately, all the females would get to mate but only the “attractive” males were allowed to mate. Poorly fit males were eventually eliminated without a chance to mate. The implementation was as described below: 1. generate a population of individuals(solutions) 2. split the population into two: males and female 3. select a male using tournament selection 4. select the next unmated female 5. return to step 3 until all females have mated 6. Replace the old population 7. Go to 2 The replacement strategy adopted in this model was actually generational (see section 2.3.4)
a. Separate the population into males and females
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b. Select a male with tournament selection
c. Select a female in order of occurrence Fig 4. Gender based selection model by Goh et al. [Goh03]. This approach was applied to known combinatorial optimisation problems e.g. The Royal Road Problem and Open Shop Scheduling problems and was found to be better than the Tournament selection in more difficult problems when no scaling is used. 2.7 Numerical Optimisation Vincent [Vin03] defines the goal of a global optimisation as navigating “the (potentially highly) complex terrain represented by the fitness surface so as to reach, or come close to, the global optimum (or other acceptable solution)”. The nature of a typical optimisation problem has been introduced in section 1.1. An objective function evaluates the quality of the solutions (new and old) in order to make a reasonable decision and is formally defined as given below [Vin03]: f: X Rn R where X is a user-defined bounded set of valid (or desirable) inputs and f is the objective function (refer to [Fon95] for further definitions). Page 38 of 98
Above describes a typical single objective problem (because it shows only one objective function). Where a single perfect (or Utopian) solution exists – the global optimum – it can be found depending on the search strategy [Fon95]. However, in reality we most often need to simultaneously optimise multiple, possibly conflicting objective functions. A suitable solution often compromises optimal performance of any one of the objectives to achieve acceptable (or Pareto optimal19) solutions in all objective dimensions [Fon95]. In practice though, these objectives need to be recombined to form a single objective function that represents the component functions with a redefinition of the required performance level in order to be able to select good solution from a family of alternatives which are otherwise considered equivalent [Coe00], [Fon95]. Generally, similar approaches to search for a better solution in the solution space apply. Formal definitions of multiobjective optimisation problems could be found in [Fon95].
19
The set of Pareto optimal decision vectors are such that any decrease in one objective must cause a simultaneous increase in at least one other objective [Coe00, Fon95, Vin03].
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3: METHODOLOGY In this section, the test problems, the algorithmic models as well as the methodology adopted in conducting this research work will be presented. The experimental conditions and the performance metrics used are also discussed. 3.1 The Test Problems The experiments in this work were performed on a number of classic optimisation problems. These problems are as described below.
3.1.1 The Rastrigin Function This is one of the commonly used numerical optimisation test functions [Whi96], [Bal04]. Here the goal is to find the solution vector with values that yields the minimum evaluation of the objective function defined below.
f(X) = 10n + ∑(xi2 – where a = 10 and b
i))········································ (1)
The search domain is: -5.12 ≤ xi ≤ 5.12, i = 1,…, n.
There are several local minima in this function but the global minima is at X* = (0,…,0), f(X*) = 0
Fig 5: Rastrigin Function Graph for n=2 [Hed06]
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3.1.2 The Schwefel function The Schwefel function is another commonly used test function [Whi96]. The goal here is to find the value of the solution vector, X, that gives the minimum evaluations of the function defined below:
f(X) = 10n + ∑(xi2 –
i))········································ (2)
The search domain is: -5.12 ≤ xi ≤ 5.12, i = 1… n. There are several local minima in this function but the global minima is at X* = (0,…,0), f(X*) = 0
Fig 6: Schwefel Function Graph for n=2 [Hed06]
3.1.3 Whitley’s F8F2 Function This is a composite function formed by the combination of the F2 (Rosenbrock’s) and F8 (Griewangk) functions which on their own have dissimilar characteristics [Whi96]. Rosenbrock’s function is a non-symmetric, non-linear function over two variables defined by:
f(xi|i=1,2) = 100(x12 – x2)2 + ······························· (3)
Fig 7: Rosenbrock Function Graph for n=2 [Hed06]
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where the search domain is: -2.048 ≤ xi ≤ 2.047 Griewangk function is a scalable, non-linear and non-separable function. This function is defined by.
This position is for the Griewangk Function ···································· (4) where the search domain is: -600 ≤ xi ≤ 600, i = 1, 2,…, n. Like others, it has several local minima with the global at X* = (0,…,0), f(X*) = 0. the graph below is shown for n = 2.
Fig 8: Griewank Function Graph for n=2 [Hed06]
F8F2 function is the composition of the Griewangk’s and Rosenbrock’s functions in the following manner:
F8F2(x1, x2, …, xn) = F8(F2(x1, x2)) + F8(F2(x2, x3)) + … + F8(F2(xn – 1, x n )) + F8(F2(xn , x 1 )) ······································································· (5) The resultant function becomes non-symmetric with many local minima. The three GA models were used to optimise each of the above functions for maximum number of evaluations 5000, 10000 and 15000 with the dimensionality of 10, 20 and 30.
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3.2 The Algorithmic Models The models used in this works are termed StandardGA, GenderedGA(1) and GenderedGA(2). These share similar schemes and configurations commonly used by researchers but differ slightly in the selection and replacement strategies. The section will describe the general characteristics of the GA used for three models as well as the individual implementations of the selection schemes adopted.
3.2.1 The general GA features The general features of the GA adopted and the authors that recommended them are as described below: i. Encoding:
real encoding20, [Dav91], [Wri91]
ii. Crossover:
BLX0.5,
[Esh91]
iii. Basic selection:
tournament
[Bli951], [Bli952]
iv. Mutation:
creep mutation, [Wri91]
v. Basic replacement: steady state,
[Dav91]
3.2.2 The “StandardGA” The StandardGA as used here is by no means widely regarded as such. However, its features are more generally applied as can be seen from the literature. The fundamental difference between this algorithm and its gendered counterparts is basically in the selection and replacement strategies. In selection, both parents are selected from the population using simple tournament selection while for replacement; the worst individual member of the population is simply replaced with the offspring, if it is better. Its features are basically as described above.
20
For the genes of the GA, the unit interval {0, 1} method suggested by Garloff [Gar03] will be used.
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3.2.3 The GenderedGA(1), GGA1 The selection strategy here is based on the work of Goh et al. [Goh03]. The population is split into males and females. A male is selected using the tournament selection while all the females are given the chance to mate (see section 2.6.7). The replacement is slightly different21: either the worst male or female is replaced depending on the gender of the offspring which is assigned randomly. Below is the description of GGA1 Step 1: randomly generate n-individuals (n = popSize)
Step 2: split the population into two22
Gender demarcation point (popsize/2)
males
females
Tournament selection
Step 3:
+
=
Step 4: randomly replace the worst male or the worst female with the new child. Step 5: re–initialise the female pointer after the female in the last position has mated. Continue with Steps 3 and 4. Fig 9: the model of the GenderedGA1, GGA1
Steps 4 and 5 defined the difference between this model and Goh’s. The selection in Step 3 was not to build the mating pool but to reproduce and replace
This work borrowed the concept and not the entire implementation as other strategies defined in the methodology differ with the one used by [Goh03] and the aim was not to confirm their work. 21
22
This should have the same effect as randomly separating them because they are randomly generated.
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3.2.4 The GenderedGA(2), GGA2 Like the GenderedGA(1), the population is split into two – males and females. The best male is chosen to mate with the best female provided they are not “too close” to each other. The motivation is from the work of Eshelman and Schaffer, [Esh91] on preventing “incest” and the mating restriction suggested by Vrajitoru [Vra02] (see section 2.6.6). The distance metric used is the Euclidean distance defined by Vincent [Vin03, p5]. With this a male and a female within a distance below a certain threshold are not allowed to mate. Fig 9 illustrates the difference in the selection stage from the GGA1 and this describes how Step 3 is implemented in GGA2. They are basically the same in order stages. Step 1: (see GGA1) Step 2: (see GGA1) Step 3: Select the best male and then the best female “not too close” to him.
Selection of the best male
Threshold Euclidean distance
Step 4: (see GGA1)
+
=
Step 5: Repeat 3 and 4 until stopping criterion is met. Fig 9: The model of the GenderedGA2, GGA2
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3.3 The Meta GA and Parameter Optimisation Setting the optimal parameters for the GA models poses a problem on its own. Another GA, Meta GA23, was used to optimise some of the configuration parameters of the underlying optimiser GA models. This approach was originally suggested by Grefenstette [Gre86]. The parameters that are to be optimised by the Meta GA are: population size, over initialization, tournament size, mutation rate, max creep and theta. 3.4 Metrics and Statistics Some performance metrics were used to compare performance of the different models in solving the test functions. Also some statistical functions were used to validate the distribution of the result data
3.4.1 Best fitness This is the fitness score of the best individual in the population. It is often mapped to the objective function value of the problem. In this study, the objective function value of any one of the functions will be used in each case.
3.4.2 Average fitness This is the mean of all the fitness scores of all the individuals in the population. It is also a measure of the solution quality as well as a diversity measure.
3.4.3 Genotypic Diversity This is the mean variation of genes about the centroid, (centre of mass) of the population [Any05]. The function definitions start with the mean value of an individual gene (say j), denoted mean(j), in population P, where Pij is the jth gene of the ith chromosome, is popSize 1 mean j Pij ······················································ (6) popSize i 1
Some of the drawbacks of the meta GA have been pointed out [Clu05], but this work intends not to focus on this as the central experiment is about the performance of the models describe. 23
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The gene j variation, var(j) is defined as: var j
popSize 1 mean j Pij popSize i 1
················································ (7)
so mean variation or diversity is given by: div( P)
1 nPars var ( j ) nPars j 1
···························································· (8)
3.4.4 Standard Deviation This is a statistical function that measures how spread out a set of data is defined as the square root of the variance [Wei06]. It is used to analyse the stability of the results over 500 trials for the validation experiments with the results from the meta GA. ························································································ (9)
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4: RESULTS This section presents the results of the experiments and analysis of the experimental data. These are categorised based on the functions, number of parameters and the run lengths for all three GA models. Discussion on the first set of experiments will be in much depth delving into general issues on the performance of a GA, the graphs, the metaGA optimised parameters and the possible impact of the parameter values. On subsequent experiments, re-occurring trends may be left out emphasizing only unexpected variations. Only those needed for illustrative descriptions in the discussions are presented in this section. 4.1 Rastrigin Function The performances of the various optimisers in solving the Rastrigin problem (see section 3.1.1) are presented in this section.
4.1.1 Rastrigin with 10 Parameters This describes the experimental results for Rastrigin on 10 parameters
GA - Models
Parameters StandardGA
24
popSize overInit tSize24 mRate theta
2 57 3 0.1671 7.7439
GenderedGA(1) 10 23 4 0.1130 2.1483
maxCreep Metrics Best Fit Avg Fit Diversity StDev
0.5164
0.1099
Performance Summary 0.3583 0.0720 0.3598 0.0730 0.0000 0.0000 0.5552 0.2248
GenderedGA(2) 48 4
0.1127
*** 0.0786 1.3247
0.0400 0.0551 0.0001 0.1509
Observe that the tournament size was not used in GGA2 as tournament selection was not implemented.
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4.1.1(a) 5000 Maximum Evaluation The standard GA, (SGA); GenderedGA1, (GGA1); and GenderedGA2 (GGA2) started each with the initial optimal popsize of 2, 10, 48 and overInit values of 57, 23, 4. The high overInit value seems to counter the imminent loss of diversity for SGA which had very low popsize. However, with very high overInit/popsize (28.5) ratio and high tsize/popsize ratio (1.5), it was expected to do a little exploration at the initialization stage and thereafter continue more of exploitative search while GGA1 and GGA2 were expected to do more exploration. SGA also had the highest mRate, theta and maxCreep (0.1671, 7.7439, and 0.5164) which ought to have improved its explorative ability but it was overtaken by GGA1 and GGA2 at about the 2000th evals suggesting that it might have got stuck in a local optima. This is not surprising considering the high tsize/popsize ratio which it was working with. SGA made best and average fitness of 0.3583 and 0.3598 respectively against GGA1’s 0.0720, 0.0730 and GGA2’s 0.0400 and 0.0551. Also the standard deviations on the final results for 500 trials from each of the algorithms indicate highest stability for GGA2 with 0.1509 against 0.2248 and 0.5552 for GGA1 and SGA respectively. In this class of experiments GGA2 performed best. Best Fitness 60
50
40
30 StandardGA GenderedGA(1) GenderedGA(2) 20
10
0 0
1000
2000
3000
4000
5000
Fig 11: Best Fitness graph Rastrigin_10_5000
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Best Fitness(b) 1
0.9
0.8 StandardGA
0.7
GenderedGA(1) GenderedGA(2)
0.6
0.5
0.4
0.3
0.2
0.1
0 3000
3400
3800
4200
4600
5000
Fig 12: Last 2000 evals Best Fitness Rastrigin_10_5000
On diversity, GGA2 again outperformed the others in maintaining appropriate level of diversity during the search and ending with near zero convergence of 0.0001. Diversity 0.01
0.008
0.006 StandardGA GenderedGA(1) GenderedGA(2) 0.004
0.002
0 0
1000
2000
Fig 13: Diversity Graph Rastrigin_10_5000
3000
4000
5000
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4.1.1(b) 10000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
GA - Models StandardGA GenderedGA(1) GenderedGA(2) 3 11 58 51 72 67 5 4 *** 0.1205 0.1623 0.1796 7.5243 9.9007 9.9608 0.1199 0.2091 0.1130 Performance Summary 0.019013 0.013936 0.007960 0.019013 0.013938 0.007960 0.000000 0.000000 0.000000 0.148526 0.117015 0.088724
In this set of experiments SGA again started with low popsize of 3 and very high overinit/popize ratio of 17 against 6.54 and 1.15 for GGA1 and GGA2 respectively. This may have accounted for its starting off with good average fitness which it lost due to poor exploration after initialisation. The GGAs are more likely to explore better due to relatively higher popSize, mRate, theta and maxCreep more. Average Fitness 80
60 StandardGA GenderedGA(1) GenderedGA(2)
40
20
0 0
200
400
600
800
1000
Fig 13: First 1000 evals Average fitness Graph Rastrigin_10_10000 Also an overInit of 5 in the SGA with popSize of 3 will make the algorithm more exploitative despite the rather high mRate and theta, because many evaluations may see only one individual producing all the offspring. GGA1s poor
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performance may have to do with the high mRate and theta (0.1623 and 9.9007) on popSize of 11 which might have been disruptive. Noteworthy though is the fact that the percentage improvement on the best fitness score was generally high i.e. SGA, GGA1 and GGA2 improved by 94.7%, 80.7% and 80.8% making SGA the most improved. GGA2, however got the best solution of 0.00796 and the StDev shows that it was the most stable. Best Fitness 80
70
60
50
StandardGA GenderedGA(1) GenderedGA(2)
40
30
20
10
0 0
2000
4000
Fig 14: Best fitness Graph Rastrigin_10_10000
6000
8000
10000
Best Fitness 0.1
0.08
0.06 StandardGA GenderedGA(1) GenderedGA(2)
0.04
0.02
0 7000
7600
8200
8800
9400
10000
Fig 15: Last 3000 evals Graph for Best fitness Rastrigin_10_10000
On diversity, GGA2 still held sufficient level of diversity during the search and eventually converged even below GGA1. SGA lost diversity at an early stage which is expected considering the low popSize and high tSize/popSize ratio.
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Diversity 0.01
0.008
0.006 StandardGA GenderedGA(1) GenderedGA(2)
0.004
0.002
0 0
2000
4000
Fig 16: Diversity Graph for Rastrigin_10_10000
6000
8000
10000
4.1.1(c) 15000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
StandardGA 10 41 4 0.1398 10.0000 0.1148 0.000000 0.000000 0.000000 0.000000
GA - Models GenderedGA(1) GenderedGA(2) 11 83 92 37 4 *** 0.1675 0.1234 10.0000 10.0000 0.1045 0.1110 Performance Summary 0.001990 0.003980 0.001990 0.003986 0.000000 0.000000 0.044496 0.062864
SGA showed an extremely good performance in this class of experiments. The length of the run may have helped but obviously the comparatively higher popSize and reduced tSize/popSize ratio seem to have enabled it to explore better. The overInit/popSize (4.1) ratio is also sufficiently high to boost the average quality before the search started. Also, fairly high mRate, theta and maxCreep values (0.1398, 10 and 0.1148 respectively) would have improved the exploration.
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Avg Fitness 100
80
60 StandardGA GenderedGA(1) GenderedGA(2)
40
20
0 0
200
400
600
800
1000
Fig 17: First 1000 evals Avg Fitness Graph for Rastrigin_10_15000 GGA1 had an almost similar set of initial configuration and even the highest overInit/popSize ratio (8.3636) – GGA2 was quite low (0.4458) – which explains why GGA1 started off with the best average fitness. A zero (or near zero) difference between the best fitness and the average fitness in all three cases show that they were all converging to a local minimum before the end of search, with all ending with a diversity of 0.000000. However, GGA2 maintained the diversity more than the rest during the search. Diversity 0.005
0.004
0.003 StandardGA GenderedGA(1) GenderedGA(2)
0.002
0.001
0 0
3000
6000
9000
Fig 18: Diversity Graph for Rastrigin_10_15000
12000
15000
Fig 19 further emphasized the explorative nature of GGA2. The last 2500 evaluations obviously saw GGA1 ‘holed’ in a local minimum with best fitness value of 0.001990 and SGA hitting approximately 0.000000 in but GGA2 made a percentage improvement 66.6%
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Best Fitness 0.012
0.0096
StandardGA GenderedGA(1)
0.0072
GenderedGA(2)
0.0048
0.0024
0 12500
13100
13700
14300
14900
Fig 19: Last 2500 evals Best fitness Graph for Rastrigin_10_15000 Perhaps it is in the stability of results across the 500 trials that the SGA defeated the two GGAs. SGA had a standard deviation of approximately 0.000000 against the 0.044496 and 0.06286 from GGA1 and GGA2 respectively.
4.1.2 Rastrigin with 20 Parameters This section presents the classes of the Rastrigin experiments with 20 parameters
4.1.2(a) 5000 Maximum Evaluation GA - Models Parameters popSize overInit tSize mRate theta
StandardGA 2 47 4 0.0604 0.2360
GenderedGA(1) 10 18 3 0.0587 3.4144
GenderedGA(2) 55 13 *** 0.0400 1.2890
maxCreep
0.1122
0.3137
0.2182
Metrics Best Fit Avg Fit Diversity
1.2048 1.2067 0.0000
StDev
0.6496
Performance Summary 2.7224 2.7336 0.0001 1.4832
2.4357 2.6548 0.0004 1.6798
The overInit/popSize ratio for SGA, GGA1 and GGA2 (23.5, 1.8 and 0.236 respectively) compares almost well with what was observed in the 10 parameter counterpart (28.5, 2.3, 0.833). So starting with a reasonably good set of 2
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chromosomes and high tSize/popSize ratio, (2), its average fitness dropped faster and overtook the rest within the first 2000 evals. Average Fitness 50
40
30 StandardGA GenderedGA(1) GenderedGA(2) 20
10
0 0
1000
2000
3000
4000
Fig 20: Average Fitness Graph for Rastrigin_20_5000
5000
mRate, theta and maxCreep was considerably lowered in SGA (63.9%, 97% 78.5% respectively) reducing its explorative strength. So ending with very low diversity was not surprising. As for GGA1, 48% drop in mRate was compensated by a gain of 58% in theta and 185% in maxCreep. This, and of course its higher popSize may have helped it in managing the diversity better. Diversity
0.02
0.016
0.012 StandardGA GenderedGA(1) GenderedGA(2)
0.008
0.004
0 0
1000
2000
3000
4000
5000
Fig 21: Diversity Graph Rastrigin_20_5000
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Best Fitness 10
8 StandardGA GenderedGA(1)
6
GenderedGA(2)
4
2
0 3500
3800
4100
4400
4700
5000
Fig 22: Last 1500 Best Fitness Graph Rastrigin_20_5000
With GGA2 a 49% drop in mRate and 2.69% and 93.6% gains in theta and maxCreep respectively suggested that it may not mutate as often but its mutation could be more disruptive. But so far it seems to have inherent ability to manage diversity. Even in the last 1000 evals, it made the highest improvement in the best fitness i.e. 57.8% (5.7785 to 2.4356) followed closely by SGA with 57.5% (2.8316 to 1.2047) and finally GGA1 with 49.6% Expectedly, the increase in parameter generally worsened the results but the results showed that the GGAs suffered most which will be further discussed in the section for the ‘damage’ due to increased dimensionality. The standard deviation indicates that the SGA gave the most stable results.
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4.1.2(b) 10000 Maximum Evaluation GA - Models Parameters popSize overInit tSize mRate theta maxCreep
StandardGA 3 84 2 0.0514 2.0819
GenderedGA(1) 10 44 5 0.1251 9.9830
GenderedGA(2) 49 47 *** 0.0335 0.9153
0.1235
0.2229
0.1147
Metrics Best Fit Avg Fit Diversity
Performance Summary 0.129261 1.136915 0.129335 1.171897 0.000000 0.000118
0.127823 0.150688 0.000051
StDev
0.253206
0.266675
1.048514
In this class of experiments, SGA took off with an even higher overInit/popSize ratio (28) than it was with parameter-10 (17). Although the tSize/popSize ratio is reduced suggesting possible improved exploration, the mRate was more than halved and theta drastically reduced as well but with slight increase in the maxCreep. Its diversity measure indicates that it was already converging prematurely almost as in parameter-10. Best Fitness 200
160 StandardGA GenderedGA(1) GenderedGA(2)
120
80
40
0 0
2000
4000
Fig 23: Best Fitness Graph Rastrigin_20_10000
6000
8000
10000
GGA1 had reduced overInit/popSize ratio and slightly varied mRate, theta and maxCreep from its parameter-10 counterpart. GGA2 also had a slightly reduced overInit/popSize ratio but highly reduced mRate and theta. It is however interesting to not how closely the SGA and GGA2 moved together throughout the search
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process (see fig 24). They were all losing diversity but GGA2 gave the best result while SGA gave the most stable set of results. Best Fitness 2.5
2 StandardGA GenderedGA(1) GenderedGA(2)
1.5
1
0.5
0 8000
8400
8800
9200
9600
10000
Fig 24: Best Fitness Graph Rastrigin_20_10000
Diversity 0.005
0.004 StandardGA GenderedGA(1) GenderedGA(2)
0.003
0.002
0.001
0 0
2000
4000
Fig 25: Diversity Graph Rastrigin_20_10000
6000
8000
10000
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4.1.2(c) 15000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
StandardGA 3 80 4 0.0709 4.9372 0.4135 0.140148 0.140159 0.000000 0.353874
GA - Models GenderedGA(1) GenderedGA(2) 10 50 48 23 4 *** 0.0523 0.0817 2.6540 6.8586 0.1089 0.2021 Performance Summary 0.030561 0.101313 0.030787 0.109480 0.000000 0.000017 0.158960 0.330347
A similar trend was observed in the tSize-popSize distribution. The GGA1 performed best in this class of experiment across board i.e. best result and more stable set of trial results. It also optimised with fairly lower mRate, theta and maxCreep values and having started with an overInit/popSize ratio of 4.8 and a popSize 10, its explorative-exploitative ability seemed balanced which may have contributed to it producing better and more stable result. Best Fitness 2
1.6
1.2 StandardGA GenderedGA(1) GenderedGA(2)
0.8
0.4
0 6600
8400
10200
12000
13800
Fig 26: Last 3000 evals for Best fitness Graph Rastrigin_20_15000 On the other hand SGA had very low popSize (3) and very high overInit/popSize ratio (26.67) which must have imposed a drastic reduction in diversity even before it started. It is therefore not surprising that it converged prematurely; as a result slowing down search process. Also, GGA2 had a poor start with low overInit/popSize ratio (0.46).
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4.1.3 Rastrigin with 30 Parameters The discussion on the results for 30 parameters on Rastrigin is presented in this section.
4.1.3(a) 5000 Maximum Evaluation GA - Models Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity
StandardGA 2 0 4 0.0694 9.2613
GenderedGA(1) 10 15 2 0.0798 9.6276
GenderedGA(2) 66 89 *** 0.0211 0.0689
0.4131
0.3289
0.2134
Performance Summary 9.73293 10.17325 9.73791 10.18808 0.00001 0.00007
StDev
3.2642
8.39188 8.84619 0.00055
3.6122
3.3975
For SGA, a different trend is observed with the overInit/popSize ratio which was 0 while GGA1 and GGA2 had 1.5 and 1.35 respectively. GGA2 managed to overtake the SGA and GGA1 just before the last 1000 evals (see fig 28). Best Fitness
400
300 StandardGA GenderedGA(1) GenderedGA(2) 200
100
0 0
1000
2000
3000
4000
5000
Fig 27: Best Fitness Graph Rastrigin_30_5000 It does appear that the high population and large population size in GG2 enabled it to explore for good solutions and consistently improved on that. Fig 28 shows the last 2000 evals for this class of experiments. While the level of stability remains about the same, the SGA produced a more stable set of results.
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Best Fitness
23
StandardGA
20
GenderedGA(1) GenderedGA(2) 17
14
11
8
5 3000
3400
3800
4200
4600
5000
Fig 28: Last 2000 evals for Best Fitness Graph Rastrigin_30_5000
4.1.3(b) 10000 Maximum Evaluation GA - Models Parameters
StandardGA
GenderedGA(1)
GenderedGA(2) 89
popSize
2
11
overInit
14
76
0
3
4
***
mRate
0.0375
0.0286
0.0310
theta
2.1774
1.6990
0.0243
0.2276
0.2272
0.1093
tSize
maxCreep Metrics
Performance Summary
Best Fit
1.553529
1.464783
0.876532
Avg Fit
1.554799
1.468822
0.955155
Diversity
0.000007
0.000014
0.000099
StDev
1.028718
0.964526
1.668083
Consistently, GGA2 works more with large popSize and at times large overInit in Rastrigin compared to the rest. Best Fitness 400
320
240 StandardGA GenderedGA(1) GenderedGA(2)
160
80
0 0
400
800
1200
1600
2000
Fig 29: First 2000 evals for Best Fitness Graph Rastrigin_30_10000 Page 62 of 98
The high overInit/popSize ratio that SGA and GGA1 (7 and 7.6 respectively) started with may suggest why they moved faster at the beginning (see fig 29). However, at the end of the search GGA2 emerged the winner (fig 30) while GGA1 had the lowest value for the standard deviation. Convergence was faster with SGA and GGA1 which may be due to low popSize. Best Fitness(b) 20
15 StandardGA GenderedGA(1) GenderedGA(2)
10
5
0 4000
5000
6000
7000
8000
9000
10000
Fig 30: Last 6000 evals for Best Fitness Graph Rastrigin_30_10000 Diversity 0.015
0.012 StandardGA GenderedGA(1) GenderedGA(2)
0.009
0.006
0.003
0 0
2000
4000
6000
8000
10000
Fig 31: Diversity Graph Rastrigin_30_10000 GGA2 seems to have an inherent ability to manage diversity which may be as a result of the mating restrictions implemented in it.
Page 63 of 98
4.1.3(c) 15000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
GA - Models GenderedGA(1) GenderedGA(2) 11 100 57 86 3 *** 0.0373 0.0340 2.7297 2.2614 0.2093 0.2057 Performance Summary 0.136634 0.348116 0.370237 0.137399 0.348165 0.442805 0.000004 0.000000 0.000085 0.249666 0.524410 0.546763
StandardGA 6 65 5 0.0335 1.5811 0.1114
15000 maxEvals seems to generally demand an increase in popSize or overInit/popSize ratio especially from the SGA. Fig 32 shows a good performance till the 6000th evals.
Best Fitness
300
240
180 StandardGA GenderedGA(1) GenderedGA(2)
120
60
0 0
3000
6000
9000
Fig 32: Best Fitness Graph Rastrigin_30_15000
12000
15000
Best Fitness
20
16
12 StandardGA GenderedGA(1) GenderedGA(2)
8
4
0 3000
5400
7800
10200
12600
15000
Fig 33: Last 9000 evals Best Fitness Graph Rastrigin_30_15000 Page 64 of 98
Diversity 0.01
0.008
0.006 StandardGA GenderedGA(1) GenderedGA(2)
0.004
0.002
0 0
3000
6000
9000
12000
15000
Fig 34: Diversity Graph Rastrigin_30_15000
Again GGA2 demonstrated a yet-to-be equalled capacity to manage diversity. The fact that it converges successfully by the end of the search is a good case for it. SGA and GGA1 drops sharply in diversity in the first few evals. However, SGA generally produced more stable results.
Page 65 of 98
4.2 Schwefel Function The performance of the various optimisers in solving the Schwefel problem (see section 3.1.2) is presented in this section,
4.2.1 Schwefel with 10 Parameters This describes the experimental results for Schwefel on 10 parameters
4.2.1(a) 5000 Maximum Evaluation GA - Models
Parameters StandardGA
GenderedGA(1)
GenderedGA(2)
popSize overInit tSize mRate theta
2 54 5 0.1051 1.0022
10 65 2 0.1003 0.7891
50 39 *** 0.0792 1.1913
maxCreep
0.8035
0.7546
0.7999
Metrics
Performance Summary
Best Fit Avg Fit Diversity
-4174.467268 -4174.220903 0.000112
-4170.410895 -4155.057650 0.000640
-4165.346505 -4167.741212 0.002853
StDev
29.784388
36.161934
58.798802
The SGA, GGA1, and GGA2 took off with an optimal popSize of 2, 10, and 50 with overInit values of 54, 65 and 39. The optimisers all used high overInit values to select the best samples for the population. Best Fitness 0
1000
2000
3000
4000
5000
-1800
-2300 StandardGA GenderedGA(1) GenderedGA(2) -2800
-3300
-3800
-4300
Fig 35: Best fitness Graph Schwefel_10_5000
Page 66 of 98
However the SGA had a high overInit/popSize ratio (27.0) and a tSize/popSize (2.5) ratios, which means that it is expected to do some exploration at the initialisation stage and more of exploitative search, while the others were likely to do the contrary. 0
1000
2000
Diversity
3000
4000
5000
0.15
0.12
StandardGA GenderedGA(1) GenderedGA(2) 0.09
0.06
0.03
0
Fig 36: Diversity Graph Schwefel_10_5000 SGA used the highest mutation rate and creep value while GGA2 had the least overInit/popSize ratio (0.78), least tSize/popSize ratio (0.04) and minimum mutation rate. This suggests a gradual optimisation and although it might not register the top best fitness score it will achieve a very good average fitness score. It also suggests that the GGA2 will achieve top best fitness scores with more evaluations. The GGA1 was between the SGA and GGA2 in the overInit/popSize (6.5) and tSize/popSize ratios (0.5) but had the least theta and maxCreep, further suggesting that the two gender GAs require little mutation. The SGA got the top best fitness score (-4174.4673) with an almost close average fitness score (-4174.2209) suggesting the superiority in this class of experiments and the near convergence. The standard deviations on the final results for 500 trials from each of the algorithms indicate highest stability for SGA (29.7843) against GGA1 (36.1619) and GGA2 (58.7988). The GGA2 got the highest diversity as well as Standard deviation, showing poorer convergence and stability. SGA performed best in this class of experiments.
Page 67 of 98
4.2.1(b) 10000 Maximum Evaluation GA – Models
Parameters StandardGA
GenderedGA(1)
GenderedGA(2)
popSize
2
17
54
overInit
98
90
100
tSize
4
4
***
mRate
0.1157
0.0700
0.0657
theta
3.7598
2.9400
2.2715
maxCreep
0.7871
0.7679
0.7658
Metrics
Performance Summary
Best Fit
-4187.149041
-4184.548303
-4186.050036
Avg Fit
-4187.145796
-4184.054333
-4185.352226
Diversity
0.000001
0.000196
0.000223
Std Dev
17.385692
23.926168
20.242144
The SGA, GGA1, and GGA2 used with an optimal popsize of 2, 17, and 54. The overInit values of 98, 90 and 100. In this class of experiments, the optimisers also used high overInit values to select the best samples for the population, with the SGA using very high overInit/popSize (48.0) and tSize/popSize (2.0) ratios. This means that it is expected to do some exploration at the initialisation stage and more of exploitative search later in the run, while the others will do the opposite with the GGA1 using overInit/popSize ratio (5.2941) and a tSize/popSize (0.2353) ratios and the GGA2 using overInit/popSize ratio (1.8519) and a tSize/popSize (0) ratios. Best Fitness 0
2000
4000
6000
8000
10000
-1800
-2300 StandardGA GenderGA(1) GenderGA(2) -2800
-3300
-3800
-4300
Fig 37: Best Fitness Graph Schwefel_10_10000
Page 68 of 98
Furthermore, the SGA used the highest mutation rate (0.1157) and creep value (0.7871). It also suggests that the GGA2 will achieve top best fitness scores with more evaluations. Best Fitness -4160 6000
6800
7600
8400
9200
10000
-4168
StandardGA GenderGA(1) -4176 GenderGA(2)
-4184
-4192
-4200
Fig 38: Last 5000 evals Best fitness Graph Schwefel_10_10000 The SGA got the top best fitness score (-4187.1490) with an almost close average fitness score (-4187.1458) and a standard deviation of 17.3857 suggesting its superiority in this class of experiments and the near convergence. The standard deviations on the final results for 500 trials from each of the algorithms indicate highest stability for SGA (17.3857) against GGA1 (23.9262) and GGA2 (20.2421) respectively. SGA performed best in this class of experiments.
Diversity
0.15
0.12
0.09 StandardGA GenderedGA(1) GenderedGA(2) 0.06
0.03
0 0
2000
4000
6000
8000
10000
Fig 39: Diversity Graph Schwefel_10_10000
Page 69 of 98
4.2.1(c) 15000 Maximum Evaluation
Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
GA - Models GenderedGA(1) GenderedGA(2) 3 13 42 56 45 81 2 3 2 0.1572 0.1667 0.1390 10.0000 10.0000 9.9733 0.7692 0.8115 0.8138 Performance Summary -4187.332574 -4187.22304 -4187.910765 -4187.242764 -4187.222889 -4187.386596 0.000045 0.000000 0.000189 16.822558 17.390301 14.876716
StandardGA
The SGA, GGA1, and GGA2 used with an optimal popsize of 3, 13, and 42 with overInit values of 56, 45 and 81.The optimisers all used high overInit values to select the best samples for the population. The SGA maintained its low popSize but used a relatively lower overInit value compared with the previous two experiments in this problem. The optimisers made use of overInit/popSize ratios of SGA (18.6667), GGA1 (3.4615) and GGA2 (1.9286), and tSize/popSize ratios of SGA (0.6667), GGA1 (0.2308) and GGA2 (0.0476). It is unclear what these figures may suggest but an almost balanced performance is observed with the best fitness graph for all three (Fig 39). Best Fitness -2300
-2700
StandardGA -3100
GenderedGA(1) GenderedGA(1)
-3500
-3900
-4300 0
3000
6000
Fig 40: Best fitness Graph Schwefel_10_15000
9000
12000
15000
Page 70 of 98
However, GGA1 also used a fairly high mRate suggesting that it might have explored more. The GGA2 used the highest maxCreep value and the lowest theta. It may be suggested that metaGA may have recognised the need for mutation in this class of experiments and almost maximised the mutationrelated parameters.
This might suggest near convergence across all the
optimisers and therefore almost nil diversity. Diversity 0.02
0.016 StandardGA GenderedGA(1) GenderedGA(2)
0.012
0.008
0.004
0 0
3000
6000
9000
12000
15000
Fig 41: Diversity Graph Schwefel_10_15000 It is noteworthy, however, that although GGA2 is not leading in preserving diversity as much as it did with Rastrigin, it remains at the top even with Schwefel but loses its diversity almost in the same manner.
Page 71 of 98
4.2.2 Schwefel with 20 Parameters 4.2.2(a) 5000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fitness Avg Fitness Diversity StDev
StandardGA 2 41 5 0.1229 9.7803 1.0000 -7908.3994 -7901.8291 0.0012 202.2669
GA - Models GenderedGA(1) GenderedGA(2) 10 99 71 0 5 *** 0.0876 0.0443 9.9196 0.4936 0.9644 0.9428 Performance Summary -7849.1020 -8110.4711 -7840.6907 -8079.8446 0.0013 0.0029 223.9009 199.5837
The initial popSize distribution from the Meta GA – very low for SGA; moderate for GGA1 and very high for GGA2 – is not very much different from what has been observed in the previous experiments. GGA2, however, usually settles with comparatively very low theta. GGA2 can be said to have performed best in this class of experiments considering its final result, standard deviation and diversity management. Best fitness 0
1000
2000
3000
4000
5000
-2500
-3500 StandardGA GenderedGA(1) GenderedGA(2) -4500
-5500
-6500
-7500
-8500
Fig 42: Best fitness Graph Schwefel_20_5000
Page 72 of 98
4.2.2(b) 10000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fitness Avg Fitness Diversity StDev
GA - Models GenderedGA(1) GenderedGA(2) 10 93 21 52 5 *** 0.0447 0.0423 0.6782 0.7708 0.7707 0.8269 Performance Summary -8349.0728 -8346.6086 -8336.0870 -8348.7022 -8345.3465 -8322.3461 0.0000 0.0002 0.0018 37.4995 43.2916 71.7398
StandardGA 2 33 4 0.0457 0.6244 0.7711
Again the same pattern on popSize distribution from the Meta GA – very low for SGA; moderate for GGA1 and very high for GGA2. However, all the three arrived around the same average values for mRate, theta and maxCreep. SGA was best in terms of the solution (-8349.0728) and standard deviation (37.4995) followed by GGA1 and GGA2 in that order.
4.2.2(c) 15000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
StandardGA 2 96 5 0.0523 1.6040 0.7588 -8373.8351 -8373.7769 0.0000 18.4599
GA - Models GenderedGA(1) GenderedGA(2) 10 100 67 0 3 *** 0.0324 0.0378 1.2244 1.2008 0.7628 0.8094 Performance Summary -8371.4760 -8368.8910 -8370.9687 -8363.6587 0.0001 0.0007 23.2198 32.0791
Page 73 of 98
But for gain in the overInit values for SGA and GGA1 and loss for GGA, the pattern is the same as in 10000 evals. Their best fitness graphs (shown below) also look quite similar. Best Fitness -3500
-4500 StandardGA GenderedGA(1) GenderedGA(2) -5500
-6500
-7500
-8500 0
2000
4000
Fig 43: Best fitness Graph Schwefel_20_10000
6000
8000
10000
Best Fitness -3500
-4500 StandardGA GenderedGA(1) GenderedGA(2)
-5500
-6500
-7500
-8500 0
3000
6000
9000
12000
15000
Fig 43: Best fitness Graph Schwefel_20_15000
Page 74 of 98
4.2.3 Schwefel with 30 Parameters This describes the experimental results for Schwefel on 30 parameters
4.2.3(a) 5000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fitness Avg Fitness Diversity StDev
GA - Models StandardGA GenderedGA(1) GenderedGA(2) 3 10 96 75 97 20 5 5 *** 0.0746 0.0230 0.0187 8.6841 0.0958 0.0557 0.9778 1.0000 0.7700 Performance Summary -11506.1911 -11547.5555 -11674.6794 -11499.1652 -11526.9584 -11626.9025 0.0009 0.0020 0.0031 284.1659 279.8213 323.3766
Perhaps the high value of theta for SGA may have reduced the exploitative effect of the usual high overInit/popSize ratio and high tSize/popSize ratio too. SGA got the worst best fitness score, after GGA2 and GGA1, but it was the most and recorded the best convergence. Best Fitness
-4000 0
1000
2000
3000
4000
5000
-5600 StandardGA GenderedGA(2) GenderedGA(1)
-7200
-8800
-10400
-12000
Fig 44: Best fitness Graph Schwefel_30_5000
Page 75 of 98
4.2.3(b) 10000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
GA - Models GenderedGA(1) GenderedGA(2) 2 10 100 3 18 0 5 5 *** 0.0294 0.0276 0.0280 0.0839 0.0000 0.0000 0.7773 0.7645 0.7605 Performance Summary -12,398.1616 -12,365.8560 -12,390.1532 -12,395.7234 -12,359.2287 -12,367.0923 0.0002 0.0007 0.0015 95.9000 114.5794 155.9839
StandardGA
Interestingly, SGA got low overInit value here and generally higher value for the mutation parameters and also high tSize value. GGA2 insists on high popSize but their performances all closed in. SGA had the best result and best stability while others – GGA1 and GGA2 – followed in that order. GGA2’s convergence did not make it below three 0’s decimal level. Best Fitness -4500 0
2000
4000
6000
8000
10000
-6100 StandardGA GenderedGA(1) GenderedGA(2)
-7700
-9300
-10900
-12500
Fig 45: Best fitness Graph Schwefel_30_10000
Page 76 of 98
4.2.3(c) 15000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
GA - Models GenderedGA(1) GenderedGA(2) 10 100 97 56 4 *** 0.0286 0.0316 0.6050 0.5560 0.7685 0.7641 Performance Summary -12519.6088 -12514.0901 -12517.6104 -12518.8677 -12511.5344 -12504.8368 0.0001 0.0003 0.0010 48.4495 55.5475 61.6814
StandardGA 3 63 4 0.0321 0.6941 0.7813
These optimal parameters set distribution i.e. low popSize for high overInitpopSize ratio for SGA, moderate size for GGA1 and very high popSize for GGA2 has been recurrent. The mutation parameters’ ranges in this experiment are quite close as well. It is not clear yet what influences the value of theta which had varied widely especially for SGA. SGA was best – result, stability and convergence - because it would have started very well high overInit and it generally does well with high maxEvals. GGA2 got the next best result but did not get a stable set of results and also did not converge as well as others. Although considering the population it worked with, it was not a very bad performance. Best Fitness -5000 0
3000
6000
9000
12000
15000
-6600 StandardGA GenderedGA(1) GenderedGA(2)
-8200
-9800
-11400
-13000
Fig 46: Best fitness Graph Schwefel_30_15000 Page 77 of 98
4.3 F8F2 Function The section will highlight key varying issues with the performance of the GA models on the F8F2 problem (see section 3.1.3).
4.3.1 F8F2 with 10 Parameters This describes the experimental results for F8F2 on 10 parameters
4.3.1(a) 5000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
StandardGA 15 57 2 0.4066 1.8893 0.0373 0.0954 0.1493 0.0031 0.1117
GA - Models GenderedGA(1) GenderedGA(2) 27 96 82 14 2 *** 0.4826 0.3879 3.9547 3.2717 0.0485 0.0788 Performance Summary 0.0909 0.2027 0.1734 0.3106 0.0037 0.0025 0.1028 0.1601
One remarkable difference may be the data here is the relatively high popSize for SGA and GGA1 whose overInit/popSize ratio were equally high (3.8000 and 3.0370). The GGA2 kept its popSize but its overInit/popSize ratio (0.1458) may have affected the starting average quality and by extension the overall performance which was worse almost across board. GGA1 performed best in this experiment in terms of solution quality, stability but not in convergence. GGA2 lost convergence faster than others. Best Fitness
4
3.2 StandardGA GenderedGA(1) GenderedGA(1)
2.4
1.6
0.8
0 0
1000
2000
Fig 47: Best fitness Graph F8F2_10_5000
3000
4000
5000
Page 78 of 98
4.3.1(b) 10000 Maximum Evaluation Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
StandardGA 40 63 3 0.3574 0.6687 0.0272 0.0192 0.0413 0.0021 0.0419
GA - Models GenderedGA(1) GenderedGA(2) 55 11 20 67 3 *** 0.4327 0.4153 2.6686 3.1860 0.0365 0.0774 Performance Summary 0.0178 0.1453 0.0534 0.1952 0.0027 0.0035 0.0443 0.1489
Surprisingly, the trend in the distribution of optimal parameters seems to have reversed. GGA2 has low popSize and high overInit/popSize ratio (6.0909) while both SGA and GGA1 have high popSize and overInit/popSize ratios of 1.575 and 0.3636 respectively. GGA2 performed badly all through while GGA1 made best results but worse than SGA in stability. The graphs below show that GGA2 actually dropped into a local minimum and evidently got stagnated there. Best Fitness 20
16 StandardGA GenderedGA(1) GenderedGA(2)
12
8
4
0 1
0
2000
4000
Best Fitness
6000
8000
10000
0.8 StandardGA GenderedGA(1) GenderedGA(2)
0.6
0.4
0.2
0 0
2000
4000
6000
8000
10000
Fig 48: Best fitness Graph and zoom on fitness < 1F8F2_10_10000 Page 79 of 98
4.3.1(c) 15000 Maximum Evaluation
Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
StandardGA 70 88 2 0.2730 0.7499 0.0377 0.0050 0.0106 0.0012 0.0185
GA - Models GenderedGA(1) GenderedGA(2) 67 47 99 90 1 *** 0.2912 0.3147 0.9847 0.0108 0.0326 0.0609 Performance Summary 0.0061 0.0905 0.0142 0.1014 0.0013 0.0005 0.0195 0.0906
SGA has a surprisingly high popSize and all three had overInit values to start with. GGA2 has the lowest popSize a very low theta too. SGA’s results was the best across board followed quite closely by GGA1 while GGA2 performed poorly in all cases based on the data shown above. Best Fitness 4
3.2 StandardGA GenderedGA(1) GenderedGA(2) 2.4
1.6
0.8
0 0.8
0
3000
6000
Best Fitness
9000
12000
15000
0.64 StandardGA GenderedGA(1) GenderedGA(2) 0.48
0.32
0.16
0 0
3000
6000
9000
12000
15000
Fig 49: Best fitness Graph and zoom on fitness < 1 F8F2_10_15000
Page 80 of 98
4.3.2 F8F2 with 20 Parameters This describes the experimental results for F8F2 on 20 parameters
4.3.2(a) 5000 Maximum Evaluations Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
StandardGA 3 57 3 0.2268 5.1587 0.0154 0.0573 0.0589 0.0004 0.0594
GA - Models GenderedGA(1) GenderedGA(2) 10 15 27 100 5 *** 0.1950 0.1620 7.1020 6.8161 0.0361 0.0275 Performance Summary 0.0665 0.0812 0.0675 0.0816 0.0002 0.0001 0.0594 0.1252
In this experiment, all three models started with a relatively lower popSize although they all have above unit overInit/popSize ratio i.e. 19.0, 2.7 and 6.667 for SGA, GGA1 and GGA2 respectively. SGA maintained the lead in best solution but had the same stability with GGA1. Best Fitness 320
256
192 StandardGA GenderedGA(1) GenderedGA(2)
128
64
0 0.2
0
600
1200
Best Fitness 1800
2400
3000
0.16 StandardGA GenderedGA(1) GenderedGA(2)
0.12
0.08
0.04
0 4000
4200
4400
4600
4800
5000
Fig 50: Best fitness Graph; zoom on fitness < 0.2 and last 1000 evals F8F2_20_5000 Page 81 of 98
4.3.2(b) 10000 Maximum Evaluations Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
StandardGA 45 2 5 0.3260 9.4969 0.0147 0.0346 0.0357 0.0004 0.0372
GA - Models GenderedGA(1) GenderedGA(2) 23 24 34 87 3 *** 0.2136 0.4280 6.5940 10.0000 0.0097 0.0122 Performance Summary 0.0312 0.0477 0.0317 0.0478 0.0003 0.0001 0.0581 0.1205
SGA started with a population that is likely to have a poor average fitness due to low overInit value on high popSize. The GGAs had above unit overInit/popSize ratio and therefore led in average fitness at the beginning. The mutation parameters seem equally distributed across the models with generally high theta and low maxCreep. GGA1 had the best result but was less stable than SGA. Evidently, from the graph below, GGA2 got stuck in the first approximately 4000 evals and was overtaken by SGA and GGA1 Best Fitness 400
320 StandardGA GenderedGA(1) GenderedGA(2)
240
160
80
0 0.5
0
2000
4000
Best Fitness 6000
8000
10000
0.4 StandardGA GenderedGA(1) GenderedGA(2)
0.3
0.2
0.1
0 4000
5200
6400
7600
8800
10000
Fig 51: Best fitness Graph; zoom on fitness < 0.5 and last 6000 evals F8F2_20_10000 Page 82 of 98
4.3.2(c) 15000 Maximum Evaluations GA - Models
Parameters StandardGA 90 64 3 0.3108 1.1898
GenderedGA(1) 44 82 3 0.0722 8.6508
GenderedGA(2) 28 48 *** 0.5000 8.1551
maxCreep
0.0150
0.0352
0.0120
Metrics Best Fit Avg Fit Diversity
0.0249 0.0539 0.0026
StDev
0.0336
popSize overInit tSize mRate theta
Performance Summary 0.0265 0.0269 0.0001
0.0428 0.0429 0.0002
0.0329
0.0921
The pattern that seems to emerge with the SGA optimal popSize is that higher maxEvals require high popSize. Again, the starting average fitness for SGA is expected to be worse than those of GGA1 and GGA2 due to its low overInit/popSize ratio. SGA got the best result but lost in stability and convergence to GGA1 whose result was next best. GGA2, up to this point has not done well with F8F2. The diversity graph below shows how poorly GGA2 manages diversity in this function.
Average Fitness
2000
1600
1200 StandardGA GenderedGA(1) GenderedGA(2)
800
400
0 0
300
600
Diversity
900
1200
1500
0.25
0.2
0.15 StandardGA GenderedGA(1) GenderedGA(2)
0.1
0.05
0 0
3000
6000
9000
12000
15000
Fig 52: Average fitness & Diversity Graph F8F2_20_15000 Page 83 of 98
4.3.3 F8F2 with 30 Parameters This describes the experimental results for F8F2 on 30 parameters
4.3.3(a) 5000 Maximum Evaluations Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
StandardGA 2 95 5 0.5000 5.9266 0.0146 0.0919 0.1031 0.0012 0.0681
GA - Models GenderedGA(1) GenderedGA(2) 10 10 58 26 5 *** 0.1840 0.3687 6.7089 9.9246 0.0377 0.0229 Performance Summary 0.1313 0.0761 0.1394 0.0813 0.0006 0.0006 0.0858 0.0962
In this class of experiments, SGA began with very low popSize and a high overInit/popSize ratio (48.5) against 5.8 and 2.6 for GGA1 and GGA2. So SGA is expected to have explored well at the initialisation stage in order to take off well but the effect of the high tSize/popSize ratio (2.50) may drag it towards exploitation and so hamper its performance. GGA2 improved in this experiment with the best result and good diversity management but was the least stable.
Best Fitness
400
Best Fitness
4
3.2
StandardGA
320
GenderedGA(1)
2.4
GenderedGA(2)
240
StandardGA
1.6
GenderedGA(1) GenderedGA(2)
0.8
160
0 2000
2600
3200
3800
4400
5000
80
0 0
1000
2000
Diversity
3000
4000
5000
0.01
0.008
0.006 StandardGA GenderedGA(1) GenderedGA(2)
0.004
0.002
0 0
1000
2000
3000
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4.3.3(b) 10000 Maximum Evaluations Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
StandardGA 2 67 5 0.3465 7.1159 0.0085 0.0211 0.0213 0.0003 0.0273
GA - Models GenderedGA(1) GenderedGA(2) 10 51 74 91 4 *** 0.1273 0.4715 6.7983 8.5587 0.0294 0.0058 Performance Summary 0.0276 0.0150 0.0277 0.0154 0.0001 0.0003 0.0324 0.0220
In this class of experiments, GGA2 showed a good performance in the result, diversity and stability. SGA and GGA1 seem to be moving together. GGA1 has a relatively lower mRate and theta but its maxCreep (0.0294) was much higher than those of SGA (0.0085) and GGA2 (0.0220). From the graphs, diversity in SGA and GGA2 dropped very sharply at the beginning of the search but reduced gradually thereafter. GGA1 lost its diversity slower but converged more eventually.
Best Fitness Best Fitness
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4.3.3(c) 15000 Maximum Evaluations
Parameters popSize overInit tSize mRate theta maxCreep Metrics Best Fit Avg Fit Diversity StDev
GA - Models GenderedGA(1) GenderedGA(2) 13 44 97 63 5 *** 0.3087 0.3732 7.7685 6.3877 0.0113 0.0029 Performance Summary 0.0185 0.0189 0.0167 0.0187 0.0190 0.0172 0.0002 0.0002 0.0002 0.0241 0.0245 0.0658
StandardGA 30 27 5 0.1575 7.3766 0.0093
Again GGA1 shows that it requires higher maxCreep value (0.0113) than others – 0.0093 and 0.0029 for SGA and GGA2 respectively. This will hopefully balance the effect of high overInit/popSize ratio (7.4615) and also high tSize. The overInit/popSize ratios in SGA (0.9) and GGA2 (1.4318) were comparatively much lower than that of GGA1. This may explain why GGA1 made an early progress in the search before yielding to the effects of the mutation operators. At the expense of stability, GGA2 has consistently got the best solution in the 30-parameter F8F2 series. GGA2 has poor diversity management for its population to drop to a diversity of 0.0008 in the first 600 evals (see fig 55 below).
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Fig 55: Average fitness & Diversity Graph F8F2_30_15000. (Inset: last 2000 evals. of Best fitness)
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4.4 Robustness In the analysis presented above, different GA models got the best result at different points and under different circumstances. In this work, the robustness of a GA is seen as a measure of how well it performs across a range of problems or metrics. In this section, the results obtained by all the GA models on all the three functions are presented and analysed. On the tables, the best scores are printed in bold blue while the worst scores are in red italics. The summary of points is got by assigning the points 0, 0.5 and 1 to the worst, middle and best scores respectively and adding up the values in each column.
4.4.1 Rastrigin With the Rastrigin function, GGA2 made the overall best performance on the best fitness and average fitness scores. The table shows that SGA converged most and gave the most stable results. The diversity measure only shows the last values obtained. It does not really indicate good diversity management because very low diversity with poor fitness suggests premature convergence. The graphs indicate that GGA2 performance on preserving diversity was desirable.
4.4.2 Schwefel SGA performed best on the Schwefel function. GGA1 and GG2 dropped in rating from the Rastrigin results. Convergence, as well as stability, by GGA2 worsened comparatively but SGA’s stability was maintained.
4.4.3 F8F2 The SGA also took the lead in solving F8F2 problem. The performance of GGA2 seemed worsened with lower parameter but it took over the parameter30 experiments with this function suggesting that it might be better with higher dimensionality for F8F2. Also the convergence (diversity) score rose because it appears to lose diversity much more rapidly with lower parameters. In general, the result differences in most of the solutions are quite insignificant suggesting that they all arriving at about the same solution.
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5: CONCLUSION AND FUTURE RESEARCH DIRECTIONS This dissertation has presented the results of the study of the effect of gender restricted strategies on GAs that solve numerical optimisation problems. In this concluding section, the overview of the research motivations and aims; methodology and findings will be presented. The limitations of the study as well as future research directions are also discussed. 5.1 Overview of the research aims The motivation for this work was drawn from the need to explore the possibilities of adopting a key concept in natural evolution and population genetics, i.e. gender, in order to improve the performance of a typical GA. Gender is a major concept in biogenetics and natural evolution upon which evolutionary algorithms are modelled. In nature, gender separation play a useful role in not only preserving diversity but also in encouraging specialisation among the main gender groups [San031]. However, a review of the works in EC indicates that only a few researchers have studied this concept. The hypothesis held in this work is that gendered GAs will perform better because they improve diversity, which enables them to explore the search space better without getting trapped in local minimum. Although most of the researcher that have studied gendered GAs reported successful performance of their models, the class and the nature of problems they have been tried on are quite limited. Most of them have been on multiobjective optimisation [All92], [Lis96] and some others on combinatorial optimisation [Goh03]. Also, there has not been a standardized approach to such designs and so different models have been introduced. This study originally aims at studying the application of gendered strategies for GAs with numerical optimisation problems. This will be a contribution to the class of problems currently being solved with such algorithms. Also, other designs of the gendered strategy implemented GA versions with the canonical
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GA feature such as binary encoding [Lis96] or generational replacement [Goh03]. This work implemented a more recent variety of GAs with real parameter encoding, BLX-0.5, creep mutation, and steady state replacement. The reason for using GAs with these improved features is to know if gender will still have any impact on the reported improvement they already have. 5.2 The experiments conducted Several experiments were conducted in this research work in investigating the effect of gender on GAs. Three models of the GA were used: the standard GA (SGA), genderedGA1 (GGA1), genderedGA2 (GGA2). The investigations were carried out on the performance of these GA models in solving the classical optimisation function; Rastrigin, Schwefel and F8F2 (section 3.1). Each of the GA models solved each of the problems for nine times with different run lengths (5000, 10000, 15000) and a different dimensionality (10, 20, 30) giving a total of 81 GA runs and 27 comparisons to make on the results from each GA. The comparisons involved the analysis of the average results by each model on any class of the problems over 500 trials. The metrics: best fitness, average fitness, diversity and standard deviation were used in the analysis of the performance of each GA model. The parameters used by the GA models were optimised with a meta GA that had a run length of 1000 for 50 trials. The GA parameters optimised with the meta GA are: population size, over initialization, tournament size, mutation rate, theta and maximum creep. 5.3 Research findings The findings from this work indicate that a gendered approach to selection in a GA has an impact on its performance. The different models were observed to have performed differently on various classes of problems. Results, for instance, show that GGA2 had better results on Rastrigin compared to others while SGA did better with Schwefel (see the General Performance Summary table). GGA1 Page 89 of 98
was just dragging on but made a couple of good results too. Also, it was observed that diversity was superbly managed by GGA2 in those functions. SGA needed very low population size in solving Rastrigin and Schwefel but high in F8F2. For GGA2 the reverse was the case – high population size in Rastrigin and Schwefel but low with F8F2. GGA1 did not maintain a clear pattern in that regard. Other optimised parameter did not actually show any clear behaviour across board. However, with F8F2, a different trend was observed. GGA1 and SGA did better than GGA2 at lower dimensionality. Surprisingly, diversity management with GGA2 was so bad that it lost its diversity in the first few hundred evals. But for high dimensionality, GGA2 picked up and defeated the others. This agrees with the No free lunch theorem for search by Wolpert and Macready [Wol97] which basically states that no one algorithm performs better than all others across different classes of problems. 5.4 Limitations of this study A lot of time and effort have been spent on this project in other to get it to this level but a lot more could still be done because the concept of gender modelling is indeed wide and complex. There was not enough literature on the topic as it is relatively new. There was also no standard way of modelling the gendered strategy and so different models were attempted and dropped. As a student project, only a limited amount of time was available for it. Besides, a lot of time was lost in framing the research question and scoping the project to fall within the expectations of the assessors. All these pose major obstacles to making a more explorative and in-depth study in this area. Also time was lost to technical hitches encountered during the running of the experiments. There was the problem of “insufficient memory” on the student’s profile login used for saving data got from this study for which a lot of time and data were wasted.
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Because of the above, this work cannot claim to have exhaustively dealt with all the issue in this research. For instance, the meta GA’s run length (1000) and number of trials (50) appears small and may have caused sampling errors. Also a significance test on the data using the T-Test ought to have been used to test for the significance of some of the data as well as the significance to other sets. This could not be achieved due to lack of time and limited space reporting. 5.5 Future research direction This work did establish that gendered strategies work in GAs. However, since they were modelled on top of an existing GA, it may be good to use them without some of the operators in standard GAs. This will hopefully help to distil out the actual effect of the gendered strategy from the noise that come from other operators. Future research directions may involve implementing gendered GA without mutation implementing gendered GA without recombination implementing gendered GA with varying demarcation point implementing hybrid gendered GA and standard GA Also, the standardisation of this approach and the derivation of some scientific theorem to back it up will be necessary for future research in this area.
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