Confidence Based Robust Algorithm.pdf

Confidence-based Robust Optimisation of Engineering Design Problems

Seyedali Mirjalili MS,BS

School of Information and Communication Technology Faculty of Engineering and Information Technology Griffith University

Submitted in fulfilment of the requirements of the degree of Doctor of Philosophy

November 2015

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Abstract Robust optimisation refers to the process of combining good performance with low sensitivity to possible perturbations. Due to the presence of different uncertainties when optimising real problems, failure to employ robust optimisation techniques may result in finding unreliable solutions. Robust optimisation techniques play key roles in finding reliable solutions when considering possible uncertainties during optimisation. Evolutionary optimisation algorithms have become very popular for solving real problems in science and industry mainly due to simplicity, gradient-free mechanism, and flexibility. Such techniques have been employed widely as very reliable alternatives to mathematical optimisation approaches for tackling difficulties of real search spaces such as constraints, local optima, multiple objectives, and uncertainties. Despite the advances in considering the first three difficulties in the literature, there is significant room for further improvements in the area of robust optimisation, especially combined with multi-objective approaches. Finding optimal solutions that are less sensitive to perturbations requires a highly systematic robust optimisation algorithm design process. This includes designing challenging robust test problems to compare algorithms, performance metrics to measure by how much one robust algorithm is better than another, and computationally cheap robust algorithms to find robust solutions for optimisation problems. The first two phases of a systematic algorithm design process, developing test functions and performance metrics, are prerequisite to the third phase, algorithm development. Firstly, this thesis identifies the current gaps in the literature relating to each of these phases to establish a systematic robust algorithm design process as follows: • The need for more standard and challenging robust test functions for both single- and multi-objective algorithms. • The need for more standard performance metrics for quantifying the performance of robust multi-objective algorithms. • The need for more investigation and analysis of the current robustness metrics. • High computational cost of the current robust optimisation techniques that rely on additional function evaluations. • Low reliability of the current robust optimisation techniques that rely on the search history (sampled points during optimisation). Secondly, the current robustness metrics are investigated and analysed in details. Thirdly, several test functions and performance metrics are proposed to fill

ii out the first two above-mentioned gaps in the literature. Fourthly, a novel metric called the confidence measure is proposed to reduce the computational cost and increase the reliability of the current robust optimisation methods. Lastly but most importantly, the proposed confidence metric is employed to establish novel and cheap approaches for finding robust optimal solutions in single- and multi-objective search spaces called confidence-based robust optimisation and confidence-based robust multi-objective optimisation. The most well-regarded evolutionary population-based algorithms such as Genetic Algorithm (GA), Particle Swarm Optimisation (PSO), and Multi-Objective Particle Swarm Optimisation (MOPSO) are modified as the first confidence-based robust optimisation algorithms. Several experiments are conducted using the proposed benchmark problems and performance metrics to evaluate the proposed confidence-based robust algorithms qualitatively and quantitatively. The thesis also considers the application of the proposed techniques in designing a marine propeller problem to emphasise the applicability of the confidence-based optimisation in practice. The results show that the proposed confidence-based algorithms mainly benefit from high reliability and low computational cost when solving the benchmark problems. The merits of the proposed benchmark problems and performance metrics in comparing different algorithms are evidenced by the results of the test beds as well. The results of real applications demonstrate that the proposed method is able to confidently and reliably find robust optimal solutions without significant extra computational burden for real problems with unknown search spaces.

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Certificate of Originality

This work has not previously been submitted for a degree or diploma in any university. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made in the thesis itself.

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Approval Name: Title: Submission Date:

Seyedali Mirjalili Confidence-based Robust Optimisation of Engineering Design Problems 7 November, 2015

Supervisor:

Dr Andrew Lewis Griffith University Australia

Co-supervisor:

Dr Ren´e Hexel Griffith University Australia

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Acknowledgements Commencing, pursuing and completing this dissertation, like any other project, required an abundance of resources as well as strong motivation which would not have been possible without guidance and support of a group of people. Therefore, I would like to express my gratitude to the people below. I should first like to thank my principal supervisor and mentor, Dr. Andrew Lewis whose advice, guidance, patience and support has been always available for me throughout my academic journey in my Ph. D. candidature. I should also like to thank Dr. Ren´e Hexel for his generous support and guidance as my associate supervisor and Dr. Seyed Ali Mohammad Mirjalili for his invaluable advice.

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To my mother and father

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List of Outcomes Arising from this Thesis From chapter 4: 1. S. Mirjalili, A. Lewis, Novel Frameworks for Creating Robust Multiobjective Benchmark Problems, Information Sciences 300 (2015): 158-192. http://dx.doi.org/10.1016/j.ins.2014.12.037 2. S. Mirjalili, A. Lewis, Obstacles and Difficulties for Robust Benchmark problems: A Novel Penalty-based Robust Optimization Method, Information Sciences 328 (2016): 485-509. http://dx.doi.org/10.1016/j.ins.2015.08.041 3. S. Mirjalili, A. Lewis, Hindrances for Robust Multi-objective Test Problems, Applied Soft Computing 35 (2015): 333-348. http://dx.doi.org/10.1016/j.asoc.2015.05.037 4. S. Mirjalili, Shifted Robust Multi-objective Test Problems, Structural and Multidisciplinary Optimization 52 (2015): 217-226. http://dx.doi.org/10.1007/s00158-014-1221-9 From Chapter 5: 5. S. Mirjalili, A. Lewis, Novel Performance Metrics for Robust Multiobjective Optimization Algorithms, Swarm and Evolutionary Computation 21 (2015): 1-23. http://dx.doi.org/10.1016/j.swevo.2014.10.005 From Chapters 6 and 7: 6. S. Mirjalili, A. Lewis, and S. Mostaghim. Confidence measure: A novel metric for robust meta-heuristic optimisation algorithms. Information Sciences 317 (2015): 114-142. http://dx.doi.org/10.1016/j.ins.2015.04.010 7. S. Mirjalili, A. Lewis, A Reliable and Computationally Cheap Approach for Finding Robust Optimal Solutions, GECCO 2015 : 1439-1440. http://dx.doi.org/10.1145/2739482.2764640 From Chapter 9: 8. S. Mirjalili, A. Lewis, Multi-objective Optimization of Marine Propellers, Procedia Computer Science 51 (2015): 2247-2256. http://dx.doi.org/10.1016/j.procs.2015.05.504

viii 9. S. Mirjalili, T. Rawlings, J. Hettenhausen, A, Lewis, A comparison of multi-objective optimisation metaheuristics on the 2D airfoil design problem, ANZIAM journal, 2013 Volume 54, Pages C345 - C360. http://dx.doi.org/10.0000/anziamj.v54i0.6154 Copyrith and permission notice: Reprinted from Information Sciences, Vol. 317, Seyedali Mirjalili, Andrew Lewis, and Sanaz Mostaghim, Confidence measure: A novel metric for robust metac 2015, with heuristic optimisation algorithms, Pages No. 114-142, Copyright permission from Elsevier. Reprinted from Information Sciences, Vol. 300, Seyedali Mirjalili and Andrew Lewis, Novel frameworks for creating robust multi-objective benchmark probc 2015, with permission from Elsevier. lems, Pages No. 158-192, Copyright Reprinted from Information Sciences, Vol. 328, Seyedali Mirjalili and Andrew Lewis, Obstacles and difficulties for robust benchmark problems: A novel penaltyc 2016, with based robust optimisation method, Pages No. 485-509, Copyright permission from Elsevier. Reprinted from Applied Soft Computing, Vol. 35, Seyedali Mirjalili and Andrew Lewis, Hindrances for robust multi-objective test problems, Pages No. 333-348, c 2015, with permission from Elsevier. Copyright Reprinted from Swarm and Evolutionary Computation, Vol. 21, Seyedali Mirjalili and Andrew Lewis, Novel performance metrics for robust multi-objective c 2015, with permission optimization algorithms, Pages No. 1-23, Copyright from Elsevier. Structural and Multidisciplinary Optimization, Shifted robust multi-objective c 2015 test problems, Vol. 52, Pages No. 217-226, Seyedali Mirjalili, Copyright Springer-Verlag Berlin Heidelberg, With permission of Springer.

Contents 1 Introduction 1.1 Problem Background . . . . . . . . 1.2 Problem Statement and Objectives 1.3 Scope and Significance . . . . . . . 1.4 Organisation of the thesis . . . . .

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2 Related work 2.1 Evolutionary single-objective optimisation . . . . . . . . . . . . . 2.2 Evolutionary Multi-objective optimisation . . . . . . . . . . . . . 2.3 Robust single-objective optimisation . . . . . . . . . . . . . . . . 2.3.1 Preliminaries and definitions . . . . . . . . . . . . . . . . . 2.3.2 Expectation measure . . . . . . . . . . . . . . . . . . . . . 2.3.3 Variance measure . . . . . . . . . . . . . . . . . . . . . . . 2.4 Robust multi-objective optimisation . . . . . . . . . . . . . . . . . 2.4.1 Preliminaries and definitions . . . . . . . . . . . . . . . . . 2.4.2 Current expectation and variance measures . . . . . . . . . 2.5 Benchmark problems . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Benchmark problems for single-objective robust optimisation 2.5.2 Benchmark problems for multi-objective robust optimisation 2.6 Performance metrics . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Convergence performance indicators: . . . . . . . . . . . . 2.6.1.1 Generational Distance (GD): . . . . . . . . . . . 2.6.1.2 Inverted Generational Distance (IGD): . . . . . . 2.6.1.3 Delta Measure: . . . . . . . . . . . . . . . . . . . 2.6.1.4 Hypervolume metric: . . . . . . . . . . . . . . . . 2.6.1.5 Inverse hypervolume metric: . . . . . . . . . . . . 2.6.2 Coverage performance indicators: . . . . . . . . . . . . . . 2.6.2.1 Spacing (SP): . . . . . . . . . . . . . . . . . . . 2.6.2.2 Radial coverage metric: . . . . . . . . . . . . . . 2.6.2.3 Maximum Spread (M ): . . . . . . . . . . . . . . 2.6.3 Success performance indicators: . . . . . . . . . . . . . . . 2.6.3.1 Error Ratio (ER): . . . . . . . . . . . . . . . . . 2.6.3.2 Success counting (SCC): . . . . . . . . . . . . . . ix

1 4 6 7 8 11 13 21 29 30 33 35 38 38 44 50 54 56 60 62 62 62 63 63 63 63 63 64 64 65 65 65

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CONTENTS 2.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3 Analysis 3.1 Benchmark problems . . . . . . . . . . . 3.2 Performance metrics . . . . . . . . . . . 3.3 Robust algorithms . . . . . . . . . . . . 3.4 Systematic robust optimisation algorithm 3.5 Objectives and plan . . . . . . . . . . . . 3.6 Contributions and scope . . . . . . . . . 3.7 Significance of Study . . . . . . . . . . . 3.8 Summary . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . design . . . . . . . . . . . . . . . .

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4 Benchmark problems 4.1 Benchmarks for robust single-objective optimisation . . . . . . . 4.1.1 Framework I . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Framework II . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Framework III . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Obstacles and difficulties for single-objective robust benchmark problems . . . . . . . . . . . . . . . . . . . . . . . 4.1.4.1 Desired number of variables . . . . . . . . . . . 4.1.4.2 Biased search space . . . . . . . . . . . . . . . . 4.1.4.3 Deceptive search space . . . . . . . . . . . . . . 4.1.4.4 Multi-modal search space . . . . . . . . . . . . 4.1.4.5 Flat search space . . . . . . . . . . . . . . . . . 4.2 Benchmarks for robust multi-objective optimisation . . . . . . . 4.2.1 Framework 1 . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Framework 2 . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Framework 3 . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Hindrances for robust multi-objective test problems . . . 4.2.4.1 Biased search space . . . . . . . . . . . . . . . . 4.2.4.2 Deceptive search space . . . . . . . . . . . . . . 4.2.4.3 Multi-modal search space . . . . . . . . . . . . 4.2.4.4 Flat (non-improving) search space . . . . . . . 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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93 93 95 96 97 99 101 102 106 109 111 112 113 115 118 119

5 Performance measures 5.1 Robust coverage measure (Φ) . . . . . . . . . . . . . . . . . . . 5.2 Robust success ratio (Γ) . . . . . . . . . . . . . . . . . . . . . . 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Improving robust optimisation techniques 136 6.1 Confidence measure . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Confidence-based robust optimisation . . . . . . . . . . . . . . . . 140

CONTENTS

6.3

6.4

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6.2.1 Confidence-based relational operators . . . . . . . 6.2.2 Confidence-based Particle Swarm Optimisation . 6.2.3 Confidence-based Robust Genetic Algorithms . . Confidence-based robust multi-objective optimisation . . 6.3.1 Confidence-based Pareto optimality . . . . . . . . 6.3.2 Confidence-based Robust Multi-Objective Particle Optimisation . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Confidence-based robust optimisation 7.1 Behaviour of CRPSO on benchmark problems . . 7.2 Comparative Results for Confidence-based Robust 7.3 Comparative Results for Confidence-based Robust 7.4 Comparison of CRPSO and CRGA . . . . . . . . 7.5 Summary . . . . . . . . . . . . . . . . . . . . . .

. . . PSO GA . . . . . .

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8 Confidence-based robust multi-objective optimisation 168 8.1 Behaviour of CRMOPSO on benchmark problems . . . . . . . . . 168 8.2 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9 Real world applications 9.1 Marine Propeller design and related works . . . . . . . . . . 9.1.1 Propeller design . . . . . . . . . . . . . . . . . . . . . 9.1.2 Related work . . . . . . . . . . . . . . . . . . . . . . 9.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . 9.2.1 Approximating the Pareto Front using the algorithm 9.2.2 Number of blades . . . . . . . . . . . . . . . . . . . . 9.2.3 Revolutions Per Minute (RPM) . . . . . . . . . . . . 9.2.4 Post analysis of the results . . . . . . . . . . . . . . . 9.2.5 Effects of uncertainties in operating conditions on the jectives . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.6 Uncertainties in the structural parameters . . . . . . 9.3 Confidence-based Robust optimisation of marine propellers . 9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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185 . 185 . 185 . 188 . 189 . 190 . 191 . 191 . 194 . . . .

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10 Conclusion 202 10.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . 202 10.2 Achievements and significance . . . . . . . . . . . . . . . . . . . . 207 10.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Index

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Bibliography

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Appendices

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A Single-objective robust test functions A.1 Test functions in the literature . . . . . . . . . . . . . A.1.1 TP1 . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 TP2 . . . . . . . . . . . . . . . . . . . . . . . . A.1.3 TP3 . . . . . . . . . . . . . . . . . . . . . . . . A.1.4 TP4 . . . . . . . . . . . . . . . . . . . . . . . . A.1.5 TP5 . . . . . . . . . . . . . . . . . . . . . . . . A.1.6 TP6 . . . . . . . . . . . . . . . . . . . . . . . . A.1.7 TP7 . . . . . . . . . . . . . . . . . . . . . . . . A.1.8 TP8 . . . . . . . . . . . . . . . . . . . . . . . . A.1.9 TP9 . . . . . . . . . . . . . . . . . . . . . . . . A.2 Test functions generated by the proposed framework I . A.2.1 TP10 . . . . . . . . . . . . . . . . . . . . . . . . A.3 Proposed biased test functions . . . . . . . . . . . . . A.3.1 TP11 - biased 1 . . . . . . . . . . . . . . . . . . A.3.2 TP12 - biased 2 . . . . . . . . . . . . . . . . . . A.4 Proposed deceptive test functions . . . . . . . . . . . . A.4.1 TP13 . . . . . . . . . . . . . . . . . . . . . . . . A.4.2 TP14 . . . . . . . . . . . . . . . . . . . . . . . . A.4.3 TP15 . . . . . . . . . . . . . . . . . . . . . . . . A.5 Proposed multi-modal robust test functions . . . . . . A.5.1 TP16 . . . . . . . . . . . . . . . . . . . . . . . . A.5.2 TP17 . . . . . . . . . . . . . . . . . . . . . . . . A.6 Proposed flat robust test function . . . . . . . . . . . . A.6.1 TP18 . . . . . . . . . . . . . . . . . . . . . . . . A.7 Test functions generated by the proposed frameworks II A.7.1 TP19 . . . . . . . . . . . . . . . . . . . . . . . . A.7.2 TP20 . . . . . . . . . . . . . . . . . . . . . . . .

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B Multi-objective robust test functions B.1 Deb’s test functions . . . . . . . . . . B.1.1 RMTP1 . . . . . . . . . . . . B.1.2 RMTP2 . . . . . . . . . . . . B.1.3 RMTP3 . . . . . . . . . . . . B.1.4 RMTP4 . . . . . . . . . . . . B.1.5 RMTP5 . . . . . . . . . . . . B.2 Gaspar Cunha’s functions . . . . . . B.2.1 RMTP6 . . . . . . . . . . . . B.2.2 RMTP7 . . . . . . . . . . . . B.2.3 RMTP8 . . . . . . . . . . . . B.2.4 RMTP9 . . . . . . . . . . . .

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B.2.5 RMTP10 . . . . . . . . . . . . . . . . . . . . . Test functions generated by the proposed frameworks 1, B.3.1 RMTP11 . . . . . . . . . . . . . . . . . . . . . B.3.2 RMTP12 . . . . . . . . . . . . . . . . . . . . . B.3.3 RMTP13 . . . . . . . . . . . . . . . . . . . . . B.3.4 RMTP14 . . . . . . . . . . . . . . . . . . . . . B.3.5 RMTP15 . . . . . . . . . . . . . . . . . . . . . B.3.6 RMTP16 . . . . . . . . . . . . . . . . . . . . . B.3.7 RMTP17 . . . . . . . . . . . . . . . . . . . . . B.3.8 RMTP18 . . . . . . . . . . . . . . . . . . . . . B.3.9 RMTP19 . . . . . . . . . . . . . . . . . . . . . B.3.10 RMTP20 . . . . . . . . . . . . . . . . . . . . . B.3.11 RMTP21 . . . . . . . . . . . . . . . . . . . . . B.3.12 RMTP22 . . . . . . . . . . . . . . . . . . . . . B.3.13 RMTP23 . . . . . . . . . . . . . . . . . . . . . B.3.14 RMTP24 . . . . . . . . . . . . . . . . . . . . . B.3.15 RMTP25 . . . . . . . . . . . . . . . . . . . . . Extended version of current test functions . . . . . . . B.4.1 RMTP26 . . . . . . . . . . . . . . . . . . . . . B.4.2 RMTP27 . . . . . . . . . . . . . . . . . . . . . B.4.3 RMTP28 . . . . . . . . . . . . . . . . . . . . . B.4.4 RMTP29 . . . . . . . . . . . . . . . . . . . . . B.4.5 RMTP30 . . . . . . . . . . . . . . . . . . . . . B.4.6 RMTP31 . . . . . . . . . . . . . . . . . . . . . B.4.7 RMTP32 . . . . . . . . . . . . . . . . . . . . . B.4.8 RMTP33 . . . . . . . . . . . . . . . . . . . . . B.4.9 RMTP34 . . . . . . . . . . . . . . . . . . . . . B.4.10 RMTP35 . . . . . . . . . . . . . . . . . . . . . Proposed deceptive test functions . . . . . . . . . . . . B.5.1 RMTP36 . . . . . . . . . . . . . . . . . . . . . B.5.2 RMTP37 . . . . . . . . . . . . . . . . . . . . . B.5.3 RMTP38 . . . . . . . . . . . . . . . . . . . . . Proposed multi-modal robust test functions . . . . . . B.6.1 RMTP39 . . . . . . . . . . . . . . . . . . . . . B.6.2 RMTP40 . . . . . . . . . . . . . . . . . . . . . B.6.3 RMTP41 . . . . . . . . . . . . . . . . . . . . . Proposed flat robust test functions . . . . . . . . . . . B.7.1 RMTP42 . . . . . . . . . . . . . . . . . . . . . B.7.2 RMTP43 . . . . . . . . . . . . . . . . . . . . . B.7.3 RMTP44 . . . . . . . . . . . . . . . . . . . . .

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C Complete results 259 C.1 Robust Pareto optimal fronts obtained by CRMOPSO, IRMOPSO, and ERMOPSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

List of Figures 1.1

Organisation of the thesis (Purple: literature review and related works, Red: analysis of the literature and current gaps, Green: proposed systematic robust algorithm design process, Blue: results on the test beds and real case study, and Orange: conclusion and future works) . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1

Different components of an optimisation system: inputs, outputs, operating conditions, and constraints . . . . . . . . . . . . . . . . Example of a search landscape with two variables and several constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic population-based optimisers consider the system as black box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual-based versus population-based stochastic optimisation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pareto dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . Pareto optimal set versus Pareto optimal front . . . . . . . . . . . A priori method versus a posteriori methods [42] . . . . . . . . . Different categories of uncertainties and their effects on a system: Type A, Type B, and Type C . . . . . . . . . . . . . . . . . . . . Conceptual model of a robust optimum versus a non-robust optimum. The same perturbation level (δ) in the parameter (p) 0 results in different changes (∆ and ∆ ) in the objective (f ) . . . . Search space of an expectation measure versus its objective function Conceptual model of infeasible regions when employing a variance measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concepts of robustness and a robust solution in multi-objective search space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Four possible robust Pareto optimal fronts with respect to the main Pareto optimal front . . . . . . . . . . . . . . . . . . . . . . Collected current test functions in the literature for robust singleobjective optimisation. The details can be found in Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test problems proposed by Deb and Gupta in 2006 [44] . . . . . .

2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

2.10 2.11 2.12 2.13 2.14

2.15

xv

14 15 16 17 23 24 26 31

32 34 36 40 42

55 57

xvi

LIST OF FIGURES

2.16 Test problem proposed by Gaspar-Cunha et al. in 2013 . . . . . . 58 3.1

3.2 3.3 3.4 4.1

4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

4.11 4.12 4.13 4.14 4.15 4.16

4.17 4.18 4.19

An example of the failure of archive-based methods in distinguishing robust and non-robust solutions. Note that the variances shown are the actual variances, not those detected by the sampling. Test functions and performance metrics are essential for systematic algorithm design . . . . . . . . . . . . . . . . . . . . . . . . . Gaps targeted by the thesis . . . . . . . . . . . . . . . . . . . . . Scope and contributions of the thesis . . . . . . . . . . . . . . . .

74 76 78 79

Proposed function with adjustable local optima robustness parameter. The parameter α changes the landscape significantly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Effect α of on the robustness of the global optimum . . . . . . . . 88 Shape of the search landscape with controlling parameters constructed by framework II . . . . . . . . . . . . . . . . . . . . . . . 90 Effect of parameter λ on the shape of search landscape . . . . . . 91 An example of the search space that can be constructed by the framework III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Search space of the function G(~x) . . . . . . . . . . . . . . . . . . 94 Search space becomes larger proportional to N and without any change in the main search landscape . . . . . . . . . . . . . . . . 94 Density of solutions when θ < 1, θ = 1, θ > 1 (p = 0) . . . . . . . 95 Conversion of an un-biased search space to a biased search space . 96 50,000 randomly generated solutions reveal there is low density toward the robust optimum in the biased test function, while the density is uniform in the un-biased test function . . . . . . . . . . 97 Proposed deceptive robust test problem . . . . . . . . . . . . . . . 98 Proposed multi-modal robust test function (M = 10) . . . . . . . 99 Proposed flat robust test function . . . . . . . . . . . . . . . . . . 100 Search space and objective space constructed by proposed framework 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Effect of α on the robustness of the global Pareto optimal front’s valley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Shape of parameter space and Pareto optimal fronts when: β = 0.5, β = 1, and β = 1.5. Note that the red curve indicates the robustness of the robust front and black curves are the front. . . 104 Changing the shape of global and robust Pareto optimal front with β 105 Shape of the parameter space and its relation with the objective space constructed by the framework 2 . . . . . . . . . . . . . . . . 107 Effect of λ on both parameter and objective spaces. Note that the red curve indicates the robustness of the robust front and black curves are the fronts. . . . . . . . . . . . . . . . . . . . . . . . . . 108

LIST OF FIGURES 4.20 Effect of γ on the fronts. Note that the red curve indicates the robustness of the robust front and black curves are the fronts. . . 4.21 Effect of ζ on the parameter and objective spaces. Note that the red curve indicates the robustness of the robust front and black curves are the fronts. . . . . . . . . . . . . . . . . . . . . . . . . . 4.22 Parameter and objective spaces constructed by the third framework. The red curve indicates the robustness of the robust front and black curves are the fronts. . . . . . . . . . . . . . . . . . . . 4.23 A non-biased objective space versus a biased objective space (50,000 random solutions). The proposed bias function requires the random points to cluster away from the Pareto optimal front. . . . . 4.24 Bias of the search space is increased inversely proportional to ψ . 4.25 There are four deceptive non-robust optima and one robust optimum in the function H(x) . . . . . . . . . . . . . . . . . . . . . . 4.26 Different shapes of Pareto fronts that can be obtained by manipulating β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.27 H(x) creates one robust and 2M global Pareto optimal fronts . . 4.28 Parameter space and objective space of the proposed multi-modal robust multi-objective test problem . . . . . . . . . . . . . . . . . 4.29 Different shapes of Pareto fronts that can be obtained by manipulating β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.30 H(x) makes two global optima close to the boundaries . . . . . . 4.31 H(x) makes two global optima close to the boundaries . . . . . . 5.1 5.2

xvii

109

110

111

112 113 114 115 117 117 118 119 120

Schematic of the proposed coverage measure (Φ) . . . . . . . . . . 124 Effect of the number of occupied robust segments on the proposed coverage measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.3 Zero effect of occupied non-robust segments on the proposed coverage measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4 Segments that are partially robust do not count when calculating Φ127 5.5 The accuracy of the proposed coverage measure is increased proportional to the number of segments . . . . . . . . . . . . . . . . 127 5.6 Effect of the minimum robustness on the number of robust segments and Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.7 All segments are converted to robust and counted when Rmin > max (R(robustness curve)) . . . . . . . . . . . . . . . . . . . . . . 128 5.8 Conceptual model of the proposed success ratio measure . . . . . 129 5.9 An example of a probable problem in case of using diagonal segments when calculating Γ . . . . . . . . . . . . . . . . . . . . . . 130 5.10 Success ratio is zero if there is no robust solution in the set of solutions obtained . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.11 Example of the success ratio for a set that contains only robust solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

xviii

LIST OF FIGURES

5.12 Success ratio increases proportional to the number of robust solutions obtained . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Success ratio is inversely proportional to the number of non-robust solutions obtained . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Effect of minimum robustness on success ratio when Rmin < min (R(robustness curve)) . . . . . . . . . . . . . . . . . . . . . 5.15 Effect of minimum robustness on success ratio when Rmin > max (R(robustness curve)) . . . . . . . . . . . . . . . . . . . . . 6.1 6.2 7.1 7.2 7.3

7.4 8.1 8.2

8.3

9.1 9.2 9.3 9.4 9.5 9.6

. 132 . 132 . 133 . 134

Confidence measure considers the number, distribution, and distance of sampled point from the current solution . . . . . . . . . . 138 Flow chart of the general framework of the proposed confidencebased robust optimisation . . . . . . . . . . . . . . . . . . . . . . 142 Behaviour of CRPSO1 finding the robust optima of TP1, TP2, TP3, TP4, and TP5 . . . . . . . . . . . . . . . . . . . . . . . . . Behaviour of CRPSO1 finding the robust optima of TP6, TP7, TP8, TP9, and TP10 . . . . . . . . . . . . . . . . . . . . . . . . . Search history of GA and IRGA. GA converges towards the global non-robust optimum, while IRGA failed to determine the robust optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Search history of CRGA1 and CRGA2. The exploration of CRGA2 is much less than that of CRGA1 . . . . . . . . . . . . . . . . . .

152 153

165 165

Robust fronts obtained for RMTP1, RMTP7, RMTP9, and RMTP27, one test case per row. . . . . . . . . . . . . . . . . . . . . . . . . 170 Robust fronts obtained for RMTP13 to RMTP16 and RMTP19, one test case per row. Note that the dominated (local) front is robust and considered as reference for the performance measures. 179 Robust fronts obtained for RMTP21 to RMTP25 one test case per row. Note that the worst front is the most robust and considered as reference for the performance measures. . . . . . . . . . . . . 181 Airfoils along the blade define the shape of the propeller (NACA a = 0.8 meanline and NACA 65A010 thickness) . . . . . . . . . . 188 Propeller used as the case study . . . . . . . . . . . . . . . . . . . 189 (left) Pareto optimal front obtained by the MOPSO algorithm (6 blades), (right) Pareto optimal fronts for different numbers of blades190 (left) Best Pareto optimal fronts obtained for different RPM (right) Optimal RPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 PF obtained when varying RPM compared to PFs obtained with different RPM values . . . . . . . . . . . . . . . . . . . . . . . . . 193 Optimal RPM coordinates . . . . . . . . . . . . . . . . . . . . . . 194

LIST OF FIGURES

xix

9.7

Pareto optimal solutions in case of (left) δRP M = +1 , (right) δRP M = −1 fluctuations in RPM (right). Original values are shown in blue, perturbed results in red. . . . . . . . . . . . . . . . 195 9.8 Pareto optimal solutions in case of (left) δ = +1.5% (right) δ = −1.5% perturbations in parameters. Original values are shown in blue, perturbed results in red. . . . . . . . . . . . . . . . . . . . . 196 9.9 Robust front obtained by CRMOPSO versus global front obtained by MOPSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 9.10 Global and robust Pareto optimal fronts obtained by MOPSO and CRMOPSO when RPM is also a variable . . . . . . . . . . . . . . 199 9.11 Optimal and robust optimal values for RPM (note that there are 98 robust Pareto optimal solutions and 100 global optimal solutions)200 10.1 Gaps filled by the thesis . . . . . . . . . . . . . . . . . . . . . . . 203 10.2 Contributions of the thesis . . . . . . . . . . . . . . . . . . . . . . 207 C.1 Robust fronts obtained for RMTP1 to RMTP5. . . . . . . . . . . 260 C.2 Robust fronts obtained for RMTP6 to RMTP10. . . . . . . . . . . 261 C.3 Robust fronts obtained for RMTP11 to RMTP15. Note that the dominated (local) front is robust and considered as reference for the performance measures. . . . . . . . . . . . . . . . . . . . . . . 262 C.4 Robust fronts obtained for RMTP16 to RMTP19. Note that the dominated (local) front is robust and considered as reference for the performance measures. . . . . . . . . . . . . . . . . . . . . . . 263 C.5 Robust fronts obtained for RMTP20 to RMTP22. Note that the worst front is the most robust and considered as reference for the performance measures. . . . . . . . . . . . . . . . . . . . . . . . . 264 C.6 Robust fronts obtained for RMTP23 to RMTP25. Note that the worst front is the most robust and considered as reference for the performance measures. . . . . . . . . . . . . . . . . . . . . . . . . 265 C.7 Robust fronts obtained for RMTP26 and RMTP27. . . . . . . . . 265 C.8 Robust fronts obtained for RMTP28 and RMTP32. . . . . . . . . 266 C.9 Robust fronts obtained for RMTP33 and RMTP38. Note that in RMTP36, RMTP37, and RMTP38, the worst front is the most robust and considered as the reference for the performance measures.267 C.10 Robust fronts obtained for RMTP39 and RMTP44. . . . . . . . . 268

List of Tables 7.1 7.2 7.3 7.4 7.5 7.6 8.1 8.2 8.3 8.4 8.5 8.6 8.7 9.1

9.2

Number of times that the confidence operators and confidence measure triggered . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical results of the RPSO algorithms over 30 independent runs: (ave ± std(median)) . . . . . . . . . . . . . . . . . . . . . Results of Wilcoxon ranksum test for RPSO algorithms . . . . . Statistical results of the RGA algorithms over 30 independent runs: (ave ± std(median)) . . . . . . . . . . . . . . . . . . . . . Results of Wilcoxon ranksum test for RGA algorithms . . . . . Number of times that CRGA2 makes confident and risky decisions over 100 generations . . . . . . . . . . . . . . . . . . . . . . . . Statistical results of RMOPSO algorithms using IGD . . . . . . P-values of Wilcoxon ranksum test for the RMOPSO algorithms in Table 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical results of RMOPSO algorithms using Φ . . . . . . . . P-values of Wilcoxon ranksum test for the RMOPSO algorithms in Table 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical results of RMOPSO algorithms using Γ . . . . . . . . P-values of Wilcoxon ranksum test for the RMOPSO algorithms in Table 8.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of times that the proposed confidence-based Pareto dominance prevented a solution entering the archive . . . . . . . . .

. 154 . 157 . 158 . 161 . 162 . 164 . 171 . 172 . 173 . 174 . 175 . 176 . 183

Fuel consumption discrepancy in case of perturbation in all of the structural parameters for both PS obtained by MOPSO and RPS obtained by CRMOPSO . . . . . . . . . . . . . . . . . . . . . . . 198 Fuel consumption discrepancy in case of perturbation in RPM for both PS obtained by MOPSO and RPS obtained by CRMOPSO . 200

xx

Nomenclature Γ

Robust success ratio

Φ

Robust coverage measure

ACO

Ant Colony Optimisation

BWA

Bang-Bang Weighted Aggregation

C

Confidence Measure

CFD

Computational Fluid Dynamics

CRGA

Confidence-based Robust Genetic Algorithm

CRMO

Confidence-based Robust Multi-objective Optimisation

CRMOPSO

Confidence-based Robust Multi-Objective Particle Swarm Optimisation

CRO

Chemical Reaction Optimisation

CRO

Confidence-based Robust Optimisation

CRPSO

Confidence-based Robust Particle Swarm Optimisation

DE

Differential Evolution

DMOO

Dynamic Multi-Objective Optimisation

DWA

Dynamic Weighted Aggregation

EA

Evolutionary Algorithm

EMA

Evolutionary Multi-objective Algorithm

EMOO

Evolutionary Multi-Objective Optimisation

EP

Evolutionary Programming

ER

Error Ratio xxi

xxii ERGA

Explicit Averaging Robust Genetic Algorithm

ERMOPSO

Explicit Averaging Robust Multi-Objective Particle Swarm Optimisation

ERPO

Explicit Averaging Robust Particle Swarm Optimisation

ES

Evolution Strategy

GA

Genetic Algorithm

gBest

Global Best

GD

Generational Distance

GSA

Gravitational Search Algorithm

IGD

Inverted Generational Distance

IMOO

Interactive Multi-Objective Optimisation

IRGA

Implicit Averaging Robust Genetic Algorithm

IRMOPSO

Implicit Averaging Robust Multi-Objective Particle Swarm Optimisation

IRPO

Implicit Averaging Robust Particle Swarm Optimisation

LHS

Latin Hypercube Sampling

MOEA/D

Multi-Objective Evolutionary Algorithm based on Decomposition

MOPSO

Multi-Objective Particle Swarm Optimisation

N/A

Not Applicable

NACA

National Advisory Committee for Aeronautics

NSGA

Non-dominated Sorting Genetic Algorithm

O

Big O

PAES

Pareto Archived Evolution Strategy

pBest

Personal Best

PDE

Pareto-frontier Differential Evolution

PF

Pareto Optimal Front

xxiii PS

Pareto Optimal Set

PSO

Particle Swarm Optimisation

R

Robustness Measure

RMOO

Robust Multi-Objective Optimisation

RMTP

Robust Multi-objective Test Problem

RNSGA

Robust Non-dominated Sorting Genetic Algorithm

RPF

Robust Pareto Optimal Front

RPM

Revolutions Per Minute

RPS

Robust Pareto Optimal Set

RPSGA

Reduced Pareto Set Genetic Algorithm

SA

Simulated Annealing

SCC

Success Counting

SP

Spacing

SPEA

Strength-Pareto Evolutionary Algorithm

TP

Test Problem

TS

Tabu Search

ZDT

ZitzlerDebThiele

Chapter 1 Introduction In the past, the computational engineering design process used to be mostly experimentally based [103]. This meant that a real system first had to be designed and constructed to be able to do experiments. In other words, the design model was an actual physical model. For instance, an actual airplane or prototype would have to put in a massive wind tunnel to investigate the aerodynamics of the aircraft [1]. Obviously, the process of design was very tedious, expensive, and slow. After the development of computers, engineers started to simulate models in computers to compute and investigate different aspects of real systems. This was a revolutionary idea since there was no need for an actual model in the design phase anymore. Another advantage of modelling problems in computers was the reduced time and cost. It was no longer necessary to build a wind tunnel and real model to compute and investigate the aerodynamics of an aircraft. The next step was to investigate not only the known characteristics of the problem but also explore and discover new features. Exploring the search space of the simulated model in a computer allowed designers to better understand the problem and find optimal values for design parameters. Despite the use of computer in modelling, a designer still had to manipulate the parameters of the problem manually. After the first two steps, people started to develop and utilise computational/optimisation algorithms to use the computer itself to find optimal solutions of the simulated model for a given problem. Thus, the computer manipulated and chose the parameters with minimum human involvement. This was the birth of automated and computer-aided design fields. Evolutionary Algorithms (EA) also became popular tools in finding the optimal solutions for optimisation 1

2

1. Introduction

problems. Generally speaking, EAs mostly have very similar frameworks. They first start the optimisation process by creating an initial set of random, trial solutions for a given problem. This random set is then iteratively evaluated by objective function(s) of the problem and evolved to minimise or maximise the objective(s). Although this framework is very simple, optimisation of real world problems requires considering and addressing several issues of which the most important ones are: local optima, expensive computational cost of function evaluations, constraints, multiple objectives, and uncertainties. Real problems have mostly unknown search spaces that may contain many sub-optimal solutions. Stagnation in local optima is a very common phenomenon when using EAs. In this case, the algorithm is trapped in one of the local solutions and assumes it to be the global solution. Although the stochastic operators of EAs improve the local optima avoidance ability compared to deterministic mathematical optimisation approaches, local optima stagnation may occur in any EAs as well. EAs are also mostly population-based paradigms. This means they iteratively evaluate and improve a set of solutions instead of a single solution. Although this improves the local optima avoidance as well, solving expensive problems with EAs is not feasible sometimes due to the need for a large number of function evaluations. In this case, different mechanisms should be designed to decrease the required number of function evaluations. Constraints are another difficulty of real problems, in which the search space may be divided into two regions: feasible and infeasible. The search agents of EAs should be equipped with suitable mechanisms to avoid all the infeasible regions and explore the feasible areas to find the feasible global optimum. Handling constraints requires specific mechanisms and has been a popular topic among researchers. Real engineering problems often also have multiple objectives. Optimisation in a multi-objective search space is quite different and needs special considerations compared to a single-objective search space. In a single-objective problem, there is only one objective function to be optimised and only one global solution to be found. However, in multi-objective problems there is no longer a single solution for the problem, and a set of solutions representing the trade-offs between the multiple objectives, the Pareto optimal set, must be found. Last but not least, another key concept in the optimisation of real engineer-

1. Introduction

3

ing problems is robustness. Robust optimisation refers to the process of finding optimal solutions for a particular problem that have least variability in response to probable uncertainties. Uncertainties are unavoidable in the real world and can be classified in three categories: those affecting parameters, operating conditions, and outputs. One of the most common uncertainties is perturbation of parameters, in which the design parameters may vary. Such uncertainties mostly occur during the manufacturing process due to the resolution and imprecision of devices. These inaccuracies always exist, so a design that does not consider them is prone to be unreliable. An example is the length of a bar in a truss. If the length goes over a certain threshold due to manufacturing perturbations, it may cause collapse of the entire structure. Since parameters are primary inputs of a problem, this type of uncertainty is of the highest importance. Uncertainties also may happen in the secondary inputs of a system: the operating conditions. An example is the change in fuel consumption of a car when varying speed. There is an optimal speed for a car to have the least fuel consumption: the car consumes more fuel if it goes slower or faster. Of course, the primary parameters and uncertainties in the design of the car play the key role in fuel consumption. However, varying speed also has significant impact on the consumption. This type of uncertainty has been a main cause of airplane crashes in history: due to icing and wind. The last type of uncertainty occurs in the outputs themselves. They usually come from the approximate and simulated models in the computer, which are unavoidable. Systems with time varying outputs (dynamic systems) also fall into this category. Once again, failure to consider such perturbation during the design process may result in failure of the entire system to produce the desired output. Although uncertainties are small perturbations in different components of a system, they usually have substantial impacts on the desired outputs. Without considering uncertainties during the design process, a system has the potential to show undesirable outputs. This is critical for systems on which humans rely. For instance, a small perturbation in the shape of an aircraft’s wing, operating conditions, or simulated model may result in a crash. Uncertainties are undesirable inputs that always exist when the system is operating in a real environment. Therefore, it is essential to consider and handle them using robust techniques

4

1. Introduction

during the design process to avoid or minimise their negative consequences on the output(s) of the entire system. Robust optimisation is essential when solving real problems since failure to consider uncertainty can eclipse all the efforts a team put into designing and implementing a system. In addition, this may bring a substantial waste of money and time for stakeholders. Similar to conventional optimisation, robust optimisation in a single-objective search space is also different from that in a multi-objective search space. In a single-objective search space, there is one robust solution with the best performance and least variations. In a multi-objective search space, however, robust optimal solutions belong to a set of optimal solutions called the robust Pareto optimal set representing the robust trade-offs between the multiple objectives. There are several works in the literature that employed multi-objective metaheuristics to perform robust optimisation [27, 88, 136, 44]. In 2006, Deb and Gupta [44] investigated two different approaches for robust optimisation in multi-objective search spaces: expectation-based and variance-based methods. In the former method an expectation measure, which is calculated by averaging a representative set of neighbouring solutions, is optimised instead of the main objective function. In the latter method, however, the main objective functions are optimised with an additional constraint (variance measure) which limits the optimisation process in terms of the robustness of search agents. In contrast to other branches of multi-objective optimisation, unfortunately, robust multi-objective optimisation has not gained deserved attention [81, 80]. As evidence, a publication report was conducted from 1994 to 2014 in ISI Web of Knowledge with the keywords “multi-objective optimisation” and “robust optimisation”. It was found that only about 0.5% of publications on these topics over the past decade contained both keywords. Among current works, there is a considerable number of studies that focused on robustness in single-objective search spaces (e.g. [176].) However, there are fewer works on the investigation of robustness in multi-objective search spaces [69, 75].

1.1

Problem Background

In the literature, robust meta-heuristic optimisation is performed with a wide range of robust measures [93], of which the most well-regarded ones are: expectation and variance. Generally speaking, these measures are used for observing

1. Introduction

5

the behaviour in objective space in the neighbourhood of a particular solution to confirm robustness. Robust optimisation using expectation measures was named Type I robust optimisation by Deb and Gupta [44]. In this kind of optimisation, the objective function(s) are replaced by the expectation measure(s). Then, the expectation measure(s) are optimised. Technically speaking, a finite set of H solutions are chosen randomly or with a structured procedure in the hypervolume ([−δ, δ]) around the solution ~x, and then the expectation measure(s) of all samples are optimised by heuristic optimisation algorithms. Since the proposal of this kind of optimisation, several researchers proposed different expectation measures and tried to perform robust optimisation [74, 12, 13, 168, 94, 159]. Another way of robust optimisation using expectation measures is to consider them as separate objective functions [135]. In this case the Pareto front finally obtained would indicate trade-offs between objective(s) and robustness. So the production for decision makers of a wide range of solutions with different degrees of robustness could be considered as the advantage of this method. As a drawback, however, considering expectation measures as additional objectives increases the computational complexity of a problem, as discussed by Brockhoff et al. [24, 23]. The second method of robust optimisation, using a variance measure, does not replace the main objective functions. An additional constraint is added to the problem in order to handle uncertainties. This constraint controls the variance of objectives of solutions in the objective space based on the local perturbations around the solution in the parameter space. A set of random solutions is generated around the solutions in the parameter space and the variance of their corresponding objective values limited by a pre-defined threshold. Violation of this constraint assists us in distinguishing between a robust solution and a non-robust solution. Deb and Gupta named this method Type II robustness handling [44]. There are also other, different variance measures proposed in the literature [69, 74]. Robust optimisation using expectation measures and variance measures are both able to consider uncertainties during optimisation and prevent an algorithm from finding solutions that are sensitive to possible perturbations in real environments. The question here is how effective each of theses methods are and if both of them are worthy of further improvements. Robust optimisation using

6

1. Introduction

an expectation measure (Type I) benefits from the separation of the robustness measure from the algorithm’s structure. This method directly integrates the robustness measure in the objective function, so there is no need to modify the structure of an algorithm or employ specific operators to handle uncertainties. However, this method changes the shape of the search landscape that might impact the computational time, as usually a function evaluation is replaced by several (function evaluations). The variance measures (Type II) do not change the shape of the search space, which can be considered as an advantage because the optimiser searches the actual search space. However, they add another difficulty to the search space, which is infeasible regions. This means that an algorithm should be equipped with a constraint handling technique to be able to work with a variance measure. A variance measure has the potential to increase the amount of infeasible regions, which will definitely need special considerations. A search space with a large infeasible portion is very likely to result in having many infeasible search agents in each iteration. The problem here is that, by default, most of the metaheuristics discard infeasible solutions and only rely on feasible solutions to drive the search agents towards optimal solution(s). Therefore, a powerful constraint handling method should be utilised when solving such problems.

1.2

Problem Statement and Objectives

The disadvantages of type I and type II robust optimisation methods are related to the additional difficulty that needs to be considered as well as a large number of function evaluations. However, they are able to find robust solutions for a given problem, subject to creating enough sampled points or additional function evaluations. Both types of measure quantify the robustness of solutions, which is fruitful when comparing the solutions during optimisation. No matter how expensive this is, these measures can confirm the robustness of solutions. With this confirmation, an algorithm can then reliably favour robust solutions and discard non-robust solutions to drive the whole optimisation process towards better robust solutions. In order to alleviate the disadvantages of both methods, several steps need to be taken systematically. All of these steps are essential and will be discussed in detail in the following chapters. Obviously, a solution to the problem of

1. Introduction

7

both methods start with hypotheses. However, any idea needs to be tested, evaluated, compared, and verified to reliably and confidently prove that it is beneficial. This process is done in different branches of optimisation including global optimisation, dynamic optimisation, interactive optimisation, and so on as well. Despite the advances in all of the above-mentioned branches and importance of considering uncertainties during optimisation of real problems, there is significant room for further improvements in the area of robust optimisation, especially combined with multi-objective approaches. There are few works in this field as I will show in the next chapter. Finding optimal solutions that are less sensitive to perturbations needs a systematic design approach. Therefore: A highly systematic robust optimisation algorithm design process is essential to design reliable robust algorithms. There is a need for a systematic design process in the field of robust optimisation to better and conveniently alleviate the drawbacks of the current robust optimisation techniques and/or propose new ones. Consequently, the aim of this study is to investigate and fill one of the most substantial current gaps in robust heuristic optimisation techniques, with emphasis on development of tools that could assist in real-world applications. The main research question can be stated as follows: How can a systematic design process be established to systematically test, evaluate, and propose computationally cheap robust optimisation techniques? Associated research questions, objectives, and plan are discussed in detail in Chapter 3.

1.3

Scope and Significance

The presence of a systematic design process will allow designers in the field of robust optimisation to reliably and confidently test, evaluate, and propose new algorithms or improve the current ones. The first phase of a systematic design process will provide test environment for designers. A set of suitable test beds is an essential in any kind of experimental studies and can benchmark different ideas. I will only concentrate on

8

1. Introduction

unconstrained single- and multi-objective optimisation in this phase since they are the main foundation of most of research branches in this field. After the first phase, a systematic design process needs to evaluate the performance of a given idea to be able to compare with others. Qualitative and quantitative evaluation criteria are very important in this phase because they show us how and how much better an algorithm is. Since quantitative performance evaluators are more accurate compared to qualitative ones, I will contribute to the quantitative evaluators in the second phase. In addition, the focus will be on multi-objective performance indicators due to the importance and complexity of performance evaluation in multi-objective optimisation. In the third phase of the systematic design process, a series of ideas are proposed to alleviate the drawbacks of the current robust optimisation techniques. The contributions will be in both fields of robust single- and multi-objective optimisation. A set of algorithms will be proposed that are reliable and do not need additional function evaluation. These make them highly suitable for solving expensive real world problems. All of the methods are unconstrained, but they can readily be applied to constrained problems as well.

1.4

Organisation of the thesis

The organisation of the thesis is illustrated in Fig. 1.1. It may be seen in this figure that Chapter 2 investigates the state-of-the-art of heuristic optimisation algorithms, multi-objective optimisation techniques, robust optimisation in single-objective search spaces, robust optimisation in multi-objective search spaces, current single- and multi-objective robust test functions, current performance metrics, explicit and implicit methods, and relevant criticism of the current works. The state-of-the-art is analysed and current gaps are identified in detail in Chapter 3. Chapter 4 proposes the first phase of the systematic design process for testing robust single- and multi-objective optimisation techniques. Chapter 5 proposes the second phase of the systematic optimisation process for evaluating and comparing the performance of robust multi-objective optimisation algorithms. The confidence measure, confidence-based relational operators, confidence-based robust optimisation, confidence-based Particle Swarm Optimisation (PSO), confidence-based Genetic Algorithms (GA), confidence-based Pareto optimality, confidence-based robust multi-objective optimisation, and

1. Introduction

9

confidence-based Multi-Objective Particle Swarm Optimisation (MOPSO) are proposed and discussed theoretically in Chapter 6 as the last phase of the systematic robust algorithm design process. The results of the confidence-based single-objective robust optimisation techniques on the robust single objective test functions are presented and discussed in Chapter 7. Chapter 8 demonstrates and analyses the results of confidence-based multi-objective robust optimisation techniques on the robust multi-objective test functions. The real application of the proposed confidence-based robust optimisation perspective is presented and discussed in Chapter 9. Finally, Chapter 10 concludes the thesis, describes the achievements of the work, and suggests several research directions for future studies.

10

1. Introduction

Single-objective optimisation Multi-objective optimisatoin Chapter 2: related works Robust optimisation Chapter 3: Analysis

Proposed systematic design process Chapter 4: Benchmark problems

Robust multi-objective optimisation Robust single objective test functions Robust multi-objective test functions Coverage measure

Chapter 5: Performance measures Success ratio

Core Chapter 6: Improving robust optimisation techniques

Confidence measure Confidence-based robust singleobjective optimisation Confidence-based robust multiobjective optimisatoin

Chapter 7: Confidence-based robust optimisation

Results

Chapter 8: Confidence based robust multiobjective optimisation

Results

Chapter 9: Real application

CRPSO

CRGA CRMOPSO

Multi-objective optimisation of marine propellers Robust multi-objective optimisation of marine propellers Conclusion

Chapter 10: Conclusion

Achievements and significance Future works

Figure 1.1: Organisation of the thesis (Purple: literature review and related works, Red: analysis of the literature and current gaps, Green: proposed systematic robust algorithm design process, Blue: results on the test beds and real case study, and Orange: conclusion and future works)

Chapter 2 Related work In recent years meta-heuristic algorithms have been used as primary techniques for obtaining the estimated optimal solutions of real engineering design optimisation problems [16, 17, 79]. Such algorithms mostly benefit from stochastic operators [15] that make them distinct from deterministic approaches. A deterministic algorithm [36, 6, 109] reliably determines the same answer for a given problem with a similar initial starting point. However, this behaviour results in local optima entrapment, which can be considered as a disadvantage for deterministic optimisation techniques [150]. Local optima stagnation refers to the entrapment of an algorithm in local solutions and consequent failure in finding the true global optimum. Since real problems may have a large number of local solutions, deterministic algorithms lose their reliability in finding the global optimum. Stochastic optimisation (meta-heuristic) algorithms [152] refer to the family of algorithms with stochastic operators including evolutionary algorithms [7]. Randomness is the main characteristic of stochastic algorithms [91]. This means that they utilise random operators when looking for global optima in search spaces. Although the randomised nature of such techniques might make them unreliable in obtaining a similar solution in each run, they are able to avoid local solutions much more easily than deterministic algorithms. The stochastic behaviour also results in obtaining different solutions for a given problem in each run [101]. Meta-heuristic and evolutionary algorithms search for the global optimum in a search space by creating one or more random solutions for a given problem [156]. This set is called the set of candidate solutions. The set of candidates 11

12

2. Related work

is then improved iteratively until satisfaction of a terminating condition. The improvement can be considered as finding a more accurate approximation of the global optimum than the initial random guesses. This mechanism brings evolutionary algorithms several intrinsic advantages: independency of problem, independency of derivatives, local optima avoidance, and simplicity. Problem and derivation independencies originate from the consideration of problems as a black box. Evolutionary algorithms only utilise the objective function for evaluating the set of candidate solutions. The main process of optimisation is independent of the problem and based on the inputs provided and outputs received. Therefore, the nature of the problem is not a concern, yet the representation is the key step when utilising evolutionary algorithms. This is the same reason why evolutionary algorithms do not need to find derivatives of functions in the problem to obtain its estimated global optimum. As another advantage, local optima avoidance is high due the stochastic nature of evolutionary algorithms. If an evolutionary algorithm is trapped in a local optimum, stochastic operators lead to random changes in the solution and eventual escape from the local optimum. Although there is no guarantee of resolving this issue, stochastic algorithms have much higher probability to escape from local optima compared to deterministic methods. Very accurate approximation of the global optimum also is not guaranteed, but running an evolutionary algorithm several times increases the probability of obtaining a better solution. Lastly, simplicity is another characteristic of evolutionary algorithms. Natural evolutionary or collective behaviours are the main inspirations for the majority of algorithms in this field. While exhibiting sophisticated behaviour, their basic mechanics are often inherently quite simple. In addition, evolutionary algorithms follow a general and common framework, in which a set of randomly created solutions is enhanced or evolved iteratively. What makes algorithms different in this field is the method of improving this set. Some of the most popular algorithms in this field are: Genetic Algorithms (GA) [89, 90], Particle Swarm Optimisation (PSO) [60], Ant Colony Optimisation (ACO) [35], Differential Evolution (DE) [154], Evolutionary Programming (EP) [70, 175], and Evolution Strategy (ES) [139, 138]. Although these algorithms are able to solve many real and challenging problems, the so-called No Free Lunch theorem [169] allows researchers to propose new algorithms. Ac-

2. Related work

13

cording to this theorem, all algorithms perform equally when averaged across all possible optimisation problems. Therefore, one algorithm can be very effective in solving one set of problems and not effective on a different set of problems. This is the foundation of many works in this field. Despite the popularity and simplicity of evolutionary algorithms, optimisation using these techniques requires several considerations and has its own challenges. There are also different types of optimisation in this field, of which the most important ones are single-objective, multi-objective, unconstrained, constrained, dynamic, robust, and interactive optimisation. Single-objective optimisation is the simplest and fundamental expression of the optimisation process. It deals with varying parameters, seeking to satisfy an objective. As such, this kind of optimisation is the foundation for consideration of new, generally applicable methods and ideas. This thesis concentrates on development of effective robust optimisation and as a starting point, must consider single-objective approaches. In the real world, most optimisation problems have multiple, often competing objectives. For the methods proposed to be useful, widely applicable and effective, they must take into consideration this multi-objective nature of the majority of problems to be addressed. Multi-objective optimisation deals with extending and developing approaches to solve these kind of problems. So, for the contributions of this thesis to be broadly applicable, methods must be developed and tested for single-objective optimisation and multi-objective optimisation. This chapter reviews the literature of single-objective optimisation, multiobjective optimisation, robust single-objective optimisation, and robust multiobjective optimisation. Due to the scope of the thesis, a large portion of this chapter covers robust optimisation.

2.1

Evolutionary single-objective optimisation

This section first covers the preliminaries and definitions of optimisation. The mechanisms and challenges of stochastic/heuristic optimisation techniques are then discussed. As its name implies, single-objective optimisation deals with optimising only one objective. Handling multiple objectives requires special considerations and mechanisms [179] and will be discussed in the next section.

14

2. Related work

In addition to the objective, other elements involved in the single-objective optimisation process are parameters and constraints. Parameters are the variables of optimisation problems (systems) that have to be optimised. As Fig. 2.1 shows, variables can also be considered as inputs of systems and constraints are the limitations applied to the system. In fact, the constraints define the feasibility of the obtained objective value. Examples of constraints are stress constraints when designing aerodynamic systems or the range of variables. Operating/environmental conditions

Variables (inputs)

System

Constraints

Objective (output)

Feasibility

Figure 2.1: Different components of an optimisation system: inputs, outputs, operating conditions, and constraints Other inputs of a system that may affect its output are operating (environmental) conditions. Such inputs are considered as secondary inputs that are defined when a system is operating in the simulated/final environment. Examples of such conditions are: temperature/density of fluid when a propeller is turning or the angle of attack when an aircraft is flying. These types of inputs are not optimised by the optimisers but definitely have to be considered during optimisation since they may have significant impacts on the outputs. Without loss of generality, a single-objective optimisation can be formulated as a minimisation problem as follows: M inimise :

f (x1 , x2 , x3 , ..., xn−1 , xn )

(2.1)

Subject to :

gi (x1 , x2 , x3 , ..., xn−1 , xn ) ≥ 0, i = 1, 2, ..., m

(2.2)

2. Related work

15

hi (x1 , x2 , x3 , ..., xn−1 , xn ) = 0, i = 1, 2, ..., p

(2.3)

lbi ≤ xi ≤ ubi , i = 1, 2, ..., n

(2.4)

where n is number of variables, m indicates the number of inequality constraints, p shows the number of equality constraints, lbi is the lower bound of the i-th variable, and ubi is the upper bound of the i-th variable. As can be seen in Equations 2.2 and 2.3, there are two types of constraints: inequality and equality. The set of variables and constraints construct a search space, and the objectives define the search landscape for a given problem. Unfortunately, it is usually impossible to draw the search space due to the highdimensionality of the variables. However, an example of a search space constructed by two variables and several constraints is shown in Fig. 2.2.

Figure 2.2: Example of a search landscape with two variables and several constraints It may be observed in Fig. 2.2 that the search space can have multiple local optima, but one of them is the global optimum (or some of them in case of a flat landscape). The constraints create gaps in the search space and occasionally split it into various separated regions. In the literature, infeasible regions refer to the areas of the search space that violate constraints. The search space of a real

16

2. Related work

problem can be very challenging. Some of the difficulties of the real search spaces are discontinuity, a massive number of local optima, high infeasibility, global optimum located on the boundaries of constraints, deceptive valleys toward local optima, and isolation of the global optimum. When formulating a problem, an optimiser would be able to tune its variables based on the outputs and constraints. As mentioned in the introduction of this chapter, one of the advantages of evolutionary algorithms is that they consider a system as a black box. Fig. 2.3 shows that the optimisers only provide the system with variables and observe the outputs. The optimisers then iteratively and stochastically change the inputs of the system based on the feedback (output) obtained so far until satisfaction of an end criterion. The process of changing the variables based on the history of outputs is defined by the mechanism of an algorithm. For instance, PSO saves the best solutions obtained so far and encourages new solutions to relocate around them. Operating/environmental conditions

Variables

Objective System (black box)

Feasibility

Optimiser

Figure 2.3: Stochastic population-based optimisers consider the system as black box A general classification of the algorithms in this field is based on the number of candidate solutions that is improved during optimisation. An algorithm may start and perform the optimisation process by single or multiple random solutions. In the former case the optimisation process begins with a single random

2. Related work

17

solution, and it is iteratively improved over the iterations. In the latter case, a set of solutions (more than one) is created and improved during optimisation. These two families are called individual-based and population-based algorithms and illustrated in Fig. 2.4.

(a) Individual-based stochastic optimisation

(b) Population-based stochastic optimisation

Figure 2.4: Individual-based versus population-based stochastic optimisation algorithms There are several advantages and disadvantages for each of these families. Individual-based algorithms need less computational cost and function evaluation but can suffer from premature convergence. Premature convergence refers to the stagnation of an optimisation technique in local optima, which prevents it from convergence towards the global optimum. Fig. 2.4 shows that the single candidate solution becomes entrapped in the local optimum which is very close the the global optimum. In contrast, population-based algorithms have a greater ability to avoid local optima since a set of solutions are involved during optimisation. Fig. 2.4 illustrates how the collection of candidate solutions result in finding the global optimum. In addition, information can be exchanged between the candidate solutions and assist them to overcome the above-mentioned difficulties of search spaces. However, high computational cost and the need for more function evaluations are two major drawbacks of population-based algorithms. The well-known algorithms in the individual-based family are: Tabu Search (TS) [70, 77, 78], hill climbing [41], Iterated Local Search (ILS) [117], and Sim-

18

2. Related work

ulated Annealing (SA) [102, 25]. TS is an improved local search technique that utilises short-term, intermediate-term, and long-term memories to ban and truncate unpromising/repeated solutions. Hill climbing is also another local search and individual-based technique that starts optimisation from a single solution. This algorithm then iteratively attempts to improve the solution by changing its variables. ILS is an improved hill climbing algorithm to decrease the probability of entrapment in local optima. In this algorithm, the optimum obtained at the end of each run is retained and considered as the starting point in the next iteration. Initially, the SA algorithm tends to accept worse solutions proportionally to a variable called the cooling factor. This assists SA to promote exploration of the search space and prevents it becoming trapped in local optima when it does search them. Although different improvements of individual-based algorithms promote local optima avoidance, the literature shows that population-based algorithms are better in handling and alleviating this problem. Regardless of the differences between population-based algorithms, the common characteristic is the separation of the optimisation process into two, conflicting goals: exploration versus exploitation [62]. Exploration encourages candidate solutions to change abruptly and stochastically. This mechanism improves the diversity of solutions and causes greater exploration of the search space. In PSO, for instance, the inertial weight maintains the tendency of particles toward their previous directions and emphasises exploration. In GA, a high probability of cross-over causes more combination of individuals and is the main mechanism for exploration. In contrast, exploitation aims at improving the quality of solutions by locally searching around the promising solutions obtained in the exploration. In exploitation, candidate solutions are obliged to change less suddenly and search locally. In PSO, for instance, a low inertial rate causes low exploration and a higher tendency toward the best personal/global solutions obtained. Therefore, the particles converge toward best points instead of churning around the search space. The mechanism that brings GA exploitation is mutation. Mutation causes slight random changes in the individuals and local search around the candidate solutions. Exploration and exploitation are two conflicting goals where promoting one generally results in degrading the other [4]. A correct balance between these two goals can guarantee a very accurate approximation of the global optimum

2. Related work

19

using population-based algorithms. On the one hand, mere exploration of the search space prevents an algorithm from finding an accurate approximation of the global optimum. On the other hand, mere exploitation results in local optima stagnation and low quality of the approximated optimum. Due to the unknown shape of the search landscape for optimisation problems, in addition, there is no clear and accurate timing for transition between these two goals. Therefore, population-based algorithms balance exploration and exploitation to firstly find a rough approximation of the global optimum, and then improve its accuracy. The general framework of population-based algorithms is almost identical. ~ = {X ~1, X ~ 2 , ..., X~n } The first step is to generate a set of random initial solutions ( X ). Each of these solutions is considered as a candidate solution for a given problem, assessed by the objective function, and assigned an objective value ~ = {O1 , O2 , ..., On }). The algorithm then combines/moves/updates the can(O didate solutions based on their fitness values with the hope to improve them. The solutions created are again assessed by the objective function and assigned their relevant fitness values. This process is iterated until satisfaction of an end condition. At the end of this process, the best solution obtained is reported as the best approximation for the global optimum. Recently, many population-based algorithms have been proposed. They can be classified into three main categories based on the source of inspiration: evolution, physical, or swarm. Evolutionary algorithms are those who mimic the evolutionary processes in nature. Some of the most popular proposed evolutionary algorithms are GA [90, 89], DE [154], ES [139, 138], and EP [70, 175]. There are also many swarm-based algorithms. Two of the most popular ones are ACO [35] and PSO [60]. The third class of algorithms is inspired from physical phenomena in nature. The most recent algorithms in this category are: Gravitational Search Algorithm (GSA) [134] and Chemical Reaction Optimisation (CRO) [108]. In addition to the above-mentioned algorithms, there are also other population-based algorithms with different sources of inspiration [16, 17, 79]. As the above paragraphs show, there are many algorithms in this field, which indicates the popularity of these techniques in the literature. If we consider the hybrid, multi-objective, discrete, and constrained methods, the number of publications will increase dramatically. The reputation of these algorithms is due to several reasons. Firstly, simplicity is the main advantage of the population-based

20

2. Related work

algorithm. The majority of algorithms in this field follow a simple framework and have been inspired from simple concepts. Secondly, these algorithms consider problems as black boxes, so they do not need derivative information of the search space in contrast to mathematical optimisation algorithms. Thirdly, local optima avoidance of population-based stochastic optimisation algorithms is very high, making them suitable for practical applications. Lastly, population-based algorithms are highly flexible, meaning that they are readily applicable for solving different optimisation problems without structural modifications. In fact, the problem representation becomes more important than the optimiser when using population-based algorithms. The application of these algorithms can be found in science and industry as well [39, 30]. Despite the merits of these optimisers, there is a fundamental question here as whether there is any optimiser for solving all optimisation problems. According to the NFL theorem [169], there is no algorithm for solving all optimisation problems. This means that an optimiser may perform well in a set of problems and fail to solve a different set of problems. In other words, the average performance of optimisers is equal when considering all optimisation problems. Therefore, there are still problems that can be solved by new optimisers better than the current optimisers. Although meta-heuristics are very efficient, optimisation of real problems is not as easy as just applying algorithms and involves many difficulties that should be considered: expensive computational cost of function evaluations, constraints, multiple objectives, and uncertainties. Since most of the population-based algorithms search for a single solution to maximise of minimise an objective, they cannot be used for solving problems with multiple objectives. However, most real problems have more than one objective to be optimised. A single-objective algorithm can be applied to such problems, but multiple objectives must first be aggregated to a single objective, and then a single objective optimisation algorithm needs to be run multiple times to find the best trade-offs between objectives. Another drawback of this method is that it cannot solve all types of multi-objective problems as will be discussed and explained in detail in the following sections.

2. Related work

2.2

21

Evolutionary Multi-objective optimisation

There are different challenges in solving real engineering problems, which need specific tools to handle them. One of the most important characteristics of real problems is multi-objectivity. A problem is called multi-objective if there is more than one objective to be optimised. There are two common approaches for handling multiple objectives: a priori versus a posteriori [119, 22]. The former class of optimisers combines the objectives of a multi-objective problem to a single-objective with a set of weights (provided by decision makers), which defines the importance of each objective, and employs a single-objective optimiser to solve it. The unary-objective nature of the combined search spaces allows finding a single solution as the optimum. In contrast, a posterior methods maintain the multi-objective formulation of the multi-objective problems, allowing exploration of the behaviour of the problems across a range of design parameters and operating conditions compared to a priori approaches [42]. In this case, decision makers will eventually choose one of the solutions obtained based on their needs. There is also another way of handling multiple objectives called the progressive method, in which decision makers’ preferences about the objectives are considered during optimisation [21]. In contrast to single-objective optimisation, there typically might be no single solution when considering multiple objectives as the goal of the optimisation process. In this case, a set of optimal solutions, which represents various tradeoffs between the objectives, is the “solution” of a multi-objective problem [31]. Before 1984, mathematical multi-objective optimisation techniques were popular among researchers in different fields of study such as applied mathematics, operation research, and computer science. Since the majority of the conventional approaches (including deterministic methods) suffered from stagnation in local optima, however, such techniques were not as widely applicable as they are nowadays. In 1984, a revolutionary idea was proposed by David Schaffer [32]. He introduced the concept of multi-objective optimisation in stochastic (including evolutionary and heuristic) optimisation techniques. Since then a significant number of researches has been dedicated to developing and evaluating multi-objective evolutionary/heuristic algorithms. The advantages of stochastic optimisation techniques such as their gradient-free mechanism and local optima avoidance

22

2. Related work

made them readily applicable to real problems as well. Nowadays, the application of multi-objective optimisation techniques can be found in many different fields of studies: e.g. mechanical engineering [100], civil engineering [118], chemistry [133], and other fields [30]. Without loss of generality, multi-objective optimisation can be formulated as a minimisation problem as follows: M inimise :

F (~x) = {f1 (~x), f2 (~x), ..., fo (~x)}

(2.5)

Subject to :

gi (~x) ≥ 0, i = 1, 2, ..., m

(2.6)

hi (~x) = 0, i = 1, 2, ..., p

(2.7)

lbi ≤ xi ≤ ubi , i = 1, 2, ..., n

(2.8)

where o is the number of objective functions, m is the number of inequality constraints, p is the number of equality constraints, g shows the inequality constraints, h indicates the equality constraints, and [lbi , ubi ] are the boundaries of the i-th variable. In single-objective optimisation, solutions can be compared easily due to the unary objective function. For minimisation problems, solution x is better than y if and only if x < y. However, the solutions in a multi-objective space cannot be compared by the inequality relational operators due to multiple comparison metrics. In this case, a solution is better than (dominates) another solution if and only if it shows better or equal objective value on all of the objectives and provides a better value in at least one of the objective functions. The concepts of comparison of two solutions in multi-objective problems were first proposed Francis Ysidro [61] and then extended by Vilfredo Pareto [130]. Without loss of generality, the mathematical definition of Pareto dominance for a minimisation problem is as follows [29] Definition 2.2.1 (Pareto Dominance): Suppose that there are two vectors such as: ~x = (x1 , x2 , ..., xk ) and ~y = (y1 , y2 , ..., yk ).

2. Related work

23

Vector ~x dominates vector ~y (denote as ~x ≺ ~y ) iff: ∀i ∈ (1, 2, ..., o) [fi (~x) ≤ fi (~y )] ∧ [∃i ∈ 1, 2, ..., o : fi (~x) < fi (~y )]

f2

Minimise

Fig. 2.5 illustrates the concept of Pareto dominance. It can be seen in this

f1 Minimise

Figure 2.5: Pareto dominance figure that the circles dominate some of the other solutions (squares) since they show lesser values on both of the objectives. However, a circle shows a lesser value on one objective and greater value on another, compared to other circles, meaning that it cannot dominate them. The definition of Pareto optimality is as follows [125] Definition 2.2.2 (Pareto Optimality): A solution ~x ∈ X is called Paretooptimal iff: {6∃ ~y ∈ X|~y ≺ ~x} Definition 2.2.3 (Pareto optimal set): The set of all Pareto-optimal solutions: P S := {~x, ~y ∈ X|6∃ ~y ≺ ~x} A set containing the corresponding objective values of Pareto optimal solutions in the Pareto optimal set is called the Pareto optimal front. The definition of the Pareto optimal front is as follows:

24

2. Related work

Definition 2.2.4 (Pareto optimal front): A set containing the value of objective functions for Pareto solutions set: ∀i ∈ (1, 2, ..., o) P F := {fi (~x)|~x ∈ P S}

f2

x2

Minimise

The Pareto optimal set and Pareto optimal front of Fig. 2.5 are shown in Fig. 2.6.

x1 Pareto set: {

,

,

,

,

,

f1 ,

}

Pareto front: {

,

,

,

,

,

,

}

Minimise

Figure 2.6: Pareto optimal set versus Pareto optimal front The ultimate goal of multi-objective optimisation algorithms (via a posteriori methods) is to find a very accurate approximation of the true Pareto optimal solutions with the highest diversity. This allows decision makers to have a diverse range of design options. As mentioned above, in the past, the solution of multi-objective problems would have been undertaken by a priori aggregation of objectives into a single objective. However, this method has two main drawbacks [38, 99, 121]: • In order to find the Pareto optimal front, different weights with a proper distribution should be employed. However, an even distribution of the weights does not necessarily guarantee finding Pareto optimal solutions with an even distribution. • This method is not able to find the non-convex regions of the Pareto optimal front because negative weights are not allowed and the sum of all

2. Related work

25

the weights should be constant. In other words, the convex sum of the objectives is usually used in conventional aggregation methods There are some works in the literature that tried to improve this method. For example Parsopoulos and Vrahatis used two dynamic weighted aggregations [131]: • Dynamic Weighted Aggregation (DWA): In this method, the weights are changed gradually over the course of iterations. • Bang-Bang Weighted Aggregation (BWA): The weights are abruptly changed as the iteration index increases. It should be noted here that Chebyshev decomposition solves the non-convex issue and is used in many modern decomposition-based approaches. However, aggregation methods still need to be run many times to approximate the whole set of Pareto optimal solutions because there is only one best solution obtained in each run. This method is illustrated in Fig. 2.7 (left). According to Deb [42], the multi-objective optimisation process utilising meta-heuristics deals with overcoming many difficulties such as infeasible areas, local fronts, diversity of solutions, and isolation of the optimum. It can be seen in Fig. 2.7 (left) that an a priori method should deal with all these difficulties in each run. However, maintaining the multi-objective formulation of problems brings some advantages. First, information about the search space is exchanged between the search agents, as illustrated in Fig. 2.7 (right). It can be seen that the information exchanges bring quick movement towards the true Pareto optimal front. Second, the multi-objective approaches assist in approximating the whole true Pareto optimal front in a single run. Finally, maintaining the multi-objective formulation of a problem allows the exploration of the behaviour of the problems across a range of design parameters and operating conditions, but requires the use of more complex meta-heuristics and a need to address conflicting objectives. In general, the majority of the most well-known heuristic algorithms have been extended to solve multi-objective problem. In the following paragraphs the most popular and recent ones are briefly presented. Early years of multi-objective stochastic optimisation saw conversion of different single-objective optimisation techniques to multi-objective algorithms. Some of the most well-known stochastic optimisation techniques proposed so far are: • Strength-Pareto Evolutionary Algorithm (SPEA) [180, 183]

26

2. Related work Local front

Infeasible area

Pareto front

Local front

Infeasible area

Pareto front

Initial start points in each run

f2

Maximise

f2

Maximise

Initial population Pareto solutions Information exchange Quick movement

f1

f1

Maximise

Maximise

Figure 2.7: A priori method versus a posteriori methods [42] • Non-dominated Sorting Genetic Algorithm [153] • Non-dominated sorting Genetic Algorithm version 2 (NSGA-II) [45] • Multi-Objective Particle Swarm Optimisation (MOPSO) [28] • Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) [177] • Pareto Archived Evolution Strategy (PAES) [104] • Pareto-frontier Differential Evolution (PDE) [2] The literature shows that the most popular multi-objective meta-heuristic is Non-dominated Sorting GA (NSGA-II) [45], which is a multi-objective version of the well-known GA algorithm [89, 82]. This algorithm was proposed to alleviate the three problems of the first version [153]. These problems are: high computational cost of non-dominated sorting, lack of considering elitism, and lack of a sharing parameter (different from niching). In order to alleviate the aforementioned problems, NSGA-II utilises a fast non-dominated sorting technique, an elite-keeping technique, and a new niching operator which is parameterless as follows: • Fast non-dominated sorting: The non-dominated sort of NSGA is of order O(M N 3 ) where N and M are the number of individuals in the population

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27

and the objective functions, respectively. This order is due to the comparison of all individuals based on all the objective functions together (M N 2 ) and sorting in non-dominated levels (M N 2 × N ). In the new method a hierarchical model of non-dominated levels has been proposed to reduce this computational cost whereby we do not need to compare all the dominated individuals after the first non-dominated level (O(M N 2 )). There are two counters for each individual which show how many individuals dominate it and how many individual it dominates. These counters help to build the domination levels. • Elite-keeping technique: Elitism is automatically achieved due to the comparison of the current population with the previously found best nondominated solutions. • New niching operator (crowding-distance): The nearest neighbour density estimates the perimeter of a rectangle (cube or hypercube) neighbourhood, which is formed by the nearest neighbours. The individuals with a higher value for this measure are selected as the leaders. In the niching technique a diameter (σshare ) should be defined, and the results are highly dependent on the diameter. However, there is no parameter to define in the proposed operator. The NSGA-II algorithm starts with a random population. The individuals are grouped based on the non-dominated sorting method. The fitness of each individual is defined based on its non-domination level. The second population is created by selection, recombination, and mutation operators. Both populations create a large new population. This new population is then sorted again by the non-dominated sorting approach. The higher the non-domination level, the higher the priority to be selected as a new individual for the final population. The process of selecting the non-dominated individuals should be repeated until a population with the same size as the initial population is constructed. Finally, these steps are run until the satisfaction of an end criterion. The second most popular multi-objective meta-heuristic is Multi-Objective Particle Swarm Optimisation (MOPSO). The MOPSO algorithm was proposed by Coello Coello [28, 33]. Following the same concepts as PSO, it employs a number of particles, which fly around in the search space to find the best solution. Meanwhile, they all trace the best location (best solution) in their paths [147].

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In contrast to PSO, there is, of course, no single “best” solution to track. In other words, particles must consider their own non-dominated solutions (pbest) as well as one of the non-dominated solutions the swarm has obtained so far (gbest) when updating position. An external archive is generally used for storing and retrieving the Pareto-optimal solutions obtained. In addition, a mutation operator called turbulence is also embedded in MOPSO in some cases to increase randomness and promote diversity of trial solutions. A comprehensive survey of the PSO-based multi-objective optimisers can be found in [140]. The external archive of MOPSO is similar to the adaptive grid in Pareto Archived Evolution Strategy (PAES) [104] as it has been designed to save the non-dominated solutions obtained so far. It has two main components: an archive controller and a grid. The former component is responsible for deciding if a solution should be added to the archive or not. If a new solution is dominated by one of the archive members it should be omitted immediately. If the new solution is not dominated by the archive members, it should be added to the archive. If a member of the archive is dominated by a new solution, it has to be replaced by the new solution. Finally, if the archive is full the adaptive grid mechanism is triggered. The grid component is responsible to keep the archive solutions as diverse as possible. In this method the objective space is divided into several regions. If a newly obtained solution lies outside the grid, all the grid locations should be recalculated to cover it. If a new solution lies within the grid, it is directed to the portion of the grid with the lowest number of particles. The main advantage of this grid is the low computational cost compared to niching (In worst case when the grid must be updated in each iteration it is the same as niching O(N 2 ) ). MOPSO has a very fast convergence speed which could make it prone to premature termination with a false Pareto optimal front in multi-objective optimisation. The mutation strategy is helpful in this case. The mutation strategy randomly affects not only particles in the swarm but also the design variables of problems. The mutation rate decreases over the course of iterations. The MOPSO algorithm handles constraints whenever two solutions are being compared. In comparing two feasible solutions the non-dominance comparison is applied directly. In comparing a feasible and an infeasible solution the feasible solution is selected. Among two infeasible solutions the solution with less constraint violation is chosen.

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The MOPSO algorithm starts by randomly placing the particles in a problem space. Over the course of iterations, the velocities of particles are calculated. After defining the velocities, the position of particles can be updated. All the non-dominated solutions are added to the archive. Finally, the search process is terminated by satisfaction of a stopping criterion. In summary, the discussions in this subsection showed that optimisation in a multi-objective search space is much more challenging that a single-objective search space. Multi-objective optimisation has its owns difficulties that might not exist in single-objective optimisation: local fronts, high number of nondominated solutions, coverage of solution across the front, and so on. However, the two types of optimisation have several difficulties in common as well. One of the most important challenges is the existence of uncertainties in both search spaces. Uncertainties do exist as undesirable inputs in real problems with any number of objectives and have to be considered during the optimisation process to prevent the optimal solutions obtained from showing undesirable behaviours in real environments. Although uncertainties are a common difficulty in both single- and multi-objective problems, they require different approaches in order to be handled due to the different nature of these two types of problems. Therefore, the following two sections discuss single-objective and multi-objective robust optimisation methods respectively.

2.3

Robust single-objective optimisation

One of the key concepts in the optimisation of real problems is robustness. Robust optimisation refers to the process of finding optimal solutions for a particular problem that have least variability to probable uncertainties. Uncertainties are unavoidable in the real world and occur in different aspects of a system: operating/environmental conditions, parameters, outputs, and constraints. Generally speaking, robust optimisation refers to the process of considering any types of uncertainties during optimisation. In the literature, this term has mostly referred to handling uncertainties in the parameters of a problem. In this thesis, however, the term “robust optimisation” is used for considering any type of uncertainties. There are different classifications in the literature for categorising uncertainties [14, 176]. The classification provided by Beyer and Sendhoff [14] is utilised, in which the uncertainties are categorised based on

30

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their sources, as follows: 1. Type A: this uncertainty occurs in the environmental and operating conditions. Perturbation in speed, temperature, moisture, angle of attack in airfoil design, and speed of the vehicle in propeller design are some examples of this type of uncertainty. 2. Type B: in this case the parameters of the problem may change. One of the major sources of this kind of uncertainty is manufacturing tolerance. 3. Type C: in this case the system itself produces noises. The uncertainty of the outputs of a system is caused by Type A and Type C uncertainties. It might be due to sensory measurement errors or randomised simulations. Time-varying (dynamic) systems are also considered as having type C uncertainty. It should be noted that computer models (e.g. CFD) do produce errors, but, being deterministic, they do not produce noisy outputs. The source for these errors can be found either in the failure to consider uncertain parameters of type A or B during simulation, or to errors in the models, caused by a number of issues. Real world systems do produce noisy outputs, but these again are the effects of type A and B uncertainties. Fig. 2.8 shows where these three types of uncertainties happen during and after optimisation. Another very important classification is between aleatory (i.e. random) and epistemic uncertainty (i.e. due to lack of knowledge) [165].

2.3.1

Preliminaries and definitions

There are different types of uncertainties, but some of the most common types are the manufacturing errors and production perturbations. In this thesis the focus is on this type of uncertainty. In this type, the variables of a particular problem may vary after finding the optimum. This undesired fluctuation might degrade the output of the cost function significantly. Without loss of generality, a robust optimisation problem with respect to the perturbation in the variables (parameters) is formulated as follows: M inimise :

f (~x + ~δ)

(2.9)

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Parameters Type B: parameters

Type A: operating conditions

Environment

Type C: outputs

System

Type D: constraints

Figure 2.8: Different categories of uncertainties and their effects on a system: Type A, Type B, and Type C

Subject to :

gi (~x + ~δ) ≥ 0, i = 1, 2, ..., m

(2.10)

hi (~x + ~δ) = 0, i = 1, 2, ..., p

(2.11)

lbi ≤ xi + δi ≤ ubi , i = 1, 2, ..., n

(2.12)

where ~x is the set of parameters, ~δ indicates the uncertainty vector corresponding to each variable in ~x, o is the number of objective functions, m is the number of inequality constraints, p is the number of quality constraints, [lbi , ubi ] are the boundaries of the i-th variable. It should be noted that ~δ in Equation 2.9 is a stochastic (random) variable with a given (known or unknown) probability density function, and not a deterministic variable. The general concepts of a robust solution that is not sensitive to uncertainties are illustrated in Fig. 2.9. In this figure, the horizontal axis shows a parameter and the vertical axis is the objective function. There are two valleys (one global and one local) in this figure and the objective is the minimisation of f . The left valley is the global optimum, whereas the right valley is the robust optimum. The reason for the greater robustness of the right valley

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f

Δ’ Δ Robust solution

Non-robust solution p

δ

δ

δ

δ

Figure 2.9: Conceptual model of a robust optimum versus a non-robust optimum. The same perturbation level (δ) in the parameter (p) results in different changes 0 (∆ and ∆ ) in the objective (f ) is its lesser sensitivity to δ error in the variable x compared to the left valley. Note that a robust solution should be an acceptable solution as well. Fig. 2.9 0 clearly shows that δ in the parameter causes ∆ and ∆ in the left and right 0 valleys, respectively. What makes the right valley robust is that ∆ > ∆ . In robust optimisation such solutions are fruitful. The same concepts are valid when considering uncertainties in operation conditions. In such circumstances, perturbations in the operating conditions might cause greater change (∆) or lesser 0 change ∆ in the output of the system. Handling uncertainties in parameters is mostly undertaken, in the literature, by investigating the behaviour of a neighbourhood of solutions in the the objective space. In the literature of population-based stochastic optimisation techniques, the robustness of each individual should be verified in every iteration. There are two main approaches proposed so far in order to require populationbased stochastic algorithm to handle uncertainties in parameters as follows [44] • Replacing objective functions with an expectation measure

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33

• Adding a new constraint called the variance measure • Formulating a multi-objective problem considering both expectation and variance measures The first two approaches are discussed with their recent developments in the following subsections. It should be noted that there are numerous examples of optimisation in the presence of uncertainty being dealt with by solving multiobjective (expectation vs. variance) optimisation problems as well [98, 76, 146, 59, 107].

2.3.2

Expectation measure

As discussed above, an expectation measure defines the robustness of search agents during optimisation for stochastic optimisation algorithms. So, search agents are no longer evaluated by the main objective function, and the evaluator is the expectation measure. The mathematical formulation of robust optimisation using an expectation measure is as follows [44] Z 1 f (~y )dy (2.13) M inimise : E(~x) = |Bδ (~x)| ~y∈Bδ (~x) where Bδ (~x) shows the δ-radius neighbourhood of the solution ~x and |Bδ (~x)| indicates the hypervolume of the neighbourhood. It should be noted here that in the literature, the uncertainty sets and scenario sets are referred to in order to explicitly acknowledge the potentially asymmetric, and location dependent, nature of the uncertainty at any design location. Due to the existence of Bδ (~x) in Equation 2.13, however, this definition does not allow for this and is therefore limited to a subset of design problems. It may be inferred from Equation 2.13 that the expectation measure is the analytical integration of the main objective function over the maximum possible perturbation in the parameters. This equation is applicable to problems with known integration of the search space. For real problems with unknown search space, however, the analytical integration of the search space is impossible to calculate. In this case, the integration is approximated by the Monte Carlo method as follows: E(~x) =

H 1 X f (~x + δ~i ) H i=1

(2.14)

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where H is the number of samples. This method approximates the integration by perturbing the variables and calculating the average of the objective values. Fig. 2.10 illustrates the search space of an objective function and its corresponding expectation measure. This figure shows how a non-robust global optimum is considered as a local optimum when using the expectation measure. In contrast, a robust optimum has the potential to behave as the global optimum using an expectation measure as the right valley in Fig. 2.10 shows.

f

Expectation measure (E)

Main objective function (f)

p

δ

δ

δ

δ

Figure 2.10: Search space of an expectation measure versus its objective function Deb and Gupta named this method “Type I” robust optimisation [44]. In this case, the robust optimisation starts by creating a set of random candidate solutions for a particular problem. Every candidate solution is evaluated by the average of H generated random solutions around it. The random solutions are created in the hypervolume of δmax around the solutions where δmax indicates the maximum possible perturbation. In the literature, there are also other different expectation measures for improving the performance of robust meta-heuristics [43, 74, 94, 106, 159, 9]. Expectation measures have been optimised mostly instead of the main objective function. Some studies [135, 115, 5], however, consider the expectation measure

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(or other similar robustness indicators) as an additional objective and convert the problem to a multi- or many-objective problem. In this case, the primary objective is to minimise/maximise a cost function, and the secondary objective is to minimise the perturbation using the expectation measure. The Pareto optimal front obtained represents trade-offs between the objective function and expectation measure.

2.3.3

Variance measure

In this method a constraint is employed to confirm the robustness of the solutions during optimisation. A variance measure that indicates the variance of H generated random solutions in the neighbourhood of a search agent is utilised mostly in the literature. In this case, a robust algorithm optimises the original objective function, but it is subject to satisfying the robustness constraint calculated by the variance measure. Technically speaking, H random solutions are generated in the hypervolume of δ around the solutions in the parameter space where δ indicates the maximum possible perturbation. Then, the variance of the corresponding objective values should not exceed a threshold (η). The mathematical formulation of robust optimisation using a variance measure is as follows [44]: M inimise :

f (~x)

Subject to :

V (~x) =

(2.15)

||F (~x) − f (~x)|| ≤η ||f (~x)||

(2.16)

where F (~x) can be selected as the effective mean or worst function value among the H selected solutions and η is a vector of thresholds in [0,1]. Note that || || in this equation can be any norm measure. It may be seen in the Equation 2.16 that the normalised fluctuation of the objective is considered as the constraint. The robust solutions are favoured as η decreases. Fig. 2.11 illustrates the effects of the variance measure on the search space as a constraint. This figure shows that the regions of the search space that show greater fluctuations are considered as infeasible. Therefore, a solution becomes infeasible if it enters such regions.

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f

Non-robust (infeasible) regions

p

δ δ

δ

δ

Figure 2.11: Conceptual model of infeasible regions when employing a variance measure Deb and Gupta first named methods that utilise variance measures as “Type II” robust optimisation. After the proposal of this type of robust optimisation [44], different variance measures were proposed in order to improve the performance of robust meta-heuristics [74, 94]. The disadvantage of this method is that a robust meta-heuristic should be equipped with a constraint handling method to be able to find the robust optimum. In summary, both measures quantify the robustness of solutions during optimisation. This assists optimisation techniques to quantitatively compare the solutions and favour robust ones. The expectation measures change the shape of the search space and smooth out the global optimum. Some of them use Monte Carlo approximation to investigate the landscape around a solution. The advantage of these methods is replacement of the main objective by the expectation measure and separation of robust measure from the optimisation algorithm. However, they may change the shape of the search landscape and affect the expected behaviour of an algorithm. Considering expectation measure as an additional objective gives us solutions with different levels of robustness but increases the difficulty of optimisation due to the additional objective. Note that

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there are also other, cheaper approximations for expectation than Monte Carlo in the literature such as Polynomial Chaos [173, 59], Collocation Methods [172, 63], etc. Variance measures do not change the search space but bring an additional constraint. This means that an algorithm should be equipped with a constraint handling technique to be able to work with a variance measure. A variance measure has the potential to make a search space with dominated infeasible regions, which will definitely need special considerations. A search space with a large infeasible portion is very likely to result in having many infeasible search agents in each iterations. The problem here is that by default, most of the metaheuristics discard infeasible solutions and only rely on feasible solutions to drive the search agents towards optimal solution(s). Therefore, a powerful constraint handling method should be utilised when solving such problems. Despite the success of both expectation and variance measures in assisting algorithms for finding robust solutions, they both suffer from the need for additional function evaluation. All of the current measures become unreliable as the number of additional sampled points decreases. This is the main gap in the literature at present. In addition, most of the current work only concentrates on single-objective search spaces and there should be more work in the literature about computationally cheap robust optimisation in multi-objective search spaces. Comparison of solutions with these two metrics are different when considering single or multiple objectives. In a single-objective search space, there is only one objective, so the solutions can be compared easily with the value of robustness indicators. Due to the nature of single-objective problems, there is one global robust optimum. In a multi-objective problem, however, the solutions cannot be compared with the robustness indicators across only one objective due to the presence of multiple objectives. In this case robustness should be calculated across all objectives and then the solutions can be compared using Pareto dominance operators. Due to the nature of such problems, there is a set of robust solutions (robust Pareto optimal solutions) as the robust designs. Considering robustness and multiple objectives make the whole optimisation process very challenging. The following section presents the preliminaries and reviews the literature of robust multiobjective optimisation approaches.

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Robust optimisation is important in both single- and multi-objective search spaces. There are still many single-objective problems, which are used widely in science and industry. Needless to say, failure in considering uncertainties in such problems may result in undesired output(s). Considering multiple objectives is also essential, but requires special considerations. A robust multi-objective optimiser should look for the best trade-offs between objectives while considering their robustness. More challenges when finding robust solutions in a multiobjective search space does not mean it is less important than a single-objective search space.

2.4

Robust multi-objective optimisation

This section reviews the literature of Robust Multi-Objective Optimisation (RMOO).

2.4.1

Preliminaries and definitions

As mentioned above, there are three main types of uncertainty in a system, of which perturbation in the parameters can be considered as the most important one. In the following paragraphs, this type of uncertainty (Type B) is discussed in the context of multi-objective search spaces. There is a set of robust solutions for a multi-objective problem because of its nature. Without loss of generality, the robust multi-objective optimisation considering uncertainties in the parameters is formulated as a minimisation problem as follows: M inimise :

F (~x + ~δ) = {f1 (~x + ~δ), f2 (~x + ~δ), ..., fo (~x + ~δ)}

(2.17)

Subject to :

gi (~x + ~δ) ≥ 0, i = 1, 2, ..., m

(2.18)

hi (~x + ~δ) = 0, i = 1, 2, ..., p

(2.19)

lbi ≤ xi ≤ ubi , i = 1, 2, ..., n

(2.20)

2. Related work

39

where ~x is the set of parameters, ~δ indicates the uncertainty vector corresponding to each variable in ~x, which is a stochastic (random) variable with a given (known or unknown) probability density function, o is the number of objective functions, m is the number of inequality constraints, p is the number of quality constraints, [lbi , ubi ] are the boundaries of the i-th variable. In robust single-objective optimisation, there is a single robust solution that might be either the global or a local optimum. The ultimate goal is to find the best solution that is not sensitive to the probable uncertainties. Since there is one comparison criterion (the objective function), solutions can be compared easily with inequality/equality operators. In robust multi-objective optimisation, however, two solutions cannot be compared with similar operators as in robust single-objective optimisation. This is due to the fact that two solutions in a multi-objective search space might be incomparable (non-dominated) with respect to each other. In this case, there is a new concept of comparison called robust Pareto dominance. Without loss of generality, the mathematical definition of robust Pareto dominance for a minimisation problem is as follows: Definition 2.4.1 (Robust Pareto Dominance): Suppose that there are two vectors such as: ~x = (x1 , x2 , ..., xk ) and ~y = (y1 , y2 , ..., yk ). Vector ~x dominates vector ~y (denote as ~x ≺ ~y ) iff: ∀i ∈ (1, 2, ..., o) [fi (~x + ~δ) ≤ fi (~y + ~δ)] ∧ [∃i ∈ (1, 2, ..., o) : fi (~x + ~δ) < fi (~y + ~δ)] where k is the number of variables, o is the number of objectives, and ~δ = {δ1 , δ2 , ..., δk }. It should be noted that tolerances are not constant across design space in definition 2.4.1. This definition shows that a solution is able to dominate another if and only if it shows better or equal values on all objectives of which at least one of them is better considering perturbations in the parameters. With this definition, robust Pareto optimality can be defined as follows: Definition 2.4.2 (Robust Pareto Optimality): A solution ~x ∈ X is called Paretooptimal iff: {6∃ ~y ∈ X|~y ≺ ~x}

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A set containing all the non-dominated robust solutions (robust Pareto optimal solutions) is the robust answer to a multi-objective problem and defined as follows [81]: Definition 2.4.3 (Robust Pareto optimal set): The set of all Pareto-optimal solutions: RP S := {~x, ~y ∈ X|6∃ ~y ≺ ~x} The projection of the robust Pareto optimal set in the objective space is called the robust Pareto optimal front and defined as follows: Definition 2.4.4 (Robust Pareto optimal front): A set containing the value of objective functions for Pareto solutions set: ∀i ∈ (1, 2, ..., o) RP F := {fi (~x)|~x ∈ RP S}

Minimize (f1)

As may be inferred from the first two definitions, a solution is able to robustly dominate another if and only if it is compared to all perturbations and found to be better or equal under all of them. The concepts of robustness in a multiobjective search space are illustrated in Fig. 2.12.

S1

f2

x2

Robust solution S2

S4 S3

x1

f1 Minimize (f2)

Figure 2.12: Concepts of robustness and a robust solution in multi-objective search space

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41

This figure shows the mapping from a 2D parameter space to a 2D objective space. It may be observed that equal perturbations in the parameters x1 and x2 result in different variations in the objective space. As shown in Fig. 2.12, solution S2 has the least sensitivity to perturbations, meaning that this solution is the most robust solution. However, there are other robust solutions (S1, S3, and S4 ) with different sensitivities to the perturbations. This figure shows that the Robust Pareto optimal Front (RPF) may consist of both non-dominated and/or dominated solutions. Depending on the robustness of solutions, the RPF may also consist of dominated solutions completely. Generally speaking, there are four possible scenarios for the RPF in comparison with the Pareto optimal front [44, 69, 75, 83]: (a) The Pareto optimal front is totally robust (b) A part of the Pareto optimal front is robust (c) The Pareto optimal front is partially robust (d) The RPF is completely dominated by the Pareto optimal front These four situations are illustrated in Fig. 2.13. In order to handle uncertainties in parameters, three main methods are used in the literature. Firstly, the average of fitness functions in a pre-defined neighbourhood around a solution is calculated and optimised as the expected fitness function instead of the original objective functions. Secondly, the variation of the fitness function is investigated in a pre-defined neighbourhood around a solution and considered an additional constraint for the problem. This assists robust optimisers to ignore non-robust solutions with high variation in objectives (which become infeasible) during optimisation. Thirdly, the expected fitness function or any other robustness indicators are added to a problem as new objectives to be optimised. In this case the final Pareto optimal front will represent different trade-offs between other objectives and robustness. The first type of robustness handling (type I) is formulated as follows [44]: M inimise :

f1ef f (~x), f2ef f (~x), .., foef f (~x)

(2.21)

Subject to :

~x ∈ S

(2.22)

f2

Minimise

2. Related work

f2

Minimise

42

Robust front Pareto front and robust front Pareto front f1

f1

Minimise

Minimise

f2

b) A part of Pareto font is robust

Minimise

f2

Minimise

a) Pareto front is totally robust

Robust front Pareto front

Pareto front Robust front f1

f1

Minimise

Minimise

c) A part of Pareto front is robust, but there are other robust solutions

d) The Pareto front is not robust at all, so the robust front consists the local front(s)

Figure 2.13: Four possible robust Pareto optimal fronts with respect to the main Pareto optimal front

gi (~x) ≥ 0, i = 1, 2, ..., m

(2.23)

hi (~x) = 0, i = 1, 2, ..., p

(2.24)

lbi ≤ xi ≤ ubi , i = 1, 2, ..., n

(2.25)

2. Related work

where :

fief f

H 1 X fi (~x + δ~j ) = H j=1

43

(2.26)

where S is the feasible search space, H is the number of samples, o is the number of objectives, ~x is the set of parameters, ~δ indicates the uncertainty vector corresponding to each variable in ~x, o is the number of objective functions, m is the number of inequality constraints, p is the number of equality constraints, [lbi , ubi ] are the boundaries of the i-th variable. As may be seen in these equations, H random solutions are generated in the neighbourhood of a solution in order to investigate its robustness. These random solutions can be created systematically or obtained from previously sampled points during optimisation. In fact, this method tries to approximate the integration of the main objective(s) along the perturbations (δ) by a Monte Carlo sampling approach. If the mathematical formulation of the search space is known, the robustness can be confirmed by analytical integration as follows: Z 1 ef f f (~y )dy (2.27) M inimise : f (~x) = |Bδ (~x)| y∈Bδ (~x)

Subject to :

~x ∈ S

(2.28)

where S is the feasible search space, Bδ (~x) is the neighbourhood of the solution ~x within δ radius, and |Bδ (~x)| is the hypervolume of the neighbourhood. The second method of handling type B uncertainties adds an extra constraint (variance measure) to a problem as follows: M inimise :

f1 (~x), f2 (~x), .., fo (~x)

(2.29)

Subject to :

~x ∈ S

(2.30)

||f p (~x) − f (~x)|| ≤η ||f (~x)||

(2.31)

gi (~x) ≥ 0, i = 1, 2, ..., m

(2.32)

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2. Related work

hi (~x) = 0, i = 1, 2, ..., p

(2.33)

lbi ≤ xi ≤ ubi , i = 1, 2, ..., n

(2.34)

where S is the feasible search space, f p (x) can be selected as the effective mean or worst function value among H selected solutions in the neighbourhood, η is a threshold that defines the level of robustness for solutions, o is the number of objectives, ~x is the set of parameters, o is the number of objective functions, m is the number of inequality constraints, p is the number of equality constraints, [lbi , ubi ] are the boundaries of the i-th variable. In these equations, robust solutions are favoured as η decreases. Equation 2.31 is called the variance measure because it defines the normalised variation of the objectives in a neighbourhood. The η threshold can be chosen as a single value for all objectives or a different value for each objective according to decision makers’ preferences.

2.4.2

Current expectation and variance measures

The expectation measure can be any effective mean of the objective function(s). Technically speaking, a finite set of H solutions are chosen randomly or with a structured procedure in the hypervolume ([−δi , δi ]) around the solution x, and then the effective mean objectives of all samples are optimised by optimisation algorithms. Another similar method is called degree of robustness, proposed by Barrico et al. [12, 13]. Basically, the degree of robustness refers to the analysis of a solution with respect to its neighbours to find the robust solution(s). A set of solutions in the neighbourhood (kδ) around a solution (~x) is selected. This selection is subject to keeping the images of these solutions in a predefined neighbourhood ∆ around f (~x) in objective space. A hyperbox (neighbourhood) of radius δ is assumed around the solution ~x, and the radius of this hyperbox is increased in steps of 2δ, 3δ, 4δ, etc. Random solutions (of number H) are generated and analysed inside these hyperboxes. This process continues until the percentage of solutions that lies in the projection in objective space (hypersphere with the

2. Related work

45

radius of ∆) is not greater than a pre-defined threshold. The degree of robustness is proportional to the number of hyperbox enlargements required. In 2002, Ray handled perturbations in parameters by adding two external objective functions to the main objective function: 1- mean performance of neighbouring solutions and 2- standard deviation of neighbouring solutions [135]. A new constraint-handling method which is based on considering the individual feasibility and its feasibility of neighbours has been proposed as well [135]. The average and standard deviation were calculated by re-sampling. The first variance measure was also proposed by Deb and Gupta [44]. In this measure, a set of randomly created solutions in the δ−neighbourhood of x should not exceed a certain pre-defined neighbourhood ∆ around f (x) in objective space. Violation of this constraint assists us in distinguishing between a robust solution and a non-robust solution. The mathematical model is as follows:

V (~x) =

||F (~x) − f (~x)|| ≤ η , ~xS ||f (~x)||

(2.35)

where η is a threshold in [0,1], S shows the feasible search space, and F (~x) can be selected as the effective mean or worst function value among the H selected solutions. This constraint is called the variance measure because it defines the deviation of the objective function(s) in the neighbourhood of a solution. In 2003, Jin and Sendhoff used the current information of individuals in each iteration to estimate the robustness of the individuals of the next iteration, so there were no additional fitness evaluations (re-sampling) [94]. Their work is formulated to permit different uncertainty levels in design space. After estimating the robustness, they used it as another objective function to be optimised in addition to the main objective function(s). Eventually the Pareto optimal front provided the trade-offs between the performance and robustness. They proposed two methods to optimise the original function and minimise the variance. The difference between this method and other methods is that it considers the variances of design variables in addition to the variance of objective functions. The variances were calculated on the members of a neighbourhood around the individuals within distance δ. In both methods the robustness of individuals were calculated by dividing the standard deviation of neighbourhood individual’s

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fitness values and the standard deviation of variables as follows: N 1 X σfj Vi (~x) = N j=1 σxj

(2.36)

where fj indicates the objective of j-th neighbouring solution, N is the number of neighbouring solutions within the desired radius from the solution xi , σfj is the standard deviation of objectives of the j-th neighbouring solution, and σxj shows the standard deviation of the variables of the j-th neighbouring solution. In order to decrease the computational cost, the neighbourhood members were selected from the current population. Each of these estimated functions were used as another objective function to be optimised by multi-objective algorithms. The non-dominated front obtained contained the trade-offs between the performance and robustness. There are also some studies in the literature which propose hybrids of type I and type II in order to handle uncertainties as follows: A study of different approaches of handling parameters’ uncertainties for modification of canonical PSO, fully-informed PSO, multi-swarm PSO and charged PSO was performed by Dippel [58]. The effects of different topologies such as ring and fully-connected were also investigated. Both re-sampling and archiving (previously sampled points) approaches were used in this study. In the latter case, the expectation measure was as follow: N P

E(~x) =

w(x~j ).f (x~j )

j=1 N P

(2.37) w(x~j )

j=1

where x~j shows the j-th solution, N is the number of desirable solutions in the archive within radius δ from the solution ~x, w(x~j ) ∼ pdf (~δ) is a weighting function that weights the importance of the previously sampled points in terms of their distributions within δ. The sampled points that had not been used for a pre-defined period of time should be removed from the archive in this method. In 2011, Saha et al. improved the handling of type B uncertainties proposed by Deb in terms of reducing the number of function evaluations [143]. In this method the vicinity of a point in the search space was defined based on the

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47

previously evaluated solutions and an archive of size P × 100 was employed to keep the most recent unique solutions where P is the maximum number of search agents. In other words, for every search agent in the current population, there were 100 points in the archive to decide its robustness. For every search agent, H neighbourhood points were created by the Latin Hypercube Sampling (LHS) method. This set was called the reference set. Then, the closest solutions in the archive to the reference points were chosen. If H solutions close to the reference points could not be found, they had to be created by true function evaluations. After building the full neighbourhood, the robust measure could be calculated. There was also a simple modification in the variance measure as follows:

V (~x) =

max (

m=1,2,...,M

||Fm (~x) − fm (~x)|| )≤η ||fm (~x)||

(2.38)

where M is the number of objectives, Fm (x) is the mean value of m-th objectives of neighbouring solutions, and η is a threshold in [0, 1]. In 2008, Gaspar-Cunha and Covas proposed two measures for handling type B uncertainties [74]. The efficiency of combining both previously proposed metrics of type I and II robustness handling was also investigated. The measures proposed were [74, 69]: Type I: N P |f˜(x~j ) − f˜(~ xi )|

E(~ xi ) = (1 −

j=0

N

)f (~ xi )

(2.39)

Type II: N 1 X f˜(x~j ) − f˜(~ xi ) V (~ xi ) = | | , di,j < dmax N j=0 x~j − x~i

(2.40)

where di,j is the euclidean distance between agent i and j, N is the number of (x~i )−fmin those agents having distances less than dmax , f˜(~ xi ) = ffmax for maximisation, −fmin f ( x ~ )−f i min and f˜(~ xi ) = 1 − fmax −fmin for minimisation. The authors also investigated the efficiency of these robust metrics in finding robust frontiers for multi-objective problems. In order to do this, it was suggested that the metrics for each of objective functions be calculated one by one. Two

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methods for calculating the final robustness measurement in this case are effective mean and worst function value: M 1 X Vm (~ xi ) V 1(~ xi ) = M m=1

(2.41)

V 2(~ xi ) =

(2.42)

max Vm (~ xi )

m=1,...,M

where x~i is the i-th solution, Vm is the variance of neighbouring solutions in m-thobjective, and M is the maximum number of objectives. The authors applied these concepts to the Reduced Pareto Set Genetic Algorithm (RPSGA) and examined it over five test problems. The combinations of both methods were: c1 = f + V where f is the fitness function and V is the variance measure in Equation 2.40 (this combination duplicates the number of objectives), c2 = f + V 1, c3 = f + V 2, E, and f . The evaluation criteria were the percentage of peaks detected, the ability to find the fittest and more robust solutions, and the accuracy of results. The results show that both robust measures were better than previous measures. The best results of combinations were those of c1 = f + V . However, this method increases the complexity of the problem, which is a factor that should be considered. The authors recommended it for moderately sized problems. For bigger problems, the authors recommended c1 = f + V 1, based on the results. The superior results of this combination were due to the effective mean of robustness measures which included good knowledge of the landscape around the current position of particles. The measure for type II robustness introduced showed good results because the authors considered both the distance between objective values in the objective space (f˜(x~j ) − f˜(~ xi )) and variables in the parameter space (x~j − x~i ). This helped to differentiate between the samples around a particle based on the ratio of the distances in both spaces. One of the best comparative studies in hybrid methods was made by Branke in 1998 [18]. Branke introduced 10 different methods for handling uncertainties of design parameters (type B) divided into 4 different groups. These methods were applied to EA with the island model and compared over 2 benchmark functions. The four groups of robust handling approaches were: 1. Average of several evaluations: In this approach several random points in a pre-defined neighbourhood of a particular solution were created and

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49

evaluated. The average of their fitnesses would indicate the robustness. However, this approach increased the computational time due to evaluating random points. The distribution of the random points was also important. A suggestion could be to use the same distribution as the expected noise in the real environment. Note that this method is called explicit averaging in some references. 2. Single distributed evaluation: In this method the disturbed input was evaluated instead of evaluating multiple samples. 3. Re-evaluating just the best solutions: In this method all the solutions were evaluated once without perturbations. The robustness of some of the best solutions was then evaluated using re-sampling several times. So the computational time was less in this method, and time was not wasted computing robustness of useless solutions. 4. Using sampled points over past iterations: In this method the weighted mean of the previously sampled points around a solutions over the course of iterations was considered as the robustness measure. This method did not entail additional computational cost, but needed memory to save the sampled points. Note that this method is called implicit averaging in some references. Branke concluded that previously sampled points are able to provide very useful information about the robustness of solutions during optimisation. Therefore, an algorithm is able to find the robust optimum without the need for extra function evaluations subject to proper use of previously sampled points. However, this method is not reliable due to the stochastic nature of stochastic algorithms. For improving the reliability of such techniques, one of the current mechanism in the literature is to generate new neighbouring solutions and evaluate them by true function evaluations. However, true function evaluations directly increase the computational cost of an algorithm, which is a vital issue when solving real expensive problems. This section showed that the process of considering and handling uncertainties in a multi-objective search space is very challenging. This difficulty is perhaps one of the reasons of the lesser popularity of this field. The literature review of this section and its preceding indicated that there are two main robustness measures in the literature: expectation and variance.

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On one hand, expectation measures do not add additional borders to the optimiser but may change the shape of the search space. In addition, considering expectation measure as an additional objective would increase the difficulty of problem. On the other hand, a variance measure maintains the original shape of the search space, but requires a suitable constraint handling method. It should be noted that a variance measure can be considered as an objective as well. Despite the effectiveness of both of methods, they suffer from unreliability in the case of using previously sampled points and high computational cost in the case of using new sampled points. Currently, the literature lacks (simultaneously) reliable and cheap robust multi-objective optimisation approaches in both single and multi-objective search spaces. In addition, the literature lacks specific test functions and performance metrics for robust optimisation. Suitable benchmark problems allow us to compare different algorithms effectively. performance metrics are useful for quantifying the performance of algorithms. Without performance metrics all the analysis can only be made from qualitative results, which are not as accurate as quantitative results. In the field of robust optimisation, there is a negligible number of test functions and literally no specific performance metrics. This is the motivation of proposing several test problems and performance metrics in this thesis. The next two sections review the current benchmark functions and performance metrics in both global and robust optimisation fields.

2.5

Benchmark problems

Generally speaking, benchmark problems are essential for testing and challenging algorithms from different perspectives. They are involved directly in the design process of optimisation techniques. A set of suitable benchmark problems can verify the performance of an algorithm confidently and reliably. Although solving a real problem can prove the applicability of an algorithm better, a real search space is usually expensive and has a mix of characteristics which prevents us from observing different abilities of an algorithm conveniently and independently. Therefore, researchers in this field should be aware of the types and characteristics of the benchmark problems. Due to the known search space of such test problems, the behaviour of optimisation algorithms can be clearly observed and confirmed. Needless to say, it is not possible to simulate all the difficulties of real

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search spaces in one benchmark problem. This section reviews the benchmark functions in the fields of single- and multi-objective optimisation with a focus on those for robust optimisation. Generally speaking, the design process of a test problem has two goals. Firstly, a test problem should be simple and modular in order to allow researchers to observe the behaviour of meta-heuristics and benchmark their performance from different perspectives. Secondly, a test function should be difficult to be solved in order to provide a challenging environment similar to that of real search spaces for meta-heuristics. These two characteristics are in conflict where oversimplification makes a test function readily solvable for meta-heuristics and the relevant comparison inefficient. In contrast, although a very difficult test function is able to effectively mimic real search spaces, it may be very difficult to solve so that the performance of algorithms cannot be clearly observed and compared. These two conflicting issues make the development of test problems very challenging. In single-objective problems there are several important characteristics for an algorithm: exploration, exploitation, local optima avoidance, and convergence speed. Unimodal [57], multimodal [122], and composite [116] test functions have been designed in order to benchmark these abilities. These three types of test functions have been extensively utilised in the literature. Basically, the conceptual approach of benchmark design is creating different difficulties of real search spaces to challenge an algorithm. The single-objective test problems are relatively simple due to the existence of only one global optimum. They are mostly equipped with obstacles to benchmark the accuracy of an algorithm in finding the global optimum, and its convergence speed. Challenging test functions with multiple local solutions are able to benchmark the accuracy of an algorithm due to the high likeliness of local optima entrapment. However, test functions with no local solutions but different slopes and saturations are able to test the convergence speed of a single-objective algorithm. In multi-objective optimisation, however, there are different performance characteristics for an algorithm. In addition to the above-mentioned characteristics for single-objective optimisation, other important performance features are convergence towards the global front, and diversity (coverage) of the Pareto optimal solutions obtained. In order to benchmark the first characteristic, a test problem should have a Pareto optimal set located in a unimodal, multi-modal, or

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composite search space. In addition, a test function should have different shapes of Pareto optimal front such as linear, convex, concave, and discontinuous in order to benchmark the coverage capability of a multi-objective meta-heuristic. Due to the concepts of Pareto optimality and multi-optimal nature of multiobjective search spaces, developing multi-objective test problems is significantly more challenging than single-objective test functions. Since the proposal of evolutionary multi-objective optimisation by David Schaffer in 1984 [32], a significant number of test functions were developed up to 1998. Some of them are as follows (classified based on the shape of the true Pareto optimal front): • Convex Pareto optimal front: Osyczka et al. in 1995 [128], ValenzuelaRend´o et al. in 1997 [160], and Laumanns et al. in 1998 [110] • Concave Pareto optimal front: Fonseca and Fleming in 1993 [72] and in 1995 [71], and Murata and Ishibuchi in 1995 [124] • Discontinuous Pareto optimal front: Osyczka et al. in 1995 [128] and Vlennet et al. in 1996 [164] In 1998, Van Veldhuizen and Lamont argued that the majority of the test functions proposed until then could not be considered as standard test problems [161]. They sifted the test functions and chose three of them as standard test functions due to their large search space, high dimensionality, multiple objectives, and global optimum composed of a shape of bounded complexity. A generic framework for creating a two-objective test function was first proposed by Deb in 1999 [46]. The mathematical model of this framework is as follows: M inimise :

f1 (~x) = f1 (x1 , x2 , ..., xm )

M inimise :

f2 (~x) = g(xm+1 , xm+2 , ..., xN )×

(2.43)

(2.44)

h (f1 (x1 , x2 , ..., xm ), (xm+1 , xm+2 , ..., xN )) The main idea of this framework was to break up a test function into different controllable components to systematically benchmark multi-objective algorithms. In this framework, f1 (~x) controls the distribution of true Pareto optimal solutions and benchmarks the coverage ability of an Evolutionary Multi-objective

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Algorithm (EMA). The g function was proposed to provide multi-modal, deceptive, and isolated search spaces, in which different challenging test beds can be constructed in order to benchmark the convergence of EMAs. The last component of this framework, h, is to define the shape of true/local Pareto optimal fronts. Deb showed that convex, non-convex, and discontinuous Pareto optimal fronts could easily be achieved by modifying this component [46]. As can be seen in this framework, different characteristics of a real search space can be simulated by the proposed components. These components may be modified individually or simultaneously in order to mimic different characteristics of real problems and eventually benchmark the performance of EMAs from different perspectives. In 1999, Zitzler et al. proposed the first so-called ZitzlerDebThiele (ZDT) standard set of test functions [181] using the proposed framework of Deb [46] and mimicked six difficulties in real search spaces (convexity, concavity, discontinuity, multi-modality, deceptiveness, and non-uniformity) within six test problems. The authors compared 8 different algorithms on the ZDT test problems and observed their behaviours from two perspectives: convergence and coverage. Almost none of the proposed test functions available up to 2001 were extendable to a desired number of objectives. In 2001, Deb et al. proposed three systematic methods for creating scalable multi-objective test problems [56, 55]. The three proposed methods were: multiple single-objective functions, bottomup, and constraint surface. The first method was to combine different singleobjective test problems as a multi-objective test problem. The advantage of this method is the ease of construction of a multi-objective test problem. In the second method, a Pareto optimal front is first created in an n-dimensional space, and then the search space beyond that is constructed. The advantage of this method is that the shape of the Pareto optimal front is known and controlled by the designer. Finally, the last method starts by creating a simple search space (hyperbox) and applying a set of constraints to define the shape of the Pareto optimal front. The advantage of this method is its simplicity but the mathematical formulation of the final Pareto optimal front is difficult to express. Deb et al. employed the last two approaches to propose the DTLZ test problems. In 2006, Huband et al. provided a review of multi-objective test problems and proposed a scalable test problem toolkit [92]. They also provided similar recommendations to those of Deb et al. [56, 55]. The recommended features for the multi-objective test problems were: scalable number of parameters, scalable

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number of objectives, dissimilar parameter domains, dissimilar trade-off ranges, known Pareto optimal set, known Pareto optimal front, different shapes of Pareto optimal front, parameter dependencies, bias, many-to-one mapping, and multimodality. Another key issue in multi-objective test problems that was first argued by Okabe et al. in 2004 is the shape and complexity of the Pareto optimal set [126]. Before 2004, the majority of researchers had tried to construct test functions concentrating on the shape of Pareto optimal front. Okabe et al., however, suggested a method for controlling the shape of the Pareto optimal set and created several test problems. Despite the merits of their work, the proposed test functions were too simple, mapping a 2-D search space to a 2-D objective space. There is also another method of generating complicated Pareto optimal sets called variable linkage [123], in which a method is designed to create dependencies among variables. Linkage was also investigated by Deb et al. [54] and Huband et al. [92]. Despite providing complicated search spaces using linkage methods, in 2009, Li and Zhang identified that the shape of the Pareto optimal set and linkage are two different aspects of multi-objective problems [114]. According to them, a test function with linkage properties may have a very simple Pareto optimal set. Therefore, they proposed a general class of multi-objective test problems with known, complicated Pareto optimal sets. Finally, a set of test functions considering all the above-mentioned characteristics were prepared in a CEC 2009 special session [178].

2.5.1

Benchmark problems for single-objective robust optimisation

There are not many robust benchmark functions in the literature [106, 105]. Some studies utilised the dynamic benchmark problems in addition to the current robust benchmark functions as discussed in [18, 58]. This thesis collects the majority of the robust test functions from [18, 19, 20, 58, 105, 106, 129, 163] and analyses them. These test functions are illustrated in Fig. 2.14. Generally speaking, the benchmark functions are divided into four groups in terms of the location of robust and global optima [58], as follows: • Identical global and robust optima: in this case the robust and global

x

y

x

TP3

x

y

y

TP7

TP8

y

x

TP10

x

f(x,y)

f(x,y)

f(x,y)

x

TP9

TP4

x

TP6

f(x,y)

TP5

y

x

f(x,y)

f(x,y)

y

x

y

y

TP2

f(x,y)

TP1

f(x,y) y

x

f(x,y)

y

f(x,y)

55

f(x,y)

f(x,y)

2. Related work

y

x

TP11

y

TP12

x

Figure 2.14: Collected current test functions in the literature for robust singleobjective optimisation. The details can be found in Appendix A. optima are same. • Neighbouring global and robust optima: The global and robust optima are at the same peak (valley). • Local-global robust and global optima: The global and robust optima are at different peaks (valleys), and the robust optimum is a local optimum. • Max-min robust and global optima: The robust minimum/maximum are in a non-robust maximum/minimum As may be seen in Fig. 2.14, the test functions are very simple. For instance, TP8 and TP11 have a stair-shaped search space and the robust optimum of TP10 is very wide. Simplicity can also be observed in the other test functions. Another major drawback is lack of scalability. The majority of these test functions cannot be scaled to more than 2 or 5 dimensions. The number of variables is one of the key factors for increasing the difficulty and effectively benchmarking the performance of meta-heuristics. The majority of the current test functions have

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few non-robust local optima as well. It may also be noticed that there are no deceptive or flat test functions. The last gap here is the lack of specific test functions with alterable parameters for defining the degree of difficulty. All these drawbacks make the current test functions inefficient and readily solvable by robust meta-heuristics. Therefore, the performance of the robust meta-heuristics cannot be benchmarked effectively.

2.5.2

Benchmark problems for multi-objective robust optimisation

The literature shows that different branches of multi-objective optimisation need specific or adapted test functions in order to observe the performance of algorithms in detail. For instance, there are specific sets of constrained test functions for constrained multi-objective meta-heuristics [52, 128, 158, 160], reliability-based optimisation [40, 47, 51], and dynamic multi-objective test problems [3, 37, 67, 68, 85, 97, 113, 120] . In the field of robust multi-objective optimisation, however, there is little in the literature on the development of robust multi-objective problems. The first robust multi-objective test problems were proposed by Deb and Gupta in 2006 [44]. These test functions are illustrated in Fig. 2.15. In this figure, the Pareto front as well as robust fronts with different perturbation levels in parameters are provided for each test function. The left subplots show the search landscape made by both objectives for each test problem. The right subplots include the Pareto optimal front and expectation of Pareto optimal fronts (robust fronts) considering different levels of perturbations. In RMTP1, the local fronts are the robust front when δ = 0.007, 0.008, 0.009, 0.01 from the bottom to the top. In RMTP2, the local fronts become the robust front when δ = 0.004, 0.005, 0.006, 0.007. The fronts for RMTP3, RMTP4, RMTP5, and RMTP6 show the nominal value and expectation of both global and local fronts. Deb and Gupta proposed this set of test functions in order to simulate the four possible different situations (as illustrated in Fig. 2.13) of a robust Pareto optimal front with respect to the main Pareto optimal front. As may be seen in Fig. 2.15, the robust and global Pareto optimal fronts are identical in RMTP1. The RMTP2 test function simulates the second type of robust front, in which a part of the global Pareto optimal front is robust. The robust Pareto optimal

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front of RMTP3 is completely dominated by the global Pareto optimal front. Finally, RMTP4 has a robust Pareto optimal front that is partially identical to the main Pareto optimal front and local fronts. The rest of the test functions are the extended three-objective version of RMTP1 and RMTP3. These test functions provide very challenging test beds, as investigated in [43]. 3

2.5

2

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f2

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f2

f1, f2

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f1, f2, f3

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Objective space RMTP6

Figure 2.15: Test problems proposed by Deb and Gupta in 2006 [44] Deb and Gupta noted that the analytical robust front of each of these benchmark functions is known, so they can be utilised to benchmark both type I and II robustness handling methods. Although this set of test functions is able to simulate different types of robust Pareto optimal front with respect to the global Pareto optimal front, there are other issues when solving real problems, such as discontinuous robust/global Pareto optimal fronts, convex/concave robust/global Pareto optimal fronts, and multi-modality. These issues have been discussed and addressed to some extent by GasparCunha et al. in 2013 [75]. Their five new benchmark problems are illustrated

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1

1

0.8

0.8

0.6

0.6

0.4

f2

f1, f2

f2

f1, f2

in Fig. 2.16. Note that the robustness curves in this figure (red lines) are the cumulative value for the robustness of f 1 and f 2 and that it is plotted for given values of f 1 (i.e. a point on the Pareto front has its robustness plotted at the same x position).

0

0.4 0.2

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1

Objective space RMTP8

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f1, f2

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Parameter space

Objective space RMTP11

Figure 2.16: Test problem proposed by Gaspar-Cunha et al. in 2013 As can be seen in this figure, the shapes of the test functions are very different to that of Deb and Gupta [44]. The robust regions of the main Pareto optimal front are on the convex section of RMTP7, whereas the robust areas lie on concave regions of the Pareto optimal front in RMTP9. RMTP10 and RMTP11 were proposed in order to design separated robust regions in the robust Pareto optimal fronts. As the robustness curves in Fig. 2.16 suggest (the red lines), the robustness of the separated regions decreases from left to right in RMTP10, while the robustness is equal for the three discontinuous parts in RMTP11. Note that the robustness of the Pareto optimal front is calculated by averaging re-sampled points in the neighbourhood, so a low value in the robustness curve shows low

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59

fluctuation and sensitivity to perturbations in the corresponding region of the Pareto optimal set. For instance, the middle region of RMTP7’s Pareto front is the most robust area because the robustness curve shows the lowest fluctuation in both objectives in case of perturbations in the parameters. As discussed by Gaspar-Cunha et al., RMTP10 is able to test convergence of an algorithm toward more robust regions of a search space. In addition, RMTP11 is suitable for benchmarking the ability of an algorithm in terms of converging to distinct regions of the Pareto optimal front. Another set of test functions was proposed by Goh et al. in 2010 [81]. They extended three single-objective robust test functions proposed by Branke [18, 19] and Paenke et al. [129] and compared them with those of Deb et al. [44]. They argued that the search spaces of these earlier test functions have a bias toward the robust Pareto front, so it is hard to distinguish whether the robust measure assists the EMA to find the robust Pareto front or the robust Pareto front is obtained because of the failure of the algorithm in finding the Pareto optimal front. Therefore, they proposed a Gaussian landscape generator to integrate different parametric sensitivities in deterministic search spaces. They designed five test functions (GTCO) which show different characteristics based on the Gaussian landscape generator. There is also a recent set of test functions (BZ) proposed by Bader and Zitzler [11]. The lack of specific multi-objective test problems in the literature was reported and an early attempt made to design a standard set. Bader and Zitzler proposed six test functions with different robust characteristics. The focus was mostly on the proposal of a multi-modal parameter space and multifrontal objective space. The shapes of the robust Pareto optimal fronts were linear, concave, or convex in the proposed test functions. Although the proposed test functions in the literature provide different test beds for benchmarking the performance of robust meta-heuristics, there is a lack of general-purpose test functions (or frameworks) with control variables for adjusting the complexity. In addition, there are few multi-modal benchmark problems, which can provide very similar test beds to the real search spaces. Another gap in the literature is the lack of test functions with disconnected, biased, and flat robust Pareto optimal fronts. In summary, this section showed that the test functions in the field of global optimisation are not suitable for benchmarking the performance of robust al-

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gorithms. This is because such test functions are not seeking to test robustness of the solutions obtained. There might not even be a robust optimum in a test function that has been design for testing a global optimiser. It was also observed that what specific test functions there are for testing robust algorithms are very simple and limited. They mostly have few local solutions, symmetric search spaces, and low numbers of variables. On one hand, they allow us to observe some of the behaviours of a robust algorithm. On the other hand, they are readily solvable by most of the algorithms. The robust multi-objective benchmark problems also suffer from the same drawbacks despite their use in different studies. The current gaps for both robust single-objective and multi-objective test functions are lack of other difficulties such as bias, deceptiveness, flatness, large number of local solutions (fronts), and large number of variables.

2.6

Performance metrics

Benchmark problems are the main tools for testing the ability of different algorithms in this field. Performance metrics quantify and measure the performance of algorithms on benchmark and other problems. Without such tools, we can only compare different optimisation techniques qualitatively, which is not an accurate analysis. We ask ourselves, “How do we determine the extent to which changes in algorithms are beneficial”. In other words, a qualitative analysis can show us which method is better, while a quantitative analysis shows how much better a technique is. Similarly to benchmark functions, there should be different performance metrics for indicating the performance of a robust algorithm since the goal is to find the robust optimal solution(s) and not necessarily global optimal solution(s). This section only covers the current performance metrics in the fields of single- and multi-objective optimisation. The literature shows that the performance of single-objective algorithms is quantified mostly by two main metrics: an accuracy indicator and convergence rate. As its name implies, the former performance indicator measures the discrepancy of the approximated global optimum from the true optimum. The latter performance indicator measures the convergence speed of an algorithm in approximating the global optimum. These two metrics are the main performance measures in the literature of single-objective optimisation. For the single-objective robust algorithms, the above-mentioned performance

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indicators (accuracy and convergence speed) can be utilised. This includes the accuracy of the robust optimum obtained and convergence speeds towards the robust optimum instead of the global optimum. In addition, the sensitivity of the solution obtained to the possible uncertainties could be considered as a specific metric of robustness. This can be highlighted by solving the problem with both global and robust optimisers to show the robustness of both solutions obtained. The main purpose of a performance indicator in Evolutionary Multi-Objective Optimisation (EMOO) is to quantify the performance from a specific point of view. Generally speaking, the ultimate goal in EMOO is to find a very accurate approximation and large number of the true Pareto optimal solutions with uniform distribution across all objectives [181]. Therefore, the current performance measures can be classified into three main categories: convergence, coverage, and success metrics. The first class of performance measures quantifies the closeness of the solutions obtained to the true Pareto front [142, 141], and the second class of metrics defines how well the solutions obtained “cover” the range of each of the objectives [66]. In addition, the number of Pareto optimal solutions obtained is important [184] (the success ratio), which provides decision makers with more designs from which to choose. Another classification in the literature is between unary [160, 170] and binary [182, 84] performance indicators. The former class of metrics only accepts one input and provides a real value, whereas the latter metrics have two inputs and one output. According to Zitzler et al. [184], each of these types have their own disadvantages. The drawback of the unary performance indicators is that there should be more than one measure to assess the performance of the algorithms, and it has been proven by Zitzler et al. that designing an effective general-purpose unary performance measure to evaluate the overall performance of an algorithm (convergence, coverage, and success ratio) is impossible [184]. In addition, binary performance measures provide n(n − 1) different values when comparing n algorithms, whereas unary metrics provide n values. As the main drawback, this makes the interpretation, analysis, and presentation of the binary measures more challenging. It should be noted that an important characteristic of a performance metric is Pareto-compliance [73]. A performance metric is Pareto-compliant if it does not contradict the order enforced by the Pareto dominance relation. Despite the limitations of the unary performance indicators, there is no doubt

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that they have been the most popular performance assessors in the literature. This is probably due to their simplicity and ease of analysis. This thesis concentrates on the unary performance measures, but addressing the three, major aspects of performance already specified. In the following subsections a review of the current convergence, coverage, and success ratio (number of Pareto optimal solutions obtained) metrics is provided.

2.6.1

Convergence performance indicators:

This subsection covers the most widely-used convergence metrics in the literature. 2.6.1.1

Generational Distance (GD):

This metric was proposed by Veldhuizen in 1998 [161]. GD calculates the distance of Pareto optimal solutions obtained from a selected reference set in the Pareto optimal front. The mathematical formulation is as follows: pPno 2 i=1 di (2.45) GD = n where no is the number of obtained Pareto optimal solutions and di indicates the Euclidean distance between the i-th Pareto optimal solution obtained and the closest true Pareto optimal solution in the reference set. Note that the Euclidean distance is calculated in the objective space. 2.6.1.2

Inverted Generational Distance (IGD):

The mathematical formulation of IGD is similar to that of GD. This modified measure was proposed by Sierra and Coello Coello in 2005 [148]. qP 0 2 nt i=1 (di ) IGD = (2.46) n 0

where nt is the number of true Pareto optimal solutions and di indicates the Euclidean distance between the i-th true Pareto optimal solution and the closest Pareto optimal solution obtained in the reference set. The Euclidean distance between solutions obtained and the reference set is different here. In IGD, the Euclidean distance is calculated for every true solution with respect the nearest Pareto optimal solution obtained, in the objective space.

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63

Delta Measure:

A similar metric was proposed by Deb et al. in 2002 [45] as follows: PN PH i=1 j=1 di,j Υ= NH

(2.47)

where N is the number of Pareto optimal solutions obtained, H is the number of solutions selected in the reference set which is different for each solution, and d(i,j) shows the Euclidean distance from the i-th solution obtained to the j-th reference point. A lower value of this metric indicates closeness of the Pareto optimal front obtained to the true Pareto optimal front. 2.6.1.4

Hypervolume metric:

This metric is for quantifying the convergence behaviour of MOEAs, designed by Zitzler [183, 180]. The idea is to calculate the area/volume of the objective space that is dominated by the non-dominated Pareto optimal solutions obtained. Note that this performance indicator is called Size of Space Covered (SCC) in some references [157]. 2.6.1.5

Inverse hypervolume metric:

As the name implies, this metric is the inverse of the hypervolume metric, in which the area/volume of the objective space that is not dominated by the Pareto optimal front obtained is calculated with respect to a reference set [112].

2.6.2

Coverage performance indicators:

This subsection presents the most-widely used coverage metrics in the literature. 2.6.2.1

Spacing (SP):

The spacing metric was first proposed by Schott in 1995 [144]. The main idea of this metric was to calculate the variance of the Pareto optimal solutions obtained. The mathematical expression of SP is as follows [28]: v u n u 1 X t (d¯ − di )2 (2.48) SP , n − 1 i=1

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where d¯ is the average of all di , n is the number of Pareto optimal solutions obtained, and di = minj (|f1 (~ xi ) − f1 (x~j )|+|f2 (~ xi ) − f2 (x~j )|) for all i, j = 1, 2, ..., n. A low value for this measure shows a greater number of, and more equally spread solutions along the Pareto optimal front obtained. Another similar metric was proposed by Deb et al. in 2002 [45]. This method averages the Euclidean distance between the neighbouring Pareto optimal solutions obtained as the spread of the solutions. Note that this metric is calculated with respect to at least two extreme solutions that define the maximum extent of each objective based on the true Pareto optimal front. This metric is as follows: P −1 ¯ df + dl + N i=1 |di − d| ∆= df + dl + (N − 1)d¯

(2.49)

where N is the number of Pareto optimal solutions obtained, d¯ is the average of Euclidean distances, di is the Euclidean distance of the i-th solution and its consecutive solution in the Pareto optimal set obtained, and df , dl are the Euclidean distance between the boundary of solutions obtained and the extreme solutions. 2.6.2.2

Radial coverage metric:

This metric was proposed by Lewis et al. in 2009 [112] where the objective space is divided into radial sectors originating from the origin. Then the number of segments that are occupied by at least one Pareto optimal solution obtained is calculated as the coverage of an algorithm. The mathematical expression of this metric is as follows: Pn ψi Ψ = i=1 (2.50) N    1 (Pi ∈ P F ∗ ) ∧ αi−1 ≤ tan f1 (x) ≤ αn f2 (x) ψi = 0 otherwise 2.6.2.3

(2.51)

Maximum Spread (M ):

This method was proposed by Zitzler and is as follows [180]: v u o uX M =t max (d(ai , bi )) i=1

(2.52)

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where o is the number of objectives, and d() calculates the Euclidean distance, ai is the maximum value in the i-th objective, and b is the minimum value in the i-th objective As this equation shows, this metric defines a hyperbox/hypercube using the Pareto optimal front obtained and finds the maximum diagonal distance.

2.6.3

Success performance indicators:

The number of success metrics is substantially less than convergence and coverage measures in the literature. This subsection discusses two of the most popular ones. 2.6.3.1

Error Ratio (ER):

This metric counts the number of Pareto optimal solutions obtained that belong to the set of true Pareto optimal solutions and divides it by the total number of solutions found. The formulation of this metric was proposed by Veldhuizen in 1999 [160] as follows: Pn ei (2.53) ER = i=1 n  0 P ∈ P F ∗ i ei = 1 otherwise

(2.54)

where n is the number of Pareto optimal solutions obtained and Pi is the i-th Pareto optimal solution obtained. The lower value of this measure shows the better approximation of the true Pareto optimal solutions. 2.6.3.2

Success counting (SCC):

This measure was proposed by Sierra and Coello Coello [148]. This measure counts the number of solutions obtained that are members of the true Pareto optimal set. The mathematical formula proposed is as follows: SCC =

n X i=1

si

(2.55)

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 1 P ∈ P F ∗ i si = 0 otherwise

(2.56)

where n is the number of Pareto optimal solutions obtained and Pi is the i-th Pareto optimal solution obtained. In contrast to ER, a high value for this measure shows better performance. The above-mentioned multi-objective performance indicators have been widely used in the literature to compare different algorithms. Regardless of the advantages and disadvantages of unary and binary measures, they are all highly suitable for the quantitative analysis of results from different perspectives. However, none of them are able to measure the performance of robust multi-objective algorithms effectively. There is literally no specific performance metric to measure the robustness of the Pareto optimal solutions obtained. Therefore, the comparison of the current robust multi-objective algorithms are qualitative and obviously not accurate enough, while this is essential when evaluating the performance of algorithms. This is the motivation of the work in this thesis where several specific performance metrics will be proposed to quantify the performance of robust multi-objective algorithms for the first time.

2.7

Summary

This chapter first provided a comprehensive review of the evolutionary singleobjective optimisation methods. Different types of such optimisation techniques, drawbacks, advantages, and the state-of-the-art were discussed in detail. After that, multi-objective optimisation using evolutionary algorithms was discussed as one of the most practical and popular branches in this field. The essential definitions, recent advances, different techniques, difficulties, most popular algorithms, benchmark problems, and performance metrics in this field were the main discussions. The chapter also included two further sections: single-objective robust optimisation and multi-objective robust optimisation. The former section was dedicated to the literature review of the current robust single-objective optimisation techniques, benchmark problems, and robustness measures in single-objective search spaces. This section also identified diverse types of uncertainties and

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their impacts on the real systems/problems. In the latter section, the literature of robust multi-objective optimisation was reviewed in detail. Similarly to robust single-objective optimisation section, preliminaries, essential definitions, current robust multi-objective optimisation methods, performance metrics, benchmark problems, and robustness measures were covered.

Chapter 3 Analysis The preceding chapter reviewed the literature of robust optimisation and related fields. Similarly to global optimisation, a robust optimisation process includes four main phases: benchmark development or preparation, development or preparation of performance metrics, proposing and testing an algorithm or improvement, and applying the algorithm to an actual industrial problem to be optimised. This chapter analyses each of these phases and identifies their current gaps.

3.1

Benchmark problems

Benchmark problems are essential for testing different algorithms. They provide test beds with different difficulties to challenge optimisation algorithms. Without benchmark problems, we would have to compare algorithms on real problems that are usually expensive and have unknown search spaces. The computationally expensive nature of real problems makes the design process of an algorithm significantly longer. However, benchmark problems are computationally cheap and allow us to compare algorithms conveniently. Real problems also have unknown search spaces and true optimal solution(s). These prevent an algorithm designer from testing the different ability of the algorithm independently. With an unknown optimum, we are never sure if we have reached it. There is not a way to measure how close we are to it. A real search space is a mixture of difficulties, so an algorithm has to overcome all of them simultaneously to estimate the optimal solution(s). Benchmark problems isolate one of a set of difficulties that an algorithm may face in a real search 68

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space. This assists us to test the abilities of an algorithm dealing with isolated difficulties. In the literature of global optimisation, there is a significant number of test suits, test cases, and frameworks for testing algorithms. The number, quality, and popularity of the test functions in this field assure a designer that comparisons on them are reliable. In order to design a robust algorithm, there is a need for adequate and proper test functions as well. Some of the challenges that an algorithm may face when searching for robust solutions in a real search space are: non-robust local optimal solutions, robust local optimal solutions, slow convergence, deceptive non-robust optimal solutions, robust optimal solutions close to the boundary of the search space, and so on. Unfortunately, a limited number of these difficulties have been implemented in test functions. The current test problems are very simple and limited, so they are not efficient in benchmarking the ability of robust algorithms. They mostly have few local solutions, symmetric search spaces, and a low number of variables. Although such test problems allow us to observe some of the behaviours of a robust algorithm, they are readily solvable by most of the algorithms. The robust multiobjective benchmark problems also suffer from the same drawbacks despite their use in different studies. The current gaps for both robust single-objective and multi-objective test functions are lack of difficulties such as bias, deceptiveness, flatness, large number of local solutions (fronts), and large number of variables. In addition, there is no framework with alterable parameters to allow a designer to generate new test functions based on their needs. Therefore: We must design more challenging test functions, and frameworks to alter their difficulties.

3.2

Performance metrics

After finding or designing a suitable set of test functions in the systematic design process, the next step is to find appropriate performance metrics. Test functions allow us to test and observe the behaviour of an algorithm. In order to compare different algorithms, however, we need to use performance metrics. There are two types of performance metrics in the literature: qualitative versus quantitative. Qualitative metrics are usually illustrative in this field and determine if an algorithm is better than another qualitatively. Although these metrics determine

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the superiority of an algorithm, they cannot measure how much an algorithm is better than another. In other words, a quantitative metric defines the degree of superiority. Similarly to benchmark problems, there is a substantial number of works for evaluating and proposing performance metrics specifically for multi-objective global optimisers. However, there is no work in the literature to investigate and propose performance metrics for evaluating robust algorithms. All of the current works borrow the qualitative metrics from the field of global optimisation and there is no specific quantitative performance metrics to measure how robust the solutions obtained are. Therefore: We must propose and utilise performance metrics to determine the extent to which minor or major changes in algorithms are beneficial.

3.3

Robust algorithms

The above-discussed two phases of a systematic algorithm design process, developing benchmark problems and performance metrics, are prerequisite to the last phase, algorithm invention or improvement. Due to the existence of standard tests function and performance metrics in the field of global optimisation, there is again a remarkable number of works in the literature on improving, hybridising, and proposing different algorithms. However, more works should be done in the literature of robust optimisation. The number of works is dramatically less than in the field of global optimisation. A global optimiser cannot determine robust solutions, so designers usually use additional constraints to guarantee the robustness of solutions. The advantage of this method is that there is no need to develop robust algorithms and a global optimiser can be applied to the problem directly. The disadvantage is that adding constraints increases the difficulty of problem and changes the boundaries of the search space. A better way is to use specific operators to look for the robust optimal solution(s) instead of global solution(s). On one hand, this method does not require adding additional constraints and the optimiser is searching the main search space. On the other hand, suitable mechanisms should be integrated into the robust algorithm to avoid non-robust, often global optimal solution(s) and search for robust solutions.

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The majority of current methods suffer from significant additional computational costs due to the need for additional function evaluations to confirm the robustness of solutions, making them impractical for solving real problems [96, 75]. As an example, antenna design problems are usually very expensive and may easily take up to 60,000 CPU hours (roughly 357 week or 6 years) to optimise them [111]. If we want to find the robust solutions using an algorithm that needs 4 sampled points for every solution, the robust optimisation would take up to 240,000 CPU hours (1,432 weeks or 27 years). This example shows how impractical a robust optimisation that relies on additional sampled points is. Therefore: We must find a way to reduce the computation time of robust optimisation algorithms. There have been two main approaches proposed for reducing the computational costs (true function evaluations) of robustness handling methods, as follows: 1. Archive-based methods, in which previously sampled solutions are saved and re-used during the optimisation in order to define the robustness of meta-heuristics’ search agents [143, 18] 2. Surrogate-based techniques where a meta-model is employed to approximate the search space in the neighbourhood of solutions [127, 137] Generally speaking, surrogate models are approximations of the real search spaces and computationally cheaper than the real model. They allow designers to have a rough image of the search landscape and make the design process faster. However, one of the major problems is that a surrogate model might not be accurate enough. This means that an optimiser (or a designer) always investigates the approximated model of the search space instead of the actual one. Therefore, utilising surrogate models reduces the reliability of the whole design process although sometimes they can lead us to the real solution faster. This deteriorates when searching for robust solutions due to the intrinsic uncertainties involved in meta-models. The errors from meta-models can be considered mistakenly as the sensitivity of a solution to other uncertainties. In this case, a solution that is not sensitive to the probable uncertainties can be discarded by an optimiser due to the noise from the meta-model.

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Another disadvantage of surrogate models is that they mostly use one or more than one [162, 151] model for the entire search space, while a real search space usually has regions with diverse shapes. This is because surrogate models are constructed from limited (local) information about the search space, which works well for some regions but may provide deceptive information to the optimiser about other areas of the search space. Therefore, surrogate-assisted algorithms may become unreliable because they can be deceived by the surrogate models in some regions of the search space. Due to inaccuracy and unreliability, these techniques are not investigated in this thesis and considered out of scope, so interested readers are referred to the comprehensive review by Jin [95]. The archive-based methods, which are the focus of this work, rely on previously evaluated solutions during robust optimisation. The main advantage of these methods compared to surrogate-assisted approaches is the use of the real search space. Utilising a real search space prevents an optimiser from making unreliable comparisons between robust and non-robust solutions. It is worth mentioning here that the reliability of archive-based methods can be improved by making more true function evaluations around the solution. This process becomes computationally cheaper every year with the improvement of hardware. It seems both surrogate-assisted and archive-based algorithms have their own advantages and drawbacks. On one hand, surrogate-assisted algorithms do not solve the real search space, have intrinsic errors, and rely on local information extracted from the search space. However, they are cheap, so additional function evaluations have no substantial additional cost. On the other hand, archivebased algorithms solve the actual search space but suffer from unreliability because of the stochastic nature of meta-heuristics. The solutions contained in the archive are the product of the algorithms’ operations, not a systematic sampling of the search space. Due to the importance of solving the actual search space (and not the surrogate model), the archive-based methods are investigated and improved in this thesis. The unreliability of such methods is targeted for improvement as the only disadvantage. The usefulness of archive-based methods was investigated and confirmed by a number of studies [18, 94, 58, 143]. They experimentally proved that previously sampled points could reduce the number of true function evaluations significantly and are able to provide good information about the robustness of solutions. According to the central limit theorem, in addition, a large number of samples

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from a population is able to yield a mean equal to that of the population itself. Therefore, the average of a large number of sampled points in the neighbourhood of a solution gives us the average of the real search landscape to determine the robustness. Although previously evaluated points can provide very useful information about the robustness of new solutions without the need for additional evaluation [18], the stochastic nature of meta-heuristics prevents this method from providing the highest reliability. In effect, the reliability of the archive-based methods is decreased as the number of archive members and true function evaluations are reduced. Deb et al. studied the effects of neighbourhood solutions in terms of finding the analytical robust front and found that finding the robust front becomes more challenging when the number of sampled solutions (or neighbouring solutions) decreases [43]. In addition, archive-based methods that only use previously sampled points are very unstable in the initial steps of optimisation due to fewer sampled points. All these reasons reduce the confidence of designers and decision makers in the performance of the archive-based methods and the quality of robust designs obtained. What make these methods unreliable are: lack of sufficient previously sampled points in the archive, lack of a good distribution of sampled points around particular solutions, and lack of appropriate sampled solutions in a certain radius around particular solutions. One might think that the unreliability of the archive-based approaches is resolved after the initial steps of meta-heuristic optimisation, but the stochastic nature of meta-heuristics and the unknown shape of the search space prevent the archive-based methods from making confident decisions throughout the whole optimisation process. Fig. 3.1 shows an example of an archive-based method providing misleading information when relying on previously sampled points. In this figure the solution S2 is more robust than S1. In the archive, however, there is one sampled point for S1 and three for S2 to confirm the robustness. Since the sampled solution and S1 are very close, the robustness measure indicates high robustness. However, the robustness measure for S2 show less robustness due to the distribution of the solutions around S2 in parameter and objective spaces. In this case, an archive-based method assumes that S1 is more robust than S2, while it is not. Such circumstances can happen throughout robust optimisation, which results in guiding the search agent of meta-heuristics toward misleading,

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Previously Sampled points

f2

x2

S1 S2

x1

f1

Figure 3.1: An example of the failure of archive-based methods in distinguishing robust and non-robust solutions. Note that the variances shown are the actual variances, not those detected by the sampling. non-robust regions of search spaces. Therefore, it seems these robustness measures are not particularly reliable metrics for confirming the robustness of solutions when using previously sampled points. In multi-objective robust optimisation, the Pareto dominance based on robustness measure is also premature since there are possibilities that a nonrobust solution dominates a robust solution only because of the unreliability of the archive-based methods. The reason for this unreliability of the robustness measures proposed so far is that the status of previously sampled points in parameter space is not considered. Only the magnitude of changes in the objective space are considered.

3.4

Systematic robust optimisation algorithm design process

As discussed above, the main lack is of a standard and systematic design process including suitable robust test functions, robust performance metrics, and robust algorithms. This thesis attempts to propose the first systematic robust optimisation process in the literature. It proposes frameworks for generating challenging test functions with different levels of difficulty, performance metrics to quantify the performance of algorithms and compare them, and techniques to improve the reliability of archive-based methods. The unreliability originates from the

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lack of confidence that we have in the sampled points inside the archive. Therefore, if a method improves our confidence in the values of the archive members, we can design more reliable algorithms. This is what the literature lacks and the specific motivation of this research, in which a novel method is proposed to measure the confidence level of search agents during optimisation to alleviate the unreliability of the archive-based algorithms. It should be noted here that we cannot achieve 100% reliability because a method that improves our confidence is only an estimate, as it is derived from samples. The current robust test problems suffer from simplicity and are not able to test robust algorithms efficiently. There is no performance metric for evaluating the ability of robust algorithms. In addition, the majority of current methods suffer from significant additional computational costs due to the need for additional function evaluations to confirm the robustness of solutions, making them impractical for solving real problems [96]. To fill these gaps, I propose the systematic design process in the following chapters. This includes designing challenging robust test problems to compare algorithms, performance metrics to measure how much a robust algorithm is better than another, and computationally cheap robust algorithms to find robust solutions for optimisation problems. As Fig. 3.2 shows, the first two phases of this systematic process, the development of test functions and performance metrics, are prerequisite to the third phase, algorithm development.

3.5

Objectives and plan

To answer the main research question, this thesis proposes robust test functions, robust performance metrics, and computationally cheap robust optimisation techniques. These three components are the foundation of the systematic process and can be used by other researchers. For creating challenging test problems, several frameworks are proposed that allow creating test functions not only for the purpose of this thesis but also for other research in future. Two performance metrics are also proposed to quantify the performance of robust multi-objective algorithms. Again, the performance metrics can be used to compare any two algorithms in the literature. In addition to benchmark functions and performance metrics, a new metric called the Confidence (C ) measure is proposed to consider the status of previ-

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Robust test function design

Robsut performance metric design

Robust algorithm design

Figure 3.2: Test functions and performance metrics are essential for systematic algorithm design ously sampled points in the parameter space in order to improve the reliability of robustness measures. Confidence-based relational operators and confidencebased Pareto optimality/dominance are then proposed using both robustness and confidence metrics in order to make confident and reliable comparison between solutions in both single- and multi-objective search spaces. Two novel and cheap approaches called Confidence-based Robust optimisation (CRO) and Confidence-based Robust Multi-objective optimisation (CRMO) are established. The proposed approach improves the reliability of archive-based methods without additional computational costs and assists designers to confidently rely on points previously evaluated during optimisation. In addition, the proposed approach allows designing different confidence-based methods for finding robust Pareto solutions. The objectives of this thesis are: 1. To propose a confidence measure for quantifying and measuring the confidence level of robust solutions 2. To evaluate the current robust single-objective test functions and propose frameworks to generate more challenging ones 3. To evaluate the current robust multi-objective test functions and propose

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frameworks to generate more challenging ones 4. To propose novel performance metrics for evaluating and comparing robust multi-objective algorithms 5. To propose a confidence-based robust optimisation approach for finding the robust solutions of single-objective problems reliably and without extra computational cost 6. To propose confidence-based robust multi-objective optimisation approach for finding the robust solutions of multi-objective problems reliably and without extra computational cost 7. To propose the first systematic robust design process including standard test functions, performance metrics, and computationally cheap robust algorithms 8. To investigate the application of the proposed confidence-based robust optimisation approaches in finding robust solutions of real world problems The gaps that are going to be filled by the above-mentioned objectives and consequently where the contributions of this thesis fit are illustrated in Fig. 3.3.

3.6

Contributions and scope

In addition and as a result of this analysis, further contributions and scope of this thesis are shown in Fig. 3.4. The first contribution is the development of robust single-objective and multi-objective test functions. The focus will be on the proposal of frameworks with alterable parameters for generating test functions with different difficulties. In addition, various test functions will be proposed with diverse characteristics to benchmark the performance of robust algorithms from different perspectives. All the test functions will be unconstrained, but can easily be equipped with constraints. Proposed robust multi-objective performance measures can be used to compare any two sets of robust Pareto optimal solutions. However, the robustness of the true front should be known a priori. Therefore, all the robustness curves are defined for the test functions. The performance metrics are illustrated by using them to quantify the performance of algorithms on bi-objective problems.

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Population-based stochastic optimisation methods

Single-objective optimisation

2

4

1

7 Robust optimisation 5

3 Multi-objective optimisation

6 8

Engineering problems: marine propeller design

Figure 3.3: Gaps targeted by the thesis The confidence measure will be proposed for handling type B uncertainties only (in parameters). However, perturbations in operating conditions can be handled in case of parameterising them as well. Such cases will be considered when solving the real-world test case. Confidence-based relational operators and confidence-based operators are only applicable to robust single-objective optimisation algorithms. The focus will be on unconstrained robust optimisation. The proposed confidence-based robust optimisation will be demonstrated only on PSO and GA. The proposed confidence-based Pareto optimality is applicable to compare two sets of solutions for problems with two or more objectives. Confidence-

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Confidence measure

Implicit mathods

Confidence-based relational operators

Explicit methods

Confidence-based robsust optimisation

Population-based stochastic robust optimisaiton methods

CRPSO Single-objective robust optimisation

CRGA Benchmark problems Confidence-based Pareto optimality Implicit methods Confidence-based robust multi-objective optimisation

CRMOPSO

Explicit methods Multi-objective robust optimisation Benchmark problem

Performance metrics

Figure 3.4: Scope and contributions of the thesis based operators can also be integrated into any multi-objective optimisation algorithm, but this thesis only proposes a confidence-based MOPSO algorithm called Confidence-based Robust MOPSO (CRMOPSO). A real problem is chosen from the field of propeller design. To be exact, the shape of several marine/submarine propellers is optimised using MOPSO and CRMOPSO.

3.7

Significance of Study

The confidence measure allows us to define the confidence that we have when relying on previously sampled points. The proposed confidence measure will consider the status of the previously sampled points from different perspectives: the number of sampled points, the distance from the main solutions, and their distribution. Integrating the confidence measure into the relational operators (second objective) allows us to perform confidence-based comparison between the search agents of single-objective algorithms. In addition, confidence-based relational operators are integrable into different operators of the meta-heuristics,

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so specific mechanisms can be constructed for different meta-heuristics for doing confidence-based robust optimisation. The confidence-based Pareto dominance concept allows confidence-based comparison of solutions in multi-objective search spaces. The proposed confidencebased Pareto dominance is integrable into different components of multi-objective meta-heuristics as well. Therefore, it can be utilised to design specific operators for different meta-heuristics and consequently perform confidence-based robust multi-objective optimisation. This thesis also proposes several challenging frameworks and test functions for single-objective and multi-objective robust algorithms. Due to the lack of such difficult test functions in the literature, these contributions can be considered as one of the seminal attempts in designing standard single- and multi-objective robust test problems. The thesis also considers the proposal of two specific novel performance measures for comparing the robust multi-objective algorithms. There are no performance metrics in the field of robust multi-objective optimisation, so the proposed metrics are very important and can be used to quantify the performance of robust multi-objective algorithms for the first time. In addition, the proposed systematic robust algorithm design process allows designers to reliably and confidently propose new algorithms or improve the current ones. The confidence measure and confidence-based robust optimisation perspectives only utilise previously sampled points during the optimisation process. Therefore, they would not increase the computational burden of the algorithms. For example, a real problem, which needs 1 month to be optimised by an algorithm, needs 6 months time to be optimised robustly by a re-sampling method (supposing 5 new points were re-sampled around each solution). However, the proposed confidence-based robust optimisation process requires the same period of time (1 month) to determine the robust solution(s). The systems designed by computer scientists usually have very powerful computational foundations, but there is often little focus on real application. The investigated case studies of this thesis are real problems, so there will be an emphasis on real applications in addition to theoretical works. Several propellers will be optimised by MOPSO for the first time. In addition, this thesis is the first research which tries to design a robust propeller.

3. Analysis

3.8

81

Summary

The discussions of this chapter showed that the state-of-the-art in the field of single-objective and multi-objective evolutionary optimisation are very mature since there is considerable research in these two fields. There are many algorithms, benchmark problems, performance metrics, and constraint handling techniques. The applications of optimisation techniques in both fields can be found widely in different branches of science and industry. Although most of the real problems have multiple objectives, the importance of single-objective optimisation should not be underestimated. Such techniques are very essential in solving and analysing real problems with one objective. Robust optimisation is also important in both areas. It does not matter if we look for one solution or a set of solutions, the presence of uncertainties in real environments is always a substantial threat for the stability and reliability of the optimal solution(s) obtained. Robust optimisation in a multi-objective search space seems to be more challenging and critical than a single-objective search space although it is essential when solving real problems of any type. Finding optimal solutions that are less sensitive to perturbations requires a highly systematic robust optimisation algorithm design process. This includes designing challenging robust test problems to compare algorithms, performance metrics to measure how much a robust algorithm is better than another, and computationally cheap robust algorithms to find robust solutions for optimisation problems. The first two phases of a systematic algorithm design process, developing test functions and performance metrics, are prerequisite to the third phase, algorithm development. Benchmark functions provide test beds for challenging and testing different algorithms. They are the foundation of a systematic algorithm design and without them benchmarking of the algorithms is not possible. Despite the large number of test functions proposed for benchmarking global optimisers, there are inadequate benchmark functions to effectively test the performance of robust algorithms. Also, comparing algorithms on a benchmark function requires qualitative and quantitative metrics. Despite the significant advancements in multi-objective performance metrics, it was observed that the literature substantially lacks performance metrics to quantify the performance of robust multi-objective algo-

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3. Analysis

rithms. For robust single-objective algorithms, the current performance indicators can be employed easily due to similar performance measures: accuracy and convergence speed. For robust multi-objective algorithms, however, there are no performance metrics at all. All of the current works in the literature are qualitative, and not as accurate as quantitative research. Therefore, this gap is even more significant than the lack of test problems in the literature of robust optimisation. With suitable benchmark problems and performance metrics, different ideas can be implemented and evaluated to propose new/improved techniques as the main phase of systematic algorithm design. Due to the existence of diverse and standard test problems and performance metrics, there are a substantial number of algorithms in the field of global multi-objective optimisation. However, robust optimisation techniques are in the minority in this field. In addition, the literature review of this thesis revealed that the current robust optimisation techniques suffer from high computational cost and low reliability. The lack of systematic robust algorithm design is a main motivation for this thesis. This gap is the main reason why robust optimisation field lags behind the field of global optimisation and even some relevant fields such as dynamic optimisation or noisy optimisation. As discussed in the preceding paragraphs, the main components of a systematic design framework are: test function design, performance metric design, and algorithm design. This thesis contributes to these areas to establish the first systematic robust optimisation algorithm design process. In the next chapters, firstly, a variety of challenging test problems will be proposed to mimic the challenges in real search space when searching for robust solutions. The test functions are proposed in both single- and multi-objective optimisation fields. With these challenging test problems, obviously, the performance of a robust algorithm can be benchmarked effectively and reliably from different perspectives. Secondly, a set of performance indicators will be proposed for the first time in the literature of robust multi-objective optimisation. The performance measures are proposed only for robust multi-objective algorithms since the current singleobjective metrics can be utilised for quantifying the performance of robust singleobjective algorithms. The proposed performance measures allow efficient and reliable quantitative comparison of algorithms for the first time.

3. Analysis

83

Lastly, a novel indicator called confidence measure is proposed to improve the reliability of the archive-based methods when searching for robust solutions. The proposed confidence metric is designed for both single- and multi-objective algorithms. Therefore, it allows establishing two novel robust optimisation approaches called confidence-based robust optimisation and confidence-based robust multi-objective optimisation to search for robust solutions of real problems with single and multiple objectives.

Robust test function design

Robsut performance metric design

Chapter 4

Robust algorithm design

Benchmark problems

The first phase of a systematic design process is to find or design suitable benchmark problems. Benchmark problems are essential for challenging algorithms to observe and verify their performance. Although solving a real problem can better challenge an algorithm, we do not know how good an algorithm performs due to the unknown position of the optimum, and computational time is one of the main issues of real optimisation problems. In addition, a real search space has diverse difficulties combined that prevent us from observing and testing different ability of an algorithm independently. If suitable test functions exist in the literature, we have to chose a combination of different test functions to effectively benchmark the performance of algorithms. If there is no suitable challenging benchmark problem available in the literature, we have to propose some to reliably test and verify the performance of a given algorithm. Proposal of test functions is important, but having a framework to generate them with different degree of difficulty can facilitate the benchmark design process significantly. Frameworks mostly have alterable parameters that allow designers to generate test functions with a desired level of difficulty. For instance, a framework for creating multi-modal test functions has to offer a parameter to define the number of local optima. Using standard frameworks makes us confident about the quality of the test functions generated and allows difficulties to challenge an algorithm to be presented in isolation. This chapter proposes several frameworks that allow generating test functions with diverse characteristics and difficulties at desired levels. 84

4. Benchmark problems

85

The design process of a test problem includes two goals. On one hand, a test problem should be simple and modular in order to allow researchers to observe the behaviour of meta-heuristics and benchmark their performances from different perspectives. On the other hand, a test function should be difficult to be solved in order to provide challenging environments similar to those of real search spaces for meta-heuristics. These two characteristics are in conflict where greater simplicity makes a test function readily solvable for meta-heuristics and the relevant comparison inefficient. In contrast, although a very difficult test function is able to effectively mimic the real search space, it may be very difficult to be solved so that the performance of algorithms cannot be clearly observed and compared. These two conflicting issues make the development of test problems challenging. In this chapter several frameworks are proposed and utilised to design test functions for benchmarking robust single- and multi-objective meta-heuristics. For designing the frameworks and test functions, the guidelines suggested by Whitley et al. [166] are followed for creating standard test suites: • Standard test sets should contain test problems that are resistant to simple optimisation methods. • Standard test sets should include test problems with non-linear, non-separable, and non-symmetric search spaces. • Standard test sets should have scalable test problems. • Standard test sets should have test problems with scalable evaluation cost. • Standard test sets should contain test problem that are of canonical form, meaning that they should be independent of problem representation. These essentials were extended by B¨ack and Michalewicz [8, 10] (e.g. having few unimodal and highly multi-modal test functions). However, these guidelines are very generic and mostly applicable to single-objective test problems. In the literature, Deb et al. suggested specific recommendations for making multiobjective problems as follows [10, 55]: • Simplicity is one of the main factors of a multi-objective test problem. • There should be scalability in variables to allow making a desirable number of parameters.

86

4. Benchmark problems • There should be scalability in objectives for making a desirable number of objectives. • The exact shape and location of the Pareto optimal front should be known and easy to understand. • The Pareto optimal set in the parameter space should also be known and understandable. • There should be controllable mechanisms to provide different level of challenges for an algorithm to approach the Pareto optimal front (for instance the number of local Pareto optimal fronts). • There should be different shapes for the Pareto optimal front and discontinuity in order to benchmark the ability of an algorithm in finding well-distributed Pareto optimal solutions.

A suitable test suite is one that provides different test functions with a variety of the above-mentioned features. However, capturing all the possible combinations of these features is impractical, as discussed by Huband et al. [92] . In the following subsections, therefore, these features are captured as much as possible within various frameworks. The focus is on single-objective and multi-objective test problems due to the difficulty of many-objective test problems and scope of the thesis. In addition, the proposed frameworks only generate unconstrained test problems. Due to the standard modularity of the proposed frameworks, however, any kind of constraints in the multi-objective test problems proposed in [52] and other works in the literature can easily be integrated in the proposed test functions.

4.1

Benchmarks for robust single-objective optimisation

This section proposes three novel frameworks for generating different test functions with specific characteristics for effectively benchmarking the performance of robust single-objective algorithms.

4. Benchmark problems

4.1.1

87

Framework I

This framework is for creating a bi-modal parameter space with two optima. One of the optima is robust and the other is not robust. The mathematical formulation of this test function is as follows: x−1.5 2 x−0.5 2 1 2 f (x) = √ e−0.5( 0.5 ) + √ e−0.5( α ) 2π 2π

(4.1)

where α defines the width (robustness) of the global optimum. This function is illustrated in Fig. 4.1. This figure shows how parameter α defines the shape of the global valley without changing the fitness values of both local and global optima. 0

-0.01

f(x)

-0.02

-0.03

 = 0.01  = 0.05  = 0.1  = 0.15  = 0.2  = 0.25  = 0.3

-0.04

-0.05

-0.06

0

0.5

1 x

1.5

2

Figure 4.1: Proposed function with adjustable local optima robustness parameter. The parameter α changes the landscape significantly The two dimensional version of this function is defined as follow: 

1 −0.5 f (x, y) = √ e 2π

(x−1.5)2 +(y−1.5)2 0.5

2



2 −0.5 +√ e 2π

(x−0.5)2 +(y−0.5)2 α

2

(4.2)

Fig. 4.2 shows that the parameter α has a similar effect on the robustness of the global optimum using Equation 4.2. Finally, framework I is defined as follows:

 Pn

M inimise : f (~x) =

1 −0.5 √ e 2π

2 i=1 (xi −1.5) 0.5

2 !

 Pn

+

1 −0.5 √ e 2π

2 i=1 (xi −0.5) α

2 !

f(x,y)

f(x,y)

4. Benchmark problems

f(x,y)

88

y

x

y

x

y

x

y

x

y

x

x

y

x

y

x

f(x,y)

f(x,y) f(x,y)

f(x,y)

y

f(x,y)

x

f(x,y)

y

Figure 4.2: Effect α of on the robustness of the global optimum (4.3)

where : 0 ≤ xi ≤ 2

(4.4)

where n is the maximum number of variables. The local optimum is always located at (0.5, 0.5, ..., 0.5) and the global optimum is at (1.5, 1.5, ..., 1.5). This framework creates a global optimum with alterable degree of robustness that allows benchmarking the performance of a robust meta-heuristic in terms of favouring a robust solution. By changing the robustness of the global optimum, the resistance of a robust meta-heuristic dealing with a non-robust global optima can be observed. In addition, it may be seen

4. Benchmark problems

89

in Equation 4.3 that this framework is able to generate scalable test functions with a desirable number of variables. The characteristics of the test functions generated by this framework are summarised as follows: • Test functions are not readily solvable by simple optimisation methods. • The search space is non-linear, non-separable, and non-symmetric. • Test functions are scalable. • The robustness of the global optimum is alterable • The robustness of the global optimum does not affect the optimal values of both local and global optima. • Both local and global optima have the potential to be the robust optimum based on the value of the parameter α.

4.1.2

Framework II

The second framework generates a desirable number of local non-robust optima. In other words, a multi-modal search space with one global optimum, one robust optimum, and several local non-robust optima can be created by this framework. The mathematical formulation of this framework is as follows: M inimise :

f (~x) = −G(~x)H(x1 )H(x2 ) + ω

2

where :

e−2x sin λ × 2π(x + H(x) = 3 PN

G(~x) = 1 + 10

i=3

N

xi

π ) 4λ




− xβ

(4.5)

+ 0.5

(4.6)

(4.7)

0 ≤ xi ≤ 1

(4.8)

λ>0

(4.9)

90

4. Benchmark problems

f(x1,x2)

optima

x2

x1

Figure 4.3: Shape of the search landscape with controlling parameters constructed by framework II

β>0

(4.10)

As may be seen in Fig. 4.3, this framework allows generating (λ + 1)2 local optima in the search space. The effect of this parameter on the shape of the search space can be observed in Fig. 4.4. This figure shows that the search space becomes more challenging as λ increases. Another characteristic of this test framework is its parameter scalability. The function G(~x) is responsible for supporting three or more variables. Since G(~x) is a kind of penalty function, an algorithm should find zero values for variables x3 − xn in order to find the best robust optimum. The characteristics of the test functions generated by this framework are summarised as follows: • Test functions are not readily solvable by simple optimisation methods. • The search space is non-linear, separable, and non-symmetric. • Test functions are scalable. • The number of local optima can be adjusted. • The last, worst local optimum is the most robust optimum and has the highest distance from the global optimum.

91

x2

x1

x2

x1

x2

x1

x1

f(x1,x2)

f(x1,x2)

f(x1,x2)

x2

f(x1,x2)

f(x1,x2)

f(x1,x2)

4. Benchmark problems

x2

x1

x2

x1

Figure 4.4: Effect of parameter λ on the shape of search landscape

It should be noted here that test problems (including those proposed in this thesis) that are framed using one subset of decision variables to move around a surface (or front) and another subset that vary distance to that surface (front) are effectively separable as these sub-components can be solved separately, and are therefore biased toward algorithms which propagate this sub-vectors (and, if these are all at a boundary, those which truncate at boundaries). This is why rotation matrices are incorporated into the DTLZ problems used in the CEC multi-objective test suite. A modification with a rotational matrix is required to convert the resulting problems into non-separable ones.

4.1.3

Framework III

This framework was inspired by some of the current test functions in the field of global optimisation. It divides the search space into four sections and allows defining different functions in each section. The mathematical formulation is as

92

4. Benchmark problems

follows:

M inimise :

   f1 (x, y)     f (x, y) 2 f (x, y) =   f3 (x, y)     f4 (x, y)

(x ≤ 0) ∧ (y ≥ 0) (x ≥ 0) ∧ (y ≤ 0)

(4.11)

(x > 0) ∧ (y > 0) (x < 0) ∧ (y < 0)

f(x,y)

Any type of functions with robust and non-robust optima can be utilised as f1 to f4 . For instance, Fig. 4.5 shows a search space constructed using spherical, Ackley, Rastrigin, and pyramid-shaped functions. It is evident from the figure that the spherical function has the most robust optimum.

y

f(x,y)

y

x

x

x

Figure 4.5: An example of the search space that can be constructed by the framework III In order to provide scalability for this framework, there can be two possibilities. Each of the sub-functions can be chosen with a different number of variables or the function G(~x), which was integrated with the second proposed framework, can be multiplied by the results of each function as follows:    f1 (x, y) × G(~x) (x ≤ 0) ∧ (y ≥ 0)     f (x, y) × G(~x) (x ≥ 0) ∧ (y ≤ 0) 2 M inimise : f (x, y) = (4.12)   f (x, y) × G(~ x ) (x > 0) ∧ (y > 0) 3     f4 (x, y) × G(~x) (x < 0) ∧ (y < 0) PN where :

G(~x) = 1 + 10

i=3

N

xi

(4.13)

4. Benchmark problems

93

The characteristics of the test functions generated by this framework are summarised as follows: • Test functions are not readily solvable by simple optimisation methods. • The search space is non-linear, non-separable, and non-symmetric. • Test functions are scalable. • There can be a desired number of local, global, and robust optima.

4.1.4

Obstacles and difficulties for single-objective robust benchmark problems

Although the proposed frameworks are able to generate very challenging test beds, there are other difficulties when solving real problems that should be considered and simulated. In this subsection, five obstacles and difficulties such as desired number of variables, bias, deceptiveness, multi-modality, and flatness are introduced/employed to increase the difficulties of current test problems and propose several new test beds. 4.1.4.1

Desired number of variables

The majority of robust test problems are of low dimension. In addition, the robust optimum is moved when the dimensions change (for instance TP1, TP2, TP3 and TP4 in Fig. 2.14. This sub-section is inspired by the method of adding multiple variables when designing multi-objective test problems proposed by Deb et al. [45] and Zitzler et al. [181]. In this method a function called G(~x) is employed to handle all the variables except x1 and x2 as follows: ! N X G(~x) = (4.14) 50x2i + 1 i=3

where N is the desired number of variables. The shape of the function G(~x) is illustrated in Fig. 4.6. This figure shows that this function is a spherical function with an optimum located at the origin (xoptimum = [0, 0, ..., 0]). In order to handle multiple variables for a problem, it is multiplied by the original function as follows: F (~x) = f (~x)G(~x)

(4.15)

94

4. Benchmark problems

Figure 4.6: Search space of the function G(~x) This equation allows defining the shape of the search space by f (~x) and handling multiple variables by G(~x). To find the optimum of F (~x), an optimisation algorithm should find the optimal values for x1 and x2 because these two parameters define the search space of the f (~x) function in Equation 4.15. Then, it has to find the optimal values for x3 and xN which are all equal to 0. It should be noted that this method works for minimisation problems, yet the negation/inverse of G(~x) function can make it applicable for maximisation problems. In fact, the function G(~x) defines a similar search space to the f (~x) above the main landscape. A set of 10,000 random solutions is generated for a 10-variable version of TP1 using Equation 4.15 and illustrated in Fig. 4.7 . It may be seen that the size of search space (range of variables) increases proportional to the number of variables without changing the shape of the search landscape. In other words, an unlimited number of parallel layers (surfaces) with shape similar to f (~x) are constructed above it. N=5

N=10

N=15

N=30

N=50

Figure 4.7: Search space becomes larger proportional to N and without any change in the main search landscape The advantage of this method is the ease of applicability to any test functions without changing their shape.

4. Benchmark problems 4.1.4.2

95

Biased search space

Density of solutions in the search space is another important characteristic of a challenging test problem. If the search space has an intrinsic bias towards its robust optimum, it would not be clear to distinguish whether an algorithm approximates the robust optimum successfully or it obtains the robust optimum because of failure to find the global optimum. The following function is proposed to be multiplied by the test functions (if applicable) to bias the search space away from the robust optimum. !θ n X B(~x) = |xi + p| (4.16) i=1

where p indicates the point, the bias is defined toward or away from it, and θ defines the density. Equation 4.16 shows that the density of solutions in the search space can be adjusted by a parameter called θ. The density is uniform when θ = 1, towards the point p when θ > 1, and away from the point p when θ < 1. The effect of θ can be observed in Fig. 4.8.

1.2

 = 0.14286

10

Density

8

0.8

B(x)

B(x)

1 0.6 0.4

10

Uniform density

0 x

10

6 4

0 -10

x 10

4

=5

8

2

0.2 0 -10

=1

B(x)

1.4

6 4

Density

2 0 x

0 -10

10

0 x

10

Figure 4.8: Density of solutions when θ < 1, θ = 1, θ > 1 (p = 0) These functions can be multiplied or added to the current test problems to bias their search spaces. The function B(x) can bias the search space away from the robust optimum when θ < 1 and bias the search space toward non-robust optima when θ > 1. As an example of the usage of the function B(x), the search space of TP1 is biased with the following modifications:    1 − QN H(xi ) + 1 PN x2 G(~x) x2 + x2 < 25 1 2 i=1 i=1 i 1000 f (~x) = (4.17) B(~x)G(~x) 2 2 x + x ≥ 25 1

2

96

4. Benchmark problems

where B(~x) =

P2


θ

i=1 |xi |

, G(~x) =

 0 x < 0 i N 2 50x +1, and H(x ) = i i i=3 1 otherwise

P



Fig. 4.9 illustrates the transformation of the shape of the search space. This figure shows that the boundaries of the search space are first extended and then multiplied by the function B(~x). This maintains the original position and shape of the robust optimum while having bias away from it. It should be noted here the range of the search space changes due to the multiplication of the function for bias as shown in the objective function in Equation 4.17.

Figure 4.9: Conversion of an un-biased search space to a biased search space To experimentally observe the change in the bias of solutions in the search space, Fig. 4.10 is provided. This figure shows 50,000 randomly generated solutions of the un-biased and biased versions of TP1 function. It may be observed that the solutions are almost uniformly distributed throughout the un-biased search space. However, the density of solutions is decreased towards the robust optimum of the biased search space. 4.1.4.3

Deceptive search space

In a deceptive search space, a large region of the search space favours the undesirable optimum. In global optimisation, a deceptive search space tends towards local solutions. In this case, the search agents of meta-heuristics are deceived and converge toward the local optima [48, 167]. There is no deceptive robust test function in the literature, so the first one is proposed in this subsection. For constructing a deceptive function, generally speaking, there should be at least two optima: a deceptive optimum versus a true optimum [46]. The shape of the search space should be designed to favour the deceptive optimum.

4. Benchmark problems

97

Uniform density High density Low density

Figure 4.10: 50,000 randomly generated solutions reveal there is low density toward the robust optimum in the biased test function, while the density is uniform in the un-biased test function The proposed mathematical formulation of a deceptive robust test function is as follows: M inimise :

where :

G(~x) =

f (~x) = (H(x1 ) + H(x2 )) × G(~x) − 1

x−0.2 2

H(x) = 0.5−0.3e−( 0.004 ) −0.5e−(

N X

x−0.5 2 ) 0.05

(4.18)

x−0.8 2

−0.3e−( 0.004 ) +sin(πx) (4.19)

! 50x2i

+1

(4.20)

i=3

where N is the number of variables. The shape of this test function in illustrated in Fig. 4.11. This figure shows that there are four deceptive global optima on the corners of the search space. In addition, the search space includes four deceptive local optima. It may be observed that the entire search space favours global and local optima, while the robust optimum is located at [0.5, 0.5, 0, 0, ..., 0]. 4.1.4.4

Multi-modal search space

Multiple local solutions are another type of difficulty for test problems. Real search spaces often have a massive number of local solutions that make them very hard to optimise. In the literature of global evolutionary optimisation,

98

4. Benchmark problems Robust optima density

Local optima

Global optima density

Figure 4.11: Proposed deceptive robust test problem multi-modal test functions are very popular. There are different test functions in this field with exponentially increasing numbers of local optima [116, 122, 155, 174, 175, 57]. Since there is no robust test function with many local optima, one is proposed in this subsection as follows: M inimise :

where :

f (~x) = (H(x1 ) + H(x2 ))) × G(~x) − 1.399

H(x) = 1.5 − 0.5e

)2 −( x−0.5 0.04

(4.21)

 M  X 2 x−(0.6+0.02i) 2 − −( x−0.02i ) ( ) 0.004 − 0.8e 0.004 + 0.8e i=0

(4.22)

G(~x) =

N X

50x2i

(4.23)

i=3

where N is the number of variables and M indicates the number of non-robust local optima. Fig. 4.12 shows the shape of the search space that is created by Equations 4.21, 4.22, and 4.23. This figure shows that the robust optimum is the optimum located at [0.5, 0.5, 0, 0, ..., 0]. The function f has 4(M + 1)2 global optima, which are all non-robust. The number of non-robust global optima can easily be defined by the parameter M . In should be noted that the search space includes 4(M + 1) local robust optima in addition to the robust and global optima as well. The function G also extend these optima over other dimensions, therefore, a very difficult search space is constructed to challenge robust algorithms. An algorithm should avoid all non-robust local and global optima to

4. Benchmark problems

99

approximate the single robust optimum. Note that for very small values of δ, the middle panel is no longer robust in Fig. 4.12. So, the δ > 0.05 should be considered for this test function. Robust optimum Local optimum

Global optima

Figure 4.12: Proposed multi-modal robust test function (M = 10)

4.1.4.5

Flat search space

In non-improving or flat test beds, very little information about the possible location of the optimum solution can be extracted from the search space. A flat search space might wrongly be assumed very simple because of the very small number of local solutions. However, it is not very simple since the majority of meta-heuristics fail in solving such problems, especially if the first random individuals are all located on the flat regions. It this case, all the individuals are assigned equal fitness values, so evolutionary operators become ineffective. For instance, the PSO algorithm fails to update gBest and pBest effectively for guiding the particles. This deteriorates when searching for the robust optimum because of the high and consistent robustness level of the flat regions. Therefore, a robust algorithm may mistakenly assume the flat regions to be the robust optimum, while the best robust optimum can be somewhere else in the search space. There is no robust test problem with a flat search space, so the first one is proposed as follows: M inimise :

where :

f (~x) = (H(x1 ) + H(x2 )) × G(~x) − 2

H(x) = 1.2 − 0.2e−(

x−0.95 2 ) 0.03

− 0.2e−(

x−0.05 2 ) 0.01

(4.24)

(4.25)

100

G(~x) =

4. Benchmark problems

N X

50x2i

(4.26)

i=3

The shape of the search space constructed by Equations 4.24, 4.25, and 4.26 is illustrated in Fig. 4.13 This figure shows that there are only four optima near the corners of the search space. One of them (located at [0.05, 0.05]) has the least robustness, while the optimum located at [0.95, 0.95] has the highest robustness. The robustness of the optima positioned on [0.05, 0.95] and [0.95, 0.05] have high robustness along x2 and x1 respectively. It should be noted that the fitness value of all these optima are equal to 0. This function is deliberately required to have such optima to challenge robust algorithms in finding optima with equal fitness but different degree of robustness. In addition, the flat search space proposed provides very little information about the location of optima.

Figure 4.13: Proposed flat robust test function With the proposed framework and difficulties, desirable test functions can be constructed. Several test functions are created for the purpose of this thesis, which can be found in Appendix A. Note that TP1 to TP9 are taken from the literature and TP10 to TP20 are proposed by the above discussed frameworks and difficulties. This set of test functions provides very challenging environments for robust algorithms. The proposed test functions may be theoretically effective for benchmarking the performance of robust meta-heuristics due to the following reasons: • Scalable test functions allow defining a desired number of variables, increase the dimension of search agents of meta-heuristics, and boost the difficulty of robust test problems.

4. Benchmark problems

101

• The proposed scaling method does not change the shape of test functions, so it is readily applicable to any test function. • Biased test functions change the density of solutions toward non-robust regions of the search space, so the performance of robust algorithms can effectively be benchmarked. In other words, a robust algorithm should resist the intrinsic tendency of solutions toward the non-robust optimum. • Deceptive test functions entirely mislead search agents of meta-heuristics toward non-robust optima, so robust algorithms should be equipped with powerful operators to handle such issues. • Multi-modal test functions are highly suitable for benchmarking the performance of robust algorithms in terms of avoiding local and less robust solutions. Since the worst local optimum is the most robust, the benchmark functions of this class are very challenging. • Flat test functions provide the search agents of robust algorithms with very little information about the robust optimum, so meta-heuristics should search for the robust optimum without relying on the information provided by the flat regions. • Several optima with equal fitness but different degree of robustness can challenge robust algorithms in finding partial or complete robust optimal solutions. • Since the proposed test functions are difficult, scalable, independent of problem representations, and non-linear, they can be considered as standard test functions as per the guidelines of Whitley et al. [167].

4.2

Benchmarks for robust multi-objective optimisation

This section proposes three frameworks and several difficulties for creating challenging robust multi-objective test problems.

102

4.2.1

4. Benchmark problems

Framework 1

This framework is for creating a bi-modal parameter space and a bi-frontal objective space. The core part of this framework is the following mathematical function: x−1.5 2 x−0.5 2 2 1 f (x) = √ e−0.5( 0.5 ) + √ e−0.5( α ) 2π 2π

(4.27)

where α defines the width (robustness) of the global optimum. This function is identical to the function utilised in the previous section and may be seen in Fig. 4.1. This function is employed to propose a multi-objective framework. A similar framework to that of Deb in 1999 [46] is utilised, which consists of different controllable components. Without loss of generality, the framework can be formulated as follows: M inimise :

f1 (~x) = x1

(4.28)

M inimise :

f2 (~x) = H(x2 ) × {G(~x) + S(x1 )} + ω

(4.29)

where :

G(~x) =

x−1.5 2 x−0.5 2 1 2 H(x) = √ e−0.5( 0.5 ) + √ e−0.5( α ) 2π 2π

N X

50x2i

(4.30)

(4.31)

i=3

S(x) = −xβ

(4.32)

α>0

(4.33)

where α and β are two parameters for adjusting the robustness of the global Pareto optimal front and the shape of the fronts, and ω indicates a threshold for moving f2 (~x) (Pareto optimal fonts) up and down.

4. Benchmark problems

103

The shapes of parameter and objective spaces constructed by this framework are illustrated in Fig. 4.14. Note that in the following figures, the robustness curve (red line) indicates the robustness of points lying on the same vertical line (i.e. with the same value of f1 ), using as axis the same axis used for f2 (x) itself. The robustness is calculated by means of an average in a given neighborhood with two neighboring solutions. 1.5

G(x) G(x)

α

f2

1

G(x)

0.5

0

-0.5

0

0.5

1

1.5

2

f1

Figure 4.14: Search space and objective space constructed by proposed framework 1 It may be observed in this figure that the proposed method is able to provide two Pareto optimal fronts: global and robust. The robustness of the global front can be defined by α in H(x). The interesting characteristic of the function H(x) is that the Pareto optimal front does not move when the robustness of the left valley in the search space is varied. Therefore, the specific behaviour of robust meta-heuristics while changing the robustness of the global Pareto optimal front can be observed. The other component of this framework, G(~x), is responsible for providing an unlimited number of variables for this framework. As the formulation of this function shows, a weighted addition of all variables except x1 and x2 is calculated by this function and multiplied by H(x) in the framework. By increasing the value of H(x), both Pareto optimal fronts are converted to local fronts. In other words, the G(~x) function drives both fronts away from their optimal positions. A meta-heuristic has to find zeroes for x3 to xN in order be able to reach the best trade-offs between f1 (~x) and f2 (~x). Fig. 4.15 shows the changing shape of the search space with altered values for α. This figure shows the global valley has the potential to be even more robust than the local valley with some values of the proposed parameter.

α = 0.01

f(x1,x2)

f(x1,x2)

f(x1,x2)

f(x1,x2)

4. Benchmark problems

f(x1,x2)

104

α = 0.2

α = 0.1

α = 0.4

α = 0.3

Figure 4.15: Effect of α on the robustness of the global Pareto optimal front’s valley

f(x1,x2)

f(x1,x2)

f(x1,x2)

Another important feature of the proposed framework is the ability to generate linear, convex, and concave global and robust Pareto optimal fronts. The function S(x) is responsible for defining the shape of the bases of both valleys in the search space and consequently the fronts in the objective space. An adjustable function, S(x), is proposed in order to generate different shapes with different complexity for both global and robust optimal fronts. As may be seen in Equation 4.32, the parameter β defines the shape of both Pareto optimal fronts. The effects of different values of this parameter on the shape of parameter and objective spaces are illustrated in Fig. 4.16.

1.5

1.5

1.5

1

1 1

f2

f2

f2

0.5 0.5

0 0.5 0

0

0

0.5

1

1.5

2

-0.5

-0.5

0

0.5

1

f1

(

1.5

2

-1

0

0.5

1

)

(

1.5

2

f1

f1

)

(

)

Figure 4.16: Shape of parameter space and Pareto optimal fronts when: β = 0.5, β = 1, and β = 1.5. Note that the red curve indicates the robustness of the robust front and black curves are the front. The curvature of both concave and convex fronts is changed proportionally

4. Benchmark problems

105 2

1.4

0

1.2

-2

1

-4

f2

f2

1.6

0.8

-6

0.6

-8

0.4

-10

0.2

0

0.5

1 f1

1.5

2

-12

0

0.5

1 f1

1.5

2

Figure 4.17: Changing the shape of global and robust Pareto optimal front with β to the value of β. Fig. 4.17 shows how the parameter β allows having different shapes with different robustness for both global and robust Pareto optimal fronts. With the first proposed framework, a designer is able to create a bi-modal search space with an unlimited number of parameters that maps to a twodimensional objective space with various linear, concave, and convex global and robust Pareto optimal fronts. However, the proposed framework generates global and robust Pareto front with similar shapes. In order to provide a more flexible framework, it is necessary to create different shapes for each of the fronts. The generalised version of the framework is formulated as follows: M inimise :

f1 (~x) = x1

M inimise :

 H(x ) × {G(~x) + S (x )} + ω 2 1 1 f2 (~x) = H(x2 ) × {G(~x) + S2 (x1 )} + ω

where :

G(~x) =

x−1.5 2 x−0.5 2 2 1 H(x) = √ e−0.5( 0.5 ) + √ e−0.5( α ) 2π 2π

N X

50x2i

(4.34)

if x2 < 0.8

(4.35)

if x2 ≥ 0.8

(4.36)

(4.37)

i=3

S1 (x) = −xβ1

(4.38)

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4. Benchmark problems

S2 (x) = −xβ2

(4.39)

where α defines the robustness of the global valley, β1 defines the shape of the global Pareto optimal front, β2 defines the shape of the robust Pareto optimal front, and ω indicates a threshold for moving f2 (~x) (Pareto optimal front) up and down. As can be inferred from these equations, the parameter space is divided into two parts x2 < 0.8 and x2 ≥ 0.8. Since the global valley is located in x2 < 0.8, the global Pareto optimal front’s shape follows S1 (x). In addition, the local/robust valley is in x2 ≥ 0.8 and obeys S2 (x). The parameters β1 and β2 allow adjustment of the shape of the global and robust Pareto optimal fronts independently. With this mechanism, nine different combinations of linear, convex and concave global and robust Pareto optimal fronts can be constructed. This combination allows designers to investigate the behaviour of different robust meta-heuristics dealing with different shapes of global and robust fronts. Benchmark functions with different shapes for robust and global optima would generally be more difficult to solve because a robust algorithm needs to adapt to a very different Pareto optimal front with different shape when transferring from the global/local Pareto optimal front to the robust Pareto optimal front(s). This was recommended by Deb for multi-objective benchmark problems [46].

4.2.2

Framework 2

In order to provide a more challenging multi-modal test set, another framework is also proposed in this work. Generally speaking, multi-modal test problems have a large number of local optima (local Pareto optimal solutions), which make them suitable for benchmarking the exploration and local optima avoidance ability of an algorithm. The framework 1 is modified to propose a new framework that allows designers to construct a search space with some desired number of local Pareto optimal fronts. The multi-modal framework is formulated as follows: M inimise :

f1 (~x) = x1

(4.40)

M inimise :

f2 (~x) = H(x2 ) × {G(~x) + S(x1 )} + ω

(4.41)

4. Benchmark problems

107

2

where :

G(~x) =

e−x cos(λ × 2πx − x) + 0.5 H(x) = γ

N X

(4.42)

50x2i

(4.43)

i=3

S(x) = −xβ

(4.44)

γ ≥ 1.3

(4.45)

λ≥1

(4.46)

As may be seen in Equation 4.42, there is a new formulation for the H function with two new control parameters: λ and γ. Fig. 4.18 shows the shapes of the search space and Pareto optimal fronts. local fronts

valleys 1

Robust front

0.8

f2

0.6

Local fronts

0.4 0.2 0

Global front 0

0.2

0.4

0.6

0.8

1

f1

Figure 4.18: Shape of the parameter space and its relation with the objective space constructed by the framework 2 Fig. 4.18 shows that the search space has an incremental wave-shaped curvature along x2 . The proposed exponential-based equation of H(x) allows each valley to have different robustness. The robustness is proportional to the value

108

4. Benchmark problems

f(x1,x2)

f(x1,x2)

f(x1,x2)

f(x1,x2)

f(x1,x2)

of x2 , in which the first value is the least robust valley and the last valley has the best robustness. The objective space illustrated in Fig. 4.18 shows that each of the valleys corresponds to a front, so the robustness of the fronts also increase from bottom to top along f2 . There are four control parameters for this framework: β, ω, λ, and γ. The role of β and ω are identical to those of the first framework, in which β defines the shape of the Pareto optimal front and ω is a threshold for moving f2 (~x) (Pareto optimal fronts) up and down and locating them in a desirable range. The λ parameter defines the number of valleys in the search space and consequently the number of local Pareto optimal fronts in the objective space. This control parameter allows designers to provide a multi-modal search space with λ − 1 local fronts. It should be noted that the robustness of local fronts increases as they become farther from the global Pareto optimal front. The effect of this control parameter on parameter and objective spaces is shown in Fig. 4.19.

1

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.6

f2

f2

f2

f2

1 0.8

f2

1 0.8

0.4

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0.2

0

0

0

0

0

0.2

0.4

0.6

0.8

1

0

0.5

1

0

0.2

0.4

0.6

f1

f1

f1

λ=2

λ=4

λ=8

0.8

1

0

0.2

0.4

0.6

f1

λ = 16

0.8

1

0

0

0.5

1

f1

λ = 32

Figure 4.19: Effect of λ on both parameter and objective spaces. Note that the red curve indicates the robustness of the robust front and black curves are the fronts. The last control parameter, γ, defines the distance of the robust Pareto optimal front from the line f2 = 1. The greater the value of γ, the flatter the shape of the robust Pareto optimal front, and the greater the distance of local and robust fronts from the global Pareto optimal front. In addition, this control parameter controls the distance between fronts, as fronts become closer as γ is increased. The effects of this parameter are illustrated in Fig. 4.20. The second proposed framework allows control of the multi-modality of benchmark problems and the number of local Pareto optimal fronts. The shapes of the fronts are similar in this framework. However, as with the first proposed

4. Benchmark problems

109

1

1

1

1

0.5

0.5

0.5

0.5

0.5

0

0

-0.5

0

0.5

1

-0.5

0

0.5

1

-0.5

0

0.5

f1

)

(

f2 0

0

f1

(

f2

f2

f2

f2

1

1

-0.5

0

0

0.5

f1

)

(

1

-0.5

0

0.5

)

(

1

f1

f1

)

(

)

Figure 4.20: Effect of γ on the fronts. Note that the red curve indicates the robustness of the robust front and black curves are the fronts. framework, there is the possibility of changing the shape of each front using Equation 4.35. This capability has not been integrated with this framework in order to maintain its simplicity. It is worth mentioning that cos(λ × 2πx) in H(x) can be replaced with cos(deλ e × 2πx) in order to have an exponentially increasing number of local fronts.

4.2.3

Framework 3

In addition to the shape of Pareto optimal fronts and multi-modality, there is another issue when solving real engineering problem called discontinuity. This refers to possible gaps (dominated regions) in the Pareto optimal fronts of a problem that provide more complexity compared to continuous Pareto optimal fronts. In this work the ZDT test functions introduced by Deb et al. are modified to propose a framework that allows creation of a desired number of discontinuous robust regions in the main Pareto optimal front. The mathematical formulation of this framework is as follows: M inimise :

M inimise :

f1 (~x) = x1

  r x1 x1 − sin(ζ × 2πx1 ) H(x2 ) + ω f2 (~x) = G(~x) × 1 − G(~x) G(~x) (4.48)

2

where :

(4.47)

H(x) =

e−2x sin(λ × 2π(x + γ

π )) 4λ

−x

+ 0.5

(4.49)

110

G(~x) =

4. Benchmark problems

N X

50x2i

(4.50)

i=3

γ≥1

(4.51)

λ≥1

(4.52)

1

0

0

0.5

1

f(x1,x2)

f(x1,x2) 4

4

4

3

3

3

3

2

2

2

2

f2

f2

2

4

f2

f2

3

f2

4

f(x1,x2)

f(x1,x2)

f(x1,x2)

Similarly to the previous framework H(x) is responsible for making a waveshaped plane along x2 and making the search space multi-modal with λ − 1 local valleys. In this case, objective space also has λ − 1 local fronts. The parameter ω works similarly to the previous frameworks and is for moving the fronts in a desirable range. Parameter γ is for making fronts closer or farther apart, but a low value of γ now makes fronts closer, in contrast to the previous framework. The new parameter here is ζ, which is responsible for defining the number of discontinuous regions of the Pareto optimal fronts. The effects of this parameter on both parameter space and objective space are illustrated in Fig. 4.21.

1

1

1

0

0

0

0

-1

-1

-1

-1

0

0.2

0.4

0.6

f1

f1

ζ=1

ζ=2

0.8

1

0

0.2

0.4

0.6

f1

ζ=4

0.8

1

1

0

0.2

0.4

0.6

f1

ζ=6

0.8

1

0

0.2

0.4

0.6

0.8

1

f1

ζ=8

Figure 4.21: Effect of ζ on the parameter and objective spaces. Note that the red curve indicates the robustness of the robust front and black curves are the fronts. This figure shows that there are ζ discontinuous efficient regions in the main Pareto optimal front. By adjusting λ ad ζ, different test functions can be constructed. The key point of this framework is that the robustness of discontinuous regions decreases from left-top to right-bottom. Therefore, the most robust front

4. Benchmark problems

111

is the last local front and its leftmost region is the most robust area. As Fig. 4.22 shows, the search space is very challenging; a robust meta-heuristic should avoid optimal regions of the search space and move toward the most non-optimal areas, which are robust. valleys

local fronts

discontinuous regions

Robust front 4 3

Least robust regions f2

2 1 0

valleys

Most Robust regions -1

0

0.2

0.4

0.6

0.8

1

Global front

f1

Robustness

Figure 4.22: Parameter and objective spaces constructed by the third framework. The red curve indicates the robustness of the robust front and black curves are the fronts. Note that all the functions proposed in this subsection are separable. As mentioned above, a rotational matrix is required to make them non-separable similar to those in the DTLZ problems used in the CEC multi-objective test suite.

4.2.4

Hindrances for robust multi-objective test problems

According to Deb [42], a multi-objective optimisation process utilising metaheuristics deals with overcoming many difficulties such as infeasible areas, local fronts, diversity of solutions, and isolation of optima. Difficulties dramatically increase when searching for robust Pareto optimal solutions. Therefore, a robust multi-objective algorithm has to be equipped with proper operators to handle several difficulties in addition to the above-mentioned challenges: robust local fronts, multiple non-robust fronts, non-improving search space, isolation of robust fronts, deceptive non-robust fronts, different shapes of robust fronts, and robust fronts with separate robust regions. Designing an algorithm to handle all these difficulties in a real search space requires challenging test beds during development.

112

4. Benchmark problems

In order to simulate these difficulties, the following difficulties are introduced for proposing new challenging test functions and integration with the current test functions. 4.2.4.1

Biased search space

The first and simplest method of increasing the difficulty of a multi-objective test function is to bias its search space [46]. Bias refers to the density of the solutions in the search space. Almost all of the current test beds in the literature of robust multi-objective optimisation suffer from a biased search space towards robust regions [81]. Fig. 4.23 (a) illustrates 50,000 randomly generated points in the first two dimensions of the RMTP10 in Fig. 2.16 to have an image of the bias of the search space. This figure shows that the bias is toward the righthand corner of the front. Therefore, it is hard to determine whether the robust measure assists an algorithm to find the robust front or the robust front is obtained because of the failure of the algorithm to find the global front.

(a)

(b)

Figure 4.23: A non-biased objective space versus a biased objective space (50,000 random solutions). The proposed bias function requires the random points to cluster away from the Pareto optimal front. In order to prevent such issues, the following equation is introduced for the function g(~x) in the test functions to bias the search space away from the robust front: Pn  ψ i=2 xi g(~x) = 1 + 10 + (4.53) n−1

4. Benchmark problems

113

where ψ defines the degree of bias (ψ < 1 causes bias away from the PF) and n is the maximum number of variables. In Equation 4.53, ψ is responsible for defining the bias level of the search space. Fig. 4.23 (b) shows 50,000 random solutions in the same search space of Fig. 4.23 (a) but with ψ = 0.3. This figure shows that density of solutions is very low close to the robust Pareto front, and increased further from the front. This behaviour in a test function effectively assists benchmarking the performance of robust algorithms in approximating the robust Pareto optimal solutions. To further observe the effect of ψ on the density of solutions in the search space, 50,000 random solutions with different values for ψ are illustrated in Fig. 4.24. This figure shows that the search space is biased inversely proportional to ψ. In other words, the density of solutions is increased as ψ decreases.

𝜓 = 1/2

𝜓 = 1/4

𝜓 = 1/8

𝜓 = 1/16

Figure 4.24: Bias of the search space is increased inversely proportional to ψ

4.2.4.2

Deceptive search space

According to Deb [46], there are at least two optima in a deceptive search space: the deceptive optimum and the true optimum. The search space should be designed in such a way to entirely favour the deceptive optimum. Such problems are very challenging for evolutionary algorithms since the search agents are directed automatically towards the deceptive optimum by the search space while the global optimum is somewhere else [48, 167]. To date there has been no deceptive robust multi-objective test problem: the first is proposed in this subsection. The proposed mathematical formulation for generating deceptive test function is as follows: M inimise :

f1 (~x) = x1

(4.54)

114

4. Benchmark problems

f2 (~x) = H(x2 ) × {G(~x) + S(x1 )}

M inimise :

where :

G(~x) =

x−0.2 2

H(x) = 0.5−0.3e−( 0.004 ) −0.5e−(

N X

(4.55)

x−0.5 2 ) 0.05

x−0.8 2

−0.3e−( 0.004 ) +sin(πx) (4.56)

50x2i

(4.57)

i=3

S(x) = 1 − xβ

(4.58)

It may be observed that the framework is similar to those of ZDT [181] and DTLZ [55, 56]. However, the function H is modified as shown in Fig. 4.25. This figure shows that the proposed H function has two non-robust deceptive local optima, two non-robust global optima, and one true robust (which is local) optimum. The element sin(πx) at the end of this function causes deceptiveness of the search space, in which the entire search space deceptively favours the non-robust optima. 1.5

Robust non-robust optimum

1.4 1.3 1.2

H(x2)

1.1 1

Local non-robust optimum

0.9 0.8

Global non-robust optimum

0.7 0.6 0.5

0

0.2

0.5 x2

0.8

1

Figure 4.25: There are four deceptive non-robust optima and one robust optimum in the function H(x) The combination of H, G, and S functions constructs a deceptive multiobjective test problem as illustrated in Fig. 4.26. It should be noted that the

4. Benchmark problems

115

f(x1,x2)

f(x1,x2)

f(x1,x2)

function S is also modified to define the shape of non-robust and robust fronts. The newly added parameter β defines the shape of the fronts. It may be observed that the fronts are concave when β < 1 and convex when β > 1. Fig. 4.26 also shows that the proposed test function has two overlapped non-robust global fronts, two non-robust local fronts, and one robust local front. The entire search space favours the non-robust regions, so the non-robust fronts are highly deceptive.

β = 0.5

β=1

β=2

Figure 4.26: Different shapes of Pareto fronts that can be obtained by manipulating β The deceptive non-robust fronts are very attractive for the search agents of meta-heuristics. Therefore, these test problems have the potential to challenge robust algorithm significantly. The ability of an algorithm to avoid deceptive non-robust regions can be benchmarked. Also, the performance of an algorithm in approximating robust fronts with convex, linear, and non-convex shapes is benchmarked. 4.2.4.3

Multi-modal search space

Although the first two hindrances introduced can mimic the difficulties of real search spaces and challenge robust algorithms, there is another important characteristic called multi-modality. Real search spaces may have many local solutions that make them very challenging to solve. In the field of evolutionary singleobjective and multi-objective optimisation, there is a considerable number of

116

4. Benchmark problems

test problems with local optima. However, there is no multi-modal robust multiobjective test problem in the literature. The following test function is proposed in order to fill this gap: M inimise :

f1 (~x) = x1

(4.59)

M inimise :

f2 (~x) = H(x2 ) × {G(~x) + S(x1 )}

(4.60)

where :

H(x) = 1.5 − 0.5e

−( x−0.5 )2 0.04

 M  X 2 x−(0.6+0.02i) 2 − −( x−0.02i ( ) ) 0.004 − 0.8e 0.004 + 0.8e i=0

(4.61)

G(~x) =

N X

50x2i

(4.62)

i=3

S(x) = 1 − xβ

(4.63)

where N is the number of variables and M indicates the number of non-robust valleys in the parameter space and non-robust fronts in the objective space. It may be seem in the formulation that the framework is again identical to those of ZDT and DTLZ, yet the function H is different. To see the search space, the shape of function H is shown in Fig. 4.27. This figure shows that the robust optimum is a local optimum, while there are many non-robust global fronts. Note that the number of global fronts can be defined by adjusting the parameter M in the H function. To see how the parameter space and objective space of the proposed multimodal test problem look, Fig. 4.28 is provided. This figure shows that the same shape and number of optima are created along x2(f 2). So, the parameter space has many non-robust globally optimal valleys and one robust locally optimal valley. The objective space shows that the global and local valleys create only two fronts: a local front and a global front. It should be noted that there are actually more than two fronts, all the global fronts are overlapped. Obviously,

4. Benchmark problems

117

1.6 1.4

H(x2)

1.2 1 0.8

0

0.5 x2

1

f(x1,x2)

Figure 4.27: H(x) creates one robust and 2M global Pareto optimal fronts

Figure 4.28: Parameter space and objective space of the proposed multi-modal robust multi-objective test problem

the local front is the robust front, which should be approximated by robust algorithms. Similarly to the deceptive test functions, the function S is required to define the shape of the fronts as well. There is again a parameter β that is able to change the shape of search space and Pareto optimal fronts as shown in Fig. 4.29. This mechanism challenges robust algorithms to approximate different shapes for non-robust and robust Pareto optimal fronts. This set of test functions provides non-robust fronts as hindrances for robust multi-objective test functions. The search agents of robust algorithms should avoid all the local fronts to entirely approximate the robust front.

f(x1,x2)

f(x1,x2)

4. Benchmark problems

f(x1,x2)

118

β = 0.5

β=2

β=1

Figure 4.29: Different shapes of Pareto fronts that can be obtained by manipulating β 4.2.4.4

Flat (non-improving) search space

As mentioned above, a large portion of such landscapes is featureless, so there is no useful or deceptive information about the location of optima. For creating a robust multi-objective test function with a flat search space, the following equations are proposed: M inimise :

f1 (~x) = x1

(4.64)

M inimise :

f2 (~x) = H(x2 ) × {G(~x) + S(x1 )}

(4.65)

where :

G(~x) =

H(x) = 1.2 − 0.2e−(

N X

50x2i

x−0.95 2 ) 0.03

− 0.2e−(

x−0.05 2 ) 0.01

(4.66)

(4.67)

i=3

S(x) = 1 − xβ

(4.68)

4. Benchmark problems

119

This function H is modified to construct this test function. As illustrated in Fig. 4.30, this function has two global optima close to the boundaries. This function deliberately provides two optima to challenge robust algorithms in terms of favouring fronts with different degrees of robustness. In addition, a large portion of the function H is flat, so no information about the location of fronts can be extracted from the search space. Another challenge here would be the robustness of the flat region. Since the variation is consistent in the second objective, a robust algorithm may be trapped in the flat region, assuming it as the robust optimum, and failing to approximate the actual robust front.

1.3

H(x2)

1.2

Non-robust optimum

1.1 1 0.9 0.8

Robust optimum 0.05

0.95 x2

Figure 4.30: H(x) makes two global optima close to the boundaries Again the test function is equipped with a parameter called β which is responsible for defining the shape of the fronts. Three variations of this test function are constructed as shown in Fig. 4.31. Therefore, different shapes for both fronts are also another challenge for robust algorithms when solving these test functions. All the proposed robust multi-objective test functions are provided in Appendix B.

4.3

Summary

This section tackled the lack of suitable and challenging test functions in the literature of robust optimisation as the first phase of a systematic robust optimisation process. Three frameworks were first proposed to generate different single-objective test functions. The frameworks allowed creation of test func-

f(x1,x2)

4. Benchmark problems

f(x1,x2)

f(x1,x2)

120

Figure 4.31: H(x) makes two global optima close to the boundaries tions with desired levels of difficulty: optima with alterable robustness level (Framework I), multiple local non-robust solutions (Framework II), and alterable number of variables (Framework III). These frameworks allow us to create test functions with diverse difficulties to challenge robust algorithms. A framework is more beneficial than a single test function because it allows creating new test functions. It is easy to use, reliable, and creates test functions with single feature varying in degree of difficulty. In other words, frameworks assist testing a specific ability of an algorithm at different levels of difficulty. In addition, diverse difficulties were integrated with the test functions: desired number of variables, biased search space, deceptive non-robust local solutions, multiple non-robust local solutions, and flat search space. The characteristics of each test function and difficulty were investigated theoretically, by generating random solutions, and observing the shape of search space. A set of 11 test functions were proposed including very challenging test functions as the first test suite in the literature of robust single-objective optimisation. The details of these test functions can be found in Appendix A (TP9 to TP20). The second part of this chapter covered the proposal of three multi-objective frameworks for creating robust multi-objective benchmark functions and integration of several hindrances with the current test functions. There is no framework in the literature, so these three frameworks are the first. Framework 1 allowed us to create test functions with a robust front with different degree of robustness.

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121

Framework 2 was designed to create a search space with a desired number of non-robust local fronts. Framework 3 was for designing a robust front with disconnected regions. These difficulties are the main challenges that an algorithm may face when solving real problems and can be simulated efficiently by the frameworks proposed. Similarly to the single-objective benchmarks, test functions with the following difficulties were proposed for the first time: multi-modal search space, multiple local non-robust fronts, deceptive non-robust local front, robust front with different shapes (concave, convex, and linear), disconnected robust front, biased search space, and non-improving (flat) search space. The characteristics of each test function were inspected and confirmed theoretically in detail. This chapter also considered increasing the level of difficulty of current test problems. The main contribution was the proposal of frameworks as the core for the first phase of a systematic robust algorithm design process, but the current test functions were also improved. This is due to the drawbacks of the current robust test functions: low number of local solutions, low number of local fronts, low number of variables, and simplicity. The frameworks allow us to create challenging test functions to efficiently challenge robust algorithms. They can also be used by other researchers to generate test problems with different levels of difficulty. The improved test functions are also more challenging and better mimic the difficulties of the real search spaces compared to the current ones. Although the frameworks are beneficial in terms of generating test functions, there should be a common test suite as well to allow comparing research from different contributors. Therefore, a total of 34 challenging robust multi-objective test problems were proposed as the most difficult test functions in the literature. The proposed test suite will provide suitable test beds to compare the main method of this thesis with other methods in the literature. The details of these test functions can be found in Appendix B. Due to the difficulties and diversity of the proposed frameworks and benchmark problems, they can confidently be used in the first phase of a systematic robust algorithm design process to assure testing and challenging new algorithms, including the method proposed of this thesis.

Robust test function design

Robsut performance metric design

Chapter 5

Robust algorithm design

Performance measures

The second phase of a systematic robust algorithm design process is performance metric selection or proposal. Benchmark functions provide test beds, but performance metrics allow comparing algorithms quantitatively. Performance metrics are essential when evaluating an algorithm during the design process [34, 149] because they measure how much better an algorithm is. In addition they determine the extent to which minor or major changes in algorithms are beneficial. If suitable performance metrics are available in the literature, we can easily employ some of them to be able to evaluate and compare algorithms. If such metrics are not available, however, we have to propose new ones or extend the current ones. Due to the lack of performance metrics for quantifying the performance of robust algorithms, this chapter proposes two for the first time. For robust single-objective algorithms, the current performance indicators can be employed easily due to similar performance measures: accuracy and convergence speed. However, the current performance metrics in the field of multi-objective optimisation cannot be employed to measure the performance of a robust multi-objective algorithm. In this chapter, therefore, specific performance metrics are proposed only for robust multi-objective algorithms. As discussed in the related work, the literature shows that there are a considerable number of performance metrics in the field of EMOO [184]. Due to the complexity of the optimisation process and multi-objectivity in the EMOO field, there should be several performance metrics when comparing algorithms in order to provide a fair and objective comparison [112, 145]. The literature shows that different branches of EMOO also need specific or 122

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123

adapted metrics for effectively quantifying the performance of algorithms. For instance, Dynamic Multi-Objective Optimisation (DMOO) [49] and Interactive Multi-Objective Optimisation (IMOO) [50] need their own modified metrics as discussed in [86, 87] and [53] respectively. To date, however, there is no performance metric in the field of RMOO despite its significant importance. This is the motivation of this chapter, in which two novel performance metrics are proposed for robust multi-objective algorithms. Similar to the performance metrics in EMOO, there should be more than one unary metric in order to efficiently evaluate and compare the performance of an algorithm. The three main performance characteristics for an algorithm when finding an approximation of the robust front are: convergence, distribution, and the number of robust Pareto optimal solutions obtained. The first characteristic refers to the convergence of an algorithm towards the true robust Pareto optimal solutions. In this case, the ultimate goal is to find very accurate approximations of the robust Pareto optimal solutions. The second feature of performance is the ability of an algorithm in finding uniformly distributed robust Pareto optimal solutions. Finally, the number of robust and non-robust Pareto optimal solutions obtained is also important. A robust multi-objective algorithm should be able to find robust Pareto optimal solutions as much as possible and avoid returning non-robust Pareto optimal solutions. Obviously, all these performance characteristics cannot be measured effectively by one unary metric. For the convergence, this thesis employs the IGD to measure how close the solutions are to the expectation of the Pareto optimal front considering the possible perturbations. However, the current convergence and success ratio measures are not efficient because they do not consider the robustness of different regions of the expectation of the front. In the following subsections two novel metrics are proposed in order to measure coverage and the success ratio of robust multi-objective algorithms. It is assumed that the robustness of the true Pareto optimal front has been previously calculated (and is represented by the grey curves in the figures). Technically speaking, if the robustness has not been previously defined for each true Pareto optimal solution, it can be calculated easily by generating and averaging a set of random solutions (perturbing parameters with probable δ error) in the neighbourhood of the true Pareto solutions. Another assumption is that a smaller value of a robustness metric for a solution means less sensitivity to perturbation.

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5.1

5. Performance measures

Robust coverage measure (Φ)

The main idea of the proposed coverage measure of this thesis has been inspired by that of Lewis et al. in [112]. The general concepts of the proposed coverage measure are illustrated in Fig. 5.1. As may be seen in this figure, the objective Robust segments S1

S2

S3

S4

S5 S6 Robust segments S7

Pareto front S8

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Minimise (f2)

S9

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Figure 5.1: Schematic of the proposed coverage measure (Φ) space is divided into several, equal sectors from the origin in the proposed measure. The sectors are divided into two groups: robust and non-robust sectors. Needless to say, the robustness of the entire regions of the Pareto front should be known in order to identify the robust and non-robust sectors. After defining the number of robust segments, the number of segments that contain at least one Pareto optimal solutions obtained should be counted and divided by the total number of robust segments. The mathematical formulation is as follows (note that this performance measure only works for bi-objective problems): Φ=

n 1 X φn N n=1

 f1 (~ x) 1 ∃~x ∈ P S, α ≤ αn , R(Pn ) ≤ Rmin n−1 ≤ tan f2 (~ x) φn = 0 otherwise

(5.1)

(5.2)

where N is the number of robust sectors, ~x is an approximation of the Pareto optimal solutions obtained, αn is the angle of the right line of a segment, Pn

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125

indicates the closet true Pareto optimal solution to ~x, and Rmin is a minimum robustness value defined by the user. Note that there should be an exception when there is no robust segment in order to prevent division by zero. The accuracy of this measure is increased by the number of segments. It should be noted that some of the segments can be partially robust and partially non-robust (S8 in Fig. 5.1). It is assumed that a segment is robust if and only if it is completely robust even if the solution obtained lies on the robust part of the segment. In the example of Fig. 5.1, there are 10 segments and 9 Pareto optimal solutions obtained. The segments S1, S8, S9, and S10 are not robust, whereas S2 to S7 segments are robust. Among the six robust sectors, three of them are occupied by at least one solution. Therefore, the coverage measure is Φ = 63 = 0.5, meaning that 50% of the robust segments (approximately the robust Pareto optimal front) are covered by the given Pareto optimal solutions obtained. Some comments on the theoretical effectiveness of the proposed measure are written in the following paragraphs. Note that the grey line determines the robustness of points on the front; it gives no details regarding points away from the front. It also assumes one-to-one mappings, whereas in many real-world problems there are many-to-one mappings, which may have different robustnesses. In the following figures, a segment bounded on at least one side with a red line is not robust. By contrast, a segment with two blue lines is robust. • The greater the number of occupied robust segments, the higher the coverage (see Fig. 5.2):

N X n=1

φ1n

>

N X

φ2n −→ Φ1 > Φ2

(5.3)

n=1

• The higher the value of Φ, the greater the coverage of an algorithm on robust regions of a Pareto optimal front. • The number of the robust segments are considered by Φ, so the proposed measure indicates the coverage of robust regions (without considering nonrobust segments) (see Fig. 5.3).

126

5. Performance measures  = 0.33333

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Figure 5.3: Zero effect of occupied non-robust segments on the proposed coverage measure • Since the segments with partially robust regions are omitted, they have no effect on the final value of Φ (see Fig. 5.4). • Since Φ counts the number of segments (not the solutions in the segments), a large number of solutions obtained does not necessarily increase Φ. For example, an algorithm that finds a large set of solutions in a single segment shows low coverage by the proposed measure. • A greater number of segments results in greater accuracy of the coverage measure. In addition, increasing the number of segments does not increase the value of Φ since the occupied robust segments are divided by the number of robust segments (see Fig. 5.5).

5. Performance measures

127 =0

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Figure 5.4: Segments that are partially robust do not count when calculating Φ  = 0.33333

 = 0.090909

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Figure 5.5: The accuracy of the proposed coverage measure is increased proportional to the number of segments • Since the number of occupied segments is divided by the total number of robust segments, the values of Φ always lie in [0, 1]. • Use of the minimum robustness (Rmin ) assists in defining the degree of robustness when calculating Φ. • The coverage measure cannot be calculated when Rmin < min (R(robustness curve)), and this is considered an exceptional case (see Fig. 5.6). • The Φ measure converts to a normal coverage measure (all the segments are assumed as robust) when Rmin > max (R(robustness curve)) (see Fig. 5.7).

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Figure 5.6: Effect of the minimum robustness on the number of robust segments and Φ  = 0.73684 1 0.8 0.6

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Figure 5.7: All segments are converted to robust and counted when Rmin > max (R(robustness curve))

5.2

Robust success ratio (Γ)

With IGD and the measure proposed so far, the convergence and coverage of a given set of robust Pareto optimal solutions can be measured. However, none of these measures can quantify the success of an algorithm in terms of finding a number of robust solutions and avoiding non-robust solutions. Although different numbers of robust Pareto optimal solutions obtained affects the proposed measures, it seems there should be another measure that specifically assesses the number of robust and non-robust solutions. This is the motivation of a novel measure called robustness success ratio (Γ) in this subsection. A conceptual model of the Gamma measure is depicted in Fig. 5.8. As can be seen in this figure, the first step of calculating this measure is

5. Performance measures

129 Robust segments

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Maximise (f2)

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Minimum robustness

Maximise (f1)

Figure 5.8: Conceptual model of the proposed success ratio measure very similar to that of the coverage measure. The objective space is divided into vertical segments. The reason of choosing vertical sectors instead of diagonal segments is the importance of the robustness of each robust optimal solution obtained. In other words, a vertical line is drawn from each point to intersect the robustness curve. If the intersection point lies below the minimum desired robustness, the solution is robust. When calculating the Φ measure, the coverage of a given set of solutions is important no matter how far they are from the true robust Pareto optimal solutions. So a solution can be non-robust itself but in a robust segment due to its corresponding robust reference point. An example of this phenomenon is illustrated in Fig. 5.9. This figure shows that the solution obtained is located in a robust segment when calculating Φ and in a non-robust segment when defining the Γ measure. After vertically dividing the objective search into N segments, the division of the number of solutions obtained in the robust sectors by the number of solutions obtained in the non-robust sectors is calculated as the success ratio of an algorithm. The success ratio is mathematically expressed as: Γ=

γR 1 + γN R

where :

γR =

(5.4)

M X m=1

pR m

(5.5)

5. Performance measures

Pareto front

Pareto front

Robustness

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Minimum robustness

Maximise (f2)

Maximise (f2)

130

Minimum robustness

Maximise (f1)

Maximise (f1)

Figure 5.9: An example of a probable problem in case of using diagonal segments when calculating Γ

 1 ∃~x ∈ P S, β x) ≤ βm , R(Pn ) ≤ Rmin m−1 ≤ f1 (~ R pm = 0 otherwise

γN R =

M X

R pN m

(5.6)

(5.7)

m=1

R pN m

 1 ∃~x ∈ P S, β x) ≤ βm , R(Pn ) > Rmin m−1 ≤ f1 (~ = 0 otherwise

(5.8)

where M is the number of Pareto optimal solutions obtained, ~x is an approximation of the Pareto optima solutions obtained, βn is the f1 value of the right line of a segment, Pn is the closet true Pareto optimal solution to ~x, and Rmin is the minimum robustness value. In addition to the different segmentation mechanism, another difference of this measure compared to the proposed coverage measure is that Γ counts the number of solutions obtained in the robust segments, whereas the Φ measures counts the number of segments that have at least one solution. Note that γ N R is incremented by one in Equation 5.4 in order to prevent division by zero in case an algorithm does not find any non-robust solutions. In order to see how the proposed measure can be effective some highlights are:

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131

• The success ratio of an algorithm equals zero if there are no robust solutions found (see Fig. 5.10): γ R = 0 −→ Γ = 0

(5.9) =0 1

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Figure 5.10: Success ratio is zero if there is no robust solution in the set of solutions obtained • The success ratio of an algorithm with no non-robust solutions obtained is equal to the number of robust solutions obtained (see Fig. 5.11): γ N R = 0 −→ Γ = γ R

(5.10) =4

1 0.8

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Figure 5.11: Example of the success ratio for a set that contains only robust solutions • The greater the number of robust solutions obtained, the higher the success ratio (see Fig. 5.12): (γ1N R = γ2N R ) ∧ (γ1R > γ2R ) −→ Γ1 > Γ2

(5.11)

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 = 1.5 1

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Figure 5.12: Success ratio increases proportional to the number of robust solutions obtained • The fewer non-robust solutions obtained, the higher the success ratio (see Fig. 5.13): (γ1R = γ2R ) ∧ (γ1N R > γ2N R ) −→ Γ1 < Γ2

(5.12)

 = 0.75 1

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Figure 5.13: Success ratio is inversely proportional to the number of non-robust solutions obtained • The relative success ratios of two algorithms without non-robust solutions are defined by the respective number of solutions obtained in robust segments. (γ1N R = γ2N R = 0) ∧ (γ1R > γ2R ) −→ Γ1 > Γ2

(5.13)

5. Performance measures

133

(γ1N R = γ2N R = 0) ∧ (γ1R < γ2R ) −→ Γ1 < Γ2

(5.14)

• The use of minimum robustness (Rmin ) assists in defining the degree of robustness when calculating Γ. • The success ratio equals zero when Rmin < min (R(robustness curve)) (see Fig. 5.14). =0 1 0.8

f2

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Figure 5.14: Effect of minimum robustness on success ratio when Rmin < min (R(robustness curve)) • The Γ measure counts all Pareto optimal solutions obtained and converts to a normal success measure (all the segments are assumed as robust) when Rmin > max (R(robustness curve)) (see Fig. 5.15). • A greater number of segments results in greater accuracy of the Γ measure. It should be noted the proposed performance measures are designed for theoretical studies, in which the robust Pareto front of test functions are known. However, these measures can also be employed to quantify the performance of algorithms in solving real problems subject to availability of a known true Pareto optimal front. If the true Pareto optimal front is unknown (which is generally the case in real problems), the following steps should be completed to be able to use the proposed performance measures: • Solve the problem with a robust algorithm using the maximum possible number of search agents, iterations, and sampling points to find an accurate approximation of the true robust Pareto optimal front.

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Figure 5.15: Effect of minimum robustness on success ratio when Rmin > max (R(robustness curve))

• Calculate the robustness of each obtained solution in the first step by resampling n perturbed solutions around it.

After these two steps, robust multi-objective algorithms can be employed to approximate the robust front and then compared with each other quantitatively using the reference set obtained and the proposed performance measures. Another point that may be noted is that the proposed performance measures are helpful for quantifying the performance of robust multi-objective algorithms. They can be employed to analyse the results of any general multi-objective optimisation algorithms. An algorithm need not be Pareto-based to be measured by the proposed performance indicators. For instance, objective aggregation-based algorithms can also be employed to approximate the robust Pareto optimal front and then the performance measures applied to the set of solutions obtained. These measures cannot be utilised directly to improve the performance of robust multi-objective algorithms, but they are helpful for comparing the results of different algorithms after the optimisation process. The main point of the proposed performance metrics is that they can work with any set of reference points. No matter if the reference point is in the Pareto optimal front defined by the expectations or nominal objective functions, the performance metrics quantify the performance of an algorithm.

5. Performance measures

5.3

135

Summary

Due to the lack of performance measures in the field of robust optimisation, we cannot systematically design a robust algorithm. As the second phase of a systematic robust algorithm design process, this chapter proposed two performance measures for quantifying the performance of robust multi-objective algorithms for the first time. Both proposed measures are able to evaluate the performance of a robust multi-objective algorithm from different perspectives quantitatively. For one, the coverage measure proposed quantifies the distribution of Pareto optimal solutions obtained by algorithms along the robust front. For another, the proposed success ratio allows calculating the number of robust and non-robust solutions obtained. For each of the proposed performance indicators, several tests were conduced on manually created robust Pareto optimal fronts. The tests showed that: the coverage and success measures are able to quantify the spread of the robust Pareto optimal solutions across the robust regions and the number of robust and non-robust solutions obtained. There is no specific performance measure in the field of robust multi-objective optimisation, so the proposed measures in this chapter fill this substantial gap. Without these measures, we can only observe which algorithm is better in a qualitative sense. However, the proposed measures allow us to reliably investigate and confirm how much better an algorithm is. In addition, they are helpful in determining the extent to which changes in algorithms are beneficial. With such measures, therefore, systematic robust algorithm design process is possible not only in this thesis but also in other works in future. The remarks for each of the measures suggested that the proposed measures allow designers to benchmark their algorithms effectively and quantitatively. They will be the main comparison measures in this thesis as well. In Chapter 8, experimental results will demonstrate the effectiveness of the proposed measures in practice.

Robust test function design

Robsut performance metric design

Chapter 6

Robust algorithm design

Improving robust optimisation techniques The last phase of a systematic robust design process is algorithm design. Without benchmark problems and performance metrics, we cannot compare and find out which ideas are better than others quantitatively. The proposed phases in Chapter 4 and 5, benchmark problems and performance metrics, allow us to reliably and confidently compare and evaluate new ideas. Although algorithm evaluation/verification requires test functions and performance metrics, the algorithm design itself includes several steps. An algorithm design or improvement process starts with new ideas. An idea might be to hybridise algorithms, to integrate new operators in an algorithm, or to propose a novel approach. Currently, the two approaches of robust optimisation in this field, explicit versus implicit, suffer from two main drawbacks: high computational cost and low reliability. Therefore, they are less applicable to real problems with computationally expensive cost function(s). In addition, unreliability of implicit methods prevents us from finding robust solution(s) confidently, which is very critical for real problems. In order to alleviate such shortcomings, this chapter proposes several new ideas and establishes novel approaches for finding robust solution(s) in both single-objective and multi-objective search spaces reliably without the need for additional true function evaluations. Firstly, a confidence measure is proposed to define the degree of robustness of solutions during optimisation when using meta-heuristics. Secondly, confidencebased relational operators are proposed to establish Confidence-based Robust 136

6. Improving robust optimisation techniques

137

Optimisation (CRO) using single-objective optimisation algorithms. The proposed operators have been integrated with PSO and GA as the first confidencebased robust algorithms in the literature. Finally, confidence-based Pareto optimality is proposed to establish Confidence-based Robust Multi-objective Optimisation (CRMO) using multi-objective algorithms. The mechanism of performing CRMO using the MOPSO algorithm is introduced and discussed at the end.

6.1

Confidence measure

In this section a novel metric called the Confidence (C ) measure is proposed in order to calculate the confidence we have in the robustness of particular solutions based on the location of solutions already known in the neighbourhood. Later, different confidence-based relational operators are proposed utilising the C measure for robust meta-heuristics. Finally, the confidence-based operators are employed to establish a novel approach for finding robust solutions, called CRO. As case studies, some different methods of performing CRO using PSO and GA are introduced to demonstrate the general applicability of the proposed approach. In order to be useful, the confidence metric should be able to define the confidence level of a robust solution. Needless to say, the highest confidence is achieved when there are a large number of solutions available with greatest diversity within a suitable neighbourhood around the solution (search agent) in the parameter space. These three descriptive factors can be mathematically expressed as follows: • C ∝ the number of sampled points in the neighbourhood (n) • C ∝ the inverse of the radius of the neighbourhood (r) • C ∝ the inverse of the distribution of the available points in the neighbourhood (σ) The proposed confidence equations are as follows: n C= r.σ + 1 sP σ=

n ¯ i=1 (d −

n−1

di )2

(6.1)

(6.2)

138

6. Improving robust optimisation techniques x +Std.y

P1

P2

-Std.x

+Std.x

y

r P3 Current solution Sampled solution Average point Std. boundaries -Std.y

Figure 6.1: Confidence measure considers the number, distribution, and distance of sampled point from the current solution where n ≥ 2, d¯ is the average of the distance between the current solution and all the sampled points within the neighbourhood, and di is the Euclidean distance of the i-th sampled point to the current solution. Note that due to the stochastic nature of meta-heuristics, we assume that the distribution of sampled points within r radius around a solution is approximately uniform. Therefore, if the sampled points are closer to the solution, they give better confidence about the robustness. The concepts of the proposed metric and components involved are illustrated in Fig. 6.1. This figure shows that the proposed confidence measure defines and assumes a neighbourhood with radius r around every solution during optimisation. Defining this radius allows investigating different level of perturbations in the parameter space. By assuming a neighbourhood, an algorithm is able to differentiate between the confidence level of neighbouring solutions closer to or farther from the main solutions. Obviously, previously sampled solutions closer to the main solutions are able to better assist us in confirming the robustness of the main solution. The proposed confidence measure formulation considers this fact by dividing the factors of number of solutions and distribution by r. Fig. 6.1 also shows that the proposed confidence measure considers the number of previously sampled points in the neighbourhood as well as their distribution. This figure shows that considering both of these factors is essential

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139

because the number of solutions is not able to show the status of neighbouring solutions in terms of distribution. One solution may have more sampled points but without broad distribution. As may be seen in Fig. 6.1, the distance of each previously sampled solution is first calculated with respect to the main solution. The standard deviation is then employed to indicate the dispersion of the previously sampled solutions. Technically, in order to calculate the confidence measure there is a need to calculate the Euclidean distance of a particular position and each previously sampled point for finding those in a desirable distance. Therefore, the computational complexity of the proposed metric is of O(ns d) where ns is the number of previously sampled points and d indicates dimension. This is the computational complexity of calculating Euclidean distance between each previously sampled solutions and the main solution. To see how the proposed C metrics can be theoretically efficient some remarks are: • The confidence level of a search agent without previously evaluated neighbouring solutions is equal to zero: n = 0 =⇒ C = 0

(6.3)

• The confidence level of those solutions with the same number of neighbouring samples evaluated within equal radii are differentiated based on the dispersion: n1 = n2 ∧ rr = r2 ∧ σ1 > σ2 =⇒ C1 < C2

(6.4)

• The confidence levels of two solutions with an equal number of neighbouring solutions and similar distributions are defined based on the radii of their neighbourhoods. The closer the neighbourhood, the higher the level of confidence: n1 = n2 ∧ σ1 = σ2 ∧ r1 > r2 =⇒ C1 < C2

(6.5)

• The greater the value of C, the greater the level of confidence. • Due to using Euclidean distance, the C measure can easily be extended to problems with diverse numbers of dimensions

140

6. Improving robust optimisation techniques

• The proposed measure is independent of the objective function(s), so it is compatible with any kind of robustness measure. • The overall computational complexity of calculating C is O(ns d), so it is considered a cheap metric. • Confidence measure is easy to implement.

6.2

Confidence-based robust optimisation

As discussed, the confidence measure is able to calculate the confidence level of solutions with respect to the location of neighbouring evaluated solutions. This measure has the potential to be integrated with other mathematical operators for single- and multi-objective optimisation. In this thesis five new confidencebased relational operators are proposed as follows. Please note that the term robustness indicator (R) is utilised to refer to any kind of robustness measure. So, R(x) and C(x) calculates the robustness and confidence of the solution x respectively.

6.2.1

Confidence-based relational operators

~x is said to be confidently less than ~y (denoted by ~x C(~y ))

(6.6)

~x is said to be confidently less than or equal to ~y (denoted by ~x ≤c ~y ) iff : (R(~x) ≤ R(~y )) ∧ (C(~x) ≥ C(~y ))

(6.7)

~x is said to be confidently greater than ~y (denoted by ~x >c ~y ) iff : (R(~x) > R(~y )) ∧ (C(~x) > C(~y ))

(6.8)

~x is said to be confidently greater than or equal to ~y (denoted by ~x ≥c ~y ) iff :

(R(~x) ≥ R(~y )) ∧ (C(~x) ≥ C(~y ))

(6.9)

~x is said to be confidently equal to ~y (denoted by ~x =c ~y ) iff : (R(~x) = R(~y )) ∧ (C(~x) = C(~y ))

(6.10)

6. Improving robust optimisation techniques

141

As can be seen in the proposed confidence-based relational operators, the comparison is not solely based on the robustness indicator. This gives us the opportunity to design different mechanisms to confidently evaluate the search agent of meta-heuristics and consequently direct them toward the robust regions of the search space.

6.2.2

Confidence-based Particle Swarm Optimisation

The proposed operators have the possibility to be integrated with any kind of meta-heuristics. Various combinations of these operators can be incorporated into different components of meta-heuristic algorithms. The main idea of performing CRO is to employ confidence-based relational operators to design confidence-based components for meta-heuristics and guide the search agents confidently towards robust solution(s). Depending on the structure of a metaheuristic, different CRO scenarios can be defined. In EA, for instance, confidencebased reproduction processes (the crossover component) can be designed to generate offspring of an individual with robustness of highest confidence. Moreover, elitism can be re-designed totally based on the confidence level of robustness of individuals. The general framework of the proposed confidence-based robust optimisation is illustrated in Fig. 6.2. The most important steps of robust optimisation that are influenced by the proposed methodology are highlighted in grey. It may be seen that the initial random population is first evaluated. Before modifying solutions, which is based on the mechanism of the algorithm, the confidence level of each solution is calculated based on the current status of previously sampled points. By calculating the confidence level of all solutions, they can be compared by the confidence-based relational operators proposed in the preceding subsection. These operators allow us to decide whether a solution is confidently better than another. The two actions that can be taken are illustrated in Fig. 6.2. On one hand, the confidently better solutions are normally modified/evolved/combined by the operators of the algorithm. On the other hand, the non-confident solutions can either be discarded or randomly initialised/modified. These steps are repeated until the satisfaction of an end criterion. It is evident from this flow chart diagram that the proposed confidence-based robust optimisation prevents non-confident solutions from participating in improving the population. Therefore, the reliability of an algorithm can be improved signifi-

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Figure 6.2: Flow chart of the general framework of the proposed confidence-based robust optimisation cantly when utilising the previously sampled points. The proposed method is readily applicable for handling uncertainties in operating conditions as well. However, the process of handling such uncertainties

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is subject to one condition. Since operating conditions are secondary inputs for a system and not usually considered as a parameter to be optimised (they are always considered as a fixed value), they have to be parametrised first and then may be optimised by a confidence-based robust optimiser. This means that the operating conditions will be changed and optimised by the optimiser in addition to other parameters. In this thesis, the PSO algorithm is chosen as the first case study. Later, the CRO method will be applied using GA. With the proposed metric, generally speaking, there would be two metrics to find robust solutions: robustness and confidence metrics. The former metric defines the robustness of the search agent of meta-heuristics, whereas the latter metric defines how confident we are in the robustness of the solution. In the simple robust PSO algorithm, the particles are compared as follows: ~x < ~y ⇐⇒ R(~x) < R(~y )

(6.11)

where R(.) is the robustness indicator. The gBest and pBests are updated when a particle finds a better solution in the search space. With the proposed confidencebased operators, however, two new Confidence-based Robust PSOs (CRPSO) are proposed as follows: In CRPSO1, a particle is replaced with the best particle obtained so far if and only if it is confidently better than the best solution as follows: For minimisation: gBest := ~x ⇐⇒ ~x ≤c gBest

(6.12)

where ~x is a particle. For maximisation: gBest := ~x ⇐⇒ ~x ≥c gBest

(6.13)

where ~x is a particle. In order to implement this the confidence of the best solution obtained so far (gBest) is stored in a new variable called cgBest. It is clear that the gBest is updated if and only if the confidence level and robustness indicator are both better in CRPSO1. In CRPSO2, the personal best solutions (pBest) found so far are also updated (in addition to the gBest) based on the confidence level of solutions as follows:

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For minimisation: pBesti := ~x ⇐⇒ ~x ≤c pBest

(6.14)

For maximisation: pBesti := ~x ⇐⇒ ~x ≥c pBest

(6.15)

A new vector called cpBest is defined to store the best confidence level of particles over the course of iterations. As may be inferred from these equations, two confidence-based update procedures were designed to perform CRO using CRPSO. The two CRPSOs proposed show how the proposed confidence-based operators establish a new way of robust optimisation using meta-heuristics.

6.2.3

Confidence-based Robust Genetic Algorithms

In a Robust GA (RGA) every individual is evaluated based on its corresponding robustness measure and allowed to participate in the production of following generations. The greater the robustness of an individual, the higher the probability of mating. When RGA uses previously sampled points in the neighbourhood to define the robustness of each individual, it is prone to favour non-robust individuals, especially in the initial iterations because there is not enough sampled points to decide on the robustness of individuals. In this thesis, the confidence measure is employed to prevent such circumstances and drive the individuals toward robust optima. In order to integrate the confidence measure and confidence operators in the RGA, the following confidence-based components are proposed: 1. Confidence-based elitism: This component gives higher priority for an individual with high confidence. Without loss of generality, the mathematical model for a minimisation problem is as follows: E := Ii ⇐⇒ ∀k | Ii ≤c Ik

(6.16)

where E is the elite and Ii is the i-th individual. The purpose of this component is to save and replicate the most confidently robust solutions throughout generations and guide the rest of the individuals towards it.

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2. Confidence-based cross-over: this component prevents children with low confidence to replace their parents in the next generation. In other words, the parents are replaced by their children if and only if they are confidently more robust than them. The mathematical model for a minimisation problem is as follows: Pi := CHi ⇐⇒ CHi ≤c Pi

(6.17)

where Pi is i-th parent and CHi indicates the i-th child. The purpose of this component is to preserve highly confidently robust individual(s) in each generation and allow them to guide other individuals toward promising highly confidently robust regions of search spaces. These two components are incorporated in the simple RGA and named as Confidence-based Robust Genetic Algorithms (CRGA). In the first version, CRGA1, the best individual in each generation is selected by using Equation 6.16 and moved directly to the next generation based on the confidence-based elitism component. Note that if two solutions are better than the elite, the first one replaces the elite. If the second solution has a higher confidence and better fitness, the elite is updated again. In the second version, CRGA2, the individuals of each generation are allowed to move to the next generation subject to Equation 6.17.

6.3

Confidence-based robust multi-objective optimisation

6.3.1

Confidence-based Pareto optimality

With the Confidence measure, the concepts of Pareto dominance, Pareto optimality, Pareto solution set, and Pareto front for robust optimisation can now be modified as follows (Please note that the term robustness indicator (R) is utilised to refer to any kind of robustness measure. So, R(x) and C(x) calculates the robustness and confidence of the solution x respectively. ): Definition 6.3.1 (Confidence-based Pareto Dominance for minimisation): Suppose that there are two vectors: ~x = (x1 , x2 , ..., xk ) and ~y = (y1 , y2 , ..., yk ).

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Vector ~x confidently dominates vector ~y (denoted as ~x ≺c ~y ) if and only if: ∀i ∈ (1, 2, ..., o) [Ri (~x) ≤ Ri (~y )] ∧ [∃i ∈ (1, 2, ..., o) : Ri (~x) < Ri (~y )] ∧ [C(~x) ≥ C(~y )] Note that confidence-based Pareto dominance in a maximisation problem can be achieved by converting ≤ and < to ≥ and >. Since the concept of confidence measure does not change in minimisation and maximisation problems, the confidence-based Pareto dominance is defined as follows: Definition 6.3.2 (Confidence-based Pareto Dominance for maximisation): Suppose that there are two vectors: ~x = (x1 , x2 , ..., xk ) and ~y = (y1 , y2 , ..., yk ). Vector ~x confidently dominates vector ~y (denoted as ~x c ~y ) if and only if: ∀i ∈ (1, 2, ..., o) [Ri (~x) ≥ Ri (~y )] ∧ [∃i ∈ (1, 2, ..., o) : Ri (~x) > Ri (~y )] ∧ [C(~x) ≥ C(~y )] This definition results in the expression [C(xi ) ≥ C(yi )] being identical in maximisation and minimisation problems. Definition 6.3.3 (Confidence-based Pareto Optimality): A solution ~x ∈ X is called confidence-based Pareto optimal if and only if: {6∃ ~y ∈ X|~y ≺c ~x} Definition 6.3.4 (Confidence-based Pareto set): The set of all confidence-based Pareto optimal solutions: CPs = {~x ∈ X|6∃ ~y ∈ X, ~y ≺c ~x} Definition 6.3.5 (Confidence-based Pareto front): The set containing the value of objective functions for confidence-based Pareto solutions: ∀i ∈ (1, 2, ..., o) CPf = {Ri (~x)|~x ∈ CPs }

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Note that Definition 6.3.5. means that it is possible to have a ‘thick’ front (as in e.g. probabilistic domination). Some comments on the proposed Pareto optimality concepts are: • A solution is not able to confidently dominate another if it has less confidence: ~x ≺ ~y ∧ C(~x) < C(~y ) =⇒ ~x 6≺c ~y

(6.18)

• If the confidence of a solution is greater than another, the confidence-based Pareto dominance become equivalent to the normal Pareto dominance for that particular solution: C(~x) > C(~y ) =⇒≺c ≡≺

(6.19)

• If two solutions have equal confidence, the confidence-based Pareto dominance becomes equivalent to the normal Pareto dominance for that particular solution: C(~x) = C(~y ) =⇒≺c ≡≺

(6.20)

• If two solutions are non-dominated with respect to each other, they are also confidently non-dominated with respect to each other: ~x 6≺ ~y ∧ ~y 6≺ ~x =⇒ (~x 6≺c ~y ) ∧ (~y 6≺c ~x)

(6.21)

• The confidence-based Pareto solution set contains all the confident solutions and none of them can confidently dominate another. The proposed confidence-based Pareto optimality, dominance, set, and front can be integrated with different meta-heuristics. In other words, various combinations of these confidence-based concepts can be incorporated into different modules of meta-heuristic algorithms in order to perform a reliable robust optimisation. As mentioned above, this type of robust optimisation is named CRMO.

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6. Improving robust optimisation techniques

Confidence-based Robust Multi-Objective Particle Swarm Optimisation

In this thesis the MOPSO algorithm is chosen as the case study. Confidencebased Pareto dominance is integrated with the MOPSO algorithm in order to allow it to perform confidence-based robust multi-objective optimisation. The archive controller module of MOPSO is targeted as the main tool for integration. In the MOPSO algorithm, the archive controller module is responsible for deciding if a solution should be added to the archive or not. If a new solution is dominated by one of the archive member it should be omitted immediately. If the new solution is not dominated by the archive members, it should be added to the archive. If a member of the archive is dominated by a new solution, it is removed. Finally, if the archive is full an adaptive grid mechanism is triggered. In the proposed Confidence-based Robust MOPSO (CRMOPSO), however, the following rules are proposed to provide a confidence-based archive controller: • If a new solution is confidently dominated by one of the archive member it should be omitted immediately. • If the new solution is not confidently dominated by the archive members, it should be added to the archive. • If a member of the archive is confidently dominated by a new solution, it is removed. • If the archive is full the adaptive grid mechanism is triggered. There is also a modification to the archive itself in order to maintain the reliability of the archive. The CRMOPSO algorithm is required to update the confidence level of archive members at each iteration. This mechanism allows archive members to improve their confidence level based on the current status of the previously sampled solutions. In this case, an archive member may be omitted if its confidence level does not improve over the iterations. Note that this method should be employed when saving and using all the previously sampled points. If CRMOPSO only utilises a set of recent previously sampled points, this method would not be effective since the highly confident robust solutions will also be prone to being omitted from the archive.

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It also may be noticed that the dominance of a solution in CRMOPSO is identical to that of RMOPSO, and the confidence-based dominance is only applied to the archive controller. This prevents CRMOPSO from showing degraded exploration.

6.4

Summary

This chapter began with the proposal of a novel confidence measure for calculating the confidence level of robust solutions. Five new confidence-based relational operators were defined using the confidence measure. In addition, a new approach of robust optimisation called CRO was established in order to employ the confidence measure and relational operators for performing confident robust heuristic optimisation. The novel approach was employed to design two new variants of robust PSO and GA. Several theoretical comments and discussions were made about the potential success of the proposed concepts in finding robust optimal solutions at the end of the first part. The second part of this chapter was dedicated to the proposal of confidencebased Pareto optimality concepts using the confidence metric. The confidencebased Pareto dominance was integrated with the archive update mechanism of RMOPSO as a case study. The main contribution of the second part was the proposal and establishment of a novel perspective called CRMO. Similarly to the first part, several remarks were discussed to investigate and theoretically prove the effectiveness of the proposed CRMO approach in finding robust Pareto optimal solutions without extra function evaluations. In summary, the proposed concepts in this chapter have the potential to assist different algorithms in finding robust solutions in single- and multi-objective search spaces reliably and without extra function evaluations. In the following chapters the CRPSO, CRGA, and CRMOPSO algorithms are employed to solve the test functions proposed in Chapter 4 as well as a real problem. These algorithms are compared with a diverse range of algorithms in the literature quantitatively and qualitatively using the proposed benchmark problems and performance metrics in Chapter 4 and 5.

Chapter 7 Confidence-based robust optimisation The algorithm design is the last phase in a systematic design process. It can be considered as the main phase since the other two phases are essential for starting the last phase. Although an idea can be expressed in this phase without the need for the other two phases, we need test functions and performance metrics to investigate and prove the usefulness of the idea. To prove the merits of the ideas proposed in Chapter 6, a number of experiments are systematically undertaken in this chapter and the next chapter utilising tools proposed in chapters 4 and 5.

7.1

Behaviour of CRPSO on benchmark problems

The PSO algorithm was inspired by the social behaviour of bird flocking. It uses a number of particles (candidate solutions) which fly around in the search space to find the best solution. Meanwhile, they all trace the best location (best solution) in their paths. In other words, particles consider their own best solutions as well as the best solution the whole swarm has obtained so far. The confidence-based robust version of this algorithm is proposed in this thesis and examined in the following paragraphs. To see the behaviour of the proposed CRPSO algorithms Fig. 7.1 and Fig. 7.2 are provided. These figures show the behaviour of the proposed CRPSO1 dealing 150

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with some of the benchmark functions. The experiment was conducted using five particles over 100 iterations. The other initial parameters were as follows: • C1 = 2 • C2 = 2 • w was decreased linearly from 0.9 to 0.4 • Topology: fully connected • Initial velocity: 0 • Maximum velocity: 6 The radii of neighbourhoods were considered fixed and equal to the maximum perturbation in parameters. The maximum perturbation of each benchmark problem is available in Appendix A. Note that a simple expectation measure, which is based on the mean of the neighbouring solutions, is calculated as the robustness indicator. The first column of Fig. 7.1 and Fig. 7.2 shows the benchmark function, the search history, and the optimum finally obtained. The number of sampled points detected around each particle in each iteration is provided in the second column. The Cbests, expectation measure (mean of particle and its neighbouring solutions), and convergence curves are provided in the last three columns. Other results demonstrated are available in Table 7.1 which were obtained by CRMOPSO1 when the maximum number of iterations was increased to 500. This table shows the number of times that the confidence operators and confidence measures were triggered over the course of iteration. This table reveals how many times the proposed measure and operators assisted CRPSO1 to make confident decisions (updated gBest) during optimisation. In Fig. 7.1 and Fig. 7.2, the second columns of Test Problems (TP) show that the number of sampled points increased over the course of iterations. As mentioned in the previous chapters, the confidence measure is highly proportional to the number of neighbouring solutions. Moreover, the PSO algorithm tends to search mainly around the global best solution obtained so far. These are the reasons for the incremental behaviour in cBest curves of TP1 to TP10. However, this incremental trend stopped for a period of iterations occasionally. This shows that there was no confidence to update gBest in those iterations.

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Figure 7.1: Behaviour of CRPSO1 finding the robust optima of TP1, TP2, TP3, TP4, and TP5

For instance, the Cbest curve of TP1 shows that the confidence of gBest remains unchanged in almost half of the iterations. This shows that the PSO algorithm did not confidently find better robust solutions to replace at least one of them with gBest from nearly the 25-th to 75-th iterations. The same pattern can be observed in the figures for TP2, TP3, TP6, TP7, TP8, TP9, and TP10. The results of Table 7.1 show that gBest was confidently updated 29, 23, and 34

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Table 7.1: Number of times that the confidence operators and confidence measure triggered Test function Pi ≤c gBest C(Pi ) > C(gBest) TP1 29/500 130/500 TP2 23/500 69/500 TP3 34/500 180/500 TP4 63/500 272/500 TP5 9/500 434/500 TP6 12/500 166/500 TP7 13/500 243/500 TP8 21/500 85/500 TP9 20/500 217/500 TP10 9/500 95/500

solving all the benchmark functions approximately 190 times (on average). This is almost 40% of iterations on average for all the benchmark functions, showing that a normal PSO algorithm was prone to make non-confident decisions on these iterations when not using a confidence metric. This suggests the merit of the proposed confidence measure, confidence-based operators, and CRO in guiding search agents of CRPSO (or any other meta-heuristics) toward robust optima confidently.

Another interesting point that can be inferred from Table 7.1 is the correlation between making a confident decision and the number of local robust/nonrobust optima. The third column of Table 7.1 shows that TP4, TP6, TP8, TP9, and TP10 have the least number of confident decisions compared to TP1, TP2, TP3, and TP5. Other points worth noting can be observed in TP6, TP7, and TP9 of Fig. 7.2. The convergence curves on these benchmark functions show that the CRPSO1 algorithm faces degrading fitness functions over the course of iterations. In TP6, TP7, and TP9 this phenomenon happens near iteration 40, 30, and 10 respectively. This shows that the proposed method assists CRPSO1 to converge to robust optima located at non-global valleys/peaks. In addition, the search agents in CRPSO1 are able to diverge from non-robust solutions by using the proposed confidence measure.

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7.2

155

Comparative Results for Confidence-based Robust PSO

In this section the proposed CRPSO1 and CRPSO2 are compared with an Implicit averaging RPSO (IRPSO) and Explicit averaging RPSO (ERPSO). The CRPSO1, CRPSO2, and IRPSO algorithms utilise the previously sampled points when calculating the robustness measure. The robustness is calculated by an expectation measure with simple averaging of the neighbouring solutions. Since the robustness of a candidate solution is evaluated based on the available solutions in the neighbourhood during optimisation, IRPSO shows less reliability and its use will allow comparison of CRPSO1 and CRPSO2 in terms of improved reliability. In ERPSO, as the expectation measure for all algorithms, the H number of sampled solutions is created by the Latin Hypercube Sampling (LHS) method around every candidate solution to investigate and confirm robustness during optimisation. This provides the highest reliability and is helpful for verifying the performance of the proposed CRPSO1 and CRPSO2 algorithms as the most reliable reference. It does, however, markedly increase the computational load per iteration. In order to provide a fair comparison and see if the proposed method is reliable and effective, the same number of function evaluations (1000) is used for each of the algorithms. The number of trial solutions in the population was 5 for all algorithms, so the maximum iterations for CRPSO1, CRPSO2, and IRPSO was 200. Since ERPSO uses explicit averaging, however, 4 re-sampling points and 40 iterations are used in order to achieve the total number of 1000 function evaluations. Each algorithm is run 30 times on the benchmark functions in Appendix A and the statistical results (average, standard deviation, and median) are provided in Table 7.2. Note that the results are expected values and presented in the form of (ave ± std(median)). In addition, the Wilcoxon ranksum test was conducted at 5% significance level to determine the significance of discrepancy in the results. The p-values that are less than 0.05 could be considered as strong evidence against the null hypothesis. The results of this statistical test are provided in Table 7.3. For the statistical test, the best algorithm in each test function is chosen and compared with other algorithms independently. For

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example, if the best algorithm is CRPSO1, the pairwise comparison is done between CRPSO1/CRPSO2, CRPSO1/IRPSO, and CRPSO1/ERPSO. The same approach is followed throughout the thesis. As the results of Table 7.2 and Table 7.3 suggest, generally speaking the proposed CRPSO algorithms are able to successfully outperform IRPSO on the majority of benchmark functions. The statistical results show that this superiority is statistically significant in some of the cases. This shows that the reliability of the IRPSO can be improved with the concepts of confidence measure proposed. The results of CRPSO algorithms are very competitive compared to ERPSO as the algorithm with highest reliability, which again show the merits of the proposed algorithms. It is worth discussing the higher performance of CRPSO1 compared to CRPSO2. Table 7.2 show that CRPSO1 outperforms CRPSO2 on the majority of test functions. The results of these two algorithms are very close on some of the test problems. This shows that the idea of just confidently updating the global best obtained so far is better than confidently updating global best and personal bests found so far. A possible reason for this is that systematic exploration is weakened when CRPSO2 is required to confidently update the pBests obtained so far as well as the gBest. However, CRPSO1 allows the particles to update their pBests normally and only restricts update of the gBest. This assists the particles to search freely and find promising regions of a search space while they have to find a highly confident solution in order to be able to update gBest. However, it is very hard for particles of CRPSO2 (especially in initial iterations) to find confident solutions and update their pBests, so the particles tend to randomly fly around the search space rather being somewhat anchored around their pBests. In order to provide further analysis in terms of exploration and exploitation, the benchmark functions can be divided into two groups: uni-modal versus multimodal. The uni-modal functions have one global robust optimum and are highly suitable for benchmarking the exploitation of the robust algorithm. However, multi-modal functions have many local optima (robust or non-robust) with the number increasing with dimension, making them suitable for benchmarking the exploration of the robust algorithm. Note that TP1, TP2, TP5, and TP7 are uni-modal, whereas the rest of benchmark functions are multi-modal. The results on uni-modal benchmark problems show that the proposed algo-

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Table 7.2: Statistical results of the RPSO algorithms over 30 independent runs: (ave ± std(median)) Algorithm CRPSO1 CRPSO2 IRPSO ERPSO Algorithm CRPSO1 CRPSO2 IRPSO ERPSO Algorithm CRPSO1 CRPSO2 IRPSO ERPSO Algorithm CRPSO1 CRPSO2 IRPSO ERPSO Algorithm CRPSO1 CRPSO2 IRPSO ERPSO Algorithm CRPSO1 CRPSO2 IRPSO ERPSO Algorithm CRPSO1 CRPSO2 IRPSO ERPSO

TP1 0.516 ± 0.011(0.519) 0.521 ± 0.0138(0.527) 0.52 ± 0.0138(0.523) 0.251 ± 0.0888(0.24) TP4 0.47 ± 0.0673(0.477) 0.504 ± 0.0914(0.569) 0.525 ± 0.0534(0.522) 0.485 ± 0.0708(0.476) TP7 −3.88 ± 1.23(−4.1) −3.97 ± 0.483(−3.94) −3.65 ± 0.658(−3.55) −9.41 ± 1.13(−9.38) TP10 2.23 ± 0.52(2.48) 2.47 ± 0.63(2.56) 1.85 ± 0.26(1.86) 2.81 ± 0.65(2.67) TP13 35.1 ± 1.27(35) 36.1 ± 2.45(36.1) 50.5 ± 49.2(35) 36 ± 0.05(36) TP16 1.4 ± 0.342(1.31) 1.55 ± 0.493(1.42) 1.56 ± 0.551(1.42) 1.14 ± 0.139(1.07) TP19 −0.27 ± 0.088(−0.29) −0.28 ± 0.060(−0.29) −0.26 ± 0.049(−0.23) −0.37 ± 0.0021(−0.39)

TP2 1.01 ± 0.0224(0.999) 0.992 ± 0.0525(0.967) 1.01 ± 0.0238(1) 1.02 ± 0.0374(1.01) TP5 −11.7 ± 1.32(−11.9) −6.44 ± 1.41(−6.23) −6.63 ± 1.28(−6.75) −6.22 ± 1.24(−6.74) TP8 −0.652 ± 0.0584(−0.626) −0.838 ± 0.0523(−0.84) −0.623 ± 0.00589(−0.623) −0.599 ± 0.0806(−0.623) TP11 6358.8 ± 3553.3(7267.6) 5027.1 ± 2433.2(4720.1) 7092.8 ± 2982.7(7447.5) 3326.7 ± 3599.5(1082.7) TP14 50.2 ± 19.1(45.1) 43.4 ± 2.02(43.3) 60.3 ± 49.8(44.3) 44.3 ± 1.11(44.3) TP17 0.558 ± 0.333(0.458) 4.4 ± 11.8(0.768) 0.645 ± 0.518(0.72) 0.86 ± 0.62(0.857) TP20 −1.81 ± 0.898(−1.43) −0.589 ± 2.04(−0.711) −0.498 ± 1.76(−0.906) −1.38 ± 2.01(−1.76)

TP3 0.525 ± 0.126(0.603) 0.526 ± 0.127(0.604) 0.526 ± 0.126(0.603) 0.565 ± 0.185(0.63) TP6 −9.29 ± 1.39(−8.87) −3.99 ± 0.937(−4.04) −3.07 ± 1.42(−2.75) −3.42 ± 1.32(−3.28) TP9 −30.4 ± 1.6(−30.3) −29.7 ± 1.24(−30) −30 ± 2.3(−29.8) −27 ± 1.6(−27.1) TP12 9.5E3 ± 1.9E3(8.8E3) 1.3E4 ± 3.5E3(1.4E4) 1.1E4 ± 3.8E3(9.0E3) 2.2E5 ± 1.4E5(2.3E5) TP15 434.89 ± 5.006(433.33) 458.70 ± 55.61(433.81) 433.08 ± 0.99(433.22) 449.44 ± 30.30(435.08) TP18 0.107 ± 0.068(0.0858) 1.06 ± 2.9(0.133) 0.127 ± 0.0802(0.0863) 0.35 ± 0.123(0.403)

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Table 7.3: Results of Wilcoxon ranksum Test function CRPSO1 CRPSO2 TP1 0.000182672 0.000182672 TP2 0.10410989 N/A TP3 N/A 0.570750388 TP4 N/A 0.520522883 TP5 N/A 0.000182672 TP6 N/A 0.000182672 TP7 0.000182672 0.000182672 TP8 0.000246128 N/A TP9 N/A 0.570750388 TP10 0.140465048 0.025748081 TP11 0.10410989 0.140465048 TP12 N/A 0.037635314 TP13 N/A 0.185876732 TP14 0.212293836 N/A TP15 0.79133678 0.570750388 TP16 0.031209013 0.014019277 TP17 N/A 0.007566157 TP18 N/A 0.001706249 TP19 0.000182672 0.000182672 TP20 N/A 0.088973012

test for RPSO IRPSO 0.000182672 0.16197241 0.623176224 0.16197241 0.000182672 0.000182672 0.000182672 0.000182672 0.733729996 N/A 0.064022101 0.212293836 0.000182672 0.185876732 N/A 0.011329697 0.27303634 0.733729996 0.000182672 0.088973012

algorithms ERPSO N/A 0.241321593 0.10410989 0.969849977 0.000182672 0.000182672 N/A 0.000182672 0.000768539 0.002827272 N/A 0.000182672 0.344704222 0.384673063 0.000182672 N/A 0.04515457 0.212293836 N/A 0.909721889

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rithms are not significantly better than IRPSO. This originates from the fact the unreliabilty of the IRPSO is a bonus in a unimodal search space since this algorithm quickly converges towards the global optimum, which is most of the cases the robust optimum as well. Obviously, this leads the IRPSO algorithm towards a local solution in a multi-modal search space. However, the proposed CRPSO algorithm are limited in terms of updating global best and personal bests, which slows down the convergence speed of these algorithms. This originates from the proposed confidence measure that prevents CRPSO1 and CRPSO2 from premature exploitation and consequently stagnation in local robust/global optima. In contrast to the results on the uni-modal functions, those on multi-modal robust benchmark functions are different. The results on TP3, TP4, TP6, TP8, TP9, and TP10 to TP20 show that CRPSO1 and CRPSO2 tend to provide much better results compared to the uni-modal test functions. The significance of these results can be observed in the p-values in Table 7.3. The results suggest that the proposed algorithms are very capable of avoiding local robust/global optima. This originates from the proposed confidence measure that assists CRPSO1 and CRPSO2 to find promising robust area(s) of the search space confidently. The better results of CRPSO1 compared to CRPSO2 are again due to the greater exploration of CRPSO1. To sum up, firstly, the results demonstrated that the proposed confidence measure is able to assist optimisation algorithms and improves their reliability. Secondly, the results of CRPSO1 and CRPSO2 revealed the merits of the proposed confidence-based relational operators and new CRO approach in finding robust optima. Thirdly, the results on the uni-modal benchmark functions showed that both proposed algorithms show slow convergence behaviour, preventing them from easily stagnating in local solutions (robust or non-robust) especially in the initial iterations. Finally, the results on the multi-modal functions indicate that the proposed methods are able to avoid local robust and non-robust optima as well. In the next section the proposed confidence measure and relational operators are applied to a GA in order to further investigate the applicability of these novel concepts to different meta-heuristics.

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Comparative Results for Confidence-based Robust GA

These CRGA1 and CRGA2 algorithms are applied to the benchmark functions and the results are reported in Table 7.4. The CRGA1 and CRGA2 algorithms are compared with an Implicit averaging RGA (IRGA) and Explicit averaging RGA (ERGA). Similarity to the algorithms in the preceding section, the CRGA1, CRGA2, and IRGA algorithms utilise the previously sampled points when calculating the robustness measure. The robustness is calculated by an expectation measure with simple averaging of the neighbouring solutions. Since the robustness of a candidate solution is evaluated based on the available solutions in the neighbourhood during optimisation, IRGA shows less reliability and its use will allow comparison of CRGA1 and CRGA2 in terms of improved reliability. In ERGA, as the expectation measure for all algorithms, the H number of sampled solutions is created by the LHS method around every candidate solution to investigate and confirm robustness during optimisation. This provides the highest reliability and is helpful for verifying the performance of the proposed CRGA1 and CRGA2 algorithms as the most reliable reference. It does, however, markedly increase the computational load per iteration. In order to provide a fair comparison and see if the proposed method is reliable and effective, the same number of function evaluations (1000) is used for each of the algorithms. The number of trial solutions in the population was 5 for all algorithms, so the maximum number of iterations for CRGA1, CRGA2, and IRGA was 200. Since ERGA uses explicit averaging, however, 4 re-sampling points and 40 iterations are used in order to achieve the total number of 1000 function evaluations. Each algorithm is run 30 times and the statistical results (average, standard deviation, and median) are provided in Table 7.4. Also, the Wilconxon ranksum test was conducted at 5% significance level to decide on the significance of discrepancy in the results. The p-values that are less than 0.05 could be considered as strong evidence against the null hypothesis. Table 7.5 shows the results of the Wilcoxon ranksum test. Table 7.4 Table 7.5 suggests that IRGA provides the worst results on the majority of benchmark problems. This is evidence that utilising previously eval-

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Table 7.4: Statistical results of the RGA algorithms over 30 independent runs: (ave ± std(median)) Algorithm CRGA1 CRGA2 IRGA ERGA Algorithm CRGA1 CRGA2 IRGA ERGA Algorithm CRGA1 CRGA2 IRGA ERGA Algorithm CRGA1 CRGA2 IRGA ERGA Algorithm CRGA1 CRGA2 IRGA ERGA Algorithm CRGA1 CRGA2 IRGA ERGA Algorithm CRGA1 CRGA2 IRGA ERGA

TP1 2.36 ± 0.799(2.15) 0.443 ± 0.359(0.278) 2.35 ± 0.96(2.17) 1.92 ± 0.332(1.92) TP4 0.817 ± 0.158(0.788) 0.468 ± 0.0682(0.443) 0.74 ± 0.167(0.712) 0.712 ± 0.124(0.692) TP7 −5.13 ± 1.15(−5.02) −9.81 ± 0.461(−9.64) −6.62 ± 0.844(−6.23) −5.87 ± 1.33(−5.46) TP10 4.27 ± 1.44(3.98) 1.76 ± 0.421(1.61) 4.17 ± 1.03(4.35) 3.87 ± 1.77(3.67) TP13 81.5 ± 10.9(81.7) 218 ± 43.2(211) 259 ± 46.3(270) 224 ± 49.5(236) TP16 340.99 ± 89.11(343.56) 245.17 ± 65.42(250.17) 229.27 ± 117.56(208.46) 36.88 ± 13.41(38.67) TP19 −0.368 ± 0.0608(−0.395) −0.21 ± 0.181(−0.228) −0.211 ± 0.151(−0.233) −0.216 ± 0.151(−0.232)

TP2 2.02 ± 0.256(2.07) 1.17 ± 0.162(1.13) 2.09 ± 0.217(2.04) 2.05 ± 0.175(2.04) TP5 −9.06 ± 1.39(−8.67) −12.5 ± 1.01(−12.3) −9.44 ± 1.49(−9.3) −8.98 ± 1.65(−8.49) TP8 −0.876 ± 0.0323(−0.879) −0.615 ± 0.113(−0.617) −0.564 ± 0.137(−0.531) −0.532 ± 0.146(−0.494) TP11 1.3E4 ± 7.9E3(1.3E4) 1.2E4 ± 7.8E3(1.2E4) 1.8E4 ± 9.6E3(1.9E4) 5.1E2 ± 3.9E2(3.3E2) TP14 254.97 ± 40.81(264.21) 226.76 ± 51.62(219.80) 239.51 ± 59.23(229.67) 88.15 ± 10.87(91.28) TP17 341.21 ± 150.36(313.37) 257.94 ± 96.35(259.78) 287.02 ± 97.76(271.76) 31.01 ± 15.60(28.53) TP20 +2.53 ± 1.16(2.49) +2.3 ± 0.958(2.2) +2.38 ± 0.97(2.25) −1.32 ± 0.64(−1.45)

TP3 1.43 ± 0.38(1.42) 0.682 ± 0.128(0.722) 1.56 ± 0.317(1.8) 1.6 ± 0.23(1.67) TP6 −5.32 ± 1.14(−5.24) −10.9 ± 1.65(−10.6) −6.64 ± 0.67(−6.49) −5.29 ± 1.66(−5.23) TP9 −27.6 ± 1.61(−27.8) −23.2 ± 1.72(−23.2) −23.8 ± 2.51(−24.1) −23.3 ± 2.06(−22.3) TP12 2.2E6 ± 2.8E6(1.3E6) 2.2E8 ± 1.8E8(1.6E8) 4.1E8 ± 2.8E8(3.6E8) 2.4E8 ± 8.8E7(2.2E8) TP15 673.37 ± 66.86(671.51) 665.84 ± 35.17(674.43) 704.03 ± 67.23(708.44) 510.41 ± 25.51(518.36) TP18 238.01 ± 43.49(260.77) 39.53 ± 14.49(38.61) 284.93 ± 107.78(300.44) 223.70 ± 42.16(228.25)

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Table 7.5: Results of Wilcoxon ranksum Test function CRGA1 CRGA2 TP1 0.000182672 N/A TP2 0.000182672 N/A TP3 0.000182672 N/A TP4 0.000246128 N/A TP5 0.00058284 N/A TP6 0.000182672 N/A TP7 0.000182672 N/A TP8 N/A 0.000329839 TP9 N/A 0.000439639 TP10 0.000329839 N/A TP11 0.000182672 0.00058284 TP12 N/A 0.000182672 TP13 N/A 0.000182672 TP14 0.000182672 0.000182672 TP15 0.000182672 0.000182672 TP16 0.000182672 0.000182672 TP17 0.000182672 0.000182672 TP18 0.000182672 N/A TP19 N/A 0.002827272 TP20 0.000182672 0.000182672

test for RGA IRGA 0.000182672 0.000182672 0.000182672 0.000246128 0.00058284 0.000182672 0.000182672 0.000182672 0.001314945 0.000329839 0.000182672 0.000182672 0.000182672 0.000182672 0.000182672 0.000182672 0.000182672 0.000182672 0.007284557 0.000182672

algorithms ERGA 0.000182672 0.000182672 0.000182672 0.000329839 0.000246128 0.000182672 0.000182672 0.000182672 0.004586392 0.000768539 N/A 0.000182672 0.000182672 N/A N/A N/A N/A 0.000182672 0.009108496 N/A

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uated points may give satisfactory knowledge about the robustness of search space, but does not guarantee having reliable neighbouring solutions with uniform distribution during optimisation. This is the reason for the poor results of IRGA, in which individuals might crossover with non-robust individuals and evolve toward non-robust regions of search space. An individual might become very robust in case of a small number of neighbouring solutions since an average of the neighbourhood is considered as the fitness of individual in IRGA. The comparison of individuals based on effective mean fitness becomes more unfair when the neighbouring solutions are very close to the individual. These phenomena happened quite often during optimisation especially in the initial generations when there are few evaluated points. Table 7.4 and Table 7.5 show that the results of CRGA1 and CRGA2 are better than those of IRGA on the majority of benchmark functions. Generally speaking, the superiority of the results is due to the confidence measure employed. The confidence measure assists CRGA1 and CRGA2 to consider not only the robustness measure but also the number of neighbouring solutions, the radius of the neighbourhood, and the distribution of neighbouring solutions in distinguishing robust solutions. Therefore, the possibility of favouring a nonrobust solution due to few and close neighbouring solutions is much less than the previous methods. The role of the confidence measure in driving the individuals toward robust areas of search space is significant, especially in the initial generations when the individuals have very few neighbouring solutions to prove their robustness. As may be observed in Table 7.4 and Table 7.5, the CRGA2 algorithm outperforms CRGA1 on the majority of the benchmark problems. The CRGA1 algorithm has a confidence-based elitism component, in which the most robust and confident individual obtained so far is saved and allowed to move without any modification to the next generation(s). The reason for better results of this algorithm compared to IRGA is that the confidence-based component assists CRGA1 to favour robust solutions. The confidence-based elitism also prevents the best individual (elite) from corruption by the mutation operator. The advantage of this method is that there is no significant loss of exploration since all the individuals except one are selected without considering the confidence measure. However, the confident and robust elite is only one individual in the population so it is able to crossover with one of the other individuals. There are

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Table 7.6: Number of times that CRGA2 makes confident and risky decisions over 100 generations Function TP1 TP2 TP3 TP4 TP5 TP6 TP7 TP8 TP9 TP10

Child < P arent 125 120 130 133 145 64 133 96 156 77

Confident decision 78 54 59 58 82 58 57 45 59 73

Risky decision 47 66 71 75 63 6 76 51 97 4

n − 1 other individuals in the populations that might not be confidently robust, but are allowed to participate in the generation of the next population. In contrast to CRGA1, the CRGA2 algorithm compares each child produced with its corresponding parents based on the confidence measure. The advantage of this method is that the overall confidence level of individuals increases over the course of generations. This guarantees that there is no possibility of selecting a non-confident robust solution in the production process. In other words, two parents keep mating until they produce a confidently better robust child. Although this method maintains the high confidence level of the population and produces confidently robust children, the exploration of search space is decreased. However, the confidence measure is essential in preventing risky crossover of individuals at each generation. The number of confident and risky crossovers during optimisation for the first 10 test functions is shown in Table 7.6. This table shows that the confidence measure assists CRGA2 to make confident decisions in more than half of the total decisions. For instance, CRGA2 makes 78 confident decisions out of 125 when solving the first test function. This shows that CRGA2 would be prone to choose non-confidently robust solutions 47 times if the confidence measure were not used. The same trend can be observed in the rest of the benchmark problems. The higher performance of CRGA2 is due to these confident decisions, in which non-confidently robust solutions are discarded in order to prevent risky crossovers between individuals. To further observe the effects of the confidence measure and operators in driving individuals toward robust regions of search space, Fig. 7.3 and Fig. 7.4 are

7. Confidence-based robust optimisation Obtained optimum

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165 Obtained optimum

Figure 7.3: Search history of GA and IRGA. GA converges towards the global non-robust optimum, while IRGA failed to determine the robust optimum

Exploited region Obtained optimum

Exploited region Obtained optimum

Figure 7.4: Search history of CRGA1 and CRGA2. The exploration of CRGA2 is much less than that of CRGA1 provided. Note that the fourth benchmark function is chosen as the test bed, so there are one global, two local, and one robust optima. The history of evaluated points in the GA shows that this algorithm is able to find the global optimum with satisfactory exploration of the search space. Fig. 7.3 shows individuals of IRGA tend to move toward the robust region of the fourth benchmark function. However, the best robust solutions found are near local optimum on the right. This figure shows that there is no guarantee of finding robust optima despite the tendency of individuals to move toward the robust regions of search space in IRGA. The search history of CRGA1 and CRGA2 are shown in Fig. 7.4. It may be seen that the exploration of CRGA2 is much less than that of CRGA1.

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Although the exploration of CRGA2 is less, there is a good exploitation of the robust part of the test function. This is the reason for more accurate results from CRGA2 compared to CRGA1. The search history of CRGA1 shows that this algorithm shows high exploration with the tendency toward robust optima. However, there are a lot of risky crossovers during the search which result in finding a solution far from the real robust solution. These results show that the performance of IRGA can also be increased by using the proposed confidence measure. In addition, the confidence-based operators, crossover and elitism components are readily incorporable in evolutionary algorithms. In summary, the results of this section demonstrate that the proposed confidence measure and operators are applicable to different algorithms.

7.4

Comparison of CRPSO and CRGA

Although PSO and GA have different structures that prevent us from distinguishing whether the superior results of one algorithm are due to the confidence measure employed or its mechanism, the results and behaviour of both algorithms are compared in this section. At first glance, from the table of results in the preceding sections it is apparent that the results of RGAs are generally worse than those of RPSOs. This is due to the simple version of GA used and initial parameters. The GAs and CRGAs employed are real coded with single-point crossover operators, selection rate of 0.5, mutation rate of 0.3, and population size of 20. Fine-tuning these parameters is beyond the scope of this thesis. Another reason for the lower performance originates from the nature of GAs (and evolutionary algorithms in general). In contrast to PSO, the crossover mechanisms in GA, RGA, and CRGAs cause abrupt changes in the candidate solutions that result in enhancing the exploration ability. So the probability of having previously evaluated points around the best individual obtained so far is much less than with PSO. Moreover, the GA, RGA, and CRGA algorithms are not equipped with adaptive parameters, so there is no particular emphasis on exploitation as the number of iterations increases, in contrast to PSO with the adaptive inertial weight. This section does not compare PSO and GA in detail since both of these algorithms have their own advantages/disadvantages, and the focus of this chapter is to investigate the applicability of the proposed confidence measure/operators to a different algorithm.

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Summary

This chapter experimentally investigated the performance of the proposed confidencebased robust optimisation. The CRPSO and CRGA were employed to solve the challenging test beds proposed in Chapter 4. The results demonstrated the value of the proposed concepts: • The proposed C measure can be employed to design different confidencebased relational operators for meta-heuristics. • The proposed confidence-based relational operators give us the opportunity to design different CRO mechanisms for meta-heuristics. • A CRO meta-heuristic can be highly efficient in terms of guiding its search agents towards robust optima. • The proposed CRO approach is able to give the meta-heuristic accelerated convergence behaviours. • The accelerated convergence prevents meta-heuristics from easily stagnating in local robust or non-robust solutions. • Since CRO utilises both robustness and confidence measures, it is more reliable than normal robust optimisation techniques. • CRO is computationally very cheap, so its is highly suitable for computationally expensive real engineering problems. The results of this chapter also showed the merits of the proposed challenging test problems when comparing different algorithms. Due to the similar characteristics of the proposed test functions compared to the real search spaces, the performance of the CRO algorithms were verified and confirmed confidently. Also, it can be stated that the CRO techniques are able to find robust optima of real problems.

Chapter 8 Confidence-based robust multi-objective optimisation The previous chapter proved the merits of the confidence measure and confidencebased robust optimisation techniques systematically. This chapter presents and discusses the results of CRMOPSO as the first CRMO technique when solving the current and proposed test functions in Chapter 4. The performance metrics proposed in Chapter 5 are employed to compare the algorithms. Therefore, this chapter systematically investigates and proves the merits of the confidence measure and confidence-based robust optimisation in multi-objective search spaces.

8.1

Behaviour of CRMOPSO on benchmark problems

Three versions of RMOPSO are implemented and employed: an Explicit averaging Robust MOPSO (ERMOPSO), Implicit averaging Robust MOPSO (IRMOPSO), and Confidence-based Robust MOPSO (CRMOPSO). The method of explicit averaging in ERMOPSO is identical to that of RNSGA-II proposed by Deb and Gupta [44]. In this method H number of sampled solutions is created by the Latin Hypercube Sampling (LHS) method around every candidate solution to investigate and confirm robustness during optimisation. This provides the highest reliability and is helpful for verifying the performance of the proposed algorithm as the most reliable reference. It does, however, markedly increase the computational load per iteration. In contrast, IRMOPSO utilises the previously 168

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sampled solutions to confirm robustness. In this case, robustness of a candidate solution is evaluated based on the available solutions in the neighbourhood during optimisation. All the sampled solutions are saved during optimisation, in a manner similar to Branke [18] and Dippel [58]. As was discussed in the gaps and motivation for this thesis, IRMOPSO shows less reliability and its use will allow comparison of CRMOPSO in terms of improved reliability. Robustness is calculated by an expectation measure with simple averaging of the neighbouring solutions for all algorithms. All three optimisers aim to minimise the expectation of the objective functions in a neighborhood of the nominal solution (with maximum noise of 10% of the range). In order to provide a fair comparison and see if the proposed method is reliable and effective, the same number of function evaluations (100,000) is used for each of the algorithms. The number of trial solutions in the population was 100 for all algorithms, so the maximum number of iterations for IRMOPSO and CRMOPSO was 1000. Since ERMOPSO uses explicit averaging, however, 4 resampling points and 250 iterations are used in order to achieve the total number of 100,000 function evaluations. Another assumption was 10% fluctuation in the parameters to simulate uncertainties in parameters. Each algorithm was run 30 times and the statistical results of performance metrics are reported in Table 8.1, 8.3, and 8.5. For quantifying the convergence of algorithms, the IGD performance measure is utilized in Table 8.1. For the coverage and success ratio, the proposed performance measures in this thesis are used to collect and present the results in Table 8.3 and 8.5. Also, the Wilconxon ranksum test was conducted at 5% significance level to decide on the significance of discrepancy in the results. The p-values that are less than 0.05 could be considered as strong evidence against the null hypothesis. Table 8.2, 8.4, and 8.6 show the results of the Wilcoxon ranksum test for IGD, Φ, and Γ respectively. Note that all test functions have 10 variables and the quantitative results are presented in the form of (average ± standard deviation). For the performance measures, 10% noise was considered and Rmin was calculated using the equation minrobustness + (maxrobustness − minrobustness × 0.1) where minrobustness is the minimum of the robustness curve and maxrobustness indicates the maximum of robustness in the test functions. In addition, the best robust fronts obtained by each algorithm are illustrated

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in Fig. 8.1, 8.2 and 8.3. Note that the results on some of the test functions are illustrated in this chapter, but all the results are presented in Appendix C. The results are not compared with other meta-heuristics since the different mechanisms of the algorithms would prevent us from distinguishing whether the superior results of one algorithm were due to CRMO or the algorithm’s underlying mechanism.

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8.2

Discussion of results

It may be observed in Fig. 8.1 that the most well-distributed robust front for RMTP1 is that of IRMOPSO. This is because the robust front is identical to the main Pareto front (the first case shown in Fig. 2.13). Since IRMOPSO uses

8. Confidence-based robust multi-objective optimisation

Table 8.1: Statistical results of Test function CRMOPSO RMTP1 0.338 ± 0.169 RMTP2 0.112 ± 0.106 RMTP3 0.035 ± 0.03 RMTP6 0.017 ± 0.007 RMTP7 0.038 ± 0.028 RMTP8 0.033 ± 0.009 RMTP9 0.037 ± 0.027 RMTP10 0.034 ± 0.018 RMTP11 0.037 ± 0.015 RMTP12 0.034 ± 0.014 RMTP13 0.024 ± 0.007 RMTP14 0.03 ± 0.012 RMTP15 0.039 ± 0.013 RMTP16 0.027 ± 0.008 RMTP17 0.041 ± 0.021 RMTP18 0.06 ± 0.026 RMTP19 0.033 ± 0.002 RMTP20 0.039 ± 0.042 RMTP21 0.037 ± 0.059 RMTP22 0.052 ± 0.122 RMTP23 0.072 ± 0.016 RMTP24 0.064 ± 0.014 RMTP25 0.069 ± 0.013 RMTP26 0.038 ± 0.013 RMTP27 0.006 ± 0.028 RMTP33 0.021 ± 0.007 RMTP34 0.04 ± 0.015 RMTP35 0.113 ± 0.03 RMTP36 0.011 ± 0.008 RMTP37 0.015 ± 0.012 RMTP38 0.015 ± 0.015 RMTP39 0.01 ± 0.001 RMTP40 0.014 ± 0.002 RMTP41 0.056 ± 0.087 RMTP42 0.004 ± 0.003 RMTP43 0.005 ± 0.005 RMTP44 0.005 ± 0.005

RMOPSO algorithms using IGD IRMOPSO ERMOPSO 0.061 ± 0.05 7.261 ± 6.528 0.21 ± 0.13 8.057 ± 5.329 0.002 ± 0.001 0.106 ± 0.206 0.004 ± 0.006 0.013 ± 0.012 0.01 ± 0.004 0.007 ± 0.007 0.011 ± 0.007 0.007 ± 0.006 0.004 ± 0.004 0.022 ± 0.026 0.004 ± 0.003 0.014 ± 0.017 0.032 ± 0.019 0.939 ± 1.343 0.049 ± 0.016 0.079 ± 0.098 0.034 ± 0.009 0.249 ± 0.347 0.026 ± 0.015 1.197 ± 1.743 0.056 ± 0.008 0.533 ± 0.825 0.042 ± 0.009 0.695 ± 1.408 0.06 ± 0.032 0.814 ± 1.112 0.062 ± 0.043 0.59 ± 1.699 0.034 ± 0.004 0.447 ± 0.843 0.007 ± 0.005 0.094 ± 0.215 0.004 ± 0.003 0.051 ± 0.087 0.007 ± 0.006 0.024 ± 0.036 0.124 ± 0.025 0.084 ± 0.016 0.1 ± 0.013 0.08 ± 0.014 0.104 ± 0.009 0.087 ± 0.01 0.028 ± 0.041 0.008 ± 0.004 0.007 ± 0.003 0.044 ± 0.061 0.001 ± 0 0.081 ± 0.06 0.013 ± 0.008 0.047 ± 0.027 0.007 ± 0.01 0.105 ± 0.042 0.026 ± 0.001 0.05 ± 0.078 0.036 ± 0.008 0.235 ± 0.274 0.019 ± 0.023 0.555 ± 0.739 0.037 ± 0.033 0.202 ± 0.251 0.087 ± 0.064 0.127 ± 0.314 0.082 ± 0.05 0.68 ± 1.206 0.002 ± 0.002 0.032 ± 0.043 0.001 ± 0.001 0.265 ± 0.788 0.001 ± 0.002 0.51 ± 0.91

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Table 8.2: P-values of Wilcoxon ranksum test for the Table 8.1 Test function CRMOPSO IRMOPSO RMTP1 0.032983954 N/A RMTP2 N/A 0.212293836 RMTP3 0.021133928 N/A RMTP6 0.001706249 N/A RMTP7 0.000439639 0.000439639 RMTP8 0.000182672 0.27303634 RMTP9 0.000246128 N/A RMTP10 0.000246128 N/A RMTP11 0.520522883 N/A RMTP12 N/A 0.003447042 RMTP13 N/A 0.025748081 RMTP14 0.850106739 N/A RMTP15 N/A 0.04515457 RMTP16 N/A 0.010410989 RMTP17 N/A 0.031209013 RMTP18 N/A 0.427355314 RMTP19 N/A 0.79133678 RMTP20 0.001706249 N/A RMTP21 0.005707503 N/A RMTP22 0.121224503 N/A RMTP23 N/A 0.000768539 RMTP24 N/A 0.00058284 RMTP25 N/A 0.000439639 RMTP26 0.000182672 1.0000 RMTP27 N/A 0.049721889 RMTP33 0.000182672 N/A RMTP34 0.000182672 N/A RMTP35 0.000182672 N/A RMTP36 N/A 0.049849977 RMTP37 N/A 0.020522883 RMTP38 N/A 0.79133678 RMTP39 N/A 0.017257456 RMTP40 N/A 0.001858767 RMTP41 N/A 0.677584958 RMTP42 0.384673063 N/A RMTP43 0.427355314 N/A RMTP44 0.472675594 N/A

RMOPSO algorithms in ERMOPSO 0.000246128 0.000182672 0.000329839 0.005795359 N/A N/A 0.005390255 0.185876732 0.185876732 0.002827272 0.000246128 0.007284557 0.001706249 0.001402210 0.021133928 0.307489457 0.002827272 0.000850106 0.002202220 0.677584958 0.04515457 0.021224503 0.012293836 N/A 0.000246128 0.000182672 0.000182672 0.000182672 0.017257456 0.003610514 0.009108496 0.000472675 0.000246128 0.04515457 0.000182672 0.001706249 0.000182672

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Table 8.3: Statistical results of RMOPSO algorithms using Φ Test function CRMOPSO IRMOPSO ERMOPSO RMTP1 0.3 ± 0.09 0.6 ± 0.05 0.2 ± 0.2 RMTP2 0.3 ± 0.1 0.4 ± 0.2 0.1 ± 0.1 RMTP3 0.4 ± 0.1 0.5 ± 0.1 0.5 ± 0.2 RMTP6 0.4 ± 0.1 0.5 ± 0.1 0.5 ± 0.1 RMTP7 0.3 ± 0.1 0.2 ± 0.05 0.4 ± 0.1 RMTP8 0.19 ± 0.1 0.2 ± 0.1 0.34 ± 0.08 RMTP9 0.46 ± 0.15 0.48 ± 0.12 0.53 ± 0.15 RMTP10 0.34 ± 0.12 0.44 ± 0.08 0.56 ± 0.14 RMTP11 0.388 ± 0.159 0.417 ± 0.113 0.215 ± 0.151 RMTP12 0.442 ± 0.18 0.488 ± 0.162 0.642 ± 0.245 RMTP13 0.469 ± 0.117 0.465 ± 0.206 0.458 ± 0.224 RMTP14 0.248 ± 0.057 0.285 ± 0.102 0.183 ± 0.159 RMTP15 0.591 ± 0.089 0.551 ± 0.103 0.391 ± 0.331 RMTP16 0.533 ± 0.212 0.463 ± 0.138 0.575 ± 0.204 RMTP17 0.381 ± 0.092 0.356 ± 0.138 0.11 ± 0.112 RMTP18 0.515 ± 0.123 0.46 ± 0.075 0.235 ± 0.238 RMTP19 0.838 ± 0.077 0.625 ± 0.155 0.704 ± 0.22 RMTP20 0.385 ± 0.136 0.366 ± 0.114 0.44 ± 0.265 RMTP21 0.202 ± 0.045 0.225 ± 0.083 0.264 ± 0.109 RMTP22 0.639 ± 0.208 0.689 ± 0.131 0.7 ± 0.109 RMTP23 0.596 ± 0.147 0.378 ± 0.072 0.548 ± 0.124 RMTP24 0.543 ± 0.184 0.333 ± 0.103 0.471 ± 0.141 RMTP25 0.586 ± 0.127 0.448 ± 0.075 0.633 ± 0.149 RMTP26 0.42 ± 0.1 0.33 ± 0.12 0.41 ± 0.24 RMTP27 0.53 ± 0.22 0.37 ± 0.15 0.54 ± 0.18 RMTP33 0.257 ± 0.075 0.333 ± 0.08 0.18 ± 0.104 RMTP34 0.223 ± 0.052 0.483 ± 0.102 0.366 ± 0.105 RMTP35 0.17 ± 0.035 0.565 ± 0.106 0.12 ± 0.035 RMTP36 0.557 ± 0.063 0.611 ± 0.028 0.524 ± 0.127 RMTP37 0.2 ± 0.258 0.2 ± 0.35 0.1 ± 0.211 RMTP38 0.311 ± 0.187 0.605 ± 0.165 0.427 ± 0.309 RMTP39 0.174 ± 0.208 0.636 ± 0.049 0.321 ± 0.171 RMTP40 0±0 0±0 0.1 ± 0.211 RMTP41 0.103 ± 0.125 0.573 ± 0.219 0.519 ± 0.316 RMTP42 0.447 ± 0.13 0.642 ± 0.051 0.405 ± 0.136 RMTP43 0.25 ± 0.354 0.55 ± 0.497 0.05 ± 0.158 RMTP44 0.432 ± 0.262 0.711 ± 0.134 0.668 ± 0.289

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Table 8.4: P-values of Wilcoxon ranksum test for the Table 8.3 Test function CRMOPSO IRMOPSO RMTP1 0.000329839 N/A RMTP2 0.212293836 N/A RMTP3 0.313298369 N/A RMTP6 0.317062499 N/A RMTP7 0.104109894 0.000439639 RMTP8 0.231826742 0.27303634 RMTP9 0.053902557 0.121224503 RMTP10 0.011329697 0.185876732 RMTP11 0.520522883 N/A RMTP12 0.344704222 0.909721889 RMTP13 N/A 0.025748081 RMTP14 0.850106739 N/A RMTP15 N/A 0.04515457 RMTP16 0.969849977 0.10410989 RMTP17 N/A 0.031209013 RMTP18 N/A 0.047584958 RMTP19 N/A 0.03133678 RMTP20 0.890465048 0.850106739 RMTP21 0.522022235 0.570750388 RMTP22 0.520522883 0.677584958 RMTP23 N/A 0.000768539 RMTP24 N/A 0.00058284 RMTP25 0.212293836 0.000439639 RMTP26 N/A 0.10410989 RMTP27 0.909721889 0.16197241 RMTP33 0.018267235 N/A RMTP34 0.001526336 N/A RMTP35 0.000182672 N/A RMTP36 0.969849977 N/A RMTP37 N/A 0.520522883 RMTP38 0.009108496 N/A RMTP39 0.017257456 N/A RMTP40 0.053902557 0.185876732 RMTP41 0.05515457 N/A RMTP42 0.384673063 N/A RMTP43 0.427355314 N/A RMTP44 0.046721824 N/A

RMOPSO algorithms in ERMOPSO 0.000246128 0.000182672 0.4621133928 0.4635795359 N/A N/A N/A N/A 0.185876732 N/A 0.022886128 0.007284557 0.001706249 N/A 0.003763531 0.027355314 0.05108496 N/A N/A N/A 0.04515457 0.121224503 N/A 1.00000 N/A 0.0063344 0.003843352 0.000182672 0.911329697 0.003610514 0.79133678 0.472675594 N/A 0.677584958 0.000182672 0.001706249 0.472675594

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Table 8.5: Statistical results of RMOPSO algorithms using Γ Test function CRMOPSO IRMOPSO ERMOPSO RMTP1 42.2 ± 16.6 6.2 ± 7.8 9.2 ± 6.1 RMTP2 41.7 ± 18.4 61.8 ± 42.4 9.2 ± 9.7 RMTP3 20.8 ± 11.3 39.6 ± 22.8 28.1 ± 25.4 RMTP6 1.2 ± 0.6 2.1 ± 0.9 1.8 ± 0.5 RMTP7 1.7 ± 1 0.9 ± 0.6 0.7 ± 0.4 RMTP8 0.61 ± 0.3 0.49 ± 0.2 0.62 ± 0.2 RMTP9 0.75 ± 0.29 7.72 ± 5.05 5.85 ± 1.68 RMTP10 0.64 ± 0.25 1.98 ± 0.97 2.51 ± 0.71 RMTP11 1.35 ± 0.503 1.193 ± 0.463 1.986 ± 1.49 RMTP12 3.718 ± 10.994 1.413 ± 4.013 8.368 ± 15.047 RMTP13 1.173 ± 0.47 2.378 ± 1.941 1.924 ± 1.399 RMTP14 0.984 ± 0.268 2.57 ± 3.674 2.474 ± 2.482 RMTP15 6.652 ± 3.888 8.118 ± 2.414 3.187 ± 3.585 RMTP16 9.316 ± 19.185 3.749 ± 10.981 1.209 ± 1.525 RMTP17 0.852 ± 0.216 1.128 ± 0.503 2.893 ± 2.777 RMTP18 9.574 ± 5.173 7.786 ± 4.993 2.955 ± 5.589 RMTP19 3.636 ± 2.595 6.318 ± 17.476 2.143 ± 1.993 RMTP20 20.753 ± 19.771 21.183 ± 16.183 24.79 ± 26.788 RMTP21 7.964 ± 6.86 5.918 ± 4.399 4.23 ± 1.62 RMTP22 2.624 ± 3.218 0.984 ± 1.2 3.789 ± 7.222 RMTP23 8.034 ± 3.164 20.64 ± 14.587 14.808 ± 13.073 RMTP24 15.722 ± 10.593 14.833 ± 8.38 17.97 ± 12.006 RMTP25 39.5 ± 21.671 28.25 ± 9.39 44.9 ± 20.102 RMTP26 21.14 ± 13.65 7.75 ± 6.99 6.53 ± 6.61 RMTP27 7.52 ± 8.09 0.45 ± 0.44 1.54 ± 1.98 RMTP33 2.466 ± 1.56 0.245 ± 0.123 2.415 ± 3.936 RMTP34 0.98 ± 0.447 2.408 ± 1.225 1.167 ± 0.458 RMTP35 0.612 ± 0.67 0.275 ± 0.07 0.268 ± 0.089 RMTP36 57.2 ± 33.923 41.359 ± 29.695 6.729 ± 2.836 RMTP37 0.04 ± 0.028 0.019 ± 0.009 0.033 ± 0.021 RMTP38 1.213 ± 2.088 10.933 ± 30.247 20.729 ± 24.349 RMTP39 14.469 ± 25.914 21.712 ± 7.661 4.404 ± 2.369 RMTP40 0.019 ± 0.031 0.031 ± 0.01 0.021 ± 0.019 RMTP41 6.408 ± 8.04 21.562 ± 24.385 9.573 ± 8.91 RMTP42 29.412 ± 14.821 34.107 ± 22.813 5.926 ± 3.245 RMTP43 0.021 ± 0.017 0.028 ± 0.015 0.031 ± 0.022 RMTP44 6.475 ± 19.515 0.239 ± 0.133 3.659 ± 4.194

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Table 8.6: P-values of Wilcoxon ranksum test for the Table 8.5 Test function CRMOPSO IRMOPSO RMTP1 N/A 0.000329839 RMTP2 0.212293836 N/A RMTP3 0.000329839 N/A RMTP6 0.001706249 N/A RMTP7 N/A 0.000439639 RMTP8 0.27303634 0.000182672 RMTP9 0.000246128 N/A RMTP10 0.000246128 0.011329697 RMTP11 0.241321593 0.185876732 RMTP12 0.909721889 0.344704222 RMTP13 0.025748081 N/A RMTP14 0.007284557 N/A RMTP15 0.427355314 N/A RMTP16 N/A 0.044022101 RMTP17 0.021133928 0.031209013 RMTP18 N/A 0.04273553 RMTP19 0.79133678 N/A RMTP20 0.140465048 0.850106739 RMTP21 N/A 0.00220222 RMTP22 0.677584958 0.520522883 RMTP23 0.000768539 N/A RMTP24 0.121224503 0.00058284 RMTP25 0.212293836 0.000439639 RMTP26 N/A 0.000182672 RMTP27 N/A 0.000246128 RMTP33 N/A 0.000182672 RMTP34 0.000182672 N/A RMTP35 N/A 0.000273036 RMTP36 N/A 0.069849977 RMTP37 N/A 0.003610514 RMTP38 0.009108496 0.03108496 RMTP39 0.472675594 N/A RMTP40 0.000246128 N/A RMTP41 0.04515457 N/A RMTP42 0.384673063 N/A RMTP43 0.021133928 0.041706249 RMTP44 N/A 0.000182672

RMOPSO algorithms in ERMOPSO 0.005795359 0.000182672 0.021133928 0.005795359 0.000182672 N/A 0.121224503 N/A N/A N/A 0.10410989 0.850106739 0.001706249 0.001041098 N/A 0.003074894 0.002827272 N/A 0.001750388 N/A 0.00220222 N/A N/A 0.000104109 0.16197241 0.000768539 0.000182672 0.000182672 0.000017257 0.520522883 N/A 0.017257456 0.185876732 0.067758495 0.000182672 N/A 0.042675594

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an implicit averaging method, the convergence is greater than other algorithms. However, the greatest density of solutions is in the middle of the robust front, which has less robustness compared to the left end of the main front. Fig. 8.1 shows that the robust front obtained by CRMOPSO for RMTP1 is slightly more widely distributed than that of ERMOPSO. In addition, a high density of solutions can be observed on the left end of the robust font. Despite the apparent similarity of CRMOPSO and ERMOPSO in Fig. 8.1, the above tables show that the CRMOPSO algorithm provided much better statistical results compared to ERMOPSO. This shows that the confidence-based Pareto dominance operators allow CRMOPSO to provide more reliable performance within an equal number of function evaluations. The statistical results of IRMOPSO for this test problem were better that CRMOPSO, in terms of convergence, due to the nature of RMTP1. However, the results show that the confidence-based operators were able to provide high reliability without significant negative impact on convergence. The results for RMTP7 are slightly different to those of RMTP1, for which ERMOPSO showed the highest convergence. Fig. 8.1 shows that ERMOPSO approximated the entire Pareto optimal front and there was no tendency to favour robust regions of the front. This resulted in this algorithm showing the worst coverage and success ratio on RMTP7. In contrast, CRMOPSO and IRMOPSO found better approximations of the robust front. The solutions obtained by CRMOPSO were clustered on the left side of the robust front and it seems there is resistance to generating any solutions on the less robust regions of the front. Although the robust Pareto optimal solutions obtained by IRMOPSO followed a similar pattern, some of them are located on the least robust regions of the front. The RMTP9 test function has three separate robust regions. The robustness curve shows that the robust regions are around f 1 = 0, 0.4, 1. The results in Table 8.1 demonstrate that the convergence of CRMOPSO was again slightly worse than IRMOPSO and ERMOPSO. The shape of the approximated robust Pareto front in Fig. 8.1 shows that ERMOPSO behaved similarly on this test function, in contrast to RMTP7 where there was no particular tendency to favour robust regions of the Pareto optimal front. A slight bias to robust regions can be observed in the robust Pareto optimal solutions obtained by IRMOPSO. The approximated robust Pareto optimal front of CRMOPSO, however, shows that

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this algorithm had a greater ability to guide its solutions toward robust regions of the Pareto optimal front. There is a gap between the solutions obtained on the least robust region of the Pareto optimal front. The behaviour of algorithms in terms of finding robust regions of the Pareto optimal front can be observed more clearly with RMTP27. In fact, this test function is arranged deliberately to have a stair-shaped Pareto optimal front in order to benchmark the ability of algorithms in terms on converging toward robust regions of the front and refraining from finding non-robust solutions. The quantitative results of the algorithms on RMTP27 show that the results of CRMOPSO are significantly better than IRMOPSO and ERMOPSO. The CRMOPSO algorithm only approximates the first two stairs, which are considered the most robust regions of the Pareto optimal front, as shown in Fig. 8.1. However, the robust solutions obtained by IRMOPSO and ERMOPSO tend to be distributed on other, less robust regions as well. Test functions RMTP11 to RMTP19 are bi-modal. The local front is the robust front, so the behaviour of algorithms in favouring a robust front instead of a global front can be investigated. These test functions are designed with different shapes of robust and global fronts in order to extensively test the performance of the algorithms. In RMTP13 the shape of both local and global fronts is convex. In contrast to the quantitative results in Fig. 8.1, the CRMOPSO algorithm showed the highest convergence on this test function. This shows that although the convergence of the proposed method is slightly low, this may be favourable when solving multi-modal test functions. The coverage and success ratio of CRMOPSO were also very good on this test function. The best robust solutions obtained for all algorithms are again illustrated in Fig. 8.2. It may be seen that the best front obtained was from CRMOPSO. This algorithm did not approximate even one non-robust solution for most of the test functions, proving the merits of the proposed method. IRMOPSO was not able to approximate the robust front over 30 runs for this problem showing the potential unreliability of using previous samples without a confidence measure. The ERMOPSO algorithm also mostly tended to approximate the global front, but there were some solutions found on the robust front. As Fig. 8.2 shows, RMTP15 and RMTP16 have identical linear global fronts that make the approximation of the global front easy. Deliberately, these two test functions have been arranged to have linear global fronts to observe what

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times during the optimisation that fewer neighbouring solutions mislead an algorithm toward non-robust regions. The archive-based mechanism of IRMOPSO deteriorated on these problems, with cases in which a non-robust solution entered the archive and was never dominated by robust solutions outside the archive. In contrast ERMOPSO which has an explicit averaging mechanism never favoured a non-robust solution. This caused a well-distributed robust front in Fig. 8.2 for ERMOPSO on RMTP15. The robust solutions obtained by CRMOPSO are very competitive with those found by ERMOPSO. Tables 8.1 and 8.3 show that CRMOPSO had superior convergence and coverage on RMTP15. The approximate solutions of the algorithms on RMTP16 were somewhat different; the coverage of solutions obtained by ERMOPSO was very low and the CRMOPSO algorithm also found non-robust solutions. Generally speaking, approximation of a concave front is more challenging, and this phenomenon can be seen for the RMTP16 test function. The IRMOPSO algorithm again failed to approximate the robust front, whereas CRMOPSO and ERMOPSO tended to find robust solutions. RMTP19 has two fronts with opposite shapes: a convex global front and a concave robust/local front. The results of algorithms on this test function show the difficulty of RMTP16 as none of the algorithms approximate the entire robust front. In Fig. 8.2, some resolution of the robust front can be observed in the solutions obtained by both CRMOPSO and ERMOPSO. The CRMOPSO algorithm shows a slightly greater tendency to correct selection in this case. Generally speaking, benchmark functions with different shapes for robust and global optima would be more difficult to solve because a robust algorithm needs to adapt to a very different Pareto optimal front with different level of robustness when transferring from the global/local Pareto optimal front to the robust Pareto optimal front(s). Deb asserted this for multi-objective benchmark problems [46], and it is the reason for the poor performance of all algorithms on RMTP19. The last two test functions, RMTP24 and RMTP25, have multiple discontinuous local fronts. They provide the most challenging test cases for the algorithms. It should be noted that the robustness increases from right to left and bottom to top in the figure. The results of Tables 8.1, 8.3, and 8.5 and Fig. 8.3 show that the proposed CRMOPSO algorithm was the best in terms of favouring robust regions of the fronts. The Pareto optimal solutions obtained by CRMOPSO for RMTP24 indicate that there is more coverage on the most

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Figure 8.3: Robust fronts obtained for RMTP21 to RMTP25 one test case per row. Note that the worst front is the most robust and considered as reference for the performance measures. robust region of the local fronts. However, IRMOPSO and ERMOPSO do not display such behaviours. The results of algorithms on RMTP25, which has more discontinuous robust regions, were also consistent with those of RMTP24: the CRMOPSO algorithm was able to find more robust solutions. It should be noted that the quantitative results of RMTP26 to RMTP44 (biased test functions) in Tables 8.1, 8.3, and 8.5 as well as obtained robust Pareto optimal fronts in Appendix C also show that the CRMOPSO algorithm outperforms IRMOPSO and ERMOPSO on the majority of the biased test functions. Theses results are consistent with those of other test functions and confirm the merits of the proposed CRMO approach in solving difficult problems. Overall, CRMOPSO and IRMOPSO show better results on 18 and 16 test

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functions respectively considering the IGD values. As the p-values show, however, the superiority was statistically significant in 13 cases for CRMOPSO but 10 cases for IRMOPSO. Another fact is that IRMOPSO shows better results mostly on unimodal test functions, which is due to the unreliably high exploitation and convergence speed of this algorithm. The proposed confidence measure prevents CRMOPSO from converging towards the global front, which is advantageous in multi-modal test functions. Considering the results of the coverage measure, it may be seen that the CRMOPSO shows statistically better results in 7 out of 9 test functions. However, the results of IRMOPSO are statistically better in 7 out of 16 test functions. This is again due to the impacts of the confidence measure on movement of particles in CRMOPSO. It seems that the lesser movement of particles decreases the coverage of solutions, but the results show that CRMOPSO is still competitive compared to the IRMOPSO. The results of algorithms on the success ratio measure show CRMOPSO yields statistically better results in 12 out of 37 functions. The IRMOPSO is better in 14 out of 37 test functions. However, the results were only statistically significant in 9 cases. This shows that CRMOPSO is slightly more reliable in terms of finding robust solutions and avoiding non-robust ones. To further investigate the merits of the proposed method, Table 8.7 determines the number of times that a solution dominated an archive member but was not allowed to enter the archive with the proposed confidence measure on some of the test functions. This table shows that a significant number of times during optimisation (approximately 19% of attempts) the proposed confidence-based Pareto optimality concepts prevented solutions with low confidence from being added to the archive. Firstly, these results demonstrate that often IRMOPSO was likely to make unreliable decisions and consequently favour non-robust solutions. Secondly, the proposed method is able to prevent unreliable decisions throughout optimisation, providing more reliability than the current archivebased robustness-handling methods. In summary, the results show that the convergence of CRMOPSO is slower than IRMOPSO because less confident solutions were ignored over the course of iterations. Although this prevents CRMOPSO from providing superior results on uni-modal test functions, it can be very helpful in optimising real problems since CRMOPSO has less probability of premature convergence toward non-robust

8. Confidence-based robust multi-objective optimisation

183

Table 8.7: Number of times that the proposed confidence-based Pareto dominance prevented a solution entering the archive Test function (Pi ≺ Archivem ) ∧ (Pi 6≺c Archivem ) RMTP1 18,918 / 100,000 RMTP7 28,210 / 100,000 RMTP9 19,240 / 100,000 RMTP27 17,157 / 100,000 RMTP13 13,938 / 100,000 RMTP14 14,703 / 100,000 RMTP15 15,768 / 100,000 RMTP16 17,054 / 100,000 RMTP24 26,566 / 100,000 RMTP25 17,570 / 100,000 regions compared to IRMOPSO. The results of CRMOPSO were also superior to ERMOPSO on the majority of test functions. The proposed performance indicators quantitatively showed the greater coverage and success ratio of the CRMOPSO algorithm. In addition, the proposed algorithm showed very good convergence on the robust regions of challenging, multi-modal test cases.

8.3

Summary

This chapter was dedicated to the results and discussion of the proposed CRMO approach. The CRMOPSO algorithm was employed to solve the proposed challenging test functions in Chapter 4 and compared to other algorithms in the literature qualitatively and quantitatively (using the performance measures proposed in Chapter 5). The results showed that the novel approach proposed is able to deliver very promising results in terms of convergence, coverage, and proportion of robust Pareto optimal solutions obtained. The findings demonstrated the value of the proposed concepts: • The proposed confidence-based Pareto optimality provides the opportunity to design different CRMO mechanisms for meta-heuristics. • The proposed confidence-based Pareto optimality improves the reliability of archive-based robust optimisation techniques. • CRMO is able to confidently find robust solutions.

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• CRMO is computationally cheap since there is no need for additional true function evaluations. • CRMO is able to approximate different types of true robust Pareto optimal fronts. • CRMO is suitable for computationally expensive real engineering problems. The results of this chapter also showed the merits of the proposed challenging robust multi-objective test problems when comparing different algorithms. Due to the different characteristics of the proposed test functions, the performance of the CRMOPSO algorithm was observed and investigated in detail. It is also worth mentioning here that the proposed performance measures allowed quantitatively comparing different algorithms in this chapter for the first time in the literature.

Chapter 9 Real world applications In the previous two chapters, the merits of the ideas proposed in Chapter 6 have been investigated systematically. This chapter demonstrates the application of the proposed CRMOPSO to the design of propellers using a reduced-order model. The case study is a propeller design problem. In the following sections, this problem is first solved by a MOPSO algorithm. The results are then analysed mostly in terms of the effects of uncertainties on structural parameters and operating conditions. Finally, the CRMOPSO algorithm is employed to determine the robust Pareto optimal front for this problem.

9.1 9.1.1

Marine Propeller design and related works Propeller design

Due to the relatively high density of water, the efficiency of propellers for marine vehicles is very important. The efficiency of propellers refers to the amount of the power of the motor(s) that is converted to thrust. In addition to the efficiency, this conversion should be done with a minimum level of vibration and noise. The third characteristic of a good propeller is low surface erosion which is caused by cavitation. Finding a balance between these three features is a challenging task which should be considered during the design process of a propeller. The main part of a propeller is its blades. The geometric shape of these blades should satisfy all the above-mentioned requirements. The propeller adds velocity (∆v) to an incoming velocity (v) of the surrounding fluid. This acceleration is created in two places: the first half in front of the 185

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9. Real world applications

propeller and the second half behind the propeller. A propeller is rotated, which swirls the outflow. The amount of this swirl is based on the rotation speed of the motor and energy loss. Efficient propellers lose 1% to 5% of their power because of swirl. The thrust of propellers is calculated as follows [26]: T =

∆v π 2 D (v + )ρ∆v 4 2

(9.1)

where T is thrust, D is the propeller diameter, v is the velocity of the incoming flow, ∆v is the additional velocity which is created by the propeller, and ρ is the density of the fluid. It may be seen in Equation 9.1 that the final thrust depends on the volume of the incoming stream which has been accelerated per unit of time, the amount of this acceleration, and the density of the medium. Power is defined as force times distance per time. The required power to drive a vehicle with a velocity of v using the available thrust is calculated as follows: Pa = T v

(9.2)

One of the objectives of optimisation in propellers is to create as much thrust as possible with the smallest amount of power. This is the efficiency of propellers which can be expressed as follows: η=

Pa Pengine

=

Tv Pengine

(9.3)

where Pengine is the engine power. The efficiency of a propeller can be calculated as follows: η(x) =

JKT (x) 2πKQ (x)

(9.4)

where J is the advance number, KT is the propeller thrust coefficient, and KQ is the propeller torque coefficient. J is defined as follows: J=

Va nD

(9.5)

where Va is the axial velocity, n is rotational velocity, and D is the diameter of the propeller.

9. Real world applications

187

By substitution of terms the efficiency can also be presented as follows: η(x) =

Va KT (x) 2πnD KQ (x)

(9.6)

The thrust coefficient (KT ) and torque coefficient (KQ ) are calculated as follows: KT =

39 X n=1

KQ =

47 X n=1

CTn (J)sn (

P tn Ae un ) ( ) (Z)vn D Ao

(9.7)

P tn Ae un ) ( ) (Z)vn D Ao

(9.8)

CQn (J)sn (

where P/D is the pitch ratio, Ae /Ao is the disk ratio of the propeller, Z is the number of blades, and CTn , CQn , sn , tn , un , vn are corresponding regression coefficients. There is another issue in propellers called cavitation. When the blades of a propeller move through water at high speed, low pressure regions form as the water accelerates and moves past the blades. This can cause bubbles to form, which collapse and can cause strong local shockwaves which result in erosion of propellers. The sensitivity of the propeller to cavitation is calculated as follows:

σn,0.8 =

(pa + pgh0.8 − pv ) 0.5ρ(πnD)2

(9.9)

where pa is the atmospheric pressure, pv indicates the vapour pressure of water, g is the acceleration due to gravity, and h0.8 shows immersion of 0.8 blade radius when the blade is at the position of 12:00. The ultimate goal here is to design a propeller with the highest efficiency and the lowest cavitation sensitivity. In order to find the final geometrical shape of the blade, standard National Advisory Committee for Aeronautics (NACA) airfoils are selected as shown in Fig. 9.1. It may be seen in this figure that two parameters define the shape of the airfoil: maximum thickness and chord length. In this chapter ten airfoils are considered along the blade, so the total number of parameters is 20. The final parameter vector is as follows: ~ = (T1 , C1 , T2 , C2 , ..., T10 , C10 ) X

(9.10)

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9. Real world applications

Maximum thickness

Chord length

Figure 9.1: Airfoils along the blade define the shape of the propeller (NACA a = 0.8 meanline and NACA 65A010 thickness) where Ti and Ci indicates the thickness and chord length of the i-th airfoil along the blade. Finally, the problem can be formulated as follows: ~ = (Ti , Ci ), i = 1, 2, ..., 10 Suppose : X

(9.11)

~ M aximise : η(X)

(9.12)

~ M inimise : Vc (X)

(9.13)

Subject to : T hrust ≥ 40000

(9.14)

where Vc is a function to calculate the cavitation. In order to calculate the objective functions, a freeware called OpenProp is utilised as the simulator. Details of the model are provided in the following sources [65, 64].

9.1.2

Related work

Work using a heuristic algorithm to optimise the shape of a B-series propeller has been reported in the literature [171]. The NSGA-II algorithm was employed to optimise the shape of a propeller with specific performance in given conditions.

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189

Designing a propeller for ships was considered as a multi-objective problem with two objectives: minimising propeller efficiency and maximising the thrust coefficient. The author considered two main constraints for this problem. These constraints were wake friction and thrust deduction. The author did not specify the exact number of variables. NGSA-II provided 15 Pareto optimal solutions. Finally, a decision making technique was used to select one of the Pareto solution as the best solution. One other study has investigated the application of the NSGA-II multi-objective optimisation algorithm to the design of marine propellors [132]. Difficulties were reported with the nature of the design space: constraints applied isolated feasible results into “small islands” and the optimisation algorithm failed to converge.

9.2

Results and discussion

A MOPSO algorithm is employed to estimate the Pareto optimal front. A population of 100 search agents and maximum number of 200 iterations are chosen for MOPSO. The main case study is a ship propeller with 2 metre diameter as shown in Fig. 9.2.

2 meters

Figure 9.2: Propeller used as the case study The   experiments undertaken using MOPSO are as follow: 1. Observing the behaviour of MOPSO in finding an accurate approximation and well-spread Pareto optimal solutions 2. Observing the effect of the number of blades on the efficiency and cavitation of the propeller

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9. Real world applications

3. Finding the optimal number of blades 4. Observing the effects of Revolutions Per Minute (RPM) on the efficiency and cavitation of the propeller 5. Finding the optimal values (range) for RPM 6. Post analysis of the result to extract the possible physical behaviour and impacts of the parameters on the efficiency and cavitation of the propeller 7. Observing the effects of uncertainties in operating conditions (RPM) on the the Pareto optimal fronts obtained by MOPSO 8. Observing the effects of uncertainties in structural parameters on the Pareto optimal fronts obtained by MOPSO The following subsections present and discuss the results for each of these experiments.

9.2.1

Approximating the Pareto Front using the algorithm

The MOPSO algorithm was run 4 times on the problem and the best estimate of Pareto Front obtained is illustrated in Fig. 9.3. Note that RPM=200 in this experiment. The blue points in the figure, at left, show that the MOPSO al20000 function evaluations

20000 function evaluations

7 blades 6 blades 5 blades

4 blades

3 blades

Figure 9.3: (left) Pareto optimal front obtained by the MOPSO algorithm (6 blades), (right) Pareto optimal fronts for different numbers of blades

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191

gorithm was able to find a set of highly distributed estimations for the Pareto optimal solutions across both objectives. The search history of points sampled by particles during optimisation is illustrated by black points. The search history also shows that the MOPSO algorithm explored and exploited the search space efficiently, which results in obtaining this highly distributed and accurately converged estimation for the Pareto optimal front. The accurate convergence of the solutions obtained is due to the intrinsic high exploitation of the MOPSO algorithm around the selected leaders, gBest and pBests, in each iteration. The uniform distribution originates from the selection mechanism of leaders in MOPSO. Since particle guides were selected from the less populated parts of the archive, there was always a high tendency toward finding Pareto optimal solutions along the regions of the Pareto optimal front with lower distribution.

9.2.2

Number of blades

The effects of the number of blades on efficiency and cavitation were investigated. Five problems were first formulated by altering the number of blades from 3 to 7. The MOPSO algorithm was then employed to approximate the Pareto optimal fronts. The MOPSO algorithm was run 4 times on each of the problems and the best Pareto optimal fronts obtained are illustrated in Fig. 9.3, at right. This figure shows that the efficiency increases with to the number of blades, up to a limit of 5 blades. Beyond this number, efficiency decreases. The figure also shows that cavitation decreases in proportion to the number of blades. The reason why the majority of ship propellers are made of 5 or 6 blades is due to the shape of the fronts in Fig. 9.3. The highest efficiency, which is the main objective in ship propellers, is achieved by 5 or 6 blades. Therefore, 5 blades are chosen by default unless cavitation is a major issue.

9.2.3

Revolutions Per Minute (RPM)

RPM is one of the most important operating conditions for propellers. In order to observe the effects of this parameter on the efficiency and cavitation of the propeller, a 5-blade version of the propeller investigated in the previous subsection was selected. The RPM considered was limited to the range of 150 to 250. Since changing the RPM changes the operating conditions of the propeller significantly, two types of experiments were done, as follows:

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9. Real world applications

1. Finding the Pareto optimal front for the propeller at RPM increments of 10. 2. Parametrising the RPM and finding the optimal front for it using MOPSO. The MOPSO algorithm was employed to find the Pareto optimal front for the propeller at each of the 11 RPM varying from 150 to 250. The algorithm was run 4 times on each case and the best Pareto optimal fronts obtained are illustrated in Fig. 9.4, at left. This figure first shows that there is no feasible Pareto optimal solution when RPM = 150, 160, or 250. For the remaining RPMs, it may be observed that increasing RPM generally results in decreasing efficiency and increasing cavitation. Although increasing the RPM seems to increase the thrust, these results show that high RPM is not very effective and risks increased damage to the propeller in long term use due to the high cavitation. The peak of the high efficiency and low cavitation occurred between RP M = 170 and RP M = 180. Therefore, such RPM rates can be recommended when using a 5-blade version of the ship propeller investigated. RPM=150,160 all infeasible RPM=170 RPM=180 RPM=190 RPM=200 RPM=210 RPM=220 RPM=230 RPM=240 RPM=250 all infeasible

Figure 9.4: (left) Best Pareto optimal fronts obtained for different RPM (right) Optimal RPM To find the optimal values for the RPM, this operating condition was parametrised and optimised by MOPSO as well. The number of parameters increases to 21 when considering RPM as a parameter, but the same number of particles and iteration were chosen to approximate the Pareto optimal front. The best Pareto optimal front is illustrated in Fig. 9.4, at right.

9. Real world applications

193

The Pareto optimal front estimated shows that the approximations of Pareto optimal solutions mostly tend to the best Pareto optimal front found for RP M = 170. Almost 20% of the solutions are distributed between the Pareto optimal fronts for RP M = 170 and RP M = 180. The search history of the MOPSO algorithm is also illustrated in Fig. 9.4 to make sure that all of the Pareto fronts obtained in the previous experiment have been explored. The search history clearly illustrates that the fronts have been found by MOPSO, but all of them are dominated by the Pareto optimal front for RP M = 170 and the solutions between RP M = 170 and RP M = 180. This can be seen in Fig. 9.5.

Figure 9.5: PF obtained when varying RPM compared to PFs obtained with different RPM values A parallel coordinates visualisation of the solutions from the Pareto optimal front in Fig. 9.4(right) is shown in Fig. 9.6, for RPM between 170 and 180. Parallel coordinates are a very helpful tool for analysing high-dimensional data. The parallel coordinates in Fig. 9.6 show all the parameters and objectives. Each line indicates one of the Pareto optimal solutions obtained in Fig. 9.4. It may be observed in this figure that the range of the RPM is between 170 to 180. However, the density of solutions is higher close to RP M = 170.

194

9. Real world applications 𝑀𝑖𝑛 𝑅𝑃𝑀 = 170, 𝑀𝑎𝑥 𝑅𝑃𝑀 = 180

Normalized values

1 0.5 0 -0.5 -1

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11 P12 Parameters

P13

P14

P15

P16

P17

P18

P19

P20

RPM

Eff Cav objectives

Figure 9.6: Optimal RPM coordinates

Other features can be seen in this representation as clustering of solutions near particular parameter values.

9.2.4

Post analysis of the results

Since the Pareto optimal front obtained by MOPSO contains the best tradeoffs between cavitation and efficiency, some of the characteristics and physical rules applied to the propeller can possibly be inferred. One of the best tools for identifying and observing such behaviours is parallel coordinates (Fig. 9.6). The first pattern that can be seen in the parallel coordinates is the relatively uniform distribution of solutions over P 1 − P 6. These first pairs of parameters define the shape of the first three airfoils starting from the shaft of the propeller. This shows that the first three airfoils do not play very important roles in defining the final efficiency and cavitation. In contrast, parameters P 7 − P 10 are not distributed uniformly across the vertical lines. This shows that the shape of the fourth and fifth airfoils is very critical for designing the propeller. This is also consistent with the fact that the middle part of a blade has the greatest width and is consequently significantly involved in generating thrust, and cavitation. Although similar behaviour can be seen in the rest of the parameters, the distribution of P 7 − P 10 is much less than others, showing again the importance of these structural parameters. Other features evident in Fig. 9.6 also bear further investigation, a topic for future work.

9. Real world applications

9.2.5

195

Effects of uncertainties in operating conditions on the objectives

This experiment was to investigate the effects of uncertainties in the RPM on the efficiency/cavitation of the propeller. To do this, the best Pareto optimal front obtained for the 5-blade propeller in Fig. 9.3 was selected as the main front. The efficiency and cavitation of the Pareto optimal solutions in this front were then re-calculated by changing the RPM as the most important environmental condition. The projections of the solutions are illustrated in Fig. 9.7. Note that the perturbation considered is δRP M = ±1, which has been recommended by an expert in the field of mechanical engineering.

Figure 9.7: Pareto optimal solutions in case of (left) δRP M = +1 , (right) δRP M = −1 fluctuations in RPM (right). Original values are shown in blue, perturbed results in red. As Fig. 9.7 (left) shows, the efficiencies of all the Pareto optimal solutions obtained decrease when δRP M = +1 perturbations occur. The cavitation of Pareto optimal solutions is also increased. A similar behaviour for the efficiency can be observed in Fig. 9.7 (right). This figures shows that the efficiencies of Pareto optimal solutions decrease when δRP M = −1. However, the cavitation is decreased, which is obviously due to the lower rate of RPM. These results shows that perturbations in RPM can have significant negative impacts on the expected and desired efficiencies. The cavitation can also vary substantially with

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9. Real world applications

uncertainties in RPM.

9.2.6

Uncertainties in the structural parameters

Uncertainties may occur in the structural parameters as well. This type of uncertainty mostly originates from manufacturing errors. This subsection considers the maximum permitted errors, according to ISO 484/2-1981, that can alter the optimal values obtained by MOPSO. Note that the perturbation considered is δ = 1.5% of the nominal values. The Pareto optimal solutions obtained in Fig. 9.3 are first selected. Maximum positive and negative perturbations are then applied to parameters (thickness and chord). Finally, the objectives of the Pareto optimal solutions obtained are re-calculated. The results are illustrated in Fig. 9.8.

Figure 9.8: Pareto optimal solutions in case of (left) δ = +1.5% (right) δ = −1.5% perturbations in parameters. Original values are shown in blue, perturbed results in red. The trend is similar to the results of the preceding subsection, in that the uncertainties in parameters also degrade the expected efficiency significantly. In addition, the results show that the cavitation can vary dramatically in case of uncertainties in parameters. In summary, these results strongly show the remarkably negative impacts of perturbations on the performance of marine propellers and emphasise the

9. Real world applications

197

importance of considering such undesirable inputs when designing propellers. As a further illustration of the effect on efficiency, it may be noted that the perturbations in structural parameters gave rise to reductions in efficiency of about 0.25%. This translates directly to increased fuel consumption, the biggest cost in marine shipping. For the vessels for which the propeller tested is suited, generally those up to 100 tonnes displacement, the difference may be an increase of 40 litres per day. Scaling the effect to typical container ships operating under normal conditions, the increased fuel usage could be over half a tonne of bunker oil a day, increasing not only costs but also environmental emissions.

9.3

Confidence-based Robust optimisation of marine propellers

The preceding section showed the significant negative impacts of perturbations in parameters on both objectives of the propeller design problem. The proposed confident-based robust multi-objective perspective in this thesis has been designed to handle such perturbations without the need for extra function evaluation. Therefore, this section employs the proposed CRMOPSO algorithm for finding robust Pareto optimal solutions for the propeller design problem. The experimental set up is identical to that of the preceding section, in which 100 particle is utilised and allowed to search for the robust Pareto optimal solutions over 200 iterations. The maximum level of perturbation is also considered as 1.5% as per ISO 484/2-1981. The CRMOPSO algorithm was run 5 times and the best approximated robust front is illustrated in Fig. 9.9. Note that the nominal objective values for the robust Pareto optimal solutions obtained by the CRMOPOS algorithm are illustrated in this figure as well as the best Pareto optimal solutions obtained by MOPSO for comparison. It may be seen in this figure that the robust Pareto optimal front is completely dominated by the global Pareto optimal front. The coverage of the robust front is also lower than the global front. In order to see the significance of the results, the amount of excessive fuel consumptions that can be raised in case of 1.5% perturbation during manufacturing are calculated for the solutions obtained by both algorithms and presented in Table. 9.1. This shows how much the designs obtained increased the fuel consumption of the motor if 1.5% perturbation

198

9. Real world applications PF obtained by MOPSO

20000 function evaluations -25.55 -25.6 -25.65

-cavitation

-25.7

Robust PF obtained by CRMOPSO

-25.75 -25.8 -25.85 -25.9 -25.95 0.68

0.685

0.69

0.695

efficiency

Figure 9.9: Robust front obtained by CRMOPSO versus global front obtained by MOPSO Table 9.1: Fuel consumption discrepancy in case of perturbation in all of the structural parameters for both PS obtained by MOPSO and RPS obtained by CRMOPSO Algorithm average min max MOPSO 0.1735 0.1676 0.1851 CRMOPSO 0.0825 0.0805 0.0863

occurs. This table shows that the discrepancy of fuel consumption is much lower for the designs obtained by CRMOPSO algorithm. The average, minimum, and maximum of the excessive fuel consumption is almost half for the solutions obtained by the CRMOPSO algorithm. Due to the difficulty of the propeller design problem and importance of uncertainties, these results strongly evidence the merits of the proposed confidence-based robust optimisation. It is worth highlighting here that the robust Pareto optimal solutions are obtained without even a single extra function evaluation. To further investigate the effectiveness of the proposed CRMOPSO, this algorithm is employed to find the robust optimal values for RPM as well. In the previous subsections, the global Pareto optimal front for this problem was found

9. Real world applications

199

with parameterising the RPM. An equal number of particles and iterations are employed to determine the robust front and robsut optimal values for RPM. The results are illustrated in Fig. 9.10. This figure also shows the obtained Pareto optimal front by the MOPSO algorithm for comparison. PF obtained by MOPSO

20000 function evaluations 0 -2 -4

-cavitation

-6 -8 -10 -12 -14

Robust PF obtained by CRMOPSO

-16 -18 -20 0.685

0.69

0.695

0.7

efficiency

Figure 9.10: Global and robust Pareto optimal fronts obtained by MOPSO and CRMOPSO when RPM is also a variable As may be seen in Fig. 9.10 the major part of the robust Pareto front is dominated by the Pareto optimal front. The distributions of both fronts are almost identical. A small potion of robust front overlaps with global front. Therefore, the robust front is of type C, which means that a part of Pareto front is robust, but there are other robust solutions. To observe the range of RPMs in both fronts, Fig. 9.11 is provided. This figure shows that the range of RPM obtained by both algorithms is evidently different. It may be seen that the range of RPM tends to be higher in the Pareto optimal solutions obtained by CRMOPSO. Another interesting pattern is that half of the Pareto optimal solutions obtained by MOSPO have RPM of 170. However, the results of CRMOPO in Fig. 9.11 show that 170 is not a robust RPM since a few of the solutions have this value for their RPMs. The range of RPMs in CRMOPSO is wider than that of MOPSO. This is due to the intrisic higher exploration of the CRMOPSO. Since less non-dominated solutions

200

9. Real world applications 190 CRMOPSO MOPSO

188 186 184

RPM

182 180 178 176 174 172 170

0

20

40

60

80

100

Sorted Pareto optimal solutions

Figure 9.11: Optimal and robust optimal values for RPM (note that there are 98 robust Pareto optimal solutions and 100 global optimal solutions) Table 9.2: Fuel consumption discrepancy in case of perturbation in RPM for both PS obtained by MOPSO and RPS obtained by CRMOPSO Algorithm average min max MOPSO 0.0718 0.0136 0.0927 CRMOPSO 0.0521 0.0018 0.0904 are alloweded to enter the archive, the exploitation is less and exploration is higher than the normal MOPO, which results in finding a wider range of RPMs. To observe the impacts of the perturbation on the Pareto optimal solutions obtained by both of the algorithms, Table. 9.2 is provided. Note that the RPM is only perturbed in this experiment and other parameters are fixed. This table shows that the average, maximum, and minimum fuel consumption discrepancy of the robust Pareto optimal solutions obtained by CRMOPSO algorithm is better that those of MOPSO. These results again strongly prove the merits of the proposed confidence-based robust optimisation approach in finding robust solutions that are less sensitive to perturbations in parameters.

9.4

Summary

In this chapter, the shape of several ship propellers were optimised considering two objectives: efficiency versus cavitation. MOPSO was first employed to find

9. Real world applications

201

the best approximation of the true Pareto optimal front for the propeller. It was observed that the MOPSO algorithm showed very good convergence and was able to find a uniformly distributed Pareto optimal front. The MOPSO algorithm was then employed to undertake several experiments, investigating the effect of the number of blades, RPM, and uncertainties in manufacturing and operating parameters. The results of MOPSO were also analysed to identify the possible physical behaviour of the propeller. The results showed that the best efficiency and cavitation can be achieved by having five or six blades, since any other number of blades significantly degrades one of the objectives. The best RPM for the propeller was also found by the MOPSO algorithm. It was observed that the best Pareto optimal front can be obtained when the propeller is operating at RPM = 170 to 180. However, the results of the impact of uncertainties on RPM show that the optimal RPM is very sensitive to perturbation: efficiency and cavitation can be degraded significantly by a small amount of uncertainty. Simulation of manufacturing perturbations also revealed that both of the objectives for the Pareto optimal solutions obtained also vary dramatically. In addition, post analysis of the results showed that the most important parameters of the propeller are the maximum thickness and chord length of the fourth and fifth NACA airfoils along the blade. The results of CRMOPSO on the propeller design problem showed that this algorithm is able to find robust optimal values for parameters and RPM that are not sensitive to perturbations. It was observed that the fuel consumption discrepancy is much less for robust designs obtained by CRMOPSO in case of perturbations in parameters and RPM. Since the propeller design problem is a challenging real problem with a large number of constraints, this chapter strongly demonstates and supports the practicality of the proposed CRMO approach in finding robust solutions for real problems.

Chapter 10 Conclusion 10.1

Summary and conclusions

This thesis concentrates on robust optimisation in single- and multi-objective search spaces using population-based meta-heuristics. It was observed that the literature mainly lacks a systematic design process, which is essential for designing new algorithms or improving the current ones. The current main gaps in the literature in each phase of a systematic design process were identified as lack of standard single/multi-objective robust test problems, lack of specific performance metrics for quantifying the performance of robust multi-objective meta-heuristics, high computational cost of explicit methods, and low reliability of implicit methods. After identifying the gaps the following steps were taken to fill them as shown in Fig. 10.1. A set of novel frameworks were proposed with alterable parameters to design test functions with different level of difficulty and characteristics. The frameworks generated test problems with multiple local/global non-robust solutions. It was also possible to design a bi-modal search space with desired robustness level for the robust and global optima. In addition, several difficulties such as bias, deceptiveness, multi-modality, and flatness were integrated to different functions to design challenging single-objective robust test problems. Three frameworks were proposed to design robust multi-objective test problems. The first framework allowed designing bi-frontal search spaces with a parameter to control the level of robustness. This framework was equipped with a parameter that defines the shape of the fronts as well. Therefore, several test functions with different shapes for robust and global Pareto optimal front were 202

10. Conclusion

203



 

Proposal of two performance measures to quantify the performance of robust multi-objective algorithms



 Robust test function design

Proposal of three frameworks for single-objective robust optimisation Proposal of thee frameworks for multi-objective robust optimisation Integrating several hindrances, obstacles, and difficulties to the current test functions in both fields Establishing the first standard and challenging test suits for both fields

Robsut performance metric design

 



Proposal of confidence measure Establishment of confidencebased robust optimisation Establishment of confidencebased robust multi-objective optimisation Proposal of CRGA and CRPSO



Proposal of CRMOPSO



Robust algorithm design



Finding robust design for marine propellers reliably without extra functions evaluation

Figure 10.1: Gaps filled by the thesis constructed. The second framework generated multi-frontal search spaces with different degrees of robustness to benchmark the performance of robust multiobjective algorithms in terms of non-robust local fronts avoidance. The third framework created test functions with separated fronts. Each region of the front had its own level of robustness, so the ability of an algorithm in finding such types of fronts could be benchmarked. Several test functions with different levels of difficulty were proposed. In addition, the current robust multi-objective benchmarks proposed in the literature were improved and extended to mimic the characteristics of real search spaces better. The thesis also considered the introduction of several hindrances for robust multi-objective test functions: bias, a large number of local fronts, deceptive search spaces, and flat search spaces.

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10. Conclusion

A set of novel performance measures was proposed to quantify the performance of robust multi-objective algorithms. The first performance measure was a coverage measure, in which the distribution of the solutions along the robust regions of the true Pareto optimal front was measured and quantified. The proposed coverage measure allowed comparing the algorithm based on their ability in finding uniformly distributed robust Pareto optimal solutions. The second proposed performance measure, success ratio, quantified the performance of a robust algorithm in finding robust and non-robust Pareto optimal solutions. This measure counted the number of robust and non-robust solutions and measured how successful an algorithm was in finding robust Pareto optimal solutions and avoiding non-robust Pareto optimal solutions. A novel measure called confidence measure was proposed to improve the reliability of the implicit robust optimisation methods. In this measure the status of previously sampled points in the parameter space were considered and utilised to measure the confidence level of solutions during optimisation. The proposed confidence measure was integrated with inequality operators. As a result, several confidence-based relational operators were proposed to compare the search agents of meta-heuristics confidently. In addition, a novel robust optimisation approach called confidence-based robust optimisation was established. The proposed confidence-based robust optimisation was integrated with two well-known algorithms: PSO and GA. In the confidence-based PSO, particles updated their positions in the search space normally, but the process of updating gBest and pBests were done by the proposed relational confidence-based operator. The GA was also modified by the proposed confidence-based relational operators. Two confidence-based operators were proposed for GA: confidence-based cross-over and confidence-based elitism. In the former method, the individuals were compared and allowed to mate if and only if they were confidently better. In the latter operator, the elite was only updated based on the proposed confidencebased relational operators. The proposed confidence measure was employed to design a confidence-based Pareto dominance operator. The proposed confidence-based dominance allowed designing a novel approach for finding robust Pareto optimal solutions called confidence-based robust multi-objective optimisation. This novel approach could be integrated with any meta-heuristics. The MOPSO algorithm was chosen as one of the best algorithms in the literature and converted to confidence-based

10. Conclusion

205

robust MOPSO. In this algorithm the particles updated their positions normally, but they were added to the archive if and only if they confidently dominated one of the archive members or were confidently non-dominated compared to the archive members. The shape of a ship propeller was optimised by the proposed approach to find its robust designs. The thesis first considered the investigation of this problem in terms of the shape of Pareto optimal front, effect of number of blades, effect of RPM, and negative impacts of uncertainties on efficiency and cavitation. The proposed CRMOPSO was then employed to find the robust designs for the case study. The results of the proposed CRPSO and CRGA on the proposed test functions first revealed the merits of the proposed confidence measure. It was observed that the confidence measure prevents the algorithms from favouring nonconfident solutions and making risky decisions during optimisation. The results showed that the confidence-based algorithms were able to outperform other robustness handling techniques in the literature. The results also proved the merits of the proposed test functions, in that they provide very challenging test beds and allow benchmarking the performance of different algorithms effectively. The results of the proposed CRMOPSO proved that the confidence measure and confidence-based robust multi-objective optimisation approach can provide very promising results. It was observed that confidence-based Pareto dominance prevents non-confident particles from entering the archive. This assisted CRMOPSO to always have confident solutions in the archive as the leaders to guide other particles toward robust regions of the search space. The qualitative and quantitative results indicated that CRMOPSO was able to outperform other robust algorithms in the literature. In addition, the comparative results of algorithms on the proposed benchmark functions showed that the proposed robust multi-objective test functions provide very challenging test beds with diverse characteristics and allow designers to benchmark and compare different algorithms efficiently. The results of the MOPSO algorithm on the real case study first showed the optimal values for structural parameters and operating conditions. The importance of uncertainties in the parameters was also revealed, in which both efficiency and cavitation fluctuated noticeably. The results of CRMOPO on the case study proved that this algorithms is able to find robust solutions for

206

10. Conclusion

expensive, challenging problems efficiently. These results strongly demonstrated the merits of the confidence-based robust optimisation perspective proposed in this thesis. The following main conclusions can therefore be made: • The proposed test functions provide very challenging test beds for benchmarking the performance of roust algorithms. • The proposed robust performance measures allow quantifying the performance of robust multi-objective algorithms and facilitate relative comparisons. • The proposed confidence measure is able to effectively quantify the confidence level that we have when relying on the previously sampled points. • The proposed confidence-based relational operators are able to efficiently compare the robust solutions during optimisation and favour the more confident ones. • The proposed confidence-based Pareto dominance operator also assist designers to find non-dominated robust Pareto optimal solutions by considering their confidence levels. • The proposed confidence-based robust optimisation prevents algorithms from favouring non-robust solutions during optimisation and consequently allows search agents with high levels of confidence to guide other search agents. • The proposed two confidence-based robust optimisation approaches improve the reliability of current robust optimisation algorithms that only rely on previously sampled points. • The proposed confidence-based robust optimisation apporaches are readily applicable for solving real problems. • The proposed systematic design process allows contributors to reliably and conveniently design and compare robust algorithms.

10. Conclusion

10.2

207

Achievements and significance

A number of contributions have been made to robust single-objective and multiobjective optimisation as summarised in Fig. 10.2. The highlighted boxes in Confidence measure

Implicit mathods

Confidence-based relational operators

Explicit methods

Confidence-based robsust optimisation

Population-based stochastic robust optimisaiton methods

CRPSO Single-objective robust optimisation

CRGA Benchmark problems Confidence-based Pareto optimality Implicit methods

CRMOPSO Confidence-based robust multi-objective optimisation

Explicit methods

Real application

Multi-objective robust optimisation Benchmark problem

Performance metrics

Efficiency

Marine propeller design

Single objective optimisation

Cavitation

Thrust

Efficiency vs. thrust

Multi-objective optimisation

Efficiency vs. cavitation

Robust multi-objective optimisation

Impacts of uncertainties

Figure 10.2: Contributions of the thesis green indicate where the following contributions fit:

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10. Conclusion

• A systematic design process has been proposed to design robust algorithms in the fields of robust single- and multi-objective optimisation. • Due to the lack of standard test functions in the literature of robust singleobjective optimisation, the thesis also considers the collection/investigation of the current robust test functions and the proposal of more challenging ones. • Several test functions were proposed, which can be considered as the first standard set of test functions in the literature of single-objective robust optimisation. • Three frameworks were proposed to generate robust single-objective test functions with desired characteristics and level of difficulty, another novel contribution to the field. • Several multi-objective test functions were proposed, which can be considered as the first standard set of test functions in the literature of multiobjective robust optimisation. • Three frameworks were proposed to generate robust multi-objective test functions with desired characteristics and level of difficulty. • Two standard test suites were proposed for the first time, to be used by other researchers. • Two performance measures for quantifying the performance of robust multiobjective algorithms were proposed. There were no performance measures in the literature that consider robustness of Pareto optimal solutions. Therefore, the proposed set of performance measures is another significant achievement of this research. • A systematic attempt has been undertaken to improve the reliability of robust algorithms that rely on previously sampled points during optimisation. • A novel metric called confidence measure was proposed to quantify the confidence level of solutions based on the status of previously sampled points in the parameter space. This is no such metric in the literature, so it can be considered as a substantial contribution to the field.

10. Conclusion

209

• With the proposal of the confidence measure, the relational operators and Pareto dominance have been re-defined to confidently compare the algorithms in search spaces with single and multiple objectives respectively. The proposed operators were also a fresh contribution to the literature. • The most significant contribution and achievement of this research was the proposal of two novel robust optimisation approaches: confidence-based robust optimisation and confidence-based robust multi-objective optimisation. These two approaches established two new research branches in the fields of robust single-objective and multi-objective optimisation. The experimental results presented in the thesis proved that these two concepts are able to find robust solutions with a very high level of reliability without addition function evaluations. • Two novel robust algorithms based on PSO and GA were proposed, which can be considered as the first two confidence-based robust algorithm in the field of single-objective optimisation. • One robust multi-objective algorithm based on MOPSO was proposed, which utilises the proposed confidence-based robust multi-objective optimisation approach. • A problem of propeller design was solved by MOPSO considering two conflicting objectives (efficiency versus cavitation), and optimal values for RPM, number of blades, and parameters were found for the first time. • Investigating and proving the significant negative impacts of uncertainty on the obtained optimal values for parameters and RPM were other substantial outcomes of this research. • Robust optimal values for propeller design parameters and RPM were found by the proposed CRMOPSO for the first time. In this thesis I have proposed and implemented the necessary components of a systematic robust optimisation design process and demonstrated its ability to reliably find robust solutions for challenging design problems without additional computational cost.

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10.3

10. Conclusion

Future work

This thesis opens up many research directions. The main and most interesting future work would be the investigation of all the proposed concepts in the field of constrained robust optimisation. The effectiveness of the proposed confidence measure, relational operators, Pareto dominance, and confidence-based robust optimisation can all be applied to constrained problems. However, constrained search spaces have their own difficulties that should be considered and require special mechanisms. Due to the similarity of robust optimisation and dynamic optimisation, investigating the concept of the confidence measure and confidence-based robust optimisation in dynamic optimisation may also prove advantageous. The high computational cost of explicit methods and low reliability of implicit methods in the field of dynamic optimisation have the potential to be improved by the concepts proposed in this thesis. The proposed confidence measure can co-operate alongside any kind of robustness indicators to consider the confidence level of solutions during optimisation. Therefore, another research direction would be the investigation of the confidence measure to improve different type I and type II robustness handling techniques in the literature. The proposed confidence-based relational operators and Pareto dominance can be employed to compare search agents of different algorithms in the literature, which is another research direction. All of the benchmark problems in this thesis are unconstrained. For future work, therefore, integration of different types of constraints for designing constrained robust single-objective and multi-objective test functions is worth consideration. Investigating the suitability of the current binary measure or proposing new ones for RMOO is recommended for future work as well. Another important research avenue is to propose confidence-based operators and mechanisms to find robust Pareto optimal solutions of many-objective problems. In the field of many-objective optimisation, designing unconstrained and constrained robust many-objective robust test problems would also be a valuable contribution. Investigating special mechanisms for CRMOPSO algorithms to handle ‘thick’ fronts is important as well.

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Appendix A Single-objective robust test functions Some of the test problems of this work have been adopted from [18, 58, 163, 105]. The full description of the benchmark functions utilised and proposed are as follows:

A.1 A.1.1

Test functions in the literature TP1 N Y

N

1 X 2 f (~x) = 1 − H(xi ) + x 100 i=1 i i=1

(A.1)

 0 x < 0 i H(xi ) = 1 otherwise

(A.2)

Search space: ~x ∈ [−10, 10]N ~ (−1, 1) Input noise: ~δ ∼ U Robust optimum fitness (nominal value) ≈ 0.02(2D), 0.05(5D) Robust optimum fitness (expected value) ≈ 0.026(2D), 0.066(5D) Robust optimum location 2D(1, 1), 5D(1, 1, 1, 1, 1) The 5D version of this function has been used in the experimental results. 230

A. Single-objective robust test functions

A.1.2 f (~x) =

231

TP2 p

||~x|| + e−5||~x||

2

(A.3)

Search space: ~x ∈ [−4, 4]N ~ (−0.5, 0.5) Input noise: ~δ ∼ U Robust optimum fitness (nominal value) ≈ 0.924(2D), 1.135(5D) Robust optimum fitness (expected value) ≈ 1.005(2D), 0.967(5D) Robust optimum location 2D(0.735, 0.0), 5D(−0.334, −0.056, −0.024, −0.023, 0.074) The 5D version of this function has been used in the experimental results. Note that despite the fact that the inventor of this test function suggested the robust optima mentioned above, a more robust optimum is found in (0.735, 0.0). This point has an expectation of 1.005 and a nominal value of 0.924. For the 5variable problem, a more robust optimum is located in (−0.334, −0.056, −0.024, −0.023, 0.074). This point has an expectation of 0.967 and a nominal value of 1.135.

A.1.3

TP3

f (~x) =

5 √ − max{f0 (~x), f1 (~x), f2 (~x), f3 (~x)} 5− 5

(A.4)

f0 (~x) =

1 0.5||~x|| e 10

(A.5)

f1 (~x) =

5 √ (1 − 5− 5

s

||~x + 5|| √ ) 5 N

(A.6)

f2 (~x) = c1 (1 − (

||~x + 5|| 4 √ )) 5 N

(A.7)

f3 (~x) = c2 (1 − (

||~x − 5|| d2 √ ) ) 5 N

(A.8)

c1 =

625 , c2 = 1.5975, d2 = 1.1513 624

(A.9)

232

A. Single-objective robust test functions

Search space: ~x ∈ [−10, 10]N ~ (−1, 1) Input noise: ~δ ∼ U Robust optimum fitness (nominal value) ≈ 0.2115(2D), 0.2115(5D) Robust optimum fitness (expected value) ≈ 0.33(2D), 0.34(5D) Robust optimum location 2D(5, 5), 5D(5, 5, 5, 5, 5) The 5D version of this function has been used in the experimental results.

A.1.4

TP4

N 1 X f1 (xi ) f (~x) = c − N i=1

(A.10)

 −(x + 1)2 + 1 −2 ≤ x < 0 i i f1 (xi ) = c.2−8|xi −1| 0 ≤ xi < 2

(A.11)

c = 1.3

(A.12)

Search space: ~x ∈ [−2, 2]N ~ (−0.5, 0.5) Input noise: ~δ ∼ U Robust optimum fitness (nominal value) ≈ 0.3(2D), 0.3(5D) Robust optimum fitness (expected value) ≈ 0.37(2D), 0.366(5D) Robust optimum location 2D(−1, −1), 5D(−1, −1, −1, −1, −1) The 5D version of this function has been used in the experimental results.

A.1.5 f (~x) =

TP5 N X

f1 (xi )

(A.13)

i=1

 1 −0.5 ≤ x < 0.5 i f1 (xi ) = 0 otherwise Search space: ~x ∈ [−0.5, 0.5]N

(A.14)

A. Single-objective robust test functions

233

~ (−0.2, 0.2) Input noise: ~δ ∼ U Robust optimum fitness (nominal value) ≈ −2.0(2D), −20.0(20D) Robust optimum fitness (expected value) ≈ −2.0(2D), −20.0(20D) Robust optimum location 2D(0, 0), 20D(0, 0, ..., 0) The 20D version of this function has been used in the experimental results.

A.1.6 f (~x) =

TP6 N X

f1 (xi )

(A.15)

i=1

   1 −0.6 ≤ xi < −0.2   f1 (xi ) = 1.25 (0.2 ≤ xi < 0.36) ∨ (0.44 ≤ xi < 0.6)    0 otherwise

(A.16)

Search space: ~x ∈ [−1.5, 1.5]N ~ (−0.2, 0.2) Input noise: ~δ ∼ U Robust optimum fitness (nominal value) ≈ −2(2D), −20(20D) Robust optimum fitness (expected value) ≈ −2.0(2D), −20(20D) Robust optimum location 2D(−0.4, −0.4), 20D(−0.4, −0.4, ..., −0.4) The 20D version of this function has been used in the experimental results.

A.1.7

TP7

f (~x) = 1 −

N 1 X f1 (xi ) N i=1

 x + 0.8 −0.8 ≤ x < 0.2 i i f1 (xi ) = 0 otherwise

(A.17)

(A.18)

Search space: ~x ∈ [−1, 1]N ~ (−0.2, 0.2) Input noise: ~δ ∼ U Robust optimum fitness (nominal value) ≈ 0(2D), 0(20D) Robust optimum fitness (expected value) ≈ −1.6(2D), −16.0(20D) Robust optimum location 2D(0.2, 0.2), 20D(0.2, 0.2, ..., 0.2) The 20D version of this function has been used in the experimental results.

234

A.1.8

A. Single-objective robust test functions

TP8

N 1 X f (~x) = − g(x) N i=1

(A.19)

 p e−2ln2( x−0.1 )2 0.8 |sin(5πxi )| 0.4 < xi ≤ 0.6 g(xi ) = x−0.1 2 e−2ln2( 0.8 ) sin6 (5πxi ) otherwise

(A.20)

Search space: ~x ∈ [0, 1]N ~ (−0.0625, 0.0625) Input noise: ~δ ∼ U Robust optimum fitness (nominal value) ≈ −1(2D), −1(20D) Robust optimum fitness (expected value) ≈ −0.91(2D), −0.91(20D) Robust optimum location 2D(0.1, 0.1) The 2D version of this function has been used in the experimental results.

A.1.9

TP9

   cos( 21 xi ) + 1 −2π ≤ xi < 2π   f (~x) = 1.1cos(xi + π) + 1.1 2π ≤ xi < 4π    0 otherwise

(A.21)

Search space: ~x ∈ [−2π, 4π]N ~ (−2, 2) Input noise: ~δ ∼ U Robust optimum fitness (nominal value) ≈ −4(2D), −40(20D) Robust optimum fitness (expected value) ≈ −3.68(2D), −36.8(20D) Robust optimum location 20D(0, 0, ..., 0) The 20D version of this function has been used in the experimental results.

A. Single-objective robust test functions

A.2

235

Test functions generated by the proposed framework I

A.2.1

TP10

   f1 (~x)      f2 (~x) f (~x) =   f3 (~x)     f4 (~x)

(x1 ≤ 0) ∧ (x2 ≥ 0) (x1 ≥ 0) ∧ (x2 ≤ 0)

(A.22)

(x1 > 0) ∧ (x2 > 0) (x1 < 0) ∧ (x2 < 0)

2 f1 (~x) = ΣN i=1 xi

(A.23)

f2 (~x) = maxi {|xi |, 1 ≤ i ≤ N }

(A.24)

2 f3 (~x) = ΣN i=1 [xi − 10cos(2πxi ) + 10]

(A.25)

√1 N 2 1 N f4 (~x) = −20e(−0.2 N Σi=1 xi ) − e( N Σi=1 cos(2πxi )) + 20 + e

(A.26)

Search space: ~x ∈ [−4, 4]N ~ (−1, 1) Input noise: ~δ ∼ U Robust optimum fitness (nominal value) ≈ 0.8(2D) Robust optimum fitness (expected value) ≈ 1.46(2D) Robust optimum location 2D(−2, 2) The 2D version of this function has been used in the experimental results.

A.3 A.3.1

Proposed biased test functions TP11 - biased 1

 Y (~x)G(~x) (x2 + x2 > 25) 1 2 f (~x) = G(~x)B(~x) (x2 + x2 ≤ 25) 1 2

(A.27)

236

A. Single-objective robust test functions

N Y

N

1 X 2 H(xi ) + x Y (~x) = 1 − 100 i=1 i i=1

(A.28)

 0 x < 0 i H(x) = 1 otherwise

(A.29)

2 G(~x) = ΣN i=3 50xi

(A.30)

θ B(~x) = (ΣN i=1 |xi |)

(A.31)

θ = 0.1 Search space: ~x ∈ [−10, 10]N ~ (−1, 1) Input noise: ~δ ∼ U Robust optimum fitness (nominal value): 0.02 Robust optimum fitness (expected value): 3.58 Robust optimum location (1, 1, 0, 0, ..., 0) The 10D version of this function has been used in the experimental results.

A.3.2

TP12 - biased 2

2 1 X f (~x) = (c − y(xi ))G(~x)B(~x) N i=1

(A.32)

 −(x + 1)2 + 1 −2 ≤ x < 0 i i y(xi ) = c × 2xi −1 0 ≤ xi < 2

(A.33)

2 G(~x) = ΣN i=3 50xi

(A.34)

θ B(~x) = (ΣN i=1 |xi |)

(A.35)

A. Single-objective robust test functions

237

θ = 0.1 c = 1.3 Search space: ~x ∈ [−10, 10]N ~ (−1, 1) Input noise: ~δ ∼ U Robust optimum fitness (nominal value): 0.2663 Robust optimum fitness (expected value): 8720 Robust optimum location (5, 5, 0, 0, ..., 0) The 10D version of this function has been used in the experimental results.

A.4 A.4.1

Proposed deceptive test functions TP13

f (~x) = (H(x1 ) + H(x2 )) × G(~x) − 1

H(x) =

x−0.4 2 x−0.5 2 x−0.6 2 1 − 0.3e−( 0.004 ) − 0.5e−( 0.05 ) − 0.3e−( 0.004 ) + sin(πx) 2

2 G(~x) = (ΣN i=3 50xi ) + 1

(A.36)

(A.37)

(A.38)

Search space: ~x ∈ [0.2, 0.8]N ~ (−0.01, 0.01) Input noise: ~δ ∼ U Robust optimum fitness (nominal value): 0.2663 Robust optimum fitness (expected value): 1.0395 Robust optimum location (0.5, 0.5, 0, 0, ..., 0) The 10D version of this function has been used in the experimental results.

A.4.2

TP14

f (~x) = (H(x1 ) + H(x2 )) × G(~x) − 1

H(x) =

(A.39)

x−0.5 2 x−0.04 2 1 −( x−0.04 )2 0.004 − 0.5e−( 0.05 ) − Σ11 + 0.3e−( 0.004 ) + sin(πx)) (A.40) i=1 (0.3e 2

238

A. Single-objective robust test functions

2 G(~x) = (ΣN i=3 50xi ) + 1

(A.41)

Search space: ~x ∈ [0.22, 0.78]N ~ (−0.01, 0.01) Input noise: ~δ ∼ U Robust optimum fitness (nominal value): 1 Robust optimum fitness (expected value): 1.0396 Robust optimum location (0.5, 0.5, 0, 0, ..., 0) The 10D version of this function has been used in the experimental results.

A.4.3

TP15

   (H(x1 ) + H(x2 )) × G(~x) − 1     (H(2 − x ) + H(x − 2)) × G(~x) − 1 1 2 f (~x) =   (H(2 − x1 ) + H(x2 )) × G(~x) − 1     (H(x1 ) + H(2 − x2 )) × G(~x) − 1

(x1 ≤ 1) ∧ (x2 ≤ 1) (x1 > 1) ∧ (x2 > 1)

(A.42)

(x1 > 1) ∧ (x2 ≤ 1) (x1 ≤ 1) ∧ (x2 > 1)

H(x) = e−3x sin(8πx) + x

(A.43)

2 G(~x) = (ΣN i=3 50xi ) + 1

(A.44)

Search space: ~x ∈ [0.79, 1.21]N ~ (−0.01, 0.01) Input noise: ~δ ∼ U Robust optimum fitness (nominal value): 1.721 Robust optimum fitness (expected value): 1.7451 Robust optimum location (1.1, 1.1, 0, 0, ..., 0), (1.1, 0.9, 0, 0, ..., 0), (0.9, 0.9, 0, 0, ..., 0), (0.9, 1.1, 0, 0, ..., 0) The 10D version of this function has been used in the experimental results.

A.5 A.5.1

Proposed multi-modal robust test functions TP16

f (~x) = (H(x1 ) + H(x2 )) × G(~x) − 1.399

(A.45)

A. Single-objective robust test functions

H(x) =

x−(1−0.0063i) 2 x−0.5 2 3 −( x−0.0063i )2 0.004 − 0.5e−( 0.04 ) − Σ16 + 0.8e−( 0.004 ) ) i=0 (0.8e 2

2 G(~x) = (ΣN i=3 50xi )

239

(A.46)

(A.47)

Search space: ~x ∈ [0, 1]N ~ (−0.01, 0.01) Input noise: ~δ ∼ U Robust optimum fitness (nominal value): 0.601 Robust optimum fitness (expected value) ≈ 0.6484 Robust optimum location (0.5, 0.5, 0, 0, ..., 0) The 10D version of this function has been used in the experimental results.

A.5.2

TP17

f (~x) = (H(x1 ) + H(x2 )) × G(~x) − 1.399

H(x) =

x−(1−0.0063i) 2 x−0.5 2 3 )2 −( x−0.0063i 0.004 − 0.8e−( 0.04 ) − Σ16 + 0.5e−( 0.004 ) ) i=0 (0.5e 2

2 G(~x) = (ΣN i=3 50xi )

(A.48)

(A.49)

(A.50)

Search space: ~x ∈ [0, 1]N ~ (−0.01, 0.01) Input noise: ~δ ∼ U Robust optimum fitness (nominal value): 0.001 Robust optimum fitness (expected value): 0.529 Robust optimum location (0.5, 0.5, 0, 0, ..., 0) The 10D version of this function has been used in the experimental results.

A.6 A.6.1

Proposed flat robust test function TP18

f (~x) = (H(x1 ) + H(x2 )) × G(~x) − 2

(A.51)

240

H(x) =

A. Single-objective robust test functions

x−0.95 2 x−0.05 2 1 − (0.2e−( 0.03 ) + 0.2e−( 0.01 ) ) 2

2 G(~x) = (ΣN i=3 50xi ) + 1

(A.52)

(A.53)

Search space: ~x ∈ [0, 1]N ~ (−0.01, 0.01) Input noise: ~δ ∼ U Robust optimum fitness (nominal value): 0 Robust optimum fitness (expected value): 0.0412 Robust optimum location (0.95, 0.95, 0, 0, ..., 0) The 10D version of this function has been used in the experimental results.

A.7

Test functions generated by the proposed frameworks II and III

A.7.1

TP19

1 −0.5( Σni=1 (xi −1.5)2 )2 1 −0.5( Σni=1 (xi −0.5)2 )2 0.5 0.01 √ √ f (~x) = ( e e )+( ) 2π 2π

(A.54)

Search space: ~x ∈ [0, 2]N ~ (−0.2, 0.2) Input noise: ~δ ∼ U Robust optimum fitness (nominal value): −0.3989 Robust optimum fitness (expected value): −0.3981 Robust optimum location (1.5, 1.5) The 2D version of this function has been used in the experimental results.

A.7.2

TP20

f (~x) = −H(x1 ) × H(x2 ) + G(~x)

2

e2x sin(8π(x + H(x) = 3

π )) 16

−x

(A.55)

+ 0.5

(A.56)

A. Single-objective robust test functions

G(~x) = 10

ΣN i=3 xi N

241

(A.57)

Search space: ~x ∈ [0, 1]N ~ (−0.02, 0.02) Input noise: ~δ ∼ U Robust optimum fitness (nominal value): −3.83 Robust expected fitness (expected value): −3.65 Robust optimum location (0.8, 0.8, 0, 0, ..., 0) The 10D version of this function has been used in the experimental results.

Appendix B Multi-objective robust test functions B.1

Deb’s test functions

B.1.1

RMTP1

f1 (~x) = x1

(B.1)

f2 (~x) = h(x1 ) + g(~x)S(x1 )

(B.2)

subject to : 0 ≤ x1 ≤ 1, −1 ≤ xi ≤ 1, i = 2, 3, ..., n

(B.3)

where : h(x1 ) = 1 − x21

(B.4)

g(~x) =

n X

10 + x2i − 10cos(4πxi )

(B.5)

i=2

S(x1 ) =

1 + x21 0.2 + x1

(B.6) 242

B. Multi-objective robust test functions

B.1.2

243

RMTP2

f1 (~x) = x1

(B.7)

f2 (~x) = h(x1 ) + g(~x)S(x1 )

(B.8)

subject to : 0 ≤ x1 ≤ 1, −1 ≤ xi ≤ 1, i = 2, 3, ..., n

(B.9)

where : h(x1 ) = 1 − x21

g(~x) =

n X

(B.10)

10 + x2i − 10cos(4πxi )

(B.11)

i=2

S(x1 ) =

1 + 10x21 0.2 + x1

B.1.3

RMTP3

(B.12)

f1 (~x) = x1

(B.13)

f2 (~x) = h(x2 ) + (g(~x) + S(x1 ))

(B.14)

subject to : 0 ≤ x1 , x2 ≤ 1, −1 ≤ xi ≤ 1, i = 3, 4, ..., n

(B.15)

where : h(x1 ) = 2 − 0.8e−(

g(~x) =

n X

50x2i

x2 −0.35 2 ) 0.25

− e−(

x2 −0.85 2 ) 0.03

(B.16)

(B.17)

i=3

S(x1 ) = 1 −

√

x1

(B.18)

244

B.1.4

B. Multi-objective robust test functions

RMTP4

f1 (~x) = x1

(B.19)

f2 (~x) = x2

(B.20)

f3 (~x) = h(x1 , x2 ) + g(~x)S(x1 , x2 )

(B.21)

subject to : 0 ≤ x1 , x2 ≤ 1, −1 ≤ xi ≤ 1, i = 3, 4, ..., n

(B.22)

where : h(x1 , x2 ) = 2 − x21 − x22

(B.23)

g(~x) =

n X

(10 + x2i − 10cos(4πxi ))

(B.24)

i=3

S(x1 , x2 ) =

B.1.5

0.75 0.75 + 10x81 + + 10x82 0.2 + x1 0.2 + x2

(B.25)

RMTP5

f1 (~x) = x1

(B.26)

f2 (~x) = x2

(B.27)

f3 (~x) = h(x3 )(g(~x) + S(x1 , x2 ))

(B.28)

subject to : 0 ≤ x1 , x2 , x3 ≤ 1, −1 ≤ xi ≤ 1, i = 4, 5, ..., n

(B.29)

B. Multi-objective robust test functions

where : h(x3 ) = 2 − 0.8e−(

g(~x) =

n X

x3 −0.35 2 ) 0.25

− e−(

x3 −0.85 2 ) 0.03

(10 + x2i − 10cos(4πxi ))

245

(B.30)

(B.31)

i=3

S(x1 , x2 ) = 1 −

B.2

√

x1 −

√

x2

(B.32)

Gaspar Cunha’s functions

B.2.1

RMTP6

f1 (~x) = x1

(B.33)

f2 (~x) = h(x1 ) + g(~x)S(x1 )

(B.34)

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.35)

where : h(x1 ) =

(x1 − 0.6)3 − 0.43 −0.63 − 0.43

(B.36)

Pn g(~x) =

S(x1 ) =

i=2 (xi )

n−1

1 0.2 + x1

(B.37)

(B.38)

246

B.2.2

B. Multi-objective robust test functions

RMTP7

f1 (~x) = cos(

πx1 ) 2

f2 (~x) = g(~x)sin(

πx1 ) 2

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.39)

(B.40)

(B.41)

Pn

i=2 (xi )

g(~x) = 1 + 10

B.2.3

n−1

(B.42)

RMTP8

f1 (~x) = 1 − x21

(B.43)

f2 (~x) = g(~x)sin(

πx1 ) 2

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.44)

(B.45)

Pn

i=2 (xi )

g(~x) = 1 + 10

B.2.4

n−1

(B.46)

RMTP9 x1

f1 (~x) =

e −1 e−1

(B.47)

sin(4πx1 ) − 15x1 + 1] 15

(B.48)

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.49)

f2 (~x) = g(~x)[

Pn g(~x) = 1 + 10

i=2 (xi )

n−1

(B.50)

B. Multi-objective robust test functions

B.2.5

247

RMTP10

f1 (~x) = x1

(B.51)

sin(4πx1 ) − 15x1 + 1] 15

(B.52)

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.53)

f2 (~x) = g(~x)[

Pn g(~x) = 1 + 10

B.3

i=2 (xi )

n−1

Test functions generated by the proposed frameworks 1, 2, and 3

B.3.1

RMTP11

Framework 1 α = 0.1 ω = 1.5 β1 = 1 β2 = 1

B.3.2

RMTP12

Framework 1 α = 0.1 ω = 1.5 β1 = 1.5 β2 = 1.5

B.3.3

(B.54)

RMTP13

Framework 1 α = 0.1

248

B. Multi-objective robust test functions

ω = 1.5 β1 = 0.5 β2 = 0.5

B.3.4

RMTP14

Framework 1 α = 0.1 ω = 1.5 β1 = 0.5 β2 = 1

B.3.5

RMTP15

Framework 1 α = 0.1 ω = 1.5 β1 = 1 β2 = 0.5

B.3.6

RMTP16

Framework 1 α = 0.1 ω = 1.5 β1 = 1 β2 = 1.5

B.3.7

RMTP17

Framework 1 α = 0.1 ω = 1.5 β1 = 1.5 β2 = 1

B. Multi-objective robust test functions

B.3.8

RMTP18

Framework 1 α = 0.1 ω = 1.5 β1 = 1.5 β2 = 0.5

B.3.9

RMTP19

Framework 1 α = 0.1 ω = 1.5 β1 = 0.5 β2 = 1.5

B.3.10

RMTP20

Framework 2 γ=3 λ=4 ω=1 β=1

B.3.11

RMTP21

Framework 2 γ=3 λ=4 ω=1 β = 0.5

B.3.12

RMTP22

Framework 2 γ=3 λ=4

249

250

B. Multi-objective robust test functions

ω=1 β = 1.5

B.3.13

RMTP23

Framework 2 ζ=2 λ=6 γ=3 ω = 0.5

B.3.14

RMTP24

Framework 2 ζ=4 λ=6 γ=3 ω = 0.5

B.3.15

RMTP25

Framework 2 ζ=8 λ=6 γ=3 ω = 0.5

B.4 B.4.1

Extended version of current test functions RMTP26

f1 (~x) = x1

(B.55)

f2 (~x) = g(~x) − C(x1 )

(B.56)

B. Multi-objective robust test functions

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

251

(B.57)

Pn

i=2 (xi )

g(~x) = 1 + 10

n−1

  0       0.25   C(x1 ) = 0.5      0.75     1

B.4.2

(B.58)

0 ≤ x1 ≤ 0.2 0.2 < x1 ≤ 0.4 0.4 < x1 ≤ 0.6

(B.59)

0.6 < x1 ≤ 0.8 0.8 < x1 ≤ 1

RMTP27 x1

f1 (~x) =

e −1 e−1

(B.60)

f2 (~x) = g(~x) − C(x1 )

(B.61)

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.62)

Pn g(~x) = 1 + 10

  0       0.25   C(x1 ) = 0.5      0.75     1

i=2 (xi )

n−1

(B.63)

0 ≤ x1 ≤ 0.2 0.2 < x1 ≤ 0.4 0.4 < x1 ≤ 0.6 0.6 < x1 ≤ 0.8 0.8 < x1 ≤ 1

(B.64)

252

B.4.3

B. Multi-objective robust test functions

RMTP28

f1 (~x) = x1

(B.65)

f2 (~x) = x2

(B.66)

 x1 x1 f3 (~x) = g(~x) × 1− − sin(ζ × 2πx1 ) g(~x) g(~x) (B.67)     r x2 x2 − sin(ζ × 2πx2 ) + H(x3 ) +ω + H(x1 ) 1− g(~x) g(~x) 

r

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

2

e−2x sin(λ × 2π(x + where : H(x) = γ

PN g(~x) = 1 + 10

i=2

xi

N

γ, λ ≥ 1

B.4.4

(B.68)

π )) 4λ

−x

+ 0.5

(B.69)

(B.70)

(B.71)

RMTP29

f1 (~x) = x1

(B.72)

f2 (~x) = x2

(B.73)

f3 (~x) = g(~x) − C(x1 , x2 )

(B.74)

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.75)

B. Multi-objective robust test functions

253

Pn

i=2 (xi )

g(~x) = 1 + 10

n−1

  0      0.25   C(x1 ) = 0.5      0.75     1

B.4.5

(B.76)

0(≤ x1 ≤ 0.2) ∨ (0 ≤ x2 ≤ 0.2) (0.2 < x1 ≤ 0.4) ∨ (0.2 < x2 ≤ 0.4) (0.4 < x1 ≤ 0.6) ∨ (0.4 < x2 ≤ 0.6)

(B.77)

(0.6 < x1 ≤ 0.8) ∨ (0.6 < x2 ≤ 0.8) (0.8 < x1 ≤ 1) ∨ (0.8 < x2 ≤ 1)

RMTP30

f1 (~x) = x1

(B.78)

f2 (~x) = x2

(B.79)

sin(4πx1 ) − 15x1 + 1] 15

(B.80)

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.81)

f3 (~x) = g(~x)[

Pn g(~x) = 1 + 10

B.4.6

i=2 (xi )

n−1

(B.82)

RMTP31 x1

f1 (~x) =

e −1 e−1

(B.83)

f2 (~x) =

ex2 − 1 e−1

(B.84)

f3 (~x) = g(~x)[

sin(4πx1 ) − 15x1 + 1] 15

(B.85)

254

B. Multi-objective robust test functions

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.86)

Pn

i=2 (xi )

g(~x) = 1 + 10

B.4.7

(B.87)

n−1

RMTP32

f1 (~x) = x1

(B.88)

f2 (~x) = x2

(B.89)

f3 (~x) = g(~x)[

sin(4πx1 x2 ) − 15x1 x2 + 1] 15

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.90)

(B.91)

Pn g(~x) = 1 + 10

B.4.8

i=2 (xi )

(B.92)

n−1

RMTP33

f1 (~x) = cos(

πx1 ) 2

f2 (~x) = g(~x)sin(

(B.93)

πx1 ) 2

(B.94)

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

 g(~x) = 1 + 10

ψ = 1/3

Pn

i=2 (xi )

n−1

(B.95)

ψ (B.96)

B. Multi-objective robust test functions

B.4.9

255

RMTP34

f1 (~x) = x1

(B.97)

f2 (~x) = h(x1 ) + g(~x)S(x1 )

(B.98)

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.99)

where : h(x1 ) =

(x1 − 0.6)3 − 0.43 −0.63 − 0.43

 Pn

i=2 (xi )

g(~x) = 1 +

ψ (B.101)

n−1

1 0.2 + x1

S(x1 ) =

(B.100)

(B.102)

ψ = 1/3

B.4.10

RMTP35 x1

f1 (~x) =

e −1 e−1

(B.103)

sin(4πx1 ) − 15x1 + 1] 15

(B.104)

subject to : 0 ≤ xi ≤ 1, i = 1, 2, ..., n

(B.105)

f2 (~x) = g(~x)[

 g(~x) = 1 + 10

ψ = 1/3

Pn

i=2 (xi )

n−1

ψ (B.106)

256

B.5 B.5.1

B. Multi-objective robust test functions

Proposed deceptive test functions RMTP36

f1 (~x) = x1

(B.107)

f2 (~x) = H(x2 ) × g(~x) + S(x1 )

(B.108)

H(x) =

x−0.2 2 x−0.5 2 x−0.8 2 1 − 0.3e−( 0.004 ) − 0.5e−( 0.05 ) − 0.3e−( 0.004 ) + sin(πx) 2

(B.109)

2 g(~x) = ΣN i=3 50xi

(B.110)

S(x) = 1 − xβ

(B.111)

β = 1/2

B.5.2

RMTP37

Same as RMTP36 but with β = 1.

B.5.3

RMTP38

Same as RMTP36 but with β = 2.

B.6 B.6.1

Proposed multi-modal robust test functions RMTP39

f1 (~x) = x1

(B.112)

f2 (~x) = H(x2 ) × {g(~x) + S(x1 )}

(B.113)

B. Multi-objective robust test functions

257

M

 X x−(0.6+0.02i) 2 3 −( x−0.5 )2 −( x−0.02i )2 −( ) 0.04 0.004 0.004 H(x) = − 0.5e − 0.8e − 0.8e 2 i=0

(B.114)

2 g(~x) = ΣN i=3 50xi

(B.115)

S(x) = 1 − xβ

(B.116)

M = 20 β = 1/2

B.6.2

RMTP40

Same as RMTP39 but with β = 1.

B.6.3

RMTP41

Same as RMTP39 but with β = 2.

B.7 B.7.1

Proposed flat robust test functions RMTP42

f1 (~x) = x1

(B.117)

f2 (~x) = H(x2 ) × {g(~x) + S(x1 )}

(B.118)

H(x) =

x−0.05 2 x−0.95 2 1 − 0.2e−( 0.03 ) − 0.2e−( 0.01 ) 2

2 g(~x) = (ΣN i=3 50xi ) + 1

(B.119)

(B.120)

258

B. Multi-objective robust test functions

S(x) = 1 − xβ

β = 1/2

B.7.2

RMTP43

Same as RMTP42 but with β = 1.

B.7.3

RMTP44

Same as RMTP42 but with β = 2.

(B.121)

Appendix C Complete results C.1

Robust Pareto optimal fronts obtained by CRMOPSO, IRMOPSO, and ERMOPSO

259

260

C. Complete results

2

2

2

1

f2

3

1

0

0

0

-1

-1

0

0.5 f1

1

0

0.5 f1

2

2

f2

2 1

1 0

0.5

-1

1

1 0

0

0.5

f1

-1

1

0

CRMOPSO

2 1.5

f2

2 1.5

f2

2

0.5

1

0

1

0.5

1 0.5

0.5

0

1

f1

ERMOPSO

IRMOPSO

1.5

0

0.5

f1

1

1

ERMOPSO 3

0

0.5 f1

IRMOPSO

f2

f2

0

3

-1

f2

1

3

0

RMTP3

1

-1

CRMOPSO

RMTP2

ERMOPSO

IRMOPSO 3

f2

RMTP1

f2

CRMOPSO 3

0

1

0.5

0

f1

f1

CRMOPSO

IRMOPSO

1

0.5

0

f1

ERMOPSO 2

8

8

4

4

f3

f3

f3

RMTP4

1.5

6

6

2

2

0 1

0 1 1

0.5

f2

0

0.5 0

f2

f1

0

IRMOPSO

14

14

12

12

12

f3

14

8

10 8

1 0.5

0.5

1

f1

0

f2

6 0

0

0.5

1

f2

ERMOPSO 16

10

1

f1

16

6 0

0.5

f1

16

f3

f3

CRMOPSO

RMTP5

0 0

1

0.5

0.5 0

1 0.5

10 8

1 0.5

0.5

1

f1

0

f2

6 0

1 0.5

0.5

1

f1

0

f2

Figure C.1: Robust fronts obtained for RMTP1 to RMTP5.

C. Complete results

261

CRMOPSO 0.8

0.8

0.8

0.6

0.6

0.6

f2

f2

1

0.4

0.4

0.4

0.2

0.2

0.2

0

0

1

0

CRMOPSO 0.8

0.8

0.6

0.5 f1

1

0.8

0.4 0.2

0

0

1

0

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0.5

1

0

0.5

0

1

IRMOPSO 0.8

0.8

0.6

0.6

0.6

f2

f2

0.8

f2

1

0.4

0.4

0.4

0.2

0.2

0.2

0.5 f1

0

1

0

CRMOPSO

0.5 f1

0

1

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

1

1

f2

f2

1 0.8

f2

1 0.8

f1

0.5 f1

ERMOPSO

1

0.5

0

IRMOPSO

0.8

0

1

ERMOPSO

1

0

0.5

f1

1

0

0

f1

Gap

1

f2

1 0.8

f2

1 0.8

f2

1

0

0.5 f1

ERMOPSO

0.8

CRMOPSO

RMTP10

0.5 f1

IRMOPSO

f1

RMTP9

0.6

0.2

CRMOPSO

RMTP8

ERMOPSO

0.4

0

1

f1

Obtained less robust solutions

f2

f2

Density in the robust region

0.5

1

0.6

0.4

0

0

IRMOPSO 1

0

0

1

f1

1

0.2

0.5

f2

0.5

0

f1

RMTP7

ERMOPSO

1

f2

RMTP6

IRMOPSO

1

0

0.5

f1

1

0

0

0.5

f1

Figure C.2: Robust fronts obtained for RMTP6 to RMTP10.

1

262

C. Complete results

CRMOPSO

1

1

1

f2 0

f2

1.5

0.5

0.5

0

0.5

1

1.5

0

2

0.5

0

1

0.5

f1

f2

1

f2

f2

1 0.5

1

1.5

-0.5

2

0.5

0

0.5

1.5

1

-0.5

2

CRMOPSO

IRMOPSO

1

1.5

0

2

0

0.5

CRMOPSO

1 f1

1.5

0

2

1

1

0.5

0.5

1

1.5

0

2

0

0.5

1

1.5

0

2

CRMOPSO

IRMOPSO

2

1.5

2

0.5

0.5

1.5

1

f2

1

f2

1

f2

1

1 f1

0.5

ERMOPSO 1.5

0.5

2

f1

1.5

0

0

f1

1.5

0

1.5

0.5

f1

0.5

1 f1

f2

1

f2

1.5

f2

1.5

0

0.5

ERMOPSO

1.5

0

0

IRMOPSO

0.5

2

0.5

0.5

1 f1

1.5

f2

1 f2

1 f2

1.5

0.5

1

ERMOPSO

1.5

0

0.5

f1

1.5

0

0

f1

0.5

2

0

0

0.5

1.5

ERMOPSO

1 0.5

1

f1 1.5

0

0.5

IRMOPSO

f1

RMTP15

0

1.5

-0.5

RMTP14

0

2

1.5

0

RMTP13

1.5

f1

CRMOPSO

RMTP12

ERMOPSO

1.5

f2

RMTP11

IRMOPSO

1.5

0

0

0.5

1 f1

1.5

2

0

0

0.5

1 f1

1.5

2

Figure C.3: Robust fronts obtained for RMTP11 to RMTP15. Note that the dominated (local) front is robust and considered as reference for the performance measures.

C. Complete results

263

CRMOPSO

1

1

1

0

f2

f2

1.5

0.5

0.5

0

0.5

1 f1

1.5

0

2

0.5

0

1 f1

0.5

CRMOPSO

1

1

1

0.5

0.5

1

1.5

0

2

0

1

0.5

CRMOPSO

0

2

0

1

1

f2

1

0

0.5 0

1

1.5

-0.5

2

0

0.5

1.5

1

-0.5

2

0

0.5

1

1.5

f1

f1

CRMOPSO

IRMOPSO

ERMOPSO

1

1

0.5

1 f1

1.5

2

2

f2

f2

1 f2

1.5

0.5

2

0

1.5

0

1.5

0.5

1.5

0

1

f1

0.5

2

ERMOPSO 1.5

0.5

0.5

IRMOPSO

0.5

1.5

f1

1.5

f2

f2

1.5

f1

1.5

0

1 f1

0.5

f1

-0.5

0.5

f2

f2

1.5

0

0

ERMOPSO

1.5

0

RMTP19

0

2

1.5

0.5

RMTP18

1.5

IRMOPSO

f2

RMTP17

ERMOPSO

1.5

f2

RMTP16

IRMOPSO

1.5

0

0.5

0

0.5

1 f1

1.5

2

0

0

0.5

1 f1

1.5

2

Figure C.4: Robust fronts obtained for RMTP16 to RMTP19. Note that the dominated (local) front is robust and considered as reference for the performance measures.

264

C. Complete results

CRMOPSO

1

0.8

0.8

0.8

0.6

0.6

0.6 f2

f2

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

0.5 f1

1

0

CRMOPSO

0

1

1

1

1

0.8

0.8

0.6

0.6

0.4

0.2

0.2

0.2

0

0

1

0

0.5 f1

0

1

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

1

1

f2

f2

0.8

f2

1

0.5 f1

0.5 f1

ERMOPSO

1

0

0

IRMOPSO

1

0

1

f2

f2

0.4

0.5 f1

0.5 f1

0.6

0.4

0

0

ERMOPSO

0.8

CRMOPSO

RMTP22

0.5 f1

IRMOPSO

f2

RMTP21

ERMOPSO

1

f2

RMTP20

IRMOPSO

1

0

0

0.5 f1

1

0

0

0.5 f1

1

Figure C.5: Robust fronts obtained for RMTP20 to RMTP22. Note that the worst front is the most robust and considered as reference for the performance measures.

C. Complete results

265

IRMOPSO

2

2

2

1

-1

f2

3

0

1 0

0.5 f1

0

-1

1

0.5 f1

0

1

3

3

2

2

1

0

0

0

-1

-1

0.5 f1

1

0.5 f1

0

CRMOPSO

1

2

2

0

f2

2 f2

3

1

1 0

0.5 f1

-1

1

1

0.5 f1

1

ERMOPSO

3

0

0

IRMOPSO

3

-1

0.5 f1

1

-1

0

0

ERMOPSO

f2

f2

f2

2

f2

-1

1

IRMOPSO

3

RMTP25

1 0

CRMOPSO

RMTP24

ERMOPSO

3

f2

RMTP23

f2

CRMOPSO 3

1 0

0.5 f1

0

-1

1

0.5 f1

0

1

Figure C.6: Robust fronts obtained for RMTP23 to RMTP25. Note that the worst front is the most robust and considered as reference for the performance measures.

CRMOPSO

RMTP27

ERMOPSO

0.8

0.8

0.8

0.6

0.6

0.6

f2

f2

1

f2

1

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

0.5

1

0

0.5

0

1

0.5

f1

f1

CRMOPSO

IRMOPSO

ERMOPSO

1

1

1

0.8

0.8

0.8

0.6

0.6 f2

f2

0.4

0.4

0.2

0.2

0.2

0

0

0.5 f1

1

1

0.6

0.4

0

0

f1

f2

RMTP26

IRMOPSO

1

0

0.5 f1

1

0

0

0.5 f1

1

Figure C.7: Robust fronts obtained for RMTP26 and RMTP27.

266

C. Complete results

CRMOPSO

4

2

2

0

f3

4

2

f3

4

f3

RMTP28

0 0

-2 0

2

1

1

1

f3

3

f3

f3

3 2

0

0

0

-1 0

-1 0

-1 0

1

0.5

0

1

0.5

f2

f1

CRMOPSO

0.5

1

0.5

0

1.5

1

1

0.5

0.5

0

1.5

1.5

1

0.5

1

0.5

0

1

0.5

0

f1

1

1

f1

0.5

f1

1

0.5

0

1

f2

f1

ERMOPSO 1.5

1

1

f3

1.5

f3 f2

0 0

1

0.5

0.5

0.5

0

1

IRMOPSO

1

1

0.5

f2

f1

1.5

0.5

0.5

ERMOPSO 2

0 0

1

1

f1

f2

f1

f3

f3

f3

1

2

f3

0 0

1

2

CRMOPSO

0 0

0.5

0

0.5

f2

RMTP32

1

IRMOPSO

CRMOPSO

0.5

0.5

0

0.5

f2

f1

1

1

ERMOPSO

1.5

0 0

1

1.5

0.5

0.5

f2

0.5

f2

0 0

1

f1

f3

1 0.5

RMTP31

0.5

0

f3

f3

1.5

0 0

1

IRMOPSO

2

RMTP30

ERMOPSO

3

f2

f1

1

1

f2

IRMOPSO

2

0.5

0.5

0.5

f1

1

1

f2

CRMOPSO

0

-2 0

0.5

0.5

f1

1

1

f2

0 0

-2 0

0.5

0.5

RMTP29

ERMOPSO

IRMOPSO

0 0

0.5

0.5

f2

1

0.5

0

f1

1

0 0

0.5

f2

1

0.5

0

1

f1

Figure C.8: Robust fronts obtained for RMTP28 and RMTP32.

C. Complete results

267

CRMOPSO

1

0.8

0.8

0.8

0.6

0.6

0.6

f2

f2

1

f2

RMTP33

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

0.5 f1

1

0

CRMOPSO

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.5 f1

0

1

0

0.8

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

1

0

0.5 f1

0

1

0.8

0.8

0.6

0.6

0.6

f2

f2 0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

CRMOPSO

0.5 f1

0

1

0.8

0.8

0.6

0.6

f2

f2 0.4

0.4

0.4

0.2

0.2

0.2

0

0

1

0

CRMOPSO

0.5 f1

0

1

1

0.5 f1

1

0.6

0.6

f2

0.6

f2

1 0.8

f2

1 0.8

0.4

0.4

0.4

0.2

0.2

0.2

0

0

1

0.5 f1

ERMOPSO

1

0.5 f1

0

IRMOPSO

0.8

0

1

f2

0.8 0.6

0.5 f1

0.5 f1

ERMOPSO 1

0

0

IRMOPSO 1

1

1

f2

0.8

1

0.5 f1

ERMOPSO 1

0.5 f1

0

IRMOPSO 1

0

0

f2

0.8

0.6

f2

0.8

f2

1

0.5 f1

1

ERMOPSO

1

CRMOPSO

RMTP38

0

1

IRMOPSO

1

RMTP37

0.5 f1

1

0

0.5 f1

f2

0.8

0.6

f2

0.8

0.6

f2

0.8

0

0

ERMOPSO 1

CRMOPSO

RMTP36

0

1

1

0

RMTP35

0.5 f1

IRMOPSO

1

RMTP34

IRMOPSO

ERMOPSO

1

0

0.5 f1

1

0

0

0.5 f1

1

Figure C.9: Robust fronts obtained for RMTP33 and RMTP38. Note that in RMTP36, RMTP37, and RMTP38, the worst front is the most robust and considered as the reference for the performance measures.

268

C. Complete results

CRMOPSO

1

0.8

0.8

0.8

0.6

0.6

0.6

f2

f2

1

f2

RMTP39

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

0.5 f1

1

0

CRMOPSO

0.8

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

1

0

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

1

0

0.5 f1

0

1

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

0.5 f1

0

1

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

0.5 f1

0

1

0.6

0.6

0.6

0.5 f1

1

f2

f2

1 0.8

f2

1 0.8

0.4

0.4

0.4

0.2

0.2

0.2

0

0

1

1

ERMOPSO

1

0.5 f1

0

IRMOPSO

0.8

0

0.5 f1

f2

f2

0.8

f2

1

1

1

ERMOPSO

1

0.5 f1

0

IRMOPSO

1

0

0.5 f1

f2

f2

0.8

f2

1

1

1

ERMOPSO

1

0.5 f1

0

IRMOPSO

1

0

0.5 f1

f2

f2

1 0.8

f2

1 0.8

0.5 f1

0

ERMOPSO

1

CRMOPSO

RMTP44

0

1

IRMOPSO

CRMOPSO

RMTP43

0.5 f1

0.8

0

1

f2

0.8

0.6 f2

0.8

0.5 f1

0.5 f1

ERMOPSO 1

0

0

IRMOPSO

CRMOPSO

RMTP42

0

1

1

CRMOPSO

RMTP41

0.5 f1

1

f2

RMTP40

ERMOPSO

IRMOPSO

1

0

0.5 f1

1

0

0

0.5 f1

1

Figure C.10: Robust fronts obtained for RMTP39 and RMTP44.