本研究結合多目標修正螢火蟲演算法(Multi-Objective Modified Firefly Algorithm)與作為偏好策略的拆解式方法，提出偏好多目標修正螢火蟲演算法(Preference based Multi-Objective Modified Firefly Algorithm)，並將其應用於機構最佳化設計問題。本研究首先以四個多目標測試函數為範例，將偏好多目標修正螢火蟲演算法與包含多目標修正螢火蟲演算法(Multi-Objective Modified Firefly Algorithm)、多目標粒子群演算法（Multi- Objective Particle Swarm Optimization）、拆解式多目標演化演算法(Multi-Objective Evolutionary Algorithm Based on Decomposition)、多目標蟻獅演算法（Multi-Objective Ant Lion Optimization）與非支配排序基因演算法III (Nondominated Sorting Genetic Algorithm III)之五個多目標演算法做比較，測試偏好多目標修正螢火蟲演算法在多目標最佳化的收斂性與分布性。再以兩種路徑產生機構Klann Linkage與Jansen Mechanism的最佳化範例證明偏好多目標修正螢火蟲演算法於機構最佳化的應用優於其他演算法。最佳化機構之目標函數分別為軌跡誤差、機構面積占比與輸入扭矩波動值，而軌跡誤差為優先滿足之目標函數。Klann Linkage與Jansen Mechanism採用軌跡誤差搭配控制機構大小與耗能的目標函數，呈現本研究之偏好多目標演算法在機構最佳化的能力，最佳化之機構不但有與理想軌跡足夠相似度的軌跡誤差數值且能夠節省機構運作佔用的空間與耗能。

This study combines the multi-objective modified firefly algorithm with the decomposition-based method as a preference strategy. The proposed preference-based multi-objective modified firefly algorithm (Pref. MOMFA) is applied to design path generating mechanisms. This study first takes four multi-objective test functions as examples. There are five algorithms used in the study for the purpose of comparison, the multi-objective modified firefly algorithm (MOMFA), the multi-objective particle swarm algorithm (MOPSO), multi-objective evolutionary algorithm based on decomposition (MOEA/D), multi-objective ant lion optimization (MOALO) and non-dominated sorting genetic algorithm III (NSGAIII). To compare the five multi-objective algorithms and test the convergence and distribution of preference-based MOMFA, optimal designs of path generating mechanisms are used to prove that preference-based MOMFA is the best in the application of mechanism optimization among six multi-objective algorithms. The tracking error, linkage size, and input torque fluctuation value are used to optimize the gait, size of planar linkages, and energy consumption of mechanism, respectively. The results show that the preference-based MOMFA in this study can synthesize path generating mechanisms with particular requests of objective values successfully.

[1] A. Zhou, B.-Y. Qu, H. Li, S.-Z. Zhao, P. N. Suganthan, and Q. Zhang, "Multiobjective evolutionary algorithms: A survey of the state of the art," Swarm and Evolutionary Computation, vol. 1, no. 1, pp. 32-49, 2011.
[2] N. Srinivas and K. Deb, "Muiltiobjective optimization using nondominated sorting in genetic algorithms," Evolutionary Computation, vol. 2, no. 3, pp. 221-248, 1994.
[3] C. A. C. Coello, "A comprehensive survey of evolutionary-based multiobjective optimization techniques," Knowledge and Information Systems, vol. 1, no. 3, pp. 269-308, 1999.
[4] V. Pareto, G. H. Bousquet, and G. Busino, Political economy course(Cours d'économie politique). Droz, 1964.
[5] K. Miettinen, Nonlinear multiobjective optimization. Springer Science & Business Media, 2012.
[6] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, 2002.
[7] K. Deb and H. Jain, "An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints," IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, 2013.
[8] K. Deb and T. Goel, "Controlled elitist non-dominated sorting genetic algorithms for better convergence," in International Conference on Evolutionary Multi-Criterion Optimization, 2001: Springer, pp. 67-81.
[9] J. Kennedy and R. Eberhart, "Particle swarm optimization," in Proceedings of ICNN'95-International Conference on Neural Networks, 1995, vol. 4: IEEE, pp. 1942-1948.
[10] X. Li, "A non-dominated sorting particle swarm optimizer for multiobjective optimization," in Genetic and Evolutionary Computation Conference, 2003: Springer, pp. 37-48.
[11] Q. Zhang and H. Li, "MOEA/D: A multiobjective evolutionary algorithm based on decomposition," IEEE Transactions on Evolutionary Computation, vol. 11, no. 6, pp. 712-731, 2007.
[12] Q. Zhang, W. Liu, and H. Li, "The performance of a new version of MOEA/D on CEC09 unconstrained MOP test instances," in 2009 IEEE Congress on Evolutionary Computation, 2009: IEEE, pp. 203-208.
[13] L. Rachmawati and D. Srinivasan, "Preference incorporation in multi-objective evolutionary algorithms: A survey," in 2006 IEEE International Conference on Evolutionary Computation, 2006: IEEE, pp. 962-968.
[14] C. M. Fonseca and P. J. Fleming, "Genetic algorithms for multiobjective optimization: formulation discussion and generalization," in ICGA Journal (The Journal of the Computer Games Community), 1993, vol. 93: Citeseer, pp. 416-423.
[15] M. Sakawa and K. Kato, "An interactive fuzzy satisficing method for general multiobjective 0–1 programming problems through genetic algorithms with double strings based on a reference solution," Fuzzy Sets and Systems, vol. 125, no. 3, pp. 289-300, 2002.
[16] J. Branke and K. Deb, "Integrating user preferences into evolutionary multi-objective optimization," in Knowledge Incorporation in Evolutionary Computation: Springer, 2005, pp. 461-477.
[17] K. Deb and J. Sundar, "Reference point based multi-objective optimization using evolutionary algorithms," in Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, 2006, pp. 635-642.
[18] K. Deb and A. Kumar, "Interactive evolutionary multi-objective optimization and decision-making using reference direction method," in Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation, 2007, pp. 781-788.
[19] K. Deb and S. Chaudhuri, "I-EMO: An interactive evolutionary multi-objective optimization tool," in International Conference on Pattern Recognition and Machine Intelligence, 2005: Springer, pp. 690-695.
[20] D. A. Van Veldhuizen, Multiobjective evolutionary algorithms: classifications, analyses, and new innovations. 1999.
[21] E. Zitzler and L. Thiele, "Multiobjective optimization using evolutionary algorithms—a comparative case study," in International Conference on Parallel Problem Solving from Nature, 1998: Springer, pp. 292-301.
[22] E. Zitzler and S. Künzli, "Indicator-based selection in multiobjective search," in International Conference on Parallel Problem Solving from Nature, 2004: Springer, pp. 832-842.
[23] J. Bader and E. Zitzler, "HypE: An algorithm for fast hypervolume-based many-objective optimization," Evolutionary Computation, vol. 19, no. 1, pp. 45-76, 2011.
[24] A. Elhossini, S. Areibi, and R. Dony, "Strength Pareto particle swarm optimization and hybrid EA-PSO for multi-objective optimization," Evolutionary Computation, vol. 18, no. 1, pp. 127-156, 2010.
[25] D. A. Van Veldhuizen and G. B. Lamont, "Multiobjective evolutionary algorithm research: A history and analysis," Citeseer, 1998.
[26] D. Yang, L. Jiao, and M. Gong, "Adaptive multi‐objective optimization based on nondominated solutions," Computational Intelligence, vol. 25, no. 2, pp. 84-108, 2009.
[27] A. Lara, G. Sanchez, C. A. C. Coello, and O. Schutze, "HCS: A new local search strategy for memetic multiobjective evolutionary algorithms," IEEE Transactions on Evolutionary Computation, vol. 14, no. 1, pp. 112-132, 2009.
[28] H. Ishibuchi and T. Murata, "A multi-objective genetic local search algorithm and its application to flowshop scheduling," IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), vol. 28, no. 3, pp. 392-403, 1998.
[29] S. F. Adra, T. J. Dodd, I. A. Griffin, and P. J. Fleming, "Convergence acceleration operator for multiobjective optimization," IEEE Transactions on Evolutionary Computation, vol. 13, no. 4, pp. 825-847, 2009.
[30] Y. Wang, Z. Cai, G. Guo, and Y. Zhou, "Multiobjective optimization and hybrid evolutionary algorithm to solve constrained optimization problems," IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 37, no. 3, pp. 560-575, 2007.
[31] H. Wang, M. Olhofer, and Y. Jin, "A mini-review on preference modeling and articulation in multi-objective optimization: current status and challenges," Complex & Intelligent Systems, vol. 3, no. 4, pp. 233-245, 2017.
[32] K.-W. Jee, D. L. McShan, and B. A. Fraass, "Lexicographic ordering: intuitive multicriteria optimization for IMRT," Physics in Medicine & Biology, vol. 52, no. 7, p. 1845, 2007.
[33] M. J. Rentmeesters, W. K. Tsai, and K.-J. Lin, "A theory of lexicographic multi-criteria optimization," in Proceedings of ICECCS'96: 2nd IEEE International Conference on Engineering of Complex Computer Systems (held jointly with 6th CSESAW and 4th IEEE RTAW), 1996: IEEE, pp. 76-79.
[34] R. Cheng, Y. Jin, M. Olhofer, and B. Sendhoff, "A reference vector guided evolutionary algorithm for many-objective optimization," IEEE Transactions on Evolutionary Computation, vol. 20, no. 5, pp. 773-791, 2016.
[35] H.-L. Liu, F.-Q. Gu, and Y.-M. Cheung, "T-MOEA/D: MOEA/D with objective transform in multi-objective problems," in 2010 International Conference of Information Science and Management Engineering, 2010, vol. 2: IEEE, pp. 282-285.
[36] H. Sato, "Inverted PBI in MOEA/D and its impact on the search performance on multi and many-objective optimization," in Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation, 2014, pp. 645-652.
[37] A. López Jaimes and C. A. Coello Coello, "Study of preference relations in many-objective optimization," in Proceedings of the 11th Annual Conference on Genetic and Evolutionary Computation, 2009, pp. 611-618.
[38] G.-Z. Fu, Y.-F. Li, Y. Tao, and H.-Z. Huang, "An interactive preference-based evolutionary algorithm for multi-criteria satisficing optimization," Journal of Intelligent & Fuzzy Systems, vol. 34, no. 4, pp. 2503-2511, 2018.
[39] G. Jeantet and O. Spanjaard, "Computing rank dependent utility in graphical models for sequential decision problems," Artificial Intelligence, vol. 175, no. 7-8, pp. 1366-1389, 2011.
[40] L. R. Pedro and R. H. Takahashi, "Modeling decision-maker preferences through utility function level sets," in International Conference on Evolutionary Multi-Criterion Optimization, 2011: Springer, pp. 550-563.
[41] J.-P. Brans, P. Vincke, and B. Mareschal, "How to select and how to rank projects: The PROMETHEE method," European Journal of Operational Research, vol. 24, no. 2, pp. 228-238, 1986.
[42] J.-P. Brans and B. Mareschal, "The PROMCALC & GAIA decision support system for multicriteria decision aid," Decision Support Systems, vol. 12, no. 4-5, pp. 297-310, 1994.
[43] K. Deb and S. Tiwari, "Multi-objective optimization of a leg mechanism using genetic algorithms," Engineering Optimization, vol. 37, no. 4, pp. 325-350, 2005.
[44] M. Khorshidi, M. Soheilypour, M. Peyro, A. Atai, and M. S. Panahi, "Optimal design of four-bar mechanisms using a hybrid multi-objective GA with adaptive local search," Mechanism and Machine Theory, vol. 46, no. 10, pp. 1453-1465, 2011.
[45] M. Russo, S. Herrero, O. Altuzarra, and M. Ceccarelli, "Kinematic analysis and multi-objective optimization of a 3-UPR parallel mechanism for a robotic leg," Mechanism and Machine Theory, vol. 120, pp. 192-202, 2018.
[46] D. Fedorov and L. Birglen, "Geometric optimization of a self-adaptive robotic leg," Transactions of the Canadian Society for Mechanical Engineering, vol. 42, no. 1, pp. 49-60, 2018.
[47] 陳妍綺, 多目標修正螢火蟲演算法於路徑產生機構之最佳化尺寸合成, 國立成功大學機械工程學系碩士學位論文, 2019.
[48] J. Zheng, G. Yu, Q. Zhu, X. Li, and J. Zou, "On decomposition methods in interactive user-preference based optimization," Applied Soft Computing, vol. 52, pp. 952-973, 2017.
[49] 王秋閔, 修正螢火蟲演算法於路徑產生機構之最佳化尺寸合成, 國立成功大學機械工程學系碩士學位論文, 2018.
[50] X.-S. Yang, Nature-inspired metaheuristic algorithms. Luniver press, 2010.
[51] X.-S. Yang and S. Deb, "Cuckoo search via Lévy flights," in 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC), 2009: IEEE, pp. 210-214.
[52] X.-S. Yang, "Firefly algorithms for multimodal optimization," in International Symposium on Stochastic Algorithms, 2009: Springer, pp. 169-178.
[53] M. Laumanns, L. Thiele, K. Deb, and E. Zitzler, "Combining convergence and diversity in evolutionary multiobjective optimization," Evolutionary Computation, vol. 10, no. 3, pp. 263-282, 2002.
[54] M. Laumanns, L. Thiele, and E. Zitzler, "An adaptive scheme to generate the Pareto front based on the epsilon-constraint method," in Dagstuhl Seminar Proceedings. 04461 - Practical Approaches to Multi-Objective Optimization(Schloss Dagstuhl - Leibniz-Zentrum für Informatik), Germany, Europe, 2005.
[55] V. Pareto, Political economy course(Cours d'économie politique). Librairie Droz, 1964.
[56] A. P. Wierzbicki, "The use of reference objectives in multiobjective optimization," in Multiple Criteria Decision Making Theory and Application: Springer, 1980, pp. 468-486.
[57] K. Deb, "Multi-objective genetic algorithms: Problem difficulties and construction of test problems," Evolutionary Computation, vol. 7, no. 3, pp. 205-230, 1999.
[58] K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, "Scalable test problems for evolutionary multiobjective optimization," in Evolutionary Multiobjective Optimization: Springer, 2005, pp. 105-145.
[59] C. C. Coello and M. S. Lechuga, "MOPSO: A proposal for multiple objective particle swarm optimization," in Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No. 02TH8600), 2002, vol. 2: IEEE, pp. 1051-1056.
[60] S. Mirjalili, P. Jangir, and S. Saremi, "Multi-objective ant lion optimizer: a multi-objective optimization algorithm for solving engineering problems," Applied Intelligence, vol. 46, no. 1, pp. 79-95, 2017.
[61] P. A. Bosman and D. Thierens, "The balance between proximity and diversity in multiobjective evolutionary algorithms," IEEE Transactions on Evolutionary Computation, vol. 7, no. 2, pp. 174-188, 2003.
[62] L. While, L. Bradstreet, and L. Barone, "A fast way of calculating exact hypervolumes," IEEE Transactions on Evolutionary Computation, vol. 16, no. 1, pp. 86-95, 2011.
[63] W. Wang and M. Sebag, "Hypervolume indicator and dominance reward based multi-objective Monte-Carlo Tree Search," Machine Learning, vol. 92, no. 2-3, pp. 403-429, 2013.
[64] 黃盈嘉, 創新八連桿足型爬梯機構設計與軌跡最佳化之研究, 國立成功大學機械工程學系碩士學位論文, 2017.
[65] 林孟弦, 模仿人類爬梯步態之創新八連桿足型機構最佳設計與多足爬梯機器人設計之研究, 國立成功大學機械工程學系碩士學位論文, 2016.
[66] J. C. Klann, "Walking device," ed: Google Patents, 2001.
[67] J. C. Klann, "Walking device," ed: Google Patents, 2002.
[68] S. Nansai, N. Rojas, M. R. Elara, and R. Sosa, "Exploration of adaptive gait patterns with a reconfigurable linkage mechanism," in 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2013: IEEE, pp. 4661-4668.