Abstract
An enhanced nature-inspired metaheuristic optimization algorithm, called the modified firefly algorithm (MFA) is proposed for multidimensional structural design optimization. The MFA incorporates metaheuristic components, namely logistic and Gauss/mouse chaotic maps, adaptive inertia weight, and Lévy flight with a conventional firefly algorithm (FA) to improve its optimization capability. The proposed MFA has several advantages over its traditional FA counterpart. Logistic chaotic maps provide a diverse initial population. Gauss/mouse maps allow the tuning of the FA attractiveness parameter. The adaptive inertia weight controls the local exploitation and the global exploration of the search process. Lévy flight is used in the exploitation of the MFA. The proposed MFA was evaluated by comparing its performance in solving a series of benchmark functions with those of the FA and other well-known optimization algorithms. The efficacy of the MFA was then proven by its solutions to three multidimensional structural design optimization problems; MFA yielded the best solutions among the observed algorithms. Experimental results revealed that the proposed MFA is more efficient and effective than the compared algorithms. Therefore, the MFA serves as an alternative algorithm for solving multidimensional structural design optimization problems.
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References
Adekanmbi O, Green P (2015) Conceptual comparison of population based metaheuristics for engineering problems. Sci World J 2015:9
Akhtar S, Tai K, Ray T (2002) A socio-behavioural simulation model for engineering design optimization. Eng Optim 34(4):341–354
Alberdi R, Murren P, Khandelwal K (2015) Connection topology optimization of steel moment frames using metaheuristic algorithms. Eng Struct 100:276–292
Baykasoglu A (2012) Design optimization with chaos embedded great deluge algorithm. Appl Soft Comput 12(3):1055–1067
Baykasoğlu A, Akpinar Ş (2015) Weighted superposition attraction (WSA): a swarm intelligence algorithm for optimization problems – part 2: constrained optimization. Appl Soft Comput 37:396–415
Baykasoğlu A, Ozsoydan FB (2015) Adaptive firefly algorithm with chaos for mechanical design optimization problems. Appl Soft Comput 36:152–164
Benfratello S, Palizzolo L, Tabbuso P (2015) Optimization of structures with unrestricted dynamic shakedown constraints. Struct Multidiscip Optim 52(3):431–445
Bernardino HS, Barbosa IJC, Lemonge A (2007) A hybrid genetic algorithm for constrained optimization problems in mechanical engineering. IEEE Congress on Evolutionary Computation, IEEE, Piscataway, p 646–653
Cheng M-Y, Prayogo D (2014) Symbiotic organisms search: a new metaheuristic optimization algorithm. Comput Struct 139:98–112
Chickermane H, Gea HC (1996) Structural optimization using a new local approximation method. Int J Numer Methods Eng 39(5):829–846
Chou J-S, Ngo N-T (2016) Time series analytics using sliding window metaheuristic optimization-based machine learning system for identifying building energy consumption patterns. Appl Energy 177:751–770
Chou J-S, Ngo N-T, Pham A-D (2016) Shear strength prediction in reinforced concrete deep beams using nature-inspired metaheuristic support vector regression. J Comput Civ Eng 30(1):04015002
Civicioglu P (2013) Backtracking search optimization algorithm for numerical optimization problems. Appl Math Comput 219(15):8121–8144
Coelho L d S, Mariani VC (2013) Improved firefly algorithm approach applied to chiller loading for energy conservation. Energ Buildings 59:273–278
Črepinšek M, Liu S-H, Mernik M (2013) Exploration and exploitation in evolutionary algorithms: a survey. ACM Comput Surv 45(3):1–33
Dos Santos Coelho L, Coelho AAR (2009) Model-free adaptive control optimization using a chaotic particle swarm approach. Chaos, Solitons Fractals 41(4):2001–2009
Dugré A, Vadean A, Chaussée J (2016) Challenges of using topology optimization for the design of pressurized stiffened panels. Struct Multidiscip Optim 53(2):303–320
Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, IEEE, Piscataway, p 39–43
Fesanghary M, Mahdavi M, Minary-Jolandan M, Alizadeh Y (2008) Hybridizing harmony search algorithm with sequential quadratic programming for engineering optimization problems. Comput Methods Appl Mech Eng 197(33–40):3080–3091
Fister I, Fister I Jr, Yang X-S, Brest J (2013a) A comprehensive review of firefly algorithms. Swarm Evol Comput 13:34–46
Fister I, Yang X-S, Brest J, Fister I Jr (2013b) Modified firefly algorithm using quaternion representation. Expert Syst Appl 40(18):7220–7230
Fister I Jr, Perc M, Kamal SM, Fister I (2015) A review of chaos-based firefly algorithms: perspectives and research challenges. Appl Math Comput 252:155–165
Fleury C, Braibant V (1986) Structural optimization: a new dual method using mixed variables. Int J Numer Methods Eng 23(3):409–428
Friedman M (1937) The use of ranks to avoid the assumption of normality implicit in the analysis of variance. J Am Stat Assoc 32(200):675–701
Friedman M (1940) A comparison of alternative tests of significance for the problem of m rankings. Ann Math Stat 11(1):86–92
Gandomi AH, Yang X-S, Alavi AH (2011) Mixed variable structural optimization using firefly algorithm. Comput Struct 89(23–24):2325–2336
Gandomi A, Yang X-S, Alavi A (2013a) Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng Comput 29(1):17–35
Gandomi AH, Yang XS, Talatahari S, Alavi AH (2013b) Firefly algorithm with chaos. Commun Nonlinear Sci Numer Simul 18(1):89–98
Geisel T, Nierwetberg J, Zacherl A (1985) Accelerated diffusion in Josephson junctions and related chaotic systems. Phys Rev Lett 54(7):616–619
Gharooni-fard G, Moein-darbari F, Deldari H, Morvaridi A (2010) Scheduling of scientific workflows using a chaos-genetic algorithm. Procedia Comput Sci 1(1):1445–1454
Gold S, Krishnamurty S (1997) Trade-offs in robust engineering design. Proceeding of the ASME design engineering technical conferences, ASME, New York
Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley Longman Publishing Co., Inc., Boston
Gomes H (2012) A firefly metaheuristic structural size and shape optimisation with natural frequency constraints. Int J Metaheuristics 2(1):38–85
Guirguis D, Hamza K, Aly M, Hegazi H, Saitou K (2015) Multi-objective topology optimization of multi-component continuum structures via a Kriging-interpolated level set approach. Struct Multidiscip Optim 51(3):733–748
Haklı H, Uğuz H (2014) A novel particle swarm optimization algorithm with Levy flight. Appl Soft Comput 23:333–345
He S, Prempain E, Wu QH (2004) An improved particle swarm optimizer for mechanical design optimization problems. Eng Optim 36(5):585-605
Hedar A-R, Fukushima M (2006) Derivative-free filter simulated annealing method for constrained continuous global optimization. J Glob Optim 35(4):521–549
Holden AV (1986) Chaos. Manchester University Press, Manchester
Hong W-C, Dong Y, Chen L-Y, Wei S-Y (2011) SVR with hybrid chaotic genetic algorithms for tourism demand forecasting. Appl Soft Comput 11(2):1881–1890
Iman RL, Davenport JM (1980) Approximations of the critical region of the fbietkan statistic. Commun Stat -Theory Methods 9(6):571–595
Jaberipour M, Khorram E (2010) Two improved harmony search algorithms for solving engineering optimization problems. Commun Nonlinear Sci Numer Simul 15(11):3316–3331
Jamil M, Yang X-S (2013) A literature survey of benchmark functions for global optimization problems. Int J Math Model Numer Optim 4(2):150–194
Karaboga D, Akay B (2009) A comparative study of artificial bee colony algorithm. Appl Math Comput 214(1):108–132
Karaboga D, Basturk B (2007) A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. J Glob Optim 39(3):459–471
Kazem A, Sharifi E, Hussain FK, Saberi M, Hussain OK (2013) Support vector regression with chaos-based firefly algorithm for stock market price forecasting. Appl Soft Comput 13(2):947–958
Kripka M, Medeiros GF, Lemonge ACC (2015) Use of optimization for automatic grouping of beam cross-section dimensions in reinforced concrete building structures. Eng Struct 99:311–318
Lamberti L, Pappalettere C (2011) Metaheuristic design optimization of skeletal structures: a review. Comput Technol Rev 4:1–32
Lee KS, Geem ZW (2005) A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice. Comput Methods Appl Mech Eng 194(36–38):3902–3933
Lemonge ACC, Barbosa HJC (2004) An adaptive penalty scheme for genetic algorithms in structural optimization. Int J Numer Methods Eng 59(5):703–736
Liu B, Wang L, Jin Y-H, Tang F, Huang D-X (2005) Improved particle swarm optimization combined with chaos. Chaos, Solitons Fractals 25(5):1261–1271
Luo Q, Tong L (2015) Structural topology optimization for maximum linear buckling loads by using a moving iso-surface threshold method. Struct Multidiscip Optim 52(1):71–90
Mahdavi M, Fesanghary M, Damangir E (2007) An improved harmony search algorithm for solving optimization problems. Appl Math Comput 188(2):1567–1579
May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261(5560):459–467
Meng X-B, Gao XZ, Liu Y, Zhang H (2015) A novel bat algorithm with habitat selection and Doppler effect in echoes for optimization. Expert Syst Appl 42(17–18):6350–6364
Nickabadi A, Ebadzadeh MM, Safabakhsh R (2011) A novel particle swarm optimization algorithm with adaptive inertia weight. Appl Soft Comput 11(4):3658–3670
Pal SK, Rai CS, Singh AP (2012) Comparative study of firefly algorithm and particle swarm optimization for noisy Non-linear optimization problems. Int J Intell Syst Appl 4(10):50–57
Pavlyukevich I (2007) Lévy flights, non-local search and simulated annealing. J Comput Phys 226(2):1830–1844
Peri D, Tinti F (2012) A multistart gradient-based algorithm with surrogate model for global optimization. Commun Appl Ind Math 3(1):1-22
Rao SS (1996) Engineering optimization: theory and practice. John Wiley & Sons, Chichester
Rojas-Labanda S, Stolpe M (2015) Benchmarking optimization solvers for structural topology optimization. Struct Multidiscip Optim 52(3):527–547
Roque CMC, Martins PALS (2015) Differential evolution for optimization of functionally graded beams. Compos Struct 133:1191–1197
Saka MP, Dogan E (2012) Recent developments in metaheuristic algorithms: a review. Comput Technol Rev 5:31–78
Saka MP, Geem ZW (2013) Mathematical and metaheuristic applications in design optimization of steel frame structures: an extensive review. Math Probl Eng 2013:33
Saka MP, Dogan E, Aydogdu I (2013) Review and analysis of swarm-intelligence based algorithms. Swarm intelligence and bio-inspired computation. Elsevier, London, pp 25–47
Sergeyev YD, Kvasov DE (2015) A deterministic global optimization using smooth diagonal auxiliary functions. Commun Nonlinear Sci Numer Simul 21(1–3):99–111
Shi Y, Eberhart R (1998) A modified particle swarm optimizer. The IEEE International Conference on Evolutionary Computation, Piscataway, p 69–73
Solomon TH, Weeks ER, Swinney HL (1994) Chaotic advection in a two-dimensional flow: Lévy flights and anomalous diffusion. Physica D: Nonlinear Phenom 76(1–3):70–84
Storn R, Price K (1997) Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359
Tilahun SL, Ong HC (2012) Modified firefly algorithm. J Appl Math 2012:12
Triguero FH (2016) Statistical inference in computational intelligence and data mining. http://sci2s.ugr.es/sicidm
Wang GG (2003) Adaptive response surface method using inherited latin hypercube design points. J Mech Des 125(2):210–220
Wang Y, Cai Z, Zhang Q (2011) Differential evolution with composite trial vector generation strategies and control parameters. IEEE Trans Evol Comput 15(1):55–66
Wang Y, Li H-X, Huang T, Li L (2014) Differential evolution based on covariance matrix learning and bimodal distribution parameter setting. Appl Soft Comput 18:232–247
Weeks E, Solomon TH, Urbach J, Swinney H (1995) Observation of anomalous diffusion and Lévy flights. In: Shlesinger M, Zaslavsky G, Frisch U (eds) Lévy flights and related topics in physics. Springer Berlin Heidelberg, p 51–71
Yang X-S (2008) Firefly algorithm. Luniver Press, Bristol
Yang X-S (2010a) Firefly algorithm, Lévy flights and global optimization. In: Bramer M, Ellis R, Petridis M (eds) Research and development in intelligent systems XXVI. Springer, London, pp 209–218
Yang X-S (2010b) A new metaheuristic bat-inspired algorithm. In: González J, Pelta D, Cruz C, Terrazas G, Krasnogor N (eds) Nature inspired cooperative strategies for optimization (NICSO 2010). Springer, Berlin, pp 65–74
Yang X-S (2014a) Analysis of algorithms. Nature-inspired optimization algorithms. Elsevier, Oxford, pp 23–44
Yang X-S (2014b) Chapter 8 - firefly algorithms. In: Yang X-S (ed) Nature-inspired optimization algorithms. Elsevier, Oxford, pp 111–127
Yang X-S, Deb S (2009) Cuckoo search via Lévy flights. World Congress on Nature & Biologically Inspired Computing, NaBIC 2009. p 210–214
Yang D, Liu Z, Zhou J (2014) Chaos optimization algorithms based on chaotic maps with different probability distribution and search speed for global optimization. Commun Nonlinear Sci Numer Simul 19(4):1229–1246
Zhang M, Luo W, Wang X (2008) Differential evolution with dynamic stochastic selection for constrained optimization. Inf Sci 178(15):3043–3074
Zhang J, Liang C, Huang Y, Wu J, Yang S (2009) An effective multiagent evolutionary algorithm integrating a novel roulette inversion operator for engineering optimization. Appl Math Comput 211(2):392–416
Zhou G, Ma Z-D, Cheng A, Li G, Huang J (2015) Design optimization of a runflat structure based on multi-objective genetic algorithm. Struct Multidiscip Optim 51(6):1363–1371
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Chou, JS., Ngo, NT. Modified firefly algorithm for multidimensional optimization in structural design problems. Struct Multidisc Optim 55, 2013–2028 (2017). https://doi.org/10.1007/s00158-016-1624-x
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DOI: https://doi.org/10.1007/s00158-016-1624-x