Abstract
Whale optimization algorithm (WOA) is a recently developed swarm intelligence-based algorithm which is inspired from the social behavior of humpback whale. This algorithm mimics the bubble-net hunting strategy of whales and has been applied to optimization problems. But the algorithm suffers from the problem of poor exploration and local optima stagnation. In this paper, three different modified algorithms of WOA have been proposed to improve its explorative ability. The modified versions are based on the concepts of opposition-based learning, exponentially decreasing parameters and elimination or re-initialization of worst particles. These properties have been added to improve the explorative properties of WOA by maintaining diversity among the search agents. The proposed algorithms have been tested on CEC2005 benchmark problems for variable population and dimension sizes. Statistical testing and scalability testing of the best algorithm have been carried out to prove its significance over other algorithms such as with well-known algorithms such as bat algorithm, bat flower pollinator, differential evolution, firefly algorithm, flower pollination algorithm. It has been found from the experimental results that the performance of all the proposed versions is better than the original WOA. Here, opposition- and exponential-based WOA is the best among all the proposed variants. Statistical testing and convergence profiles further validate the results.
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Coello, C.A.C.: Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput. Methods Appl. Mech. Eng. 191(11), 1245–1287 (2002)
Marler, R.T.; Arora, J.S.: Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim. 26(6), 369–395 (2004)
Saremi, S.; Mirjalili, S.; Lewis, A.: Grasshopper optimisation algorithm: theory and application. Adv. Eng. Softw. 105, 30–47 (2017)
Spall, J.C.: Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control, vol. 65. Wiley, London (2005)
Dasgupta, D.; Michalewicz, Z.: Evolutionary algorithms—an overview. In: Evolutionary Algorithms in Engineering Applications, pp. 3–28. Springer, Berlin (1997)
Kaur, K.; Singh, U.; Salgotra, R.: An enhanced moth flame optimization. Neural Comput. Appl. 2018, 1–35 (2018)
Holland, J.H.: Genetic algorithms. Sci. Am. 267, 66–72 (1992)
Storn, R.; Price, K.: Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11, 341–359 (1997)
Simon, D.: Biogeography-based optimization. IEEE Trans. Evol. Comput. 12(6), 702–713 (2008)
Mirjalili, S.: The ant lion optimizer. Adv. Eng. Softw. 83, 80–98 (2015)
Mirjalili, S.: Moth-flame optimization algorithm: a novel nature-inspired heuristic paradigm. Knowl. Based Syst. 89, 228–249 (2015)
Kennedy, J.; Eberhart, R.: Particle swarm optimization. In: Proceedings, IEEE International Conference on Neural Networks, 1995, pp. 1942–1948 (1995)
Dorigo, M.; Birattari, M.; Stutzle, T.: Ant colony optimization. IEEE Comput. Intell. Mag. 1(4), 28–39 (2006)
Yang, X.S.; Deb, S.: Cuckoo search via Lévy flights. In: NaBIC 2009. World Congress on Nature and Biologically Inspired Computing, pp. 210–214. IEEE, New York (Dec. 2009)
Yang, X.S.: Firefly algorithm, stochastic test functions and design optimisation. Int. J. Bioinspired Comput. 2(2), 78–84 (2010)
Mirjalili, S.: Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems. Neural Comput. Appl. 27(4), 1053–1073 (2016)
Mirjalili, S.; Mirjalili, S.M.; Lewis, A.: Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61 (2014)
Yao, X.; Liu, Y.: Fast evolutionary programming. Evol. Program. 3, 451–460 (1996)
Hansen, N.; Müller, S.D.; Koumoutsakos, P.: Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). Evol. Comput. 11(1), 1–18 (2003)
Watkins, W.A.; Schevill, W.E.: Aerial observation of feeding behavior in four baleen whales: eubalaena glacialis, Balaenoptera borealis, Megaptera novaeangliae, and Balaenoptera physalus. J. Mammal. 60(1), 155–163 (1979)
Salgotra, R.; Singh, U.: A novel bat flower pollination algorithm for synthesis of linear antenna arrays. Neural Comput. Appl. 30, 1–14 (2016)
Salgotra, R.; Singh, U.: Application of mutation operators to flower pollination algorithm. Expert Syst. Appl. 79, 112–129 (2017)
Javadi, M.; Marzband, M.; Funsho Akorede, M.; Godina, R.; Saad Al-Sumaiti, A.; Pouresmaeil, E.: A centralized smart decision-making hierarchical interactive architecture for multiple home microgrids in retail electricity market. Energies 11(11), 3144 (2018)
Abuamer, I.M.; Silgu, M.A.; Celikoglu, H.B.: Micro-simulation based ramp metering on Istanbul freeways: an evaluation adopting ALINEA. In: 2016 IEEE 19th International Conference on Intelligent Transportation Systems (ITSC), pp. 695–700. IEEE, New York (Nov. 2016)
Silgu, M.A.; Celikoglu, H.B.: Clustering traffic flow patterns by fuzzy C-means method: some preliminary findings. In: International Conference on Computer Aided Systems Theory, pp. 756–764. Springer, Cham (Feb. 2015)
Valinejad, J.; Marzband, M.; Funsho Akorede, M.; Elliott, I.D.; Godina, R.; Matias, J.; Pouresmaeil, E.: Long-term decision on wind investment with considering different load ranges of power plant for sustainable electricity energy market. Sustainability 10(10), 3811 (2018)
Celikoglu, H.B.: A dynamic network loading process with explicit delay modelling. Transp. Res. Part C Emerg. Technol. 15(5), 279–299 (2007)
Rashedi, E.; Nezamabadi-Pour, H.; Saryazdi, S.: GSA: a gravitational search algorithm. Inf. Sci. 179(13), 2232–2248 (2009)
Green, R.: The electricity contract market in England and Wales. J. Ind. Econ. 47(1), 107–124 (1999)
Hulagu, S.; Celikoglu, H.B.: An integer linear programming formulation for routing problem of university bus service. In: New Trends in Emerging Complex Real Life Problems, pp. 303–311. Springer, Cham (2018)
Marzband, M.; Azarinejadian, F.; Savaghebi, M.; Pouresmaeil, E.; Guerrero, J.M.; Lightbody, G.: Smart transactive energy framework in grid-connected multiple home microgrids under independent and coalition operations. Renew. Energy 126, 95–106 (2018)
Salgotra, R.; Singh, U.; Saha, S.: Improved cuckoo search with better search capabilities for solving CEC2017 benchmark problems. In: 2018 IEEE Congress on Evolutionary Computation (CEC), pp. 1–7. IEEE, New York (July 2018)
Mirjalili, S.; Lewis, A.: The whale optimization algorithm. Adv. Eng. Softw. 95, 51–67 (2016)
Goldbogen, J.A.; Friedlaender, A.S.; Calambokidis, J.; Mckenna, M.F.; Simon, M.; Nowacek, D.P.: Integrative approaches to the study of baleen whale diving behavior, feeding performance, and foraging ecology. Bioscience 63(2), 90–100 (2013)
Kaveh, A.; Ilchi Ghazaan, M.: Enhanced whale optimization algorithm for sizing optimization of skeletal structures. Mech. Based Des. Struct. Mach. (2016). https://doi.org/10.1080/15397734.2016.1213639
Aljarah, I.; Faris, H.; Mirjalili, S.: Optimizing connection weights in neural networks using the whale optimization algorithm. Soft. Comput. 22(1), 1–15 (2018)
Bentouati, B.; Chaib, L.; Chettih, S.: A hybrid whale algorithm and pattern search technique for optimal power flow problem. In: 2016 8th International Conference on Modelling, Identification and Control (ICMIC), pp. 1048–1053. IEEE (2016)
Hu, H.; Bai, Y.; Xu, T.: A whale optimization algorithm with inertia weight. WSEAS Trans. Comput. 15, 319–326 (2016)
Horng, M.F.; Dao, T.K.; Shieh, C.S.; Nguyen, T.T.: A multi-objective optimal vehicle fuel consumption based on whale optimization algorithm. In: Advances in Intelligent Information Hiding and Multimedia Signal Processing: Proceeding of the 12th International Conference on Intelligent Information Hiding and Multimedia Signal Processing, Nov., 21–23, 2016, Kaohsiung, Taiwan, Volume 2, pp. 371–380. Springer, Cham (2017)
Dao, T.K.; Pan, T.S.; Pan, J.S.: A multi-objective optimal mobile robot path planning based on whale optimization algorithm. In: 2016 IEEE 13th International Conference on Signal Processing (ICSP), pp. 337–342. IEEE, New York (2016, Nov.)
Mirjalili, S.; Mirjalili, S.M.; Saremi, S.; Mirjalili, S.: Whale optimization algorithm: theory, literature review, and application in designing photonic crystal filters. In: Nature-Inspired Optimizers, pp. 219–238. Springer, Cham (2020)
Bui, Q.T.; Pham, M.V.; Nguyen, Q.H.; Nguyen, L.X.; Pham, H.M.: Whale optimization algorithm and adaptive neuro-fuzzy inference system: a hybrid method for feature selection and land pattern classification. Int. J. Remote Sens. 40, 1–16 (2019)
Bozorgi, S.M.; Yazdani, S.: IWOA: an improved whale optimization algorithm for optimization problems. J. Comput. Des. Eng. (2019). https://doi.org/10.1016/j.jcde.2019.02.002
Aljarah, I.; Faris, H.; Mirjalili, S.: Optimizing connection weights in neural networks using the whale optimization algorithm. Soft Comput. 22, 1–15 (2016)
Reddy, P.D.P.; Reddy, V.V.; Manohar, T.G.: Whale optimization algorithm for optimal sizing of renewable resources for loss reduction in distribution systems. Renew. Wind Water Solar 4(1), 3 (2017)
Mostafa, A.; Hassanien, A.E.; Houseni, M.; Hefny, H.: Liver segmentation in MRI images based on whale optimization algorithm. Multimed. Tools Appl. 76, 1–24 (2017)
Zhou, Y.; Ling, Y.; Luo, Q.: Lévy flight trajectory-based whale optimization algorithm for global optimization. IEEE Access (2017)
Trivedi, I.N.; Bhoye, M.; Bhesdadiya, R.H.; Jangir, P.; Jangir, N.; Kumar, A.: An emission constraint environment dispatch problem solution with microgrid using whale optimization algorithm. In: Power Systems Conference (NPSC), 2016 National, pp. 1–6. IEEE, New York (Dec. 2016)
Hassanien, A.E.; Elfattah, M.A.; Aboulenin, S.; Schaefer, G.; Zhu, S.Y.; Korovin, I.: Historic handwritten manuscript binarisation using whale optimisation. In: 2016 IEEE International Conference on Systems, Man, and Cybernetics (SMC), pp. 003842–003846. IEEE, New York (Oct. 2016)
Mafarja, M.; Mirjalili, S.: Whale optimization approaches for wrapper feature selection. Appl. Soft Comput. 62, 441–453 (2018)
Hasanien, H.M.: Performance improvement of photovoltaic power systems using an optimal control strategy based on whale optimization algorithm. Electr. Power Syst. Res. 157, 168–176 (2018)
El Aziz, M.A.; Ewees, A.A.; Hassanien, A.E.; Mudhsh, M.; Xiong, S.: Multi-objective whale optimization algorithm for multilevel thresholding segmentation. In: Advances in Soft Computing and Machine Learning in Image Processing, pp. 23–39. Springer, Cham (2018)
Li, L.L.; Sun, J.; Tseng, M.L.; Li, Z.G.: Extreme learning machine optimized by whale optimization algorithm using insulated gate bipolar transistor module aging degree evaluation. Expert Syst. Appl. 127, 58–67 (2019)
Mukherjee, V.; Mukherjee, A.; Prasad, D.: Whale optimization algorithm with wavelet mutation for the solution of optimal power flow problem. In: Handbook of Research on Predictive Modeling and Optimization Methods in Science and Engineering, pp. 500–553. IGI Global, Harrisburg (2018)
Ala’M, A.Z.; Faris, H.; Hassonah, M.A.: Evolving support vector machines using whale optimization algorithm for spam profiles detection on online social networks in different lingual contexts. Knowl. Based Syst. 153, 91–104 (2018)
Wolpert, D.H.; Macready, W.G.: No free lunch theorems for optimization. IEEE Trans. Evol. Comput. 1(1), 67–82 (1997)
Rahnamayan, S.; Tizhoosh, H.R.; Salama, M.M.: Opposition-based differential evolution. IEEE Trans. Evol. Comput. 12(1), 64–79 (2008)
Črepinšek, M.; Liu, S.H.; Mernik, M.: Exploration and exploitation in evolutionary algorithms: a survey. ACM Comput. Surv. (CSUR) 45(3), 35 (2013)
Chen, G.; Huang, X.; Jia, J.; Min, Z.: Natural exponential inertia weight strategy in particle swarm optimization. In: The 6th World Congress on Intelligent Control and Automation, 2006. WCICA 2006. vol. 1, pp. 3672–3675. IEEE, New York (June 2006)
Viswanathan, G.M.; Afanasyev, V.; Buldyrev, S.V.; Havlin, S.; Da Luz, M.G.E.; Raposo, E.P.; Stanley, H.E.: Lévy flights in random searches. Physica A 282(1), 1–12 (2000)
Suganthan, P.N.; Hansen, N.; Liang, J.J.; Deb, K.; Chen, Y.P.; Auger, A.; Tiwari, S.: Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL Report, 2005005 (2005)
Yang, X.S.: Flower pollination algorithm for global optimization. In: International Conference on Unconventional Computing and Natural Computation, pp. 240–249. Springer, Berlin (Sept. 2012)
Fister, Jr, I.; Fister, D.; Yang, X.S.: A hybrid bat algorithm. Preprint (2013). arXiv:1303.6310
Wang, Y.; Cai, Z.; Zhang, Q.: Enhancing the search ability of differential evolution through orthogonal crossover. Inf. Sci. 185(1), 153–177 (2012)
Draa, A.; Bouzoubia, S.; Boukhalfa, I.: A sinusoidal differential evolution algorithm for numerical optimisation. Appl. Soft Comput. 27, 99–126 (2015)
Singh, U.; Salgotra, R.: Synthesis of linear antenna arrays using enhanced firefly algorithm. Arab. J. Sci. Eng. 44, 1–16 (2018)
Derrac, J.; García, S.; Molina, D.; Herrera, F.: A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol. Comput. 1(1), 3–18 (2011)
Acknowledgements
Rohit Salgotra acknowledges the support of Inspire Fellowship (IF-160215) DST Govt. of India and Dr. Sriparna Saha gratefully acknowledges the support from SERB Women in Excellence Award 2018 of Science and Engineering Research Board (SERB) of Department of Science & Technology, Govt. of India for conducting this research.
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Salgotra, R., Singh, U. & Saha, S. On Some Improved Versions of Whale Optimization Algorithm. Arab J Sci Eng 44, 9653–9691 (2019). https://doi.org/10.1007/s13369-019-04016-0
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DOI: https://doi.org/10.1007/s13369-019-04016-0