A novel algorithm to solve fixed points of various functions by the Bisection-Moth-Flame optimization algorithm

Document Type : Research Paper


Department of Applied Mathematics, Iran University of Science and Technology, Tehran, Iran


The essential purpose of this paper is to obtain the fixed point of different functions by using a modern repetitive method. We incorporate concepts suggested in the Bisection method and the Moth-Flame Optimization algorithm. This algorithm is more impressive for finding fixed point functions. We also implement this method for four functions and finally compare the current method with other methods such as ALO, MVO, SSA, SCA algorithms. the proposed method shows a decent functionality than the other four methods.


[1] A.Y. Abdelaziz, E.S. Ali and S.M. Abd Elazim, Flower pollination algorithm and loss sensitivity factors for optimal sizing and placement of capacitors in radial distribution systems, J. Electr. Power Energy Syst. 78 (2016), 207–214.
[2] A. Alizadegan, B. Asady and M. Ahmadpour, Two modified versions of artificial bee colony algorithm, Appl. Math. Comput. 225 (2013), 601–609.
[3] A.A. Alderremy, R.A. Attia, J.F. Alzaidi, D. Lu and M. Khater, Analytical and semi-analytical wave solutions for longitudinal wave equation via modified auxiliary equation method and Adomian decomposition method, Therm. Sci. 23 (2019), 1943–1957.
[4] E. Atashpaz-Gargari and C. Lucas, Imperialist competitive algorithm: an algorithm for optimization inspired by imperialistic competition, IEEE Cong. Evol. Comput., IEEE, 2007, pp. 4661–4667.
[5] Z. Bayraktar, M. Komurcu and D.H. Werner, Wind Driven Optimization (WDO): A novel nature-inspired optimization algorithm and its application to electromagnetics, IEEE Antennas Propag. Soc. Int. Symp., IEEE, 2010, pp. 1–4.
[6] R.L. Burden and J. Douglas, Faires. Numerical analysis, BROOKS/COLE, 1985.
[7] A. Colorni, M, Dorigo and V. Maniezzo, Distributed optimization by ant colonies, Proc. First Eur. Conf. Artific. Life, 142 (1991), 134–142.
[8] E. Cuevas, M. Cienfuegos, D. Zald´─▒var and M.A. Prez-Cisneros, A swarm optimization algorithm inspired in the behavior of the social-spider, Expert Syst. Appl. 40 (2013), no. 18, 6374–6384.
[9] G. David Edward, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesle, 2002.
[10] L.N. De Castro, L.N. Castro and J. Timmis, Artificial immune systems: A new computational intelligence approach, Springer Science and Business Media, 2002.
[11] K. Deb, Optimization for engineering design: Algorithms and examples, PHI Learning Pvt. Ltd, 2012.
[12] A. Djerioui, A, Houari, M. Machmoum and M. Ghanes, Grey wolf optimizerbBased Predictive torque control for electric buses applications, Energies 13 (2020), no. 19, 5013.
[13] M. Dorigo, M. Birattari and T. Stutzle, Ant colony optimization, IEEE Comput. Intell. Mag. 1 (2006), no. 4, 28–39.
[14] M. Eusuff, K. Lansey and F. Pasha, Shuffled frog-leaping algorithm: a memetic meta-heuristic for discrete optimization, Eng. Optim. 38 (2006), no. 2, 129–154.
[15] F. Glover, Future paths for integer programming and links to artificial intelligence, Comput Oper Res. 13 (1986), no. 5, 533–549.
[16] S. Harifi, M. Khalilian, J. Mohammadzadeh and S. Ebrahimnejad, Emperor Penguins Colony: a new metaheuristic algorithm for optimization, Evol. Intell. 12 (2019), no. 2, 211–226.
[17] J.H. Holland, Adaptation in natural and artifficial systems, Ann Arbor, 1975.
[18] W. Jomaa, M. Eddaly and B. Jarboui, Variable neighborhood search algorithms for the permutation flowshop scheduling problem with the preventive maintenance, Oper. Res. 21 (2021), no. 4, 2525–2542.
[19] D. Karaboga, An idea based on honey bee swarm for numerical optimization, Technical Report-tr06, Erciyes University, Engineering Faculty, Computer Engineering Department, 2005.
[20] D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: Artificial bee colony (ABC) algorithm, J. Glob. Optim. 39 (2007), no. 3, 459–471.
[21] A.H. Kashan, League Championship Algorithm (LCA): An algorithm for global optimization inspired by sport
championships, Appl. Soft Comput. 16 (2014), 171–200.
[22] J. Kennedy and R. Eberhart, Particle swarm optimization, Proc. ICNN’95-Int. Conf. Neural Networks, IEEE, Vol. 4, 1995, pp. 1942–1948.
[23] J. Kennedy, R.C. Eberhart and Y. Shi, Swarm Intelligence, Morgan Kaufmanns Academic Press, San Francisco, 2001.
[24] S.R. Kenneth and R.M. Storn, Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim. 11 (1997), no. 4, 341–359.
[25] M. Khater, R.A. Attia and D. Lu, Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions, Math. Comput. Appl. 24 (2018), no. 1, 1.
[26] M.M. Khater, C. Park, D. Lu and R.A. Attia, Analytical, semi-analytical, and numerical solutions for the CahnAllen equation, Adv. Differ. Equ. 2020 (2020), no. 1, 1–12.
[27] S. Kirkpatrick, C.D. Gelatt Jr and M.P. Vecchi, Optimization by simulated annealing, Science 220 (1983), no. 4598, 671–680.
[28] H. Ma and D. Simon, Blended biogeography-based optimization for constrained optimization, Eng. Appl. Artif. Intell. 24 (2011), no. 3, 517–525.
[29] A. Maheri, S. Jalili, Y. Hosseinzadeh, R. Khani and M. Miryahyavi, A comprehensive survey on cultural algorithms, Swarm Evol. Comput. 62 (2021), 100846.
[30] P. Mansouri, B. Asady and N. Gupta, The Bisection-artificial bee colony algorithm to solve fixed point problems,
 Appl. Soft Comput. 26 (2015), 143–148.
[31] F. Merrikh-Bayat, The runner-root algorithm: a metaheuristic for solving unimodal and multimodal optimization problems inspired by runners and roots of plants in nature, Appl. Soft Comput. 33 (2015), 292–303.
[32] P. Mills and E. Tsang, Guided local search for solving SAT and weighted MAX-SAT problems, J. Autom. Reason.24 (2000), no. 1, 205–223.
[33] S. Mirjalili, Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems, Neural. Comput. Appl. 27 (2016), no. 4, 1053–1073.
[34] S. Mirjalili, Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm, Knowledge-Based Syst. 89 (2015), 228–249.
[35] S. Mirjalili, The ant lion optimizer, Adv. Eng. Softw. 83 (2015), 80–98.
[36] S. Mirjalili, SCA: A Sine Cosine Algorithm for solving optimization problems, A School of Information and Communication Technology, Griffith University, Nathan Campus, Brisbane, QLD, 4111, 2016.
[37] S. Mirjalili and A. Lewis, The whale optimization algorithm, Adv. Eng. Softw. 95 (2016), 51–67.
[38] S. Mirjalili, S.M. Mirjalili and A. Hatamlou, Multi-verse optimizer: A nature-inspired algorithm for global optimization,Neural. Comput. Appl. 27 (2016), no. 2, 495–513.
[39] M. Misaghi and M. Yaghoobi, Improved invasive weed optimization algorithm (IWO) based on chaos theory for optimal design of PID controller, J. Comput. Des. Eng. 6 (2019), no. 3, 284–295.
[40] A. Ochoa, L. Margain, A. Hernandez, J. Ponce, A. De Luna, A. Hernandez and O. Castillo, Bat Algorithm toimprove a financial trust forest, World Cong. Nature Bio, Inspired Comput., IEEE, 2013, pp. 58–62.
[41] S. Olariu and A.Y. Zomaya, Biology-derived algorithms in engineering optimization, Handbook of Bioinspired Algorithms and Applications, Chapman and Hall/CRC, 2005.
[42] T. Rahkar Farshi, Battle royale optimization algorithm, Neural. Comput. Appl. 33 (2021), no. 4, 1139–1157.
[43] R. Rajabioun, Cuckoo optimization algorithm, Appl. Soft Comput. 11 (2011), no. 8, 5508–5518.
[44] E. Rashedi, H. Nezamabadi-Pour and S. Saryazdi, GSA: a gravitational search algorithm, Inf. Sci. Lett. 179 (2009), no. 13, 2232–2248.
[45] R.V. Rao, V.J. Savsani and D.P. Vakharia, Teaching–learning-based optimization: a novel method for constrained mechanical design optimization problems, Computer-aided Design 43 (2011), no. 3, 303–315.
[46] H. Rezazadeh, M. Younis, M. Eslami, M. Bilal and U. Younas, New exact traveling wave solutions to the (2+1) dimensional Chiral nonlinear Schrodinger equation, Math. Model. Nat. Phenom. 16 (2021), 38.
[47] H. Salimi, Stochastic fractal search: a powerful metaheuristic algorithm, Knowledge-Based Syst. 75 (2015), 1–18.
[48] S. Saremi, S. Mirjalili and A. Lewis, Grasshopper optimisation algorithm: theory and application, Adv. Eng. Software 105 (2017), 30–47.
[49] H. Shah-Hosseini, The intelligent water drops algorithm: a nature-inspired swarm-based optimization algorithm, Int. J. Bio-Inspit. Com. 1 (2009), no. 1-2, 71–79.
[50] J. Vahidi, S.M. Zekavatmand and H. Rezazadeh, An efficient method for solving hyperbolic partial differential equations, Fifth Nat. Conf. New Approach. Educ. Res., Mahmudabad, 2020.
[51] J. Vahidi, S.M. Zekavatmand, A. Rezazadeh, M. Inc, M.A. Akinlar and Y.M. Chu, New solitary wave solutions to the coupled Maccaris system, Results Phys. 21 (2021), 103801.
[52] J. Vahidi, S.M. Zekavatmand and S.M.S. Hejazi, A modern procedure to solve fixed point functions using BisectionSocial Spider Algorithm, Sixth Nat. Conf. New Approach. Educ. Res., Mahmudabad, 2021.
[53] J. Vahidi, S.M. Zekavatmand and S.M.S. Hejazi, A novel way to obtain fixed point functions using sine cosine algorithm, Sixth Nat. Conf. New Approach. Educ. Res., Mahmudabad, 2021.
[54] A.M. Wazwaz, A sine-cosine method for handlingnonlinear wave equations, Math. Comput. Model. Dyn. Syst. 40 (2004), no. 5-6, 499–508.
[55] A.M. Wazwaz, The tanh method and the sine-cosine method for solving the KP-MEW equation, Int. J. Comput. Math. 82 (2005), no. 2, 235–246.
[56] X.S. Yang, Firefly algorithms for multimodal optimization, Int. Symp. Stoch. Algorithms, Springer, Berlin, Heidelberg, 2009, pp. 169–178.
[57] X.S. Yang, Nature-inspired metaheuristic algorithms, Luniver press, 2010.
[58] S.W. Yao, S.M. Zekavatmand, H. Rezazadeh, J. Vahidi, M.B. Ghaemi and M. Inc, The solitary wave solutions to the Klein-Gordon-Zakharov equations by extended rational methods, AIP Adv. 11 (2021), no. 6, 065218.
[59] A. Yokus, H. Durur and H. Ahmad, Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system, FU= Math Inf. 35 (2020), no. 2, 523–531.
[60] Y. Yonezawa and T. Kikuchi, Ecological algorithm for optimal ordering used by collective honey bee behavior, MHS’96 Proc. Seventh Int. Symp. Micro Machine Human Sci., IEEE, 1996, pp. 249–256.
[61] S.M. Zekavatmand, H. Rezazadeh, M. Inc, J. Vahidi and M.B. Ghaemi, The new soliton solutions for long and short-wave interaction system, J. Ocean Engin. Sci. In Press (2021), https://doi.org/10.1016/j.joes.2021.09.020. JOES, 2021.
[62] S.M. Zekavatmand, H. Rezazadeh and M.B. Ghaemi, Exact travelling wave solutions of nonlinear Cahn-Allen equation, 5th Nat. Conf. Modern Approach. Educ. Res., Mahmudabad, 2020.
[63] S.M. Zekavatmand, J. Vahidi and M.B. Ghaemi, Obtain the fixed point of nonlinear equations through the Whale optimization algorithm, Sixth Nat. Conf. New Approach. Educ. Res., Mahmudabad, 2021.
Volume 14, Issue 1
January 2023
Pages 2999-3010
  • Receive Date: 20 April 2022
  • Revise Date: 02 September 2022
  • Accept Date: 02 September 2022