This article's innovation and novelty are introducing a new metaheuristic method called mother optimization algorithm (MOA) that mimics the human interaction between a mother and her children. The real inspiration of MOA is to simulate the mother's care of children in three phases education, advice, and upbringing. The mathematical model of MOA used in the search process and exploration is presented. The performance of MOA is assessed on a set of 52 benchmark functions, including unimodal and high-dimensional multimodal functions, fixed-dimensional multimodal functions, and the CEC 2017 test suite. The findings of optimizing unimodal functions indicate MOA's high ability in local search and exploitation. The findings of optimization of high-dimensional multimodal functions indicate the high ability of MOA in global search and exploration. The findings of optimization of fixed-dimension multi-model functions and the CEC 2017 test suite show that MOA with a high ability to balance exploration and exploitation effectively supports the search process and can generate appropriate solutions for optimization problems. The outcomes quality obtained from MOA has been compared with the performance of 12 often-used metaheuristic algorithms. Upon analysis and comparison of the simulation results, it was found that the proposed MOA outperforms competing algorithms with superior and significantly more competitive performance. Precisely, the proposed MOA delivers better results in most objective functions. Furthermore, the application of MOA on four engineering design problems demonstrates the efficacy of the proposed approach in solving real-world optimization problems. The findings of the statistical analysis from the Wilcoxon signed-rank test show that MOA has a significant statistical superiority compared to the twelve well-known metaheuristic algorithms in managing the optimization problems studied in this paper.

Department of Mathematics Faculty of Science University of Hradec Králové Rokitanského 62 500 03 Hradec Králové Czech Republic

Faculty of Chemical Technology Institute of Applied Physics and Mathematics University of Pardubice 532 10 Pardubice Czech Republic

See more in PubMed

Dehghani M, et al. A spring search algorithm applied to engineering optimization problems. Appl. Sci. 2020;10(18):6173. doi: 10.3390/app10186173. DOI

Dehghani M, et al. DM: Dehghani Method for modifying optimization algorithms. Appl. Sci. 2020;10(21):7683. doi: 10.3390/app10217683. DOI

Coufal P, Hubálovský Š, Hubálovská M, Balogh Z. Snow leopard optimization algorithm: A new nature-based optimization algorithm for solving optimization problems. Mathematics. 2021;9(21):2832. doi: 10.3390/math9212832. DOI

Kvasov DE, Mukhametzhanov MS. Metaheuristic vs. deterministic global optimization algorithms: The univariate case. Appl. Math. Comput. 2018;318:245–259.

Mirjalili S. The ant lion optimizer. Adv. Eng. Softw. 2015;83:80–98. doi: 10.1016/j.advengsoft.2015.01.010. DOI

Dokeroglu T, Sevinc E, Kucukyilmaz T, Cosar A. A survey on new generation metaheuristic algorithms. Comput. Ind. Eng. 2019;137:106040. doi: 10.1016/j.cie.2019.106040. DOI

Dehghani M, et al. Binary spring search algorithm for solving various optimization problems. Appl. Sci. 2021;11(3):1286. doi: 10.3390/app11031286. DOI

Hussain K, Mohd Salleh MN, Cheng S, Shi Y. Metaheuristic research: A comprehensive survey. Artif. Intell. Rev. 2019;52(4):2191–2233. doi: 10.1007/s10462-017-9605-z. DOI

Iba K. Reactive power optimization by genetic algorithm. IEEE Trans. Power Syst. 1994;9(2):685–692. doi: 10.1109/59.317674. DOI

Lu C, Gao L, Li X, Xiao S. A hybrid multi-objective grey wolf optimizer for dynamic scheduling in a real-world welding industry. Eng. Appl. Artif. Intell. 2017;57:61–79. doi: 10.1016/j.engappai.2016.10.013. DOI

de Lima TPF, da Silva AJ, Ludermir TB, de Oliveira WR. An automatic methodology for construction of multi-classifier systems based on the combination of selection and fusion. Prog. Artif. Intell. 2014;2:205–215. doi: 10.1007/s13748-014-0053-6. DOI

Geetha TV, Deepa AJ. A FKPCA-GWO WDBiLSTM classifier for intrusion detection system in cloud environments. Knowl.-Based Syst. 2022;253:109557. doi: 10.1016/j.knosys.2022.109557. DOI

Cura T. A particle swarm optimization approach to clustering. Expert Syst. Appl. 2012;39(1):1582–1588. doi: 10.1016/j.eswa.2011.07.123. DOI

Gomez J, Leon E, Nasraoui O, Giraldo F. The parameter-less randomized gravitational clustering algorithm with online clusters’ structure characterization. Prog. Artif. Intell. 2014;2:217–236. doi: 10.1007/s13748-014-0054-5. DOI

Ahmadi R, Ekbatanifard G, Bayat P. A modified grey wolf optimizer based data clustering algorithm. Appl. Artif. Intell. 2021;35(1):63–79. doi: 10.1080/08839514.2020.1842109. DOI

Sun W, Tang M, Zhang L, Huo Z, Shu L. A survey of using swarm intelligence algorithms in IoT. Sensors. 2020;20:1420. doi: 10.3390/s20051420. PubMed DOI PMC

Al Shahrani AM, et al. An internet of things (IoT)-based optimization to enhance security in healthcare applications. Math. Probl. Eng. 2022;2022:6802967. doi: 10.1155/2022/6802967. DOI

Mehmood K, Chaudhary NI, Khan ZA, Cheema KM, Raja MAZ. Variants of chaotic grey wolf heuristic for robust identification of control autoregressive model. Biomimetics. 2023;8(2):141. doi: 10.3390/biomimetics8020141. PubMed DOI PMC

Mehmood K, Chaudhary NI, Khan ZA, Cheema KM, Raja MAZ, Milyani AH, Azhari AA. Dwarf mongoose optimization metaheuristics for autoregressive exogenous model identification. Mathematics. 2022;10(20):3821. doi: 10.3390/math10203821. DOI

Mehmood K, Chaudhary NI, Khan ZA, Raja MAZ, Cheema KM, Milyani AH. Design of aquila optimization heuristic for identification of control autoregressive systems. Mathematics. 2022;10(10):1749. doi: 10.3390/math10101749. DOI

Mehmood K, Chaudhary NI, Khan ZA, Cheema KM, Raja MAZ, Milyani AH, Azhari AA. Nonlinear hammerstein system identification: A novel application of marine predator optimization using the key term separation technique. Mathematics. 2022;10(22):4217. doi: 10.3390/math10224217. DOI

Mehmood K, Chaudhary NI, Cheema KM, Khan ZA, Raja MAZ, Milyani AH, Alsulami A. Design of nonlinear marine predator heuristics for hammerstein autoregressive exogenous system identification with key-term separation. Mathematics. 2023;11(11):2512. doi: 10.3390/math11112512. DOI

Ghasemi M, Ghavidel S, Ghanbarian MM, Gharibzadeh M, Vahed AA. Multi-objective optimal power flow considering the cost, emission, voltage deviation and power losses using multi-objective modified imperialist competitive algorithm. Energy. 2014;78:276–289. doi: 10.1016/j.energy.2014.10.007. DOI

Montazeri Z, Niknam T. Optimal utilization of electrical energy from power plants based on final energy consumption using gravitational search algorithm. Electr. Eng. Electromech. 2018;2018(4):70–73. doi: 10.20998/2074-272X.2018.4.12. DOI

Rezk H, Fathy A, Aly M, Ibrahim MNF. Energy management control strategy for renewable energy system based on spotted hyena optimizer. Comput. Mater. Continua. 2021;67(2):2271–2281. doi: 10.32604/cmc.2021.014590. DOI

Panda M, Nayak YK. Impact analysis of renewable energy distributed generation in deregulated electricity markets: A context of Transmission Congestion Problem. Energy. 2022;254:124403. doi: 10.1016/j.energy.2022.124403. DOI

Xing Z, Zhu J, Zhang Z, Qin Y, Jia L. Energy consumption optimization of tramway operation based on improved PSO algorithm. Energy. 2022;258:124848. doi: 10.1016/j.energy.2022.124848. DOI

Alsallami SA, Rizvi ST, Seadawy AR. Study of stochastic–fractional Drinfel’d–Sokolov–Wilson equation for M-shaped rational, homoclinic breather, periodic and kink-cross rational solutions. Mathematics. 2023;11(6):1504. doi: 10.3390/math11061504. DOI

Ahmad H, Seadawy AR, Khan TA. Numerical solution of Korteweg–de Vries-Burgers equation by the modified variational iteration algorithm-II arising in shallow water waves. Phys. Scr. 2020;95(4):045210. doi: 10.1088/1402-4896/ab6070. DOI

Seadawy AR, Iqbal M, Lu D. Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg–de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma. Physica A. 2020;544:123560. doi: 10.1016/j.physa.2019.123560. DOI

Seadawy AR, Rizvi STR, Ahmad S, Younis M, Baleanu D. Lump, lump-one stripe, multiwave and breather solutions for the Hunter-Saxton equation. Open Phys. 2021;19(1):1–10. doi: 10.1515/phys-2020-0224. DOI

Tala-Tebue E, Seadawy AR, Kamdoum-Tamo P, Lu D. Dispersive optical soliton solutions of the higher-order nonlinear Schrödinger dynamical equation via two different methods and its applications. Eur. Phys. J. Plus. 2018;133:1–10. doi: 10.1140/epjp/i2018-12133-8. DOI

Wolpert DH, Macready WG. No free lunch theorems for optimization. IEEE Trans. Evol. Comput. 1997;1(1):67–82. doi: 10.1109/4235.585893. DOI

Kennedy, J. & Eberhart, R. Particle swarm optimization. In Proc. ICNN'95—International Conference on Neural Networks 1942–1948 (IEEE, 1998)

Dorigo M, Maniezzo V, Colorni A. Ant system: Optimization by a colony of cooperating agents. IEEE Trans. Syst. Man Cybern. B. 1996;26(1):29–41. doi: 10.1109/3477.484436. PubMed DOI

Karaboga, D. & Basturk, B. Artificial bee colony (ABC) optimization algorithm for solving constrained optimization problems. In Foundations of Fuzzy Logic and Soft Computing. IFSA 2007. Lecture Notes in Computer Science 789–798 (Springer, 2007).

Yang, X.-S. Firefly algorithms for multimodal optimization. In International Symposium on Stochastic Algorithms 169–178 (Springer, 2009).

Mirjalili S, Mirjalili SM, Lewis A. Grey wolf optimizer. Adv. Eng. Softw. 2014;69:46–61. doi: 10.1016/j.advengsoft.2013.12.007. DOI

Dhiman G, Kumar V. Emperor penguin optimizer: A bio-inspired algorithm for engineering problems. Knowl.-Based Syst. 2018;159:20–50. doi: 10.1016/j.knosys.2018.06.001. DOI

Trojovský P, Dehghani M. Pelican optimization algorithm: A novel nature-inspired algorithm for engineering applications. Sensors. 2022;22(3):855. doi: 10.3390/s22030855. PubMed DOI PMC

Dhiman G, Garg M, Nagar A, Kumar V, Dehghani M. A novel algorithm for global optimization: Rat swarm optimizer. J. Ambient. Intell. Humaniz. Comput. 2020;12:8457–8482. doi: 10.1007/s12652-020-02580-0. DOI

Faramarzi A, Heidarinejad M, Mirjalili S, Gandomi AH. Marine predators algorithm: A nature-inspired metaheuristic. Expert Syst. Appl. 2020;152:113377. doi: 10.1016/j.eswa.2020.113377. DOI

Abdollahzadeh B, Gharehchopogh FS, Mirjalili S. African vultures optimization algorithm: A new nature-inspired metaheuristic algorithm for global optimization problems. Comput. Ind. Eng. 2021;158:107408. doi: 10.1016/j.cie.2021.107408. DOI

Zeidabadi F-A, et al. MLA: A new mutated leader algorithm for solving optimization problems. Comput. Mater. Continua. 2022;70(3):5631–5649. doi: 10.32604/cmc.2022.021072. DOI

Dehghani M, Montazeri Z, Trojovská E, Trojovský P. Coati optimization algorithm: A new bio-inspired metaheuristic algorithm for solving optimization problems. Knowl.-Based Syst. 2023;259:110011. doi: 10.1016/j.knosys.2022.110011. DOI

Kaur S, Awasthi LK, Sangal AL, Dhiman G. Tunicate swarm algorithm: A new bio-inspired based metaheuristic paradigm for global optimization. Eng. Appl. Artif. Intell. 2020;90:103541. doi: 10.1016/j.engappai.2020.103541. DOI

Minh H-L, Sang-To T, Theraulaz G, Wahab MA, Cuong-Le T. Termite life cycle optimizer. Expert Syst. Appl. 2023;213:119211. doi: 10.1016/j.eswa.2022.119211. DOI

Doumari SA, et al. A new two-stage algorithm for solving optimization problems. Entropy. 2021;23(4):491. doi: 10.3390/e23040491. PubMed DOI PMC

Zhao W, Wang L, Mirjalili S. Artificial hummingbird algorithm: A new bio-inspired optimizer with its engineering applications. Comput. Methods Appl. Mech. Eng. 2022;388:114194. doi: 10.1016/j.cma.2021.114194. DOI

Trojovská P, Dehghani M, Trojovský P. Fennec fox optimization: A new nature-inspired optimization algorithm. IEEE Access. 2022;10:84417–84443. doi: 10.1109/ACCESS.2022.3197745. DOI

Braik M, Hammouri A, Atwan J, Al-Betar MA, Awadallah MA. White shark optimizer: A novel bio-inspired meta-heuristic algorithm for global optimization problems. Knowl.-Based Syst. 2022;243:108457. doi: 10.1016/j.knosys.2022.108457. DOI

Abualigah L, Abd Elaziz M, Sumari P, Geem ZW, Gandomi AH. Reptile search algorithm (RSA): A nature-inspired meta-heuristic optimizer. Expert Syst. Appl. 2022;191:116158. doi: 10.1016/j.eswa.2021.116158. DOI

Goldberg DE, Holland JH. Genetic algorithms and machine learning. Mach. Learn. 1988;3(2):95–99. doi: 10.1023/A:1022602019183. DOI

Storn R, Price K. Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 1997;11(4):341–359. doi: 10.1023/A:1008202821328. DOI

De Castro LN, Timmis JI. Artificial immune systems as a novel soft computing paradigm. Soft. Comput. 2003;7(8):526–544. doi: 10.1007/s00500-002-0237-z. DOI

Simon D. Biogeography-based optimization. IEEE Trans. Evol. Comput. 2008;12(6):702–713. doi: 10.1109/TEVC.2008.919004. DOI

Reynolds, R. G. An introduction to cultural algorithms. In Proc. Third Annual Conference on Evolutionary Programming 131–139 (World Scientific, 1994).

Beyer H-G, Schwefel H-P. Evolution strategies—A comprehensive introduction. Nat. Comput. 2002;1(1):3–52. doi: 10.1023/A:1015059928466. DOI

Banzhaf W, Nordin P, Keller RE, Francone FD. Genetic Programming: An Introduction. Morgan Kaufmann Publishers; 1998.

Kirkpatrick S, Gelatt CD, Vecchi MP. Optimization by simulated annealing. Science. 1983;220(4598):671–680. doi: 10.1126/science.220.4598.671. PubMed DOI

Rashedi E, Nezamabadi-Pour H, Saryazdi SGSA. A gravitational search algorithm. Inf. Sci. 2009;179(13):2232–2248. doi: 10.1016/j.ins.2009.03.004. DOI

Ghasemi M, et al. A novel and effective optimization algorithm for global optimization and its engineering applications: Turbulent flow of water-based optimization (TFWO) Eng. Appl. Artif. Intell. 2020;92:103666. doi: 10.1016/j.engappai.2020.103666. DOI

Mirjalili S, Mirjalili SM, Hatamlou A. Multi-verse optimizer: A nature-inspired algorithm for global optimization. Neural Comput. Appl. 2016;27(2):495–513. doi: 10.1007/s00521-015-1870-7. DOI

Hatamlou A. Black hole: A new heuristic optimization approach for data clustering. Inf. Sci. 2013;222:175–184. doi: 10.1016/j.ins.2012.08.023. DOI

Shah-Hosseini H. Principal components analysis by the galaxy-based search algorithm: A novel metaheuristic for continuous optimization. Int. J. Comput. Sci. Eng. 2011;6(1–2):132–140.

Tayarani-N, M. H. & Akbarzadeh-T, M. R. Magnetic optimization algorithms a new synthesis. In IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence) 2659–2664 (IEEE, 2006).

Alatas B. ACROA: Artificial chemical reaction optimization algorithm for global optimization. Expert Syst. Appl. 2011;38(10):13170–13180. doi: 10.1016/j.eswa.2011.04.126. DOI

Kaveh A, Khayatazad M. A new meta-heuristic method: Ray optimization. Comput. Struct. 2012;112–113:283–294. doi: 10.1016/j.compstruc.2012.09.003. DOI

Du H, Wu X, Zhuang J, et al. Small-world optimization algorithm for function optimization. In: Jiao L, et al., editors. Advances in Natural Computation. Springer; 2006. pp. 264–273.

Kashan AH. League championship algorithm (LCA): An algorithm for global optimization inspired by sport championships. Appl. Soft Comput. 2014;16:171–200. doi: 10.1016/j.asoc.2013.12.005. DOI

Dehghani M, Mardaneh M, Guerrero JM, Malik O, Kumar V. Football game based optimization: An application to solve energy commitment problem. Int. J. Intell. Eng. Syst. 2020;13:514–523.

Moghdani R, Salimifard K. Volleyball premier league algorithm. Appl. Soft Comput. 2018;64:161–185. doi: 10.1016/j.asoc.2017.11.043. DOI

Zeidabadi FA, Dehghani M. POA: Puzzle optimization algorithm. Int. J. Intell. Eng. Syst. 2022;15(1):273–281.

Dehghani M, Montazeri Z, Givi H, Guerrero JM, Dhiman G. Darts game optimizer: A new optimization technique based on darts game. Int. J. Intell. Eng. Syst. 2020;13:286–294.

Rao RV, Savsani VJ, Vakharia D. Teaching–learning-based optimization: A novel method for constrained mechanical design optimization problems. Comput. Aided Des. 2011;43(3):303–315. doi: 10.1016/j.cad.2010.12.015. DOI

Akbari E, Ghasemi M, Gil M, Rahimnejad A, Gadsden AS. Optimal power flow via teaching-learning-studying-based optimization algorithm. Electr. Power Components Syst. 2022;49(6–7):584–601.

Zou F, Wang L, Hei X, Chen D, Yang D. Teaching–learning-based optimization with dynamic group strategy for global optimization. Inf. Sci. 2014;273:112–131. doi: 10.1016/j.ins.2014.03.038. DOI

Xu Y, et al. Improving teaching–learning-based-optimization algorithm by a distance-fitness learning strategy. Knowl.-Based Syst. 2022;257:108271. doi: 10.1016/j.knosys.2022.108271. DOI

Trojovská E, Dehghani M. A new human-based metahurestic optimization method based on mimicking cooking training. Sci. Rep. 2022;12:14861. doi: 10.1038/s41598-022-19313-2. PubMed DOI PMC

Trojovský P, Dehghani M. A new optimization algorithm based on mimicking the voting process for leader selection. PeerJ Comput. Sci. 2022;2:e976. doi: 10.7717/peerj-cs.976. PubMed DOI PMC

Dehghani M, Trojovská E, Trojovský P. A new human-based metaheuristic algorithm for solving optimization problems on the base of simulation of driving training process. Sci. Rep. 2022;12(1):9924. doi: 10.1038/s41598-022-14225-7. PubMed DOI PMC

Al-Betar MA, Alyasseri ZAA, Awadallah MA, Abu Doush I. Coronavirus herd immunity optimizer (CHIO) Neural Comput. Appl. 2021;33(10):5011–5042. doi: 10.1007/s00521-020-05296-6. PubMed DOI PMC

Borji A, Hamidi M. A new approach to global optimization motivated by parliamentary political competitions. Int. J. Innov. Comput. Inf. Control. 2009;5(6):1643–1653.

Shi, Y. Brain storm optimization algorithm. In International Conference in Swarm Intelligence 303–309 (Springer, 2011).

Ayyarao TL, et al. War strategy optimization algorithm: A new effective metaheuristic algorithm for global optimization. IEEE Access. 2022;10:25073–25105. doi: 10.1109/ACCESS.2022.3153493. DOI

Kuhn AL. The Mother's Role in Childhood Education: New England Concepts, 1830–1860. Yale University Press; 1947.

von der Lippe AL. The impact of maternal schooling and occupation on child-rearing attitudes and behaviours in low income neighbourhoods in Cairo, Egypt. Int. J. Behav. Dev. 1999;23(3):703–729. doi: 10.1080/016502599383766. PubMed DOI

Yao X, Liu Y, Lin G. Evolutionary programming made faster. IEEE Trans. Evol. Comput. 1999;3(2):82–102. doi: 10.1109/4235.771163. DOI

Awad, N., Ali, M., Liang, J., Qu, B. & Suganthan, P. Problem Definitions and Evaluation Criteria for the CEC 2017 Special Session and Competition on Single Objective Real-Parameter Numerical Optimization. Technical Report (2016).

Wilcoxon F. Individual comparisons by ranking methods. Biometr. Bull. 1945;1:80–83. doi: 10.2307/3001968. DOI

Mirjalili S, Lewis A. The whale optimization algorithm. Adv. Eng. Softw. 2016;95:51–67. doi: 10.1016/j.advengsoft.2016.01.008. DOI

Gandomi AH, Yang X-S. Benchmark problems in structural optimization. In: Koziel S, Yang XS, editors. Computational Optimization, Methods and Algorithms. Springer; 2011. pp. 259–281.

Mezura-Montes, E. & Coello, C. A. C. Useful infeasible solutions in engineering optimization with evolutionary algorithms. In Mexican International Conference on Artificial Intelligence 652–662 (Springer, 2005).

Kannan B, Kramer SN. An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design. J. Mech. Des. 1994;116(2):405–411. doi: 10.1115/1.2919393. DOI

A New Hybrid Particle Swarm Optimization-Teaching-Learning-Based Optimization for Solving Optimization Problems

A new human-based metaheuristic algorithm for solving optimization problems based on preschool education

A New Human-Based Metaheuristic Algorithm for Solving Optimization Problems Based on Technical and Vocational Education and Training

OOBO: A New Metaheuristic Algorithm for Solving Optimization Problems