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On the Mathematical Models and Applications of Swarm Intelligent Optimization Algorithms

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Abstract

With the highly increasing demand in engineering, traditional algorithms may fail to meet required performances. Recently, intelligent algorithms have been widely studied, gradually achieving successful applications in industry. Among them, swarm intelligent algorithms are a combination of intelligent algorithms and bionic swarm theory, with advantages of simple principles, high accuracy, high efficiency, wide application scenarios, solid stability, etc. However, the research of swarm intelligent algorithms is still developing, making it difficult to compare algorithms recently proposed with others. In this paper, ten swarm intelligent optimization algorithms are comprehensively discussed with principles, applications, and improvements, as well as compared and analyzed in terms of accuracy, convergence speed, and time complexity. Furthermore, a conclusion of algorithms with less parameter dependence but better performances is summarized, which can provide further insight for engineers to choose appropriate algorithms to meet actual needs.

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References

  1. Beni G, Wang J (1993) Swarm intelligence in cellular robotic systems. Robots Biol Syst Towards Now Bionics? 102:703–712

  2. Silva AAA, Pontes E, Guelfi AE et al (2012) Methodologies, tools and new development for E-learning

  3. Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern Part B 26(1):29–41

    Article  Google Scholar 

  4. Huang L, Liu S, Gao W (2012) Differential evolution with the search strategy of artificial bee colony algorithm. Control Decis 27(11):1644–1648

    MATH  Google Scholar 

  5. Banharnsakun A, Achalakul T, Sirinaovakul B (2011) The best-so-far selection in artificial bee colony algorithm. Appl Soft Comput 11(2):2888–2901

    Article  Google Scholar 

  6. Zhang H, He L, Yuan L (2021) Mobile robot path planning using improved double-layer ant colony algorithm. Control Decis 1–10 . https://doi.org/10.13195/j.kzyjc.2020.0610

  7. Tuba M, Jovanovic R (2013) Improved ACO algorithm with pheromone correction strategy for the traveling salesman problem. Int J Comput Commun Control 8(3):477–485

    Article  Google Scholar 

  8. Zhang C, Zhang F, Li F et al (2014) Improved artificial fish swarm algorithm. In: 2014 9th IEEE conference on industrial electronics and applications, pp 748–753

  9. Zhang Z, Wang K, Zhu L et al (2017) A Pareto improved artificial fish swarm algorithm for solving a multi-objective fuzzy disassembly line balancing problem. Expert Syst Appl 86(15):165–176

    Article  Google Scholar 

  10. Duan Q, Mao M, Duan P et al (2016) An improved artificial fish swarm algorithm optimized by particle swarm optimization algorithm with extended memory. Kybernetes 45(2):210–222

    Article  Google Scholar 

  11. Arora S, Singh S (2016) An improved butterfly optimization algorithm for global optimization. Adv Sci Eng Med 8(9):711–717

    Article  Google Scholar 

  12. Singh B, Anand P (2018) A novel adaptive butterfly optimization algorithm. Int J Comput Mater Sci Eng 7(4)

  13. Arora S, Singh S (2017) An improved butterfly optimization algorithm with chaos. J Intell Fuzzy Syst 32(1):1079–1088

    Article  MATH  Google Scholar 

  14. Ouaarab A, Ahiod B, Yang X (2014) Discrete cuckoo search algorithm for the travelling salesman problem. Neural Comput Appl 24:1659–1669

    Article  Google Scholar 

  15. Valian E, Tavakoli S, Mohanna S et al (2013) Improved cuckoo search for reliability optimization problems. Comput Indus Eng 64(1):459–468

    Article  Google Scholar 

  16. Wang J, Zhang M, Song H et al (2019) Improvement and application of hybrid firefly algorithm. IEEE Access 7:165458–165477

    Article  Google Scholar 

  17. Kahya M, Altamir S, Algamal Z (2019) Improving firefly algorithm-based logistic regression for feature selection. J Interdisc Math 22(8):1577–1581

    Article  Google Scholar 

  18. Salgotra R, Singh U, Sharma S (2020) On the improvement in grey wolf optimization. Neural Comput Appl 32:3709–3748

    Article  Google Scholar 

  19. Mohamed AM, Hasanien HM, Alkuhayli A (2021) A novel hybrid GWO-PSO optimization technique for optimal reactive power dispatch problem solution. Ain Shams Eng J 12(1):620–630

    Google Scholar 

  20. Elgamal SZM, Yasin NBM, Tubishat M et al (2020) An improved Harris hawks optimization algorithm with simulated annealing for feature selection in the medical field. IEEE Access 8:186638–186652

    Article  Google Scholar 

  21. Fan Q, Chen Z, Xia Z (2020) A novel quasi-reflected Harris hawks optimization algorithm for global optimization problems. Soft Comput 24:14825–14843

    Article  Google Scholar 

  22. Guedria NB (2016) Improved accelerated PSO algorithm for mechanical engineering optimization problems. Appl Soft Comput 40:455–467

    Article  Google Scholar 

  23. Zhang J, Xia P (2017) An improved PSO algorithm for parameter identification of nonlinear dynamic hysteretic models. J Sound Vib 389:153–167

    Article  Google Scholar 

  24. Tubishat M, Abushariah MAM, Idris N et al (2019) Improved whale optimization algorithm for feature selection in Arabic sentiment analysis. Appl Intell 49:1688–1707

    Article  Google Scholar 

  25. Selim A, Kamel S, Jurado F (2018) Voltage profile improvement in active distribution networks using hybrid WOA-SCA optimization algorithm. Twentieth Int Middle East Power Syst Conf 2018:1064–1068

    Google Scholar 

  26. Pandey S, Wu L, Guru SM et al (2010) A particle swarm optimization-based heuristic for scheduling workflow applications in cloud computing environments. In: 2010 24th IEEE international conference on advanced information networking and applications, pp 400–407

  27. Devarapalli R, Bhattacharyya B (2019) Application of modified Harris hawks optimization in power system oscillations damping controller design. In: 2019 8th international conference on power systems, pp 1–6

  28. Hu H, Zhang C, Liang Y (2021) A study on the automatic generation of banner layouts. Comput Electr Eng 93:107269

    Article  Google Scholar 

  29. Hu H, Kleiner M, Pernot JP (2017) Over-constraints detection and resolution in geometric equation systems. Comput Aided Des 90:84–94

    Article  Google Scholar 

  30. Hu H, Kleiner M, Pernot JP, et al (2021) Geometric over-constraints detection: a survey. Arch Comput Methods Eng 28(7):4331–4355

    Article  MathSciNet  Google Scholar 

  31. Okdem S, Karaboga D, Ozturk C (2011) An application of wireless sensor network routing based on artificial bee colony algorithm. IEEE Congress Evol Comput 2011:326–330

    Google Scholar 

  32. Ghambari S, Rahati A (2018) An improved artificial bee colony algorithm and its application to reliability optimization problems. Appl Soft Comput 62:736–767

    Article  Google Scholar 

  33. Linh NT, Anh NQ (2010) Application artificial bee colony algorithm (ABC) for reconfiguring distribution network. Second Int Conf Comput Modeli Simul 2010:102–106

    Google Scholar 

  34. Socha K, Blum C (2007) An ant colony optimization algorithm for continuous optimization: application to feed-forward neural network training. Neural Comput Appl 16:235–247

    Article  Google Scholar 

  35. Liu X, Yi H, Ni Z (2013) Application of ant colony optimization algorithm in process planning optimization. J Intell Manuf 24:1–13

    Article  Google Scholar 

  36. Wang C, Zhou C, Ma J (2005) An improved artificial fish-swarm algorithm and its application in feed-forward neural networks. Int Conf Mach Learn Cybern 2005:2890–2894

    Google Scholar 

  37. Yan W, Li M, Pan X et al (2020) Application of support vector regression cooperated with modified artificial fish swarm algorithm for wind tunnel performance prediction of automotive radiators. Appl Therm Eng 164(5)

  38. Jiang M, Yuan D (2005) Wavelet threshold optimization with artificial fish swarm algorithm. Int Conf Neural Netw Brain 2005:569–572

    Google Scholar 

  39. Ma C, He R (2019) Green wave traffic control system optimization based on adaptive genetic-artificial fish swarm algorithm. Neural Comput Appl 31:2073–2083

    Article  Google Scholar 

  40. Arora S, Singh S, Yetilmezsoy K (2018) A modified butterfly optimization algorithm for mechanical design optimization problems. J Braz Soc Mech Sci Eng 40(21)

  41. Yildiz BS, Yildiz AR, Albak EI et al (2020) Butterfly optimization algorithm for optimum shape design of automobile suspension components. Mater Testing 62(4):365–370

    Article  Google Scholar 

  42. Arora S, Anand P (2018) Learning automata-based butterfly optimization algorithm for engineering design problems. Int J Comput Mater Sci Eng 7(4)

  43. Arora S, Singh S Node localization in wireless sensor networks using butterfly optimization algorithm. Arabian J Sci Eng 42

  44. Assiri AS (2021) On the performance improvement of butterfly optimization approaches for global optimization and feature selection. PLoS ONE 16(1)

  45. Rodrigues D, Pereira LAM, Almeida TNS et al (2013) A binary cuckoo search algorithm for feature selection. IEEE Int Symp Circ Syst 2013:465–468

    Google Scholar 

  46. Salesi S, Cosma G (2017) A novel extended binary cuckoo search algorithm for feature selection. In: 2017 2nd international conference on knowledge engineering and applications, pp 6–12

  47. Tiwari V (2012) Face recognition based on cuckoo search algorithm. Indian J Comput Sci Eng 3(3):401–405

    Google Scholar 

  48. Zhang X, Wang J, Gao Y (2019) A hybrid short-term electricity price forecasting framework: cuckoo search-based feature selection with singular spectrum analysis and SVM. Energy Econ 81:899–913

    Article  Google Scholar 

  49. Yang X (2009) Firefly algorithms for multimodal optimization. Stochastic algorithms: foundations and applications. SAGA 2009. Lecture Notes in Computer Science 5792

  50. Gandomi AH, Yang X, Alavi AH (2011) Mixed variable structural optimization using firefly algorithm. Comput Struct 89(23–24):2325–2336

    Article  Google Scholar 

  51. Baykasoǧlu A, Ozsoydan FB (2015) Adaptive firefly algorithm with chaos for mechanical design optimization problems. Appl Soft Comput 36:152–164

    Article  Google Scholar 

  52. Kohli M, Arora A (2017) Chaotic grey wolf optimization algorithm for constrained optimization problems. J Comput Des Eng 5(4):458–472

    Google Scholar 

  53. Jiang T, Zhang C (2018) Application of grey wolf optimization for solving combinatorial problems: job shop and flexible job shop scheduling cases. IEEE Access 6:26231–26240

    Article  Google Scholar 

  54. Debnath MK, Mallick RK, Sahu BK (2017) Application of hybrid differential evolution—grey wolf optimization algorithm for automatic generation control of a multi-source interconnected power system using optimal fuzzy-PID controller. Electric Power Comp Syst 45(19):2104–2117

    Article  Google Scholar 

  55. Yildiz BS, Yildiz AR (2019) The Harris hawks optimization algorithm, salp swarm algorihtm, grasshopper optimization algorithm and dragonfly algorithms for structural design optimization of vehicle components. Mater Testing 61(8):744–748

    Article  Google Scholar 

  56. Abbasi A, Firouzi B, Sendur P (2021) On the application of Harris hawks optimization (HHO) algoirithm to the design of microchannel heat sinks. Eng Comput 37:1409–1428

    Article  Google Scholar 

  57. Jiao S, Chong G, Huang C et al (2020) Orthogonally adapted Harris hawks optimization for parameter estimation of photovoltaic models. Energy 203(15)

  58. Ekinci S, Izci D, Hekimoǧlu D (2020) PID speed control of DC motor using Harris hawks optimization algorithm. In: 2020 international conference on electrical, communication, and computer engineering, pp 1–6

  59. Ülker S (2008) Particle swarm optimization application to microwave circuits. Microw Opt Technol Lett 50(5):1333–1336

    Article  Google Scholar 

  60. Valle Y, Venayagamoorthy GK, Mohagheghi S et al (2008) Particle swarm optimization. IEEE Trans Evol Comput 12(2):171–195

    Article  Google Scholar 

  61. Khare A, Rangnekar S (2013) A review of particle swarm optimization and its applications in solar photovoltaic system. Appl Soft Comput 13(5):2997–3006

    Article  Google Scholar 

  62. AlRashidi MR, El-Hawary ME (2009) A survey of particle swarm optimization applications in electric power systems. IEEE Trans Evol Comput 13(4):913–918

    Article  Google Scholar 

  63. Prakash DB, Lakshminarayana C (2016) Optimal siting of capacitors in radial distribution network using whale optimization algorithm. Alex Eng J 56(4):499–509

    Article  Google Scholar 

  64. Aljarah I, Faris H, Mirjalili S (2018) Optimizing connection weights in neural networks using the whale optimization algorithm. Soft Comput 22:1–15

    Article  Google Scholar 

  65. Mostafa A, Hassanien AE, Houseni M et al (2017) Liver segmentation in MRI images based on whale optimization algorithm. Multim Tools Appl 76:24931–24954

    Article  Google Scholar 

  66. Hassan G, Hassanien AE (2018) Parameter fundus vasculature multilevel segmentation using whale optimization algorithm. SIViP 12:263–270

    Article  Google Scholar 

  67. Mafarja MM, Mirjalili S (2018) Whale optimization approaches for wrapper feature selection. Appl Soft Comput 62:441–453

    Article  Google Scholar 

  68. Karaboga D (2005) An idea based on honey bee swarm for numerical optimization

  69. Qin Q, Cheng S, Li L et al (2014) Artificial bee colony algorithm: a survey. CAAI Trans Intell Syst 9(2):127–135

    Google Scholar 

  70. Li X (2002) A new intelligent optimization method—artificial fish school algorithm

  71. Arora S, Singh S (2019) Butterfly optimization algorithm: a novel approach for global optimization. Soft Comput 23:715–734

    Article  Google Scholar 

  72. Arora S, Singh S (2015) Butterfly algorithm with \(l\acute{e}vy\) Flights for global optimization. In: 2015 international conference on signal processing, computing and control (ISPCC), pp 220–224

  73. Yang X, Deb S (2009) Cuckoo search via lévy flights. World Cong Nat Biol Inspired Comput 2009:210–214

    Article  Google Scholar 

  74. Yang X (2008) Nature-inspired metaheuristic algorithms

  75. Abbas MS (2021) Firefly feature selection and optimization. MATLAB Central File Exchange

  76. Liu C, Ye C (2011) Novel bioinspired swarm intelligence optimization algorithm: firefly algorithm. Appl Res Comput 28(9):3295–3297

    Google Scholar 

  77. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61

    Article  Google Scholar 

  78. Zhang X, Wang X (2019) Comprehensive review of grey wolf optimization algorithm. Comput Sci 46(3):30–38

    Google Scholar 

  79. Heidari AA, Mirjalili S, Faris H et al (2019) Harris hawks optimization: algorithm and applications. Futur Gener Comput Syst 97:849–872

    Article  Google Scholar 

  80. Kennedy J, Eberhart R (1995) Particle swarm optimization. IEEE Int Conf Neural Netw 4:1942–1948

    Google Scholar 

  81. Kennedy J (1997) The particle swarm: social adaptation of knowledge. IEEE Int Conf Evol Comput 303–308

  82. Mirjalili S, Lewis A (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67

    Article  Google Scholar 

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Authors and Affiliations

  1. Westlake University, Shilongshan Road, Hangzhou, 310024, Zhejiang, China

    Xiaonan Wang, Hao Hu, Yanxue Liang & Liang Zhou

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  1. Xiaonan Wang
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  2. Hao Hu
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Correspondence to Hao Hu.

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Wang, X., Hu, H., Liang, Y. et al. On the Mathematical Models and Applications of Swarm Intelligent Optimization Algorithms. Arch Computat Methods Eng 29, 3815–3842 (2022). https://doi.org/10.1007/s11831-022-09717-8

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