A Hybrid Generalized Reduced Gradient-Based Particle Swarm Optimizer for Constrained Engineering Optimization Problems

Document Type : Regular Article

Authors

1 Assistant Professor, Department of Civil Engineering, Ale Taha Institute of Higher Education, Tehran, Iran

2 Assistant Professor, Department of Civil Engineering, Bozorgmehr University of Qaenat, Qaen, Iran

3 MSc, Department of Civil Engineering, Ale Taha Institute of Higher Education, Tehran, Iran

Abstract

A hybrid algorithm is presented that combines strong points of Particle Swarm Optimization (PSO) and Generalized Reduced Gradient (GRG) algorithm to keep a good compromise between exploration and exploitation. The hybrid PSO-GRG quickly approximates the optimum solution using PSO as a global search engine in the first phase of the search process. The solution accuracy is then improved during the second phase of the search process using the GRG algorithm to probe locally for a proper solution(s) in the vicinity of the current best position obtained by PSO. The k-nearest neighbors (k-NN)-based Purely Uniform Distributed (PUD) initial swarm is also applied to increase the convergence speed and reduce the number of function evaluations (NFEs). Hybridization between both algorithms allows the proposed algorithm to accelerate throughout the early stages of optimization using the high exploration power of PSO whereas, promising solutions will possess a high probability to be exploited in the second phase of optimization using the high exploitation ability of GRG. This prevents PUD-based hybrid PSO-GRG from becoming trapped in local optima while maintaining a balance between exploration and exploitation. The competence of the algorithm is compared with other state-of-the-art algorithms on benchmark optimization problems having a wide range of dimensions and varied complexities. Appraising offered algorithm performance revealed great competitive results on the Multiple Comparison Test (MCT) and Analysis of Variance (ANOVA) test. Results demonstrate the superiority of hybrid PSO-GRG compared to standard PSO in terms of fewer NFEs, fast convergence speed, and high escaping ability from local optima.

Graphical Abstract

A Hybrid Generalized Reduced Gradient-Based Particle Swarm Optimizer for Constrained Engineering Optimization Problems

Highlights

  • The modified PSO algorithm has been hybridized with the Generalized Reduced Gradient (GRG) algorithm.
  • A novel method to enhance the initialization has been proposed by the k-NN algorithm.
  • The GRG algorithm has been employed to enhance local search in the hybrid PSO-GRG algorithm.
  • The exploration and exploitation capabilities of the PSO algorithm are balanced.
  • The suggested PSO-GRG algorithms are evaluated on benchmark mathematical and constrained engineering optimization problems including a wide range of dimensions and varied complexities.

Keywords


[1]     Lin YC. Mixed-integer constrained optimization based on Memetic Algorithm. J Appl Res Technol 2013;11:242–50. doi:10.1016/S1665-6423(13)71534-7.
[2]     Varaee H, Ghasemi MR. Engineering optimization based on ideal gas molecular movement algorithm. Eng Comput 2017;33:71–93. doi:10.1007/s00366-016-0457-y.
[3]     Barbosa TM, Bragada JA, Reis VM, Marinho DA, Carvalho C, Silva AJ. Energetics and biomechanics as determining factors of swimming performance: Updating the state of the art. J Sci Med Sport 2010;13:262–9. doi:10.1016/j.jsams.2009.01.003.
[4]     Javidy B, Hatamlou A, Mirjalili S. Ions motion algorithm for solving optimization problems. Appl Soft Comput J 2015;32:72–9. doi:10.1016/j.asoc.2015.03.035.
[5]     Molina D, Poyatos J, Ser J Del, García S, Hussain A, Herrera F. Comprehensive Taxonomies of Nature- and Bio-inspired Optimization: Inspiration Versus Algorithmic Behavior, Critical Analysis Recommendations. Cognit Comput 2020;12:897–939. doi:10.1007/s12559-020-09730-8.
[6]     Ahmadi-Nedushan B, Varaee H. Minimum cost design of concrete slabs using particle swarm optimization with time varying acceleration coefficients. World Appl Sci J 2011;13:2484–94.
[7]     Liu L, Yang S, Wang D. Particle swarm optimization with composite particles in dynamic environments. IEEE Trans Syst Man, Cybern Part B Cybern 2010;40:1634–48. doi:10.1109/TSMCB.2010.2043527.
[8]     Dorigo M, Maniezzo V, Colorni A. Ant system: Optimization by a colony of cooperating agents. IEEE Trans Syst Man, Cybern Part B Cybern 1996;26:29–41. doi:10.1109/3477.484436.
[9]     Karaboga D, Basturk B. Artificial Bee Colony (ABC) optimization algorithm for solving constrained optimization problems. Lect Notes Comput Sci (including Subser Lect Notes Artif Intell Lect Notes Bioinformatics), vol. 4529 LNAI, Citeseer; 2007, p. 789–98. doi:10.1007/978-3-540-72950-1_77.
[10]    Yang X-S. Firefly algorithm, stochastic test functions and design optimization. Int J Bio-Inspired Comput 2 2010;2:78–84. doi:10.1504/IJBIC.2010.032124.
[11]    Krishnanand KN, Ghose D. Glowworm swarm optimisation: a new method for optimising multi-modal functions. Int J Comput Intell Stud 2009;1:93. doi:10.1504/ijcistudies.2009.515637.
[12]    Breitung K. The geometry of limit state function graphs and subset simulation: Counterexamples. Reliab Eng Syst Saf 2019;182:98–106. doi:10.1016/j.ress.2018.10.008.
[13]    Yazdani M, Jolai F. Lion Optimization Algorithm (LOA): A nature-inspired metaheuristic algorithm. J Comput Des Eng 2016;3:24–36. doi:10.1016/j.jcde.2015.06.003.
[14]    Mirjalili SM, Mirjalili SM, Lewis A. Grey Wolf Optimizer. Adv Eng Softw 2014;69:46–61. doi:10.1016/j.advengsoft.2013.12.007.
[15]    Wang GG, Deb S, Cui Z. Monarch butterfly optimization. Neural Comput Appl 2019;31:1995–2014. doi:10.1007/s00521-015-1923-y.
[16]    Gandomi AH, Alavi AH. Krill herd: A new bio-inspired optimization algorithm. Commun Nonlinear Sci Numer Simul 2012;17:4831–45. doi:10.1016/j.cnsns.2012.05.010.
[17]    Wang GG, Deb S, Coelho LDS. Elephant Herding Optimization. Proc - 2015 3rd Int Symp Comput Bus Intell ISCBI 2015, 2016, p. 1–5. doi:10.1109/ISCBI.2015.8.
[18]    Rajabioun R. Cuckoo optimization algorithm. Appl Soft Comput J 2011;11:5508–18. doi:10.1016/j.asoc.2011.05.008.
[19]    Guo L, Wang GG, Gandomi AH, Alavi AH, Duan H. A new improved krill herd algorithm for global numerical optimization. Neurocomputing 2014;138:392–402. doi:10.1016/j.neucom.2014.01.023.
[20]    Wang GG, Deb S, Gao XZ, Dos Santos Coelho L. A new metaheuristic optimisation algorithm motivated by elephant herding behaviour. Int J Bio-Inspired Comput, vol. 8, 2016, p. 394–409. doi:10.1504/IJBIC.2016.081335.
[21]    Ghasemi MR, Varaee H. A fast multi-objective optimization using an efficient ideal gas molecular movement algorithm. Eng Comput 2017;33:477–96. doi:10.1007/s00366-016-0485-7.
[22]    Ghasemi MR, Ghiasi R, Varaee H. Probability-Based Damage Detection of Structures Using Surrogate Model and Enhanced Ideal Gas Molecular Movement Algorithm. Adv Struct Multidiscip Optim, vol. 4, 2018, p. 1657–74. doi:10.1007/978-3-319-67988-4_124.
[23]    Ghasemi MR, Varaee H. Damping vibration-based IGMM optimization algorithm: fast and significant. Soft Comput 2019;23:451–81. doi:10.1007/s00500-017-2804-3.
[24]    Ghasemi MR, Ghiasi R, Varaee H. Probability-Based Damage Detection of Structures Using Surrogate Model and Enhanced Ideal Gas Molecular Movement Algorithm. Adv Struct Multidiscip Optim, vol. 4, 2018, p. 1657–74. doi:10.1007/978-3-319-67988-4_124.
[25]    Ghasemi MR, Varaee H. Enhanced IGMM optimization algorithm based on vibration for numerical and engineering problems. Eng Comput 2018;34:91–116. doi:10.1007/s00366-017-0523-0.
[26]    Ghasemi MR, Varaee H. Modified Ideal Gas Molecular Movement Algorithm Based on Quantum Behavior. In: Schumacher A, Vietor T, Fiebig S, Bletzinger K-U, Maute K, editors. Adv Struct Multidiscip Optim, Cham: Springer International Publishing; 2018, p. 1997–2010. doi:10.1007/978-3-319-67988-4_148.
[27]    Wang GG, Guo L, Duan H, Wang H. A new improved firefly algorithm for global numerical optimization. J Comput Theor Nanosci 2014;11:477–85. doi:10.1166/jctn.2014.3383.
[28]    Wang GG, Guo L, Gandomi AH, Hao GS, Wang H. Chaotic Krill Herd algorithm. Inf Sci (Ny) 2014;274:17–34. doi:10.1016/j.ins.2014.02.123.
[29]    Zamani H, Nadimi-Shahraki M-H. Feature selection based on whale optimization algorithm for diseases diagnosis. Int J Comput Sci Inf Secur 2016;14:1243.
[30]    Banaie-Dezfouli M, Nadimi-Shahraki MH, Beheshti Z. R-GWO: Representative-based grey wolf optimizer for solving engineering problems. Appl Soft Comput 2021;106:107328.
[31]    Zamani H, Nadimi-Shahraki MH, Gandomi AH. CCSA: conscious neighborhood-based crow search algorithm for solving global optimization problems. Appl Soft Comput 2019;85:105583.
[32]    Shalchi Tousi M, Ghazavi M, Laali S. Optimizing Reinforced Concrete Cantilever Retaining Walls Using Gases Brownian Motion Algorithm (GBMOA). J Soft Comput Civ Eng 2021;5:1–18.
[33]    Nenavath H, Jatoth RK. Hybrid SCA–TLBO: a novel optimization algorithm for global optimization and visual tracking. Neural Comput Appl 2019;31:5497–526. doi:10.1007/s00521-018-3376-6.
[34]    Kumar P. A Modified Genetic Algorithm in C ++ for Optimization of Steel Truss Structures. J Soft Comput Civ Eng 2021;1:95–108.
[35]    Shobeiri V, Ahmadi-Nedushan B. TOPOLOGY OPTIMIZATION OF PRETENSIONED CONCRETE BEAMS CONSIDERING MATERIAL NONLINEARITY. vol. 9. 2019.
[36]    Ghasemi MR, Dizangian B. SIZE, SHAPE AND TOPOLOGY OPTIMIZATION OF COMPOSITE STEEL BOX GIRDERS USING PSO METHOD. ASIAN J Civ Eng (BUILDING Hous 2010;11:699–715.
[37]    Ferdowsi A, Hoseini S, Farzin S, Faramarzpour M, Mousavi S. Shape optimization of gravity dams using a nature-inspired approach. J Soft Comput Civ Eng 2020;4:56–69. doi:10.22115/scce.2020.224492.1196.
[38]    Ghasemi MR, Ghiasi R, Varaee H. Probability-based damage detection using kriging surrogates and enhanced ideal gas molecular movement algorithm. World Congr Struct Multidiscip Optim 2017;11:1657–74. doi:https://doi.org/10.1007/978-3-319-67988-4_124.
[39]    Ghasemi MR, Ghiasi R, Varaee H. Probability-based damage detection of structures using model updating with enhanced ideal gas molecular movement algorithm. 12th World Congr Struct Multidiscip Optim 2017;11:1657–74.
[40]    Ghasemi MR, Ghiasi R, Varaee H. Probability-Based Damage Detection of Structures Using Surrogate Model and Enhanced Ideal Gas Molecular Movement Algorithm. World Congr Struct Multidiscip Optim, Springer International Publishing; 2017, p. 1657–74. doi:https://doi.org/10.1007/978-3-319-67988-4_124.
[41]    Fattahi F, Gholizadeh S. Seismic fragility assessment of optimally designed steel moment frames. Eng Struct 2019;179:37–51. doi:10.1016/j.engstruct.2018.10.075.
[42]    Kaveh A, Talatahari S. Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures. Comput Struct 2009;87:267–83. doi:10.1016/j.compstruc.2009.01.003.
[43]    Kaveh A, Mahdavi VR. A hybrid CBO-PSO algorithm for optimal design of truss structures with dynamic constraints. Appl Soft Comput J 2015;34:260–73. doi:10.1016/j.asoc.2015.05.010.
[44]    Rezaee Manesh M, Ghasemi SH, Rezaee Manesh M. Dual Target Optimization of Two-Dimensional Truss Using Cost Efficiency and Structural Reliability Sufficiency. J Soft Comput Civ Eng 2020;4:98–111. doi:10.22115/scce.2020.244833.1252.
[45]    Heidari A, Raeisi J. Optimum design of structures against earthquake by simulated annealing using wavelet transform. J Soft Comput Civ Eng 2018;2:23–33.
[46]    Kaveh A, Maniat M. Damage detection based on MCSS and PSO using modal data. Smart Struct Syst 2015;15:1253–70. doi:10.12989/sss.2015.15.5.1253.
[47]    Law SS, Li J, Ding Y. Structural response reconstruction with transmissibility concept in frequency domain. Mech Syst Signal Process 2011;25:952–68. doi:10.1016/j.ymssp.2010.10.001.
[48]    Luh GC, Lin CY, Lin YS. A binary particle swarm optimization for continuum structural topology optimization. Appl Soft Comput J, vol. 11, 2011, p. 2833–44. doi:10.1016/j.asoc.2010.11.013.
[49]    Javidrad F, Nazari M, Javidrad HR. An Innovative Optimized Design for Laminated Composites in terms of a Proposed Bi-Objective Technique. J Soft Comput Civ Eng 2020;4:1–28.
[50]    Chakri A, Rabia XY, Mohamed K. Reliability-based design optimization using the directional bat algorithm. Neural Comput Appl 2017. doi:10.1007/s00521-016-2797-3.
[51]    Safaeian Hamzehkolaei N, Miri M, Rashki M. Reliability-based design optimization of rotating FGM cylindrical shells with temperature-dependent probabilistic frequency constraints. Aerosp Sci Technol 2017;68:223–39. doi:10.1016/j.ast.2017.05.004.
[52]    Safaeian Hamzehkolaei N, Miri M, Rashki M. An enhanced simulation-based design method coupled with meta-heuristic search algorithm for accurate reliability-based design optimization. Eng Comput 2016;32:477–95. doi:10.1007/s00366-015-0427-9.
[53]    Petrović M, Vuković N, Mitić M, Miljković Z. Integration of process planning and scheduling using chaotic particle swarm optimization algorithm. Expert Syst Appl 2016;64:569–88. doi:10.1016/j.eswa.2016.08.019.
[54]    Paiva FAP, Silva CRM, Leite IVO, Marcone MHF, Costa JAF. Modified bat algorithm with cauchy mutation and elite opposition-based learning. 2017 IEEE Lat Am Conf Comput Intell LA-CCI 2017 - Proc, vol. 2017- Novem, IEEE; 2018, p. 1–6. doi:10.1109/LA-CCI.2017.8285715.
[55]    Safaeian Hamzehkolaei N, Miri M, Rashki M. An improved binary bat flexible sampling algorithm for reliability-based design optimization of truss structures with discrete-continuous variables. Eng Comput 2018;35:641–71. doi:10.1108/EC-06-2016-0207.
[56]    Mozafari M, Tafazzoli S, Jolai F. A new IPSO-SA approach for cardinality constrained portfolio optimization. Int J Ind Eng Comput 2011;2:249–62. doi:10.5267/j.ijiec.2011.01.004.
[57]    Ahmadi M, Mojallali H. Chaotic invasive weed optimization algorithm with application to parameter estimation of chaotic systems. Chaos, Solitons and Fractals 2012;45:1108–20. doi:10.1016/j.chaos.2012.05.010.
[58]    Kaveh A, Bakhshpoori T, Afshari E. Hybrid PSO and SSO algorithm for truss layout and size optimization considering dynamic constraints. Struct Eng Mech 2015;54:453–74. doi:10.12989/sem.2015.54.3.453.
[59]    Kotinis M. Improving a multi-objective differential evolution optimizer using fuzzy adaptation and K-medoids clustering. Soft Comput 2014;18:757–71. doi:10.1007/s00500-013-1086-7.
[60]    Gupta S, Deep K. Hybrid sine cosine artificial bee colony algorithm for global optimization and image segmentation. Neural Comput Appl 2020;32:9521–43. doi:10.1007/s00521-019-04465-6.
[61]    Yildizdan G, Baykan ÖK. A new hybrid BA_ABC algorithm for global optimization problems. Mathematics 2020;8:1–36. doi:10.3390/math8101749.
[62]    Yue S, Zhang H. A hybrid grasshopper optimization algorithm with bat algorithm for global optimization. Multimed Tools Appl 2021;80:3863–84. doi:10.1007/s11042-020-09876-5.
[63]    Yue ZH, Zhang S, Xiao WD. A novel hybrid algorithm based on grey wolf optimizer and fireworks algorithm. Sensors (Switzerland) 2020;20:1–17. doi:10.3390/s20072147.
[64]    Khoshahval F, Zolfaghari A, Minuchehr H, Abbasi MR. A new hybrid method for multi-objective fuel management optimization using parallel PSO-SA. Prog Nucl Energy 2014;76:112–21. doi:10.1016/j.pnucene.2014.05.014.
[65]    Mesloub S, Mansour A. Hybrid PSO and GA for global maximization. Int J Open Probl Comput Sci Math 2009;2:597–608. doi:1998-6262.
[66]    Jayaprakasam S, Rahim SKA, Leow CY. PSOGSA-Explore: A new hybrid metaheuristic approach for beampattern optimization in collaborative beamforming. Appl Soft Comput J 2015;30:229–37. doi:10.1016/j.asoc.2015.01.024.
[67]    Jia P, Duan S, Yan J. An Enhanced Quantum-Behaved Particle Swarm Optimization Based on a Novel Computing Way of Local Attractor 2015:633–49. doi:10.3390/info6040633.
[68]    Pluhacek M, Senkerik R, Davendra D. Chaos particle swarm optimization with Eensemble of chaotic systems. Swarm Evol Comput 2015;25:29–35. doi:10.1016/j.swevo.2015.10.008.
[69]    Shahzad M, Zahid S, Farooq M. A Hybrid GA-PSO Fuzzy System for User Identification on Smart Phones Categories and Subject Descriptors n.d.
[70]    Lee H, Chen S, Kang H-Y. A Study of Generalized Reduced Gradient Method with Different Search Directions. A Study Gen Reduc Gradient Method with Differ Search Dir 2004;1:25–38.
[71]    Arora JS. More on Numerical Methods for Constrained Optimum Design. Introd to Optim Des, 2004, p. 379–412. doi:10.1016/b978-012064155-0/50011-2.
[72]    Rosen JB. The gradient projection method for nonlinear programming. Part I. Linear constraints. J Soc Ind Appl Math 1960;8:181–217.
[73]    Lasdon LS, Fox RL, Ratner MW. NONLINEAR OPTIMIZATION USING THE GENERALIZED REDUCED GRADIENT METHOD. Rev Fr Autom Inf Rech Oper 1974;8:73–103. doi:10.1051/ro/197408V300731.
[74]    Gabriele GA, Ragsdell KM. The generalized reduced gradient method: A reliable tool for optimal design. J Eng Ind 1977:394–400.
[75]    Altman NS. An introduction to kernel and nearest-neighbor nonparametric regression. Am Stat 1992;46:175–85. doi:10.1080/00031305.1992.10475879.
[76]    Kulkarni O, Kulkarni N, Kulkarni AJ, Kakandikar G. Constrained cohort intelligence using static and dynamic penalty function approach for mechanical components design. Int J Parallel, Emergent Distrib Syst 2018;33:570–88. doi:10.1080/17445760.2016.1242728.
[77]    Grandgirard J, Poinsot D, Krespi L, Nénon JP, Cortesero AM, Miettinen K, et al. Numerical comparison of some penalty-based constraint handling techniques in genetic algorithms. J Glob Optim 2003;27:427–46. doi:10.1023/A.
[78]    Parsopoulos KE, Vrahatis MN, others. Particle swarm optimization method for constrained optimization problems. Intell Technol Appl New Trends Intell Technol 2002;76:214–20.
[79]    Daniel IM. Self-adapting control parameters in particle swarm optimization. University of British Columbia, 2019.
[80]    Wang H, Cui Z, Sun H, Rahnamayan S, Yang X, Wang H. Randomly attracted firefly algorithm with neighborhood search and dynamic parameter adjustment mechanism. Soft Comput 2016. doi:10.1007/s00500-016-2116-z.
[81]    Fattahi H, Babanouri N. Predicting tensile strength of rocks from physical properties based on support vector regression optimized by cultural algorithm. J Min Environ 2017;8:467–74. doi:10.22044/jme.2016.824.
[82]    Sharma TK, Pant M, Singh VP. Adaptive Bee Colony in an Artificial Bee Colony for Solving. arxiv, 2012. doi:1211.0957.
[83]    Kaveh  a., Mahdavi VRR. Colliding Bodies Optimization method for optimum design of truss structures with continuous variables. Adv Eng Softw 2014;70:1–12. doi:10.1016/j.advengsoft.2014.01.002.
[84]    Keane AJ. Experiences with optimizers in structural design. Conf Adapt Comput Eng Des Control 1994;94:14–27.
[85]    Mishra SK. Minimization of Keane’s Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods. SSRN Electron J 2011. doi:10.2139/ssrn.983836.
[86]    Ghasemi MR, Hinton E, Wood RD. Optimization of trusses using genetic algorithms for discrete and continuous variables. vol. 16. 1999. doi:10.1108/02644409910266403.
[87]    Sapre S, Mini S. Opposition-based moth flame optimization with Cauchy mutation and evolutionary boundary constraint handling for global optimization. Soft Comput 2019;23:6023–41. doi:10.1007/s00500-018-3586-y.
[88]    Mirjalili S, Mirjalili SM, Hatamlou A. Multi-Verse Optimizer: a nature-inspired algorithm for global optimization. Neural Comput Appl 2016;27:495–513. doi:10.1007/s00521-015-1870-7.
[89]    Askarzadeh A. A novel metaheuristic method for solving constrained engineering optimization problems: Crow search algorithm. Comput Struct 2016;169:1–12. doi:10.1016/j.compstruc.2016.03.001.
[90]    Kaveh A, Dadras A. A novel meta-heuristic optimization algorithm: Thermal exchange optimization. Adv Eng Softw 2017;110:69–84. doi:10.1016/j.advengsoft.2017.03.014.
[91]    Zahara E, Kao Y-TT. Hybrid Nelder-Mead simplex search and particle swarm optimization for constrained engineering design problems. Expert Syst Appl 2009;36:3880–6. doi:10.1016/j.eswa.2008.02.039.
[92]    Wang L, Li LP. An effective differential evolution with level comparison for constrained engineering design. Struct Multidiscip Optim 2010;41:947–63. doi:10.1007/s00158-009-0454-5.
[93]    Huang F zhuo, Wang L, He Q. An effective co-evolutionary differential evolution for constrained optimization. Appl Math Comput 2007;186:340–56. doi:10.1016/j.amc.2006.07.105.
[94]    Mezura-Montes E, Coello CACC, Velazquez-Reyes J, Munoz-Davila L. Multiple trial vectors in differential evolution for engineering design. Eng Optim 2007;39:567–89. doi:10.1080/03052150701364022.
[95]    Rao SS. Engineering Optimization: Theory and Practice: Fourth Edition. John Wiley & Sons; 2009. doi:10.1002/9780470549124.
[96]    Belegundu AD, Arora JS. A study of mathematical programmingmethods for structural optimization. Part II: Numerical results. Int J Numer Methods Eng 1985;21:1601–23. doi:10.1002/nme.1620210905.
[97]    Arora J. Introduction to optimum design. Academic Press; 2004.
[98]    Coello Coello CA, Montes EM. Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Adv Eng Informatics 2002;16:193–203. doi:10.1016/S1474-0346(02)00011-3.
[99]    Zhou Y, Liu L. An effective chaotic cultural-based particle swarm optimization for constrained engineering design problems. Appl Mech Mater, vol. 20–23, Elsevier; 2010, p. 64–9. doi:10.4028/www.scientific.net/AMM.20-23.64.
[100]  Eskandar H, Sadollah A, Bahreininejad A, Hamdi M. Water cycle algorithm - A novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput Struct 2012;110–111:151–66. doi:10.1016/j.compstruc.2012.07.010.
[101]  Kaveh A, Talatahari S. A novel heuristic optimization method: charged system search. Acta Mech 2010;213:267–89. doi:10.1007/s00707-009-0270-4.
[102]  Ben Guedria N. Improved accelerated PSO algorithm for mechanical engineering optimization problems. Appl Soft Comput J 2016;40:455–67. doi:10.1016/j.asoc.2015.10.048.
[103]  Cagnina LC, Esquivel SC, Coello CAC. Solving engineering optimization problems with the simple constrained particle swarm optimizer. Bioinspired Optim Methods their Appl - Proc 3rd Int Conf Bioinspired Optim Methods their Appl BIOMA 2008, vol. 32, 2008, p. 107–20.
[104]  Garg H. A hybrid GSA-GA algorithm for constrained optimization problems. Inf Sci (Ny) 2019;478:499–523. doi:10.1016/j.ins.2018.11.041.
[105]  Hsu YL, Liu TC. Developing a fuzzy proportional-derivative controller optimization engine for engineering design optimization problems. Eng Optim 2007;39:679–700. doi:10.1080/03052150701252664.
[106]  Raj KH, Sharma RS, Mishra GS, Dua A, Patvardhan C. An evolutionary computational technique for constrained optimisation in engineering design. J Inst Eng Mech Eng Div 2005;86:121–8.
[107]  Tsai JFA. Global optimization of nonlinear fractional programming problems in engineering design. Eng Optim 2005;37:399–409. doi:10.1080/03052150500066737.
[108]  Zhang M, Luo W, Wang X. Differential evolution with dynamic stochastic selection for constrained optimization. Inf Sci (Ny) 2008;178:3043–74. doi:10.1016/j.ins.2008.02.014.
[109]  Gandomi AH, Yang XS, Alavi AH. Erratum: Cuckoo search algorithm: A metaheuristic approach to solve structural optimization problems (Engineering with Computers DOI:10.1007/s00366-011-0241-y). Eng Comput 2013;29:245. doi:10.1007/s00366-012-0308-4.