Ref. No Study Objective Variables Constraints Optimization technique [17 ] Pochtman et al., 1988 Minimum cost 5 variables ( Constraints (internal stability and eccentricity) Random search algorithm [18 ] Dembicki and Chi, 1989 Minimum weight; maximum factor of safety 8 variables (coordinates of wall) 17 constraints (external stability and geometric requirements) Monte-Carlo simulation [19 ] Sarıbaş and Erbatur, 1996 Minimum cost; minimum weight 7 variables ( 10 constraints (internal and external stability) Nonlinear programming (NLP) [20 ] Medhekar, 1990 Minimum cost 8 variables ( 11 constraints (internal and external stability, slip circle, and eccentricity) Interior penalty function [21 ] Purohit, 2014 Minimum cost; maximum factor of safety 8 variables ( Constraints (internal stability, external stability, and eccentricity) NSGA-II [22 ] Naeem, 2016 Minimum cost 9 variables ( 9 constraints (internal and external stability) GA [23 ] Rahbari, 2017 Minimum cost; maximum robustness 4 variables ( Constraints (internal and external stability) NSGA-II [24 ] Schmied and Karlsson, 2021 Minimum cost; minimum carbon emissions 7 variables ( Constraints (internal and external stability, deflection, crack width, and minimum reinforcement spacing) Pattern search (PS); GA [25 ] Bhatti, 2006 Minimum cost 5 variables ( Constraints (internal stability, external stability, and geometric requirements) Generalized reduced gradient (GRG) solver [26 ] Ahmadi-Nedushan and Varaee, 2009 Minimum cost; minimum weight 7 variables ( 10 constraints (external and internal stability) Particle swarm optimization (PSO) [27 ] Villalba et al., 2010 Minimum cost; minimum carbon emissions 20 variables ( Constraints (external stability, internal stability, and deflection) SA [28 ] Pei and Xia, 2012 Minimum cost 9 variables ( 25 constraints (internal stability, external stability, and geometric requirements) Complex method (CM); GA; PSO; SA [29 ] Papazafeiropoulos et al., 2013 Minimum volume of wall 6 variables ( Constraints (internal stability, uplift, undrained shear, and displacement) GA [30 ] Uray and Tan, 2019 Minimum factor of safety 4 variables ( 7 constraints (external stability, slope stability, and geometric requirements) HS [31 ] Al Sebai et al., 2021 Minimum cost (reliability based design optimization) 10 variables ( 23 constraints (internal stability, external stability, deflection, and geometric requirements) Covariance matrix adaptation-evolution strategy (CMS-ES); SQP [32 ] Srivastavaa et al., 2022 Minimum cost; minimum weight 12 variables ( 12 constraints (internal stability, external stability, and eccentricity) PSO [33 ] Yücel et al., 2021 Minimum cost 5 variables ( 16 constraints (internal stability, external stability, and maximum and minimum reinforcement) TLBO; JA; GA; PSO; DE [34 ] Ceranic et al., 2001 Minimum cost 7 variables ( Constraints (external stability) Simulated annealing (SA) [35 ] Chau and Albermani, 2003 Minimum cost 3 variables ( Constraints (internal stability and crack width) Genetic algorithm [36 ] Babu and Basha, 2008 Target reliability index (reliability based design optimization) — Constraints (external stability) Method of Lagrange multipliers [37 ] Yepes et al., 2008 Minimum cost 20 variables ( Constraints (internal stability, external stability, and deflection) SA [38 ] Khajehzadeh et al., 2010 Minimum cost 8 variables ( Constraints (internal and external stability) Particle swarm optimization with passive congregation (PSOPC) [39 ] Ghazavi and Bonab, 2011 Minimum cost; minimum weight 8 variables ( 20 constraints (external stability, eccentricity, geometric requirements, and maximum and minimum reinforcement) Ant colony optimization (ACO) [40 ] Kaveh and Abadi, 2011 Minimum cost 7 variables ( 9 constraints (internal and external stability) Harmony search (HS); improved harmony search (IHS) [41 ] Khajehzadeh et al., 2011 Minimum cost 8 variables ( 11 constraints (internal and external stability, eccentricity, and deflection) PSO; PSOPC; modified particle swarm optimization (MPSO) [42 ] Camp and Akin, 2012 Minimum cost; minimum weight 12 variables ( 26 constraints (internal and external stability, geometric requirements, development length, hook length, and maximum and minimum reinforcement) Big bang–big crunch (BB-BC) [43 ] Khajehzadeh and Eslami, 2012 Minimum cost 8 variables ( Constraints (internal and external stability, eccentricity, and deflection) Gravity search algorithm (GSA) [44 ] Sable and Patil, 2012 Minimum cost; minimum weight 12 variables ( 21 constraints (internal and external stability, geometric requirements, and maximum and minimum reinforcement) Interior point method (IPM) [45 ] Sable and Patil, 2012 Minimum cost; minimum weight 12 variables ( 21 constraints (internal and external stability, geometric requirements, and maximum and minimum reinforcement) Interior point method (IPM) [46 ] Kaveh and Behnam, 2013 Minimum cost 7 variables ( Constraints (internal and external stability, and maximum and minimum reinforcement) Charged system search (CSS); IHS [47 ] Kaveh et al., 2013 Minimum cost; minimum congestion 35 variables ( Constraints (internal and external stability, eccentricity, and deflection) Nonsorting genetic algorithm (NSGA-II) [48 ] Kaveh and Khayatazad, 2014 Minimum cost 7 variables ( Constraints (internal and external stability) Ray optimization (RO); PSO; CSS [49 ] Khajehzadeh et al., 2014 Minimum cost; minimum carbon emissions 8 Variables ( 11 constraints (internal and external stability, eccentricity, and deflection) Adaptive gravitational search algorithm with pattern search (AGSA-PS); GSA; BB-BC [50 ] Sheikholeslami et al., 2014 Minimum cost 11 variables ( Constraints (internal and external stability) Improved firefly harmony search (IFA-HS); ACO [51 ] Talatahari and Sheikholeslami, 2014 Minimum cost 11 variables ( Constraints (internal and external stability) Enhanced charged system search (ECSS); hybrid big bang–big crunch (HBB-BC); CSS; PSO; BB-BC [52 ] Gandomi et al., 2015 Minimum cost; minimum weight 12 variables ( Constraints (internal and external stability, geometric requirements, and maximum and minimum reinforcement) Accelerated particle swarm optimization (APSO); firefly algorithm (FA); PSO; cuckoo search (CS) [53 ] Kaveh and Mahdavi, 2015 Minimum cost 7 variables ( Constraints (internal and external stability) Democratic particle swarm optimization (DPSO); colliding bodies optimization (CBO); PSO; IHS [54 ] Singla and Gupta, 2015 Minimum cost 3 variables ( Constraints (internal and external stability, eccentricity, and maximum and minimum reinforcement) Parametric equations [55 ] Bekdaş et al., 2016 Minimum cost 10 variables ( 17 constraints (internal and external stability, spacing of bars, concrete cover, and maximum and minimum reinforcement) Teaching learning based optimization (TLBO) [56 ] Kaveh and Farhoudi, 2016 Minimum cost 7 variables ( Constraints (internal and external stability) Dolphin echo location (DEO); HS; IHS; CSS [57 ] Sheikholeslami et al., 2016 Minimum cost 11 variables ( Constraints (internal and external stability) IFA-HS; IFA; HS; ACO [58 ] Temür and Bekdas, 2016 Minimum cost 11 variables ( 29 constraints (internal and external stability, geometric requirements, and maximum and minimum reinforcement) TLBO; improved teaching learning based optimization (ITLBO); modified teaching learning based optimization (MTLBO); PSO; BB-BC; IHS [59 ] Aydogdu, 2016 Minimum cost 13 variables ( Constraints (internal and external stability, spacing of bars, development lengths, and maximum and minimum reinforcement) Biogeography based optimization (BBO); biogeography based optimization with Levy flight (LFBBO) [60 ] Kaveh and Laien, 2017 Minimum cost 7 variables ( Constraints (internal and external stability) Vibrating particle system (VPS); enhanced colliding body (ECBO); CBO [61 ] Gandomi et al., 2017 Minimum cost; minimum weight 12 variables ( 26 constraints (internal and external stability, geometric requirements, development lengths, hook lengths, and maximum and minimum reinforcement) BBO; GA; differential evolution (DE); evolutionary strategy (ES) [62 ] Gandomi et al., 2017 Minimum cost; minimum weight 12 variables ( 26 constraints (internal and external stability, geometric requirements, development lengths, hook lengths, and maximum and minimum reinforcement) Interior search algorithm (ISA) [63 ] Kumar and Suribabu, 2017 Minimum weight 7 variables ( 10 constraints (internal stability, external stability, and eccentricity) Differential evolution algorithm (DEA); PSO [64 ] Rahbari et al., 2017 Minimum cost; maximum robustness 4 variables ( Constraints (internal and external stability) NSGA-II [65 ] Ukritchon et al., 2017 Minimum cost 11 variables ( 22 constraints (internal stability, external stability, and slip circle) NLP [66 ] Kayhan and Demir, 2018 Minimum cost 7 variables ( 15 constraints (internal and external stability and maximum and minimum reinforcement) Differential genetic algorithm (DGA) [67 ] Mohammad and Ahmed, 2018 Minimum cost 6 variables ( 20 constraints (internal stability, external stability, and geometric requirements) Evolutionary algorithm (EA) [68 ] Kalateh-Ahani and Sarani, 2019 Minimum cost; minimum displacement 9 variables ( Constraints (internal and external stability, eccentricity, and maximum and minimum reinforcement) NSGA-II [69 ] Moayyeri et al., 2019 Minimum cost 12 variables ( 26 constraints (internal and external stability, geometric requirements, development lengths, hook lengths, and maximum and minimum reinforcement) PSO [70 ] Öztürk and Türkeli, 2019 Minimum cost; minimum carbon emissions 12 variables ( 25 constraints (internal and external stability, geometric requirements, development lengths, hook lengths, and maximum and minimum reinforcement) Jaya algorithm (JA) [71 ] Uray et al., 2019 Minimum weight 4 variables (, angle of footing) Constraints (external stability and geometric requirements) HS [72 ] Dagdeviren and Kaymak, 2020 Minimum cost 8 variables ( 12 constraints (internal stability, external stability, and geometric requirements) Artificial bee colony (ABC); parametric equations [73 ] Kaveh et al., 2020 Minimum cost 7 variables ( Constraints (internal and external stability) Imperialist competitive algorithm (ICA); tug of war optimization (TWO); water evaporation optimization (WEO); cyclical parthenogenesis algorithm (CPA); BB-BC; TLBO; CS; CSS; RO; VPS; ABC [74 ] Kaveh et al., 2020 Minimum cost 7 variables ( Constraints (internal and external stability) Shuffled shepherd optimization algorithm (SSOA); BB-BC; CS; CSS; ICA; RO; TWO; WEO [75 ] Kayabekir et al., 2020 Minimum cost; minimum carbon emissions 8 variables ( 16 constraints (internal and external stability, maximum and minimum reinforcement) Flower pollination algorithm (FPA); HS [76 ] Konstandakopoulou et al., 2020 Minimum cost 4 variables ( Constraints (internal and external stability) Parametric equations [77 ] Mergos and Mantoglou, 2020 Minimum cost 6 variables ( Constraints (external stability, eccentricity, and maximum and minimum reinforcement) FPA; PSO; GA [78 ] Kalemci et al., 2020 Minimum weight 12 variables ( 26 constraints (internal and external stability, geometric requirements, development lengths, hook lengths, and maximum and minimum reinforcement) Grey wolf optimization (GWO); backtracking search algorithm (BSA); BB-BC; GA; DE; ES: PSO; APSO; FA; CS; ISA [79 ] Kayabekir et al., 2020 Minimum cost 8 variables ( 16 constraints (internal and external stability, maximum and minimum reinforcement) JA [80 ] Hoang and Cong, 2020 Minimum weight 7 variables ( 12 constraints (internal and external stability, eccentricity, and geometric requirements) Differential evolution [81 ] Millán-Paramo et al., 2020 Minimum weight 12 variables ( 21 constraints (internal and external stability, geometric requirements, and maximum and minimum reinforcement) Modified simulated annealing algorithm (MSAA) [82 ] Kashani et al., 2020 Minimum cost; minimum weight 12 variables ( 26 constraints (internal and external stability, geometric requirements, development lengths, hook lengths, and maximum and minimum reinforcement) PSO; MPSO; improved particle swarm optimization (IPSO); comprehensive learning particle swarm optimization (CLPSO); heterogeneous comprehensive learning particle swarm optimization (HCLPSO); extraordinary particle swarm optimization (EPSO); fractional-order Darwinian PSO (FDPSO); improved random drift PSO (IRDPSO); autonomous particle groups for particle swarm optimization (AGPSO); time varying acceleration particle swarm optimization (TACPSO) [83 ] Uray et al., 2020 Minimum weight 7 variables ( 7 constraints (external stability, geometric requirements, and eccentricity) ABC [84 ] Ravichandran et al., 2020 Minimum cost; maximum robustness 4 variables (B, Target reliability and deflection NSGA-II [85 ] Yücel et al., 2021 Minimum cost 5 variables ( 16 constraints (internal and external stability, maximum and minimum reinforcement) Adaptive hybrid harmony search (AHHS); AHS; HS; GA; DE; PSO; FA; ABC; TLBO; FPA; GWO; JA [86 ] Kaveh et al., 2021 Minimum cost; minimum weight 7 variables ( 14 constraints (internal and external stability, geometric requirements, and angle of footing) Plasma generation optimization (PGO); CS; TLBO [87 ] Sharma et al., 2021 Minimum cost; minimum weight 12 variables ( 16 constraints (internal and external stability, geometric requirements, and maximum and minimum reinforcement) Butterfly optimization algorithm (BOA); symbiosis organism search (SOS) algorithm; h-BOASOS [88 ] Mevada et al., 2021 Minimum cost 8 variables ( 10 constraints (internal and external stability, eccentricity) FPA; NSGA [89 ] Uray and Çarbaş, 2021 Minimum cost 5 variables ( 9 constraints (external stability in static and dynamic case, geometric requirements) HS; MABC; PSO [90 ] Tousi et al., 2021 Minimum cost; minimum weight 20 variables ( 23 constraints (internal and external stability, maximum and minimum reinforcement, development lengths, stem slope control, and minimum depth of footing) Gases Brownian motion optimization algorithm (GBMOA); bacterial foraging optimization BFOA; ACO; NLP [91 ] Eroğlu et al., 2021 Minimum cost 7 variables ( 12 constraints (internal and external stability, maximum and minimum reinforcement) JA [92 ] Uray et al., 2021 Minimum factor of safety 4 variables ( 7 constraints (external stability, slope stability, and geometric requirements) HS [93 ] Linh et al., 2021 Minimum weight 5 variables ( 12 constraints (internal and external stability, eccentricity, and geometric requirements) DE-feasibility rule-based constraint-handling (FRBCH) [94 ] Dodigović et al., 2021 Minimum cost; maximum factor of safety (sliding); maximum factor of safety (bearing) 2 variables ( 4 constraints (external stability and maximum reinforcement) NSGA-II [95 ] Tutuş et al., 2021 Minimum cost; minimum weight 12 variables ( 26 constraints (internal and external stability, geometric requirements, development lengths, spacing of reinforcement, maximum reinforcement, minimum reinforcement, and minimum cover) CS [96 ] Uray et al., 2021 Minimum weight (robust design optimization) 4 variables ( 7 constraints (external stability, slope stability, and geometric requirements) Scatter search (SS) [97 ] Yücel et al., 2021 Minimum cost 5 variables ( 16 constraints (internal stability, external stability, and maximum and minimum reinforcement) Flower pollination algorithm based artificial neural network (FPA-ANN) [98 ] Tutuş et al., 2021 Minimum cost; minimum weight 12 variables ( 30 constraints (internal and external stability, geometric requirements, development lengths, spacing of reinforcement, maximum reinforcement, minimum reinforcement, and minimum cover) Improved flower pollination algorithm (IFPA); FPA, PSO, DE; GWO [99 ] Temür, 2021 Minimum cost 24 variables ( Constraints (internal and external stability, geometric requirements, and reinforcement spacing) Hybrid teaching learning based optimization (HTLBO); TLBO; BB-BC; BBO; FPA; HS; PSO; GWO; JA; Rao-1; Rao-2; Rao-3 [100 ] Shakeel et al., 2022 Minimum cost 10 variables ( 23 constraints (internal and external stability, geometric requirements, development lengths, hook lengths, and maximum and minimum reinforcement) EA [101 ] Khajehzadeh et al., 2022 Minimum cost 8 variables ( 16 constraints (internal and external stability, eccentricity, and maximum and minimum reinforcement) Particle swarm sine-cosine algorithm (PSSCA); sine-cosine algorithm (SCA); tunicate swarm algorithm (TSA); GSA; GWO [102 ] Uray et al., 2022 Minimum cost; minimum weight (robust design optimization) 12 variables ( 26 constraints (internal and external stability, geometric requirements, development lengths, hook lengths, and maximum and minimum reinforcement) Taguchi method integrated hybrid harmony search algorithm (TIHHSA); BBO; GA; DE; ES [103 ] Khajehzadeh et al., 2022 Minimum cost 8 variables ( 17 constraints (internal stability, external stability, and maximum and minimum reinforcement) Sperm swarm optimization (SWO); adaptive sperm swarm optimization (ASSO); GSA; GWO; SCA
