Introduction
Literature review
Technique and its reference number | Name of the author and year | A quick summary |
---|---|---|
Seagull optimization algorithm (SOA) [55] | Yanhui Che and Dengxu He 2022 | This paper proposed an enhanced seagull optimization algorithm to eliminate the defects of traditional seagull optimizer. The technique is tested on 12 various engineering optimization problems |
Modified group theory-based optimization algorithms for numerical optimization (GTOA) [56] | Li et al. 2022 | This paper concentrated on studying the applicability of the proposed GTOA to solve optimization problems by introducing two versions of GTOA which uses binary coding and integer coding. The performance proved to obtain better convergence rate and average accuracy |
Criminal search optimization algorithm (CSOA) [57] | Srivastava et al. 2022 | This paper introduced criminal search optimization algorithm which has been developed based on intelligence of policemen in catching a criminal. The presentation of the technique has been evaluated on standard benchmark functions—CEC-2005 and CEC-2020. Five test cases have been operated to measure the results of the suggested algorithm with other techniques and proved the good |
Crystal structure optimization approach to problem-solving in mechanical engineering design (CryStAl) [58] | Babak Talatahari et al. 2022 | The authors of this paper introduced a metaheuristic named crystal structure algorithm to discover solutions for engineering mechanics and design problems. Further, the technique has been examined on 20 benchmark mathematical functions and obtained satisfying outputs when measured with other existing methods |
African vultures optimization algorithm (AVOA) [59] | Abdollahzadeh et al. 2021 | The authors of this paper proposed African vultures optimization algorithm imitating the living style of African vultures foraging and navigation attitude. First, the method’s feat is tested on 36 benchmark functions and its applicability is announced on finding optimum solutions for 11 engineering design problems |
Flow direction algorithm (FDA) [60] | Hojat Karami et al. 2021 | This paper focused in proposing a physics-based algorithm named Flow direction algorithm imitating flow direction in a drainage basin. The method has been tested on 13, 10 and 5 classical mathematical, new mathematical benchmark functions and engineering design problems, respectively. These results proved better than other techniques results |
A new hybrid chaotic atom search optimization based on tree-seed algorithm and Levy flight for solving optimization problems [61] | Saeid et al. 2021 | The authors in this papers used combination of metaheuristic algorithms to crack 7 special engineering issues. This atom search algorithm convergence speed is enhanced by chaotic maps as well as Levy flight random walk. Furthermore, tree-seed method ties with ASO. These combinations of algorithms yield good results |
A multi-objective optimization algorithm for feature selection problems (MOHHOFOA) [62] | Abdollahzadeh et al. 2021 | The authors in this paper used three different solutions for feature selection. First, Harris hawks optimization algorithm is multiplied; second, fruit fly optimization algorithm is multiplied, and in third stage, these two algorithms have been hybridized to locate solutions for feature selection issues |
Arithmetic optimization algorithm (AOA) [63] | Laith et al. 2021 | This paper proposed arithmetic optimization algorithm and tested its performance on 29 benchmark functions 7 real-world engineering design problems. The outcomes obtained by this technique proved better among other existing methods |
Aquila optimizer (AO) [64] | Laith Abualigah et al. 2021 | The authors suggested population-based optimization method named Aquila optimizer to solve optimization problems. The technique has been evaluated on 23 benchmark functions and 7 real-life engineering design issues. The outcomes are good than other methods |
Artificial gorilla troops optimizer (GTO) [65] | Abdollahzadeh et al. 2021 | The authors in this paper proposed Artificial gorilla troops optimizer which is designed to improve the phases of exploration and exploitation. The algorithm is examined on 52 functions and 7 engineering design problems |
Binary slime mould algorithm (BSMA) [66] | Abdel et al. 2021 | This paper proposed slime mould algorithm with 4 binary versions for feature selection. All these versions were tested on 28 datasets of UCI repository |
1D SMA models (SMAs) [67] | Sonia Marfia et al. 2021 | This paper elevates the SMA 1D models to elucidate response of SMAs in thermo mechanical models |
Slime mould algorithm (SMA) [68] | Davut Izci et al. 2021 | Tested on several benchmark functions. Using PID controllers, the capability of SMA optimization is enhanced |
Hybrid improved slime mould algorithm with adaptive β hill climbing (BTβSMA) [4] | Kangjian Sun et al. 2021 | Tested on 16 benchmark functions and is suggested to lighten the unfledged global and local hunt in standard SMA |
Archerfish hunting optimizer (AHO) [69] | Farouq Zitouni et al. 2021 | Tested on 10 benchmark functions, 5 engineering problems. AHO replicates the behavior of Archerfish like jumping and shooting to find closer optimum values |
WLSSA [70] | Hao Ren et al. 2021 | Tested on 23 benchmark functions. With the combination of slap swarm and weight adaptive Levy flight have noticed finer optimum values |
Multi-temperature simulated annealing algorithm (MTSA) [71] | Shih-Wei-Lin et al. 2021 | This algorithm is developed to reduce the scheduling issues which influence the design and optimization of automated systems |
Self-adaptive salp swarm algorithm (SASSA) [72] | Rohith Salgotra et al. 2021 | Salp swarm algorithm is improved to mould it into self-adaptive by supplementing it by four modifications which possess in improving local search |
Simulated annealing with Gaussian mutation and distortion equalization algorithm (SAGMDE) [73] | Julian Lee et al. 2020 | This combined algorithm applied on different data sets yields better results in exploratory phase when only simulated annealing algorithm was applied |
Slime mould algorithm (SMA) [3] | Shimin Li et al. 2020 | Tested on 33 benchmark functions It replicated the characteristics of slime mould. SMA, intended to give better exploration capability and extending its application in kernel extreme learning machine |
Hybrid grey wolf optimization–slime mould algorithm (GWO-SMA) [5] | Zheng-Ming Gao et al. 2020 | Made 3 types of experiments resulting it did not give better results in combining GWO and SMA. SMA equations were unique and excellent and firm to progress |
Chaotic SMA–Chebyshev map [7] | Juan Zhao et al. 2020 | Tested on standard benchmark functions and noticed the results to be better and the technique performed faster with stability |
Improved slime mould algorithm with Levy flight [6] | Juan Zhao et al. 2020 | Worked to reduce the pressure of randomness and noticed SMA–Levy flight with uniform distributed parameters would give better results |
Modified slime mould algorithm via Levy flight [8] | Zhesen Cui et al. 2020 | Tested on 13 benchmark functions and 1 engineering design issue and notice the results obtained were better and steady |
Hybridized Harris hawks optimization and slime mould algorithm (HHO-SMA) [10] | Juan Zhao et al. 2020 | The research attentively made efforts on many updating discipline mostly individuals on swarms |
Improved slime mould algorithm with cosine controlling parameter [11] | Zheng-Ming Gao et al. 2020 | The research helped in finding that the controlling parameters are very essential for the technique to perform better, at the same time noticed that all parameters were not acceptably helpful. Hence, should find more apt method |
Multi-objective slime mould algorithm based on elitist non-dominated sorting (MOSMA) [13] | Manoharan Premkumar et al. 2020 | Tested on 41 various cases, constrained, unconstrained as well as on real-life engineering issues. On applying this algorithm resulted in high-quality and effectiveness solutions for tough multi-objective issues |
PSA: a photon search algorithm [74] | Y. Liu and Li 2020 | 23 functions were put to the test. The characteristics of photons in physics were the inspiration for this piece. The algorithm has strong global search and convergence capabilities |
Movable damped wave algorithm [75] | Rizk et al. 2019 | This paper proposed movable damped wave algorithm and the algorithm has been examined on 23 benchmark functions and 3 engineering design problems |
Henry gas solubility optimization: a novel physics-based algorithm (HGSO) [76] | Hashim et al. 2019 | 47 benchmark functions were used in the testing. It is modeled after Henry’s reign. HGSO, which aims to meet the check room and halt optima locale’s production and conservation capacities |
Emperor penguins colony (EPC) [77] | Sasan et al. 2019 | A new metaheuristic algorithm named emperor penguins colony is proposed in this paper and has been tested on 10 benchmark functions |
Harris hawks optimization (HHO) [78] | Heidari et al. 2019 | There were 29 benchmarks and 6 technical issues which were tested on. It is being introduced to help with various optimization chores. Nature’s cooperative behaviors, as well as the patterns of predatory birds hunting Harris’ hawks, impact the strategy |
Tree growth algorithm (TGA) [79] | Armin et al. 2018 | The authors introduced Tree growth algorithm which is inspired by trees competition for acquiring light and food. It has been examined on 30 benchmark functions and 5 engineering design problems |
Hybrid artificial bee colony with monarch butterfly optimization [80] | Waheed et al. 2018 | This paper introduced a new algorithm named hybrid ABC/MBO (HAM) and evaluated on 13 benchmark functions and proved better in outcomes |
An improved hybrid firefly algorithm for solving optimization problems (IHFA) [81] | Fazli Wahid et al. 2018 | This paper introduced a novel method called GA-FA-PS algorithm and tested on 3 benchmark functions and proved that the obtained results are better than firefly algorithm and genetic algorithm |
An improved butterfly optimization algorithm with chaos [82] | Sankalap Arora et al. 2017 | The authors in this paper improved butterfly optimization with chaos to increase its performance to avoid local optimum and convergence speed. The suggested chaotic BOAs are validated 3 benchmark functions and 3 engineering design problems |
Cuckoo search algorithm–hill climbing technique (CSAHC) [83] | Shehab et al. 2017 | The authors proposed new cuckoo search algorithm by hybridizing with hill climbing technique to solve optimization issues. It has been examined on 13 benchmark functions and proved successful |
Hybrid GWO-SCA [84] | Singh et al. 2017 | This paper proposed hybrid grey wolf optimizer and sine cosine technique and tested on 22 functions, 5 bio-medical dataset and 1 sine dataset problems |
SCA: a sine cosine algorithm for solving optimization problems [85] | Seyedali Mirjalili 2016 | The author proposed SCA for the solutions of optimization problems and its efficiency is validated on testing 19 benchmark functions |
Lion optimization algorithm (LOA): a nature-inspired metaheuristic algorithm [86] | Maziar Yazdani et al. 2016 | This paper introduced Lion optimization algorithm and has been examined on 30 benchmark functions |
Crow search algorithm (CSA) [87] | Alireza Askarzadeh 2016 | The author proposed crow search algorithm and applied to unravel 6 engineering design issues. The outputs were promising than existing methods |
Stochastic fractal search: a powerful metaheuristic algorithm (SFS) [88] | Salimi 2015 | Uni, multi, fixed functions, and engineering functions were all tested |
Moth flame optimization algorithm: a novel nature-inspired heuristic paradigm (MFO) [34] | Mirjalili 2015 | 7 engineering designs were tested, as well as 29 benchmarks. This optimizer followed navigation tactic of moth flame. The outcomes of this method stood better than existing techniques |
Solving optimization problems using black hole algorithm (BHA) [89] | Masoum Farahmandian et al. 2015 | This paper suggested black hole algorithm and has been checked on 19 benchmark functions. These results were better than PSO and GA |
Forest optimization algorithm FOA [90] | Ghaemi et al. 2014 | This technique is for determining the utmost as well as minimum value using a practical appliance, as well as demonstrating that the FOA can generally solve that are acceptable |
Grey wolf optimizer (GWO) [39] | Mirjalili, Mirjalili, and Lewis 2014 | The researchers looked at 29 BFs and 3 optimization engineering-based approaches. The image was enthused by a swarm-intelligence optimization and was inspired by grey wolves. Grey wolves’ communal structure and hunting conduct were used to develop the suggested model |
Cuckoo search algorithm using Lèvy flight: a review [91] | Sangita Roy et al. 2013 | The authors in this paper discussed about cuckoo search algorithm using Levy flight algorithm and noticed that the presentation of this method is superior to particle swarm optimizer and genetic algorithm when examined on 10 benchmark functions |
Firefly algorithm: recent advances and applications (FA) [92] | Xin-She Yang et al. 2013 | This paper suggested firefly algorithm, its fundamentals and explained the balancing of exploration and exploitation phases. In addition, the technique has been tested on higher-dimensional optimization problems |
Bat algorithm: literature review and applications (BA) [93] | Xin-She 2013 | The author presented the literature review and applications of Bat methodology which is efficient for solving optimization issues |
Penguins search optimization algorithm (PeSOA) [94] | Youcef Gheraibia et al. 2013 | This paper presented penguins search optimization algorithm and tested on 3 benchmark functions and obtained better results |
Teaching–learning-based optimization (TLBO) [40] | Rao et al. 2012 | In a power system, TLBO has two stages: a teaching stage and a student stage. Interacting by way of both is feasible only via modification, and the issue is solved |
Krill herd (KH) [95] | A. H. Gandomi et al. 2012 | This paper proposed krill herd algorithm which is a biologically inspired algorithm. Tested on several benchmark functions |
Flower pollination algorithm (FPA) [96] | Xin-She Yang 2012 | The author in this paper proposed flower pollination method which is motivated by the procedure of pollination in flowers. The technique has been tested on 10 benchmark functions and 1 nonlinear design problem. The results were better than PSO and GA methods |
A hybrid CS/PSO algorithm for global optimization [97] | Ghodrati et al. 2012 | The authors in this paper presented hybrid CS/PSO method to crack optimization issues. The technique has been examined on many benchmark functions to prove better than other techniques |
Biogeography-based optimization (BBO) [27] | Simon 2008 | 14 typical benchmark functions were used in the testing. The BBO method, which analyses the spatial distribution of biological species, may be used to derive optimization algorithms |
A new heuristic optimization technique: harmony search (HS) [98] | Geem, Kim, and Loganathan 2001 | The comparison of the music creation cycle inspired this algorithm. The starting values of the variables may not be required for HS to make a decision |
Differential evolution (DE) [99] | Storn et al. 1997 | It shows how to minimize nonlinear and non differentiable continuous space functions that are possibly nonlinear. It merely needs a few strong control variables drawn from a predetermined numerical range |
Tabu search-part I (TS) [100] | Fred Glover 1989 | This has originated as a method of resolving combinatorial real-world scheduling and covering challenges |
Background of suggested work
Literature survey on slime mould algorithm variants and simulated annealing algorithm variants
Proposed hybridized slime mould algorithm-simulated annealing algorithm
Slime mould algorithm
Mathematical modeling of slime mould algorithm
Simulated annealing algorithm
Slime mould-simulated annealing algorithm
Standard benchmark functions
Functions | Dimensions | Range | fmin |
---|---|---|---|
\(F_{1} (U) = \sum\nolimits_{m = 1}^{z} {U_{m}^{2} }\) | 30 | [− 100, 100] | 0 |
\(F_{2} (U) = \sum\nolimits_{m = 1}^{z} {\left| {U_{m} } \right|} + \mathop \prod \nolimits_{m = 1}^{z} \left| {U_{m} } \right|\) | 30 | [− 10,10] | 0 |
\(F_{3} (U) = \sum\nolimits_{m = 1}^{z} {\left( {\sum\nolimits_{n - 1}^{m} {U_{n} } } \right)^{2} }\) | 30 | [− 100, 100] | 0 |
\(F_{4} (U) = \max_{m} \{ \left| {U_{m} } \right|,1 \le m \le z\}\) | 30 | [− 100, 100] | 0 |
\(F_{5} (U) = \sum\nolimits_{m = 1}^{z - 1} {[100(U_{m + 1} } - U_{m}^{2} )^{2} + (U_{m} - 1)^{2} ]\) | 30 | [− 38, 38] | 0 |
\(F_{6} (U) = \sum\nolimits_{m = 1}^{z} {([U_{m} } + 0.5])^{2}\) | 30 | [− 100, 100] | 0 |
\(F_{7} (U) = \sum\nolimits_{m = 1}^{z} {mU_{m}^{4} } + {\text{random}}[0,1]\) | 30 | [− 1.28, 1.28] | 0 |
Multimodal bench mark functions | Dim | Range | \(f_{\min }\) |
---|---|---|---|
\(F_{8} (U) = \sum\nolimits_{m = 1}^{z} { - U_{m} } \sin (\sqrt {\left| {U_{m} } \right|} )\) | 30 | [− 500, 500] | − 418.98295 |
\(F_{9} (U) = \sum\nolimits_{m = 1}^{z} {[U_{m}^{2} - 10\cos (2\pi U_{m} ) + 10]}\) | 30 | [− 5.12, 5.12] | 0 |
\(F_{10} (U) = - 20\exp \left( { - 0.2\sqrt {\left( {\frac{1}{z}\sum\nolimits_{m = 1}^{z} {U_{m}^{2} } } \right)} } \right) - \exp \left( {\frac{1}{z}\sum\nolimits_{m = 1}^{z} {\cos (2\pi U_{m} } } \right) + 20 + d\) | 30 | [− 32, 32] | 0 |
\(F_{11} (U) = 1 + \sum\nolimits_{m = 1}^{z} {\frac{{U_{m}^{2} }}{4000}} - \prod_{m - 1}^{z} \cos \frac{{U_{m} }}{\sqrt m }\) | 30 | [− 600, 600] | 0 |
\(\begin{aligned} F_{12} (U) & = \frac{\pi }{z}\left\{ {10\sin (\pi \tau_{1} ) + \sum\nolimits_{m = 1}^{z - 1} {(\tau_{m} - 1)^{2} [1 + 10\sin^{2} (\pi \tau_{m + 1} )] + (\tau_{z} - 1)^{2} } } \right\} \\ & \quad + \sum\nolimits_{m = 1}^{z} {g(U_{m} ,10,100,4)} \\ \end{aligned}\) where \(\tau_{m} = 1 + \frac{{U_{m} + 1}}{4}\) \(g(U_{m} ,b,x,i) = \left\{ {0\begin{array}{*{20}c} {x(U_{m} - b)^{i} U_{m} > b} \\ { - b < U_{m} < b} \\ {x( - U_{m} - b)^{i} U_{m} < - b} \\ \end{array} } \right\}\) | 30 | [− 50, 50] | 0 |
\(F_{13} (U) = 0.1\left\{ {\sin^{2} (3\pi U_{m} ) + \sum\nolimits_{m = 1}^{z} {(U_{m} - 1)^{2} [1 + \sin^{2} (3\pi U_{m} + 1)] + (x_{z} - 1)^{2} [1 + \sin^{2} ]} } \right\}\) | 30 | [− 50, 50] | 0 |
Fixed-dimension (FD) benchmark functions | Dimension | Range | \(f_{\min }\) |
---|---|---|---|
\(F_{14} (U) = \left[ {\frac{1}{500} + \sum\nolimits_{n = 1}^{2} 5 \frac{1}{{n + \sum\nolimits_{m = 1}^{z} {(U_{m} - b_{mn} )6} }}} \right]^{ - 1}\) | 2 | [− 65.536, 65.536] | 1 |
\(F_{15} (U) = \sum\nolimits_{m = 1}^{11} {\left[ {b_{m} - \frac{{U_{1} (a_{m}^{2} + a_{m} \eta_{2} )}}{{a_{m}^{2} + a_{m} \eta_{3} + \eta_{4} }}} \right]}^{2}\) | 4 | [− 5, 5] | 0.00030 |
\(F_{16} (U) = 4U_{1}^{2} - 2.1U_{1}^{4} + \frac{1}{3}U_{1}^{6} + U_{1} U_{2} - 4U_{2}^{2} + 4U_{2}^{4}\) | 2 | [− 5, 5] | − 1.0316 |
\(F_{17} (U) = \left( {U_{2} - \frac{5.1}{{4\pi 2}}U_{1}^{2} + \frac{5}{\pi }U_{1} - 6} \right)^{2} + 10\left( {1 - \frac{1}{8\pi }} \right)\cos U_{1} + 10\) | 2 | [− 5, 5] | 0.398 |
\(\begin{gathered} F_{18} (U) = [1 + (U_{1} + U_{2} + 1)^{2} (19 - 14U_{1} + 3U_{1}^{2} - 14U_{2} + 6U_{1} U_{2} + 3U_{2}^{2} )] \hfill \\ x[30 + (2U_{1} - 3U_{2} )^{2} x(18 - 32U_{1} + 12U_{1}^{2} + 48U_{2} - 36U_{1} U_{2} + 27U_{2}^{2} )] \hfill \\ \end{gathered}\) | 2 | [− 2,2] | 3 |
\(F_{19} (U) = - \sum\nolimits_{m = 1}^{4} {d_{m} \exp ( - \sum\nolimits_{n = 1}^{3} {U_{mn} (U_{m} - q_{mn} )^{2} } } )\) | 3 | [1, 3] | − 3.32 |
\(F_{20} (U) = - \sum\nolimits_{m = 1}^{4} {d_{m} } \exp ( - \sum\nolimits_{n = 1}^{6} {U_{mn} (U_{m} } - q_{mn} )^{2} )\) | 6 | [0, 1] | − 3.32 |
\(F_{21} (U) = - \sum\nolimits_{m = 1}^{5} {[(U - b_{m} )(U - b_{m} )^{T} + d_{m} } ]^{ - 1}\) | 4 | [0,10] | − 10.1532 |
\(F_{22} (U) = - \sum\nolimits_{m = 1}^{7} {[(U - b_{m} )(U - b_{m} )^{T} } + d_{m} ]^{ - 1}\) | 4 | [0, 10] | − 10.4028 |
\(F_{23} (U) = - \sum\nolimits_{m = 1}^{7} {[(U - b_{m} )(U - b_{m} )^{T} } + d_{m} ]^{ - 1}\) | 4 | [0, 10] | − 10.5363 |
Parameter setting | hSMA-SA |
---|---|
Search agents | 30 |
Count of iterations for benchmark problems (unimodal, multimodal and fixed dimension) | 500 |
Count of iterations for engineering optimal designs | 500 |
Count of trial runs for each function and engineering optimal designs | 30 |
Results and analysis
Evaluation of unimodal functions (exploitation)
Function | Mean | Standard deviation | Best fitness value | Worst fitness value | Median | Wilcoxon rank sum test | t test | |
---|---|---|---|---|---|---|---|---|
p value | p value | h value | ||||||
Sphere function (F1) | 1.4E−300 | 0 | 0 | 4.2059E−299 | 0 | 0.125 | 0 | 1 |
Schwefel absolute function (F2) | 1.9E−159 | 1.0209E−158 | 0 | 5.5926E−158 | 4.6226E−200 | 1.7344E−06 | 0.32386916 | 0 |
Schwefel double sum function (F3) | 6.1E−121 | 3.309E−120 | 0 | 1.8126E−119 | 2.5788E−151 | 2.56308E−06 | 0.324043719 | 0 |
Schwefel max. function (F4) | 3.6E−163 | 2.2228E−162 | 1.0007E−262 | 1.088E−161 | 4.4998E−199 | 1.7344E−06 | 0.378435872 | 0 |
Rosenbrock function (F5) | 11.77822 | 12.89147042 | 0.011463167 | 28.35510393 | 3.286513444 | 1.7344E−06 | 2.50694E−05 | 1 |
The step function (F6) | 0.007059 | 0.004749705 | 0.001719961 | 0.021628358 | 0.005572663 | 1.73E−06 | 5.63E−09 | 1 |
Function | Best time | Average time | Worst time |
---|---|---|---|
Sphere function (F1) | 190.3906 | 219.1385417 | 297.39063 |
Schwefel absolute function (F2) | 122.7188 | 133.9958333 | 155.51563 |
Schwefel double sum function (F3) | 137.2188 | 144.546875 | 162.70313 |
Schwefel max. function (F4) | 111.875 | 176.4671875 | 307.20313 |
Rosenbrock function (F5) | 90.25 | 112.4067708 | 150.95313 |
The step function (F6) | 195.25 | 212.9197917 | 327.8125 |
Algorithm | Parameters | Unimodal Benchmark functions | |||||
---|---|---|---|---|---|---|---|
sphere function (F1) | Schwefel absolute function (F2) | Schwefel double sum function (F3) | Schwefel max. function (F4) | Rosenbrock function (F5) | The step function (F6) | ||
Lightning search algorithm (LSA) [144] | Avg | 4.81067E−08 | 3.340000000 | 0.024079674 | 0.036806544 | 43.24080402 | 1.493275733 |
S. deviation | 3.40126E−07 | 2.086007800 | 0.005726198 | 0.156233023 | 29.92194448 | 1.302827039 | |
Dragonfly algorithm (DA) [146] | Avg | 2.850E−19 | 1.490E−06 | 1.290E−07 | 9.88E−04 | 7.6 | 4.170E−17 |
S. deviation | 7.160E−19 | 3.760E−06 | 2.100E−07 | 2.78E−03 | 6.79 | 1.320E−16 | |
Battle Royale optimization algorithm (BRO) [145] | Avg | 3.0353E−09 | 0.000046 | 54.865255 | 0.518757 | 99.936848 | 2.8731E−08 |
S. deviation | 4.1348E−09 | 0.000024 | 16.117329 | 0.403657 | 82.862958 | 1.8423E−08 | |
Multi-verse optimizer (MVO) [24] | Avg | 2.08583 | 15.9247 | 453.200 | 3.12301 | 1272.13 | 2.29495 |
S. deviation | 0.64865 | 44.7459 | 177.0973 | 1.58291 | 1479.47 | 0.63081 | |
Opposition-based enhanced grey wolf optimization algorithm (OEGWO) [147] | Avg | 2.49 × 10–34 | 4.90 × 10–25 | 1.01 × 10–1 | 1.90 × 10–5 | 2.72 × 101 | 1.40 × 1000 |
S. deviation | 7.90 × 10–34 | 6.63 × 10–25 | 3.21 × 10–1 | 2.43 × 10–5 | 7.85 × 101 | 4.91 × 10–1 | |
Particle swarm optimization (PSO) [151] | Avg | 1.3E−04 | 0.04214 | 7.01256E+01 | 1.08648 | 96.7183 | 0.00010 |
S. deviation | 0.0002.0E−04 | 0.04542 | 2.1192E+01 | 3.1703E+01 | 6.01155E+01 | 8.28E−05 | |
Photon search algorithm (PSA) [74] | Avg | 15.3222 | 2.2314 | 3978.0837 | 1.1947 | 332.6410 | 19.8667 |
S. deviation | 27.3389 | 1.5088 | 3718.9156 | 1.0316 | 705.1589 | 33.4589 | |
Sine–cosine algorithm (SCA) [102] | Avg | 0.000 | 0.000 | 0.0371 | 0.0965 | 0.0005 | 0.0002 |
S. deviation | 0.000 | 0.0001 | 0.1372 | 0.5823 | 0.0017 | 0.0001 | |
Hybrid Harris hawks optimizer–pattern search algorithm (hHHO-PS) [50] | Avg | 9.2 × 10–017 | 8.31E | 5.03 × 10–20 | 6.20 × 10–54 | 2.18 × 10–9 | 3.95 × 10–14 |
S. deviation | 5E−106 | 4.46 × 10–53 | 1.12 × 10–19 | 1.75 × 10–53 | 6.38 × 10–10 | 3.61 × 10–14 | |
Ant lion optimizer (ALO) [152] | Avg | 2.59E−10 | 1.84E−06 | 6.07E−10 | 1.36E−08 | 0.3467724 | 2.56E−10 |
S. deviation | 1.65E−10 | 6.58E−07 | 6.34E−10 | 1.81E−09 | 0.10958 | 1.09E−10 | |
Spotted hyena optimizer (SHO) [46] | Avg | 0 | 0 | 0 | 7.78E−12 | 8.59E+00 | 2.46E−01 |
S. deviation | 0 | 0 | 0 | 8.96E−12 | 5.53E−01 | 1.78E−01 | |
Moth flame optimizer (MFO) [34] | Avg | 0.00011 | 0.00063 | 696.730 | 70.6864 | 139.1487 | 0.000113 |
S. deviation | 0.00015 | 0.00087 | 188.527 | 5.27505 | 120.2607 | 9.87E−05 | |
Harris hawks optimizer (HHO) [78] | Avg | 1.06 × 10–90 | 6.92 × 10–51 | 1.25 × 10–80 | 4.46 × 10–48 | 0.015002 | 0.000115 |
S. deviation | 5.82 × 10–90 | 2.47 × 10–50 | 6.63 × 10–80 | 1.70 × 10–47 | 0.023473 | 0.000154 | |
Grey wolf optimizer (GWO) [149] | Avg | 6.590E−29 | 7.180E−18 | 3.20E−−07 | 5.610E−08 | 26.8125 | 0.81657 |
S. deviation | 6.3400E−07 | 0.02901 | 7.9.1495E+01 | 1.31508 | 69.9049 | 0.00012 | |
Enhanced crow search algorithm (ECSA) [150] | Avg | 7.4323E−119 | 5.22838E−59 | 3.194E−102 | 3.04708E−52 | 7.996457081 | 0.400119079 |
S. deviation | 4.2695E−118 | 2.86361E−58 | 1.7494E−101 | 1.66895E−51 | 0.661378213 | 0.193939866 | |
Salp swarm algorithm (SSA) [148] | Avg | 0.000 | 0.2272 | 0.000 | 0.000 | 0.000 | 0.000 |
S. deviation | 0.000 | 1.000 | 0.000 | 0.6556 | 0.000 | 0.000 | |
Transient search optimization (TSO) [109] | Avg | 1.18 × 10–99 | 8.44 × 10–59 | 3.45 × 1041 | 1.28E−53 | 8.10 × 10–2 | 3.35 × 10–3 |
S. deviation | 6.44 × 10–99 | 3.93 × 10–58 | 1.26 × 10–41 | 6.58 × 10–53 | 11 | 6.82 × 10–3 | |
LF-SMA [8] | Avg | 1.58E−156 | 2.74E−171 | 5.2412 | 0.0006 | 5.90E−05 | 0.0008 |
S. deviation | 7.53E−156 | 0 | 10.229 | 0.0002 | 6.38E−05 | 0.0008 | |
Proposed algorithm hSMA-SA | Avg | 1.4E−300 | 1.9E−159 | 6.1E−121 | 3.6E−163 | 11.77822 | 0.007059 |
S. deviation | 0 | 1.0209E−158 | 3.309E−120 | 2.2228E−162 | 12.89147042 | 0.004749705 |
Function | Mean | Standard deviation | Best fitness value | Worst fitness value | Median | Wilcoxon rank sum test | t test | |
---|---|---|---|---|---|---|---|---|
p value | p value | h value | ||||||
Schwefel sine function (F8) | − 12,569.02623 | 0.43623993 | − 12,569.48529 | − 12,567.9831 | − 12,569.15434 | 1.73E−06 | 4.22E−131 | 1 |
Rastrigin function (F9) | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
The Ackley function (F10) | 8.88E−16 | 0 | 8.88E−16 | 8.88E−16 | 8.88E−16 | 4.32E−08 | 0 | 1 |
Penalized penalty#1 function (F12) | 0.012678845 | 0.012483005 | 9.69E−05 | 0.039727727 | 0.007266933 | 1.73E−06 | 5.31E−06 | 1 |
Levi N. 13 function (F13) | 0.002689783 | 0.001733379 | 0.000388677 | 0.007871366 | 0.002737656 | 1.73E−06 | 2.30E−09 | 1 |
Evaluation of a few multimodal functions (exploration)
Function | Best time | Average time | Worst time |
---|---|---|---|
Schwefel sine function (F8) | 659.2031 | 666.775 | 686.51563 |
Rastrigin function (F9) | 513.2031 | 532.9604167 | 586.29688 |
The Ackley function (F10) | 566.0469 | 575.2671875 | 599.53125 |
Penalized penalty#1 function (F12) | 2008.016 | 2101.541146 | 2290.125 |
Levi N. 13 function (F13) | 2062.094 | 2145.145313 | 2304.7031 |
Algorithm | Parameters | Multimodal benchmark functions | ||||
---|---|---|---|---|---|---|
Schwefel sine function (F8) | Rastrigin function (F9) | The Ackley function (F10) | Penalized penalty#1 function (F12) | Levi N. 13 function (F13) | ||
Lightning search algorithm (LSA) [144] | Avg | − 8001.3887 | 62.7618960 | 1.077446947 | 2.686199995 | 0.007241875 |
St. deviation | 669.159310 | 14.9153021 | 0.337979509 | 0.910802774 | 0.006753356 | |
Dragonfly algorithm (DA) [146] | Avg | − 2.860E+03 | 1.600E+01 | 2.310E−01 | 3.110E−02 | 2.200E−03 |
St. deviation | 3.840E+02 | 9.480E+00 | 4.870E−01 | 9.830E−02 | 4.630E−03 | |
Battle Royale optimization algorithm (BRO) [145] | Avg | − 7035.2107 | 48.275350 | 0.350724 | 0.369497 | 0.000004 |
St. deviation | 712.33269 | 14.094585 | 0.688702 | 0.601450 | 0.000020 | |
Multi-verse optimizer (MVO) [24] | Avg | − 1.170E+04 | 1.180E+02 | 4.070E+00 | 2.460E+00 | 2.200E−01 |
St. deviation | 9.370E+02 | 3.930E+01 | 5.500E+00 | 7.900E−01 | 9.000E−02 | |
Opposition-based enhanced grey wolf optimization algorithm (OEGWO) [147] | Avg | − 3.36 × 103 | 8.48 × 10–1 | 9.41 × 10–15 | 9.36 × 10–02 | 1.24E+00 |
St. deviation | 3.53 × 102 | 4.65E+00 | 3.56 × 10–15 | 3.95 × 10–02 | 2.09 × 10–1 | |
Particle swarm optimization (PSO) [151] | Avg | − 4.8400E+04 | 4.670E+01 | 2.760E−01 | 6.9200E−04 | 6.6800E−04 |
St. deviation | 1.1500E+04 | 1.160E+01 | 5.090E−01 | 2.6300E−03 | 8.9100E−04 | |
Photon search algorithm (PSA) [74] | Avg | 11,648.5512 | 7.3763 | 1.6766 | 0.1716 | 1.5458 |
St. deviation | 1230.4314 | 9.1989 | 0.9929 | 0.2706 | 3.3136 | |
Sine–cosine algorithm (SCA) [102] | Avg | 1.000E+00 | 0.000E+00 | 3.800E−01 | 0.000E+00 | 0.000E+00 |
St. deviation | 3.600E−03 | 7.300E−01 | 1.000E+00 | 0.000E+00 | 0.000E+00 | |
Hybrid Harris hawks optimizer–pattern search algorithm (hHHO-PS) [50] | Avg | − 12,332 | 00 | 8.88 × 10–6 | 2.94 × 10–15 | 1.16 × 10–13 |
St. deviation | 335.7988 | 0 | 0 | 3.52E−15 | 1.15E−13 | |
Ant lion optimizer (ALO) [152] | Avg | − 1.61E+03 | 7.71E−06 | 3.73E−15 | 9.75E−12 | 2.00E−11 |
St. deviation | 3.14E+02 | 8.45E−06 | 1.50E−15 | 9.33E−12 | 1.13E−11 | |
Spotted hyena optimizer (SHO) [46] | Avg | − 1.16E × 103 | 0.00E+00 | 2.48E+000 | 3.68 × 10–2 | 9.29 × 10–1 |
St. deviation | 2.72E × 102 | 0.00E+00 | 1.41E+000 | 1.15 × 10–2 | 9.52 × 10–2 | |
Moth flame optimizer (MFO) [34] | Avg | − 8.500E+03 | 8.460E+01 | 1.260E+00 | 8.940E−01 | 1.160E−01 |
St. deviation | 7.260E+02 | 1.620E+01 | 7.300E−01 | 8.810E−01 | 1.930E−01 | |
Harris hawks optimizer (HHO) [78] | Avg | − 12,561.38 | 0 | 8.88 × 10–16 | 8.92 × 10–6 | 0.000101 |
St. deviation | 40.82419 | 0 | 0 | 1.16 × 10–5 | 0.000132 | |
Grey wolf optimizer (GWO) [149] | Avg | − 6.1200E+02 | 3.1100E−02 | 1.0600E−14 | 5.3400E−03 | 6.5400E−02 |
St. deviation | − 4.0900E+02 | 4.740E+01 | 7.7800E−03 | 2.0700E−03 | 4.470E−03 | |
Enhanced crow search algorithm (ECSA) [150] | Avg | − 2332.3867 | 0 | 8.88178E−16 | 0.11738407 | 0.444690657 |
St. deviation | 223.93995 | 0 | 0 | 0.2849633 | 0.199081675 | |
Salp swarm algorithm (SSA) [148] | Avg | 5.570E−02 | 0.000E+00 | 1.950E−01 | 1.420E−01 | 8.320E−02 |
St. deviation | 8.090E−01 | 0.000E+00 | 1.530E−01 | 5.570E−01 | 7.060E−01 | |
Transient search optimization (TSO) [109] | Avg | − 12,569.5 | 00 | 8.88 × 10–16 | 1.30 × 10–4 | 7.55 × 10–4 |
St. deviation | 1.81 × 10–2 | 00 | 0 | 1.67 × 10–4 | 1.74 × 10–3 | |
LF-SMA [8] | Avg | 0.0004 | − 3.2865 | − 17.363 | 0.0130 | 1.07E−12 |
St. deviation | 5.89E−05 | 0.0536 | 2.1907 | 0.0090 | 2.29E−12 | |
Proposed algorithm hSMA-SA | Avg | − 12,569.02623 | 0 | 8.88E−16 | 0.012678845 | 0.002689783 |
St. deviation | 0.43623993 | 0 | 0 | 0.012483005 | 0.001733379 |
Evaluation of a few fixed-dimension functions
Function | Mean | Standard deviation | Best fitness value | Worst fitness value | Median | Wilcoxon rank sum test | t test | |
---|---|---|---|---|---|---|---|---|
p value | p value | h value | ||||||
Brad function (F15) | 0.00057313 | 0.000284537 | 0.000308341 | 0.001243214 | 0.000443492 | 1.7344E−06 | 6.78516E−12 | 1 |
Camel function—six hump (F16) | − 1.031628453 | 5.42E−10 | − 1.031628453 | − 1.031628451 | − 1.031628453 | 1.73E−06 | 7.16E−271 | 1 |
Branin RCOS function (F17) | 0.397887411 | 8.70E−08 | 0.397887358 | 0.397887735 | 0.397887381 | 1.73E−06 | 6.34E−195 | 1 |
Goldstein-price function (F18) | 3 | 2.77E−11 | 3 | 3 | 3 | 1.73E−06 | 0 | 1 |
Hybrid composition function #3 (F23) | − 8.732008933 | 3.082499131 | − 10.52915056 | − 5.172702813 | − 10.49417343 | 0.25 | 0.039117859 | 1 |
Function | Best time | Average time | Worst time |
---|---|---|---|
Brad function (F15) | 53.20313 | 56.91614583 | 65.640625 |
Camel function—six hump (F16) | 43.98438 | 46.20729167 | 51.90625 |
Branin RCOS function (F17) | 42.40625 | 44.60052083 | 47.453125 |
Goldstein-price function (F18) | 20.79688 | 21.53385417 | 22.5625 |
Hybrid composition function #3 (F23) | 0.078125 | 0.229166667 | 0.484375 |
Algorithm | Parameters | Fixed-dimension benchmark functions | ||||
---|---|---|---|---|---|---|
Brad function (F15) | Camel function—six hump (F16) | Branin RCOS function (F17) | Goldstein-price function (F18) | Hybrid composition function #3 (F23) | ||
Lightning search algorithm (LSA) [144] | Mean | 0.024148546 | 0.000534843 | − 1.031628453 | 3.000000000 | − 7.910438367 |
St. deviation | 0.047279168 | 0.000424113 | 0.000000000 | 3.34499E−15 | 3.596042666 | |
Enhanced crow search algorithm (ECSA) [150] | Mean | 0.000327 | − 1.03161 | 0.397993 | 3.00003 | − 10.5359 |
St. deviation | 1.24337E−05 | 2.20378E−05 | 1.16E−04 | 2.752E−05 | 4.62E−04 | |
Salp swarm algorithm (SSA) [148] | Mean | 0.0000 | 0.1952 | 0.0000 | 0.1417 | N/A |
St. deviation | 0.0000 | 0.1527 | 0.0651 | 0.5571 | N/A | |
Multi-verse optimizer (MVO) [24] | Mean | 30.00705 | 50.00061 | 190.3 | 160.5312 | N/A |
St. deviation | 48.30615 | 52.70461 | 128.6659 | 158.2887 | N/A | |
Transient search optimization (TSO) [109] | Mean | 9.01 × 10–4 | − 1.06 × 10–1 | 3.97 × 10–1 | 3.00E+000 | 10.5267 |
St. deviation | 1.06 × 10–4 | 2.86 × 10–11 | 2.46 × 10–1 | 9.05E+000 | 2.63 × 10–2 | |
Particle swarm optimization (PSO) [151] | Mean | 0.4081 | 0.6181 | 0.4694 | 0.3566 | N/A |
St. deviation | 0.8317 | 0.5347 | 0.8406 | 0.7841 | N/A | |
Photon search algorithm (PSA) [74] | Mean | 0.0077 | − 1.036 | 0.3979 | 3 | − 9.8189 |
St. deviation | 0.0224 | 2.33 × 10–7 | 1.41 × 10–7 | 1.36 × 10–5 | 1.8027 | |
Sine–cosine algorithm (SCA) [102] | Mean | 0.0230 | 0.0497 | 0.0000 | 0.0129 | N/A |
St. deviation | 0.0676 | 0.4921 | 0.1105 | 0.0134 | N/A | |
Hybrid Harris hawks optimizer–pattern search algorithm (hHHO-PS) [50] | Mean | 0.000307 | − 1.03163 | 0.397887 | 3 | − 10.5364 |
St. deviation | 1.65 × 10–13 | 1.11 × 10–16 | 00 | 2.63 × 10–15 | 7.69 × 10–15 | |
Ant lion optimizer (ALO) [152] | Mean | 14.56498 | 175.1532 | 316.0686 | 4.399206 | N/A |
St. deviation | 32.22876 | 46.50001 | 13.02047 | 1.66107 | N/A | |
Spotted hyena optimizer (SHO) [46] | Mean | 2.70 × 10–3 | − 1.0316 | 0.398 | 3.000 | − 1.68E+000 |
St. deviation | 5.43 × 10–3 | 5.78 × 10–14 | 1.26 × 10–14 | 2.66 × 10–13 | 2.64 × 10–1 | |
Moth flame optimizer (MFO) [34] | Mean | 66.73272 | 119.0146 | 345.4688 | 10.4086 | N/A |
St. deviation | 53.22555 | 28.3318 | 43.11578 | 3.747669 | N/A | |
Harris hawks optimizer (HHO) [78] | Mean | 0.00035 | − 1.03163 | 0.397895 | 3.000001225 | − 5.78398 |
St. deviation | 3.20 × 10–5 | 1.86 × 10–9 | 1.60 × 10–5 | 4.94 × 10–6 | 1.712458 | |
Grey wolf optimizer (GWO) [149] | Mean | 0.000337 | − 1.03163 | 0.397889 | 3.000028 | − 10.5343 |
St. deviation | 0.000625 | − 1.03163 | 0.397887 | 3 | − 8.55899 | |
hSMA-SA-proposed algorithm | Mean | 0.00057313 | − 1.031628453 | 0.397887411 | 3 | − 8.732008933 |
St. deviation | 0.000284537 | 5.42E−10 | 8.70E−08 | 2.77E−11 | 3.082499131 |
Engineering-based optimization design problems
Special engineering function | I beam | Multiple disk clutch brake | Rolling element bearing | Spring design | Gear train | Speed reducer | Cantilever Beam | Three-bar truss | Pressure vessel | Welded beam | Belleville spring |
---|---|---|---|---|---|---|---|---|---|---|---|
Key objective | Minimize vertical deflection | Minimize weight | Maximize dynamic load | Minimize weight | Minimize gear ratio | Minimize weight | Minimize weight | Minimize weight | Minimize cost | Minimize cost | Minimize weight |
Count of discrete variables | 4 | 5 | 10 | 3 | 4 | 7 | 5 | - | 4 | 4 | - |
Count of constraint | 4 | 8 | 9 | 4 | 1 | 11 | 1 | 3 | 4 | 7 | 5 |
Name of design | Mean | Standard deviation | Best | Worst | Median |
---|---|---|---|---|---|
Spring design | 0.013756625 | 0.001561454 | 0.012715329 | 0.017730689 | 0.012792095 |
Pressure vessel | 6182.99867 | 451.5409611 | 5885.788679 | 7318.734233 | 5959.213246 |
Multiple disk clutch brake (discrete variables) | 0.394509836 | 0.006702453 | 0.389654341 | 0.404666132 | 0.389665239 |
I beam design | 0.00662596 | 3.62923E–09 | 0.006625958 | 0.006625976 | 0.006625959 |
Speed reducer problem | 2994.491782 | 0.023016734 | 2994.474041 | 2994.595766 | 2994.486745 |
Cantilever beam design | 1.303678864 | 0.000374902 | 1.303294886 | 1.305199129 | 1.303629652 |
Three-bar truss problem | 270.2539599 | 2.361580897 | 264.2694671 | 273.4690636 | 270.9094455 |
Welded beam | 1.778021239 | 0.143521546 | 1.725134404 | 2.321842966 | 1.728195083 |
Gear train | 2.70E−11 | 7.34E−11 | 4.13E−16 | 3.10E−10 | 1.79E−12 |
Belleville spring | 6.44E+22 | 7.81E+22 | 5.251751783 | 3.78E+23 | 5.69E+22 |
Rolling element bearing | − 85,525.79232 | 36.27683994 | − 85,539.05618 | − 85,346.80626 | − 85,538.4479 |
Name of design | p value | t value | h value |
---|---|---|---|
Belleville spring | 1.72E−06 | 9.71E−05 | 1 |
Pressure vessel | 1.73E−06 | 9.20E−35 | 1 |
Spring design | 1.73E−06 | 2.98E−29 | 1 |
I beam design | 1.7344E−06 | 2.3546E−183 | 1 |
Multiple disk clutch brake (discrete variables) | 1.73E−06 | 4.24E−53 | 1 |
Three-bar truss problem | 1.73E−06 | 1.80E−61 | 1 |
Speed reducer problem | 1.73E−06 | 4.36E−150 | 1 |
Rolling element bearing | 1.73E−06 | 1.42E−99 | 1 |
Cantilever beam design | 1.73E−06 | 1.81E−104 | 1 |
Gear train | 1.73E−06 | 0.053538734 | 0 |
Welded beam | 1.73E−06 | 1.65E−33 | 1 |
Name of design | Best time | Mean time | Worst time |
---|---|---|---|
Pressure vessel | 24.89063 | 25.81302083 | 26.90625 |
Speed reducer problem | 29.5 | 31.37760417 | 34 |
Three-bar truss problem | 24.6875 | 25.58489583 | 27.265625 |
Welded beam | 28.35938 | 28.77760417 | 29.46875 |
Gear train | 40.45313 | 43.490625 | 53.875 |
Belleville spring | 54.70313 | 58.6578125 | 73.140625 |
Cantilever beam design | 45.59375 | 47.33333333 | 49.0625 |
Rolling element bearing | 31.92188 | 40.7046875 | 60.40625 |
I beam design | 44.64063 | 47.19635417 | 52.578125 |
Spring design | 24.95313 | 25.39947917 | 25.90625 |
Multiple disk clutch brake (discrete variables) | 50.28125 | 52.92135417 | 56.953125 |
Pressure vessel
Competitive techniques | Optimal values for variables | Optimum cost | |||
---|---|---|---|---|---|
Ts | Th | r | L | ||
Suggested algorithm hSMA-SA | 0.778348 | 0.3847859 | 40.328865 | 199.871477 | 5885.788679 |
BCMO [154] | 0.7789243362 | 0.3850096372 | 40.3556904385 | 199.5028780967 | 6059.714 |
ChOA [52] | 1.04375805524499 | 0.54814029437827 | 53.2363735879272 | 77.3302047573049 | 6.854064418325173+E |
G-QPSO [155] | 0.8125 | 0.4375 | 42.0984 | 176.6372 | 6059.7208 |
SMA [3] | 0.7931 | 0.3932 | 40.6711 | 196.2178 | 5994.1857 |
ACO [156] | 0.8125 | 0.4375 | 42.1036 | 176.5727 | 6059.0888 |
Branch-bound | 1.125 | 0.625 | 47.7 | 117.701 | 8129.1 |
GWO [39] | 0.8125 | 0.4345 | 42.0892 | 176.7587 | 6051.564 |
CDE [157] | 0.8125 | 0.437500 | 42.098411 | 176.637690 | 6059.7340 |
AIS-GA [158] | 0.8125 | 0.4375 | 42.098411 | 176.67972 | 6060.138 |
HHO-SCA [54] | 0.945909 | 0.447138 | 46.8513 | 125.4684 | 6393.092794 |
HS [98] | 1.099523 | 0.906579 | 44.456397 | 176.65887 | 6550.0230 |
DELC [159] | 0.8125 | 0.4375 | 42.0984455 | 176.636595 | 6059.7143 |
SiC-PSO [160] | 0.8125 | 0.4375 | 42.098446 | 176.636596 | 6059.714335 |
NPGA [161] | 0.8125 | 0.437500 | 42.097398 | 176.654047 | 6059.946341 |
HHO [78] | 0.8125 | 0.4375 | 42.098445 | 176.636596 | 6000.46259 |
CLPSO [162] | 0.8125 | 0.4375 | 42.0984 | 176.6366 | 6059.7143 |
GeneAs [163] | 0.9375 | 0.5000 | 48.3290 | 112.6790 | 6410.3811 |
GSA [20] | 1.125 | 0.625 | 55.9887 | 84.4542 | 8538.84 |
Lagrangian multiplier | 1.125 | 0.625 | 58.291 | 43.69 | 7198.043 |
MFO [34] | 0.8125 | 0.4375 | 42.0981 | 176.641 | 6059.7143 |
MVO [24] | 0.8125 | 0.4375 | 42.0907382 | 176.738690 | 6060.8066 |
SCA | 0.817577 | 0.417932 | 41.74939 | 183.57270 | 6137.3724 |
Speed reducer
Three-bar truss engineering design
Competitive techniques | Optimal values for variables | Optimum fitness | ||||||
---|---|---|---|---|---|---|---|---|
x1 | x2 | x3 | x4 | x5 | x6 | x7 | ||
Proposed hSMA-SA | 3.5 | 0.7 | 17 | 7.3 | 7.715380 | 3.350218 | 5.286654 | 2994.474041 |
HS [98] | 3.520124 | 0.7 | 17 | 8.37 | 7.8 | 3.366970 | 5.288719 | 3029.002 |
MFO [34] | 3.507524 | 0.7 | 17 | 7.302397 | 7.802364 | 3.323541 | 5.287524 | 3009.571 |
GSA [20] | 3.600000 | 0.7 | 17 | 8.3 | 7.8 | 3.369658 | 5.289224 | 3051.120 |
HHO-SCA [54] | 3.506119 | 0.7 | 17 | 7.3 | 7.99141 | 3.452569 | 5.286749 | 3029.873076 |
GA [165] | 3.510253 | 0.7 | 17 | 8.35 | 7.8 | 3.362201 | 5.287723 | 3067.561 |
PSO [164] | 3.500019 | 0.7 | 17 | 8.3 | 7.8 | 3.352412 | 5.286715 | 3005.763 |
OBSCA | 3.0879 | 0.7550 | 26.4738 | 7.3650 | 7.9577 | 3.4950 | 5.2312 | 3056.3122 |
SCA | 3.508755 | 0.7 | 17 | 7.3 | 7.8 | 3.461020 | 5.289213 | 3030.563 |
Welded beam
Competitive techniques | Optimal values for variables | Optimum weight | |
---|---|---|---|
X1 | X2 | ||
Proposed hSMA-SA | 0.767861026 | 0.470855717 | 264.2694671 |
Ray and Liew [166] | 0.788621037 | 0.408401334 | 263.8958466 |
Hernandez | 0.788 | 0.408 | 263.9 |
CS [167] | 0.789 | 0.409 | 263.972 |
Ray and Saini [168] | 0.795 | 0.398 | 264.3 |
HHO-SCA [54] | 0.788498 | 0.40875 | 263.8958665 |
Gandomi [169] | 0.78867 | 0.40902 | 263.9716 |
CSA [170] | 0.788638976 | 0.408350573 | 263.895844337 |
GWO-SA [171] | 0.789 | 0.408 | 263.896 |
MBA [169] | 0.789 | 0.409 | 263.896 |
WDE [52] | 0.515535107819326 | 0.0156341500434795 | 2.639297829829848E+02 |
ALO [152] | 0.789 | 0.408 | 263.8958434 |
DEDS [172] | 0.789 | 0.408 | 263.896 |
Raj et al. | 0.789764410 | 0.405176050 | 263.89671 |
Competitive techniques | Optimal values for variables | Optimum cost | |||
---|---|---|---|---|---|
h | l | t | b | ||
Proposed hSMA-SA | 0.205727302 | 3.471126735 | 9.035476564 | 0.205781951 | 1.725134404 |
HS [98] | 0.2442 | 6.2231 | 8.2915 | 0.2443 | 2.3807 |
PSO [164] | 0.197411 | 3.315061 | 10.00000 | 0.201395 | 1.820395 |
Approx | 0.2444 | 6.2189 | 8.2189 | 0.2444 | 2.3815 |
CDE [157] | 0.203137 | 3.542998 | 9.033498 | 0.206179 | 1.733462 |
David | 0.2434 | 6.2552 | 8.2915 | 0.2444 | 2.3841 |
GSA [20] | 0.1821 | 3.857 | 10 | 0.2024 | 1.88 |
(PSOStr) [174] | 0.2015 | 3.526 | 9.041398 | 0.205706 | 1.731186 |
HHO-SCA [54] | 0.190086 | 3.696496 | 9.386343 | 0.204157 | 1.779032249 |
MFO [34] | 0.203567 | 3.443025 | 9.230278 | 0.212359 | 1.732541 |
Gandomi et al. (FA) [175] | 0.2015 | 3.562 | 9.0414 | 0.2057 | 1.73121 |
SCA | 0.204695 | 3.536291 | 9.004290 | 0.210025 | 1.759173 |
Gear train design
Competitive techniques | Optimal values for variables | Gear ratio | Optimum fitness | |||
---|---|---|---|---|---|---|
x1 (Td) | x2 (Tb) | x3 (Ta) | x4 (Tf) | |||
Proposed hSMA-SA | 17.39759773 | 12.00546725 | 12 | 57.39404929 | NA | 2.70E−11 |
IMFO [176] | 19 | 14 | 34 | 50 | NA | 3.0498E−13 |
MARS [177] | 19 | 16 | 43 | 49 | 0.1442 | 2.7E−12 |
CSA [167] | 19.000 | 16.000 | 43.000 | 49.000 | NA | 2.7008571489E−12 |
ISA [178] | 19 | 16 | 43 | 49 | NA | 2.701E−12 |
HGA [179] | 15 | 21 | 59 | 37 | NA | 3.07E−10 |
MIBBSQP [180] | 18 | 22 | 45 | 60 | 0.146666 | 5.7E−06 |
MP [181] | 18 | 22 | 45 | 60 | 0.1467 | 5.712E−06 |
Ahga1 [179] | 13 | 24 | 47 | 46 | NA | 9.92E−10 |
IDCNLP [182] | 14 | 29 | 47 | 59 | 0.146411 | 4.5E−06 |
MBA [169] | 16 | 19 | 49 | 43 | 0.1442 | 2.7005E−0.12 |
MINSLIP [180] | 19 | 16 | 42 | 50 | NA | 2.33E−07 |
Ahga2 [179] | 13 | 20 | 53 | 34 | NA | 2.31E−11 |
ALO [152] | 19.00 | 16.00 | 43.00 | 49.00 | NA | 2.7009E−012 |
CAPSO [169] | 16 | 19 | 49 | 43 | 0.1442 | 2.701E−12 |
Belleville spring
Competitive techniques | Optimal values for variables | Optimum fitness | |||
---|---|---|---|---|---|
W1 | W2 | W3 | W4 | ||
Suggested hSMA-SA | 12.01 | 8.242835383 | 0.309690187 | 0.2 | 5.251751783 |
HHO-SCA [54] | 11.98603 | 10.0002 | 0.204206 | 0.2 | 1.98170396 |
TLBO [40] | 12.01 | 10.03047 | 0.204143 | 0.2 | 0.198966 |
MBA [169] | 12.01 | 10.030473 | 0.204143 | 0.2 | 0.198965 |
Cantilever beam design
Competitive techniques | Optimal values for variables | Optimum weight | ||||
---|---|---|---|---|---|---|
L1 | L2 | L3 | L4 | L5 | ||
Proposed hSMA-SA | 5.982032535 | 4.846178775 | 4.491073327 | 3.48171237 | 2.138830846 | 1.303294886 |
HHO-PS [50] | 5.978829 | 4.876628 | 4.464572 | 3.479744 | 2.139358 | 1.303251 |
IMFO [176] | 5.97822 | 4.87623 | 4.46610 | 3.47945 | 2.13912 | 1.30660 |
SMA [3] | 6.017757 | 5.310892 | 4.493758 | 3.501106 | 2.150159 | 1.339957 |
GCA_I [24] | 6.0100 | 5.3000 | 4.4900 | 3.4900 | 2.1500 | 1.3400 |
GWO-SA [171] | 5.9854 | 4.87 | 4.4493 | 3.5172 | 2.1187 | 1.3033 |
MMA [183] | 6.0100 | 5.3000 | 4.4900 | 3.4900 | 2.1500 | 1.3400 |
MVO [24] | 6.02394022154 | 5.30301123355 | 4.4950113234 | 3.4960223242 | 2.15272617 | 1.3399595 |
CS [184] | 6.0089 | 5.3049 | 4.5023 | 3.5077 | 2.1504 | 1.33999 |
SOS [185] | 6.01878 | 5.30344 | 4.49587 | 3.49896 | 2.15564 | 1.33996 |
HHO-SCA [54] | 5.937725 | 4.85041 | 4.622404 | 3.45347 | 2.089114 | 1.30412236 |
Rolling element bearing
Competitive algorithms | Optimal values for variables | Optimum fitness | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
r1 | r2 | r3 | r4 | r5 | r6 | r7 | r8 | r9 | r10 | ||
Proposed hSMA-SA | 125.7224792 | 21.42329479 | 11.00114251 | 0.515 | 0.515000037 | 0.471814754 | 0.615593477 | 0.300001854 | 0.098010023 | 0.607557895 | − 85,539.05618 |
SHO [46] | 125 | 21.40732 | 10.93268 | 0.515 | 0.515 | 0.4 | 0.7 | 0.3 | 0.2 | 0.6 | 85,054.532 |
HHO [78] | 125.00 | 21.00 | 11.092073 | 0.51500 | 0.51500 | 0.4000 | 0.6000 | 0.3000 | 0.050474 | 0.600 | 83,011.88329 |
WCA [187] | 125.721167 | 21.42300 | 1.001030 | 0.515000 | 0.515000 | 0.401514 | 0.659047 | 0.300032 | 0.040045 | 0.600000 | 85,538.48 |
PVS [188] | 125.719060 | 21.425590 | 11.000000 | 0.515000 | 0.515000 | 0.400430 | 0.680160 | 0.300000 | 0.079990 | 0.700000 | 81,859.741210 |
SCA [167] | 125 | 21.03287 | 10.96571 | 0.515 | 0.515 | 0.5 | 0.7 | 0.3 | 0.027780 | 0.62912 | 83,431.117 |
MFO [34] | 125 | 21.03287 | 10.96571 | 0.515 | 0.515000 | 0.5 | 0.67584 | 0.300214 | 0.02397 | 0.61001 | 84,002.524 |
MVO [24] | 125.6002 | 21.32250 | 10.97338 | 0.515 | 0.515000 | 0.5 | 0.68782 | 0.301948 | 0.03617 | 0.61061 | 84,491.266 |
I beam design
Competitive techniques | Optimal values for variables | Optimum fitness | |||
---|---|---|---|---|---|
(br) × 1 | (he) × 2 | (twi) × 3 | (tfo) × 4 | ||
Proposed hSMA-SA | 50 | 80 | 1.764705807 | 5 | 0.006625958 |
BWOA [190] | 50.00 | 80.00 | 1.76470588 | 5.00 | 0.00625958 |
SMA [3] | 49.998845 | 79.994327 | 1.764747 | 4.999742 | 0.006627 |
HHO-PS [50] | 50.00 | 80.00 | 1.764706 | 5.00 | 0.006626 |
CS [184] | 50.0000 | 80.0000 | 0.9000 | 2.3217 | 0.0131 |
MFO [34] | 50.000 | 80.000 | 1.7647 | 5.000 | 0.0066259 |
SOS [185] | 50.0000 | 80.0000 | 0.9000 | 2.3218 | 0.0131 |
CSA [167] | 49.99999 | 80 | 0.9 | 2.3217923 | 0.013074119 |
ARMS [191] | 37.05 | 80 | 1.71 | 2.31 | 0.131 |
Improved ARMS [191] | 48.42 | 79.99 | 0.9 | 2.4 | 0.131 |
Tension/compression spring design problem
Competitive techniques | Optimal values for variables | Optimum weight | ||
---|---|---|---|---|
dwi | Cdia | ACN | ||
Proposed hSMA-SA | 0.050058749 | 0.3187474 | 13.91919024 | 0.012715329 |
GA [165] | 0.05010 | 0.310111 | 14.0000 | 0.013036251 |
PSO [164] | 0.05000 | 0.3140414 | 15.0000 | 0.013192580 |
DELC [159] | 0.051689061 | 0.356717741 | 11.28896566 | 0.012665233 |
IMFO [176] | 0.051688973 | 0.356715627 | 11.289089342 | 0.012665233 |
AIS-GA | 0.0516608 | 0.3560323 | 11.329555 | 0.0126666 |
HS [98] | 0.05025 | 0.316351 | 15.23960 | 0.012776352 |
HHO-SCA [54] | 0.054693 | 0.433378 | 7.891402 | 0.012822904 |
CDE [157] | 0.051609 | 0.354714 | 11.410831 | 0.0126702 |
G-QPSO [155] | 0.051515 | 0.352529 | 11.538862 | 0.012665 |
GSA [20] | 0.05000 | 0.317312 | 14.22867 | 0.012873881 |
BCMO [154] | 0.0516597413 | 0.3560124935 | 11.3304429494 | 0.012665 |
SCA [167] | 0.050780 | 0.334779 | 12.72269 | 0.012709667 |
MALO [192] | 0.051759 | 0.358411 | 11.191500 | 0.0126660 |
MVO [24] | 0.05000 | 0.315956 | 14.22623 | 0.012816930 |
HHO-PS [50] | 0.051682 | 0.356552 | 11.29867 | 0.012665 |
MFO [34] | 0.05000 | 0.313501 | 14.03279 | 0.012753902 |
VCS [193] | 0.051685684299756 | 0.356636508703361 | 11.29372966824506 | 0.012665222962643 |
BRGA | 0.05167471 | 0.35637260 | 11.3092294 | 0.012665237 |
WCA [187] | 0.051680 | 0.356522 | 11.300410 | 0.012665 |
MBA [169] | 0.051656 | 0.355940 | 11.344665 | 0.012665 |
HEAA | 0.0516895376 | 0.3567292035 | 11.288293703 | 0.012665233 |
Multi-disk clutch brake (discrete variables)
Competitive techniques | Optimal values for variables | Optimum fitness | ||||
---|---|---|---|---|---|---|
× 1 | × 2 | × 3 | × 4 | × 5 | ||
Proposed hSMA-SA | 69.99997189 | 90 | 1.5 | 999.9999999 | 2.312785172 | 0.389654341 |
HHO [78] | 69.999999 | 90.00 | 1.00 | 1000.00 | 2.312781994 | 0.259768993 |
TLBO [50] | 70 | 90 | 3 | 810 | 1 | 0.3136566 |
WCA [187] | 70.00 | 90.00 | 1.00 | 910.000 | 3.00 | 0.313656 |
HHO-PS [50] | 76.594 | 96.59401 | 1.5 | 1000 | 2.13829 | 0.389653 |
MBFPA [195] | 70 | 90 | 1 | 600 | 2 | 0.235242457900804 |
PVS [188] | 70 | 90 | 1 | 980 | 3 | 0.31366 |
HHO-SCA [54] | 70 | 90 | 2.312785 | 1000 | 1.5 | 0.389653842 |
NSGA-II | 70 | 90 | 3 | 1000 | 1.5 | 0.4704 |
MADE [54] | 70.00 | 90 | 3 | 810 | 1 | 0.3136566 |