Real-world engineering design problems with complex objective functions under some constraints are relatively difficult problems to solve. Such design problems are widely experienced in many engineering fields, such as industry, automotive, construction, machinery, and interdisciplinary research. However, there are established optimization techniques that have shown effectiveness in addressing these types of issues. This research paper gives a comparative study of the implementation of seventeen new metaheuristic methods in order to optimize twelve distinct engineering design issues. The algorithms used in the study are listed as: transient search optimization (TSO), equilibrium optimizer (EO), grey wolf optimizer (GWO), moth-flame optimization (MFO), whale optimization algorithm (WOA), slime mould algorithm (SMA), harris hawks optimization (HHO), chimp optimization algorithm (COA), coot optimization algorithm (COOT), multi-verse optimization (MVO), arithmetic optimization algorithm (AOA), aquila optimizer (AO), sine cosine algorithm (SCA), smell agent optimization (SAO), and seagull optimization algorithm (SOA), pelican optimization algorithm (POA), and coati optimization algorithm (CA). As far as we know, there is no comparative analysis of recent and popular methods against the concrete conditions of real-world engineering problems. Hence, a remarkable research guideline is presented in the study for researchers working in the fields of engineering and artificial intelligence, especially when applying the optimization methods that have emerged recently. Future research can rely on this work for a literature search on comparisons of metaheuristic optimization methods in real-world problems under similar conditions.

Experts involved in the design, manufacturing, and repair processes of an engineering system must make managerial and technological decisions about the system under certain constraints. Optimization is an attempt to achieve the best result under existing constraints. The most important aim of the optimization process is to minimize the effort and time spent on a system or to obtain maximum efficiency. So, if the design cost of the system is expressed as a function, optimization can be defined as the attempts to reach the minimum or maximum value of this function under certain conditions [

Constraint optimization is a critical component of any engineering or industrial problem. Most real-world optimization problems include a variety of constraints that affect the overall search space. Over the last few decades, a diverse spectrum of metaheuristic approaches for solving constrained optimization problems have been developed and applied. Constrained optimization problems provide more challenges in comparison to unconstrained optimization problems, primarily because they include the association of many constraints from different types (such as equalities or inequalities) and the interdependence between the objective functions. Nonlinear objective functions and nonlinear constraints in such problem instances may exhibit characteristics of being continuous, mixed, or discrete. There are two general categories of optimization techniques for such problems or functions: mathematical programming and metaheuristic methods. To solve such problems, various mathematical programming methods have been used, such as linear programming, homogeneous linear programming, dynamic, integer, and nonlinear programming. These algorithms use gradient information to explore the solution space in the vicinity of an initial beginning point. Gradient-based algorithms converge more quickly and generate more accurate results than stochastic approaches while performing local searches. However, for these methods to be effective, the generators’ variables and cost functions must be continuous. Furthermore, for these methods to be successful, a good starting point is required. Many optimization problems require the consideration of prohibited zones, non-smooth, and side limits or non-convex cost functions. As a result, traditional mathematical programming methods are unable to solve these non-convex optimization problems. Although mixed-integer nonlinear programming or dynamic programming as well as its variants provide a limited number of possibilities for solving non-convex problems, they are computationally expensive [

Metaheuristic optimization approaches have been used as a viable alternative to conventional mathematical procedures in order to achieve global or near-global optimal solutions [

Even though there are several optimization techniques in the academic literature, no one method has been shown to universally provide the optimal answer for all optimization issues. The assertion is rationally substantiated by the “no free lunch theorem”. The aforementioned theorem has inspired by several scholars by prompting them to develop novel algorithms. Therefore, several methodologies have been lately suggested. However, there are not many studies on which of these suggested methods perform well in which areas. In this study, seventeen recently proposed and popular methods are employed to twelve constrained design problems in engineering with different constraints, objective functions, and decision variables, and performance analysis has been performed. For the problems; speed reducer, tension-compression spring, pressure vessel design, welded beam design, three-bar truss design, multiple disc clutch brake design, himmelblau’s function, cantilever beam, tubular column design, piston lever, robot gripper, and corrugated bulkhead design are investigated. The types of the problems differ considering the problem domains. The reason for this is to obtain better comparison results by comparing the optimization algorithms with each other according to their distinct capabilities in several types of problems. The algorithms used to solve these problems can be stated as; transient search optimization (TSO), equilibrium optimizer (EO), grey wolf optimizer (GWO), moth-flame optimization (MFO), whale optimization algorithm (WOA), slime mold algorithm (SMA), harris hawks optimization (HHO), chimp optimization algorithm (COA), coot optimization algorithm (COOT), multi verse optimization (MVO), arithmetic optimization algorithm (AOA), aquila optimizer (AO), sine cosine algorithm (SCA), smell agent optimization (SAO), and seagull optimization algorithm (SOA), pelican optimization algorithm (POA), and coati optimization algorithm (CA). Each algorithm’s performance is evaluated in terms of solution quality, robustness, and convergence speed.

The subsequent sections in the paper are structured in the following manner.

Metaheuristic optimization methods can be examined in five general groups: physics-based, swarm-based, game-based, evolutionary-based, and human-based. Optimization algorithms based on swarm intelligence are the methods that emerged from examining the movements of animals living in swarms. Numerous methodologies grounded in swarm intelligence have been put forward. Some of these are Red fox optimization algorithm [

Evolutionary-based metaheuristic optimization approaches have been generated by using modeling ideas in genetics, the law of natural selection, biological concepts, and random operators. GA and DE are widely utilized evolutionary algorithms that simulate the reproductive process, natural selection, and Darwin’s theory of evolution. These algorithms employ randomly selection, crossover, and mutation operators to optimize solutions. Physics-based metaheuristic optimization approaches draw inspiration from the fundamental principles of physics. Some of these are Henry gas solubility optimization [

Game-based metaheuristic optimization methods have been created by mimicking the rules and circumstances that govern different games, as well as the behavior of the participants. Some of these are World cup optimization [

There are many algorithms that solve engineering problems with metaheuristic optimization algorithms.

Reference | Method | Problem | Year | Publisher |
---|---|---|---|---|

[ |
Water cycle algorithm | Pressure vessel design |
2012 | Elsevier |

[ |
Cuckoo search algorithm | Three-bar truss design |
2013 | Springer |

[ |
Grey wolf optimizer | Pressure vessel design |
2014 | Elsevier |

[ |
Whale optimization algorithm | Pressure vessel design |
2016 | Elsevier |

[ |
Crow search algorithm | Pressure vessel design |
2016 | Elsevier |

[ |
Multi-verse optimizer | Gear train design |
2016 | Springer |

[ |
Hybrid PSO-GA | Pressure vessel design |
2016 | Elsevier |

[ |
Salp swarm algorithm | Cantilever beam design |
2017 | Elsevier |

[ |
Spotted hyena optimizer | Multiple disk clutch brake design |
2017 | IEEE |

[ |
Iterative topographical global optimization | Tension-compression spring design |
2018 | Elsevier |

[ |
Artificial algae | Gear train design |
2018 | Springer |

[ |
Harris hawks optimization | Pressure vessel design |
2019 | Elsevier |

[ |
Seagull optimization algorithm | Optical buffer design |
2019 | Elsevier |

[ |
Butterfly optimization algorithm | Welded beam design |
2019 | Springer |

[ |
Marine predators algorithm | Welded beam design |
2020 | Elsevier |

[ |
Slime mould algorithm | Welded beam structure |
2020 | Elsevier |

[ |
Chimp optimization algorithm | Heat exchanger network design (case 1) |
2020 | Elsevier |

[ |
Hybrid |
Car side crash problem |
2021 | Wiley |

[ |
Equilibrium optimizer | Welded beam design |
2021 | Elsevier |

[ |
Arithmetic optimization algorithm | Tension-compression spring design |
2021 | Elsevier |

[ |
Aquila optimizer | Pressure vessel structure |
2021 | Elsevier |

[ |
Enhanced grasshopper optimization | Three-bar truss |
2021 | Springer |

[ |
Chaotic Lévy flight distribution optimization | Gear train design |
2022 | Wiley |

[ |
Pelican optimization algorithm | Pressure vessel design problem |
2022 | MDPI |

[ |
Orca predation algorithm | Welded beam design |
2022 | Elsevier |

[ |
Coati optimization algorithm | Pressure vessel design problem |
2023 | Elsevier |

This section provides a brief description of each algorithm that is employed in this study. Only the most significant parts are described; accordingly, interested readers can get all of the information they need in the cited papers.

As a physics-based metaheuristic approach, Qais et al. introduced the transient search algorithm (TSO) in 2020. The source of motivation for this study is derived from the transient dynamics seen in switched electrical circuits having storage components, i.e., capacitance and inductance [

Electrical circuits consist of many components capable of storing energy. The components in question may be classified as inductors (L), capacitors (C), or a hybrid configuration consisting of both (LC). Typically, an electrical circuit that incorporates a resistor (R), capacitor (C), or inductor (L) exhibits a transient response as well as a steady-state response. Circuits that include both an energy storage device and a resistor are categorized as first-order circuits. When two energy storage devices are positioned next to a resistor inside a circuit, the resulting configuration is referred to as a second-order circuit. The TSO method is introduced, drawing inspiration from the transient response shown by these circuits in the vicinity of 0.

In the year 2020, Faramarzi et al. developed an equilibrium optimizer (EO), a metaheuristic algorithm that simulates the fundamental well-mixed dynamic mass balance in a control volume [

Mirjalili et al. introduced the grey wolf algorithm (GWO) in 2014, which is an optimization algorithm influenced by the population of grey wolves, their natural leading capabilities, and hunting habits [

The moth-flame optimization (MFO) is a metaheuristic method affected by the population of moths and offered by Mirjalili in 2015. The MFO is based upon a simulation of a distinctive nocturnal navigation system used by moths. It begins the optimization procedure, like other meta-heuristics. In other words, it randomly generates a set of candidate solutions. When traveling at night, the moth uses a mechanism known as transverse orientation to navigate. In the MFO method, candidate solutions are postulated as moths, while the variables of a given issue are postulated as the locations of these moths inside the search space [

Mirjalili et al. [

Li et al. proposed a new optimization algorithm inspired by the behavior of slime mould in obtaining the optimal way to bind foods [

Heidari et al. [

The chimp optimization algorithm (COA) was designed by Khishe et al. as a biology-based optimization algorithm originated by the individual intellect and sexual motives of chimps during group hunts [

The metaheuristic technique presented by Naruei et al. (2021) draws inspiration from the behavioral patterns shown by birds navigating on the surface of water. The behavior of the coot swarm on water includes three major movements [

The notion of multi-verse optimization (MVO) was presented as a metaheuristic approach by Mirjalili et al. (2016), drawing inspiration from the field of cosmology. MVO explores search spaces with the concepts of black and white holes while exploiting search spaces with wormholes. Similar to other evolutionary algorithms, this method commences the optimization procedure by generating an initial population and endeavors to enhance these solutions via a predetermined number of iterations. Enhancement of individual performance inside each population may be attained via the utilization of this algorithm, which is grounded in one of the postulations about the presence of many universes. In the context of these theories, it is conceptualized that every solution to an optimization issue represents a distinct universe, whereby each constituent item is considered a variable within the specific problem at hand. In addition, they assign to each solution an inflation rate proportional to the value of the fitness function to which the solution corresponds [

The arithmetic optimization algorithm (AOA), as presented by Abualigah et al. [

The aquila optimizer (AO) is a population-based metaheuristic approach developed by Abualigah et al. [

The sine cosine algorithm (SCA) is a recently developed meta-heuristic algorithm that is based on the properties of trigonometric sine and cosine functions [

The smell agent optimization (SAO) is a metaheuristic method responsible for implementing the relationships that exist between a smell agent and an object that evaporates a smell molecule [

The seagull optimization algorithm (SOA) is an algorithm that pulls inspiration from the natural behavior of seagulls while migrating and attacking prey [

The pelican optimization algorithm (POA) is a swarm-based optimization technique that draws inspiration from the hunting behavior and methods shown by pelicans. In POA, exploration agents are represented by pelicans that look for sustenance sources. POA is made up of two stages that are carried out consecutively in each iteration. In the first phase, there is a global objective to which all pelicans will migrate. This global target is chosen at random inside the issue space at the start of each cycle. The pelican has two options for mobility. If this goal is more desirable than the pelican’s present position, the pelican will migrate toward it. Otherwise, the pelican will flee from this location. POA employs an acceptance-rejection method. The pelican will only relocate if the new place is superior to its present location. The pelican circles its present location throughout the second phase. Although this phrase is not always applicable, it might be thought of as a local or neighborhood search. During this step, a new location is chosen at random inside the local problem space of the pelican. With each iteration, the diameter of this local issue space rapidly decreases. It indicates that the local issue space is sufficiently large to begin with, and it may be seen as an exploration. This inquiry, on the other hand, progresses from investigation to exploitation with each repetition. In addition to the process of iteration, the current location of the agent has an impact on the problem space at a local level. In its initial form, the current positioning close to zero restricts the range of the local problem domain. Similar to the first phase, the pelican only advances toward the new location if the new location is superior to its current location [

The coati optimization algorithm (CA) is a new metaheuristic method that imitates the coati’s natural behavior when it encounters and flees from predators. The process of modifying the positions of candidate solutions in the CA is derived from the emulation of two distinct behaviors shown by coatis in nature. These behaviors include: (i) coatis’ assault method on iguanas, and (ii) coatis’ predator escape strategy. As a result, the CA population is updated in two stages. The first phase of enhancing the coati population in the designated region is shown via a simulation that models their strategies for targeting iguanas. In this particular strategy, a considerable number of coatis ascend the tree in order to closely approach an iguana and elicit a startled response. A group of coatis congregates under a tree, observing the descent of an iguana to the ground. Upon the iguana’s descent, the coatis engage in aggressive behavior by launching an assault and pursuing the iguana. By using this approach, coatis demonstrate their ability to travel to different regions within the search area, hence highlighting the worldwide search capabilities of the COA within the realm of problem-solving. The subsequent phase of updating the locations of coatis inside the search space is formulated using mathematical modeling techniques, which take into account the coatis’ inherent behavior while encountering and evading predators. When a predator initiates an attack on a coati, the coati promptly vacates its position. Coati’s actions in this approach place it in a secure position near to its present location, indicating the CA’s exploitation ability to utilize local search. The iteration of a CA concludes when all coordinates of the coatis in the solution space have been modified according to the results of the first and second phases. The best solution discovered across all rounds of the method is provided as the result after CA has finished running [

In this section, the most prevalent design challenges in engineering are stated. To make the problems more understandable, the mathematical form and definition are provided. The following are the problems investigated in the study:

Speed reducer problem

Tension-compression spring design problem

Pressure vessel design problem

Welded beam design problem

Three-bar truss design problem

Multiple disk clutch brake design problem

Himmelblau’s function

Cantilever beam problem

Tubular column design problem

Piston lever

Robot gripper

Corrugated bulkhead design problem

These engineering design issues are well recognized in practical applications. In order to identify the most favorable design, it is often necessary to use an active approach for determining the ideal parameters. In order to address each issue, some settings (variables) need adjustment. Furthermore, some limitations are included to guarantee that the variables’ values stay within the designated range. The specifics of the optimization issue are presented below.

The evaluation of optimization approaches often involves selecting bound-constrained and common-constrained optimization problems. Each design vector must consistently provide a constrained solution to any engineering or optimization problem [

The bound-constrained structure incorporates the cost function into the assessment of the selected optimization process to describe all the stated constrained issues in

The above-mentioned real-world constrained problems are used to create a benchmark suite. The benchmark suite includes a total of 12 problems designed from the problems listed above.

Name | ||||
---|---|---|---|---|

Speed reducer | 7 | 11 | 0 | 2.9944244658E+03 |

Tension-compression spring design (case 1) | 3 | 3 | 0 | 1.2665232788E-02 |

Pressure vessel design | 4 | 4 | 0 | 5.8853327736E+03 |

Welded beam design | 4 | 5 | 0 | 1.6702177263E+00 |

Three-bar truss design problem | 2 | 3 | 0 | 2.6389584338E+02 |

Multiple disk clutch brake design problem | 5 | 7 | 0 | 2.3524245790E-01 |

Himmelblau’s function | 5 | 6 | 0 | −3.0665538672E+04 |

Cantilever beam | 5 | 1 | 0 | 1.3399576 |

Tubular column design | 2 | 6 | 0 | 26.486361473 |

Piston lever | 4 | 4 | 0 | 8.41269832311 |

Robot gripper | 7 | 7 | 0 | 2.5287918415E+00 |

Corrugated bulkhead design | 4 | 6 | 0 | 6.8429580100808 |

The experiments are conducted using a computing system equipped with the Windows 11 operating system, 16 GB of RAM, and an Intel (R) Core (TM) i7-10750H CPU (2.60 GHz). The comparison methods are coded using MATLAB R2021a. The aforementioned issues are inherently constrained, therefore necessitating the implementation of an external penalty approach mechanism in order to address them. The maximum number of iterations for all problems is 1000, the number of populations is 30, and the number of evaluations is chosen as 30000. Algorithm parameters are default values found in the literature and are shown in

Algorithm | Parameter settings |
---|---|

TSO | |

EO | |

GWO | |

MFO | |

WOA | |

SMA | |

HHO | |

COA | |

COOT | |

MVO | |

AOA | |

AO | |

SCA | |

SAO | |

SOA | |

POA | – |

CA | – |

This problem is simply a gearbox problem that allows the aircraft engine to rotate at maximum efficiency [

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | 3009.168854 | 399699.6461 | 1866150.765 | 597066.8458 | 15.37 |

EO | 2994.422564 | 1.12621E-12 | |||

GWO | 2999.884112 | 3006.149462 | 3014.184087 | 3.940870538 | 6.47 |

MFO | 3003.320088 | 3043.034924 | 15.1867409 | 3.28 | |

WOA | 3002.230767 | 3107.084757 | 3612.31405 | 108.8572293 | 10.77 |

SMA | 2994.422746 | 2994.424409 | 2994.428914 | 0.001656383 | 3.17 |

HHO | 3007.417731 | 3425.055497 | 5080.047182 | 563.1360869 | 12.47 |

COA | 3048.060533 | 3144.81531 | 3200.120506 | 40.11920307 | 12.60 |

COOT | 2994.422579 | 2994.750943 | 3003.757984 | 1.673399569 | 3.37 |

MVO | 3001.692079 | 3035.025167 | 3072.59999 | 16.33049716 | 8.93 |

AOA | 3080.128866 | 3160.273461 | 3222.260728 | 40.2220024 | 13.07 |

AO | 3048.786991 | 3850.139838 | 5143.275696 | 592.7613846 | 15.00 |

SCA | 3061.137019 | 3114.63376 | 3199.643275 | 37.84743803 | 11.67 |

SAO | 3252.352584 | 3915.597815 | 5763.29265 | 535.425852 | 15.63 |

SOA | 3010.99658 | 3032.883533 | 3060.656495 | 13.93473393 | 8.93 |

POA | 2994.424736 | 3000.035806 | 3007.515500 | 5.090249 | 5.30 |

COA | 2994.555032 | 3001.492670 | 3016.723380 | 6.619445 | 5.47 |

Algorithm | Parameters values | |||||||
---|---|---|---|---|---|---|---|---|

TSO | 3.50151 | 0.7 | 17 | 7.3 | 8.073032 | 3.371746 | 5.288003 | 3009.168854 |

EO | 3.49999 | 0.7 | 17 | 7.3 | 7.715319 | 3.350541 | 5.286654 | |

GWO | 3.500937 | 0.7 | 17 | 7.364591 | 7.809638 | 3.356307 | 5.288191 | 2999.884112 |

MFO | 3.49999 | 0.7 | 17 | 7.3 | 7.715319 | 3.350541 | 5.286654 | |

WOA | 3.499959 | 0.7 | 17 | 7.443682 | 7.8541 | 3.351684 | 5.291651 | 3002.230767 |

SMA | 3.499991 | 0.7 | 17 | 7.300002 | 7.715326 | 3.350541 | 5.286655 | 2994.422746 |

HHO | 3.520425 | 0.7 | 17 | 7.38595 | 7.881842 | 3.352585 | 5.286711 | 3007.417731 |

COA | 3.535505 | 0.7 | 17 | 7.3 | 8.3 | 3.428059 | 5.297065 | 3048.060533 |

COOT | 3.49999 | 0.7 | 17 | 7.3 | 7.715319 | 3.350541 | 5.286654 | 2994.422579 |

MVO | 3.500589 | 0.7 | 17 | 7.536276 | 7.764337 | 3.364298 | 5.287189 | 3001.692079 |

AOA | 3.6 | 0.7 | 17 | 8.3 | 8.3 | 3.413136 | 5.299554 | 3080.128866 |

AO | 3.531807 | 0.7 | 17 | 7.781691 | 8.162154 | 3.399743 | 5.310019 | 3048.786991 |

SCA | 3.6 | 0.7 | 17 | 8.195417 | 8.046403 | 3.364929 | 5.300037 | 3061.137019 |

SAO | 3.50601 | 0.7 | 17.19404 | 8.128777 | 8.174155 | 3.645772 | 5.469817 | 3252.352584 |

SOA | 3.516004 | 0.7 | 17 | 7.689567 | 7.84431 | 3.356036 | 5.29075 | 3010.99658 |

POA | 3.134689 | 0.773 | 18.56372 | 7.635233 | 7.529978 | 3.082312 | 5.296658 | 2994.424736 |

CA | 2.779031 | 0.732381 | 23.68085 | 7.30957 | 7.508113 | 3.72635 | 5.404868 | 2994.555032 |

The tension-compression spring design problem is a problem defined by Arora [

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | 0.01280243 | 0.013419896 | 0.014640884 | 0.000546907 | 10.77 |

EO | 0.012667047 | 0.012986249 | 0.014034056 | 0.000332122 | 7.07 |

GWO | 0.012689887 | 0.012754846 | 0.013321233 | 0.000114098 | 4.80 |

MFO | 0.012665686 | 0.013483807 | 0.017773158 | 0.001262609 | 8.67 |

WOA | 0.012672465 | 0.013835231 | 0.015996637 | 0.000988024 | 10.97 |

SMA | 0.012665885 | 0.013381332 | 0.015753456 | 0.000933324 | 8.03 |

HHO | 0.012667145 | 0.013340624 | 0.016382034 | 0.000827094 | 9.00 |

COA | 0.012735806 | 0.01340736 | 0.015791487 | 0.000755935 | 10.00 |

COOT | 0.012665238 | 0.013331169 | 0.016001858 | 0.000906209 | 8.07 |

MVO | 0.012803582 | 0.017131704 | 0.018147631 | 0.001737585 | 15.27 |

AOA | 0.013142152 | 0.014388631 | 0.030631821 | 0.004325386 | 11.27 |

AO | 0.013684295 | 0.016522416 | 0.025325732 | 0.002220195 | 15.33 |

SCA | 0.012741025 | 0.013037944 | 0.013209119 | 0.000144589 | 8.50 |

SAO | 0.013644472 | 0.018418919 | 0.026253801 | 0.002940077 | 15.93 |

SOA | 0.01272481 | 0.012841598 | 0.014484334 | 0.000310579 | 6.03 |

POA | 0.012665237 | 0.012690486 | 0.000005677 | 1.47 | |

COA | 0.012672088 | 0.012712224 | 0.000013390 | 1.83 |

Algorithm | Parameters values | |||
---|---|---|---|---|

TSO | 0.054349629 | 0.424139632 | 8.218576613 | 0.01280243 |

EO | 0.052005294 | 0.364373243 | 10.85386669 | 0.012667047 |

GWO | 0.051729218 | 0.357405988 | 11.26857001 | 0.012689887 |

MFO | 0.051846967 | 0.360528432 | 11.0690043 | 0.012665686 |

WOA | 0.051063365 | 0.341850846 | 12.21692125 | 0.012672465 |

SMA | 0.051519654 | 0.352654671 | 11.53131506 | 0.012665885 |

HHO | 0.052013772 | 0.364579812 | 10.84249563 | 0.012667145 |

COA | 0.05 | 0.317414002 | 14.04945763 | 0.012735806 |

COOT | 0.05170472 | 0.357094576 | 11.26690843 | |

MVO | 0.05 | 0.316143553 | 14.199707 | 0.012803582 |

AOA | 0.05 | 0.312114718 | 14.84271945 | 0.013142152 |

AO | 0.053425204 | 0.395649521 | 10.11767744 | 0.013684295 |

SCA | 0.052092338 | 0.365517089 | 10.84543941 | 0.012741025 |

SAO | 0.050385213 | 0.316156139 | 15 | 0.013644472 |

SOA | 0.05 | 0.317362481 | 14.03820349 | 0.01272481 |

POA | 0.528153 | 0.934051 | 6.921844 | 0.012665237 |

CA | 0.710522 | 1.026576 | 3.942698 |

The primary aim of this challenge is to optimize the cost associated with welding, as well as the expenses related to materials and the creation process of a vessel [

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | 6879.866741 | 8704.301853 | 13069.25436 | 1582.300611 | 13.93 |

EO | 6518.793086 | 7544.492518 | 512.0568854 | 6.68 | |

GWO | 6058.801906 | 7365.851023 | 356.5801053 | 3.63 | |

MFO | 6058.72346 | 6722.668915 | 7544.492518 | 508.6435023 | 8.25 |

WOA | 6192.752799 | 8192.085505 | 11684.79247 | 1274.125452 | 13.47 |

SMA | 6058.719892 | 6596.192699 | 7544.492519 | 549.8023233 | 7.10 |

HHO | 6091.615788 | 6792.936176 | 7544.492518 | 365.7904697 | 9.30 |

COA | 6340.732653 | 7846.678347 | 8548.44731 | 447.1938273 | 13.70 |

COOT | 6058.720225 | 6406.088039 | 7352.612207 | 315.2009216 | 6.10 |

MVO | 6091.584113 | 6741.27325 | 7419.29367 | 421.1743251 | 8.80 |

AOA | 7177.685796 | 10134.5141 | 21033.80706 | 3059.849954 | 15.20 |

AO | 6060.226707 | 6572.408065 | 7547.794432 | 474.7811046 | 7.63 |

SCA | 6192.682512 | 6929.151724 | 8752.385245 | 588.2808671 | 9.70 |

SAO | 8794.883762 | 13888.48817 | 21019.53969 | 3282.14935 | 16.73 |

SOA | 6059.667071 | 6358.863021 | 7546.780769 | 466.1025508 | 5.50 |

POA | 6224.114264 | 7273.278824 | 284.930973 | 3.33 | |

CA | 6249.526308 | 7273.278824 | 303.558052 | 3.93 |

Algorithm | Parameters values | ||||
---|---|---|---|---|---|

TSO | 1.0625 | 0.5 | 50.60431861 | 93.67943093 | 6879.866741 |

EO | 0.8125 | 0.4375 | 42.11479983 | 176.4340461 | |

GWO | 0.8125 | 0.4375 | 42.11775397 | 176.3995752 | 6058.801906 |

MFO | 0.8125 | 0.4375 | 42.11575702 | 176.4221977 | 6058.72346 |

WOA | 0.8125 | 0.4375 | 41.04114852 | 190.1938826 | 6192.752799 |

SMA | 0.8125 | 0.4375 | 42.11511876 | 176.4300983 | 6058.719892 |

HHO | 0.875 | 0.4375 | 45.32724166 | 140.3511195 | 6091.615788 |

COA | 0.8125 | 0.4375 | 42.23592297 | 184.6334807 | 6340.732653 |

COOT | 0.8125 | 0.4375 | 42.1150819 | 176.4305544 | 6058.720225 |

MVO | 0.875 | 0.4375 | 45.33090975 | 140.3280126 | 6091.584113 |

AOA | 0.8125 | 0.6875 | 41.09783997 | 200 | 7177.685796 |

AO | 0.8125 | 0.4375 | 42.10333757 | 176.6195632 | 6060.226707 |

SCA | 0.8125 | 0.4375 | 42.02127579 | 182.8885334 | 6192.682512 |

SAO | 1.0625 | 0.875 | 52.572008 | 86.52438795 | 8794.883762 |

SOA | 0.8125 | 0.4375 | 42.12719613 | 176.2965619 | 6059.667071 |

POA | 3.4375 | 2.0625 | 193.1607 | 36.37869 | |

CA | 4.5625 | 0.25 | 67.12759 | 188.4114 |

The primary aim of the welded beam design challenge is to optimize the cost of producing a beam while adhering to certain limitations [

0.125 ≤

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | 1.716054986 | 2.546468795 | 3.536807483 | 0.62365212 | 14.83 |

EO | 1.670408299 | 1.675227413 | 0.000899033 | 1.93 | |

GWO | 1.671333735 | 1.673168037 | 1.678231819 | 0.001793212 | 5.37 |

MFO | 1.670217701 | 1.722023637 | 1.975024976 | 0.073257707 | 6.80 |

WOA | 1.73679486 | 2.170615908 | 4.197028463 | 0.530483963 | 13.63 |

SMA | 1.670333603 | 1.674309181 | 1.726803703 | 0.010036131 | 5.03 |

HHO | 1.690460887 | 1.891009449 | 2.227283322 | 0.13871246 | 12.00 |

COA | 1.694398105 | 1.797886006 | 1.844566653 | 0.035111486 | 10.83 |

COOT | 1.670482669 | 1.697798136 | 1.820191362 | 0.037171837 | 6.83 |

MVO | 1.672676563 | 1.699051309 | 1.798001792 | 0.030387746 | 7.77 |

AOA | 1.844129548 | 2.199481858 | 2.477850773 | 0.185303016 | 15.00 |

AO | 1.725270134 | 1.953209489 | 2.220842457 | 0.122797404 | 13.30 |

SCA | 1.726200239 | 1.804224071 | 1.874302512 | 0.040114991 | 11.00 |

SAO | 1.918621628 | 3.204991211 | 6.074979783 | 0.876216532 | 16.47 |

SOA | 1.679461577 | 1.690710091 | 1.754063757 | 0.014743169 | 7.63 |

POA | 1.670217701 | 1.671045069 | 0.000156738 | 1.97 | |

CA | 1.670343953 | 1.671357022 | 0.000253049 | 2.60 |

Algorithm | Parameters values | ||||
---|---|---|---|---|---|

TSO | 0.177884 | 3.819348 | 9.111658 | 0.202596 | 1.716054986 |

EO | 0.198832 | 3.337364 | 9.192023 | 0.198832 | 1.6702177 |

GWO | 0.198818 | 3.339296 | 9.191158 | 0.198966 | 1.671333735 |

MFO | 0.198832 | 3.337364 | 9.192024 | 0.198832 | 1.670217701 |

WOA | 0.181394 | 3.819648 | 9.424811 | 0.197769 | 1.73679486 |

SMA | 0.198723 | 3.339432 | 9.192039 | 0.198833 | 1.670333603 |

HHO | 0.185601 | 3.604042 | 9.199277 | 0.199369 | 1.690460887 |

COA | 0.194345 | 3.404508 | 9.358266 | 0.198105 | 1.694398105 |

COOT | 0.198834 | 3.33681 | 9.193963 | 0.198834 | 1.670482669 |

MVO | 0.197168 | 3.372209 | 9.193376 | 0.198846 | 1.672676563 |

AOA | 0.16434 | 4.163106 | 10 | 0.196826 | 1.844129548 |

AO | 0.183829 | 3.589614 | 9.445074 | 0.199088 | 1.725270134 |

SCA | 0.174972 | 3.942854 | 9.143209 | 0.201813 | 1.726200239 |

SAO | 0.159253 | 4.713681 | 10 | 0.198437 | 1.918621628 |

SOA | 0.193833 | 3.440986 | 9.192654 | 0.199217 | 1.679461577 |

POA | 0.888043 | 6.126506 | 2.058919 | 0.910335 | 1.670217701 |

CA | 0.749239 | 6.679802 | 8.02718 | 0.969918 |

This problem is a structural optimization problem in civil engineering. The main objective of this problem introduced by Nowacki is to minimize the volume of the three-bar truss by adjusting the cross-sectional areas (

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | 263.8962383 | 264.4652982 | 270.4725405 | 1.289643914 | 10.40 |

EO | 263.8954132 | 263.8954469 | 8.56826E-06 | 3.70 | |

GWO | 263.8954858 | 263.8976257 | 263.9054651 | 0.002128135 | 5.97 |

MFO | 263.8954304 | 263.9483466 | 264.4238545 | 0.10154114 | 8.30 |

WOA | 263.8959272 | 264.9041197 | 268.7812754 | 1.309740127 | 12.37 |

SMA | 265.2278409 | 269.8360947 | 272.5174213 | 1.924796679 | 16.03 |

HHO | 263.8959527 | 263.981987 | 264.4710671 | 0.142168454 | 9.20 |

COA | 263.9057513 | 264.0519785 | 264.3448934 | 0.115646492 | 10.80 |

COOT | 263.8954747 | 263.8962154 | 0.00018314 | 3.35 | |

MVO | 263.8954144 | 263.8963591 | 263.8986298 | 0.000930608 | 5.57 |

AOA | 263.9974295 | 265.4147112 | 282.8427125 | 3.29158132 | 14.15 |

AO | 263.9107145 | 264.1310168 | 264.5574435 | 0.158136239 | 11.83 |

SCA | 263.9012431 | 264.6511963 | 282.8426491 | 3.379030311 | 10.93 |

SAO | 264.996317 | 274.5928341 | 308.6800351 | 10.15582073 | 16.17 |

SOA | 263.8985813 | 268.9840652 | 282.8427125 | 8.357099664 | 11.22 |

POA | 263.8954081 | 0.0000000 | 1.53 | ||

CA | 263.8954081 | 0.0000000 | 1.48 |

Algorithm | Parameters values | ||
---|---|---|---|

TSO | 0.788658774 | 0.40827384 | 263.8962383 |

EO | 0.788674018 | 0.408242743 | |

GWO | 0.788350591 | 0.409157785 | 263.8954858 |

MFO | 0.788498637 | 0.408739011 | 263.8954304 |

WOA | 0.787833323 | 0.410625778 | 263.8959272 |

SMA | 0.81669173 | 0.342325367 | 265.2278409 |

HHO | 0.787812977 | 0.410683581 | 263.8959527 |

COA | 0.790361227 | 0.40357838 | 263.9057513 |

COOT | 0.788672536 | 0.408246935 | |

MVO | 0.788580396 | 0.408507606 | 263.8954144 |

AOA | 0.799935949 | 0.377413758 | 263.9974295 |

AO | 0.792467196 | 0.397589981 | 263.9107145 |

SCA | 0.79032088 | 0.403647416 | 263.9012431 |

SAO | 0.799979814 | 0.387278564 | 264.996317 |

SOA | 0.790744046 | 0.402416516 | 263.8985813 |

POA | 0.613176 | 0.047329 | |

CA | 0.303088 | 0.936761 |

The primary aim of this topic is to decrease the bulk of a clutch braking system consisting of numerous disks. Inner radius (

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | 0.235247188 | 0.235301639 | 0.235512629 | 5.92468E-05 | 10.63 |

EO | 0.235242458 | 1.66533E-16 | 2.82 | ||

GWO | 0.235243674 | 0.235269454 | 0.235367508 | 2.9044E-05 | 9.77 |

MFO | 0.235242458 | 1.39792E-16 | 2.92 | ||

WOA | 0.23524247 | 0.235242629 | 3.29813E-08 | 7.37 | |

SMA | 0.2352425 | 0.235242764 | 6.85923E-08 | 7.60 | |

HHO | 0.235242458 | 1.00841E-16 | 4.55 | ||

COA | 0.235251931 | 0.235623716 | 0.236440761 | 0.000328328 | 12.73 |

COOT | 0.235242459 | 0.235242485 | 4.79376E-09 | 5.12 | |

MVO | 0.235251002 | 0.23530991 | 0.235494147 | 5.82208E-05 | 10.90 |

AOA | 0.235613936 | 0.239784504 | 0.253856183 | 0.006455082 | 15.20 |

AO | 0.2355855 | 0.236635663 | 0.241295448 | 0.001083075 | 15.03 |

SCA | 0.235257655 | 0.236260012 | 0.239379612 | 0.000916363 | 13.93 |

SAO | 0.256485329 | 0.418898864 | 0.558433209 | 0.066573468 | 17.00 |

SOA | 0.235247016 | 0.235459629 | 0.236194322 | 0.000233995 | 11.80 |

POA | 0.235242458 | 0.000000000 | 2.82 | ||

CA | 0.235242458 | 0.000000000 | 2.82 |

Algorithm | Parameters values | |||||
---|---|---|---|---|---|---|

TSO | 69.99954044 | 90 | 1 | 871.2362442 | 2 | 0.235247188 |

EO | 70 | 90 | 1 | 707.2368619 | 2 | |

GWO | 69.999836 | 90 | 1 | 152.2987899 | 2 | 0.235243674 |

MFO | 70 | 90 | 1 | 958.6228265 | 2 | |

WOA | 69.99999994 | 90 | 1 | 1000 | 2 | |

SMA | 69.99999859 | 90 | 1 | 3.854620109 | 2 | |

HHO | 70 | 90 | 1 | 579.8999954 | 2 | |

COA | 69.99907953 | 90 | 1 | 1.103077777 | 2 | 0.235251931 |

COOT | 70 | 90 | 1 | 859.8058151 | 2 | |

MVO | 69.99916981 | 90 | 1 | 999.8337472 | 2 | 0.235251002 |

AOA | 70.01155738 | 90.0370556 | 1 | 1000 | 2 | 0.235613936 |

AO | 69.96666064 | 90 | 1 | 668.7072249 | 2 | 0.2355855 |

SCA | 69.99852333 | 90 | 1 | 1000 | 2 | 0.235257655 |

SAO | 77.03512282 | 97.07407812 | 1 | 383.959645 | 2 | 0.256485329 |

SOA | 69.99955715 | 90 | 1 | 965.9255451 | 2 | 0.235247016 |

POA | 62.32327 | 94.44017 | 2.776293 | 659.5425 | 4.399259 | |

CA | 60.75571 | 108.0781 | 2.612441 | 321.4522 | 8.080696 |

The issue proposed by Himmelblau serves as a widely used benchmark problem for the analysis of nonlinear constrained optimization methods. This issue consists of a set of five variables and six nonlinear constraints. The issue is mathematically represented in the following manner.

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | −30632.46188 | −30386.82957 | −29753.31028 | 207.7208276 | 12.37 |

EO | −30665.55912 | −30665.55912 | 1.62288E-11 | 2.65 | |

GWO | −30665.12066 | −30660.90023 | −30656.45398 | 2.430583688 | 7.00 |

MFO | −30660.83703 | −30560.13316 | 19.23734071 | 2.93 | |

WOA | −30610.82885 | −29938.37317 | −29522.34584 | 232.4049417 | 15.30 |

SMA | −30665.55902 | 2.19388E-05 | 5.63 | ||

HHO | −30660.54681 | −30570.18237 | −30249.66843 | 123.1622639 | 9.93 |

COA | −30648.19157 | −30443.71357 | −30199.17052 | 104.3200577 | 12.43 |

COOT | −30665.55742 | −30665.52782 | 0.006229751 | 4.93 | |

MVO | −30662.88669 | −30532.84249 | −30174.63609 | 114.3691665 | 10.87 |

AOA | −30627.87236 | −29647.40327 | −29130.41487 | 351.1225784 | 16.10 |

AO | −30646.13075 | −30510.79641 | −30254.06331 | 104.7187249 | 11.40 |

SCA | −30625.43612 | −30487.51895 | −30254.36315 | 90.87025306 | 11.70 |

SAO | −30224.24522 | −29764.05291 | −29434.89999 | 231.1897481 | 16.03 |

SOA | −30662.92388 | −30630.3406 | −30483.17315 | 33.77070124 | 8.63 |

POA | −30665.55911 | −30665.55894 | 0.00003 | 2.78 | |

CA | −30665.55912 | −30665.55912 | 0.00000 | 2.30 |

Algorithm | Parameters values | |||||
---|---|---|---|---|---|---|

TSO | 78 | 33 | 30.03659921 | 44.19390758 | 37.07924883 | −30632.46188 |

EO | 78 | 33 | 29.99511433 | 45 | 36.77588429 | |

GWO | 78 | 33 | 29.99728969 | 45 | 36.77219738 | −30665.12066 |

MFO | 78 | 33 | 29.99511434 | 45 | 36.77588429 | |

WOA | 78 | 33 | 30.06458145 | 43.34405862 | 37.27272535 | −30610.82885 |

SMA | 78 | 33 | 29.99511546 | 45 | 36.77588291 | |

HHO | 78 | 33 | 30.02118597 | 44.98433631 | 36.72447873 | −30660.54681 |

COA | 78 | 33 | 30.0732883 | 45 | 36.65666221 | −30648.19157 |

COOT | 78 | 33 | 29.99511433 | 45 | 36.77588429 | |

MVO | 78.01538223 | 33.01207678 | 30.00686017 | 45 | 36.74166225 | −30662.88669 |

AOA | 78 | 33 | 30.22256747 | 45 | 36.22854345 | −30627.87236 |

AO | 78 | 33.02480065 | 30.08903458 | 44.8996988 | 36.61041061 | −30646.13075 |

SCA | 78 | 33.21663991 | 30.23023995 | 45 | 36.22779393 | −30625.43612 |

SAO | 79.20636734 | 33 | 31.68999069 | 42.44374862 | 33.74096769 | −30224.24522 |

SOA | 78 | 33 | 30.007539 | 45 | 36.75534739 | −30662.92388 |

POA | 87.47638 | 37.11719 | 30.29814 | 29.01444 | 34.208 | |

CA | 85.56589 | 33.91255 | 30.06284 | 44.55554 | 37.89919 |

A cantilever beam design problem is a common optimization problem faced in the area of engineering. In this problem, the minimum values of five choice variables

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | 1.3808707 | 1.541428057 | 1.850529526 | 0.091666864 | 14.70 |

EO | 1.339957787 | 1.339966064 | 1.339995913 | 8.27917E-06 | 1.47 |

GWO | 1.339966023 | 1.340019779 | 1.340214936 | 4.85162E-05 | 3.63 |

MFO | 1.340025322 | 1.340847969 | 1.342240497 | 0.000574436 | 7.83 |

WOA | 1.355727617 | 1.48076134 | 1.744840151 | 0.101969482 | 14.10 |

SMA | 1.339965208 | 1.34016411 | 1.340754168 | 0.000157703 | 5.17 |

HHO | 1.340774991 | 1.342737909 | 1.346081888 | 0.001413135 | 10.27 |

COA | 1.349655902 | 1.378232115 | 1.415968514 | 0.015606172 | 12.57 |

COOT | 1.339993404 | 1.340515489 | 1.341698811 | 0.000404625 | 6.90 |

MVO | 1.340107417 | 1.340557897 | 1.3412843 | 0.000348472 | 7.07 |

AOA | 1.416490577 | 2.611250407 | 5.635836036 | 1.06543395 | 16.17 |

AO | 1.340409794 | 1.343149904 | 1.347341023 | 0.00150102 | 10.33 |

SCA | 1.354050532 | 1.389569289 | 1.427391273 | 0.02153387 | 12.77 |

SAO | 1.560583449 | 5.787432723 | 10.70600435 | 2.740561266 | 16.70 |

SOA | 1.340122039 | 1.340921536 | 1.342383254 | 0.000666399 | 8.10 |

POA | 1.339958155 | 1.340036022 | 0.000024296 | 2.50 | |

CA | 1.339989731 | 1.340061639 | 0.000026726 | 2.73 |

Algorithm | Parameters values | |||||
---|---|---|---|---|---|---|

TSO | 5.670155785 | 6.565312011 | 4.438323127 | 3.058677027 | 2.396870184 | 1.3808707 |

EO | 6.009289686 | 5.31362623 | 4.495212338 | 3.502169327 | 2.153385515 | 1.339957787 |

GWO | 6.021199168 | 5.308142494 | 4.490878061 | 3.500059078 | 2.153535662 | 1.339966023 |

MFO | 5.982614347 | 5.350053873 | 4.500958815 | 3.497849413 | 2.143288223 | 1.340025322 |

WOA | 5.714239494 | 5.117700403 | 4.757927854 | 3.465958349 | 2.670577941 | 1.355727617 |

SMA | 6.007546137 | 5.307390778 | 4.507549834 | 3.503500314 | 2.14779716 | 1.339965208 |

HHO | 6.072162327 | 5.412647819 | 4.49417035 | 3.423292247 | 2.084505887 | 1.340774991 |

COA | 5.90969382 | 5.029028706 | 4.560689666 | 3.724153518 | 2.405535284 | 1.349655902 |

COOT | 6.008751259 | 5.322951972 | 4.506063177 | 3.475725155 | 2.160761628 | 1.339993404 |

MVO | 6.051023626 | 5.292876108 | 4.520661102 | 3.452340484 | 2.159179081 | 1.340107417 |

AOA | 5.356286396 | 6.48053284 | 4.297025888 | 4.562178276 | 2.004146109 | 1.416490577 |

AO | 6.123160098 | 5.244389814 | 4.453868547 | 3.527086297 | 2.132421426 | 1.340409794 |

SCA | 6.634403914 | 5.010629598 | 4.321109865 | 3.539258349 | 2.194126028 | 1.354050532 |

SAO | 4.999434962 | 5.003299123 | 4.999791919 | 5.000239438 | 5.006584696 | 1.560583449 |

SOA | 6.066732579 | 5.287415377 | 4.487803249 | 3.481966957 | 2.152396558 | 1.340122039 |

POA | 45.79272 | 74.26591 | 74.93475 | 69.57448 | 76.50655 | 1.339958155 |

CA | 78.03137 | 23.95073 | 64.59176 | 63.55044 | 5.513643 |

^{2}, and a density (^{3}. The column’s length (

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | 26.48723538 | 26.52215446 | 26.71670925 | 0.047627405 | 10.03 |

EO | 26.48636047 | 9.66443E-15 | 2.25 | ||

GWO | 26.48669444 | 26.48898613 | 26.4923826 | 0.001406003 | 7.80 |

MFO | 26.4863605 | 8.49963E-09 | 4.23 | ||

WOA | 26.49142814 | 26.71389042 | 27.45916398 | 0.227930742 | 13.33 |

SMA | 26.48636253 | 26.48644974 | 26.48677412 | 0.000106177 | 6.00 |

HHO | 26.48817488 | 26.53351021 | 26.6437574 | 0.041042248 | 10.73 |

COA | 26.51914471 | 26.61126269 | 26.73052204 | 0.064046766 | 13.07 |

COOT | 26.4863627 | 26.48638274 | 4.88377E-06 | 4.67 | |

MVO | 26.48693664 | 26.48836793 | 26.49001498 | 0.000941322 | 7.40 |

AOA | 26.97263461 | 27.88226718 | 28.71081678 | 0.540678369 | 16.87 |

AO | 26.51558685 | 26.64085432 | 27.01857854 | 0.113082598 | 13.23 |

SCA | 26.54809579 | 26.62667613 | 26.7792609 | 0.056449013 | 13.63 |

SAO | 26.58472642 | 27.14467325 | 28.46629505 | 0.498784512 | 15.67 |

SOA | 26.49154645 | 26.51134498 | 26.53977444 | 0.012814277 | 10.23 |

POA | 26.48636047 | 0.00000000 | 1.97 | ||

CA | 26.48636047 | 0.00000000 | 1.88 |

Algorithm | Parameters values | ||
---|---|---|---|

TSO | 5.452278337 | 0.291633911 | 26.48723538 |

EO | 5.452181285 | 0.291626342 | 26.48636047 |

GWO | 5.452331108 | 0.29161899 | 26.48669444 |

MFO | 5.452181287 | 0.291626342 | 26.48636047 |

WOA | 5.450793798 | 0.291847411 | 26.49142814 |

SMA | 5.45218243 | 0.291626231 | 26.48636253 |

HHO | 5.451684151 | 0.291705525 | 26.48817488 |

COA | 5.453818274 | 0.292090964 | 26.51914471 |

COOT | 5.452181285 | 0.291626342 | 26.48636047 |

MVO | 5.452450386 | 0.291612678 | 26.48693664 |

AOA | 5.32711782 | 0.312578548 | 26.97263461 |

AO | 5.44908817 | 0.292455043 | 26.51558685 |

SCA | 5.451693783 | 0.292826204 | 26.54809579 |

SAO | 5.463051474 | 0.292477332 | 26.58472642 |

SOA | 5.453665515 | 0.291588485 | 26.49154645 |

POA | 7.706642 | 0.767859 | |

CA | 4.573024 | 0.633764 |

Piston lever problem was first raised by Vanderplaats [

The payload is given as

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | 17.42196241 | 397.5988085 | 2093.692333 | 373.4766691 | 14.37 |

EO | 109.1507183 | 167.4727301 | 76.6501403 | 6.73 | |

GWO | 8.42233969 | 135.7896189 | 167.7832157 | 63.67587429 | 10.10 |

MFO | 119.7547204 | 167.4727301 | 72.89046371 | 6.97 | |

WOA | 8.497928904 | 100.1318812 | 512.0065077 | 118.1165766 | 10.00 |

SMA | 8.412698478 | 87.94276045 | 167.4732992 | 79.53005361 | 6.43 |

HHO | 8.457048093 | 303.2021541 | 790.7522755 | 173.8110544 | 13.93 |

COA | 8.534000505 | 9.21985017 | 0.167712838 | 5.67 | |

COOT | 8.412697954 | 123.8580549 | 203.1498737 | 76.06694022 | 9.10 |

MVO | 8.472950733 | 140.9412925 | 284.7751907 | 117.3137109 | 10.23 |

AOA | 185.0680395 | 325.5729621 | 501.3193251 | 100.46379 | 15.27 |

AO | 8.444228253 | 20.49567107 | 190.9228266 | 44.62725153 | 4.87 |

SCA | 8.625292195 | 9.313147737 | 10.70516073 | 0.404245037 | 6.97 |

SAO | 283.1874112 | 15297.60183 | 77744.75245 | 22761.39193 | 16.60 |

SOA | 8.433373601 | 19.26894305 | 170.2505714 | 40.15273808 | 4.47 |

PA | 96.997720073 | 167.472730052 | 77.825586069 | 5.48 | |

CA | 100.753413785 | 167.472730052 | 76.533411694 | 5.82 |

Algorithm | Parameters values | ||||
---|---|---|---|---|---|

TSO | 0.071101082 | 4.304278939 | 4.05075328 | 119.9216191 | 17.42196241 |

EO | 0.05 | 2.041513399 | 4.083027183 | 120 | |

GWO | 0.050132638 | 2.042506155 | 4.08410152 | 119.9542691 | 8.42233969 |

MFO | 0.05 | 2.041513399 | 4.083027183 | 120 | |

WOA | 0.05 | 2.062664782 | 4.083411529 | 119.9550817 | 8.497928904 |

SMA | 0.05 | 2.041513292 | 4.083027241 | 119.9999998 | 8.412698478 |

HHO | 0.05 | 2.04610059 | 4.089278683 | 119.6218262 | 8.457048093 |

COA | 0.05 | 2.065759358 | 4.089137843 | 120 | 8.534000505 |

COOT | 0.05 | 2.041513382 | 4.083027184 | 120 | 8.412697954 |

MVO | 0.05 | 2.046975471 | 4.092356386 | 119.9180173 | 8.472950733 |

AOA | 500 | 500 | 2.302404367 | 61.12128883 | 185.0680395 |

AO | 0.05 | 2.04729087 | 4.085132655 | 120 | 8.444228253 |

SCA | 0.05010724 | 2.072171185 | 4.104600675 | 120 | 8.625292195 |

SAO | 469.2131864 | 403.4320025 | 2.716971944 | 65.42688267 | 283.1874112 |

SOA | 0.05 | 2.045182942 | 4.084523234 | 120 | 8.433373601 |

POA | 376.3677 | 22.52272 | 347.7443 | 116.6528 | |

COA | 120.8287 | 67.23463 | 329.0836 | 14.40809 |

The difference between the robot gripper’s minimum and maximum force is used as an objective function in this challenge. The robot is involved in this challenge, which has six nonlinear design constraints and seven design variables [

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | 4.289551866 | 22872475335 | 2.26915E+11 | 4.289551866 | 14.77 |

EO | 2.558265959 | 3.225935001 | 6.055703362 | 2.558265959 | 4.67 |

GWO | 2.636395337 | 3.325110862 | 4.094105793 | 2.636395337 | 5.87 |

MFO | 3.528008708 | 5.29421131 | 12.60060882 | 3.528008708 | 11.38 |

WOA | 3.287423023 | 6.58577709 | 38.88747796 | 3.287423023 | 12.17 |

SMA | 3.50251645 | 2.545958371 | 1.60 | ||

HHO | 3.205121938 | 18.35609682 | 79.90636828 | 3.205121938 | 14.00 |

COA | 2.795786758 | 4.064341655 | 4.289317141 | 2.795786758 | 9.13 |

COOT | 2.680610563 | 3.503447897 | 4.590989339 | 2.680610563 | 6.70 |

MVO | 2.695840012 | 3.197078896 | 4.574383286 | 2.695840012 | 4.93 |

AOA | 3.690498919 | 4.998068535 | 10.2444732 | 3.690498919 | 11.43 |

AO | 4.519607097 | 22.29788938 | 104.556152 | 4.519607097 | 15.17 |

SCA | 4.123780873 | 4.447648925 | 7.017991237 | 4.123780873 | 10.62 |

SAO | 5.190176925 | 6424961932 | 93324638481 | 5.190176925 | 16.07 |

SOA | 2.621287418 | 3.38339899 | 4.289317141 | 2.621287418 | 5.43 |

POA | 2.63467106 | 3.06246168 | 3.62569846 | 0.22417816 | 4.27 |

CA | 2.63248350 | 3.12543950 | 3.89424622 | 0.24904286 | 4.80 |

Algorithm | Parameters values | |||||||
---|---|---|---|---|---|---|---|---|

TSO | 150 | 150 | 200 | 0 | 150 | 100.0027 | 2.404797 | 4.289551866 |

EO | 149.7029 | 149.5393 | 199.9916 | 0.040805 | 149.9667 | 101.4237 | 2.334378 | 2.558265959 |

GWO | 149.7173 | 149.5536 | 200 | 0 | 30.59937 | 105.1643 | 1.709293 | 2.636395337 |

MFO | 150 | 145.711 | 200 | 0 | 150 | 165.1683 | 2.566817 | 3.528008708 |

WOA | 148.0163 | 147.8544 | 160.7647 | 0 | 149.9996 | 104.385 | 2.601234 | 3.287423023 |

SMA | 150 | 149.8805 | 200 | 0.001158 | 149.2655 | 101.0626 | 2.299768 | |

HHO | 150 | 149.3707 | 183.248 | 5.26E-05 | 148.6101 | 125.2552 | 2.539905 | 3.205121938 |

COA | 150 | 149.7792 | 200 | 0 | 10 | 110.2619 | 1.623498 | 2.795786758 |

COOT | 150 | 149.8301 | 195.8189 | 0.001801 | 149.7434 | 105.4282 | 2.379592 | 2.680610563 |

MVO | 149.2102 | 142.5991 | 196.4037 | 6.458731 | 142.5376 | 103.1736 | 2.374035 | 2.695840012 |

AOA | 150 | 122.1327 | 200 | 26.25289 | 150 | 139.686 | 2.671189 | 3.690498919 |

AO | 149.2919 | 93.109 | 199.0805 | 49.46878 | 145.6248 | 160.2051 | 3.006883 | 4.519607097 |

SCA | 150 | 147.3924 | 175.7737 | 0 | 109.8487 | 155.8737 | 2.374302 | 4.123780873 |

SAO | 112.2475 | 96.3309 | 157.2066 | 15.29257 | 148.262 | 108.467 | 2.797333 | 5.190176925 |

SOA | 150 | 149.8639 | 199.681 | 0 | 148.5837 | 103.2267 | 2.341468 | 2.621287418 |

POA | 18.96377 | 117.8522 | 180.8935 | 27.38139 | 121.1049 | 176.5338 | 2.257163 | 2.63467106 |

CA | 111.7183 | 65.50549 | 165.8307 | 3.049145 | 62.06039 | 230.6625 | 2.715952 | 2.63248350 |

Corrugated bulkhead designs are frequently employed in chemical tankers and product tankers in order to aid in the efficient cleaning of cargo tanks at the loading dock [

Algorithm | Best | Mean | Worst | SD | FMR |
---|---|---|---|---|---|

TSO | 6.855395898 | 7.515017746 | 10.98772723 | 0.892030186 | 11.07 |

EO | 6.842957472 | 2.50569E-11 | 1.17 | ||

GWO | 6.84594174 | 6.849404911 | 6.857978732 | 0.00329714 | 6.40 |

MFO | 6.95886219 | 10.31538608 | 0.623291412 | 2.67 | |

WOA | 6.868197444 | 7.234972405 | 8.666867301 | 0.463809051 | 10.50 |

SMA | 7.953976548 | 11.66989827 | 12.57187707 | 1.108933488 | 16.93 |

HHO | 6.855228127 | 7.085261807 | 7.505850206 | 0.192900564 | 10.03 |

COA | 6.18061627 | 7.46789485 | 8.743988275 | 0.657396042 | 11.23 |

COOT | 6.842961718 | 6.84395948 | 6.855249262 | 0.002264306 | 4.53 |

MVO | 6.844112205 | 6.854550423 | 6.891634775 | 0.009622446 | 6.70 |

AOA | 7.147273011 | 7.94042808 | 10.50177315 | 0.889597343 | 13.67 |

AO | 6.892071425 | 7.346390448 | 8.332439081 | 0.400497798 | 11.17 |

SCA | 7.021747181 | 7.936506325 | 8.617725871 | 0.621047699 | 13.37 |

SAO | 7.020186794 | 8.536171185 | 10.79430556 | 1.006701203 | 14.33 |

SOA | 6.863150498 | 7.609158004 | 8.30725787 | 0.642814739 | 11.67 |

POA | 6.842957474 | 6.843508225 | 6.846584381 | 0.001067412 | 3.83 |

CA | 6.843177079 | 6.845076983 | 0.000488454 | 3.73 |

Algorithm | Parameters values | ||||
---|---|---|---|---|---|

TSO | 55.94375825 | 34.12553171 | 57.63213667 | 1.049990562 | 6.855395898 |

EO | 57.69229443 | 34.14762159 | 57.69229626 | 1.049999774 | |

GWO | 57.40760601 | 34.14377986 | 57.63211944 | 1.050001441 | 6.84594174 |

MFO | 57.6922948 | 34.1476216 | 57.6922968 | 1.04999978 | |

WOA | 53.75332544 | 34.11435394 | 57.67117989 | 1.050052281 | 6.868197444 |

SMA | 38.6549421 | 37.15897337 | 66.97119063 | 1.194658396 | 7.953976548 |

HHO | 57.64186443 | 34.33682599 | 57.57011798 | 1.049999674 | 6.855228127 |

COA | 4.93434E-07 | 0 | 0 | 1.050199052 | 6.18061627 |

COOT | 57.69158469 | 34.14761555 | 57.69229153 | 1.049999734 | 6.842961718 |

MVO | 57.67291664 | 34.14655471 | 57.68941522 | 1.050156185 | 6.844112205 |

AOA | 54.86760749 | 37.15326823 | 54.86760749 | 1.054086491 | 7.147273011 |

AO | 55.60404503 | 34.45085913 | 56.91844174 | 1.050288305 | 6.892071425 |

SCA | 55.21204025 | 35.58586585 | 58.65384066 | 1.067117361 | 7.021747181 |

SAO | 54.15668556 | 34.76085576 | 58.12707556 | 1.070304388 | 7.020186794 |

SOA | 57.70299259 | 34.16103858 | 57.92914589 | 1.053813338 | 6.863150498 |

POA | 66.76194 | 8.339457 | 21.65966 | 4.846303 | 6.842957474 |

CA | 68.80847 | 51.83041 | 73.10743 | 1.992981 |

This study introduces the investigation of 17 different metaheuristic optimization algorithms, which have been proposed in recent years and are popular in the literature, on 12 real-world engineering problems. In this study, an external penalty is imposed on algorithms that are used to cope with inequality and equality constraints when they are implemented. Although the use of such a method is relatively straightforward, determining the optimal values of penalty terms, particularly for optimization problems with a high number of constraints, proves to be a challenging optimization issue in and of itself.

According to the experimental results, CA produced the best optimum value in 10 problems, EO in 9 problems, and POA and MFO in 6 problems. Following these, COOT managed to produce the best optimum value in 4 problems and WOA in 3 problems. Although GWO could not find the best optimum value in any of the 12 different engineering problems, statistically, it showed the sixth-best performance among the methods. Furthermore, COOT could not find the mean value in any of the 12 different engineering problems; statistically, it showed the fourth-best performance among the methods. It is thought that statistical analysis has great importance, especially in the comprehensive examination of the performance of metaheuristic optimization methods.

Friedman statistical analysis is performed, and mean ranks are calculated to analyze the results of the investigation. According to these calculated values, the performances of metaheuristic optimization methods in all problems are presented in

Algorithm | Mean FMR | Mean manuel rank |
---|---|---|

TSO | 12.77 | 15 |

EO | 3.55 | 3 |

GWO | 6.40 | 6 |

MFO | 6.19 | 5 |

WOA | 12.00 | 13 |

SMA | 7.39 | 7 |

HHO | 10.45 | 10 |

COA | 11.23 | 12 |

COOT | 5.81 | 4 |

MVO | 8.70 | 9 |

AOA | 14.45 | 16 |

AO | 12.03 | 14 |

SCA | 11.23 | 12 |

SAO | 16.11 | 17 |

SOA | 8.30 | 8 |

POA | 3.10 | 1 |

CA | 3.28 | 2 |

The research conducted a rigorous analysis by using the Wilcoxon signed-rank test, a non-parametric statistical test, to provide a robust comparison between the suggested and competing algorithms and to provide statistical validation for the findings obtained. The statistical analyses at a significance level of 5% are shown in

Metaheuristic |
Real-world engineering design problem | |||||
---|---|---|---|---|---|---|

Problem-1 | Problem-2 | Problem-3 | Problem-4 | Problem-5 | Problem-6 | |

TSO | 1.73E-06 | 1.73E-06 | 1.92E-06 | 1,73E-06 | 1.73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

EO | – | 1.73E-06 | 4.20E-04 | – | 1.73E-06 | 1.00E+00 |

– | △ | △ | – | △ | ≈ | |

GWO | 1.73E-06 | 1.73E-06 | 4.53E-01 | 1,36E-05 | 1.73E-06 | 1.73E-06 |

△ | △ | ≈ | △ | △ | △ | |

MFO | 1.95E-03 | 2.35E-06 | 2.05E-04 | 7,69E-06 | 1.73E-06 | 1.00E+00 |

△ | △ | △ | △ | △ | ≈ | |

WOA | 1.73E-06 | 1.73E-06 | 1.73E-06 | 1,73E-06 | 1.73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

SMA | 1.73E-06 | 1.73E-06 | 1.48E-03 | 1,73E-06 | 1.73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

HHO | 1.73E-06 | 1.73E-06 | 3.41E-05 | 1,73E-06 | 1.73E-06 | 1.00E+00 |

△ | △ | △ | △ | △ | ≈ | |

COA | 1.73E-06 | 1.73E-06 | 1.73E-06 | 1,73E-06 | 1.73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

COOT | 1.73E-06 | 3.52E-06 | 2.11E-03 | 1,73E-06 | 1.22E-05 | 1.25E-01 |

△ | △ | △ | △ | △ | ≈ | |

MVO | 1.73E-06 | 1.73E-06 | 2.61E-04 | 1,73E-06 | 1.73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

AOA | 1.73E-06 | 1.73E-06 | 1.73E-06 | 1,73E-06 | 1.73E-06 | 1.72E-06 |

△ | △ | △ | △ | △ | △ | |

AO | 1.73E-06 | 1.73E-06 | 2.41E-03 | 1,73E-06 | 1.73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

SCA | 1.73E-06 | 1.73E-06 | 1.13E-05 | 1,73E-06 | 1.73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

SAO | 1.73E-06 | 1.73E-06 | 1.73E-06 | 1,73E-06 | 1.73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

SOA | 1.73E-06 | 1.73E-06 | 1.92E-01 | 1,73E-06 | 1.70E-06 | 1.73E-06 |

△ | △ | ≈ | △ | △ | △ | |

POA | 1.73E-06 | – | – | 8,13E-01 | 1.00E+00 | – |

△ | – | – | ≈ | ≈ | – | |

CA | 1.73E-06 | 1.41E-01 | 6.06E-01 | 1,11E-02 | – | 1.00E+00 |

△ | ≈ | ≈ | △ | – | ≈ |

Metaheuristic |
Real-world engineering design problem | |||||
---|---|---|---|---|---|---|

Problem-7 | Problem-8 | Problem-9 | Problem-10 | Problem-11 | Problem-12 | |

TSO | 1.73E-06 | 1,73E-06 | 1.73E-06 | 1,73E-06 | 1,73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

EO | 1.00E+00 | – | 1.00E+00 | 4,99E-03 | 4,07E-05 | – |

≈ | – | ≈ | △ | △ | – | |

GWO | 1.73E-06 | 3,18E-06 | 1.73E-06 | 4,86E-05 | 2,88E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

MFO | 2.50E-01 | 1,73E-06 | 1.73E-06 | 1,20E-03 | 1,73E-06 | 3.88E-06 |

≈ | △ | △ | △ | △ | △ | |

WOA | 1.73E-06 | 1,73E-06 | 1.73E-06 | 7,51E-05 | 1,73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

SMA | 1.73E-06 | 1,92E-06 | 1.73E-06 | 8,97E-02 | – | 1.73E-06 |

△ | △ | △ | △ | – | △ | |

HHO | 1.73E-06 | 1,73E-06 | 1.73E-06 | 2,60E-06 | 1,73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

COA | 1.73E-06 | 1,73E-06 | 1.73E-06 | 2,11E-03 | 1,73E-06 | 1.06E-04 |

△ | △ | △ | △ | △ | △ | |

COOT | 1.82E-05 | 1,73E-06 | 5.61E-06 | 2,61E-04 | 1,73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

MVO | 1.73E-06 | 1,73E-06 | 1.73E-06 | 9,32E-06 | 3,52E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

AOA | 1.73E-06 | 1,73E-06 | 1.73E-06 | 1,73E-06 | 1,73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

AO | 1.73E-06 | 1,73E-06 | 1.73E-06 | 1,20E-01 | 1,73E-06 | 1.73E-06 |

△ | △ | △ | ≈ | △ | △ | |

SCA | 1.73E-06 | 1,73E-06 | 1.73E-06 | 3,59E-04 | 1,73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | ||

SAO | 1.73E-06 | 1,73E-06 | 1.73E-06 | 1,73E-06 | 1,73E-06 | 1.73E-06 |

△ | △ | △ | △ | △ | △ | |

SOA | 1.73E-06 | 1,73E-06 | 1.73E-06 | – | 4,86E-05 | 1.73E-06 |

△ | △ | △ | – | △ | △ | |

POA | 9.38E-02 | 4,53E-04 | 1.00E+00 | 2,18E-02 | 4,45E-05 | 1.73E-06 |

≈ | △ | ≈ | △ | △ | △ | |

CA | – | 2,60E-05 | – | 6,04E-03 | 2,37E-05 | 1.92E-06 |

– | △ | – | △ | △ | △ |

When

As a result, in the study conducted, EO and POA in 7 different problems, CA in 6 different problems, SOA and MFO in 2 different problems, and GWO, HHO, COOT, and SMA in 1 different problem showed the most successful results, or the method showing the most successful results could not provide a significant superiority to these methods.

The efficient resolution of real-world engineering design optimization issues is widely acknowledged as a significant difficulty for any new metaheuristic algorithm presented to the market. Moreover, these issues include several goals and diverse variables, including integers, continuous values, and discrete elements. Additionally, they involve a range of nonlinear restrictions related to kinematic conditions, performance parameters, operational situations, and manufacturing specifications, among others. The TSO, EO, GWO, MFO, WOA, SMA, HHO, COA, COOT, MVO, AOA, AO, SCA, SAO, SOA, POA, and CA algorithms are used to address design optimization of twelve real-world engineering issues. Accordingly, their performances are compared considering the quality of solution, robustness, and convergence speed of the solutions obtained by various approaches. The outcomes reveal that EO and POA produce better optimized results against other available techniques. However, the results of statistical comparisons show that EO and POA achieve more competitive and better performance outcomes among most of the constraint problems investigated. In addition, as a consequence of the statistical examination, it was reported that the CA approach is at a level that can compete with these two methods.

This study discusses the most important subjects in engineering and artificial intelligence disciplines. Future research confidently relies on this review to investigate metaheuristic optimization approaches and engineering design challenges in greater depth in the near future. Moreover, by examining the studies in question, the researchers can more easily identify a beginning point for future researchers.

The authors thank Manisa Celal Bayar University for the use of the laboratories in the Department of Software Engineering. Especially, the devices in the laboratory established by Manisa Celal Bayar University—Scientific Research Projects Coordination Unit (MCBU–SRPCU) with Project Code 2022-134 were utilized in the study.

The authors received no specific funding for this study.

The authors confirm contribution to the paper as follows: study conception and design: Elif Varol Altay, Osman Altay; data collection: Elif Varol Altay; analysis and interpretation of results: Elif Varol Altay, Osman Altay, Yusuf Özçevik; draft manuscript preparation: Osman Altay, Yusuf Özçevik. All authors reviewed the results and approved the final version of the manuscript.

All data generated and analyzed throughout the research process are given in the published article.

The authors declare that they have no conflicts of interest to report regarding the present study.