A Novel Swarm Optimization Algorithm Based on Hive Construction by Tetragonula carbonaria Builder Bees

Abstract

This paper introduces a new optimization problem-solving method based on how the stingless bee Tetragonula carbonaria builds and regulates temperature in the hive. The Tetragonula carbonaria Optimization Algorithm (TGCOA) models three different behaviors: strengthening the structure’s hive when it is cold, building combs in a spiral pattern at medium temperatures, and stabilizing the hive when it is hot. These temperature-dependent strategies dynamically balance global exploitation and local exploration within the solution space, enabling a more efficient search. To validate the efficiency and effectiveness of the proposed method, the TGCOA algorithm was tested using ten unimodal and ten multimodal benchmark functions, twenty-eight constrained problems with dimensions set to 10, 30, 50, and 100 taken from the IEEE CEC 2017, and seven real-world engineering design challenges. Furthermore, it was compared with ten algorithms from the literature. Wilcoxon signed-rank and Friedman statistical tests were performed to assess the outcomes. The results on the benchmark problems showed that the approach outperformed 80% of the algorithms at a 5% significance level in the Wilcoxon signed-rank test and ranked first overall according to the Friedman test. Additionally, in multidimensional problems, the TGCOA was ranked first in dimensions 30, 50, and 100. Moreover, in engineering problems, the approach demonstrated a high capacity to solve constraint problems, obtaining better results than the algorithms that were compared.

1. Introduction

Optimization is the process of finding the optimal values (or near-optimal) for a set of variables that maximize or minimize an objective function. Mathematically, we want to find the vector $\overrightarrow{X}$, where $\overrightarrow{X}=[x_1,x_2,\dots ,x_D]$, and D is the dimension of the problem (number of variables), which is represented as

$$\begin{array}{cc}\mathrm{Minimize}& f\left(\overrightarrow{X}\right),\hfill \\ \mathrm{Subject}\mathrm{to}& g_i\left(\overrightarrow{X}\right)\le 0,i=1,\dots ,p,\hfill \\ & h_j\left(\overrightarrow{X}\right)=0,j=1,\dots ,q,\hfill \\ & x_k^{\left(l\right)}\le x_k\le x_k^{\left(u\right)},k=1,\dots ,D.\hfill \end{array}$$

(1)

$g_i$ represents the inequality constraints, and $h_j$ denotes the equality constraints, where p and q are specified as the quantities of inequality and equality constraints, respectively. Finally, $x_k^{\left(l\right)}$ and $x_k^{\left(u\right)}$ denote the lower- and upper-bound constraints, in that order.

Optimization is present in several areas of engineering and applied sciences, where it is used to improve designs, reduce costs, increase efficiency, or find the best possible solution under specific constraints. Many of these real-world problems are highly complex, involving multimodal functions or nonconvex search spaces. In such cases, metaheuristics offer a powerful alternative, particularly for problems with high computational demands and intricate solution landscapes. Their main advantages include the ability to efficiently explore the search space and reduce the risk of becoming trapped in local optima. Several metaheuristic algorithms have been proposed, inspired by modeling biological behavior, chemical processes, or physical phenomena. They can be classified into twelve categories [1], as shown in Figure 1, according to their fundamental principles, as follows:

• Evolutionary algorithms: Genetic Algorithm (GA) [2,3], Cultural Algorithm (CA) [4], Memetic Algorithm (MA) [5], Gene Expression Programming (GEP) [6], Evolutionary Strategy (ES) [7], Gradient Evolution Algorithm (GEA) [8], Differential Evolution (DE) [9], Liver Cancer Algorithm (LCA) [10], Evolutionary Mating Algorithm (EMA) [11], and Clustered-Vectors Optimization algorithm (CVO) [12].
• Swarm-based algorithms: Particle Swarm Optimization(PSO) [13]), Golden Jackal Optimization (GJO) [14], Black Widow Optimization (BWOA) [15], Jumping Spider optimization algorithm (JSOA) [16], Horned Lizard Optimization Algorithm(HLOA) [1], Fennec Fox Algorithm (FFA) [17], Dingo Optimization Algorithm (DOA) [18], Artificial Rabbits Optimization (ARO) [19], Tunicate Swarm Algorithm (TSA) [20], Mantis Shrimp Optimization Algorithm (MShOA) [21], Harris hawks Optimization Algorithm (HHO) [22], Bonobo optimizer (BO) [23], Salp Swarm Algorithm (SSA) [24], Ant Lion Optimizer (ALO) [25], Mexican Axolotl Optimization Algorithm (MAO) [26], and Mountain Gazelle Optimizer [27].
• Ancient-based algorithms: Giza Pyramids Construction (GPC) [28], Tianji’s horse racing optimization (THRO) [29], Al-Biruni Earth Radius (BER) Metaheuristic Search Optimization Algorithm [30], Great Wall Construction Algorithm (GWCA) [31], and Dujiangyan Irrigation System Optimization (DISO) [32].
• Physics-based algorithms: Simulated Annealing (SA) [33], Gravitational Search Algorithm (GSA) [34], Thermal Exchange Optimization (TEO) [35], Momentum Search Algorithm (MSA) [36], Light Spectrum Optimizer (LSO) [37], Kepler optimization algorithm (KOA) [38], Artificial Satellite Search Algorithm (ASSA) [39], and Vibrating Particles System Algorithm [40].
• Chemistry-based algorithms: Chemical Reaction Optimization (CRO) [41], Enzyme action optimization (EAO) [42], Crystal Structure Algorithm (CryStAl) [43], and Chemical Reaction Pathway Algorithm (CRPA) [44].
• Human-based algorithms: Teamwork Optimization Algorithm (TOA) [45], Coronavirus Mask Protection Algorithm (CMPA) [46], Sewing Training-Based Optimization (STBO) [45], Stock Exchange Trading Optimization (SETO) [47], Revolution Optimization Algorithm (ROA) [48], and Osprey Optimization Algorithm (OOA) [49].
• Plant-based algorithms: Waterwheel Plant Algorithm (WWPA) [50], Willow Catkin Optimization Algorithm (WCO) [51], and Strawberry Algorithm (SBA) [52].
• Music-based/Art-based algorithms: Stochastic Paint Optimizer [53] and Color harmony algorithm (CHA) [54].
• Sport-based algorithms: Puzzle Optimization Algorithm (POA) [55], Running City game optimizer (RCGO) [56], Alpine Skiing Optimization (ASO) [57], Basketball team optimization algorithm (BTOA) [58], and Golf Sport Inspired Search (GSIS) [59].
• Mathematical-based algorithms: Arithmetic Optimization Algorithm (AOA) [60], Newton’s Downhill Optimizer (NDO) [61], and Multi-objective exponential distribution optimizer (MOEDO) [62].
• Single-solution-based algorithms: Tabu Search (TS) [63], Single Candidate Optimizer (SCO) [64], and Variable Neighbourhood Search (VNBS) [65].
• Hybrid algorithms: Butterfly Optimization Algorithm Combined with Black Widow Optimization (BFA-BWOA) [66], Equilibrium Whale Optimization Algorithm (EWOA) [67], Cuckoo optimization algorithm and SailFish optimizer (COA-SFO) [68], Improved Dingo Optimization Algorithm (IDOA) [69], Cuckoo Search and Stochastic Paint Optimizer (CSSPO) [70], Elite opposition-based learning, Chaotic k-best gravitational search strategy, and Grey wolf optimizer (EOCSGWO) [71], Hybrid Mutualism Mechanism-inspired Butterfly and Flower Pollination Optimization Algorithm (HMMB-FPOA) [72], and Hybrid Pelican Komodo Algorithm (HPKA) [73].

Since no unique method can universally and effectively solve all optimization problems [74], the scientific community continues to propose new algorithms to address increasingly complex optimization scenarios. A general metaheuristic algorithm framework consists of four steps, as depicted in Figure 2, and is outlined as follows:

First step: A set of possible solutions (vectors) is randomly generated. They evolve in each iteration.

Second step: The entire population is evaluated by the objective function, and the value computed is named fitness. For minimization, the goal is to find the lowest fitness.

Third step: Bio-inspired functions based on modeling biological behavior, chemical processes, or physical phenomena are used to recombine vectors.

Fourth step: The fitness of each vector, in the current iteration, is compared with the best vector in the evolutionary process. The lowest fitness is updated as the best vector.

Finally, this evolutionary process is repeated until a stop condition is met, and then the best values found are reported. Note that the number of iterations and the population size influence the algorithm’s performance in this evolutionary process. Therefore, the algorithm’s efficiency implies setting these parameters to the lowest value. Moreover, the effectiveness is related to a balance between exploitation (local search) and exploration (global search) to find feasible solutions in the search space.

The contributions of this work are the following:

• This paper proposes the Tetragonula carbonaria Optimization Algorithm (TGCOA), a novel swarm-based metaheuristic inspired by the temperature regulation behaviors during hive construction. The TGCOA models three key strategies: slight fluttering motion to warm the hive, strong fluttering motion for hive cooling, and building the spiral hive at moderate temperatures. Note that there are no works in the literature that carry out the modeling presented.
• The novel approach achieves a balance between local and global search through honeybee-inspired strategies: slight fluttering motion to warm the hive modeling local exploitation, while building the spiral hive at moderate temperatures simulates global exploration.
• The TGCOA operates without additional parameters, using only a fixed population size of 30 and 200 iterations. Note that a low number of iterations contributes to the algorithm’s efficiency compared to several approaches reported in the literature that use more iterations.
• To evaluate the efficiency and efficacy of the proposed approach, twenty benchmark functions, twenty-eight IEEE CEC 2017 constrained issues, and seven real-world engineering design challenges are used.
• The Average Convergence Rate (ACR) metric is used to quantify the average speed at which the algorithm approaches an optimal (near-optimal) solution throughout the iterative process.
• Statistical analysis is conducted through Wilcoxon signed-rank and Friedman tests to benchmark the TGCOA against ten bio-inspired metaheuristic algorithms taken from the literature.
• The TGCOA’s MATLAB code is available to support this study’s findings.

The remainder of this paper’s sections are arranged as follows: Section 2 introduces the bio-inspiration behind the TGCOA, its mathematical formulation, pseudocode, and time complexity analysis. Section 3 compares the proposed approach with ten biologically inspired algorithms and evaluates its effectiveness using a variety of testbed functions by contrasting the results with Wilcoxon and Friedman statistical tests. In Section 4, the results and discussion are presented. The constraint-handling technique employed, as well as the application of the TGCOA to real-world optimization problems, are detailed in Section 5. Finally, the conclusions and recommendations for future research are summarized.

2. Tetragonula carbonaria Optimization Algorithm (TGCOA)

2.1. Biological Fundamentals

Tetragonula carbonaria, often referred to as the sugar sac bee or bush bee, is a group of eusocial insects found predominantly in Southeast Asia and Australia [75,76,77,78,79,80]. These bees are well known for their excellent honey production capacity and their ability to pollinate a wide range of crops. Tetragonula carbonaria worker bees are tiny, measuring between 3.9 and 4.3 mm in body length (see Figure 3), and they can generate approximately 2 kg of honey per year [81].

One of its outstanding characteristics is the design and construction of the hive. Tetragonula carbonaria, unlike other bees, builds a hive with a three-dimensional spiral arrangement composed of terraces placed on top of each other, as shown in Figure 4.

The architecture of the Tetragonula carbonaria hive is profoundly influenced by environmental temperature, which determines whether the bees prioritize heat generation, efficient expansion, or ventilation of the nest. At low temperatures, between 10 and 17 °C, some worker bees cluster together and perform a slight fluttering motion to warm the hive and safeguard the brood. Meanwhile, for high temperatures, ranging from 30 to 40 °C, the fluttering motion increases as a cooling regulation mechanism. Finally, when temperatures range between 18 and 29 °C, there is no fluttering movement, and bees focus on building the spiral comb [82,83]. The construction dynamic is staggered, where the bees build new cells at the edges, fill those cells with eggs, and seal them before moving on to the next level [84]. The hexagonal geometry of the cells allows an efficient distribution of space and favors air circulation, key for comb development [85]. Each bee collaborates indirectly within a decentralized system; the task to be performed is communicated through changes in the hive’s temperature-driven environment; this mechanism is known as stigmergy [86,87].

2.2. Mathematical Model and Optimization Algorithm

The research focuses on modeling the collective construction and thermoregulatory behavior of Tetragonula carbonaria. Inspired by the way bees work together in their hive, responding to temperature changes, a new algorithm is created that mimics how bees change their behavior based on their surroundings. The model reproduces the dynamic strategies used by the colony to conserve thermal stability and expand the spiral structure of the comb.

2.2.1. Slight Fluttering Motion and Cluster to Warm the Hive

Slight fluttering motion consists of subtle vibrations of the bees’ wings without actually flying. This mechanism activates thermogenesis at temperatures between 10 and 17 °C, causing the worker bees to cluster near the brood to generate heat and maintain a stable temperature inside the comb [88,89]. This mechanism is depicted in Figure 5, and, mathematically, based on Fourier’s Law [90], it is represented by the following equation:

$$Q=-{\displaystyle \frac{dT}{dx}}·A·k$$

(2)

where Q is the heat flux (in watts, W), ${\displaystyle \frac{dT}{dx}}$ represents the temperature gradient (temperature variation per unit length), A is the cross-sectional area through which heat flows, and k is the thermal conductivity of the material (in W/m·C). Equation (2) can be represented as follows:

$${\displaystyle \frac{dT}{dx}}={\Delta }_T\left(t\right)={\displaystyle \frac{{\sum }_{r=1}^{nb}\left({\overrightarrow{\psi }}_r\left(t\right)-{\overrightarrow{x}}_i\left(t\right)\right)}{nb}}$$

(3)

where ${\overrightarrow{\psi }}_r\left(t\right)-{\overrightarrow{x}}_i\left(t\right)$ represents the temperature variation. ${\overrightarrow{\psi }}_r\left(t\right)$ denotes the r-th randomly generated subset of bees at generation t, which clusters to increase the hive temperature by a slight fluttering motion, $\overrightarrow{\psi }\subset X$, where X is the population of bees. ${\overrightarrow{x}}_i\left(t\right)$ is the current i-th worker bee in the generation t. $nb$ is a randomly generated integer that indicates the number of bees that will be clustered to warm the hive. Equation (3) can be rewritten as

$${\overrightarrow{x}}_i(t+1)={\displaystyle \frac{1}{2}}\left\{{\overrightarrow{x}}_{\ast }\left(t\right)+{\Delta }_T\left(t\right)\right\}·A^2·k$$

(4)

In this model, ${\overrightarrow{x}}_i(t+1)$ represents the position of the i-th worker bee (search agent) in the solution search space at iteration $t+1$. ${\overrightarrow{x}}_{\ast }\left(t\right)$ represents the best search agent found in the current generation t. ${\Delta }_T\left(t\right)$ represents the temperature variation shown in Equation (3). A is the flutter amplitude randomly generated between 0.2 and 0.3 mm. Finally, k represents thermal conductivity set to 0.03 W/m·C (Watts per meter-Celsius). The pseudocode for this method is presented in Algorithm 1.

Algorithm 1 Slight Fluttering Motion and Cluster of Bees

1:   Input: Population size $SizePop$ 2:   Output: Updated bee positions 3:   procedure SlightFlapping 4:      Generate a random integer $nb\in [1,SizePop]$ 5:      Generate a vector of $nb$ random integer indices $r\in [1,SizePop]$ 6:      Generate the subset ${\overrightarrow{\psi }}_r\subset Population$        ▹ Bees that flutter to increase temperature 7:      Generate a random flutter amplitude $A\in [0.2,0.3]$ 8:      Update the bee position using Equation (4) 9:      return Updated bee positions 10:  end procedure

2.2.2. Strong Fluttering Motion and Clustering for Hive Cooling

The intensified fluttering movement occurs at high ambient temperatures, between 30 and 40 °C, and serves as a cooling mechanism to regulate the internal temperature of the hive [76]. This mechanism is depicted in Figure 6 and mathematically represented by the following equation:

$${\overrightarrow{x}}_i(t+1)={\overrightarrow{x}}_{\ast }\left(t\right)-{\displaystyle \frac{1}{2}}\left\{{\overrightarrow{x}}_{\ast }\left(t\right)+{\Delta }_T\left(t\right)\right\}·cf\left(t\right)·A^2·k$$

(5)

In this model, ${\overrightarrow{x}}_i(t+1)$ represents the position of the i-th worker bee (search agent) in the solution search space at iteration $t+1$. ${\overrightarrow{x}}_{\ast }\left(t\right)$ represents the best search agent found in the current generation t. ${\Delta }_T\left(t\right)$ represents the temperature variation shown in Equation (3). A is the flutter amplitude randomly generated between 0.2 and 0.3 mm. Finally, k represents thermal conductivity, set to 0.03 W/m·C (Watts per meter-Celsius).

$$cf\left(t\right)=\left[{\displaystyle \frac{(\alpha -1)t}{{\mathrm{Max}}_{\mathrm{iter}}-\alpha }}-\alpha \right]$$

(6)

$cf\left(t\right)$ is a cooling factor that dynamically modulates the step size as the algorithm progresses.  $\alpha$ represents the number of wing flutterings per second performed by a bee, set to 50. $Ma{x}_{\mathrm{iter}}$ represents the maximum number of iterations. The pseudocode of this strategy is shown in Algorithm 2.

Algorithm 2 Strong Fluttering Motion and Cluster of Bees

1:  Input: Population size $SizePop$ 2:  Output: Updated bee positions 3:  procedure  StrongFlapping 4:     Generate a random integer $nb\in [1,SizePop]$ 5:     Generate a vector of $nb$ random integer indices $r\in [1,SizePop]$ 6:     Generate the subset ${\overrightarrow{\psi }}_r\subset Population$        ▹ Bees that flutter to dissipate heat 7:     Generate a random flutter amplitude $A\in [0.2,0.3]$ 8:     Update the bee position using Equation (5) 9:     return Updated bee positions 10: end procedure

2.2.3. Building the Spiral Hive

Bees focus on optimizing comb construction when the temperature ranges between 18 °C and 29 °C, thereby minimizing their metabolic and thermoregulatory demands [91,92,93]. In these favorable thermal conditions, bees build a geometric spiral pattern through stigmergy. The movement of the bee to build the next cell is mathematically represented as follows:

$${\overrightarrow{x}}_i(t+1)={\overrightarrow{x}}_{\ast }\left(t\right)+{\displaystyle \frac{\beta }{1-\lambda }}$$

(7)

where ${\overrightarrow{x}}_i(t+1)$ represents the position of the i-th worker bee (search agent) in the solution search space at iteration $t+1$. ${\overrightarrow{x}}_{\ast }\left(t\right)$ represents the best search agent found in the current generation t. $\beta$ is a random uniform distribution value in the range $(0,1)$, introducing slight variation in the movement. $\lambda$ is a random number generated from a standard Cauchy distribution with median set to 0 and the scale parameter set to 1. This model simulates the bees’ short-range displacements around their position, enabling them to perform local searches and fine-tune the placement of new comb cells for construction.

Hive spiral geometry is based on the Archimedean spiral and is represented using polar angles in the polar coordinate system, as shown in the following equation:

$$\Delta x=a+b·\theta ·cos\left(\theta \right),\Delta y=a+b·\theta ·sin\left(\theta \right),$$

(8)

Thereby, Equation (8) can be re-defined as follows:

$${\overrightarrow{x}}_i(t+1)={\overrightarrow{x}}_{\ast }\left(t\right)+\left|{\displaystyle \frac{arctg\left(\frac{\Delta y}{\Delta x}\right)}{1.58}}\right|·{\overrightarrow{x}}_i\left(t\right)·1.19$$

(9)

In this model, ${\overrightarrow{x}}_i(t+1)$ represents the position of the i-th worker bee (search agent) in the solution search space at iteration $t+1$. ${\overrightarrow{x}}_{\ast }\left(t\right)$ represents the best search agent found in the current generation t. a represents the distance from the origin to the starting point of the spiral, set to 0, and b defines how far the spiral expands from the origin for each unit of angular increase, set to 10. The value 1.58 corresponds to a normalization factor that regulates the influence of the spiral angle.  ${\overrightarrow{x}}_i\left(t\right)$ is the current i-th worker bee in the generation t. The average distance between cells in the comb structure is reflected by the adjustment coefficient, which stands at 1.19 mm.

The angular parameter $\theta$ controls the spiral tightening and is computed as follows:

$$\theta =\left(90-\left(90·{\displaystyle \frac{t}{{\mathrm{Max}}_{\mathrm{iter}}}}\right)\right)·{\displaystyle \frac{\pi }{180}}$$

(10)

where t denotes the current iteration, ${\mathrm{Max}}_{\mathrm{iter}}$ is the maximum number of iterations, and  ${\displaystyle \frac{\pi }{180}}$ converts the angle from degrees to radians.

Note that the hive construction phase is based on Equations (7) and (10), which are randomly selected as shown in Algorithm 3. Equation (7) represents the construction of cell around the bee, while Equation (10) corresponds to the spiral construction. The whole strategy of building the spiral hive is depicted in Figure 7. Additionally, the pseudocode is presented in Algorithm 3.

All strategies are summarized in a flowchart, as shown in Figure 8, as well as in Algorithm 4.

Algorithm 3 Building the Spiral Hive

1:   Input: Current population of bees 2:   Output: Updated bee positions 3:   procedure  SpiralHive 4:     Identify the best solution vector ${\overrightarrow{x}}_{\ast }\left(t\right)$ from the current population 5:     Generate a random value $\beta \in (0,1)$ from a uniform distribution 6:     Generate $\lambda \sim \mathrm{Cauchy}(0,1)$ 7:     if $rand\le 0.2$ then 8:         Construct the cell around the bee using Equation (7) 9:     else 10:         Construct hive spiral based on Archimedean spiral using Equation (9) 11:     end if 12:     return Updated bee positions 13: end procedure

2.3. Pseudocode for TGCOA

Algorithm 4 outlines the pseudocode for the TGCOA.

Algorithm 4Tetragonula carbonaria Optimization Algorithm (TGCOA)

1:   Input: Search agents, maximum number of iterations, population size 2:   Output: Best solution $x_{\ast }$ 3:   procedure  TGCOA 4:     Initialize parameters: define search agents, max iterations, and population size 5:     Generate the initial population randomly 6:     while iteration < MaxIterations do 7:         Generate a random temperature in the range [10, 40] °C 8:         if Temperature between 10 °C and 17 °C then 9:            Slight fluttering motion and cluster of bees Equation (4) 10:       else if Temperature between 18 °C and 29 °C then         ▹ Building the spiral hive 11:          if random $\le 0.2$ then 12:              Construct the cell around the bee Equation (7) 13:          else 14:              Construct hive spiral based on Archimedean spiral Equation (9) 15:          end if 16:       else 17:          Strong fluttering motion and clustering for hive cooling Eq.5 18:       end if 19:       Calculate $x_{\mathrm{new}}$, the fitness value of the new search agents 20:       if $x_{\mathrm{new}}<x_{\ast }$ then 21:          $x_{\ast }\leftarrow x_{\mathrm{new}}$ 22:       end if 23:       iteration ← iteration + 1 24:   end while 25:   Display $x_{\ast }$, the best optimal solution 26: end procedure

2.4. The TGCOA’s Time Complexity

Computational complexity is defined by a function that correlates the execution time of the method with the magnitude of the problem input, and Big-O notation is used to represent it. The time complexity of the TGCOA depends on the number of iterations ($Ma{x}_{Iteration}$), the population size of bees (nBee), the problem’s dimensionality (D), and the cost of the objective function f. Based on the pseudocode shown in Algorithm 4, a detailed analysis is presented in

Table 1. Time complexity analysis of TGCOA.

Line in Algorithm 4	Big-O	Description
3	$O(nBee·D)$	Initialization: Generate $nBee$ random solutions in D dimensions.
5	$O\left(1\right)$	Random temperature generation (constant-time operation).
6–14	$O(nBee·D)$	Temperature-based behavior: Update all $nBee$ agents’ positions in D dimensions (applies to Equations (2)–(4) or (6)).
15	$O(nBee·f)$	Fitness evaluation: Compute fitness for $nBee$ agents, where f is the cost of one evaluation.
16–18	$O\left(1\right)$	Update best solution: Single comparison and assignment.
4–20	$O(Ma{x}_{\mathrm{Iteration}}·nBee·(D+f))$	Main loop: Repeat lines 5–19 for $T_{\max }$ iterations. Dominated by agent updates and fitness evaluations.
21	$O\left(1\right)$	Output the best solution (constant-time).

.

The overall time complexity of the TGCOA can be computed as follows:

$$\begin{array}{c}\hfill O\left(\mathrm{TGCOA}\right)=O(nBee.D)+O\left(1\right)+O(nBee.D)+O(nBee.f)+O\left(1\right)\\ \hfill +O(Ma{x}_{Iteration}\ast nBee\ast (\mathit{D}+f\left)\right)+O\left(1\right)\end{array}$$

(11)

In O notation, when we add several terms, the fastest-growing one dominates. Hence, the time complexity of the TGCOA can be expressed as follows:

$$O\left(\mathrm{TGCOA}\right)=O(Ma{x}_{Iteration}\ast nBee\ast (\mathit{D}+f\left)\right)$$

(12)

3. Experimental Setup

To evaluate the effectiveness and stability of the proposed TGCOA in solving numerical optimization problems, a set of 20 well-known test functions was used. The set has 10 unimodal functions and 10 multimodal functions. Single-mode functions are associated with local searches (exploitation), whereas multimodal functions relate to global searches (exploration) in the solution search space. Appendix A provides an overview of these functions. Their classification is presented in

Table A1. Classification of 20 testbench functions.

ID	Function Name	Unimodal	Multimodal	n-Dimensional	Non-Separable	Convex	Differentiable	Continuous	Nonconvex	Non-Differentiable	Separable	Random
F1	Brown	X		X	X	X	X
F2	Griewank	X		X	X			X	X
F3	Schwefel 2.20	X		X	X		X	X	X
F4	Schwefel 2.21	X		X		X		X		X	X
F5	Schwefel 2.22	X		X		X		X		X	X
F6	Schwefel 2.23	X		X		X	X	X			X
F7	Sphere	X		X		X	X	X			X
F8	Sum Squares	X		X		X	X	X			X
F9	Xin-She Yang N. 3	X		X	X		X		X
F10	Zakharov	X		X		X		X
F11	Ackley		X	X			X	X	X
F12	Ackley N. 4		X	X	X		X		X
F13	Periodic		X	X			X	X	X	X
F14	Quartic		X	X			X	X			X	X
F15	Rastrigin		X	X		X	X	X			X
F16	Rosenbrock		X	X	X		X	X	X
F17	Salomon		X	X	X		X	X	X
F18	Xin-She Yang		X	X					X	X	X
F19	Xin-She Yang N. 2		X	X	X				X	X
F20	Xin-She Yang N. 4		X	X	X				X	X

, and their description is shown in

Table A2. Description of the 20 testbench functions.

ID	Function	Dim	Interval	${\mathit{f}}_{min}$
F1	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)={\sum }_{i=1}^{n-1}{\left(x_i^2\right)}^{\left(x_{i+1}^2+1\right)}+{\left(x_{i+1}^2\right)}^{\left(x_i^2+1\right)}$	30	$[-1,4]$	0
F2	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)=1+{\sum }_{i=1}^n{\displaystyle \frac{x_i^2}{4000}}-{\prod }_{i=1}^{n}cos\left({\displaystyle \frac{x_i}{\sqrt{i}}}\right)$	30	$[-600,600]$	0
F3	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)={\sum }_{i=1}^n\left|x_i\right|$	30	$[-100,100]$	0
F4	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)={max}_{i=1,\dots ,n}\left|x_i\right|$	30	$[-100,100]$	0
F5	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)={\sum }_{i=1}^n\left|x_i\right|+{\prod }_{i=1}^n\left|x_i\right|$	30	$[-100,100]$	0
F6	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)={\sum }_{i=1}^n{x}_i^{10}$	30	$[-10,10]$	0
F7	$f\left(\mathbf{x}\right)=f\left(x_1,x_2,\dots ,x_n\right)={\sum }_{i=1}^n{x}_i^2$	30	$[-5.12,5.12]$	0
F8	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)={\sum }_{i=1}^{n}i{x}_i^2$	30	$[-10,10]$	0
F9	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)=exp\left(-{\sum }_{i=1}^n{\left(x_i/\beta \right)}^{2m}\right)-2exp\left(-{\sum }_{i=1}^n{x}_i^2\right){\prod }_{i=1}^n{cos}^2\left(x_i\right)$	30	$[-2\pi ,2\pi ],m=5$, $\beta =15$	-1
F10	$f\left(x\right)=f\left(x_1,..,x_n\right)={\sum }_{i=1}^n{x}_i^2+{\left({\sum }_{i=1}^{n}0.5i{x}_i\right)}^2+{\left({\sum }_{i=1}^{n}0.5i{x}_i\right)}^4$	30	$[-5,10]$	0
F11	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)=-a·exp\left(-b\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{x}_i^2}\right)-exp\left({\displaystyle \frac{1}{d}}{\sum }_{i=1}^{n}cos\left(c{x}_i\right)\right)+a+exp\left(1\right)$	30	$[-32,32],a=20$ $b=0.3,c=2\pi$	0
F12	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)={\sum }_{i=1}^{n-1}\left(e^{-0.2}\sqrt{x_i^2+x_{i+1}^2}+3\left(cos\left(2x_i\right)+sin\left(2x_{i+1}\right)\right)\right)$	2	$[-35,35]$	$-5.901\times {10}^{14}$
F13	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)=1+{\sum }_{i=1}^n{sin}^2\left(x_i\right)-0.1e^{\left({\sum }_{i=1}^n{x}_i^2\right)}$	30	$[-10,10]$	0.9
F14	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)={\sum }_{i=1}^{n}i{x}_i^4+random[0,1)$	30	$[-1.28,1.28]$	$0+$ random noise
F15	$f(x,y)=10n+{\sum }_{i=1}^n\left(x_i^2-10cos\left(2\pi x_i\right)\right)$	30	$[-5.12,5.12]$	0
F16	$f\left(x_1\dots x_n\right)={\sum }_{i=1}^{n-1}\left(100{\left(x_i^2-x_{i+1}\right)}^2+{\left(1-x_i\right)}^2\right)$	30	$[-5,10]$	0
F17	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)=1-cos\left(2\pi \sqrt{{\sum }_{i=1}^D{x}_i^2}\right)+0.1\sqrt{{\sum }_{i=1}^D{x}_i^2}$	30	$[-100,100]$	0
F18	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)={\sum }_{i=1}^n{ϵ}_i{\left|x_i\right|}^i$	30	$[-5,5],\epsilon$ random	0
F19	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)=\left({\sum }_{i=1}^n\left|x_i\right|\right)exp\left(-{\sum }_{i=1}^{n}sin\left(x_i^2\right)\right)$	30	$[-2\pi ,2\pi ]$	0
F20	$f\left(\mathbf{x}\right)=f\left(x_1,\dots ,x_n\right)=\left({\sum }_{i=1}^n{sin}^2\left(x_i\right)-e^{-{\sum }_{i=1}^n{x}_i^2}\right)e^{-{\sum }_{i=1}^n{sin}^2\sqrt{x_i\mid }}$	30	$[-10,10]$	-1

, where $Dim$ indicates the dimension, $Interval$ is the boundary of the search space, and $f_{min}$ is the best-known optimal value. The TGCOA algorithm was compared against ten bio-inspired algorithms taken from the literature, which are described as follows:

• Black Widow Optimization Algorithm (BWOA): The algorithm, inspired by Latrodectus spider reproductive strategies, balances exploration and exploitation through natural phenomena like mate selection, pheromone rate, and cannibalism after mating [15].
• Jumping Spider Optimization Algorithm (JSOA): This algorithm resembles the natural behavior of Salticidade spiders, mathematically modeling their hunting strategies, such as searching, jumping, and pursuing [16].
• Liver Cancer Algorithm (LCA): It is based on tumor growth dynamics, consisting of three stages: tumor clonal expansion, vascular recruitment, and metastatic leap, using adaptive mutation, neighborhood sampling, and opposition-based learning [10].
• Evolutionary Mating Algorithm (EMA): The Hardy–Weinberg equilibrium, a fundamental concept in population genetics, is the basis for modeling genetic diversity and selection in natural populations [11].
• Bonobo Optimizer (BO): The algorithm, inspired by the social and reproductive strategies of bonobos, balances exploration and exploitation through behaviors such as fission–fusion grouping, diverse mating patterns, and selective mate interaction [23].
• Salp Swarm Algorithm (SSA): Based on the swarming dynamics of salps in marine ecosystems, this algorithm mimics their coordinated movement during navigation and foraging [24].
• Ant Lion Optimizer (ALO): The algorithm simulates the hunting strategy of antlions by modeling five key stages: random wandering, trap construction, prey capture, and subsequent trap rebuilding [25].
• Mexican Axolotl Optimization Algorithm (MAO): The algorithm mimics the life cycle and adaptive traits of axolotls; the algorithm emulates key processes such as regenerative healing, reproductive strategies, and tissue restoration [26].
• Particle Swarm Optimization (PSO): Inspired by the collective behavior of flocks of birds and schools of fish, optimizing through the collaborative movement of particles in the search space, balancing exploration and exploitation [13].
• Differential Evolution (DE): The algorithm optimizes solutions through mutation, crossover, and selection, combining weighted differences between vectors in the population and replacing the worst with the best in each generation [9].

Each algorithm’s testbench function was independently executed 30 times, with a population size and number of iterations of 30 and 200, respectively. Moreover, the Wilcoxon signed-rank test was used to compare statistically significant differences in performance, while the Friedman test was used to rank all algorithms. It is important to highlight that a non-parametric Wilcoxon signed-rank test was performed to compare two related algorithms, with a significance level of 5%. The null hypothesis assumes no significant difference between the paired algorithms, while the alternative hypothesis suggests a difference; a p-value less than 0.05 leads to rejection of the null hypothesis. Additionally, the Friedman test was applied to rank all algorithms within each test instance. Note that a lower value indicates better performance and a higher ranking position.

The four algorithms with the highest ranks were then selected for additional evaluation on the IEEE CEC 2017 “Constrained Real-Parameter Optimization” benchmark suite [94]. This set includes 28 constrained optimization functions and was tested for dimensions of 10, 30, 50, and 100. The description of the functions is provided in Appendix A in

Table A3. CEC 2017 Constrained Real-Parameter Optimization.

Problem/Search Range	Type of Objective	Number of Constraints
		E	I
C01	Non Separable	0	1
${[-100,100]}^D$			Separable
C02	Non Separable	0	1
${[-100,100]}^D$			Non Separable, Rotated
C03	Non Separable	1	1
${[-100,100]}^D$		Separable	Separable
C04	Separable	0	2
${[-10,10]}^D$			Separable
C05	Non Separable	0	2
${[-10,10]}^D$			Non Separable, Rotated
C06	Separable	6	0
${[-20,20]}^D$
C07	Separable	2	0
${[-50,50]}^D$
C08	Separable	2	0
${[-100,100]}^D$
C09	Separable	2	0
${[-10,10]}^D$
C10	Separable	2	0
${[-100,100]}^D$
C11	Separable	1	1
${[-100,100]}^D$		Non Separable	Non Separable
C12	Separable	0	2
${[-100,100]}^D$			Separable
C13	Non Separable	0	3
${[-100,100]}^D$			Separable
C14	Non Separable	1	1
${[-100,100]}^D$		Separable	Separable
C15	Separable	1	1
${[-100,100]}^D$
C16	Separable	1	1
${[-100,100]}^D$		Non Separable	Separable
C17	Non Separable	1	1
${[-100,100]}^D$		Non Separable	Separable
C18	Separable	1	2
${[-100,100]}^D$			Non Separable
C19	Separable	0	2
${[-50,50]}^D$			Non Separable
C20	Non Separable	0	2
${[-100,100]}^D$
C21	Rotated	0	2
${[-100,100]}^D$			Rotated
C22	Rotated	0	3
${[-100,100]}^D$			Rotated
C23	Rotated	1	1
${[-100,100]}^D$		Rotated	Rotated
C24	Rotated	1	1
${[-100,100]}^D$		Rotated	Rotated
C25	Rotated	1	1
${[-100,100]}^D$		Rotated	Rotated
C26	Rotated	1	1
${[-100,100]}^D$		Rotated	Rotated
C27	Rotated	1	2
${[-100,100]}^D$		Rotated	Rotated
C28	Rotated	0	2
${[-50,50]}^D$			Rotated

There are 28 test cases provided, with D indicating the decision factors, I indicating inequality constraints, and E indicating equality constraints.

. Additionally, seven constrained real-world design engineering problems were solved.

The parameter configurations corresponding to each optimization method are detailed in

Table 2. Initial settings for algorithm control parameters.

Algorithm	Parameters	Values
	Population size	30
All	Max. Iterations	200
	Executions	30
TGCOA	No additional parameters
BWOA	No additional parameters
EMA	Cr	[0, 1]
	r	[0, 0.2]
JSOA	No additional parameters
LCA	f	1
BO	Scab	1.5
	r	[0, 1]
	Scsb	1.6
SSA	v	0.0
ALO	$Iratio$	10
	w	2.0–0.6
MAO	$dp$	0.5
	$rp$	0.1
	k	3
	$\lambda$	0.5
PSO	w	0.8
	$C_1$ and $C_2$	0.5
DE	F	0.5
	$Cr$	0.9

. All experiments were computed on a desktop with the following specifications: Intel Core i9-13900K processor @ 5.8 GHz, 192 GB RAM, MATLAB R2024a compiler, and Linux Ubuntu 24.04 LTS operating system.

Average Convergence Rate Analysis

The Average Convergence Rate (ACR) is a quantitative metric used to evaluate the average rate at which an optimization algorithm approaches the optimal solution during its iterative process [95,96]. It measures convergence speed in a normalized manner, making it suitable for comparing different algorithms and configurations. The ACR for a single run is defined as

$$\mathrm{ACR}={\displaystyle \frac{1}{T-1}}\sum _{t=1}^{T-1}\left|{\displaystyle \frac{f_{t+1}-f_t}{f_0-f^{\ast }}}\right|,$$

(13)

where T is the total number of iterations. The numerator $f_{t+1}-f_t$ represents the improvement between consecutive iterations. The denominator $f_0-f^{\ast }$ normalizes the improvement relative to the total possible improvement in that run. $f_t$ denotes the best objective function value obtained at iteration t, $f_0$ the initial value, and $f^{\ast }$ the best value found during the entire optimization process. Moreover, for R independent runs, the average ACR is given by the following equation:

$$\overline{\mathrm{ACR}}={\displaystyle \frac{1}{R}}\sum _{r=1}^R{\mathrm{ACR}}_r,$$

(14)

where ${\mathrm{ACR}}_r$ denotes the convergence rate for run r. A rapid convergence toward the optimum (near optimum) occurs when $\mathrm{ACR}\approx 1$. Meanwhile, stagnation occurs for $\mathrm{ACR}\approx 0$. A complete interpretation of the ACR value is shown in

Table 3. Interpretation of the Average Convergence Rate (ACR) values.

ACR Value	Interpretation
$ACR\approx 1$	Optimal
$ACR\ge 0.9$	Excellent
$0.5\le$ ACR $<0.9$	Good
$0<$ ACR $<0.5$	Slow
$ACR\approx 0$	No Improvement

. This metric was used for each algorithm’s testbench function. It was independently executed 30 times, with a population size of 30 and a number of iterations of 200.

4. Results and Discussion

This section presents the computational outcomes of the TGCOA on testbench optimization problems.

Table 4. Comparison of optimization results obtained for 20 benchmark functions.

F	Fmin	TGCOA			BWOA			EMA
		Best	Ave	Std	Best	Ave	Std	Best	Ave	Std
F1	0	$9.21\times {10}^{-307}$	$1.83\times {10}^{-284}$	$0.00\times {10}^0$	$1.63\times {10}^{-264}$	$2.60\times {10}^{-203}$	$0.00\times {10}^0$	$8.47\times {10}^{-5}$	$1.35\times {10}^{-3}$	$1.54\times {10}^{-3}$
F2	0	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$2.69\times {10}^{-2}$	$5.00\times {10}^{-1}$	$3.15\times {10}^{-1}$
F3	0	$2.64\times {10}^{-153}$	$2.74\times {10}^{-139}$	$1.49\times {10}^{-138}$	$9.74\times {10}^{-127}$	$1.52\times {10}^{-97}$	$8.34\times {10}^{-97}$	$1.00\times {10}^{-2}$	$2.01\times {10}^{-1}$	$1.17\times {10}^{-1}$
F4	0	$2.03\times {10}^{-153}$	$4.37\times {10}^{-142}$	$1.44\times {10}^{-141}$	$1.50\times {10}^{-125}$	$2.22\times {10}^{-92}$	$1.21\times {10}^{-91}$	$2.27\times {10}^1$	$5.92\times {10}^1$	$1.87\times {10}^1$
F5	0	$2.83\times {10}^{-152}$	$9.78\times {10}^{-138}$	$5.35\times {10}^{-137}$	$1.96\times {10}^{-131}$	$1.29\times {10}^{-100}$	$6.96\times {10}^{-100}$	$9.46\times {10}^{-3}$	$2.23\times {10}^{-1}$	$1.49\times {10}^{-1}$
F6	0	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$1.29\times {10}^{-5}$	$6.21\times {10}^1$	$2.54\times {10}^2$
F7	0	$7.44\times {10}^{-304}$	$1.63\times {10}^{-280}$	$0.00\times {10}^0$	$4.59\times {10}^{-268}$	$2.67\times {10}^{-200}$	$0.00\times {10}^0$	$2.94\times {10}^{-4}$	$2.94\times {10}^{-3}$	$2.99\times {10}^{-3}$
F8	0	$2.66\times {10}^{-303}$	$1.62\times {10}^{-284}$	$0.00\times {10}^0$	$2.71\times {10}^{-254}$	$9.55\times {10}^{-201}$	$0.00\times {10}^0$	$1.26\times {10}^{-2}$	$1.46\times {10}^{-1}$	$1.44\times {10}^{-1}$
F9	−1	$-1.00\times {10}^0$	$-1.00\times {10}^0$	$0.00\times {10}^0$	$-1.00\times {10}^0$	$-1.00\times {10}^0$	$0.00\times {10}^0$	$9.95\times {10}^{-1}$	$9.95\times {10}^{-1}$	$3.06\times {10}^{-16}$
F10	0	$1.46\times {10}^{-295}$	$4.79\times {10}^{-282}$	$0.00\times {10}^0$	$3.04\times {10}^{-248}$	$1.21\times {10}^{-187}$	$0.00\times {10}^0$	$2.66\times {10}^2$	$4.48\times {10}^2$	$9.11\times {10}^1$
F11	0	$-4.44\times {10}^{-16}$	$-4.44\times {10}^{-16}$	$0.00\times {10}^0$	$-4.44\times {10}^{-16}$	$-4.44\times {10}^{-16}$	$0.00\times {10}^0$	$3.08\times {10}^{-2}$	$1.42\times {10}^{-1}$	$1.02\times {10}^{-1}$
F12	−4.59	$-4.59\times {10}^0$	$-3.21\times {10}^0$	$1.50\times {10}^0$	$-4.59\times {10}^0$	$-4.35\times {10}^0$	$4.90\times {10}^{-1}$	$-4.59\times {10}^0$	$-4.50\times {10}^0$	$2.69\times {10}^{-1}$
F13	0.9	$9.00\times {10}^{-1}$	$9.00\times {10}^{-1}$	$4.52\times {10}^{-16}$	$9.00\times {10}^{-1}$	$9.00\times {10}^{-1}$	$4.52\times {10}^{-16}$	$2.16\times {10}^0$	$3.13\times {10}^0$	$5.27\times {10}^{-1}$
F14	0	$7.40\times {10}^{-6}$	$2.79\times {10}^{-4}$	$2.46\times {10}^{-4}$	$2.76\times {10}^{-5}$	$4.31\times {10}^{-4}$	$2.94\times {10}^{-4}$	$5.84\times {10}^{-2}$	$1.52\times {10}^{-1}$	$7.39\times {10}^{-2}$
F15	0	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$1.69\times {10}^{-1}$	$1.75\times {10}^1$	$1.50\times {10}^1$
F16	0	$2.89\times {10}^1$	$2.89\times {10}^1$	$3.02\times {10}^{-2}$	$2.89\times {10}^1$	$2.90\times {10}^1$	$2.46\times {10}^{-2}$	$2.86\times {10}^1$	$3.26\times {10}^1$	$4.20\times {10}^0$
F17	0	$5.44\times {10}^{-151}$	$7.17\times {10}^{-139}$	$3.90\times {10}^{-138}$	$8.97\times {10}^{-121}$	$1.46\times {10}^{-84}$	$7.99\times {10}^{-84}$	$6.03\times {10}^{-1}$	$1.08\times {10}^0$	$2.56\times {10}^{-1}$
F18	0	$1.92\times {10}^{-139}$	$7.03\times {10}^{-122}$	$3.85\times {10}^{-121}$	$4.95\times {10}^{-93}$	$4.41\times {10}^{-24}$	$2.08\times {10}^{-23}$	$2.77\times {10}^{-5}$	$3.16\times {10}^{-2}$	$6.08\times {10}^{-2}$
F19	0	$1.09\times {10}^{-7}$	$3.23\times {10}^{-5}$	$9.02\times {10}^{-5}$	$3.51\times {10}^{-12}$	$1.69\times {10}^{-7}$	$6.64\times {10}^{-7}$	$1.97\times {10}^{-11}$	$2.48\times {10}^{-11}$	$1.88\times {10}^{-12}$
F20	−1	$-1.00\times {10}^0$	$-1.00\times {10}^0$	$0.00\times {10}^0$	$-1.00\times {10}^0$	$-1.00\times {10}^0$	$0.00\times {10}^0$	$4.37\times {10}^{-13}$	$9.66\times {10}^{-13}$	$4.60\times {10}^{-13}$

,

Table 5. Comparison of optimization results obtained for 20 benchmark functions.

F	Fmin	JSOA			LCA			BO
		Best	Ave	Std	Best	Ave	Std	Best	Ave	Std
F1	0	$2.32\times {10}^{-76}$	$4.12\times {10}^{-67}$	$1.17\times {10}^{-66}$	$3.04\times {10}^{-5}$	$8.32\times {10}^{-4}$	$1.06\times {10}^{-3}$	$1.35\times {10}^0$	$1.19\times {10}^1$	$1.39\times {10}^1$
F2	0	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$8.52\times {10}^{-4}$	$3.00\times {10}^{-1}$	$2.44\times {10}^{-1}$	$1.72\times {10}^0$	$8.93\times {10}^0$	$7.28\times {10}^0$
F3	0	$0.00\times {10}^0$	$1.91\times {10}^{-39}$	$2.98\times {10}^{-39}$	$1.04\times {10}^{-1}$	$2.54\times {10}^0$	$1.65\times {10}^0$	$3.98\times {10}^1$	$1.10\times {10}^2$	$4.33\times {10}^1$
F4	0	$3.87\times {10}^{-44}$	$1.48\times {10}^{-40}$	$3.86\times {10}^{-40}$	$4.36\times {10}^{-3}$	$9.29\times {10}^{-2}$	$6.19\times {10}^{-2}$	$2.31\times {10}^1$	$3.29\times {10}^1$	$5.18\times {10}^0$
F5	0	$0.00\times {10}^0$	$7.29\times {10}^{-39}$	$2.12\times {10}^{-38}$	$4.54\times {10}^{-1}$	$2.43\times {10}^0$	$1.63\times {10}^0$	$1.25\times {10}^2$	$2.52\times {10}^2$	$7.95\times {10}^1$
F6	0	$0.00\times {10}^0$	$1.12\times {10}^{-300}$	$0.00\times {10}^0$	$5.07\times {10}^{-27}$	$6.55\times {10}^{-17}$	$2.26\times {10}^{-16}$	$4.24\times {10}^{-5}$	$3.01\times {10}^3$	$8.70\times {10}^3$
F7	0	$1.76\times {10}^{-75}$	$1.07\times {10}^{-66}$	$5.65\times {10}^{-66}$	$5.52\times {10}^{-6}$	$8.11\times {10}^{-4}$	$1.16\times {10}^{-3}$	$2.56\times {10}^{-1}$	$2.82\times {10}^0$	$2.37\times {10}^0$
F8	0	$4.03\times {10}^{-74}$	$1.45\times {10}^{-64}$	$7.47\times {10}^{-64}$	$1.53\times {10}^{-3}$	$4.31\times {10}^{-2}$	$9.57\times {10}^{-2}$	$9.73\times {10}^0$	$1.30\times {10}^2$	$1.43\times {10}^2$
F9	−1	$-1.00\times {10}^0$	$-1.00\times {10}^0$	$0.00\times {10}^0$	$-1.00\times {10}^0$	$-9.97\times {10}^{-1}$	$3.72\times {10}^{-3}$	$9.95\times {10}^{-1}$	$9.95\times {10}^{-1}$	$3.39\times {10}^{-16}$
F10	0	$6.33\times {10}^{-71}$	$3.04\times {10}^{-64}$	$1.27\times {10}^{-63}$	$1.97\times {10}^{-2}$	$1.08\times {10}^1$	$1.66\times {10}^1$	$3.86\times {10}^1$	$1.90\times {10}^2$	$1.08\times {10}^2$
F11	0	$-4.44\times {10}^{-16}$	$-4.44\times {10}^{-16}$	$0.00\times {10}^0$	$1.49\times {10}^{-2}$	$1.45\times {10}^{-1}$	$1.39\times {10}^{-1}$	$9.67\times {10}^0$	$1.32\times {10}^1$	$1.45\times {10}^0$
F12	−4.59	$-4.59\times {10}^0$	$-4.53\times {10}^0$	$2.23\times {10}^{-1}$	$-4.56\times {10}^0$	$-3.46\times {10}^0$	$8.73\times {10}^{-1}$	$-4.59\times {10}^0$	$-4.34\times {10}^0$	$5.69\times {10}^{-1}$
F13	0.9	$9.00\times {10}^{-1}$	$9.00\times {10}^{-1}$	$4.52\times {10}^{-16}$	$9.00\times {10}^{-1}$	$3.46\times {10}^0$	$4.27\times {10}^0$	$1.13\times {10}^0$	$2.33\times {10}^0$	$9.63\times {10}^{-1}$
F14	0	$8.24\times {10}^{-5}$	$4.23\times {10}^{-4}$	$3.65\times {10}^{-4}$	$2.63\times {10}^{-5}$	$1.65\times {10}^{-3}$	$1.51\times {10}^{-3}$	$4.25\times {10}^{-1}$	$1.22\times {10}^0$	$9.38\times {10}^{-1}$
F15	0	$0.00\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$4.96\times {10}^{-4}$	$7.52\times {10}^0$	$4.01\times {10}^1$	$3.73\times {10}^1$	$7.95\times {10}^1$	$1.76\times {10}^1$
F16	0	$2.15\times {10}^{-17}$	$2.12\times {10}^{-1}$	$8.62\times {10}^{-1}$	$4.42\times {10}^{-4}$	$1.38\times {10}^{-1}$	$1.68\times {10}^{-1}$	$1.65\times {10}^2$	$5.83\times {10}^2$	$4.37\times {10}^2$
F17	0	$0.00\times {10}^0$	$6.21\times {10}^{-38}$	$1.99\times {10}^{-37}$	$2.72\times {10}^{-2}$	$2.26\times {10}^{-1}$	$1.12\times {10}^{-1}$	$4.60\times {10}^0$	$7.82\times {10}^0$	$1.68\times {10}^0$
F18	0	$6.14\times {10}^{-34}$	$9.08\times {10}^{-28}$	$4.58\times {10}^{-27}$	$1.78\times {10}^{-6}$	$1.61\times {10}^{-4}$	$2.69\times {10}^{-4}$	$2.22\times {10}^1$	$3.87\times {10}^6$	$1.50\times {10}^7$
F19	0	$0.00\times {10}^0$	$2.34\times {10}^{-13}$	$8.91\times {10}^{-13}$	$3.51\times {10}^{-12}$	$3.53\times {10}^{-12}$	$4.88\times {10}^{-14}$	$1.89\times {10}^{-11}$	$2.37\times {10}^{-11}$	$2.04\times {10}^{-12}$
F20	−1	$-1.00\times {10}^0$	$-1.00\times {10}^0$	$0.00\times {10}^0$	$-9.70\times {10}^{-1}$	$-8.30\times {10}^{-1}$	$1.15\times {10}^{-1}$	$1.03\times {10}^{-13}$	$8.28\times {10}^{-13}$	$5.73\times {10}^{-13}$

,

Table 6. Comparison of optimization results obtained for 20 benchmark functions.

F	Fmin	SSA			ALO			MAO
		Best	Ave	Std	Best	Ave	Std	Best	Ave	Std
F1	0	$3.82\times {10}^{-2}$	$2.85\times {10}^0$	$4.62\times {10}^0$	$1.21\times {10}^{-5}$	$2.72\times {10}^{-2}$	$5.38\times {10}^{-2}$	$9.33\times {10}^1$	$2.68\times {10}^2$	$1.47\times {10}^2$
F2	0	$1.08\times {10}^0$	$1.45\times {10}^0$	$4.03\times {10}^{-1}$	$1.17\times {10}^0$	$9.93\times {10}^0$	$6.33\times {10}^0$	$3.40\times {10}^1$	$6.16\times {10}^1$	$1.45\times {10}^1$
F3	0	$2.75\times {10}^1$	$8.03\times {10}^1$	$2.92\times {10}^1$	$1.73\times {10}^2$	$2.19\times {10}^2$	$3.65\times {10}^1$	$2.78\times {10}^2$	$3.57\times {10}^2$	$4.38\times {10}^1$
F4	0	$1.00\times {10}^1$	$1.70\times {10}^1$	$4.35\times {10}^0$	$1.47\times {10}^1$	$2.30\times {10}^1$	$5.59\times {10}^0$	$2.59\times {10}^1$	$3.59\times {10}^1$	$4.45\times {10}^0$
F5	0	$4.20\times {10}^3$	$1.17\times {10}^{19}$	$4.18\times {10}^{19}$	$4.80\times {10}^2$	$3.42\times {10}^{24}$	$1.54\times {10}^{25}$	$4.35\times {10}^{20}$	$8.75\times {10}^{25}$	$2.39\times {10}^{26}$
F6	0	$1.01\times {10}^{-3}$	$1.67\times {10}^1$	$2.80\times {10}^1$	$8.06\times {10}^{-1}$	$3.01\times {10}^3$	$1.12\times {10}^4$	$1.26\times {10}^5$	$7.67\times {10}^5$	$6.73\times {10}^5$
F7	0	$2.29\times {10}^{-2}$	$1.32\times {10}^{-1}$	$8.38\times {10}^{-2}$	$1.06\times {10}^{-1}$	$2.52\times {10}^0$	$1.76\times {10}^0$	$1.27\times {10}^1$	$1.86\times {10}^1$	$3.71\times {10}^0$
F8	0	$7.19\times {10}^0$	$3.18\times {10}^1$	$2.17\times {10}^1$	$1.34\times {10}^1$	$1.11\times {10}^2$	$5.17\times {10}^1$	$5.66\times {10}^2$	$9.44\times {10}^2$	$2.35\times {10}^2$
F9	−1	$9.96\times {10}^{-1}$	$9.96\times {10}^{-1}$	$1.61\times {10}^{-4}$	$-1.00\times {10}^0$	$-9.83\times {10}^{-1}$	$2.85\times {10}^{-2}$	$9.99\times {10}^{-1}$	$9.99\times {10}^{-1}$	$1.01\times {10}^{-4}$
F10	0	$1.34\times {10}^2$	$2.39\times {10}^2$	$7.25\times {10}^1$	$1.74\times {10}^2$	$3.85\times {10}^2$	$9.94\times {10}^1$	$1.36\times {10}^2$	$7.07\times {10}^5$	$2.25\times {10}^6$
F11	0	$2.90\times {10}^0$	$5.38\times {10}^0$	$1.41\times {10}^0$	$7.97\times {10}^0$	$1.24\times {10}^1$	$2.09\times {10}^0$	$1.29\times {10}^1$	$1.44\times {10}^1$	$8.01\times {10}^{-1}$
F12	−4.59	$-4.59\times {10}^0$	$-4.59\times {10}^0$	$7.94\times {10}^{-12}$	$-4.59\times {10}^0$	$-4.59\times {10}^0$	$2.61\times {10}^{-11}$	$-4.59\times {10}^0$	$-3.96\times {10}^0$	$5.87\times {10}^{-1}$
F13	0.9	$1.01\times {10}^0$	$1.16\times {10}^0$	$1.51\times {10}^{-1}$	$9.40\times {10}^{-1}$	$1.07\times {10}^0$	$5.54\times {10}^{-2}$	$6.83\times {10}^0$	$8.54\times {10}^0$	$8.00\times {10}^{-1}$
F14	0	$1.38\times {10}^{-1}$	$3.52\times {10}^{-1}$	$1.58\times {10}^{-1}$	$4.80\times {10}^{-1}$	$1.04\times {10}^0$	$5.72\times {10}^{-1}$	$1.00\times {10}^0$	$2.31\times {10}^0$	$7.98\times {10}^{-1}$
F15	0	$2.43\times {10}^1$	$5.58\times {10}^1$	$2.03\times {10}^1$	$3.81\times {10}^1$	$8.90\times {10}^1$	$2.55\times {10}^1$	$2.00\times {10}^2$	$2.38\times {10}^2$	$1.49\times {10}^1$
F16	0	$4.63\times {10}^1$	$1.18\times {10}^2$	$4.82\times {10}^1$	$8.56\times {10}^1$	$3.69\times {10}^2$	$2.39\times {10}^2$	$2.18\times {10}^3$	$3.90\times {10}^3$	$1.23\times {10}^3$
F17	0	$2.90\times {10}^0$	$4.59\times {10}^0$	$7.56\times {10}^{-1}$	$5.30\times {10}^0$	$7.89\times {10}^0$	$1.49\times {10}^0$	$6.91\times {10}^0$	$8.86\times {10}^0$	$9.47\times {10}^{-1}$
F18	0	$8.17\times {10}^{-2}$	$8.98\times {10}^1$	$2.71\times {10}^2$	$6.14\times {10}^3$	$3.37\times {10}^9$	$7.40\times {10}^9$	$2.75\times {10}^1$	$2.25\times {10}^5$	$1.09\times {10}^6$
F19	0	$1.70\times {10}^{-11}$	$1.40\times {10}^{-10}$	$1.87\times {10}^{-10}$	$0.00\times {10}^0$	$3.27\times {10}^{-12}$	$6.66\times {10}^{-12}$	$2.06\times {10}^{-7}$	$3.66\times {10}^{-6}$	$4.38\times {10}^{-6}$
F20	−1	$8.25\times {10}^{-14}$	$4.88\times {10}^{-13}$	$2.67\times {10}^{-13}$	$1.02\times {10}^{-14}$	$6.56\times {10}^{-13}$	$2.72\times {10}^{-12}$	$1.65\times {10}^{-11}$	$8.97\times {10}^{-11}$	$7.30\times {10}^{-11}$

and

Table 7. Comparison of optimization results obtained for 20 benchmark functions.

F	Fmin	PSO			DE
		Best	Ave	Std	Best	Ave	Std
F1	0	$9.89\times {10}^{-7}$	$6.52\times {10}^{-5}$	$1.21\times {10}^{-4}$	$2.44\times {10}^2$	$2.52\times {10}^5$	$1.05\times {10}^6$
F2	0	$8.62\times {10}^{-4}$	$5.64\times {10}^{-2}$	$7.70\times {10}^{-2}$	$4.25\times {10}^2$	$5.50\times {10}^2$	$5.92\times {10}^1$
F3	0	$1.54\times {10}^{-1}$	$1.68\times {10}^0$	$2.26\times {10}^0$	$7.63\times {10}^2$	$1.07\times {10}^3$	$8.55\times {10}^1$
F4	0	$2.53\times {10}^0$	$4.77\times {10}^0$	$1.25\times {10}^0$	$7.93\times {10}^1$	$8.74\times {10}^1$	$3.36\times {10}^0$
F5	0	$8.48\times {10}^{-1}$	$1.16\times {10}^2$	$2.03\times {10}^2$	$6.87\times {10}^{36}$	$1.59\times {10}^{40}$	$3.98\times {10}^{40}$
F6	0	$8.70\times {10}^{-15}$	$1.11\times {10}^{-9}$	$3.84\times {10}^{-9}$	$2.62\times {10}^9$	$7.68\times {10}^9$	$3.15\times {10}^9$
F7	0	$5.29\times {10}^{-6}$	$1.01\times {10}^{-4}$	$1.50\times {10}^{-4}$	$1.22\times {10}^2$	$1.63\times {10}^2$	$2.04\times {10}^1$
F8	0	$6.07\times {10}^{-5}$	$7.24\times {10}^{-3}$	$1.88\times {10}^{-2}$	$5.14\times {10}^3$	$8.14\times {10}^3$	$1.42\times {10}^3$
F9	−1	$9.96\times {10}^{-1}$	$9.96\times {10}^{-1}$	$3.61\times {10}^{-4}$	$9.97\times {10}^{-1}$	$9.98\times {10}^{-1}$	$2.20\times {10}^{-4}$
F10	0	$8.64\times {10}^0$	$3.50\times {10}^1$	$2.01\times {10}^1$	$3.66\times {10}^2$	$2.02\times {10}^7$	$9.52\times {10}^7$
F11	0	$5.66\times {10}^{-3}$	$1.45\times {10}^0$	$6.73\times {10}^{-1}$	$1.99\times {10}^1$	$2.04\times {10}^1$	$1.77\times {10}^{-1}$
F12	−4.59	$-4.59\times {10}^0$	$-4.59\times {10}^0$	$1.81\times {10}^{-15}$	$-4.02\times {10}^0$	$-1.25\times {10}^0$	$1.74\times {10}^0$
F13	0.9	$1.00\times {10}^0$	$1.00\times {10}^0$	$3.73\times {10}^{-4}$	$7.33\times {10}^0$	$9.32\times {10}^0$	$7.49\times {10}^{-1}$
F14	0	$1.51\times {10}^{-2}$	$4.74\times {10}^{-2}$	$1.62\times {10}^{-2}$	$6.24\times {10}^1$	$1.05\times {10}^2$	$1.77\times {10}^1$
F15	0	$2.69\times {10}^1$	$5.01\times {10}^1$	$1.11\times {10}^1$	$3.10\times {10}^2$	$4.09\times {10}^2$	$2.94\times {10}^1$
F16	0	$7.95\times {10}^0$	$3.31\times {10}^1$	$1.90\times {10}^1$	$1.48\times {10}^5$	$1.96\times {10}^5$	$2.83\times {10}^4$
F17	0	$7.00\times {10}^{-1}$	$1.08\times {10}^0$	$2.26\times {10}^{-1}$	$2.16\times {10}^1$	$2.56\times {10}^1$	$1.34\times {10}^0$
F18	0	$7.21\times {10}^{-6}$	$4.43\times {10}^{-1}$	$1.26\times {10}^0$	$1.16\times {10}^8$	$4.85\times {10}^{12}$	$1.15\times {10}^{13}$
F19	0	$3.66\times {10}^{-12}$	$5.06\times {10}^{-12}$	$9.36\times {10}^{-13}$	$2.74\times {10}^{-7}$	$2.33\times {10}^{-6}$	$2.06\times {10}^{-6}$
F20	−1	$6.50\times {10}^{-18}$	$9.84\times {10}^{-17}$	$1.27\times {10}^{-16}$	$4.01\times {10}^{-9}$	$4.15\times {10}^{-8}$	$2.91\times {10}^{-8}$

present the comparative findings, including the optimal, average, and standard deviation values. Furthermore, Figure 9 and Figure 10 summarize the convergence graphs of all the functions compared to the other selected methods for this study. To evaluate the significant differences observed between the TGCOA and the other methods, a non-parametric statistical test, the Wilcoxon signed-rank test, was performed, establishing a significance level of 5%. The null hypothesis indicates that there is no significant difference between the two pairs of algorithms compared. Meanwhile, the alternative hypothesis suggests that there is. It is worth noting that, with a p-value less than 0.05, the null hypothesis is rejected. The results of the Wilcoxon rank-sum test, in

Table 8. Statistical results of the Wilcoxon signed-rank test for the TGCOA versus other algorithms (20 functions, $\alpha =5\%$).

TGCOA–BWOA		TGCOA–EMA		TGCOA–JSOA		TGCOA–LCA
$(+/=/-)$	p-value	$(+/=/-)$	p-value	$(+/=/-)$	p-value	$(+/=/-)$	p-value
2/7/11	$1.16\times {10}^{-1}$	2/0/18	$\mathbf{1}\times {\mathbf{10}}^{-\mathbf{3}}$	3/6/11	$3.63\times {10}^{-1}$	3/0/17	$\mathbf{9}\times {\mathbf{10}}^{-\mathbf{3}}$
TGCOA–BO		TGCOA–SSA		TGCOA–ALO		TGCOA–MAO
$(+/=/-)$	p-value	$(+/=/-)$	p-value	$(+/=/-)$	p-value	$(+/=/-)$	p-value
2/0/18	$\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}$	2/0/18	$\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}$	2/0/18	$\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}$	2/0/18	$\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}$
TGCOA–PSO				TGCOA–DE
$(+/=/-)$		p-value		$(+/=/-)$		p-value
2/0/18		$\mathbf{1}\times {\mathbf{10}}^{-\mathbf{3}}$		1/0/18		$\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}$

Significant results (p < 0.05) are highlighted in bold.

, demonstrate that the TGCOA outperformed the following algorithms: the Evolutionary Mating Algorithm (EMA) had a p-value of $0.001$, Liver Cancer Algorithm (LCA) $0.009$, Bonobo Optimizer (BO) 0, Salp Swarm Algorithm (SSA) 0, Ant Lion Optimizer (ALO) 0, Mexican Axolotl Optimization Algorithm (MAO) 0, Particle Swarm Optimization (PSO) 0, and Differential Evolution (DE) 0. Meanwhile, no statistically significant differences were found when compared to the Black Widow Optimization Algorithm (BWOA) and Jumping Spider Optimization Algorithm (JSOA), with p-values of $0.116$ and $0.363$, respectively. Moreover, the eleven algorithms were ranked by calculating the Friedman test for 20 testbench functions. Note that a lower value indicates better performance and a higher ranking position. The results of the algorithm rankings are shown in

Table 9. All comparison algorithms of the Friedman test for function 20.

Algorithms	Mean Rank	Overall Rank
TGCOA	2.45	1
JSOA	2.47	2
BWOA	2.71	3
LCA	4.68	4
PSO	4.95	5
EMA	6.26	6
SSA	6.66	7
ALO	7.24	8
BO	7.95	9
MAO	9.79	10
DE	10.84	11

The bold numbers in the table indicate the order of the top-ranked algorithms by the Friedman test.

; observe that the TGCOA is ranked first, with a mean rank 2.45, whereas the JSOA, BWOA, and LCA are ranked second, third, and fourth, with mean ranks of 2.47, 2.71, and 4.68, respectively. Based on these results, for the remainder of the paper, these four best-ranked algorithms are only used in IEEE CEC 2017 ”Constrained Real Parameter Optimization” and real-world applications.

The results derived from the computational study of the TGCOA on benchmark functions from the IEEE CEC 2017 “Constrained Real-Parameter Optimization” problems for dimensions 10, 30, 50, and 100 are presented in

Table 10. Results of dimension 10 for the IEEE CEC 2017 comparative tests for “Constrained Real-Parameter Optimization”.

Function	TGCOA			BWOA			JSOA			LCA
	Best	Ave	Std	Best	Ave	Std	Best	Ave	Std	Best	Ave	Std
C01	$1.159\times {10}^3$	$4.820\times {10}^3$	$2.466\times {10}^3$	$4.577\times {10}^2$	$3.599\times {10}^3$	$2.455\times {10}^3$	$1.204\times {10}^3$	$4.605\times {10}^3$	$2.230\times {10}^3$	$1.627\times {10}^3$	$1.110\times {10}^4$	$4.297\times {10}^3$
C02	$1.337\times {10}^3$	$9.914\times {10}^3$	$1.011\times {10}^4$	$1.989\times {10}^2$	$4.048\times {10}^3$	$3.257\times {10}^3$	$7.945\times {10}^2$	$5.040\times {10}^3$	$4.177\times {10}^3$	$2.887\times {10}^3$	$9.170\times {10}^3$	$4.675\times {10}^3$
C03	$4.848\times {10}^3$	$2.797\times {10}^4$	$1.683\times {10}^4$	$2.106\times {10}^3$	$8.105\times {10}^3$	$4.669\times {10}^3$	$1.997\times {10}^3$	$6.525\times {10}^3$	$3.529\times {10}^3$	$2.680\times {10}^3$	$9.408\times {10}^3$	$8.435\times {10}^3$
C04	$9.973\times {10}^1$	$1.343\times {10}^2$	$1.767\times {10}^1$	$6.369\times {10}^1$	$1.188\times {10}^2$	$2.670\times {10}^1$	$8.276\times {10}^1$	$1.138\times {10}^2$	$1.639\times {10}^1$	$1.083\times {10}^2$	$1.557\times {10}^2$	$2.535\times {10}^1$
C05	$9.878\times {10}^3$	$8.907\times {10}^7$	$1.684\times {10}^8$	$7.457\times {10}^2$	$3.440\times {10}^6$	$1.470\times {10}^7$	$1.598\times {10}^3$	$8.898\times {10}^6$	$3.327\times {10}^7$	$4.769\times {10}^3$	$1.480\times {10}^8$	$2.020\times {10}^8$
C06	$1.822\times {10}^3$	$7.581\times {10}^4$	$6.919\times {10}^4$	$6.477\times {10}^2$	$2.242\times {10}^4$	$2.362\times {10}^4$	$1.954\times {10}^3$	$6.171\times {10}^4$	$7.203\times {10}^4$	$8.682\times {10}^4$	$2.316\times {10}^5$	$9.964\times {10}^4$
C07	$-7.919\times {10}^1$	$3.859\times {10}^7$	$5.491\times {10}^7$	$1.009\times {10}^0$	$3.847\times {10}^7$	$5.207\times {10}^7$	$-1.340\times {10}^2$	$1.194\times {10}^8$	$1.065\times {10}^8$	$9.730\times {10}^4$	$1.060\times {10}^8$	$1.160\times {10}^8$
C08	$5.047\times {10}^5$	$2.928\times {10}^7$	$2.736\times {10}^7$	$3.470\times {10}^6$	$3.729\times {10}^7$	$3.123\times {10}^7$	$7.608\times {10}^6$	$2.209\times {10}^7$	$1.419\times {10}^7$	$6.290\times {10}^7$	$1.670\times {10}^8$	$5.691\times {10}^7$
C09	$1.370\times {10}^1$	$1.523\times {10}^5$	$4.276\times {10}^5$	$1.075\times {10}^1$	$2.235\times {10}^6$	$6.987\times {10}^6$	$8.438\times {10}^{-1}$	$5.110\times {10}^5$	$1.313\times {10}^6$	$7.649\times {10}^5$	$5.660\times {10}^8$	$1.080\times {10}^9$
C10	$1.099\times {10}^{10}$	$1.718\times {10}^{11}$	$1.644\times {10}^{11}$	$4.695\times {10}^8$	$1.119\times {10}^{11}$	$1.088\times {10}^{11}$	$7.511\times {10}^9$	$1.445\times {10}^{11}$	$1.052\times {10}^{11}$	$3.290\times {10}^{10}$	$2.890\times {10}^{11}$	$1.270\times {10}^{11}$
C11	$1.992\times {10}^8$	$9.143\times {10}^9$	$6.400\times {10}^9$	$7.978\times {10}^5$	$4.151\times {10}^7$	$3.250\times {10}^7$	$4.223\times {10}^6$	$7.712\times {10}^7$	$4.340\times {10}^7$	$3.729\times {10}^7$	$1.600\times {10}^8$	$4.307\times {10}^7$
C12	$2.659\times {10}^8$	$8.511\times {10}^9$	$7.557\times {10}^9$	$1.630\times {10}^8$	$3.592\times {10}^9$	$3.684\times {10}^9$	$5.480\times {10}^8$	$5.506\times {10}^9$	$5.029\times {10}^9$	$4.040\times {10}^9$	$1.630\times {10}^{10}$	$8.130\times {10}^9$
C13	$2.473\times {10}^8$	$6.560\times {10}^9$	$6.567\times {10}^9$	$4.053\times {10}^8$	$4.956\times {10}^9$	$5.027\times {10}^9$	$5.196\times {10}^8$	$8.101\times {10}^9$	$5.750\times {10}^9$	$4.040\times {10}^9$	$1.750\times {10}^{10}$	$7.390\times {10}^9$
C14	$2.114\times {10}^9$	$1.305\times {10}^{10}$	$8.615\times {10}^9$	$9.555\times {10}^7$	$7.977\times {10}^9$	$6.572\times {10}^9$	$7.728\times {10}^8$	$1.052\times {10}^{10}$	$1.025\times {10}^{10}$	$1.050\times {10}^{10}$	$3.030\times {10}^{10}$	$1.330\times {10}^{10}$
C15	$1.492\times {10}^1$	$2.346\times {10}^9$	$2.929\times {10}^9$	$1.492\times {10}^1$	$2.114\times {10}^9$	$2.311\times {10}^9$	$1.492\times {10}^1$	$2.407\times {10}^9$	$2.516\times {10}^9$	$1.540\times {10}^8$	$8.730\times {10}^9$	$5.480\times {10}^9$
C16	$7.540\times {10}^1$	$3.843\times {10}^9$	$4.709\times {10}^9$	$6.440\times {10}^1$	$1.867\times {10}^9$	$3.092\times {10}^9$	$1.096\times {10}^6$	$3.152\times {10}^9$	$3.132\times {10}^9$	$1.830\times {10}^8$	$9.290\times {10}^9$	$5.480\times {10}^9$
C17	$7.480\times {10}^9$	$2.965\times {10}^{10}$	$1.356\times {10}^{10}$	$3.819\times {10}^{10}$	$3.819\times {10}^{10}$	$2.328\times {10}^{-5}$	$2.475\times {10}^{10}$	$3.759\times {10}^{10}$	$2.568\times {10}^9$	$3.820\times {10}^{10}$	$3.820\times {10}^{10}$	$2.330\times {10}^{-5}$
C18	$5.922\times {10}^{14}$	$1.927\times {10}^{19}$	$3.736\times {10}^{19}$	$2.348\times {10}^{15}$	$1.123\times {10}^{19}$	$2.505\times {10}^{19}$	$7.607\times {10}^{15}$	$2.605\times {10}^{19}$	$3.467\times {10}^{19}$	$5.480\times {10}^{18}$	$1.250\times {10}^{20}$	$8.080\times {10}^{19}$
C19	$1.777\times {10}^{11}$	$1.780\times {10}^{11}$	$1.257\times {10}^8$	$1.774\times {10}^{11}$	$1.778\times {10}^{11}$	$1.994\times {10}^8$	$1.774\times {10}^{11}$	$1.779\times {10}^{11}$	$1.769\times {10}^8$	$1.780\times {10}^{11}$	$1.780\times {10}^{11}$	$8.186\times {10}^7$
C20	$1.216\times {10}^0$	$2.155\times {10}^0$	$4.589\times {10}^{-1}$	$1.169\times {10}^0$	$1.809\times {10}^0$	$3.217\times {10}^{-1}$	$9.451\times {10}^{-1}$	$1.777\times {10}^0$	$4.665\times {10}^{-1}$	$1.610\times {10}^0$	$2.390\times {10}^0$	$3.547\times {10}^{-1}$
C21	$2.237\times {10}^9$	$5.457\times {10}^{10}$	$4.812\times {10}^{10}$	$7.449\times {10}^8$	$4.222\times {10}^{10}$	$3.957\times {10}^{10}$	$2.731\times {10}^9$	$5.878\times {10}^{10}$	$3.518\times {10}^{10}$	$3.010\times {10}^{10}$	$1.660\times {10}^{11}$	$7.570\times {10}^{10}$
C22	$1.943\times {10}^9$	$5.099\times {10}^{10}$	$4.007\times {10}^{10}$	$5.541\times {10}^9$	$4.689\times {10}^{10}$	$5.138\times {10}^{10}$	$3.046\times {10}^9$	$6.481\times {10}^{10}$	$5.216\times {10}^{10}$	$1.870\times {10}^{10}$	$1.730\times {10}^{11}$	$7.670\times {10}^{10}$
C23	$1.060\times {10}^{10}$	$9.076\times {10}^{10}$	$6.644\times {10}^{10}$	$3.871\times {10}^8$	$7.301\times {10}^{10}$	$7.039\times {10}^{10}$	$3.735\times {10}^9$	$1.035\times {10}^{11}$	$9.417\times {10}^{10}$	$7.070\times {10}^{10}$	$2.620\times {10}^{11}$	$1.020\times {10}^{11}$
C24	$3.146\times {10}^9$	$4.588\times {10}^{10}$	$4.014\times {10}^{10}$	$2.692\times {10}^8$	$3.420\times {10}^{10}$	$3.189\times {10}^{10}$	$2.121\times {10}^1$	$4.723\times {10}^{10}$	$3.876\times {10}^{10}$	$2.900\times {10}^{10}$	$1.430\times {10}^{11}$	$6.820\times {10}^{10}$
C25	$7.069\times {10}^1$	$4.246\times {10}^{10}$	$3.952\times {10}^{10}$	$1.922\times {10}^8$	$4.211\times {10}^{10}$	$4.298\times {10}^{10}$	$1.105\times {10}^8$	$3.570\times {10}^{10}$	$4.226\times {10}^{10}$	$1.450\times {10}^{10}$	$1.260\times {10}^{11}$	$5.760\times {10}^{10}$
C26	$2.561\times {10}^9$	$6.022\times {10}^{10}$	$5.261\times {10}^{10}$	$2.412\times {10}^9$	$3.876\times {10}^{10}$	$3.465\times {10}^{10}$	$5.496\times {10}^9$	$6.228\times {10}^{10}$	$5.386\times {10}^{10}$	$4.370\times {10}^{10}$	$1.500\times {10}^{11}$	$7.110\times {10}^{10}$
C27	$6.720\times {10}^{17}$	$2.659\times {10}^{21}$	$5.812\times {10}^{21}$	$6.118\times {10}^{17}$	$9.469\times {10}^{20}$	$1.256\times {10}^{21}$	$4.273\times {10}^{17}$	$2.631\times {10}^{21}$	$4.285\times {10}^{21}$	$6.920\times {10}^{19}$	$1.600\times {10}^{22}$	$1.480\times {10}^{22}$
C28	$1.776\times {10}^{11}$	$1.780\times {10}^{11}$	$1.637\times {10}^8$	$1.774\times {10}^{11}$	$1.779\times {10}^{11}$	$2.203\times {10}^8$	$1.775\times {10}^{11}$	$1.779\times {10}^{11}$	$1.867\times {10}^8$	$1.780\times {10}^{11}$	$1.780\times {10}^{11}$	$8.969\times {10}^7$

,

Table 11. Results for dimension 30 of the IEEE CEC 2017 comparative tests for “Constrained Real-Parameter Optimization”.

Function	TGCOA			BWOA			JSOA			LCA
	Best	Ave	Std	Best	Ave	Std	Best	Ave	Std	Best	Ave	Std
C01	$1.606\times {10}^4$	$4.183\times {10}^4$	$1.385\times {10}^4$	$1.408\times {10}^4$	$2.850\times {10}^4$	$1.035\times {10}^4$	$1.119\times {10}^4$	$2.170\times {10}^4$	$6.904\times {10}^3$	$1.980\times {10}^4$	$5.318\times {10}^4$	$1.049\times {10}^4$
C02	$1.256\times {10}^4$	$5.100\times {10}^4$	$3.343\times {10}^4$	$1.427\times {10}^4$	$5.064\times {10}^4$	$4.586\times {10}^4$	$8.366\times {10}^3$	$2.670\times {10}^4$	$1.128\times {10}^4$	$1.953\times {10}^4$	$5.292\times {10}^4$	$1.353\times {10}^4$
C03	$7.452\times {10}^4$	$2.681\times {10}^5$	$1.885\times {10}^5$	$8.972\times {10}^4$	$2.166\times {10}^5$	$1.102\times {10}^5$	$4.384\times {10}^4$	$2.456\times {10}^5$	$1.335\times {10}^5$	$6.179\times {10}^4$	$3.298\times {10}^5$	$7.611\times {10}^4$
C04	$4.026\times {10}^2$	$4.599\times {10}^2$	$2.882\times {10}^1$	$3.732\times {10}^2$	$4.314\times {10}^2$	$3.053\times {10}^1$	$4.034\times {10}^2$	$4.447\times {10}^2$	$2.024\times {10}^1$	$4.469\times {10}^2$	$4.983\times {10}^2$	$1.899\times {10}^1$
C05	$2.955\times {10}^5$	$5.477\times {10}^7$	$1.030\times {10}^8$	$9.992\times {10}^4$	$9.441\times {10}^6$	$2.585\times {10}^7$	$2.668\times {10}^5$	$3.480\times {10}^6$	$8.611\times {10}^6$	$2.979\times {10}^5$	$7.444\times {10}^7$	$7.339\times {10}^7$
C06	$9.015\times {10}^3$	$2.573\times {10}^5$	$2.142\times {10}^5$	$2.452\times {10}^3$	$7.872\times {10}^4$	$9.285\times {10}^4$	$3.067\times {10}^3$	$1.240\times {10}^5$	$1.038\times {10}^5$	$1.727\times {10}^5$	$8.536\times {10}^5$	$5.028\times {10}^5$
C07	$1.570\times {10}^9$	$3.750\times {10}^9$	$1.330\times {10}^9$	$1.963\times {10}^9$	$4.423\times {10}^9$	$1.206\times {10}^9$	$2.572\times {10}^8$	$4.135\times {10}^9$	$1.787\times {10}^9$	$7.900\times {10}^8$	$2.680\times {10}^9$	$7.570\times {10}^8$
C08	$5.020\times {10}^8$	$1.740\times {10}^9$	$9.210\times {10}^8$	$6.213\times {10}^8$	$6.860\times {10}^9$	$8.426\times {10}^9$	$3.117\times {10}^8$	$1.662\times {10}^9$	$1.979\times {10}^9$	$1.280\times {10}^9$	$1.220\times {10}^{10}$	$4.900\times {10}^9$
C09	$1.230\times {10}^9$	$9.280\times {10}^{10}$	$1.640\times {10}^{11}$	$8.105\times {10}^8$	$4.017\times {10}^{10}$	$5.340\times {10}^{10}$	$4.171\times {10}^9$	$9.171\times {10}^{10}$	$6.152\times {10}^{10}$	$5.160\times {10}^{10}$	$2.500\times {10}^{11}$	$1.160\times {10}^{11}$
C10	$3.430\times {10}^{12}$	$1.590\times {10}^{13}$	$8.500\times {10}^{12}$	$2.924\times {10}^{12}$	$2.170\times {10}^{13}$	$1.213\times {10}^{13}$	$3.084\times {10}^{12}$	$6.503\times {10}^{12}$	$4.852\times {10}^{12}$	$3.690\times {10}^{12}$	$4.480\times {10}^{12}$	$2.330\times {10}^{11}$
C11	$5.290\times {10}^{10}$	$6.900\times {10}^{11}$	$7.090\times {10}^{11}$	$1.001\times {10}^9$	$2.356\times {10}^9$	$8.472\times {10}^8$	$1.115\times {10}^9$	$2.834\times {10}^9$	$8.349\times {10}^8$	$2.280\times {10}^9$	$3.820\times {10}^9$	$5.360\times {10}^8$
C12	$6.850\times {10}^{10}$	$1.940\times {10}^{11}$	$5.420\times {10}^{10}$	$9.005\times {10}^{10}$	$2.102\times {10}^{11}$	$5.657\times {10}^{10}$	$8.398\times {10}^{10}$	$1.984\times {10}^{11}$	$4.901\times {10}^{10}$	$1.710\times {10}^{11}$	$2.740\times {10}^{11}$	$5.200\times {10}^{10}$
C13	$1.110\times {10}^{11}$	$2.150\times {10}^{11}$	$4.610\times {10}^{10}$	$9.151\times {10}^{10}$	$2.187\times {10}^{11}$	$5.869\times {10}^{10}$	$9.374\times {10}^{10}$	$2.196\times {10}^{11}$	$5.148\times {10}^{10}$	$1.500\times {10}^{11}$	$2.940\times {10}^{11}$	$4.700\times {10}^{10}$
C14	$9.150\times {10}^{10}$	$3.450\times {10}^{11}$	$1.200\times {10}^{11}$	$1.904\times {10}^{11}$	$4.242\times {10}^{11}$	$1.041\times {10}^{11}$	$1.878\times {10}^{11}$	$3.678\times {10}^{11}$	$1.060\times {10}^{11}$	$3.200\times {10}^{11}$	$5.370\times {10}^{11}$	$7.120\times {10}^{10}$
C15	$3.670\times {10}^{10}$	$1.120\times {10}^{11}$	$4.080\times {10}^{10}$	$1.342\times {10}^{10}$	$1.238\times {10}^{11}$	$5.001\times {10}^{10}$	$2.950\times {10}^{10}$	$1.297\times {10}^{11}$	$4.728\times {10}^{10}$	$8.590\times {10}^{10}$	$1.830\times {10}^{11}$	$3.850\times {10}^{10}$
C16	$4.870\times {10}^{10}$	$1.190\times {10}^{11}$	$4.410\times {10}^{10}$	$6.172\times {10}^{10}$	$1.402\times {10}^{11}$	$4.341\times {10}^{10}$	$6.658\times {10}^{10}$	$1.320\times {10}^{11}$	$3.705\times {10}^{10}$	$1.090\times {10}^{11}$	$1.780\times {10}^{11}$	$3.310\times {10}^{10}$
C17	$1.090\times {10}^{12}$	$2.310\times {10}^{12}$	$4.380\times {10}^{11}$	$3.535\times {10}^{11}$	$3.535\times {10}^{11}$	$0\times {10}^0$	$3.535\times {10}^{11}$	$3.535\times {10}^{11}$	$0\times {10}^0$	$3.540\times {10}^{11}$	$3.540\times {10}^{11}$	$0\times {10}^0$
C18	$3.020\times {10}^{20}$	$1.150\times {10}^{21}$	$5.930\times {10}^{20}$	$5.653\times {10}^{20}$	$1.459\times {10}^{21}$	$5.593\times {10}^{20}$	$5.511\times {10}^{20}$	$1.593\times {10}^{21}$	$6.200\times {10}^{20}$	$1.400\times {10}^{21}$	$2.360\times {10}^{21}$	$4.110\times {10}^{20}$
C19	$1.850\times {10}^{12}$	$1.850\times {10}^{12}$	$5.720\times {10}^8$	$1.848\times {10}^{12}$	$1.850\times {10}^{12}$	$7.970\times {10}^8$	$1.849\times {10}^{12}$	$1.850\times {10}^{12}$	$6.375\times {10}^8$	$1.850\times {10}^{12}$	$1.850\times {10}^{12}$	$3.950\times {10}^8$
C20	$8.336\times {10}^0$	$9.831\times {10}^0$	$7.622\times {10}^{-1}$	$7.264\times {10}^0$	$8.833\times {10}^0$	$8.058\times {10}^{-1}$	$7.195\times {10}^0$	$9.440\times {10}^0$	$7.264\times {10}^{-1}$	$8.435\times {10}^0$	$1.031\times {10}^1$	$9.668\times {10}^{-1}$
C21	$8.440\times {10}^{11}$	$3.380\times {10}^{12}$	$1.370\times {10}^{12}$	$1.990\times {10}^{12}$	$3.687\times {10}^{12}$	$1.295\times {10}^{12}$	$9.857\times {10}^{11}$	$4.314\times {10}^{12}$	$1.539\times {10}^{12}$	$3.280\times {10}^{12}$	$7.210\times {10}^{12}$	$1.960\times {10}^{12}$
C22	$9.770\times {10}^{11}$	$3.910\times {10}^{12}$	$1.510\times {10}^{12}$	$9.452\times {10}^{11}$	$4.030\times {10}^{12}$	$1.953\times {10}^{12}$	$1.395\times {10}^{12}$	$4.733\times {10}^{12}$	$1.684\times {10}^{12}$	$3.730\times {10}^{12}$	$7.480\times {10}^{12}$	$1.710\times {10}^{12}$
C23	$2.730\times {10}^{12}$	$6.550\times {10}^{12}$	$2.220\times {10}^{12}$	$2.779\times {10}^{12}$	$6.696\times {10}^{12}$	$2.635\times {10}^{12}$	$4.252\times {10}^{12}$	$8.038\times {10}^{12}$	$2.764\times {10}^{12}$	$8.280\times {10}^{12}$	$1.420\times {10}^{13}$	$2.880\times {10}^{12}$
C24	$1.020\times {10}^{12}$	$3.180\times {10}^{12}$	$1.600\times {10}^{12}$	$1.213\times {10}^{12}$	$3.801\times {10}^{12}$	$1.375\times {10}^{12}$	$7.777\times {10}^{11}$	$3.593\times {10}^{12}$	$1.537\times {10}^{12}$	$3.310\times {10}^{12}$	$6.860\times {10}^{12}$	$1.440\times {10}^{12}$
C25	$7.600\times {10}^{11}$	$3.010\times {10}^{12}$	$1.460\times {10}^{12}$	$1.302\times {10}^{12}$	$3.230\times {10}^{12}$	$1.309\times {10}^{12}$	$1.268\times {10}^{12}$	$4.734\times {10}^{12}$	$1.843\times {10}^{12}$	$4.430\times {10}^{12}$	$6.860\times {10}^{12}$	$1.480\times {10}^{12}$
C26	$1.070\times {10}^{12}$	$3.840\times {10}^{12}$	$1.410\times {10}^{12}$	$1.759\times {10}^{12}$	$4.112\times {10}^{12}$	$1.576\times {10}^{12}$	$2.914\times {10}^{12}$	$5.017\times {10}^{12}$	$1.313\times {10}^{12}$	$3.770\times {10}^{12}$	$7.420\times {10}^{12}$	$1.570\times {10}^{12}$
C27	$1.830\times {10}^{23}$	$1.320\times {10}^{24}$	$9.440\times {10}^{23}$	$6.834\times {10}^{22}$	$9.301\times {10}^{23}$	$8.046\times {10}^{23}$	$1.508\times {10}^{23}$	$1.517\times {10}^{24}$	$8.942\times {10}^{23}$	$1.280\times {10}^{24}$	$3.580\times {10}^{24}$	$1.610\times {10}^{24}$
C28	$1.850\times {10}^{12}$	$1.850\times {10}^{12}$	$4.160\times {10}^8$	$1.848\times {10}^{12}$	$1.850\times {10}^{12}$	$7.187\times {10}^8$	$1.849\times {10}^{12}$	$1.851\times {10}^{12}$	$5.298\times {10}^8$	$1.850\times {10}^{12}$	$1.850\times {10}^{12}$	$2.850\times {10}^8$

,

Table 12. Results for dimension 50 of the IEEE CEC 2017 comparative tests for “Constrained Real-Parameter Optimization”.

Function	TGCOA			BWOA			JSOA			LCA
	Best	Ave	Std	Best	Ave	Std	Best	Ave	Std	Best	Ave	Std
C01	$4.982\times {10}^4$	$1.312\times {10}^5$	$4.760\times {10}^4$	$3.644\times {10}^4$	$1.141\times {10}^5$	$6.505\times {10}^4$	$4.818\times {10}^4$	$1.114\times {10}^5$	$4.175\times {10}^4$	$1.239\times {10}^5$	$2.983\times {10}^5$	$1.137\times {10}^5$
C02	$4.262\times {10}^4$	$1.468\times {10}^5$	$8.130\times {10}^4$	$4.015\times {10}^4$	$1.266\times {10}^5$	$8.426\times {10}^4$	$3.460\times {10}^4$	$1.488\times {10}^5$	$6.933\times {10}^4$	$1.407\times {10}^5$	$3.977\times {10}^5$	$1.050\times {10}^5$
C03	$2.401\times {10}^5$	$1.212\times {10}^6$	$1.280\times {10}^6$	$1.688\times {10}^5$	$6.875\times {10}^5$	$4.385\times {10}^5$	$1.535\times {10}^5$	$6.576\times {10}^5$	$3.546\times {10}^5$	$2.525\times {10}^5$	$7.672\times {10}^5$	$3.027\times {10}^5$
C04	$7.125\times {10}^2$	$7.901\times {10}^2$	$4.255\times {10}^1$	$7.089\times {10}^2$	$7.847\times {10}^2$	$3.759\times {10}^1$	$7.206\times {10}^2$	$7.834\times {10}^2$	$2.876\times {10}^1$	$8.224\times {10}^2$	$8.575\times {10}^2$	$1.932\times {10}^1$
C05	$6.031\times {10}^5$	$1.140\times {10}^8$	$1.340\times {10}^8$	$3.266\times {10}^5$	$5.364\times {10}^7$	$8.839\times {10}^7$	$5.232\times {10}^5$	$4.942\times {10}^7$	$8.274\times {10}^7$	$2.901\times {10}^6$	$3.260\times {10}^8$	$1.820\times {10}^8$
C06	$2.662\times {10}^4$	$3.781\times {10}^5$	$3.619\times {10}^5$	$3.199\times {10}^3$	$1.123\times {10}^5$	$1.277\times {10}^5$	$3.318\times {10}^3$	$2.011\times {10}^5$	$1.752\times {10}^5$	$6.471\times {10}^4$	$1.184\times {10}^6$	$6.344\times {10}^5$
C07	$9.300\times {10}^9$	$1.540\times {10}^{10}$	$3.560\times {10}^9$	$9.450\times {10}^9$	$1.710\times {10}^{10}$	$3.770\times {10}^9$	$5.100\times {10}^9$	$1.170\times {10}^{10}$	$5.830\times {10}^9$	$2.460\times {10}^9$	$8.280\times {10}^9$	$1.700\times {10}^9$
C08	$3.160\times {10}^9$	$1.790\times {10}^{10}$	$1.420\times {10}^{10}$	$1.740\times {10}^9$	$3.140\times {10}^{10}$	$3.680\times {10}^{10}$	$1.140\times {10}^9$	$4.650\times {10}^9$	$2.980\times {10}^9$	$1.370\times {10}^{10}$	$3.910\times {10}^{10}$	$1.100\times {10}^{10}$
C09	$1.360\times {10}^{11}$	$9.870\times {10}^{11}$	$4.480\times {10}^{11}$	$1.450\times {10}^{11}$	$8.180\times {10}^{11}$	$4.610\times {10}^{11}$	$1.880\times {10}^{11}$	$9.250\times {10}^{11}$	$4.380\times {10}^{11}$	$6.500\times {10}^{11}$	$1.420\times {10}^{12}$	$3.750\times {10}^{11}$
C10	$3.700\times {10}^{13}$	$7.170\times {10}^{13}$	$3.810\times {10}^{13}$	$1.640\times {10}^{13}$	$8.190\times {10}^{13}$	$5.330\times {10}^{13}$	$2.580\times {10}^{13}$	$7.890\times {10}^{13}$	$7.850\times {10}^{13}$	$3.990\times {10}^{13}$	$9.850\times {10}^{13}$	$3.390\times {10}^{13}$
C11	$8.110\times {10}^{11}$	$3.030\times {10}^{12}$	$1.970\times {10}^{12}$	$4.670\times {10}^9$	$7.960\times {10}^9$	$1.680\times {10}^9$	$5.190\times {10}^9$	$8.730\times {10}^9$	$1.330\times {10}^9$	$8.840\times {10}^9$	$1.050\times {10}^{10}$	$7.470\times {10}^8$
C12	$4.930\times {10}^{11}$	$7.120\times {10}^{11}$	$1.200\times {10}^{11}$	$5.110\times {10}^{11}$	$7.700\times {10}^{11}$	$1.100\times {10}^{11}$	$4.910\times {10}^{11}$	$7.910\times {10}^{11}$	$1.150\times {10}^{11}$	$8.180\times {10}^{11}$	$9.690\times {10}^{11}$	$7.260\times {10}^{10}$
C13	$5.760\times {10}^{11}$	$7.550\times {10}^{11}$	$1.190\times {10}^{11}$	$4.520\times {10}^{11}$	$7.880\times {10}^{11}$	$1.470\times {10}^{11}$	$4.570\times {10}^{11}$	$8.290\times {10}^{11}$	$1.220\times {10}^{11}$	$6.970\times {10}^{11}$	$9.960\times {10}^{11}$	$8.990\times {10}^{10}$
C14	$8.060\times {10}^{11}$	$1.400\times {10}^{12}$	$2.600\times {10}^{11}$	$9.220\times {10}^{11}$	$1.530\times {10}^{12}$	$2.620\times {10}^{11}$	$1.010\times {10}^{12}$	$1.490\times {10}^{12}$	$2.260\times {10}^{11}$	$1.350\times {10}^{12}$	$1.840\times {10}^{12}$	$1.840\times {10}^{11}$
C15	$1.630\times {10}^{11}$	$4.270\times {10}^{11}$	$1.150\times {10}^{11}$	$2.720\times {10}^{11}$	$5.160\times {10}^{11}$	$1.090\times {10}^{11}$	$3.500\times {10}^{11}$	$5.520\times {10}^{11}$	$8.480\times {10}^{10}$	$3.760\times {10}^{11}$	$6.480\times {10}^{11}$	$8.290\times {10}^{10}$
C16	$1.990\times {10}^{11}$	$4.750\times {10}^{11}$	$9.890\times {10}^{10}$	$3.340\times {10}^{11}$	$5.330\times {10}^{11}$	$9.960\times {10}^{10}$	$4.080\times {10}^{11}$	$5.600\times {10}^{11}$	$6.660\times {10}^{10}$	$5.330\times {10}^{11}$	$6.610\times {10}^{11}$	$6.910\times {10}^{10}$
C17	$6.730\times {10}^{12}$	$9.890\times {10}^{12}$	$1.320\times {10}^{12}$	$1.040\times {10}^{12}$	$1.040\times {10}^{12}$	$7.450\times {10}^{-4}$	$1.040\times {10}^{12}$	$1.040\times {10}^{12}$	$7.450\times {10}^{-4}$	$1.040\times {10}^{12}$	$1.040\times {10}^{12}$	$7.450\times {10}^{-4}$
C18	$1.800\times {10}^{21}$	$5.790\times {10}^{21}$	$1.870\times {10}^{21}$	$3.600\times {10}^{21}$	$6.470\times {10}^{21}$	$1.610\times {10}^{21}$	$2.230\times {10}^{21}$	$6.890\times {10}^{21}$	$1.920\times {10}^{21}$	$2.410\times {10}^{21}$	$9.180\times {10}^{21}$	$1.680\times {10}^{21}$
C19	$5.280\times {10}^{12}$	$5.290\times {10}^{12}$	$7.820\times {10}^8$	$5.280\times {10}^{12}$	$5.280\times {10}^{12}$	$1.280\times {10}^9$	$5.280\times {10}^{12}$	$5.280\times {10}^{12}$	$1.290\times {10}^9$	$5.280\times {10}^{12}$	$5.290\times {10}^{12}$	$6.330\times {10}^8$
C20	$1.607\times {10}^1$	$1.837\times {10}^1$	$9.595\times {10}^{-1}$	$1.498\times {10}^1$	$1.727\times {10}^1$	$1.083\times {10}^0$	$1.532\times {10}^1$	$1.764\times {10}^1$	$1.002\times {10}^0$	$1.738\times {10}^1$	$1.926\times {10}^1$	$7.810\times {10}^{-1}$
C21	$5.570\times {10}^{12}$	$9.500\times {10}^{12}$	$2.020\times {10}^{12}$	$5.030\times {10}^{12}$	$9.950\times {10}^{12}$	$1.880\times {10}^{12}$	$6.000\times {10}^{12}$	$1.030\times {10}^{13}$	$1.940\times {10}^{12}$	$9.490\times {10}^{12}$	$1.250\times {10}^{13}$	$1.270\times {10}^{12}$
C22	$5.880\times {10}^{12}$	$9.480\times {10}^{12}$	$2.120\times {10}^{12}$	$6.110\times {10}^{12}$	$9.880\times {10}^{12}$	$1.750\times {10}^{12}$	$4.190\times {10}^{12}$	$1.070\times {10}^{13}$	$2.150\times {10}^{12}$	$8.460\times {10}^{12}$	$1.270\times {10}^{13}$	$1.540\times {10}^{12}$
C23	$1.100\times {10}^{13}$	$1.900\times {10}^{13}$	$3.660\times {10}^{12}$	$1.270\times {10}^{13}$	$2.060\times {10}^{13}$	$3.420\times {10}^{12}$	$1.280\times {10}^{13}$	$1.940\times {10}^{13}$	$2.990\times {10}^{12}$	$1.920\times {10}^{13}$	$2.440\times {10}^{13}$	$2.730\times {10}^{12}$
C24	$2.730\times {10}^{12}$	$7.990\times {10}^{12}$	$2.210\times {10}^{12}$	$4.930\times {10}^{12}$	$8.890\times {10}^{12}$	$2.240\times {10}^{12}$	$5.390\times {10}^{12}$	$9.670\times {10}^{12}$	$1.750\times {10}^{12}$	$8.700\times {10}^{12}$	$1.160\times {10}^{13}$	$1.240\times {10}^{12}$
C25	$4.700\times {10}^{12}$	$8.290\times {10}^{12}$	$1.780\times {10}^{12}$	$5.360\times {10}^{12}$	$8.830\times {10}^{12}$	$2.020\times {10}^{12}$	$6.530\times {10}^{12}$	$9.600\times {10}^{12}$	$1.500\times {10}^{12}$	$7.270\times {10}^{12}$	$1.160\times {10}^{13}$	$1.370\times {10}^{12}$
C26	$5.860\times {10}^{12}$	$9.300\times {10}^{12}$	$1.800\times {10}^{12}$	$5.930\times {10}^{12}$	$1.010\times {10}^{13}$	$1.960\times {10}^{12}$	$7.990\times {10}^{12}$	$1.010\times {10}^{13}$	$1.510\times {10}^{12}$	$6.120\times {10}^{12}$	$1.260\times {10}^{13}$	$1.840\times {10}^{12}$
C27	$3.880\times {10}^{23}$	$2.720\times {10}^{24}$	$1.330\times {10}^{24}$	$1.300\times {10}^{24}$	$3.530\times {10}^{24}$	$1.570\times {10}^{24}$	$1.900\times {10}^{24}$	$3.450\times {10}^{24}$	$1.140\times {10}^{24}$	$2.360\times {10}^{24}$	$6.310\times {10}^{24}$	$1.710\times {10}^{24}$
C28	$5.280\times {10}^{12}$	$5.280\times {10}^{12}$	$7.450\times {10}^8$	$5.280\times {10}^{12}$	$5.280\times {10}^{12}$	$9.140\times {10}^8$	$5.280\times {10}^{12}$	$5.280\times {10}^{12}$	$8.440\times {10}^8$	$5.290\times {10}^{12}$	$5.290\times {10}^{12}$	$3.950\times {10}^8$

, and

Table 13. Results for dimension 100 of the IEEE CEC 2017 comparative tests for “Constrained Real-Parameter Optimization”.

Function	TGCOA			BWOA			JSOA			LCA
	Best	Ave	Std	Best	Ave	Std	Best	Ave	Std	Best	Ave	Std
C01	$2.049\times {10}^5$	$4.421\times {10}^5$	$1.508\times {10}^5$	$1.780\times {10}^5$	$4.314\times {10}^5$	$1.796\times {10}^5$	$1.934\times {10}^5$	$4.697\times {10}^5$	$1.489\times {10}^5$	$4.251\times {10}^5$	$8.209\times {10}^5$	$1.714\times {10}^5$
C02	$2.408\times {10}^5$	$4.639\times {10}^5$	$1.563\times {10}^5$	$2.137\times {10}^5$	$4.752\times {10}^5$	$2.005\times {10}^5$	$2.115\times {10}^5$	$5.569\times {10}^5$	$1.690\times {10}^5$	$7.124\times {10}^5$	$9.385\times {10}^5$	$8.130\times {10}^4$
C03	$8.554\times {10}^5$	$2.233\times {10}^6$	$1.121\times {10}^6$	$7.725\times {10}^5$	$1.520\times {10}^6$	$5.547\times {10}^5$	$6.248\times {10}^5$	$2.028\times {10}^6$	$9.764\times {10}^5$	$1.506\times {10}^6$	$3.219\times {10}^6$	$1.589\times {10}^6$
C04	$1.541\times {10}^3$	$1.707\times {10}^3$	$6.048\times {10}^1$	$1.545\times {10}^3$	$1.676\times {10}^3$	$5.845\times {10}^1$	$1.557\times {10}^3$	$1.651\times {10}^3$	$4.912\times {10}^1$	$1.688\times {10}^3$	$1.762\times {10}^3$	$3.087\times {10}^1$
C05	$1.283\times {10}^6$	$6.273\times {10}^7$	$1.130\times {10}^8$	$1.155\times {10}^6$	$1.319\times {10}^7$	$3.386\times {10}^7$	$1.266\times {10}^6$	$7.036\times {10}^6$	$1.937\times {10}^7$	$1.377\times {10}^6$	$3.590\times {10}^8$	$2.500\times {10}^8$
C06	$3.546\times {10}^4$	$5.595\times {10}^5$	$4.092\times {10}^5$	$1.266\times {10}^4$	$1.724\times {10}^5$	$2.003\times {10}^5$	$9.119\times {10}^3$	$3.647\times {10}^5$	$2.352\times {10}^5$	$1.689\times {10}^5$	$3.130\times {10}^6$	$2.105\times {10}^6$
C07	$7.160\times {10}^{10}$	$9.340\times {10}^{10}$	$1.230\times {10}^{10}$	$7.840\times {10}^{10}$	$9.600\times {10}^{10}$	$1.160\times {10}^{10}$	$2.200\times {10}^{10}$	$3.890\times {10}^{10}$	$1.760\times {10}^{10}$	$2.440\times {10}^{10}$	$3.120\times {10}^{10}$	$7.230\times {10}^9$
C08	$4.970\times {10}^{10}$	$2.280\times {10}^{11}$	$1.870\times {10}^{11}$	$4.660\times {10}^{10}$	$4.900\times {10}^{11}$	$6.580\times {10}^{11}$	$1.630\times {10}^{10}$	$7.680\times {10}^{10}$	$6.900\times {10}^{10}$	$1.800\times {10}^{11}$	$4.610\times {10}^{11}$	$1.280\times {10}^{11}$
C09	$3.600\times {10}^{12}$	$5.490\times {10}^{12}$	$1.060\times {10}^{12}$	$2.560\times {10}^{12}$	$4.800\times {10}^{12}$	$1.420\times {10}^{12}$	$2.660\times {10}^{12}$	$4.910\times {10}^{12}$	$1.130\times {10}^{12}$	$3.360\times {10}^{12}$	$6.020\times {10}^{12}$	$9.220\times {10}^{11}$
C10	$1.470\times {10}^{14}$	$3.970\times {10}^{14}$	$1.640\times {10}^{14}$	$1.270\times {10}^{14}$	$5.280\times {10}^{14}$	$2.880\times {10}^{14}$	$1.270\times {10}^{14}$	$5.330\times {10}^{14}$	$5.850\times {10}^{14}$	$2.840\times {10}^{14}$	$9.040\times {10}^{14}$	$3.820\times {10}^{14}$
C11	$4.000\times {10}^{10}$	$1.250\times {10}^{13}$	$9.750\times {10}^{12}$	$3.260\times {10}^{10}$	$4.280\times {10}^{10}$	$3.830\times {10}^9$	$3.670\times {10}^{10}$	$4.550\times {10}^{10}$	$3.950\times {10}^9$	$4.300\times {10}^{10}$	$4.840\times {10}^{10}$	$2.050\times {10}^9$
C12	$4.310\times {10}^{12}$	$5.310\times {10}^{12}$	$4.790\times {10}^{11}$	$4.490\times {10}^{12}$	$5.670\times {10}^{12}$	$4.660\times {10}^{11}$	$4.860\times {10}^{12}$	$5.730\times {10}^{12}$	$3.660\times {10}^{11}$	$5.210\times {10}^{12}$	$6.100\times {10}^{12}$	$3.310\times {10}^{11}$
C13	$4.090\times {10}^{12}$	$5.620\times {10}^{12}$	$5.140\times {10}^{11}$	$4.660\times {10}^{12}$	$5.690\times {10}^{12}$	$4.630\times {10}^{11}$	$5.210\times {10}^{12}$	$5.950\times {10}^{12}$	$3.130\times {10}^{11}$	$5.570\times {10}^{12}$	$6.300\times {10}^{12}$	$3.120\times {10}^{11}$
C14	$7.490\times {10}^{12}$	$1.060\times {10}^{13}$	$1.190\times {10}^{12}$	$8.560\times {10}^{12}$	$1.080\times {10}^{13}$	$1.010\times {10}^{12}$	$9.590\times {10}^{12}$	$1.130\times {10}^{13}$	$7.210\times {10}^{11}$	$1.050\times {10}^{13}$	$1.220\times {10}^{13}$	$5.220\times {10}^{11}$
C15	$3.100\times {10}^{12}$	$3.990\times {10}^{12}$	$4.900\times {10}^{11}$	$3.420\times {10}^{12}$	$4.130\times {10}^{12}$	$3.740\times {10}^{11}$	$3.350\times {10}^{12}$	$4.240\times {10}^{12}$	$4.150\times {10}^{11}$	$3.630\times {10}^{12}$	$4.640\times {10}^{12}$	$3.180\times {10}^{11}$
C16	$2.410\times {10}^{12}$	$4.050\times {10}^{12}$	$4.890\times {10}^{11}$	$3.550\times {10}^{12}$	$4.140\times {10}^{12}$	$3.190\times {10}^{11}$	$3.460\times {10}^{12}$	$4.370\times {10}^{12}$	$3.330\times {10}^{11}$	$3.950\times {10}^{12}$	$4.660\times {10}^{12}$	$2.570\times {10}^{11}$
C17	$5.010\times {10}^{13}$	$6.650\times {10}^{13}$	$7.390\times {10}^{12}$	$6.470\times {10}^{12}$	$6.470\times {10}^{12}$	$4.966\times {10}^{-3}$	$6.470\times {10}^{12}$	$6.470\times {10}^{12}$	$4.966\times {10}^{-3}$	$6.470\times {10}^{12}$	$6.470\times {10}^{12}$	$4.966\times {10}^{-3}$
C18	$5.440\times {10}^{22}$	$9.690\times {10}^{22}$	$1.640\times {10}^{22}$	$8.880\times {10}^{22}$	$1.080\times {10}^{23}$	$1.130\times {10}^{22}$	$9.070\times {10}^{22}$	$1.130\times {10}^{23}$	$9.260\times {10}^{21}$	$9.610\times {10}^{22}$	$1.220\times {10}^{23}$	$8.340\times {10}^{21}$
C19	$2.160\times {10}^{13}$	$2.160\times {10}^{13}$	$2.610\times {10}^9$	$2.160\times {10}^{13}$	$2.160\times {10}^{13}$	$2.420\times {10}^9$	$2.160\times {10}^{13}$	$2.160\times {10}^{13}$	$2.710\times {10}^9$	$2.160\times {10}^{13}$	$2.160\times {10}^{13}$	$1.850\times {10}^9$
C20	$3.782\times {10}^1$	$4.064\times {10}^1$	$1.339\times {10}^0$	$3.740\times {10}^1$	$3.958\times {10}^1$	$1.070\times {10}^0$	$3.589\times {10}^1$	$3.956\times {10}^1$	$1.659\times {10}^0$	$3.731\times {10}^1$	$4.179\times {10}^1$	$1.695\times {10}^0$
C21	$4.500\times {10}^{13}$	$5.840\times {10}^{13}$	$6.210\times {10}^{12}$	$4.110\times {10}^{13}$	$6.390\times {10}^{13}$	$7.060\times {10}^{12}$	$5.080\times {10}^{13}$	$6.530\times {10}^{13}$	$6.330\times {10}^{12}$	$6.200\times {10}^{13}$	$7.290\times {10}^{13}$	$4.180\times {10}^{12}$
C22	$3.690\times {10}^{13}$	$5.910\times {10}^{13}$	$8.040\times {10}^{12}$	$4.260\times {10}^{13}$	$6.330\times {10}^{13}$	$8.440\times {10}^{12}$	$4.480\times {10}^{13}$	$6.360\times {10}^{13}$	$8.990\times {10}^{12}$	$6.070\times {10}^{13}$	$7.400\times {10}^{13}$	$4.870\times {10}^{12}$
C23	$7.610\times {10}^{13}$	$1.150\times {10}^{14}$	$1.580\times {10}^{13}$	$9.810\times {10}^{13}$	$1.270\times {10}^{14}$	$1.280\times {10}^{13}$	$9.850\times {10}^{13}$	$1.290\times {10}^{14}$	$1.270\times {10}^{13}$	$1.190\times {10}^{14}$	$1.440\times {10}^{14}$	$9.590\times {10}^{12}$
C24	$3.500\times {10}^{13}$	$5.390\times {10}^{13}$	$8.550\times {10}^{12}$	$4.080\times {10}^{13}$	$6.000\times {10}^{13}$	$6.360\times {10}^{12}$	$3.940\times {10}^{13}$	$5.910\times {10}^{13}$	$7.270\times {10}^{12}$	$4.960\times {10}^{13}$	$6.710\times {10}^{13}$	$5.020\times {10}^{12}$
C25	$3.650\times {10}^{13}$	$5.530\times {10}^{13}$	$7.390\times {10}^{12}$	$4.560\times {10}^{13}$	$5.790\times {10}^{13}$	$5.590\times {10}^{12}$	$4.170\times {10}^{13}$	$5.790\times {10}^{13}$	$6.560\times {10}^{12}$	$5.480\times {10}^{13}$	$6.690\times {10}^{13}$	$4.620\times {10}^{12}$
C26	$4.230\times {10}^{13}$	$5.950\times {10}^{13}$	$7.840\times {10}^{12}$	$4.600\times {10}^{13}$	$6.340\times {10}^{13}$	$7.360\times {10}^{12}$	$4.670\times {10}^{13}$	$6.310\times {10}^{13}$	$6.590\times {10}^{12}$	$6.120\times {10}^{13}$	$7.220\times {10}^{13}$	$4.200\times {10}^{12}$
C27	$1.270\times {10}^{25}$	$2.370\times {10}^{25}$	$6.500\times {10}^{24}$	$8.250\times {10}^{24}$	$2.490\times {10}^{25}$	$6.630\times {10}^{24}$	$1.660\times {10}^{25}$	$3.050\times {10}^{25}$	$5.200\times {10}^{24}$	$2.260\times {10}^{25}$	$3.720\times {10}^{25}$	$5.890\times {10}^{24}$
C28	$2.160\times {10}^{13}$	$2.160\times {10}^{13}$	$1.650\times {10}^9$	$2.160\times {10}^{13}$	$2.160\times {10}^{13}$	$2.420\times {10}^9$	$2.160\times {10}^{13}$	$2.160\times {10}^{13}$	$2.100\times {10}^9$	$2.160\times {10}^{13}$	$2.160\times {10}^{13}$	$1.150\times {10}^9$

, respectively. These tables exhibit the maximum, average, and standard deviation calculated. A Wilcoxon signed-rank test was employed to analyze the differences among the algorithms at a significance level of 5%. Moreover, the Friedman test established the ranking of the TGCOA, BWOA, JSOA, and LCA. For 10-dimensional problems, the Wilcoxon rank-sum test in

Table 14. Wilcoxon signed-rank test results for CEC 2017 functions, dimension 10, $\alpha =5\%$.

TGCOA–BWOA		TGCOA–JSOA		TGCOA–LCA
$(+/=/-)$	p-value	$(+/=/-)$	p-value	$(+/=/-)$	p-value
23/2/3	$\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}$	15/2/11	$9.7\times {10}^{-1}$	3/0/25	$\mathbf{0}\times {\mathbf{10}}^{\mathbf{0}}$

Significant differences (p < 0.05) are in bold.

shows that the TGCOA outperformed the BWOA and LCA, with p-values of 0 for both. Meanwhile, there are no significant differences, with the JSOA having a p-value $0.97$. In

Table 15. IEEE CEC 2017 benchmark functions for dimension 10, Friedman test.

Algorithms	Mean Rank	Overall Rank
BWOA	1.46	1
JSOA	2.18	2
TGCOA	2.54	3
LCA	3.82	4

, the Friedman test ranked the TGCOA third. For 30-dimensional problems, the Wilcoxon rank-sum test in

Table 16. Wilcoxon signed-rank test results for CEC 2017 functions, dimension 30, $\alpha =5\%$.

TGCOA–BWOA		TGCOA–JSOA		TGCOA–LCA
$(+/=/-)$	p-value	$(+/=/-)$	p-value	$(+/=/-)$	p-value
11/2/15	$1.12\times {10}^{-1}$	12/2/14	$9.6\times {10}^{-2}$	4/0/24	$\mathbf{3}\times {\mathbf{10}}^{-\mathbf{3}}$

Significant differences (p < 0.05) are in bold.

shows that the TGCOA outperformed the LCA, with a p-value of 0.003. However, there were no significant differences for BWOA and JSOA, with p-values of 0.112 and 0.096, respectively. In addition, the Friedman test ranked the TGCOA first, as shown in

Table 17. IEEE CEC 2017 benchmark functions for dimension 30, Friedman test.

Algorithms	Mean Rank	Overall Rank
TGCOA	2.04	1
BWOA	2.09	2
JSOA	2.23	3
LCA	3.64	4

. The results in

Table 18. Wilcoxon signed-rank test results for CEC 2017 functions, dimension 50, $\alpha =5\%$.

TGCOA–BWOA		TGCOA–JSOA		TGCOA–LCA
$(+/=/-)$	p-value	$(+/=/-)$	p-value	$(+/=/-)$	p-value
11/1/16	$\mathbf{3}.\mathbf{2}\times {\mathbf{10}}^{-\mathbf{2}}$	12/1/15	$6.1\times {10}^{-2}$	5/0/23	$\mathbf{2}\times {\mathbf{10}}^{-\mathbf{3}}$

Significant differences (p < 0.05) are in bold.

indicate that, for 50-dimensional problems, the TGCOA outperformed the BWOA and LCA, with p-values of 0.032, and 0.002. Meanwhile, no significant differences were observed for JSOA, with a p-value of 0.061. According to

Table 19. IEEE CEC 2017 benchmark functions for dimension 50, Friedman test.

Algorithms	Mean Rank	Overall Rank
TGCOA	2.04	1
BWOA	2.09	2
JSOA	2.20	3
LCA	3.68	4

, the Friedman test ranked the TGCOA first. Finally, for 100-dimensional problems, the Wilcoxon rank-sum test in

Table 20. Wilcoxon signed-rank test statistical findings for IEEE CEC 2017 benchmark functions for dimension 100 at $\alpha =5\%$.

TGCOA–BWOA		TGCOA–JSOA		TGCOA–LCA
$(+/=/-)$	p-value	$(+/=/-)$	p-value	$(+/=/-)$	p-value
9/2/17	$\mathbf{2}.\mathbf{1}\times {\mathbf{10}}^{-\mathbf{2}}$	10/2/16	$\mathbf{4}.\mathbf{3}\times {\mathbf{10}}^{-\mathbf{2}}$	5/0/23	$\mathbf{3}\times {\mathbf{10}}^{-\mathbf{3}}$

Significant differences (p < 0.05) are in bold.

shows that the TGCOA outperformed all the algorithms, BWOA, LCA, and BO, with p-values of 0.021, 0.043, and 0.003, respectively. Furthermore, the Friedman test ranked the TGCOA first, as shown in

Table 21. IEEE CEC 2017 benchmark functions for dimension 100, Friedman test.

Algorithms	Mean Rank	Overall Rank
TGCOA	1.93	1
BWOA	2.09	2
JSOA	2.41	3
LCA	3.57	4

.

Average Convergence Rate Results

This section presents the results of the Average Convergence Rate (ACR) analysis. In Appendix A,

Table A4. ACR results for TGCOA.

Function	ACR	Category
F1	1	Optimal
F2	1	Optimal
F3	1	Optimal
F4	1	Optimal
F5	1	Optimal
F6	1	Optimal
F7	1	Optimal
F8	1	Optimal
F9	1	Optimal
F10	1	Optimal
F11	1	Optimal
F12	$6.08\times {10}^{-1}$	Good
F13	1	Optimal
F14	1	Optimal
F15	1	Optimal
F16	$8.05\times {10}^{-1}$	Good
F17	1	Optimal
F18	1	Optimal
F19	1	Optimal
F20	1	Optimal

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

,

Table A5. ACR results for JSOA.

Function	ACR	Category
F1	1	Optimal
F2	1	Optimal
F3	1	Optimal
F4	1	Optimal
F5	1	Optimal
F6	1	Optimal
F7	1	Optimal
F8	1	Optimal
F9	1	Optimal
F10	1	Optimal
F11	1	Optimal
F12	1	Optimal
F13	1	Optimal
F14	1	Optimal
F15	1	Optimal
F16	1	Optimal
F17	1	Optimal
F18	1	Optimal
F19	1	Optimal

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

,

Table A6. ACR results for BWOA.

Function	ACR	Category
F1	1	Optimal
F2	1	Optimal
F3	1	Optimal
F4	1	Optimal
F5	1	Optimal
F6	1	Optimal
F7	1	Optimal
F8	1	Optimal
F9	1	Optimal
F10	1	Optimal
F11	1	Optimal
F12	$3.99\times {10}^{-1}$	Slow
F13	1	Optimal
F14	$6.56\times {10}^{-1}$	Good
F15	1	Optimal
F16	$4.92\times {10}^{-1}$	Slow
F17	1	Optimal
F18	1	Optimal
F19	1	Optimal
F20	1	Optimal

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

,

Table A7. ACR results for LCA.

Function	ACR	Category
F1	$3.15\times {10}^{-1}$	Slow
F2	1	Optimal
F3	$5.87\times {10}^{-1}$	Good
F4	$7.15\times {10}^{-1}$	Good
F5	$4.44\times {10}^{-1}$	Slow
F6	0	No Improvement
F7	1	Optimal
F8	$3.64\times {10}^{-1}$	Slow
F9	1	Optimal
F10	0	No Improvement
F11	$2.13\times {10}^{-1}$	Slow
F12	$7.60\times {10}^{-1}$	Good
F13	1	Optimal
F14	1	Optimal
F15	1	Optimal
F16	1	Optimal
F17	1	Optimal
F18	1	Optimal
F19	1	Optimal
F20	1	Optimal

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

,

Table A8. ACR results for PSO.

Function	ACR	Category
F1	1	Optimal
F2	1	Optimal
F3	1	Optimal
F4	$4.52\times {10}^{-1}$	Slow
F5	1	Optimal
F6	1	Optimal
F7	$1.66\times {10}^{-1}$	Slow
F8	1	Optimal
F9	$3.79\times {10}^{-1}$	Slow
F10	$5.75\times {10}^{-1}$	Good
F11	$5.86\times {10}^{-1}$	Good
F12	0	No Improvement
F13	1	Optimal
F14	$3.94\times {10}^{-1}$	Slow
F15	1	Optimal
F16	$9.38\times {10}^{-1}$	Excellent
F17	1	Optimal
F18	1	Optimal
F19	$3.08\times {10}^{-1}$	Slow
F20	0	No Improvement

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

,

Table A9. ACR results for EMA.

Function	ACR	Category
F1	1	Optimal
F2	$4.57\times {10}^{-1}$	Slow
F3	$6.73\times {10}^{-1}$	Good
F4	$7.06\times {10}^{-1}$	Good
F5	$6.73\times {10}^{-1}$	Good
F6	1	Optimal
F7	$2.18\times {10}^{-1}$	Slow
F8	1	Optimal
F9	0	No Improvement
F10	$4.46\times {10}^{-1}$	Slow
F11	1	Optimal
F12	$1.72\times {10}^{-1}$	Slow
F13	$4.61\times {10}^{-1}$	Slow
F14	1	Optimal
F15	$3.45\times {10}^{-1}$	Slow
F16	1	Optimal
F17	$6.92\times {10}^{-1}$	Good
F18	1	Optimal
F19	$3.75\times {10}^{-1}$	Slow
F20	$6.18\times {10}^{-1}$	Good

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

,

Table A10. ACR results for SSA.

Function	ACR	Category
F1	1	Optimal
F2	1	Optimal
F3	$7.28\times {10}^{-1}$	Good
F4	$6.50\times {10}^{-1}$	Good
F5	1	Optimal
F6	$4.53\times {10}^{-1}$	Slow
F7	$3.52\times {10}^{-1}$	Slow
F8	$9.66\times {10}^{-1}$	Excellent
F9	$2.60\times {10}^{-1}$	Slow
F10	$5.02\times {10}^{-1}$	Good
F11	$3.38\times {10}^{-1}$	Slow
F12	1	Optimal
F13	1	Optimal
F14	$2.45\times {10}^{-1}$	Slow
F15	1	Optimal
F16	1	Optimal
F17	$6.97\times {10}^{-1}$	Good
F18	1	Optimal
F19	1	Optimal
F20	$4.54\times {10}^{-1}$	Slow

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

,

Table A11. ACR results for ALO.

Function	ACR	Category
F1	$7.23\times {10}^{-1}$	Good
F2	$6.75\times {10}^{-1}$	Good
F3	$8.35\times {10}^{-1}$	Good
F4	1	Optimal
F5	1	Optimal
F6	1	Optimal
F7	$4.83\times {10}^{-1}$	Slow
F8	$5.02\times {10}^{-1}$	Good
F9	$6.02\times {10}^{-1}$	Good
F10	1	Optimal
F11	$8.38\times {10}^{-1}$	Good
F12	1	Optimal
F13	$1.67\times {10}^{-1}$	Slow
F14	1	Optimal
F15	$6.16\times {10}^{-1}$	Good
F16	$9.54\times {10}^{-1}$	Excellent
F17	$7.06\times {10}^{-1}$	Good
F18	1	Optimal
F19	0	No Improvement
F20	1	Optimal

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

,

Table A12. ACR results for BO.

Function	ACR	Category
F1	1	Optimal
F2	1	Optimal
F3	$5.22\times {10}^{-1}$	Good
F4	$4.08\times {10}^{-1}$	Slow
F5	$5.76\times {10}^{-1}$	Good
F6	1	Optimal
F7	0	No Improvement
F8	$9.59\times {10}^{-1}$	Excellent
F9	0	No Improvement
F10	$9.57\times {10}^{-1}$	Excellent
F11	$4.57\times {10}^{-1}$	Slow
F12	0	No Improvement
F13	$5.41\times {10}^{-1}$	Good
F14	1	Optimal
F15	$4.58\times {10}^{-1}$	Slow
F16	1	Optimal
F17	$7.32\times {10}^{-1}$	Good
F18	1	Optimal
F19	0	No Improvement
F20	$8.65\times {10}^{-1}$	Good

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

,

Table A13. ACR results for MAO.

Function	ACR	Category
F1	1	Optimal
F2	$6.37\times {10}^{-1}$	Good
F3	$6.40\times {10}^{-1}$	Good
F4	$5.83\times {10}^{-1}$	Good
F5	1	Optimal
F6	1	Optimal
F7	$4.95\times {10}^{-1}$	Slow
F8	1	Optimal
F9	1	Optimal
F10	1	Optimal
F11	1	Optimal
F12	1	Optimal
F13	$4.48\times {10}^{-1}$	Slow
F14	$7.61\times {10}^{-1}$	Good
F15	$6.13\times {10}^{-1}$	Good
F16	$2.95\times {10}^{-1}$	Slow
F17	$3.93\times {10}^{-1}$	Slow
F18	1	Optimal
F19	1	Optimal
F20	1	Optimal

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

,

Table A14. ACR results for DE.

Function	ACR	Category
F1	1	Optimal
F2	$4.44\times {10}^{-1}$	Slow
F3	$4.09\times {10}^{-1}$	Slow
F4	$2.81\times {10}^{-1}$	Slow
F5	$4.39\times {10}^{-1}$	Slow
F6	1	Optimal
F7	$4.67\times {10}^{-1}$	Slow
F8	$4.06\times {10}^{-1}$	Slow
F9	$3.98\times {10}^{-1}$	Slow
F10	1	Optimal
F11	$3.83\times {10}^{-1}$	Slow
F12	$8.17\times {10}^{-1}$	Good
F13	$5.94\times {10}^{-1}$	Good
F14	$3.41\times {10}^{-1}$	Slow
F15	$7.38\times {10}^{-1}$	Good
F16	$7.03\times {10}^{-1}$	Good
F17	$4.64\times {10}^{-1}$	Slow
F18	1	Optimal
F19	1	Optimal
F20	$4.97\times {10}^{-1}$	Slow

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

summarize the results obtained for the eleven algorithms on the 20 testbench functions, and the corresponding ACR plots are shown in Figure A1. Additionally,

Table 22. Average ACR values obtained by each algorithm.

Algorithm	Average ACR Value	Category
TGCOA	$9.71\times {10}^{-1}$	Excellent
JSOA	1	Optimal
BWOA	$9.27\times {10}^{-1}$	Excellent
LCA	$7.20\times {10}^{-1}$	Good
PSO	$6.90\times {10}^{-1}$	Good
EMA	$6.42\times {10}^{-1}$	Good
SSA	$7.32\times {10}^{-1}$	Good
ALO	$7.55\times {10}^{-1}$	Good
BO	$6.24\times {10}^{-1}$	Good
MAO	$7.93\times {10}^{-1}$	Good
DE	$6.19\times {10}^{-1}$	Good

The category is assigned based on the ACR value: Optimal (ACR = 1), Excellent (ACR ≥ 0.9), Good ($0.5\le$ ACR $<0.9$), Slow ($0<$ ACR $<0.5$), or No Improvement (ACR = 0).

reveals the ACR mean values computed for all the algorithms. It is important to note that the TGCOA has an ACR value greater than or equal to 0.9, which qualifies it as excellent in terms of its ability to approach the optimal (near-optimal) solution during its iterative process.

5. Real-World Applications

The capability of the TGCOA to solve real-world problems with constraints was evaluated by solving seven problems, which are as follows: tension/compression spring design, pressure vessel design, three-bar truss design problem, design of gear train, tubular column design, welded beam design, and reinforced concrete beam design, taken from [97]. For all the engineering problems solved, the TGCOA was evaluated against the top three algorithms ranked according to the Friedman test; see Table 9.

Preparatory work for implementing the TGCOA to solve other engineering optimization problems is shown in Appendix A,

Table A15. Preparatory work for implementing TGCOA.

Step	Description
1.	Parameter definition: population size, number of iterations, and the upper and lower bounds for each variable.
2.	Provide objective function to minimize/maximize, as well as equality and inequality constraints of the problem.
3.	Define the handling constraint method to be applied.
4.	The algorithm starts by executing the script named main. The TGCOA’s MATLAB code is available to support this study’s findings.
5.	To obtain reliable results, it is recommended to run the algorithm at least 30 times using benchmark functions with known optimal values. Calculate the best value found, the mean, and the standard deviation to determine how close or far the achieved result is compared to the known optimal value. Additionally, apply non-parametric statistical tests to validate the results.

.

5.1. Constraint Handling

The TGCOA algorithm uses a penalty-based constraint handling technique, where infeasible solutions are penalized in their fitness value proportional to their average constraint violation. This directs the search toward feasible areas of the solution space during optimization. The mathematical formulation of this technique is described in Equation (15) and taken from [1].

$$F\left(\overrightarrow{x}\right)=\left\{\begin{array}{ccc}f\left(\overrightarrow{x}\right),& if& MCV\left(\overrightarrow{x}\right)\le 0\\ f_{max}+MCV\left(\overrightarrow{x}\right),& & otherwise.\end{array}\right.$$

(15)

where $f\left(\overrightarrow{x}\right)$ is the value of the fitness function for a feasible solution, whereas $f_{max}$ is the fitness function value of the worst solution in the population, and $MCV\left(\overrightarrow{x}\right)$ is the Mean Constraint Violation [1,21] represented in Equation (16).

$$MCV\left(\overrightarrow{x}\right)={\displaystyle \frac{{\displaystyle \sum _{i=1}^p}{G}_i(\overrightarrow{x})+{\displaystyle \sum _{j=1}^m}{H}_j(\overrightarrow{x})}{p+m}}$$

(16)

where $G_i\left(\overrightarrow{x}\right)$ and $H_j\left(\overrightarrow{x}\right)$ are the inequality and equality constraints, depicted by Equations (17) and (18), respectively. Note that these functions incur a penalty when they violate the constraints.

$$G_i\left(\overrightarrow{x}\right)=\left\{\begin{array}{ccc}0,& if& g_i\left(\overrightarrow{x}\right)\le 0\\ g_i\left(\overrightarrow{x}\right),& & otherwise.\end{array}\right.$$

(17)

$$H_j\left(\overrightarrow{x}\right)=\left\{\begin{array}{cc}0,\hfill & if|h_j\left(\overrightarrow{x}\right)|-\delta \le 0\hfill \\ |h_j\left(\overrightarrow{x}\right)|,\hfill & otherwise\hfill \end{array}\right.$$

(18)

It is important to mention that no penalty coefficient is used. Instead, the Mean Constraint Violation (MCV) method is applied, where, if a solution violates a constraint, its objective function is penalized based on the fitness value of the worst solution ($f_{max}$) in the population, as shown in Equations (15) and (16).

5.2. Tension/Compression Spring Design

The tension/compression spring design problem, as described in [97,98] and illustrated in Figure 11, focuses on minimizing the weight of a spring under several constraints, including limits on deflection, shear stress, surge frequency, and outer diameter, as well as bounds on the design variables. These variables are the average coil diameter $D=x_1$, the wire diameter $d=x_2$, and the number of active coils $N=x_3$. The mathematical formulation is given below:

$$\begin{array}{cc}\hfill \mathrm{Minimize}:& f\left(X\right)=\left(x_3+2\right)x_2{x}_1^2\hfill \\ \hfill \mathrm{Subject}\mathrm{to}& g_1\left(X\right)=1-{\displaystyle \frac{x_2^3{x}_3}{71785x_1^4}}\le 0\hfill \\ \hfill & g_2\left(X\right)={\displaystyle \frac{4x_2^2-x_1{x}_2}{12566\left(x_2{x}_1^3-x_1^4\right)}}+{\displaystyle \frac{1}{5108x_1^2}}-1\le 0\hfill \\ \hfill & g_3\left(X\right)=1-{\displaystyle \frac{140.45x_1}{x_2^2{x}_3}}\le 0\hfill \\ \hfill & g_4\left(X\right)={\displaystyle \frac{x_1+x_2}{1.5}}-1\le 0\hfill \\ \hfill \mathrm{Variable}\mathrm{range}:& 0.05\le x_1\le 2\hfill \\ \hfill & 0.25\le x_2\le 1.3\hfill \\ \hfill & 2\le x_3\le 15\hfill \end{array}$$

(19)

The statistical results of this test, presented in

Table 23. Best results of tension/compression spring design.

Variables	TGCOA	BWOA	JSOA	LCA
$x_1$	$5.17\times {10}^{-2}$	$5.17\times {10}^{-2}$	$5.16\times {10}^{-2}$	$6.13\times {10}^{-2}$
$x_2$	$3.57\times {10}^{-1}$	$3.57\times {10}^{-1}$	$3.54\times {10}^{-1}$	$5.50\times {10}^{-1}$
$x_3$	$1.12\times {10}^1$	$1.13\times {10}^1$	$1.15\times {10}^1$	$8.16\times {10}^0$
$g_1\left(X\right)$	$-2.1\times {10}^{-5}$	$-5\times {10}^{-6}$	$-9\times {10}^{-6}$	$-3.39\times {10}^{-1}$
$g_2\left(X\right)$	0	$-1.5\times {10}^{-5}$	$-3\times {10}^{-6}$	$-1.17\times {10}^{-1}$
$g_3\left(X\right)$	$-4.06\times {10}^0$	$-4.05\times {10}^0$	$-4.05\times {10}^0$	$-2.48\times {10}^0$
$g_4\left(X\right)$	$-7.27\times {10}^{-1}$	$-7.27\times {10}^{-1}$	$-7.30\times {10}^{-1}$	$-5.92\times {10}^{-1}$
$f\left(X\right)$	$1.27\times {10}^{-2}$	$1.27\times {10}^{-2}$	$1.27\times {10}^{-2}$	$2.10\times {10}^{-2}$

, show that all algorithms reported feasible solutions, highlighting that the TGCOA outperforms the algorithms with which it was compared, obtaining a value of 0.012665. The convergence graph is shown in Figure 12.

5.3. Pressure Vessel Design

The optimization problem addresses a cylindrical pressure vessel with hemispherical caps at both ends, as shown in Figure 13. The goal is to minimize the total manufacturing cost, considering material, forming, and welding expenses. The design depends on four decision variables: $x_1$, the thickness of the cylindrical body ($T_3$); $x_2$, the thickness of the hemispherical caps ($T_h$); $x_3$, the inner radius (R); and $x_4$, the length of the cylindrical section excluding the caps (L) [97]. The variables $x_1$ and $x_2$ must be integer multiples of 0.0625 inches, while $x_3$ and $x_4$ are continuous.

$$\begin{array}{cc}\hfill \mathrm{Minimize}:& f\left(X\right)=0.6224x_1{x}_3{x}_4+1.7781x_2{x}_3^2+3.1661x_1^2{x}_4+19.84x_1^2{x}_3,\hfill \\ \hfill \mathrm{Subject}\mathrm{to}& g_1\left(X\right)=-x_1+0.0193x_3\le 0\hfill \\ \hfill & g_2\left(X\right)=-x_2+0.00954x_3\le 0\hfill \\ \hfill & g_3\left(X\right)=-\pi x_3^2{x}_4-{\displaystyle \frac{4}{3}}\pi x_3^3+1,296,000\le 0\hfill \\ \hfill & g_4\left(X\right)=x_4-240\le 0\hfill \\ \hfill \mathrm{Variable}\mathrm{range}:& x_1,x_2\in \{n\times 0.0625\mid n=1,2,\dots ,1600\}\hfill \\ \hfill & x_3\ge 10,x_4\le 200.\hfill \end{array}$$

(20)

As shown in

Table 24. Best results of pressure vessel design.

Variables	TGCOA	BWOA	JSOA	LCA
$x_1$	$1.28\times {10}^1$	$1.25\times {10}^1$	$1.43\times {10}^1$	$2.61\times {10}^1$
$x_2$	$7.25\times {10}^0$	$6.50\times {10}^0$	$6.99\times {10}^0$	$2.45\times {10}^1$
$x_3$	$4.21\times {10}^1$	$4.20\times {10}^1$	$4.53\times {10}^1$	$5.71\times {10}^1$
$x_4$	$1.77\times {10}^2$	$1.77\times {10}^2$	$1.40\times {10}^2$	$5.25\times {10}^1$
$g_1$	$-1.20\times {10}^1$	$-1.17\times {10}^1$	$-1.34\times {10}^1$	$-2.50\times {10}^1$
$g_2$	$-6.85\times {10}^0$	$-6.10\times {10}^0$	$-6.56\times {10}^0$	$-2.39\times {10}^1$
$g_3$	$-1.36\times {10}^3$	$-4.06\times {10}^0$	$-7.71\times {10}^{-1}$	$-1.96\times {10}^4$
$g_4$	$-6.31\times {10}^1$	$-6.27\times {10}^1$	$-9.97\times {10}^1$	$-1.87\times {10}^2$
$f\left(X\right)$	$6.07\times {10}^3$	$6.07\times {10}^3$	$6.09\times {10}^3$	$1.51\times {10}^4$

, all algorithms reported feasible solutions. Note that the TGCOA outperforms the algorithms with which it was compared, obtaining a value of 6065.4263 when minimizing the objective function. Figure 14 displays the corresponding convergence graph.

5.4. Three-Bar Truss Design Problem

The objective is to minimize the total volume of the structure subjected to static loads while respecting the stress constraints ($\sigma$) in each of its elements; see Figure 15. The design variables are the cross-sectional areas $A_1=x_1$ and $A_2=x_2$, whose optimum configuration is sought by means of the corresponding mathematical formulation [97].

$$\begin{array}{cc}\hfill \mathrm{Minimize}:& f\left(X\right)=\left(2\sqrt{2}{x}_1+x_2\right)·l,\hfill \\ \hfill \mathrm{Subject}\mathrm{to}& g_1\left(X\right)={\displaystyle \frac{\sqrt{2}{x}_1+x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}}·P-\sigma \le 0,\hfill \\ \hfill & g_2\left(X\right)={\displaystyle \frac{x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}}·P-\sigma \le 0,\hfill \\ \hfill & g_3\left(X\right)={\displaystyle \frac{1}{\sqrt{2}{x}_2+x_1}}·P-\sigma \le 0,\hfill \\ \hfill & l=100\mathrm{cm},P=2\mathrm{kN}/{\mathrm{cm}}^3,\sigma =2\mathrm{kN}/{\mathrm{cm}}^3,\hfill \\ \hfill \mathrm{Variable}\mathrm{range}:& 0\le x_1,x_2\le 1.\hfill \end{array}$$

(21)

Table 25. Best results of three-bar truss design.

Variables	TGCOA	BWOA	JSOA	LCA
$x_1$	$7.89\times {10}^{-1}$	$7.89\times {10}^{-1}$	$7.89\times {10}^{-1}$	$7.66\times {10}^{-1}$
$x_2$	$4.08\times {10}^{-1}$	$4.08\times {10}^{-1}$	$4.08\times {10}^{-1}$	$4.81\times {10}^{-1}$
$g_1$	0	0	0	$-3.38\times {10}^{-3}$
$g_2$	$-1.46\times {10}^0$	$-1.46\times {10}^0$	$-1.46\times {10}^0$	$-1.39\times {10}^0$
$g_3$	$-5.36\times {10}^{-1}$	$-5.36\times {10}^{-1}$	$-5.36\times {10}^{-1}$	$-6.17\times {10}^{-1}$
$f\left(X\right)$	$2.64\times {10}^2$	$2.64\times {10}^2$	$2.64\times {10}^2$	$2.65\times {10}^2$

compares the TGCOA, BWOA, JSOA, and LCA, showing that all algorithms reported feasible solutions. The TGCOA demonstrated competitive performance to BWOA and JSOA. Meanwhile, it outperformed LCA by achieving a value of 263.8958. The convergence graph is shown in Figure 16.

5.5. Design of Gear Train

The gear train design problem taken from [97,99] is a classic case of discrete optimization in mechanical engineering. The objective is to minimize the gear ratio, defined as the ratio of the angular velocity of the output shaft to that of the input shaft. The design variables are the numbers of gear teeth $n_A=x_1$, $n_B=x_2$, $n_C=x_3$, and $n_D=x_4$. Figure 17 shows the two-dimensional model associated with this problem. The mathematical formulation is presented below.

$$\begin{array}{cc}\hfill \mathrm{Minimize}:& f\left(X\right)={\left({\displaystyle \frac{1}{6.931}}-{\displaystyle \frac{x_3{x}_2}{x_1{x}_4}}\right)}^2,\hfill \\ \hfill \mathrm{Variable}\mathrm{range}:& x_1,x_2,x_3,x_4\in \{12,13,14,\dots ,60\}.\hfill \end{array}$$

(22)

The statistical results of this test, presented in

Table 26. Best results of the gear train design.

Variables	TGCOA	BWOA	JSOA	LCA
$x_1$	$4.26\times {10}^1$	$4.34\times {10}^1$	$4.28\times {10}^1$	$4.48\times {10}^1$
$x_2$	$1.64\times {10}^1$	$1.58\times {10}^1$	$1.61\times {10}^1$	$1.57\times {10}^1$
$x_3$	$1.93\times {10}^1$	$1.94\times {10}^1$	$1.90\times {10}^1$	$1.99\times {10}^1$
$x_4$	$4.90\times {10}^1$	$4.91\times {10}^1$	$4.88\times {10}^1$	$4.87\times {10}^1$
$f\left(X\right)$	$2.70\times {10}^{-12}$	$2.70\times {10}^{-12}$	$2.70\times {10}^{-12}$	$1.00\times {10}^{-6}$

, show that all algorithms reported feasible solutions. Although the TGCOA outperformed LCA by attaining a value of $2.70\times {10}^{-12}$, it also demonstrated competitive performance compared to BWOA and JSOA. Figure 18 shows the convergence graph.

5.6. Tubular Column Design

This optimization issue focuses on designing a uniform tubular column under axial compressive pressure to minimize overall material cost while maintaining structural integrity [97]. Figure 19 depicts the design variables: mean diameter $d=x_1$ and pipe thickness $t=x_2$. The column material has a yield stress of ${\sigma }_y=500{\mathrm{kgf}/\mathrm{cm}}^2$ and a modulus of elasticity $E=0.85\times {10}^6{\mathrm{kgf}/\mathrm{cm}}^2$. The mathematical formulation of the problem includes constraints that ensure that the applied stress does not exceed the buckling and creep limits ($g_1$ and $g_2$), while additional constraints ($g_3$ to $g_6$) limit the design variables to feasible ranges. As shown below,

$$\begin{array}{cc}\hfill \mathrm{Minimize}:& f\left(X\right)=9.8x_1{x}_2+2x_1,\hfill \\ \hfill \mathrm{Subject}\mathrm{to}& g_1\left(X\right)={\displaystyle \frac{P}{\pi x_1{x}_2{\sigma }_y}}-1\le 0,\hfill \\ \hfill & g_2\left(X\right)={\displaystyle \frac{8P{L}^2}{{\pi }^{3}E{x}_1{x}_2(x_1^2+x_2^2)}}-1\le 0,\hfill \\ \hfill & g_3\left(X\right)={\displaystyle \frac{2.0}{x_1}}-1\le 0,\hfill \\ \hfill & g_4\left(X\right)={\displaystyle \frac{x_1}{14}}-1\le 0,\hfill \\ \hfill & g_5\left(X\right)={\displaystyle \frac{0.2}{x_2}}-1\le 0,\hfill \\ \hfill & g_6\left(X\right)={\displaystyle \frac{x_2}{8}}-1\le 0,\hfill \\ \hfill \mathrm{Variable}\mathrm{range}:& 2\le x_1\le 14,0.2\le x_2\le 0.8.\hfill \end{array}$$

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Table 27. Best results of the tubular design problem.

Variables	TGCOA	BWOA	JSOA	LCA
$x_1$	$5.45\times {10}^0$	$5.45\times {10}^0$	$5.45\times {10}^0$	$5.42\times {10}^0$
$x_2$	$2.92\times {10}^{-1}$	$2.92\times {10}^{-1}$	$2.92\times {10}^{-1}$	$2.97\times {10}^{-1}$
$g_1$	0	$-2.00\times {10}^{-6}$	0	$-1.98\times {10}^{-2}$
$g_2$	$-2.60\times {10}^{-5}$	$-9.50\times {10}^{-5}$	$-7.00\times {10}^{-6}$	$-2.94\times {10}^{-2}$
$g_3$	$-6.33\times {10}^{-1}$	$-6.33\times {10}^{-1}$	$-6.33\times {10}^{-1}$	$-6.31\times {10}^{-1}$
$g_4$	$-6.11\times {10}^{-1}$	$-6.11\times {10}^{-1}$	$-6.11\times {10}^{-1}$	$-6.13\times {10}^{-1}$
$g_5$	$-6.33\times {10}^{-1}$	$-6.33\times {10}^{-1}$	$-6.33\times {10}^{-1}$	$-6.31\times {10}^{-1}$
$g_6$	$-3.18\times {10}^{-1}$	$-3.18\times {10}^{-1}$	$-3.18\times {10}^{-1}$	$-3.23\times {10}^{-1}$
$f\left(x\right)$	$2.65\times {10}^1$	$2.65\times {10}^1$	$2.65\times {10}^1$	$2.66\times {10}^1$

shows that all algorithms reported feasible solutions. By attaining a value of 26.4864, the TGCOA outperformed LCA and demonstrated competitive performance for BWOA and JSOA. The convergence graph is shown in Figure 20.

5.7. Welded Beam Design

The design problem of a welded beam is a reference case initially taken from [97,100] and widely addressed in the optimization literature. As illustrated in Figure 21, the beam is subjected to a vertical load, and the main objective is to minimize the total fabrication cost. This cost considers both materials and welding processes. The model is subject to seven constraints related to stress, deflection, welding design criteria, and geometric constraints. The design variables involved are weld thickness $x_1$, beam height $x_2$, plate length $x_3$, and cross bar thickness $x_4$.

$$\begin{array}{cc}\hfill \mathrm{Minimize}:& f\left(X\right)=1.10471x_1^2{x}_2+0.04811x_3{x}_4\left(14.0+x_2\right),\hfill \\ \hfill \mathrm{Subject}\mathrm{to}& g_1\left(X\right)=\tau \left(X\right)-{\tau }_{max}\le 0,\hfill \\ \hfill & g_2\left(X\right)=\sigma \left(X\right)-{\sigma }_{max}\le 0,\hfill \\ \hfill & g_3\left(X\right)=\delta \left(X\right)-{\delta }_{max}\le 0,\hfill \\ \hfill & g_4\left(X\right)=x_1-x_4\le 0,\hfill \\ \hfill & g_5\left(X\right)=P-P_c\left(X\right)\le 0,\hfill \\ \hfill & g_6\left(X\right)=0.125-x_1\le 0,\hfill \\ \hfill & g_7\left(X\right)=1.10471x_1^2+0.04811x_3{x}_4(14.0+x_2)-5.0\le 0.\hfill \end{array}$$

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where

$$\begin{array}{cc}& \tau \left(X\right)=\sqrt{{\left({\tau }^{\prime }\right)}^2+2{\tau }^{\prime }{\tau }^{\prime \prime }{\displaystyle \frac{x_2}{2R}}+{\left({\tau }^{\prime \prime }\right)}^2}\hfill \\ & {\tau }^{\prime }={\displaystyle \frac{P}{\sqrt{2}{x}_1{x}_2}},{\tau }^{\prime \prime }={\displaystyle \frac{MR}{J}},M=P\left(L+{\displaystyle \frac{x_2}{2}}\right)\hfill \\ & R=\sqrt{{\displaystyle \frac{x_2^2}{4}}+{\left({\displaystyle \frac{x_1+x_3}{2}}\right)}^2}\hfill \\ & J=2\left[\sqrt{2}{x}_1{x}_2\left({\displaystyle \frac{x_2^2}{4}}+{\left({\displaystyle \frac{x_1+x_3}{2}}\right)}^2\right)\right]\hfill \\ & \sigma \left(X\right)={\displaystyle \frac{6PL}{x_4{x}_3^2}},\delta \left(X\right)={\displaystyle \frac{6P{L}^3}{E{x}_3^2{x}_4}}\hfill \\ & P_c\left(X\right)={\displaystyle \frac{4.013E\sqrt{\frac{x_3^2{x}_4^6}{36}}}{L^2}}\left(1-{\displaystyle \frac{x_3}{2L}}\sqrt{{\displaystyle \frac{E}{4G}}}\right)\hfill \end{array}$$

Constants:

$$\begin{array}{cc}& P=6000\mathrm{lb},L=14\mathrm{in},{\delta }_{max}=0.25\mathrm{in}\hfill \\ & E=30\times {10}^6\mathrm{psi},G=12\times {10}^6\mathrm{psi}\hfill \\ & {\tau }_{max}=13,600\mathrm{psi},{\sigma }_{max}=30,000\mathrm{psi}\hfill \end{array}$$

Variable bounds: $0.1\le x_1,x_4\le 2,0.1\le x_2,x_3\le 10$.

Figure 21. Welded beam construction showing design variations.

Figure 21. Welded beam construction showing design variations.

The statistical results of this test, presented in

Table 28. Results obtained from welded beam design.

Variables	TGCOA	BWOA	JSOA	LCA
$x_1$	$2.02\times {10}^{-1}$	$1.99\times {10}^{-1}$	$1.93\times {10}^{-1}$	$2.16\times {10}^{-1}$
$x_2$	$3.60\times {10}^0$	$3.60\times {10}^0$	$3.77\times {10}^0$	$4.71\times {10}^0$
$x_3$	$9.04\times {10}^0$	$9.09\times {10}^0$	$9.04\times {10}^0$	$9.24\times {10}^0$
$x_4$	$2.06\times {10}^{-1}$	$2.06\times {10}^{-1}$	$2.06\times {10}^{-1}$	$3.28\times {10}^{-1}$
$g_1$	$-1.48\times {10}^2$	$-2.71\times {10}^0$	$-4.26\times {10}^0$	$-3.59\times {10}^{-1}$
$g_2$	0	$-3.00\times {10}^2$	$-6.94\times {10}^{-1}$	$-1.20\times {10}^0$
$g_3$	$-3.57\times {10}^{-3}$	$-6.60\times {10}^{-3}$	$-1.27\times {10}^{-2}$	$-1.12\times {10}^{-5}$
$g_4$	$-3.38\times {10}^0$	$-3.37\times {10}^0$	$-3.37\times {10}^0$	$-2.22\times {10}^{-4}$
$g_5$	$-7.72\times {10}^{-2}$	$-7.39\times {10}^{-2}$	$-6.80\times {10}^{-2}$	$-9.12\times {10}^{-6}$
$g_6$	$-2.36\times {10}^{-1}$	$-2.36\times {10}^{-1}$	$-2.36\times {10}^{-1}$	$-2.42\times {10}^{-5}$
$g_7$	$-4.16\times {10}^0$	$-2.55\times {10}^0$	$-9.88\times {10}^{-2}$	$-1.87\times {10}^0$
$f\left(x\right)$	$1.74\times {10}^0$	$1.74\times {10}^0$	$1.74\times {10}^0$	$2.97\times {10}^0$

, show that all algorithms reported feasible solutions, highlighting that the TGCOA outperforms the algorithms with which it was compared, obtaining a value of 1.7367. Figure 22 displays the corresponding convergence graph.

5.8. Reinforced Concrete Beam Design

The optimization problem for the design of a reinforced concrete beam, taken from [97], is illustrated in Figure 23. A simply supported beam with a span of 9 m is considered, subjected to a live load of 900 kgf and a dead load of 450 kgf, including the self-weight of the beam. The compressive strength of the concrete is 5 ksi, while the yield strength of the reinforcing steel is 50 ksi. The cost of the concrete is estimated at USD 0.02/in²/linear ft, and that of the steel at USD 1.00/in ²/linear ft.

The objective is to minimize the total cost of the structure by determining the reinforcement area $A_s(=x_1)$, beam width $b(=x_2)$, and beam depth $h(=x_3)$. The design must comply with ACI 318-77 code specifications as follows: $M_u=0.9A_s{\sigma }_y\left(0.8h\right)\left(1.0-0.59{\displaystyle \frac{A_s{\sigma }_y}{0.8bh{\sigma }_c}}\right)\ge 1.4M_d+1.7M_l$

In this context, $M_u$, $M_a$, and $M_1$ represent the bending resistance, the moment due to dead load, and the moment due to live load, respectively, with values of $M_a=1350$ in-kip and $M_1=2700$ in-kip. In addition, it is imposed that the ratio between the edge and the width of the beam does not exceed the value of 4, ${\displaystyle \frac{h}{b}}\le 4$. The mathematical formulation of this optimization problem is presented below.

$$\begin{array}{cc}\hfill \mathrm{Minimize}:& f\left(X\right)=2.9x_1+0.6x_2{x}_3,\hfill \\ \hfill \mathrm{Subject}\mathrm{to}& g_1\left(X\right)={\displaystyle \frac{x_2}{x_3}}-4\le 0,\hfill \\ \hfill & g_2\left(X\right)=180+7.375{\displaystyle \frac{x_1^2}{x_3}}-x_1{x}_2\le 0,\hfill \\ \hfill \mathrm{Variable}\mathrm{range}:& x_1\in \{6,6.16,6.32,6.6,7,7.11,7.2,7.8,7.9,8,8.4\},\hfill \\ \hfill & x_2\in \{28,29,30,\dots ,40\},\hfill \\ \hfill & 5\le x_3\le 10.\hfill \end{array}$$

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As shown in

Table 29. Results obtained from reinforced concrete beam design.

Variables	TGCOA	BWOA	JSOA	LCA
$x_1$	$1.89\times {10}^{-1}$	$2.56\times {10}^{-1}$	$2.27\times {10}^{-1}$	$2.06\times {10}^{-1}$
$x_2$	$4.74\times {10}^{-1}$	$4.84\times {10}^{-1}$	$4.72\times {10}^{-1}$	$4.74\times {10}^{-1}$
$x_3$	$8.50\times {10}^0$	$8.50\times {10}^0$	$8.50\times {10}^0$	$8.58\times {10}^0$
$g_1$	0	0	0	$-3.71\times {10}^{-2}$
$g_2$	$-2.24\times {10}^{-1}$	$-2.24\times {10}^{-1}$	$-2.24\times {10}^{-1}$	$-5.45\times {10}^{-1}$
$f\left(x\right)$	$3.59\times {10}^2$	$3.59\times {10}^2$	$3.59\times {10}^2$	$3.61\times {10}^2$

, all algorithms reported feasible solutions. The TGCOA demonstrated competitive performance to BWOA and JSOA. Moreover, it surpassed LCA by attaining a value of 359.208. The convergence graph is shown in Figure 24.

6. Conclusions

This paper introduces a novel metaheuristic named the Tetragonula carbonaria Optimization Algorithm (TGCOA), inspired by the thermoregulatory and building behaviors of the stingless bee. The proposed approach models three behavior patterns based on hive temperature: The strategies involve strengthening the structure’s hive when it is cold, building combs in a spiral pattern at medium temperatures, and stabilizing the hive when it is hot. These temperature-dependent strategies are mathematically integrated into the algorithm to balance global exploration and local exploitation in the solution space. The algorithm’s performance was evaluated using twenty standard testbench functions, comprising ten unimodal and ten multimodal benchmarks. Additionally, twenty-eight constrained problems taken from the IEEE CEC 2017 “Constrained Real-Parameter Optimization" were used in four dimensions, 10, 30, 50, and 100, as well as seven real-world engineering design problems. The algorithm’s performance was compared against ten state-of-the-art bio-inspired algorithms: the BWOA, JSOA, LCA, EMA, BO, SSA, ALO, MAO, PSO, and DE. The Wilcoxon signed-rank and Friedman statistical tests were used to compare performance over all the algorithms.

The outcomes and conclusions of this work are summarized below:

• The TGCOA operates without additional parameters, using only a fixed population size of 30 and only 200 iterations. Note that a low number of iterations contributes to the algorithm’s efficiency compared to several approaches reported in the literature that use more iterations.
• According to the Wilcoxon rank-sum test results on the 20 benchmark functions, the TGCOA significantly outperformed several algorithms, including the EMA, LCA, BO, SSA, ALO, MAO, PSO, and DE. Meanwhile, no significant differences were observed compared to the BWOA and JSOA. Furthermore, the Friedman test results showed that the TGCOA was ranked first.
• For 10-dimensional problems, the Wilcoxon rank-sum test shows that the TGCOA outperformed the BWOA and LCA, with no significant differences found when compared to the JSOA. Additionally, the Friedman test ranked the TGCOA in third place. In 30-dimensional scenarios, the Wilcoxon test indicates that the TGCOA outperformed the LCA, while no statistical differences were observed regarding the BWOA and JSOA. Nevertheless, the Friedman test positioned the TGCOA as the top-performing algorithm. When analyzing 50-dimensional problems, the Wilcoxon rank-sum test indicates that the TGCOA outperformed the BWOA and LCA, while no significant difference was found compared to the JSOA. Additionally, the Friedman test ranked the TGCOA first. At 100 dimensions, the TGCOA showed statistically superior results compared to all the other algorithms according to the Wilcoxon test. According to the Friedman test ranking, the TGCOA was placed at the top. Overall, this n-dimensional analysis demonstrates the TGCOA’s superior performance, particularly in higher-dimensional cases such as 30, 50, and 100 dimensions.
• The Average Convergence Rate (ACR) results show that the TGCOA was categorized as excellent in the average speed at which it approaches an optimal solution (near-optimal) during its iterative process.
• The TGCOA demonstrated competitive performance regarding engineering design problems, including gear train design, tubular column design, and reinforced concrete beam design. Meanwhile, it outperformed problems in tension/compression spring design, pressure vessel design, three-bar truss design, and welded beam design.
• The TGCOA demonstrates the ability to effectively address constrained real-world optimization problems characterized by unknown search spaces, delivering remarkable outcomes.

The proposed algorithm demonstrated competitive performance; however, certain limitations remain, revealing opportunities for future work, such as the ability to solve multi-objective optimization problems, including a higher number of variables, optimization problems in dynamic environments, and hybrid development with other techniques, such as hyper-heuristic-based environments for designing generic methods that produce solutions of acceptable quality, incorporating low-level heuristics that improve both exploration and exploitation in the search space. Furthermore, it is necessary to analyze how population size and number of iterations affect the performance of the algorithm for Large-Scale Global Optimization Problems (LSGOs). Currently, the TGCOA is being improved for hyperparameter optimization of convolutional neural networks in medical applications.