The Gaussian Mutational Barebone Dragonfly Algorithm: From Design to Analysis

Abstract

The dragonfly algorithm is a swarm intelligence optimization algorithm based on simulating the swarming behavior of dragonfly individuals. An efficient algorithm must have a symmetry of information between the participating entities. An improved dragonfly algorithm is proposed in this paper to further improve the global searching ability and the convergence speed of DA. The improved DA is named GGBDA, which adds Gaussian mutation and Gaussian barebone on the basis of DA. Gaussian mutation can randomly update the individual positions to avoid the algorithm falling into a local optimal solution. Gaussian barebone can quicken the convergent speed and strengthen local exploitation capacities. Enhancing algorithm efficiency relative to the symmetric concept is a critical challenge in the field of engineering design. To verify the superiorities of GGBDA, this paper sets 30 benchmark functions, which are taken from CEC2014 and 4 engineering design problems to compare GGBDA with other algorithms. The experimental result show that the Gaussian mutation and Gaussian barebone can effectively improve the performance of DA. The proposed GGBDA, similar to the DA, presents improvements in global optimization competence, search accuracy, and convergence performance.

1. Introduction

The swarm intelligence optimization algorithm (SIOA) mainly simulates biological individuals’ group behavior, such as cooperation and competition, to obtain the optimal solution to complex problems. Moreover, SIOA has the benefit of an uncomplicated structure, few parameters, and uncomplicated implementations [1]. To date, varies of SIOA had been proposed by domestic and foreign scholars, namely the whale optimization algorithm (WOA) [2,3]; differential evolution [4] (DE); genetic algorithm (GA) [5]; ant colony optimization (ACO) [6,7]; particle swarm optimization (PSO) [8,9]; firefly algorithm (FA) [10]; fruit fly optimization algorithm (FOA) [11,12,13]; slime mould algorithm (SMA) [14]; moth flame optimization (MFO) [15,16,17]; grey wolf optimizer (GWO) [18,19]; bat algorithm (BA) [20,21]; grasshopper optimization algorithm (GOA) [22,23]; Harris hawks optimization (HHO) [24]; colony predation algorithm (CPA) [25]; hunger games search (HGS) [26]; Runge–Kutta optimizer (RUN) [27] and weighted mean of vectors (INFO) [28].

SIOA found its application in many fields, namely expensive optimization problems [29,30]; performance optimization [31]; object tracking [32,33]; multi-objective or many optimization problems [34,35,36]; traveling salesman problem [37]; neural network training [38]; scheduling problems [39]; big data optimization problems [40]; fault diagnosis of rolling bearings [41]; evolving deep convolutional neural networks [42]; gate resource allocation [43,44], and combination optimization problems [45]. The dragonfly algorithm (DA) is a population-based heuristic search algorithm that was first proposed by Mirjalili, S. [46] in 2015 and has since gained widespread adoption. It has a high level of performance and a broad range of applications in real life. Many applications, including parameter optimization [47], feature selection [48], load balancing [49], modeling [50], and others [51], have been effectively implemented using it. Many trials with complicated, high-dimensional, and multi-modal functions, on the other hand, demonstrated that DA had some drawbacks in some situations. For example, the DA lacks internal memory, has a poor convergence time, and is prone to falling into the local optimum when running in the background. As a result, several researchers are putting forth an attempt to increase the DA.

1.1. Related Works

When it comes to solving the challenge of numerical optimization, Sree Ranjini and colleagues [52] suggested a new memory-based hybrid DA (HMDA). The drawback of the DA was remedied by combining the advantages of the DA and the PSO together. Moreover, N. S. et al. [53] integrated the crow search algorithm (CSA) with the D-Crow optimization algorithm, presented a D-Crow optimization method, and applied this algorithm to optimize the configuration of virtual machines migrating. A method combining the dynamic analysis and the pattern search algorithm was presented by Khadanga and colleagues [54] to improve the performance and optimize the controller settings, in order to improve the control efficiency of the frequency of Microgrid. Using a trained multi-layer perceptron, Ghanem et al. [55] developed a novel hybridized metaheuristic method with improved properties in terms of attaining the best optimal value, convergence speed, avoiding local minima, and accuracy compared to previous algorithms. They created a hybrid algorithm by combining the artificial bee colony (ABC) algorithm with the distributed algorithm (DA). Shilaja and colleagues [56] used a combination of the enhanced grey wolf optimization and dynamic programming to handle the nonlinearity problems. Furthermore, it has been demonstrated to be more efficient than the conventional method. Using a dragonfly-based clustering method, CAVDO, Aadil et al. [57] proposed a solution to difficulties associated with the Internet of vehicles, such as scalability, dynamic topology changes, and finding the shortest path for routing. For the DA to be more random, Aci and colleagues [58] used the Brownian motion, which they found to be more effective. Furthermore, the results of the experiments revealed that the new DA had superior properties when compared to the old algorithm. Bao and colleagues [59] proposed a new DA that was changed using opposition-based learning. It also had a faster convergence time and a more balanced exploration–exploitation ratio, according to the results of the studies. Li et al. [60] improved the performance of DA by incorporating the adaptive learning factor and differential evolution (DE) approach into the algorithm. Sayed and colleagues [61] proposed a novel chaotic DA (CDA). In order to increase the DA, the researchers included chaotic maps in the searching iterations of the algorithm. Conforming to the experimental findings, CDA outperformed the control group in classification performance and was capable of identifying more suitable feature subsets.

Mafarja et al. [62] collected eight transfer functions (s-type function and v-type function) in BDA for evaluation, and proposed the time-varying s-type BDA, which made the algorithm have a high probability of changing the element position in the early optimization period, but with a low probability in the late optimization period. Hariharan et al. [63] proposed an improved binary dragonfly optimization algorithm (IBDFO) to solve the dimension problem and combined it with a feature extraction based on a wavelet packet to improve the accuracy of identifying the type of infant crying. Zhang et al. [64] used the DA to improve the prediction accuracy of the support vector machine (SVM) to obtain the optimal combination of parameters, and proposed the DA-SVM model to realize the short-term load prediction of the micro grid. Yuan et al. [65] tended to obtain an algorithm with better exploration capability as they combined the DA with the Coulomb force search strategy (CFSS). The resultant algorithm gained both a high accuracy and a remarkably improved convergence rate. Zhang et al. [66] quantized dragonfly behaviors to improve the search efficiency of the DA to obtain a quantized dragonfly algorithm (QDA). Furthermore, they put forward a new electric load forecasting model, based on the complete ensemble empirical mode decomposition adaptive noise, QDA, and support vector regression model, to accurately forecast the electric load. Suresh et al. [67] adopted the DA as the optimization algorithm to solve static economic dispatch incorporating solar energy. Based on the modified dragonfly algorithm (MDA) and bat search algorithm (BSA), Sureshkumar et al. [68] put forward a new method that adopted the MDABSA technique to control power flow more efficiently. In this method, MDA was used to develop the control signals of the voltage source. Xie et al. [69] adopted the DA to create a cancer classification algorithm. Furthermore, the comparative experiments proved it had a higher classification accuracy on cancer datasets. Xu et al. [70] adopted the DA and DE for color image segmentation. In this method, the DA was used for global search, and DE was used for local search.

1.2. Needs for Research

However, despite the fact that the literature discussed above made significant advances to the DA, it is not optimal enough to stabilize the algorithm’s exploration and exploitation capabilities. With the goal of further improving the exploration and exploitation exactness of the DA, as well as avoiding falling into the local optimum, this work proposes an upgraded DA that incorporates Gaussian mutation and Gaussian barebone to further improve these aspects. With the use of Gaussian mutation, we were able to update the dragonfly’s unique location while also improving the global search capabilities. Additionally, the Gaussian barebone was used to increase the local exploitation capabilities as well as the speed with which the searches could be conducted. The results of the simulations demonstrated that the algorithm’s accomplishments were superior to those of the original DA, and that its global optimization capabilities, search accuracy, and convergence performance were all greatly enhanced as a consequence. In summary, the innovations and contributions of this paper are as follows.

• An improved dragonfly algorithm (GGBDA) is proposed in this paper to further improve the global searching ability and the convergence speed of DA.
• GGBDA achieves a great improvement in the ability of exploitation and exploration.
• The performance of GGBDA is verified by comparison with some excellent algorithms.
• GGBDA is applied to optimize the engineering optimization problems.

The following is a summary of the rest of this article. Section 2 introduces the DA; Section 3 describes the enhanced DA based on Gaussian mutation and Gaussian barebone; Section 4 presents the experimental findings of the benchmark functions; and Section 5 concludes the paper and provides an overview of the previous work as well as a forecast for future work.

2. Materials and Methods

2.1. Dragonfly Algorithm (DA)

DA was inspired by two states of idealized behaviors of dragonflies in nature. There are three principles in the core mathematical backgrounds of this method.

Separation aims to prevent search individuals from collisions with others in a static state within a partial range. The following is the calculation function:

$$S_i=-{\displaystyle \sum _{j=1}^{N}X}-X_j$$

(1)

where X is the position agents, $X_j$ is j-th neighboring individual’s position, and N is neighboring individuals’ number.

Alignment is aimed at matching velocity between individuals within a partial range. The following is the calculation function:

$$A_i=\frac{{\displaystyle {\sum }_{j=1}^N{V}_j}}{N}$$

(2)

where $V_j$ is the j-th velocity of the neighboring individual.

Cohesion is aimed at making individuals move closer towards the center of swarm aggregation. The following is the calculation function:

$$C_i=\frac{{\displaystyle {\sum }_{j=1}^N{X}_j}}{N}-X$$

(3)

where X is the current individual’s position, N is neighborhoods’ number, and $X_j$ is j-th neighboring individual’s position.

The following is the attraction towards a food source:

$$F_i=X^{+}-X$$

(4)

where X is the current individual’s position and $X^{+}$ is he food source’s position.

The following is the distraction outwards an enemy source:

$$E_i=X^{-}+X$$

(5)

where X is the current individual’s position and $X^{-}$ is the enemy’s position.

Step ($\Delta X$) and position(X) are prerequisites to update and record the location of agents in the search domain. The step vector can be considered as the velocity vector in PSO. It is the direction of the agents’ motion. The following is the calculation function of the position vector:

$$\Delta X_{t+1}=s{S}_i+a{A}_i+c{C}_i+f{F}_i+e{E}_i+w{\Delta }_t$$

(6)

where $S_i$, $A_i$, $C_i$, $F_i$, $E_i$ indicates the separation, alignment, cohesion, food source and an enemy of the i-th individual’s position. s, a, c, f, e represent the weights, w is the inertia weight, i is the i-th individual, and t is the number of the current iteration. The following is the calculation function of the position vector:

$$X_{t+1}=X_t+\Delta X_{t+1}$$

(7)

Search agents have some deficiencies in terms of random behavior and exploration ability, and they also lack adjacent solutions. Therefore, Levy flight-based patterns are used to update the position of agents. The following is the function to update location:

$$X_{t+1}=X_t+Levy(d)\times X_t$$

(8)

where t is the current iteration number and d is the dimension of the position vector.

2.2. Gaussian Mutation

To improve the performance of DA, this paper used the Gaussian mutation to update the individual position of the dragonfly. Gaussian mutation has applied to many optimizers [3,16,71,72]. The following is the mutation function of the Gaussian mutation:

$$temp=X_j\ast (1+k)$$

(9)

where X is the position agents, temp is a temporary individual position, $X_j$ is j-th neighboring individual position, N is neighboring individuals’ number, and k is a random number between 0 and 1.

After updating the individual position of the dragonfly with this mutation function, whether the result of the Gaussian mutation is better than the previous result needs to be verified. If the temporary individual position can obtain a better result, it will be used as the new individual position of the dragonfly. With the population iterates, the DA may fall into local optimum. The Gaussian mutation has randomness, thereby quickening the scouting speed, avoiding slipping into the local optimum effectively, improving the global optimization capacity, and eventually obtaining the global optimum or a satisfactory solution.

2.3. Gaussian Barebone Mechanism

The speed of scouting for the optimal solution is a significant indicator of the performance of the algorithm. However, in the iteration, the scouting speed of the DA is dissatisfactory; thereby, this paper employed a Gaussian barebone to improve it. The Gaussian barebone mechanism hast been shown great potential in other optimizers [71,72]. The Gaussian barebone mechanism could help the DA scout the global optimum faster and more effectively by gathering individuals into a food source. There are two methods to gather individuals. The first method calculates the middle position between the food source and individual’s position and the distance between them. Then, it generates a random position where the values of each dimension are normally distributed based on the two calculated variables. The second method obtains the distances for each dimension of two random individuals. Additionally, it uses them and the position of the food source to calculate a new position. The following is the function:

$$V_{i,j}=\{\begin{array}{c}normal(mu+sigma),rand\left(\right)<CR\\ F{P}_j+k\ast (X_{k1,j}-X_{k2,j}),rand\left(\right)\ge CR\end{array}$$

(10)

where CR is a freely settable parameter; rand is a random number between 0 and 1; $V_{i,j}$ is a new temporary position; mu is the middle position between the food source’s position and $X_j$; sigma is the distance between the j-th dimension of the i-th neighboring individual and the j-th dimension of food source; the normrnd function generates random numbers that follow a normal distribution with the mu parameter representing the mean value and the sigma parameter representing standard deviation; $F{P}_j$ is the j-th dimension of food source; k is a random number; and $X_{k1,j}$ and $X_{k2,j}$ are j-th dimension of two random individuals in the population.

3. Proposed Method

The DA lacks internal memory, has a slow convergence speed, and quickly falls into the local optimum. As a result of these defects, this paper puts forward a new DA improved by Gaussian mutation and a Gaussian barebone named GGBDA. It uses the Gaussian barebone to gather individuals to food to quicken the speed of scouting the optimal solution and strengthen local exploitation capacities. It can update the individuals’ positions based on the position of the food source. However, Gaussian barebone could make the population fall into local optimums. Therefore, this paper employs the Gaussian mutation to improve the global search capacities, search accuracy, and convergence performance by preventing it from trapping into local optimums.

The Gaussian mutation is mainly used to randomly update individuals’ positions to escape the local optimums based on the Gaussian mutation function. The flowchart of the improved DA is shown in Figure 1. And The pseudocode of GGBDA is shown in Algorithm 1.

Algorithm 1. Pseudocode of GGBDA

Begin    Initialize the dragonflies’ population $X_i\left(i=1,2,\dots ,n\right)$    Initialize the step vectors $\Delta X_i\left(i=1,2,\dots ,n\right)$    while the end condition is not satisfied     Calculate the population fitness of all the dragonflies     Update the food source and enemy     Update w, s, a, c, f, and e     Calculate S, A, C, F, and E by Equations (1)–(5)     Update the neighboring radius     if a dragonfly has at least one neighboring dragonfly      Update the velocity and vector by Equation (6)      Update the position vector by Equation (7)     else      Update the position vector by Equation (8)     end if     Check and correct the new position according to the boundaries of the variables     Update with the Gaussian mutation and Gaussian barebone    end while  End

4. Experimental Results

In this part, the GGBDA was evaluated on CEC2014 benchmarks and practical engineering problems. To obtain unbiased results, all the experiments were carried out in the same environments, and the maximum number of iterations and the population size were set to 500 and 30, respectively. Each algorithm was run 30 times independently on each function to decrease the weight of unpredictability. Regarding the parameters that affect the algorithms involved in the comparison, we adopted the same values as in the original paper. In this paper, the average value and standard deviation of the experimental results of the optimization function were used to evaluate and analyze the potential of related technologies. To show the experimental result intuitively, the best values of each function are shown in bold.

4.1. Benchmark Functions

To compare the proposed algorithm and other algorithms, this experiment used 30 classical functions, including unimodal functions, multi-modal functions, hybrid functions, and composition functions.

These 30 functions are all taken from CEC2014 [73]. Thirty different types of benchmarks can more comprehensively estimate the performance of the proposed algorithm. The details of the thirty benchmarks are listed in

Table 1. Description of the 30 benchmark functions.

ID	Function Equation	Search Range	Optimum Value
CEC 2014 Unimodal Functions
F1	Rotated High Conditioned Elliptic Function	[−100,100]	${\mathrm{f}}_1\left\{X_{min}\right\}=100$
F2	Rotated Bent Cigar Function	[−100,100]	${\mathrm{f}}_2\left\{X_{min}\right\}=200$
F3	Rotated Discus Function	[−100,100]	${\mathrm{f}}_3\left\{X_{min}\right\}=300$
CEC 2014 Simple Multi-Modal Functions
F4	Shifted and Rotated Rosenbrock Function	[−100,100]	${\mathrm{f}}_4\left\{X_{min}\right\}=400$
F5	Shifted and Rotated Ackley Function	[−100,100]	${\mathrm{f}}_5\left\{X_{min}\right\}=500$
F6	Shifted and Rotated Weierstrass Function	[−100,100]	${\mathrm{f}}_6\left\{X_{min}\right\}=600$
F7	Shifted and Rotated Griewank Function	[−100,100]	${\mathrm{f}}_7\left\{X_{min}\right\}=700$
F8	Shifted Rastrigin Function	[−100,100]	${\mathrm{f}}_8\left\{X_{min}\right\}=800$
F9	Shifted and Rotated Rastrigin Function	[−100,100]	${\mathrm{f}}_9\left\{X_{min}\right\}=900$
F10	Shifted Schwefel Function	[−100,100]	${\mathrm{f}}_{10}\left\{X_{min}\right\}=1000$
F11	Shifted and Rotated Schwefel Function	[−100,100]	${\mathrm{f}}_{11}\left\{X_{min}\right\}=1100$
F12	Shifted and Rotated Katsuura Function	[−100,100]	${\mathrm{f}}_{12}\left\{X_{min}\right\}=1200$
F13	Shifted and Rotated HappyCat Function	[−100,100]	${\mathrm{f}}_{13}\left\{X_{min}\right\}=1300$
F14	Shifted and Rotated HGBat Function	[−100,100]	${\mathrm{f}}_{14}\left\{X_{min}\right\}=1400$
F15	Shifted and Rotated Expanded Griewank Plus Rosenbrock Function	[−100,100]	${\mathrm{f}}_{15}\left\{X_{min}\right\}=1500$
F16	Shifted and Rotated Expanded Scaffer F6 Function	[−100,100]	${\mathrm{f}}_{16}\left\{X_{min}\right\}=1600$
CEC 2014 Hybrid Functions
F17	Hybrid Function 1 (N = 3)	[−100,100]	${\mathrm{f}}_{17}\left\{X_{min}\right\}=1700$
F18	Hybrid Function 2 (N = 3)	[−100,100]	${\mathrm{f}}_{18}\left\{X_{min}\right\}=1800$
F19	Hybrid Function 3 (N = 4)	[−100,100]	${\mathrm{f}}_{19}\left\{X_{min}\right\}=1900$
F20	Hybrid Function 4 (N = 4)	[−100,100]	${\mathrm{f}}_{20}\left\{X_{min}\right\}=2000$
F21	Hybrid Function 5 (N = 5)	[−100,100]	${\mathrm{f}}_{21}\left\{X_{min}\right\}=2100$
F22	Hybrid Function 6 (N = 5)	[−100,100]	${\mathrm{f}}_{22}\left\{X_{min}\right\}=2200$
CEC 2014 Composition Functions
F23	Composition Function 1 (N = 5)	[−100,100]	${\mathrm{f}}_{23}\left\{X_{min}\right\}=2300$
F24	Composition Function 2 (N = 3)	[−100,100]	${\mathrm{f}}_{24}\left\{X_{min}\right\}=2400$
F25	Composition Function 3 (N = 3)	[−100,100]	${\mathrm{f}}_{25}\left\{X_{min}\right\}=2500$
F26	Composition Function 4 (N = 5)	[−100,100]	${\mathrm{f}}_{26}\left\{X_{min}\right\}=2600$
F27	Composition Function 5 (N = 5)	[−100,100]	${\mathrm{f}}_{27}\left\{X_{min}\right\}=2700$
F28	Composition Function 6 (N = 5)	[−100,100]	${\mathrm{f}}_{28}\left\{X_{min}\right\}=2800$
F29	Composition Function 7 (N = 3)	[−100,100]	${\mathrm{f}}_{29}\left\{X_{min}\right\}=2900$
F30	Composition Function 8 (N = 3)	[−100,100]	${\mathrm{f}}_{30}\left\{X_{min}\right\}=3000$

.

4.2. Comparison with Classical Algorithms

In order to validate the effectiveness of the improved GGBDA, there are some representative algorithms employed for comparison: OBSCA [74], m_SCA [75], SCADE [76], ASCA_PSO [77], ACWOA [78], MFO [15], SCA [79], FA [80], and DA.

In the experimental part, the parameter values of the compared algorithms were set, as shown in

Table 2. Parameter settings of the algorithms in the experiment.

Algorithms	Pop	Maximum Iterations	Others
GGBDA OBSCA m_SCA	30	1000	w ∈ [0.9 0.2]; s = 0.1; a = 0.1; c = 0.7; f = 1; e = 1
	30	1000	a = 2
	30	1000	a = 2
SCADE ASCA_PSO	30	1000	a = 2; CR = 0.8; LSF = 0.8; USF = 0.2
	30	1000	M = 4; N = 9; Vmax = 6; wMax = 0.9; wMin = 0.2; c1 = 2; c2 = 2;
ACWOA MFO SCA FA	30	1000	B = 1
	30	1000	B = 1
	30	1000	a = 2
	30	1000	alpha = 0.5; betamin = 0.2; gamma = 1;
DA GGBDA	30	1000	w ∈ [0.9 0.2]; s = 0.1; a = 0.1; c = 0.7; f = 1; e = 1
	30	1000	w ∈ [0.9 0.2]; s = 0.1; a = 0.1; c = 0.7; f = 1; e = 1

. To ensure the fairness of the experiments as far as possible, the experimental environment of algorithms stayed the same. The experimentations used 30D classical functions for comparing the proposed method and other rivals.

Table 3. Experimental results of the 30 dimensions (30Ds).

	F1		F2		F3
	Ave	Std	Ave	Std	Ave	Std
GGBDA	3.3428 × 107	2.3615 × 107	5.7799 × 107	1.32174 × 107	2.2502 × 103	1.0237 × 103
OBSCA	3.8095 × 108	1.2188 × 108	2.4577 × 1010	3.9982 × 109	5.1744 × 104	7.3043 × 103
m_SCA	7.2766 × 107	3.9039 × 107	6.4809 × 109	2.7501 × 109	2.6967 × 104	7.4237 × 103
SCADE	4.3235 × 108	1.0258 × 108	2.9383 × 1010	4.9065 × 109	5.3542 × 104	6.3130 × 103
ASCA_PSO	1.5733 × 107	7.8447 × 106	5.7234 × 108	7.6338 × 108	2.0200 × 104	5.3347 × 103
ACWOA	1.3598 × 108	5.9536 × 107	7.6372 × 109	3.3593 × 109	5.1123 × 104	8.7487 × 103
MFO	7.0131 × 107	8.4361 × 107	1.3759 × 1010	7.4030 × 109	9.8036 × 104	6.1005 × 104
SCA	2.2033 × 108	7.5726 × 107	1.6600 × 1010	3.2678 × 109	3.5442 × 104	6.2559 × 103
FA	2.5375 × 108	5.0283 × 107	1.5600 × 1010	2.0292 × 109	6.3396 × 104	9.7529 × 103
DA	8.11892 × 108	4.3376 × 108	2.2171 × 1010	2.2383 × 1010	5.9107 × 104	1.5850 × 104
	F4		F5		F6
	Ave	Std	Ave	Std	Ave	Std
GGBDA	5.9527 × 102	8.7643 × 101	5.2093 × 102	5.5872 × 10−2	6.2033 × 102	4.0175 × 100
OBSCA	2.4186 × 103	8.0598 × 102	5.2097 × 102	4.9147 × 10−2	6.3202 × 102	1.7351 × 100
m_SCA	7.5730 × 102	1.0198 × 102	5.2061 × 102	1.4096 × 10−1	6.2114 × 102	3.2807 × 100
SCADE	2.4370 × 103	5.6808 × 102	5.2094 × 102	6.3764 × 10−2	6.3419 × 102	2.3689 × 100
ASCA_PSO	5.7201 × 102	1.5123 × 102	5.2094 × 102	4.1898 × 10−2	6.2512 × 102	3.2965 × 100
ACWOA	1.0827 × 103	2.3891 × 102	5.2083 × 102	1.2246 × 10−1	6.3454 × 102	3.1803 × 100
MFO	1.4154 × 103	1.1476 × 103	5.2026 × 102	2.0197 × 10−1	6.2398 × 102	3.0738 × 100
SCA	1.4155 × 103	3.0882 × 102	5.2093 × 102	4.4333 × 10−2	6.3375 × 102	2.6530 × 100
FA	1.5386 × 103	1.7232 × 102	5.2095 × 102	5.1811 × 10−2	6.3392 × 102	6.4751 × 10−1
DA	7.2148 × 103	5.0944 × 103	5.2096 × 102	3.8523 × 10−2	6.3831 × 102	3.8669 × 100
	F7		F8		F9
	Ave	Std	Ave	Std	Ave	Std
GGBDA	7.0154 × 102	1.4093 × 10−1	8.8953 × 102	1.4755 × 101	1.0718 × 103	3.5377 × 101
OBSCA	9.1188 × 102	3.2095 × 101	1.0564 × 103	1.5937 × 101	1.2007 × 103	1.8331 × 101
m_SCA	7.5112 × 102	2.7312 × 101	9.4797 × 102	2.0587 × 101	1.0570 × 103	2.4289 × 101
SCADE	8.9697 × 102	3.1487 × 101	1.0680 × 103	1.3258 × 101	1.2072 × 103	1.7261 × 101
ASCA_PSO	7.1122 × 102	1.5224 × 101	9.5707 × 102	2.6319 × 101	1.1114 × 103	3.7255 × 101
ACWOA	7.2872 × 102	1.6207 × 101	9.9483 × 102	2.5768 × 101	1.1277 × 103	2.1651 × 101
MFO	8.1621 × 102	7.0326 × 101	9.3286 × 102	3.1243 × 101	1.1154 × 103	4.2025 × 101
SCA	8.3820 × 102	2.6572 × 101	1.0372 × 103	1.6583 × 101	1.1745 × 103	1.5443 × 101
FA	8.3255 × 102	9.9991 × 100	1.0236 × 103	1.5241 × 101	1.1575 × 103	8.8945 × 100
DA	1.0796 × 103	2.5224 × 102	1.0603 × 103	8.6112 × 101	1.1875 × 103	4.3320 × 101
	F10		F11		F12
	Ave	Std	Ave	Std	Ave	Std
GGBDA	2.1632 × 103	4.0569 × 102	4.3103 × 103	5.8058 × 102	1.2016 × 103	6.4654 × 10−1
OBSCA	6.1914 × 103	3.3800 × 102	7.3712 × 103	3.8870 × 102	1.2022 × 103	3.8380 × 10−1
m_SCA	4.2173 × 103	6.7303 × 102	4.6926 × 103	5.6709 × 102	1.2007 × 103	2.8914 × 10−1
SCADE	7.3873 × 103	2.0852 × 102	8.2043 × 103	2.8866 × 102	1.2026 × 103	2.9637 × 10−1
ASCA_PSO	5.3236 × 103	6.1947 × 102	6.0330 × 103	1.0051 × 103	1.2024 × 103	3.2840 × 10−1
ACWOA	4.3616 × 103	9.4361 × 102	6.5284 × 103	8.8174 × 102	1.2018 × 103	5.3507 × 10−1
MFO	4.2961 × 103	1.0010 × 103	5.2553 × 103	5.8399 × 102	1.2004 × 103	1.6921 × 10−1
SCA	6.9536 × 103	5.2169 × 102	8.1744 × 103	2.6469 × 102	1.2024 × 103	2.8490 × 10−1
FA	7.5877 × 103	2.4931 × 102	7.8979 × 103	2.2794 × 102	1.2024 × 103	3.1798 × 10−1
DA	7.8983 × 103	8.8564 × 102	8.2497 × 103	7.2246 × 102	1.2024 × 103	3.9165 × 10−1
	F13		F14		F15
	Ave	Std	Ave	Std	Ave	Std
GGBDA	1.3006 × 103	1.1023 × 10−1	1.4003 × 103	4.9476 × 10−2	1.5246 × 103	3.8702 × 100
OBSCA	1.3037 × 103	4.2284 × 10−1	1.4669 × 103	1.1727 × 101	1.7547 × 104	9.8027 × 103
m_SCA	1.3007 × 103	3.3452 × 10−1	1.4142 × 103	1.1462 × 101	2.2627 × 103	8.4352 × 102
SCADE	1.3038 × 103	2.5871 × 10−1	1.4902 × 103	1.1514 × 101	2.0450 × 104	8.8527 × 103
ASCA_PSO	1.3006 × 103	1.4205 × 10−1	1.4035 × 103	7.1583 × 100	1.5545 × 103	1.2124 × 102
ACWOA	1.3017 × 103	1.0761 × 100	1.4166 × 103	1.0655 × 101	1.9949 × 103	5.8404 × 102
MFO	1.3019 × 103	1.2975 × 100	1.4267 × 103	1.5955 × 101	3.3650 × 105	8.2577 × 105
SCA	1.3029 × 103	3.7934 × 10−1	1.4443 × 103	9.4586 × 100	5.0147 × 103	3.4034 × 103
FA	1.3029 × 103	1.9248 × 10−1	1.4403 × 103	4.8273 × 100	1.5752 × 104	4.4028 × 103
DA	1.3068 × 103	1.9095 × 100	1.5637 × 103	8.3347 × 101	2.4757 × 104	7.1463 × 104
	F16		F17		F18
	Ave	Std	Ave	Std	Ave	Std
GGBDA	1.6122 × 103	3.8012 × 10−1	2.1700 × 106	2.7205 × 106	1.5189 × 104	5.2280 × 104
OBSCA	1.6130 × 103	1.4281 × 10−1	1.1486 × 107	5.1039 × 106	1.9793 × 108	1.4800 × 108
m_SCA	1.6115 × 103	5.1409 × 10−1	1.5833 × 106	1.7905 × 106	3.4874 × 107	4.7812 × 107
SCADE	1.6127 × 103	1.9941 × 10−1	1.4197 × 107	6.7951 × 106	1.6517 × 108	1.1211 × 108
ASCA_PSO	1.6126 × 103	3.3022 × 10−1	1.2265 × 106	1.0213 × 106	3.6646 × 106	1.0393 × 106
ACWOA	1.6123 × 103	4.6588 × 10−1	1.6366 × 107	1.4017 × 107	4.6377 × 107	3.8096 × 107
MFO	1.6128 × 103	4.8526 × 10−1	4.0035 × 106	5.0310 × 106	3.9147 × 107	1.0322 × 108
SCA	1.6127 × 103	2.8567 × 10−1	6.9907 × 106	3.6926 × 106	1.6756 × 108	8.8211 × 107
FA	1.6129 × 103	2.3262 × 10−1	6.7491 × 106	2.2624 × 106	2.6476 × 108	7.8340 × 107
DA	1.6129 × 103	2.4315 × 10−1	8.5018 × 107	4.2101 × 107	4.0928 × 109	1.8915 × 109
	F19		F20		F21
	Ave	Std	Ave	Std	Ave	Std
GGBDA	1.9217 × 103	8.2441 × 100	2.2795 × 103	6.9312 × 101	1.9235 × 105	2.8749 × 105
OBSCA	2.0091 × 103	1.1149 × 101	3.0362 × 104	1.2377 × 104	2.3649 × 106	1.5032 × 106
m_SCA	1.9453 × 103	2.5699 × 101	1.0286 × 104	4.6386 × 103	4.6439 × 105	4.6037 × 105
SCADE	2.0209 × 103	1.7879 × 101	2.7828 × 104	1.2075 × 104	2.7903 × 106	1.0593 × 106
ASCA_PSO	1.9258 × 103	2.5713 × 101	6.0026 × 103	2.2111 × 103	3.2508 × 105	2.5701 × 105
ACWOA	2.0062 × 103	3.5162 × 101	4.0828 × 104	1.8916 × 104	5.1240 × 106	4.8145 × 106
MFO	1.9722 × 103	6.5003 × 101	6.7453 × 104	3.5593 × 104	7.3786 × 105	1.1693 × 106
SCA	1.9950 × 103	2.2940 × 101	1.7570 × 104	5.4464 × 103	1.3486 × 106	6.6249 × 105
FA	2.0029 × 103	1.1339 × 101	2.1545 × 104	8.6661 × 103	1.8937 × 106	6.2858 × 105
DA	2.2044 × 103	1.3085 × 102	7.3418 × 104	4.0133 × 104	2.4506 × 107	2.0130 × 107
	F22		F23		F24
	Ave	Std	Ave	Std	Ave	Std
GGBDA	2.6667 × 103	1.3749 × 102	2.5001 × 103	8.2205 × 10−2	2.6001 × 103	4.2571 × 10−2
OBSCA	3.0956 × 103	1.6521 × 102	2.6865 × 103	1.6694 × 101	2.6000 × 103	2.6232 × 10−4
m_SCA	2.6529 × 103	1.6213 × 102	2.6396 × 103	1.0453 × 101	2.6000 × 103	6.3375 × 10−4
SCADE	3.1130 × 103	1.5936 × 102	2.5000 × 103	0.0000 × 100	2.6000 × 103	1.0671 × 10−7
ASCA_PSO	2.7768 × 103	1.7913 × 102	2.6237 × 103	3.9400 × 100	2.6366 × 103	8.2081 × 100
ACWOA	3.0574 × 103	2.1215 × 102	2.5367 × 103	7.4780 × 101	2.6000 × 103	8.5021 × 10−6
MFO	2.9977 × 103	2.5111 × 102	2.6671 × 103	4.5312 × 101	2.6827 × 103	3.0780 × 101
SCA	2.9493 × 103	1.4065 × 102	2.6668 × 103	1.2152 × 101	2.6001 × 103	5.8342 × 10−2
FA	2.9399 × 103	1.0040 × 102	2.7354 × 103	1.4354 × 101	2.7065 × 103	4.4005 × 100
DA	1.3035 × 104	1.1958 × 104	2.8764 × 103	2.2534 × 102	2.6261 × 103	5.0498 × 100
	F25		F26		F27
	Ave	Std	Ave	Std	Ave	Std
GGBDA	2.7000 × 103	1.3977 × 10−3	2.7006 × 103	1.8063 × 10−1	2.9000 × 103	1.8895 × 10−3
OBSCA	2.7000 × 103	1.0817 × 10−3	2.7039 × 103	4.7598 × 10−1	3.2360 × 103	4.5158 × 101
m_SCA	2.7134 × 103	2.6641 × 100	2.7008 × 103	3.4050 × 10−1	3.1926 × 103	1.5161 × 102
SCADE	2.7000 × 103	0.0000 × 100	2.7037 × 103	6.1565 × 10−1	3.1829 × 103	2.6437 × 102
ASCA_PSO	2.7125 × 103	5.1192 × 100	2.7006 × 103	1.2849 × 10−1	3.5114 × 103	2.3638 × 102
ACWOA	2.7000 × 103	0.0000 × 100	2.7471 × 103	5.0332 × 101	3.6882 × 103	3.2535 × 102
MFO	2.7190 × 103	1.0042 × 101	2.7023 × 103	1.5257 × 100	3.6672 × 103	1.8397 × 102
SCA	2.7242 × 103	1.1442 × 101	2.7023 × 103	5.9638 × 10−1	3.4473 × 103	3.1999 × 102
FA	2.7342 × 103	4.0567 × 100	2.7023 × 103	2.8881 × 10−1	3.8003 × 103	2.8675 × 101
DA	2.7109 × 103	4.7079 × 100	2.7740 × 103	4.0242 × 101	4.2646 × 103	2.7086 × 102
	F28		F29		F30
	Ave	Std	Ave	Std	Ave	Std
GGBDA	3.0000 × 103	2.8946 × 10−2	3.1087 × 103	5.4266 × 100	3.5861 × 103	6.0874 × 102
OBSCA	5.3347 × 103	3.2181 × 102	1.8861 × 107	1.0186 × 107	4.5744 × 105	1.5327 × 105
m_SCA	3.9404 × 103	2.3055 × 102	1.6245 × 106	4.3077 × 106	4.6418 × 104	2.2798 × 104
SCADE	5.2213 × 103	5.2511 × 102	1.5436 × 107	7.9392 × 106	4.1012 × 105	1.8490 × 105
ASCA_PSO	4.4056 × 103	3.2811 × 102	5.1473 × 106	6.2012 × 106	4.1476 × 104	3.1316 × 104
ACWOA	4.2050 × 103	1.1911 × 103	2.1367 × 107	1.7150 × 107	3.9650 × 105	2.1202 × 105
MFO	3.9192 × 103	1.3812 × 102	2.6412 × 106	3.4748 × 106	5.7740 × 104	4.9279 × 104
SCA	4.7438 × 103	2.5806 × 102	1.1250 × 107	6.3057 × 106	2.4763 × 105	7.9063 × 104
FA	4.2782 × 103	1.8313 × 102	3.2845 × 106	1.2612 × 106	1.7562 × 105	4.0487 × 104
DA	8.5874 × 103	1.1741 × 103	2.7820 × 108	2.7534 × 108	4.1499 × 106	2.4981 × 106
	Overall Rank
	Rank	+/=/−
GGBDA	1	~
OBSCA	9	28/0/2
m_SCA	2	23/0/7
SCADE	8	27/0/3
ASCA_PSO	3	24/0/6
ACWOA	5	26/0/4
MFO	4	28/0/2
SCA	6	28/0/2
FA	7	30/0/0
DA	10	30/0/0

recorded the experimental results on 30D. Each algorithm ran independently 30 times. The average (Ave) and standard deviation (Std) of the optimal solutions obtained are shown in these tables. “AVR” expresses the average of the algorithm’s ranking results on all functions. In this experiment, the maximum number of iterations and the population size (Pop) were set to 1000 and 30. Each algorithm was performed in every function with 30 dimensions for the test of scalabilities, respectively. The symbol “+/=/−” refers to whether the performance of GGBDA is greater, equal, or worse than other algorithms compared.

4.2.1. Results on 30D Functions

F1–F7 do not have local optimal solutions. They are very suitable for measuring the exploration competence of the algorithm. In F2, F3, and F6, the results of GGBDA are far superior to all the others. Furthermore, in the rest functions, the results of GGBDA are better than most comparison algorithms. The results of F1–F7 show that GGBDA has an advantage over other algorithms in the ability to explore in the unimodal locality.

F8–F13 represents the multi-modal functions that have numerous local optimal solutions. They are very suitable for evaluating the local optimal prevention of the search ability of the algorithm. For F10 and F11, the results of GGBDA are near to the global optimal solution. However, the other comparison algorithm is easy to fall into the non-global optimal solution to different degrees. For the rest functions, GGBDA still obtains results that are better than most other algorithms. In conclusion, the experimental result verifies the global exploration ability of GGBDA.

From the convergence in Figure 2, we can estimate and evaluate the convergence performance of the algorithm. In F3, F10, F18, F20, F27, F28, F29, and F30, the convergence of GGBDA is better than other comparison algorithms in the early iterations. From the convergence of F6 and F11, GGBDA does not obtain the best adaptive in the early iteration but in the later iteration. In summary, the symbol “+/=/−” shows that GGBDA ranks first with the avg far lower than the second SCA, and the performance is even better than OBSCA, m_SCA, SCADE, ASCA_PSO, ACWOA, MFO, SCA, FA, and DA.

4.2.2. Balance Analysis

In this section, we conduct a qualitative analysis of GGBDA on the 30 functions of CEC14. The original DA was selected for comparison with GGBDA. Figure 3 shows the results of the feasibility analysis of GGBDA and the DA. There are five columns in the figure. The first column (a) is the location distribution of the GGBDA search history on the three-dimensional plane. The second column (b) is the location distribution of the GGBDA search history on the two-dimensional plane. The third column (c) is the trajectory of the first dimension of GGBDA during the iteration. The fourth column (d) shows the change of the average fitness of GGBDA during the iteration. The fifth column (e) shows the convergence curves of GGBDA and DA. In Figure 3b, the red dot represents the location of the optimal solution, and the black dot represents the search location of GGBDA. In the selected 5 function images, the black dots are denser in the area around the red dots, which shows that GGBDA has developed the area in which the optimal solution is located. In Figure 3c, we can see that the first-dimensional trajectory of GGBDA fluctuates greatly in the early period. Early volatility indicates that the algorithm has conducted extensive searches. The average fitness change of GGBDA in the whole iterative process is shown in Figure 3d. We can see that the average fitness of GGBDA dropped to a lower level in the mid-term. This shows that GGBDA has a good convergence speed. In Figure 3e, we can clearly see that the convergence curve of GGBDA is lower than that of DA, which shows that GGBDA can obtain a better solution.

The balance analysis and diversity analysis are carried out on the same functions. Figure 4 shows the results of the balanced analysis of GGBDA and DA. In Figure 4, there are three curves in each graph. As shown in the Figure, the blue curve and red curve represent exploitation and exploration, respectively. The larger value of the curve means that the corresponding behavior is dominant in the algorithm. The green curve indicates incremental–decremental. The curve can more intuitively reflect the changing trends of the two behaviors of the algorithm. When the value of the curve increases, it means that the exploration activity is dominant. Instead, exploitative behavior predominates. When the curve drops to a negative value, the curve will be set to zero. Comparing the curves of the two algorithms shows that both algorithms were dominated by exploration behavior in the early stage. This is because the swarm intelligence optimization algorithm performs a global search first, at the beginning. However, the difference between the two algorithm curves is also very obvious. The DA spends more time on exploration behavior than GGBDA. The exploration behavior of DA almost accounts for half of the entire iteration process. However, the exploitative behavior of GGBDA quickly became dominant, indicating that it spent more time exploiting the target area. This is the impact of the two mechanisms added to GGBDA on its balance.

Figure 5 is the result of the diversity analysis of GGBDA and DA. In Figure 5, the ordinate represents the population diversity. We can see that the diversity of the two algorithms is very high at the beginning. This is because the initial population of the algorithm is randomly generated. Then, in the iterative process, the algorithm continues to narrow the search range so that the diversity of the population will reduce, although the diversity curves of the two algorithms almost reached the lowest in the iteration. However, the descent process of the two algorithms is very different. We can clearly observe that the DA maintained a high diversity in the early stage. The diversity curve of the DA dropped to its lowest value very quickly in the mid-term. This change was completed in a concise time.

In contrast, the curve of GGBDA declined more gently. GGBDA only declines rapidly at the initial stage, and then the rate of decline slows down. This is obvious for F2 and F14. This shows that the two added mechanisms have an impact on the diversity of the DA. Owing to the strong search capability, the proposed GGBDA can also be applied to other optimization problems, such as fault detection [81]; metabolomic data processing [82,83]; urban road planning [84]; multivariate time series analysis [85]; gene signature identification [86]; drug target discovery [87]; drug discovery [88]; pharmacoinformatics data mining [89]; service ecosystem [90,91]; information retrieval services [92,93,94]; kayak cycle phase segmentation [95]; covert communication system [96,97,98]; location-based services [99,100]; and human motion capture [101].

4.3. Real-World Problems

4.3.1. Pressure Vessel Design (PVD) Problem

The PVD problem is a common engineering design problem. There are four constraints and four parameters in the PVD problem. The main aim is to obtain a pressure vessel that meets the conditions with relatively minimal costs.

The formula of this problem is listed below.

Consider:

$$X=\left[x_1 x_2 x_3 x_4\right]=\left[T_s T_h R L\right]$$

Range of parameters:

$$\begin{gathered} 0 \le x_1 \le 99 \\ 0 \le x_2\le 99 \\ 10 \le x_3\le 200 \\ 10 \le x_4\le 200 \end{gathered}$$

Minimize:

$$\mathrm{f}\left(\overrightarrow{x}\right)=0.6224x_1{x}_3{x}_4+1.7781x_3{x}_1^2+3.1661x_4{x}_1^2+19.84x_3{x}_1^2$$

Subject to:

$$\begin{gathered} g_1\left(X\right)=-x_1+0.0193x_3\le 0 \\ {g}_2\left(X\right)=-x_3+0.00954x_3\le 0 \\ {g}_3\left(X\right)=-\pi x_4{x}_3^2-\frac{4}{3}\pi x_3^3+1296000\le 0 \\ {g}_4\left(X\right)=x_4-240\le 0 \end{gathered}$$

Table 4. Comparison results of the PVD problem between GGBDA and other approaches.

Algorithm	Optimum Variables				Optimum Cost
	${\mathit{T}}_{\mathit{s}}$	${\mathit{T}}_{\mathit{h}}$	R	L
GGBDA	0.8125	0.4375	42.0983	176.6380	6059.7298
MFO [15]	0.8125	0.4375	42.0984	176.6366	6059.7143
BA [102]	0.8125	0.4375	42.0984	176.6366	6059.7143
HPSO [103]	0.8125	0.4375	42.0984	176.6366	6059.7143
CSS [104]	0.8125	0.4375	42.1036	176.5727	6059.0888
CPSO [105]	0.8125	0.4375	42.0912	176.7465	6061.0777
ACO [106]	0.8125	0.4375	42.1036	176.5727	6059.0888
GWO [18]	0.8125	0.4345	42.0892	176.7587	6051.5639
WOA [2]	0.8125	0.4375	42.0983	176.6390	6059.7410
MDDE [107]	0.8125	0.4375	42.0984	176.6360	6059.7017
Branch-bound [108]	1.1250	0.6250	47.7000	117.7010	8129.1036

shows the results GGBDA for the optimization for the PVD problem, compared with other peers in the literature. The results show that the optimal value obtained by the GGBDA was 6059.7298, which was better than CPSO, WOA, and Branch-bound. Moreover, GGBDA has a similar effect with MFO, HPSO, and BA.

4.3.2. Hydrostatic Thrust Bearings Design (HTBD) Problem

The goal of the HTBD problem is to minimize power loss. At the same time, the design needs to meet some constraints. There are four design variables: bearing step radius (R), recess radius (R0), oil viscosity (μ), and flow rate (Q). The mathematical model of this problem is shown as below.

Minimize:

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{{\mathrm{QP}}_0}{0.7}+{\mathrm{E}}_f$$

Subject to:

$$\begin{gathered} {\mathrm{g}}_1\left(\mathrm{x}\right)=\frac{{\mathsf{\pi }\mathrm{P}}_0}{2}\times \frac{{\mathrm{R}}^2-{\mathrm{R}}_0^2}{\ln \left(\mathrm{R}/{\mathrm{R}}_0\right)}-{\mathrm{W}}_{\mathrm{s}}\ge 0 \\ {\mathrm{g}}_2\left(x\right)=P_{max}-P_0\ge 0 \\ {\mathrm{g}}_3\left(x\right)=\Delta T_{max}-\Delta T\ge 0 \\ {\mathrm{g}}_4=h-h_{min}\ge 0 \\ {\mathrm{g}}_5\left(x\right)=R-R_0\ge 0 \\ {\mathrm{g}}_6\left(x\right)=0.001-\gamma /g{P}_0\left(Q/2\pi Rh\right)\ge 0 \\ {\mathrm{g}}_7\left(x\right)=5000-\frac{W}{\pi \left(R^2-R_0^2\right)}\ge 0 \end{gathered}$$

where

$$\begin{gathered} {\mathrm{P}}_0=\frac{6\mu Q}{\pi h^3}\ln \left(\frac{R}{R_0}\right) \\ {\mathrm{E}}_f=9336Q\gamma C\Delta T \\ \Delta T=2\left({10}^{\mathrm{P}}-560\right) \\ P=\frac{\log \left(\log \left(8.122\times {10}^6+0.8\right)\right)-C_1}{n} \\ h={\left(\frac{2\pi N}{60}\right)}^2\frac{2\pi \mu }{E_f}\left(\frac{R^4}{4}-\frac{R_0^4}{4}\right) \\ {\mathrm{C}}_1=10.04 \end{gathered}$$

n = −3.55, Pmax = 1000, Ws = 101000

$$\begin{gathered} \Delta {\mathrm{T}}_{max}=50 \\ {h}_{min}=0.001 \end{gathered}$$

g = 386.4, N = 750

$$\begin{gathered} 5\le D_e,D_i\le 15 \\ 0.01\le t\le 6 \\ 0.05\le h\le 0.5 \end{gathered}$$

Table 5. Comparison results of the hydrostatic thrust bearing problem between GGBDA and other approaches.

Algorithm	Optimum Variables				Optimum Cost
	R	R0	μ	Q
GGBDA	5.956071	5.389334	5.36 × 10−6	2.271766	19,508.7584
PSO [8]	5.956868	5.389175	5.4021 × 10−6	2.301546	19,586.5788
NDE [109]	5.955781	5.389013	5.3586 × 10−6	2.269656	19,506.0090
TLBO [110]	5.955781	5.389013	5.3586 × 10−6	2.269656	19,505.3132
SQP [111]	5.955800	5.389040	8.6332 × 10−6	8.000010	26,114.5450
GASO [112]	6.271000	12.90100	5.6050 × 10−6	2.938000	23,403.4320

shows the results of the HTBD problem. It can be seen that the optimal value of GGBDA is 19,508.76, which is better than PSO, SQP, and GASO. Moreover, GGBDA has almost the same effect as TNE and TLBO.

4.3.3. Welded Beam Design (WBD) Problem

WBD problem aims to minimize the cost of welded beams subject to the four constraints of shear stress (τ), bending stress (θ), buckling load (P_c), and deflection (δ). The variables in this problem are composed of welding seam thickness (h), welding joint length (l), beam width (t), and beam thickness (b). The mathematical model of this problem is listed as below.

Consider:

$$\overrightarrow{\mathrm{x}}=\left[{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4\right]=\left[ h l t b\right]$$

Minimize:

$$\mathrm{f}\left(\overrightarrow{\mathrm{x}}\right)=1.10471{\mathrm{x}}_1^2+0.04811{\mathrm{x}}_3{\mathrm{x}}_4\left(14.0+{\mathrm{x}}_4\right)$$

Subject to:

$$\begin{gathered} {\mathrm{g}}_1\left(\overrightarrow{\mathrm{x}}\right)=\mathsf{\tau }\left(\overrightarrow{\mathrm{x}}\right)-{\mathsf{\tau }}_{\max }\le 0 \\ {\mathrm{g}}_2\left(\overrightarrow{\mathrm{x}}\right)=\mathsf{\sigma }\left(\overrightarrow{\mathrm{x}}\right)-{\mathsf{\sigma }}_{\max }\le 0 \\ {\mathrm{g}}_3\left(\overrightarrow{\mathrm{x}}\right)=\mathsf{\delta }\left(\overrightarrow{\mathrm{x}}\right)-{\mathsf{\delta }}_{\max }\le 0 \\ {\mathrm{g}}_4\left(\overrightarrow{\mathrm{x}}\right)={\mathrm{x}}_1-{\mathrm{x}}_4\le 0 \\ {\mathrm{g}}_5\left(\overrightarrow{\mathrm{x}}\right)=\mathrm{P}-{\mathrm{P}}_{\mathrm{C}}\left(\overrightarrow{\mathrm{x}}\right)\le 0 \\ {\mathrm{g}}_6\left(\overrightarrow{\mathrm{x}}\right)=0.125-{\mathrm{x}}_1\le 0 \\ {\mathrm{g}}_7\left(\overrightarrow{\mathrm{x}}\right)=1.10471{\mathrm{x}}_1^2+0.04811{\mathrm{x}}_3{\mathrm{x}}_4\left(14.0+{\mathrm{x}}_2\right)-5.0\le 0 \end{gathered}$$

Variable range:

$$\begin{gathered} 0.1\le {\mathrm{x}}_1\le 2 \\ 0.1\le {\mathrm{x}}_2\le 10 \\ 0.1\le {\mathrm{x}}_3\le 10 \\ 0.1\le {\mathrm{x}}_4\le 2 \end{gathered}$$

where

$$\begin{gathered} \mathsf{\tau }\left(\overrightarrow{\mathrm{x}}\right)=\sqrt{{\left({\mathsf{\tau }}^{\prime }\right)}^2+2{\mathsf{\tau }}^{\prime }{\mathsf{\tau }}^{″}\frac{{\mathrm{x}}_2}{2\mathrm{R}}+{\left({\mathsf{\tau }}^{″}\right)}^2} \\ {\mathsf{\tau }}^{\prime }=\frac{\mathrm{P}}{\sqrt{2}{\mathrm{x}}_1{\mathrm{x}}_2} \\ {\mathsf{\tau }}^{″}=\frac{\mathrm{MR}}{\mathrm{J}} \\ {\mathsf{\tau }}^{″}=\frac{\mathrm{MR}}{\mathrm{J}} \\ \mathrm{R}=\sqrt{\frac{{\mathrm{x}}_2^2}{4}+{\left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_3}{2}\right)}^2} \\ \mathrm{J}=2\left\{\sqrt{2}{\mathrm{x}}_1{\mathrm{x}}_2\left[\frac{{\mathrm{x}}_2^2}{4}+{\left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_3}{2}\right)}^2\right]\right\} \\ \mathsf{\sigma }\left(\overrightarrow{\mathrm{x}}\right)=\frac{6\mathrm{PL}}{{\mathrm{x}}_4{\mathrm{x}}_3{}^2} \\ \mathsf{\delta }\left(\overrightarrow{\mathrm{x}}\right)=\frac{6{\mathrm{PL}}^3}{{\mathrm{Ex}}_3^2{\mathrm{x}}_4} \\ {\mathrm{P}}_{\mathrm{C}}\left(\overrightarrow{\mathrm{x}}\right)=\frac{4.013\mathrm{E}\sqrt{\frac{{\mathrm{x}}_3^2{\mathrm{x}}_4^6}{36}}}{{\mathrm{L}}^2}\left(1-\frac{{\mathrm{x}}_3}{2\mathrm{L}}\sqrt{\frac{\mathrm{E}}{4\mathrm{G}}}\right) \\ \mathrm{P}=60001\mathrm{b} \\ \mathrm{L}=14\in \mathrm{‥}{\mathsf{\delta }}_{\max }=0.25\in \mathrm{‥} \\ \mathrm{E}=30\times 1^6\mathrm{psi} \\ \mathrm{G}=12\times 10^6\mathrm{psi} \\ { \mathsf{\tau }}_{\max }=13600\mathrm{psi} \\ {\mathsf{\sigma }}_{\max }=30000\mathrm{psi} \end{gathered}$$

The results of the WBD problem are shown in

Table 6. Comparison results of the WBD problem between GGBDA and other approaches.

Algorithm	Optimal Values for Variables				Optimum Cost
	h	l	t	b
GGBDA	0.187156	3.615020	9.056672	0.206464	1.724527
RO [113]	0.203687	3.528467	9.004233	0.207241	1.735344
SSA [114]	0.205700	3.471400	9.036600	0.205700	1.724910
CDE [115]	0.203137	3.542998	9.033498	0.206179	1.733462
GWO [18]	0.205700	3.478400	9.036800	0.205800	1.726240
GSA [116]	0.182129	3.856979	10.00000	0.202376	1.879950
NDE [109]	0.205729	3.470488	9.903662	0.205729	1.724852

. The optimal value of GGBDA is 1.724527, which is the lowest among all the algorithms. It can be seen that GGBDA has a better effect than other peers in the experiment.

4.3.4. Tension–Compression String Design (TCSD) Problem

The TCSD problem is to design a tension–compression spring with the minimum weight and meets the constraints. The three variables in the problem are the wire diameter ($\mathrm{d}$), mean coil diameter ($\mathrm{D}$), and the number of active coils ($\mathrm{N}$). The mathematical model of this problem is listed as below.

Consider:

$$\overrightarrow{x}=\left[x_1 x_2 x_3\right]=\left[d D N\right]$$

Objective function:

$$Minimize f\left(x\right)=x_1^2{x}_2{x}_3+2x_1^2{x}_2$$

Subject to:

$$\begin{gathered} h_1\left(\overrightarrow{x}\right)=1-\frac{x_2^3{x}_3}{71785x_1^4}\le 0, \\ {h}_2\left(\overrightarrow{x}\right)=\frac{4x_2^2-x_1{x}_2}{12566\left(x_2{x}_1^3-x_1^4\right)}+\frac{1}{5180x_1^2}-1\le 0 \\ {h}_3\left(\overrightarrow{x}\right)=1-\frac{140.45x_1}{x_2^3{x}_3}\le 0 \\ {h}_4\left(\overrightarrow{x}\right)=\frac{x_1+x_2}{1.5}-1\le 0 \end{gathered}$$

Variable ranges:

$$\begin{gathered} 0.05\le x_1\le 2.00 \\ 0.25\le x_2\le 1.30, \\ 2.00\le x_3\le 15.0 \end{gathered}$$

Table 7. Comparison results of the TCSD problem between GGBDA and other approaches.

Algorithm	Optimal Values for Variables			Optimum Cost
	d	D	N
GGBDA	0.051652	0.355837	11.34081	0.012665
GA [117]	0.051480	0.351661	11.63220	0.012705
RO [113]	0.051370	0.349096	11.76279	0.012679
IHS [118]	0.051154	0.349871	12.07643	0.012671
ES [119]	0.051989	0.363965	10.89052	0.012681
GSA [116]	0.050276	0.323680	13.52541	0.012702
WOA [2]	0.051207	12.00430	0.345215	0.012676
PSO [8]	0.015728	11.24454	0.357644	0.012675
NDE [109]	0.051689	0.356718	11.28896	0.012665

shows the results of the TCSD problem. The optimal values of GGBDA and NDE are both 0.012665, which is the lowest among the algorithms. It can be seen that GGBDA still has a good effect on the TCSD problem.

5. Conclusions

The purpose of this research was to propose an enhanced DA that anticipates engineering design problems more efficiently and precisely. The Gaussian mutation and the Gaussian barebone are embedded into the DA, termed as GGBDA. The Gaussian mutation was used to prevent slipping into local optimal situations and to update the individual locations in a random manner. To further enhance local exploitation capacities, Gaussian barebone was used in conjunction with the improvement of Gaussian mutation, the global searching ability, and the convergence efficiency of GGBDA to accelerate the convergent speed and strengthen local exploitation capacities. This study compared the performance of GGBDA with other competitive peers on 30 benchmarks and 4 engineering design issues. The experimental findings demonstrate that GGBDA outperforms DA and other competing algorithms in terms of solution accuracy and convergence speed.

GGBDA’s performance and time cost will be improved in future developments. For example, we will address GGBDA’s design issues. GGBDA may also be used to anticipate and optimize the parameters for energy optimization, image segmentation, and parameter optimization of machine learning methods.