Coronavirus Mask Protection Algorithm: A New Bio-inspired Optimization Algorithm and Its Applications

Abstract

Nowadays, meta-heuristic algorithms are attracting widespread interest in solving high-dimensional nonlinear optimization problems. In this paper, a COVID-19 prevention-inspired bionic optimization algorithm, named Coronavirus Mask Protection Algorithm (CMPA), is proposed based on the virus transmission of COVID-19. The main inspiration for the CMPA originated from human self-protection behavior against COVID-19. In CMPA, the process of infection and immunity consists of three phases, including the infection stage, diffusion stage, and immune stage. Notably, wearing masks correctly and safe social distancing are two essential factors for humans to protect themselves, which are similar to the exploration and exploitation in optimization algorithms. This study simulates the self-protection behavior mathematically and offers an optimization algorithm. The performance of the proposed CMPA is evaluated and compared to other state-of-the-art metaheuristic optimizers using benchmark functions, CEC2020 suite problems, and three truss design problems. The statistical results demonstrate that the CMPA is more competitive among these state-of-the-art algorithms. Further, the CMPA is performed to identify the parameters of the main girder of a gantry crane. Results show that the mass and deflection of the main girder can be improved by 16.44% and 7.49%, respectively.

Introduction

Truss optimization under high-dimensional nonlinear constraints aims to obtain better structural behavior and decrease the design cost. In the truss optimization problem, structural optimization and optimization algorithms have attracted much attention during the past decades [1, 2]. Most of the traditional optimizers have difficulty in obtaining the global optimal solution of high-dimensional nonlinear constrained problems with low cost and high efficiency [3, 4]. To deal with these challenges, various new approaches have been developed to solve real-world engineering optimization problems.

Meta-heuristic algorithms are effective tools to maximize profits with limited resources. Mathematically, the meta-heuristic algorithms based on natural phenomena and biological systems are more efficient [5, 6]. Gharehchopogh et al. [7–14] proposed a series of optimization algorithms, which have solved practical problems such as agriculture and engineering problems. Mohammadzadeh and Gharehchopogh [15, 16] solved the problem of e-mail spam detection with an optimization algorithm. Abdollahzadeh [17] solved the multi-objective optimization problem using the hybrid optimization algorithm and obtained the global optimal solution. Ghafori and Gharehchopogh [18] conducted a comprehensive survey of Spotted Hyena Optimizer (SHO) and obtained the direction of improvement of the SHO algorithm. Meta-heuristic algorithms can provide feasible results for highly complex constrained problems, i.e., Grey Wolf Optimizer (GWO) [19], Harris Hawks Optimization (HHO) [20], Flower Pollination Algorithm (FPA) [21], Social Network Search (SNS) [22], Gravitational Search Algorithm (GSA) [23], Dragonfly Algorithm (DA) [24], Alpine Skiing Optimization (ASO) [25], Elite Opposition-based Learning and Chaotic k-best Gravitational Search Strategy based Grey Wolf Optimizer (EOCSGWO) [26], Artificial Bee Colony (ABC) [27], Teaching–Learning-Based Optimization (TLBO) [28], hybrid Moth Flame Optimization and Butterfly Optimization Algorithm (h-MFOBOA) [29].

Although these optimization algorithms can find optimal solutions, the high computational cost is still a huge challenge for researchers. Moreover, the “No Free Lunch (NFL)” theorem encourages scientific researchers to come up with new optimizers consecutively. This is because there is no single optimizer that can solve all engineering optimization problems [30]. Notably, all optimization algorithms have two common stages: exploration and exploitation [31]. In the first stage, the individuals need to be able to cover each region of the search space and effectively obtain more solutions for different regions. Therefore, levy flight is a common method in the exploration stage. In the exploitation phase, all the optimizers focus more on local search, aiming at the best solution for feasible regions. Interestingly, a good-performance optimizer can usually balance between exploitation and exploration to avoid falling into local optimum and premature convergence.

According to the “No Free Lunch” theorem, we cannot, in theory, find an optimizer as the best universal optimization algorithm. In this paper, a COVID-19 prevention-inspired optimization algorithm, named Coronavirus Mask Protection Algorithm (CMPA), is proposed based on the virus transmission of COVID-19. The equations and the mathematical model of the CMPA are introduced in detail. Moreover, the pseudocode is also given based on the principle of the CMPA. The performance of the proposed new mathematical model shows excellent performance in 23 famous popular benchmark functions, CEC2020 suite problems, and 3 truss design problems. The following most contributed to the paper, which is as follows:

• A novel swarm intelligent optimization strategy, namely Coronavirus Mask Protection Algorithm (CMPA), is proposed.

• An optimization algorithm is proposed using the mathematical model.

• The proposed CMPA is tested on 23 unconstrained benchmark functions.

• The performance of the CMPA is also examined on CEC2020 suite problems and three truss design problems.

• Results indicate that the CMPA can be used as an effective and efficient optimization algorithm to solve engineering optimization problems.

• The CMPA is employed to identify the parameters of the main girder of a gantry crane.

The rest of this paper is organized as follows. Section 2 presents the inspiration for the CMPA. The principle and pseudocode of the new mathematical optimization algorithm are presented in Sect. 3. The performance of the CMPA is evaluated using 23 benchmark functions and CEC2020 suite problems in Sect. 4. Section 5 presents the ability of the CMPA to solve engineering design problems. Section 6 shows the real-world engineering problems application of the CMPA. The concluding remarks and future work are concluded in Sect. 7.

Inspiration

The coronavirus pandemic (COVID-19) was reported in December 2019, and has erupted in countries around the world. In the early days of COVID-19, there were no specific drugs for treatment in all countries. According to the global report, 5.94 million people died of COVID-19 between January 1, 2020 and December 31, 2021. However, researchers estimate that 18.2 million people died of the COVID-19 pandemic (as measured by excess mortality) worldwide during this period [32]. With the variation of COVID-19, there is no effective and complete cure. It is worth noting that the incubation period of mutated strains in humans is also different. Therefore, masks resort as an effective tool for health protection at the moment.

To address the challenge of COVID-19, all the countries have carried out vaccine research. It is worth noting that on May 22, 2020, Lancet published human clinical data online [33]. Since then, the United States, China, Russia, and other countries have successfully developed vaccines and effectively protected human health. Although the vaccine has achieved remarkable results, COVID-19 still has rebounded and erupted1. Figure 1 shows the development trend of COVID-19.

Fig. 1.

The development trend of COVID-19

COVID-19 is mainly transmitted through droplets produced by sneezing or coughing of infected people. To reduce the spread scope of COVID-19, wearing masks is an effective protection measure [34]. According to the social distance, if healthy people do not wear masks, COVID-19 may infect them over a short distance. We will eventually defeat COVID-19, which is applied Darwinian theory about the survival-of-the-fittest principle [35]. The game process between humans and COVID-19 is similar to the exploration and exploitation performance in the optimizer, which can be divided into three stages: (i) healthy people are affected by the infected people; (ii) a small number of people died and most recovered; (iii) everyone can resist COVID-19.

Coronavirus Mask Protection Algorithm (CMPA)

Hypotheses About the CMPA

The computational efficiency and robustness of the CMPA depend on the susceptible individuals, infected individuals, and immune individuals. To ensure the universality of the CMPA algorithm, the following assumptions are made:

1. Assuming that all individuals have the same constitution and wear masks during communication.

2. It is assumed that everyone has the same chance of being infected under the same conditions. All infected individuals have the same ability to infect others.

3. It is assumed that the total number of individuals is constant during the search process; if there are some infected patients who died of COVID-19, new susceptible individuals will be added.

Coronavirus Mask Protection Optimization

The Process of Infection

Individuals are not infected at the beginning; however, if some individuals wear masks incorrectly or are too close to the infected person during communication, they may get infected easily. The process of virus transmission is shown in Fig. 2. Once there is an infected person in an area, it is possible to infect more susceptible individuals. The majority of the infected people can recover, and only a few people may die. In general, the resistance of the elderly to the virus is usually weaker than that of the young, the main reason is that they may suffer from other diseases, such as cancer, cardiovascular disease, and diabetes. Therefore, the age and physical condition of the infected person play an important role in rehabilitation.

Fig. 2.

The process of virus transmission

Virus Transmission

Infected people are unknowingly attacked by the virus, and usually have no obvious symptoms in the early stage of infection. Generally, the virus has a certain incubation period, and the symptoms and time of the virus in infected people are also different. These susceptible individuals who wear masks incorrectly and come into close contact may be infected by the infected person in susceptible populations. Figure 3 shows the process of infection in susceptible people.

Fig. 3.

The process of infection

Notably, the social distance between susceptible people and infected people is an important factor. Although masks can protect most susceptible people from infection, too close social distance can also lead to some susceptible people being attacked by the virus.

Immune Process

After the incubation period, the symptoms of infection will be gradually shown in susceptible individuals. To prevent susceptible individuals from being infected, masks can act as an effective firewall to protect them. With the improvement of individuals' awareness of self-protection, the virus will be controlled within a certain range. Because humans have a strong immune system, the virus will gradually be defeated until the infected individual recovers completely.

Generally, when the total number of immunized people reaches the population immune threshold (i.e., greater than 60%), there will be no more cases of infection in this area. Notably, some countries use controlled herd immunity to defend against the virus. For example, UK and Sweden allow the virus to spread to improve individual immunity [36–38]. Compared with other methods, herd immunity takes a longer time to achieve the same effect. Therefore, some infected people may die in the immune process. Obviously, the herd immunity strategy is not the best method. On the contrary, effective strategies (such as wearing masks) can reduce mortality.

Mathematical Model of the CMPA

The process of infection and immunity consists of three phases, including the infection stage, diffusion stage, and immune stage. Notably, social distance is an important index to estimate whether an individual is infected, which can be given by:

$$x_i\left(t+1\right)=\left\{\begin{array}{ccc}{x}_i\left(t\right)& r>r_s& \\ I\left(x_i\left(t\right)\right)& 0<r\le \frac{1}{3}{r}_s& \text{infection stage}\\ S\left(x_i\left(t\right)\right)& \frac{1}{3}{r}_s<r\le \frac{2}{3}{r}_s& \text{diffusion stage}\\ R\left(x_i\left(t\right)\right)& \frac{2}{3}{r}_s<r\le r_s& \text{immune stage}\end{array}\right)$$ 1

where $x_i\left(t+1\right)$ is the health of the ith individual in the (t + 1)th social interaction; r is the social distance, which is a random number in [0, 2]; r_s is safe social distance.

Infection stage: When the social distance between the susceptible individual and infected individual is small ($0<r\le \frac{1}{3}{r}_s$), the susceptible individual will be infected. The mathematical model of the infected stage can be given by:

$$x_i\left(t+1\right)=I\left(x_i\left(t\right)\right)$$ 2

$$I\left(x_i\left(t\right)\right)=x_i\left(t\right)+{\delta }_i\left(t\right)\times r\left(t\right)\times ∥x_i\left(t\right)-x_p\left(t\right)∥$$ 3

where ${\delta }_i\left(t\right)$ is the safety factor of mask for the ith individual in the tth social interaction; $x_p\left(t\right)$ denotes the position of the infected people in the tth social interaction.

Diffusion stage: If some susceptible individuals are infected by the virus, other susceptible individuals will increase their social distance to avoid being infected. Although these individuals pay more attention to social distance, more people will still be infected by the virus. The main reason is that the virus has an incubation period, and the symptoms of the infected individuals cannot appear immediately. The model of the diffusion stage can be calculated as follows:

$$x_i\left(t+1\right)=S\left(x_i\left(t\right)\right)$$ 4

$$S\left(x_i\left(t\right)\right)=\left(x_i\left(t\right)+{\lambda }_i\left(t\right)\times {\delta }_i\left(t\right)\times r\left(t\right)\times ∥x_i\left(t\right)-x_q\left(t\right)∥\right)\times \frac{{\varphi }^{\left(\alpha /\beta +1\right)}-1}{\varphi -1}$$ 5

where $\varphi$ is infection factor; $\alpha$ and $\beta$ are the time of social activities and virus transmission, respectively. ${\lambda }_i\left(t\right)$ is the resistance of the ith individual in the tth social interaction. $x_q\left(t\right)$ is the position of the immuned individual in the tth social interaction.

Immune stage: As more susceptible individuals are attacked by the virus, social distance will be paid more and more attention. When the social distance (r) falls in the range ($\frac{2}{3}{r}_s$,$r_s$], the virus is gradually defeated with the help of the autoimmune system. The immune stage can be presented as follows:

$$x_i\left(t+1\right)=R\left(x_i\left(t\right)\right)$$ 6

$$R\left(x_i\left(t\right)\right)=\left(x_i\left(t\right)+{\delta }_i\left(t\right)\times r\left(t\right)\times ∥x_i\left(t\right)-x_b\left(t\right)∥\right)\times \eta e^{\mu }$$ 7

where $\eta$ is the physical fitness coefficient; $\mu$ denotes the happiness index. If an individual's location exceeds the search space, we will define that the individual has been infected by the virus and died. To keep the population constant, a new individual will be randomly generated in the search space. One straightforward method is to replace the dead individual with a random individual as given in Eq. (9):

$$x_i\left(t+1\right)=lb+rand\times \left(ub-lb\right)$$ 8

where ub is the upper bound, lb is the lower bound. Also, the dead individual can be replaced using another method, which can be defined as follows:

$$x_i\left(t+1\right)=-\left(1-randn\right)\times \left(x_i\left(t\right)-median\left(X\right)\right)+x_b\left(t\right)$$ 9

where randn is the random number that produces a normal distribution; median(X) is the median of the current population position; $x_b\left(t\right)$ denotes the best solution of the current iteration. Figures 4 and 5 represent the pseudocode and the flowchart of the CMPA, respectively.

Fig. 4.

The pseudocode of the CMPA

Fig. 5.

The flowchart of the CMPA

Experimental Investigation

To investigate the performance of the CMPA, 23 famous test problems and CEC2020 test problems are selected for optimization statistical analysis. Notably, these test problems are classic optimization problems, which are widely used in the Refs. [39–42].

Benchmark Functions Set I: 23 Famous Test Problems

To verify the performance of the CMPA, six state-of-the-art optimization algorithms, i.e., flower pollination algorithm (FPA) [21], differential evolution (DE) [39], bat algorithm (BA) [40], evolution strategy with covariance matrix adaptation (CMA-ES) [41], moth-flame optimization (MFO) [42], gravitational search algorithm (GSA) [23], are selected to compare the optimization results of the CMPA. Notably, these optimization algorithms have superior competitiveness and have been widely used to solve real-world engineering problems. The parameter setting of these algorithms is reported in Table 1.

Table 1.

The parameter settings

Algorithms	Parameter setting	Values
FPA	Probability switch (p)	0.4
BA	Minimum frequency (a)	0
	Maximum frequency (b)	2
DE	Scaling factor (s)	0.5
	Crossover probability (c)	0.5
CMA-ES	Number of offsprings (u)	$10\times \left(4+⌊3\text{ln}\left(n\right)⌋\right)$
	Parent weights (w)	$\frac{\text{log}\left(\text{u +}0.5\right)\text{- log}\left(\text{u}\right)}{{\sum }_{i=1}^{u}w}$
	c_sigma ($c_{\sigma }$)	$\frac{{\text{u}}_{\text{w}}+2}{n+{\text{u}}_{\text{w}}+5}$
	d_sigma (${\text{d}}_{\sigma }$)	$1+2\text{max}\left(0,\sqrt{\frac{u_w-1}{n+1}-1}\right)+\frac{{\text{u}}_{\text{w}}+2}{n+{\text{u}}_{\text{w}}+5}$
MFO	Random number (t)	[–1, 1]
	Shape of the logarithmic spiral (b)	1
GSA	Alpha ($\alpha$)	20
	Gravitational constant (G0)	100
CMPA	Safe social distance (rs)	1
	Physical fitness coefficient ($\eta$)	0.8
	Happiness index ($\mu$)	0.7

Tables 2 and 3 report the results of the F1–F7, which are the unimodal benchmark functions. Obviously, the CMPA obtains the optimal solutions in most benchmark functions. To be specific, the CMPA can obtain the first rank on F1–F7 except for F6, and the average of rank is 1.43. The second-best results belong to the CMA-ES, followed by FPA, GSA, DE, BA, and MFO. Furthermore, the plots in Figs. 6a–b show the convergence progress of F1 and F4. It can be seen that the performance of the CMA-ES and GSA is equivalent. Notably, the convergence rate and the precision of the CMPA are higher than the other algorithms. It can concluded that the CMPA has stable search performance.

Table 2.

Results of unimodal benchmark functions (F1–F7), with 30 dimensions

Functions		FPA	BA	DE	CMA-ES	MFO	GSA	CMPA
F1	Mean	3.12e-2	3.58e-1	1.35e-3	2.24e-4	3.77e1	9.07e-4	1.08e-42
	STD	7.85e-2	1.00e0	5.90e-4	1.20e-3	1.21e2	4.56e-4	3.41e-41
	Rank	5	6	4	2	7	3	1
F2	Mean	1.32e-2	1.00e0	6.79e-3	0.00e0	3.20e1	1.50e-1	0.00e0
	STD	7.02e-4	1.00e0	2.05e-3	0.00e0	2.05e1	2.80e-2	0.00e0
	Rank	4	6	3	1	7	5	1
F3	Mean	0.00e0	1.00e0	3.97e4	3.28e-2	2.44e4	2.11e-1	0.00e0
	STD	0.00e0	1.00e0	5.94e3	3.98e-2	1.42e4	5.69e-2	0.00e0
	Rank	1	6	7	4	3	5	1
F4	Mean	3.23e-1	9.15e-1	1.16e1	3.24e-1	7.00e1	9.66e-2	2.38e-8
	STD	4.20e-1	1.00e0	2.38e0	5.12e-1	7.06e0	1.85e-2	2.01e-7
	Rank	3	5	6	4	7	2	1
F5	Mean	6.01e-3	1.00e0	1.06e2	0.00e0	7.37e3	2.75e1	1.91e0
	STD	3.13e-2	1.00e0	1.01e2	0.00e0	2.26e3	4.52e-1	3.07e-1
	Rank	2	3	6	1	7	5	4
F6	Mean	8.81e-3	1.00e0	1.45e-3	3.83e-1	2.69e3	3.14e-3	4.97e-8
	STD	1.88e-2	1.00e0	5.34e-4	2.23e-1	5.85e3	1.30e-3	3.40e-7
	Rank	4	6	2	5	7	3	1
F7	Mean	3.87e-2	1.00e0	5.58e-2	2.26e-2	4.52e0	7.30e-2	5.17e-6
	STD	4.55e-2	1.00e0	1.38e-2	7.63e-2	9.20e0	2.21e-2	3.92e-6
	Rank	3	6	4	2	7	5	1
Average of rank		3.14	5.43	4.57	2.71	6.43	4	1.43

Table 3.

Results of multimodal benchmark functions (F8–F13), with 30 dimensions

Functions		FPA	BA	DE	CMA-ES	MFO	GSA	CMPA
F8	Mean	– 6.45e3	– 5.85e3	– 6.83e3	– 8.20e3	– 8.08e3	– 2.33e3	– 8.55e3
	STD	3.04e2	1.17e3	3.95e2	1.77e2	7.87e2	2.98e2	2.96e2
	Rank	5	6	4	2	3	7	1
F9	Mean	6.57e-1	7.02e-1	1.40e1	5.74e0	1.53e1	1.94e1	2.68e-6
	STD	4.18e-1	7.52e-1	1.18e1	1.27e0	3.22e1	3.57e1	0.00e0
	Rank	2	3	5	4	6	7	1
F10	Mean	7.17e-1	9.67e-1	1.21e-2	3.30e-2	1.75e1	1.93e2	1.53e-7
	STD	3.85e-1	1.16e-1	3.31e-3	7.94e-3	4.96e1	2.44e-1	1.13e-6
	Rank	4	5	2	3	6	7	1
F11	Mean	1.97e-2	8.16e-1	1.75e-2	5.54e-5	1.62e1	5.93e2	5.69e-7
	STD	5.82e-3	1.00e0	7.21e-2	2.02e-5	5.95e0	5.51e1	3.08e-7
	Rank	4	5	3	2	6	7	1
F12	Mean	2.38e-2	6.89e-1	2.27e-3	5.58e-5	2.47e2	4.72e2	1.35e-3
	STD	1.91e-1	9.64e-1	1.72e-3	4.97e-5	1.22e3	1.54e2	2.08e-3
	Rank	4	5	3	1	6	7	2
F13	Mean	3.77e-1	1.00e0	9.13e-3	7.20e-3	2.75e7	9.60e4	1.34e-2
	STD	1.57e-1	1.00e0	1.18e-2	6.75e-3	1.05e8	1.57e4	8.07e-2
	Rank	4	5	2	1	7	6	3
Average of rank		3.83	4.67	3.17	2.17	5.67	6.83	1.50

Fig. 6.

Qualitative metrics on F1, F4, F9, F11, F16, and F17: 2D views of the functions, search history, and convergence curve; a F1, b F4, c F9, d F11, e F16, f F17

The results of multimodal benchmark functions of the seven optimization algorithms are listed in Table 3. Notably, the CMPA can obtain the first rank, which is 1.43, followed by the CMA-ES, FPA, GSA, DE, BA, and MFO in the last position. Interestingly, it can be found that the CMPA can obtain the best performances on F8, F9, F10, and F11, which correspond to four out of the six multimodal benchmark functions for best performances. The convergence performance of F9 and F11 is plotted in Fig. 6c–d, from which it is clear that the CMPA convergences faster than the other algorithms. In view of the abovementioned results and the characteristics of the benchmark functions, we conclude that the CMPA has a satisfactory exploration ability.

Table 4 reports the results of fixed-dimension multimodal benchmark functions. In these benchmark functions, the performances of the seven optimization algorithms are relatively stable. Specifically, all the optimization algorithms can obtain the same optimal solution on F17 and F18. On the F16 and F19, FPA, DE, CMA-ES, MFO, GSA, and CMPA can obtain the first rank, followed by BA. The convergence performance of F16 and F17 is plotted in Fig. 6e–f. It can be concluded that the CMPA is a competitive optimization algorithm compared to the other state-of-the-art optimization algorithms.

Table 4.

Results of fixed-dimension multimodal benchmark functions (F14–F23)

Functions		FPA	BA	DE	CMA-ES	MFO	GSA	CMPA
F14	Mean	9.98e-1	1.27e1	1.24e0	1.28e1	2.74e0	1.39e0	9.98e-1
	STD	2.00e-4	6.97e0	9.24e-1	1.82e-15	1.83e0	4.63e-1	3.39e-15
	Rank	1	6	3	7	5	4	1
F15	Mean	6.89e-4	3.00e-2	5.64e-4	3.14e-4	2.36e-3	3.48e-3	3.67e-4
	STD	1.57e-4	3.34e-2	2.82e-4	2.99e-5	4.93e-3	6.43e-2	2.32e-5
	Rank	4	7	3	1	5	6	2
F16	Mean	– 1.03e0	6.78e-16	– 1.03e0	– 1.03e0	– 1.03e0	– 1.03e0	– 1.03e0
	STD	6.78e-16	3.16e-1	6.78e-16	6.78e-16	6.78e-16	6.78e-16	6.78e-16
	Rank	1	7	1	1	1	1	1
F17	Mean	3.98e-1	3.98e-1	3.98e-1	3.98e-1	3.98e-1	3.98e-1	3.98e-1
	STD	1.69e-16	1.58e-3	1.69e-16	1.69e-16	1.69e-16	1.69e-16	1.69e-16
	Rank	1	1	1	1	1	1	1
F18	Mean	3.00e0	3.00e0	3.00e0	3.00e0	3.00e0	3.00e0	3.00e0
	STD	0.00e0	0.00e0	0.00e0	0.00e0	0.00e0	0.00e0	0.00e0
	Rank	1	1	1	1	1	1	1
F19	Mean	– 3.86e0	– 3.84e0	– 3.86e0	– 3.86e0	– 3.86e0	– 3.86e0	– 3.86e0
	STD	3.22e-15	1.42e-1	3.16e-15	3.14e-15	1.44e-3	1.24e-3	2.04e-3
	Rank	1	7	1	1	1	1	1
F20	Mean	– 3.30e0	– 3.25e0	– 3.27e-0	– 3.32e0	– 3.24e0	– 3.11e0	– 3.27e0
	STD	1.95e-2	5.89e-2	5.89e-2	1.77e-15	1.51e-1	2.92e-2	6.88e-17
	Rank	2	5	3	1	6	7	3
F21	Mean	– 5.22e-0	– 7.58e0	– 9.65e0	– 9.08e0	– 8.65e0	– 4.15e0	– 9.98e0
	STD	8.15e-3	3.51e0	1.52e0	2.54e0	1.77e0	9.20e-2	9.02e-1
	Rank	6	5	2	3	4	7	1
F22	Mean	– 5.34e0	– 9.64e0	– 9.75e0	– 9.89e0	– 1.02e1	– 9.38e0	– 9.83e0
	STD	5.37e-2	2.30e0	1.99e0	1.94e-1	7.27e-3	2.60e0	1.77e0
	Rank	7	5	4	2	1	6	3
F23	Mean	– 5.29e0	– 9.75e0	– 1.05e1	– 1.05e1	– 1.01e1	– 1.01e1	– 1.05e1
	STD	3.56e-1	2.36e0	8.88e-15	1.81e-2	1.70e0	2.61e0	– 2.43e0
	Rank	7	6	1	1	4	4	1
Average of rank		3.10	5.00	2.00	1.90	2.90	3.80	1.50

The p values of the Wilcoxon rank-sum test for F1–F23 with 30 dimensions are reported in Table 5. The values reveal the significant differences between the results of the CMPA versus the FPA, BA, DE, CMA-ES, MFO, and GSA. The Wilcoxon rank-sum with 5% is carefully performed in this section. According to the p values in Table 5, it can be detected that the observed differences in the results are statistically meaningful for all cases. In view of the abovementioned results, we conclude that the CMPA can efficiently solve medium–high-dimensional optimization problems.

Table 5.

p values of the Wilcoxon rank-sum test for F1–F23 with 30 dimensions

Functions	FPA	BA	DE	CMA-ES	MFO	GSA
F1	3.02e-11	3.31e-10	4.65e-11	3.31e-11	3.03e-11	3.31e-12
F2	3.02e-11	6.81e-12	1.24e-12	6.81e-12	6.81e-12	3.02e-11
F3	3.31e-11	7.07e-12	2.64e-11	7.07e-11	7.05e-11	3.31e-11
F4	3.02e-11	3.31e-11	4.65e-11	3.31e-11	3.01e-11	4.57e-11
F5	3.31e-11	3.02e-11	7.05e-11	7.05e-11	7.05e-11	3.02e-11
F6	1.21e-12	7.07e-11	3.02e-11	3.32e-11	7.06e-12	2.71e-11
F7	3.02e-11	1.07e-11	3.02e-11	3.31e-11	7.06e-12	7.83e-9
F8	3.05e-11	3.05e-11	4.36e-11	1.23e-11	3.05e-11	3.05e-11
F9	3.05e-11	1.35e-11	3.05e-11	1.23e-11	3.05e-11	3.05e-11
F10	1.23e-12	4.36e-12	1.23e-12	6.11e-12	2.26e-12	1.23e-12
F11	1.23e-12	1.17e-12	1.23-e12	2.26e-12	6.11e-12	4.36e-12
F12	9.57e-12	2.80e-3	3.05e-11	1.23e-11	1.08e-6	1.03e-8
F13	2.61e-11	9.17e-11	3.05e-11	3.05e-11	2.01e-6	4.78e-11
F14	8.16e-1	1.08e-3	5.21e-8	7.45e-12	9.43e-6	8.16e-6
F15	2.55e-8	2.52e-11	1.38e-7	3.36e-11	5.01e-10	5.09e-6
F16	9.15e-13	9.15e-13	1.21e-12	5.56e-3	3.02e-12	2.37e-12
F17	2.07e-12	3.02e-11	1.63e-1	5.98e-1	2.64e-3	1.63e-1
F18	5.03e-9	9.54e-12	1.94e-2	1.35e-3	1.17e-9	1.17e-9
F19	1.67e-11	8.16e-3	2.52e-11	5.02e-3	4.76e-7	8.81e-8
F20	6.16e-9	5.56e-3	5.56e-6	5.10e-6	1.17e-3	8.81e-8
F21	1.94e-8	8.16e-3	8.16e-3	6.84e-7	9.46e-6	8.36e-12
F22	2.52e-11	8.88e-7	2.84e-4	8.16e-3	8.88e-7	1.21e-9
F23	2.52e-11	1.94e-8	1.21e-12	1.74e-3	8.16e-3	2.60e-8

To evaluate the performance of the CMPA in solving high-dimensional optimization problems, the dimensions of these benchmark functions are extended to 100 dimensions to track and analyze the performance of the CMPA. The mean values, standard deviation (SD), and the ranks are reported in Table 6. Obviously, the CMPA has obvious competitive advantages over other algorithms. Specifically, the CMPA can get the first rank in 9 of the 13 benchmark functions. The second-best solution belongs to the GSA, followed by CMA-ES, FPA, DE, BA, and MFO. By comparing the p values of these optimization algorithms in Table 7, it is interesting to find that the solutions of the CMPA can significantly outperform the other optimization algorithms.

Table 6.

Results of unimodal and multimodal benchmark functions (F1–F13), with 100 dimensions

Functions		FPA	BA	DE	CMA-ES	MFO	GSA	CMPA
F1	Mean	1.40e4	2.83e5	8.27e3	1.60e-10	6.21e4	3.18e-1	0.00e0
	STD	2.82e3	1.43e4	1.33e3	1.62e-9	1.25e4	5.26e-2	0.00e0
	Rank	5	7	4	2	6	3	1
F2	Mean	1.01e2	6.01e5	1.22e2	4.32e-8	2.46e2	4.06e0	4.52e-16
	STD	9.37e0	1.19e4	2.34e1	1.46e-8	4.48e1	3.17e-1	1.31e-1
	Rank	4	7	5	2	6	3	1
F3	Mean	1.90e4	1.44e6	5.02e5	4.10e2	2.15e5	6.89e0	0.00e0
	STD	5.45e3	6.22e5	5.87e4	2.78e2	4.44e4	1.03e0	0.00e0
	Rank	4	7	6	3	5	2	1
F4	Mean	3.52e1	9.42e1	9.62e1	8.90e-1	9.32e1	2.59e-1	9.35e-2
	STD	3.37e0	1.50e0	1.00e0	9.31e-1	2.12e0	2.81e-2	6.37e-2
	Rank	4	6	7	3	5	2	1
F5	Mean	4.65e6	1.10e8	1.99e7	9.80e1	1.45e8	1.35e2	9.46e1
	STD	1.99e6	9.45e7	5.81e6	6.76e-1	7.51e7	7.30e0	1.39e-1
	Rank	4	6	5	2	7	3	1
F6	Mean	1.26e4	2.69e5	8.08e3	1.04e1	6.69e4	2.68e0	7.90e-1
	STD	2.07e3	1.26e4	1.65e3	1.06e0	1.46e4	3.95e-1	1.40e0
	Rank	5	7	4	3	6	2	1
F7	Mean	5.84e0	3.00e2	1.97e1	7.61e-3	2.57e2	1.23e0	8.19e-3
	STD	2.17e0	2.63e1	5.68e0	2.67e-3	8.90e1	2.64e-1	1.07e-3
	Rank	4	7	5	1	6	3	2
F8	Mean	– 1.28e4	– 4.05e3	– 1.19e4	– 1.68e4	– 2.31e4	– 2.85e4	– 2.60e4
	STD	4.63e2	9.56e2	5.81e2	2.63e3	1.98e3	6.92e3	6.09e3
	Rank	6	1	7	5	4	2	3
F9	Mean	8.48e2	7.95e2	1.04e3	1.04e1	8.66e2	1.78e2	0.00e0
	STD	4.02e1	6.20e1	4.03e1	9.03e0	8.02e1	9.25e0	0.00e0
	Rank	5	4	7	2	6	3	1
F10	Mean	8.22e0	1.95e1	1.23e1	1.21e-7	1.99e1	3.89e-1	5.76e-1
	STD	1.14e0	6.52e-2	8.32e-1	5.08e-8	8.59e-2	5.23e2	1.06e0
	Rank	4	6	5	1	7	2	3
F11	Mean	1.20e0	2.48e3	7.42e1	4.88e-3	5.61e2	4.51e-3	0.00e0
	STD	2.01e1	1.02e2	1.41e1	1.07e-2	1.24e2	9.72e-4	8.36e-1
	Rank	4	6	5	3	7	2	1
F12	Mean	1.55e5	2.64e8	3.91e7	2.85e-1	2.83e8	2.48e-2	8.09e-2
	STD	1.75e4	5.10e7	1.89e7	6.42e-2	1.46e8	8.91e-3	7.75e-3
	Rank	4	5	7	3	6	1	2
F13	Mean	2.77e6	5.04e9	7.20e7	6.88e0	6.69e8	5.86e0	1.06e0
	STD	1.81e6	3.49e8	2.73e7	3.34e-1	3.06e8	1.61e0	6.69e-1
	Rank	4	7	5	3	6	2	1
Average of Rank		4.38	5.85	5.54	2.46	5.92	2.31	1.46

Table 7.

p values of the Wilcoxon rank-sum test for F1–F13 with 100 dimensions

Functions	FPA	BA	DE	CMA-ES	MFO	GSA
F1	4.73e-11	7.06e-10	7.06e-11	2.69e-11	2.69e-11	5.52e-12
F2	4.73e-11	2.69e-12	7.06e-12	2.69e-12	7.06e-12	4.73e-11
F3	6.62e-11	7.06e-12	2.69e-11	6.62e-11	7.06e-11	5.52e-11
F4	4.73e-11	2.69e-11	2.69e-11	1.35e-11	3.01e-11	5.52e-11
F5	2.69e-11	4.73e-11	6.62e-11	2.69e-11	7.06e-11	3.02e-11
F6	2.69e-12	2.69e-11	4.73e-11	6.62e-11	1.35e-12	2.69e-11
F7	4.73e-11	6.62e-11	4.73e-11	1.35e-11	6.62e-12	7.06e-9
F8	5.52e-11	5.52e-11	5.52e-11	1.35e-11	3.05e-11	1.35e-11
F9	5.52e-11	1.35e-11	1.35e-11	1.35e-11	2.69e-11	6.62e-11
F10	1.35e-12	7.06e-12	9.80e-12	5.52e-12	2.69e-12	6.62e-12
F11	1.35e-12	9.80e-12	1.35-e12	2.69e-12	1.35e-12	5.52e-12
F12	9.80e-12	7.06e-3	5.52e-11	1.35e-11	7.06e-6	1.35e-8
F13	9.80e-11	9.80e-11	5.52e-11	5.52e-11	7.06e-6	4.73e-11

Benchmark Functions Set II: Ten CEC2020 Suite Problems

To verify the performance of the CMPA and investigate the capabilities of the exploration, exploitation, and local optimum avoidance, the CEC’ 2020 suite is selected as the benchmark function library, which is widely used to verify new optimization algorithms. In this section, the Two-Stage differential evolution algorithm with Mutation Strategy Combination (TS-MSCDE) [43], Distance-based parameter adaptation for Success-History-XX (DISH-XX) [44], Harris Hawk Optimization using Opposition-Based Learning (HHO-OBL) [45], SHADE with Linear Population Reduction (LSHADE) [46], and Salp Swarm Algorithm using Opposition-Based Learning (SSA-OBL) [45] are used to compare with the CMPA. The results of the CEC’2020 functions with Dim = 20 are reported in Table 8.

Table 8.

The results of different algorithms on the CEC’2020 functions with Dim = 20

Functions		TS-MSCDE	DISH-XX	HHO-OBL	LSHADE	SSA-OBL	CMPA
F1	Mean	0.00e0	0.00e0	0.00e0	0.00e0	0.00e0	0.00e0
	STD	0.00e0	0.00e0	0.00e0	0.00e0	0.00e0	0.00e0
	Rank	1	1	1	1	1	1
F2	Mean	2.44e0	8.66e1	2.82e0	2.11e0	5.14e-1	1.83e0
	STD	2.54e0	1.11e2	1.67e0	1.66e0	7.15e-1	1.29e0
	Rank	4	6	5	3	1	2
F3	Mean	1.68e0	2.04e1	2.09e1	2.12e1	2.06e1	0.00e0
	STD	3.66e0	0.00e2	5.94e-1	5.94e-1	1.27e-1	2.18e-1
	Rank	2	3	5	6	4	1
F4	Mean	2.24e-1	1.45e-1	3.56e-1	3.24e-1	1.44e-1	0.00e0
	STD	1.55e-1	5.47e-2	3.64e-2	2.91e-2	5.98e-2	0.00e0
	Rank	4	3	6	5	2	1
F5	Mean	6.02e1	5.63e1	4.58e1	4.75e1	1.09e1	1.12e1
	STD	5.46e1	6.65e1	5.82e1	5.88e1	4.29e3	4.16e1
	Rank	6	5	3	4	1	2
F6	Mean	1.73e-1	1.50e1	3.85e-1	3.51e-1	3.09e-1	7.80e-2
	STD	9.66e-2	3.58e1	8.64e-2	7.74e-2	8.21e-2	1.38e-1
	Rank	2	6	5	4	3	1
F7	Mean	7.02e0	5.11e1	9.65e-1	1.13e0	1.12e0	8.13e-1
	STD	8.32e0	6.46e0	1.67e-1	1.56e0	1.35e0	3.53e0
	Rank	5	6	2	4	3	1
F8	Mean	7.52e1	1.00e2	1.00e2	1.00e2	9.74e1	1.00e2
	STD	3.27e1	1.41e-7	2.09e-9	1.40e-12	1.99e1	0.00e0
	Rank	5	1	1	1	6	1
F9	Mean	9.68e1	4.05e2	4.05e2	4.04e2	9.69e1	4.05e2
	STD	1.84e1	2.57e0	9.31e-1	9.32e-2	1.93e1	1.88e0
	Rank	5	2	2	1	6	2
F10	Mean	2.73e2	4.14e2	4.14e2	4.15e2	4.01e2	4.14e2
	STD	1.48e2	3.60e-2	7.34e-3	7.0.35e-3	6.23e-1	5.79e-3
	Rank	1	3	3	6	2	3
Average of Rank		3.5	3.6	3.3	3.5	2.9	1.5

From Table 8, the CMPA has obvious competitive advantages over the TS-MSCDE, DISH-XX, HHO-OBL, LSHADE, and SSA-OBL. Specially, TS-MSCDE and LSHADE obtain the same rank. Notably, the CMPA obtains the first rank. The second-best solution belongs to the SSA-OBL, followed by HHO-OBL, TS-MSCDE, and LSHADE. The DISH-XX ranks in the last position.

The mean time value for the CEC’ 2020 suits is listed in Table 9. It can be seen that the CMPA uses less time to obtain the optimal solution than the other optimization algorithms, followed by HHO-OBL, TS-MSCDE, LSHADE, SSA-OBL, and DISH-XX. The scores of these optimization algorithms are plotted in Fig. 7. It is indicated that the CMPA can achieve the highest score, the second-best score belongs to the SSA-OBL, followed by the HHO-OBL, TS-MSCDE, LSHADE, and DISH-XX. Notably, the scores of the CMPA and DISH-XX are 100 and 60.6, respectively. In view of the abovementioned results, we conclude that CMPA can significantly outperform the other optimization algorithms.

Table 9.

The mean time value for the CEC’2020 functions with Dim = 20

Functions	TS-MSCDE	DISH-XX	HHO-OBL	LSHADE	SSA-OBL	CMPA
F1	9.87e-1	2.38e0	8.89e-1	1.22e0	1.75e0	7.64e-1
F2	9.85e-1	1.95e0	8.80e-1	1.25e0	1.76e0	7.59e-1
F3	9.86e-1	1.89e0	8.84e-1	1.20e0	1.76e0	7.61e-1
F4	9.97e-1	1.93e0	8.84e-1	1.23e0	1.73e0	7.69e-1
F5	9.86e-1	1.97e0	8.83e-1	1.23e0	1.72e0	7.60e-1
F6	9.90e-1	1.92e0	8.87e-1	1.25e0	1.75e0	7.58e-1
F7	9.89e-1	1.85e0	8.86e-1	1.24e0	1.77e0	7.59e-1
F8	9.87e-1	1.94e0	8.85e-1	1.26e0	1.79e0	7.93e-1
F9	9.94e-1	1.99e0	8.80e-1	1.19e0	1.74e0	7.68e-1
F10	9.82e-1	2.05e0	8.79e-1	1.19e0	1.76e0	7.81e-1

Fig. 7.

The score of the six optimization algorithms

Engineering Design Problems

A 25-Bar Space Truss Structure

This is a well-known engineering case, namely the 25-bar truss optimization design problem, which has been widely used in Refs. [46–48]. The configurations of the 25-bar are illustrated in Fig. 8. To ensure the fairness of the comparison results, the same parameters as those in the literature are selected in this example.

Fig. 8.

Configurations of the 25-bar truss

For the 25-bar truss, the results of the optimized cross-sectional areas are represented in Table 10. Specifically, all these optimization algorithms can achieve optimal solutions with the same mass of 219.92 kg. It is worth noting that these algorithms do not always get the optimal solution when solving this truss model. A comparison of the mean values of these six optimization algorithms reveals that the CMPA ranks highest, followed by the FFA, AVOA, CMA-ES, DA, and ECBO. By comparing the results of the six optimization algorithms, it can be concluded that the CMPA has a satisfactory exploration ability.

Table 10.

Optimized cross-sectional areas of the 25-bar space truss

Design variables (mm2)	ECBO [46]	CMA-ES [47]	DA [48]	FFA [49]	AVOA [50]	CMPA
A1 (S1)	64.52	64.52	64.52	64.52	64.52	64.52
A2 (S2–5)	193.55	193.55	193.55	193.55	193.55	193.55
A3 (S6–9)	2193.54	2193.54	2193.54	2193.54	2193.54	2193.54
A4 (S10–11)	64.52	64.52	64.52	64.52	64.52	64.52
A5 (S12–13)	1354.84	1354.84	1354.84	1354.84	1354.84	1354.84
A6 (S14–17)	645.16	645.16	645.16	645.16	645.16	645.16
A7 (S18–21)	322.580	322.580	322.580	322.580	322.58	322.580
A8 (S22–25)	2193.54	2193.54	2193.54	2193.54	2193.54	2193.54
Best (kg)	219.92	219.92	219.92	219.92	219.92	219.92
Worst (kg)	220.22	220.26	220.33	220.23	220.23	220.23
Mean (kg)	220.38	220.03	220.06	220.01	220.02	220.00

$S_{\text{i}}\left(i=1,2,3,···,25\right)$ denotes the cross-sectional area of the ith member.

A 52-Bar Truss Design

In this section, the purpose of the 52-bar truss is to optimize the size and shape as plotted in Fig. 9. For the 52-bar truss, the nonstructural mass of 50 kg is added to the movable nodes. The boundary conditions are also the same as the literature [51–54], i.e., all the free nodes are in the range of ± 2 m. Moreover, during the optimization process of the 52-bar truss, its symmetry should be satisfied. The optimal solutions of the different optimization algorithms are reported in Table 11.

Fig. 9.

Configurations of the 52-bar truss

Table 11.

Optimized outcomes of the 52-bar space truss

Design variables (mm2)	ReDE [51]	HS [52]	CSS-BBBC [53]	HALC-PSO [54]	MGO [55]	CMPA
ZA	6.02	4.74	5.33	5.94	6.02	6.00
XB	2.30	1.56	2.13	2.24	2.30	2.31
ZB	3.74	3.74	3.72	3.73	3.74	3.73
XF	3.99	3.49	3.94	3.96	3.96	4.00
ZF	2.50	2.63	2.50	2.50	2.50	2.50
A1	1.00	1.01	1.00	1.00	1.00	1.00
A2	1.09	1.50	1.31	1.17	1.08	1.08
A3	1.20	1.39	1.42	1.23	1.23	1.20
A4	1.45	1.35	1.39	1.43	1.43	1.45
A5	1.42	1.68	1.43	1.39	1.39	1.42
A6	1.00	1.37	1.00	1.00	1.00	1.00
A7	1.56	1.41	1.56	1.60	1.56	1.56
A8	1.39	1.94	1.45	1.41	1.39	1.39
Best (kg)	193.20	214.94	197.31	194.85	194.20	193.20
Worst (kg)	–	–	–	–	202.45	202.39
Mean (kg)	195.43	229.88	–	196.85	198.80	198.73

Table 11 reports the optimal solutions obtained by ReDE, HS, SS-BBBC, HALC-PSO, MGO, and the CMPA. It can be seen that the CMPA, ReDE and MGO provide the best solution with an optimum weight 193.20 kg, followed by HALC-PSO and HS. From the mean values, it can be concluded that the robustness of the ReDE is better than the other optimization algorithms, the main reason for this is that the ReDE is an improved optimization algorithm, which includes other search strategies.

A 72-Bar Space Truss Structure

The last example is a 72-bar space truss, the purpose is to optimize the size of the truss. Notably, the 72-bar space truss has been widely used as a classical optimization problem in Refs. [56–59]. The configurations of the 72-bar are illustrated in Fig. 10. In this case, the material parameters for the 72-bar truss are chosen to be the same as those in Refs. [56–59].

Fig. 10.

Configurations of the 72-bar truss

Table 12 compares optimal cross-sectional areas of the truss obtained by SGA, HPSO, DHPSACO, MBA, AGTO, and CMPA. Obviously, CMPA and AGTO can achieve the optimum of 389.33 lb, which performs the first rank among these optimizers. The second rank belongs to the MBA, followed by DHPSACO, SGA, and HPSO. Notably, the design variables are discrete variables, which are selected from Table 12. By considering these results in combination with the three cases, we can conclude that the CMPA can be used to efficiently solve space truss engineering problems.

Table 12.

Optimized outcomes of the 72-bar space truss

Design variable (area in.2)	SGA [56]	HPSO [57]	DHPSACO [58]	MBA [59]	AGTO [60]	CMPA
A1 (S1–4)	0.196	4.970	1.800	0.196	0.196	1.990
A2 (S5–12)	0.602	1.228	0.442	0.563	0.563	0.563
A3 (S13–16)	0.307	0.111	0.141	0.442	0.141	0.111
A4 (S17–18)	0.766	0.111	0.111	0.602	0.111	0.111
A5 (S19–22)	0.391	2.88	1.228	0.442	1.457	1.228
A6 (S23–30)	0.391	1.457	0.563	0.442	0.442	0.563
A7 (S31–34)	0.141	0.141	0.111	0.111	0.111	0.111
A8 (S35–36)	0.111	0.111	0.111	0.111	0.111	0.111
A9 (S37–40)	1.800	1.563	0.563	1.266	0.563	0.563
A10 (S41–48)	0.602	1.228	0.563	0.563	0.563	0.442
A11 (S49–52)	0.141	0.111	0.111	0.111	0.111	0.111
A12 (S53–54)	0.307	0.196	0.250	0.111	0.111	0.111
A13 (S55–58)	1.563	0.391	0.196	1.800	1.563	0.196
A14 (S59–66)	0.766	1.457	0.563	0.602	0.602	0.563
A15 (S67–70)	0.141	0.766	0.442	0.111	0.442	0.391
A16 (S71–72)	0.111	1.563	0.563	0.111	0.563	0.563
Weight (lb)	427.20	933.09	393.38	390.73	389.33	389.33
Worst weight (lb)	–	–	–	399.49	396.78	393.97
Meant weight (lb)	–	–	–	395.43	391.32	390.04
Standard eviation	–	–	–	3.04	2.41	2.17

$S_{\text{i}}\left(i=1,2,3,···,25\right)$ denotes the area of the bar in the ith number

Parameters Identification of Main Girder of Gantry Crane

Gantry crane is one of the basic equipment in the manufacturing industry and plays an important role in national economic construction. The structure of the gantry crane is usually composed of truss, which has the advantages of light weight, large span, low steel consumption, and simple stress system. The structural diagram of the gantry crane is shown in Fig. 11.

Fig. 11.

Configuration of the gantry crane

To simplify the model, the main beam of the gantry crane is simplified to a plane structure. The self-weight of the main beam, the effect of goods and accessories on the main beam can be equivalent, and the corresponding force can be applied, i.e., $F_1=50000N$, $F_2=F_3=20000N$. The simplified model for the main girder of the gantry crane is shown in Fig. 12.

Fig. 12.

The simplified model for main girder of gantry crane

In this engineering optimization problem, the minimum total mass of the main beam is selected as the optimization objective, the cross-sectional areas of the top beam are A₁, and the cross-sectional areas of the chord and bottom beam are A₂ and A₃, respectively. The optimization mathematical model can be given by:

$$\left\{\begin{array}{c}find\begin{array}{c}\end{array}\left\{\mathbf{x}\right\}=\left[x_1,x_2,\cdots ,x_{12}\right]=\left[a_1,b_1,c_1,h_1\cdots ,a_3,b_3,c_3,h_3\right]\\ min\begin{array}{c}\end{array}f\left(\left\{\mathbf{x}\right\}\right)=\sum _{i=1}^n\rho ·A_i·l_i\begin{array}{c}\end{array}\left(i=1,2,\cdots ,n\right)\\ \\ \begin{array}{c}\begin{array}{c}\begin{array}{c}s.t.\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}0<s·{\sigma }_i<{\left[\sigma \right]}_i\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}{x}_{imin}<x_i<x_{imax}\begin{array}{c}\end{array}i=1,2,\cdots ,n\\ \end{array}\end{array}\\ \end{array}\right)$$ 10

where x is the design variable, f is the system mass, L_i is the length of the ith member, $\rho$ is the material density, x_i is the cross-sectional area of the ith member, s is the safety factor, $\sigma$ and $\left[\sigma \right]$ represent the member strength and yield strength, respectively; x_min and x_max are the lower and upper bounds values of the design variables, respectively. The results of the main girder on the gantry crane are listed in Table 13, where " + ” and “–" represent an increase and a decrease, respectively.

Table 13.

Results of the main girder on the gantry crane

Name	Initial value	Optimal value	Improved/%
$x_1$/m	1.50 × 10–2	1.45 × 10–2	– 3.33%
$x_2$/m	1.20 × 10–2	1.13 × 10–2	– 5.83%
$x_3$/m	2.60 × 10–1	2.36 × 10–1	– 9.23%
$x_4$/m	2.60 × 10–1	2.13 × 10–1	– 18.08%
$x_5$/m	3.80 × 10–2	3.92 × 10–2	+ 3.16%
$x_6$/m	2.6 × 10–2	2.41 × 10–2	– 7.31%
$x_7$/m	3.60 × 10–1	3.32 × 10–1	– 7.78%
$x_8$/m	3.60 × 10–1	3.29 × 10–1	– 8.61%
$x_9$/m	1.50 × 10–2	1.35 × 10–2	– 10.00%
$x_{10}$/m	1.20 × 10–2	1.26 × 10–2	+ 5.00%
$x_{11}$/m	2.00 × 10–1	2.03 × 10–1	+ 1.50%
$x_{12}$/m	2.00 × 10–1	1.87 × 10–1	– 6.50%
Mass/kg	7203.03	6018.82	16.44
Deflection /m	3.07 × 10–3	2.84 × 10–3	7.49

Table 13 reports the results of the gantry crane. It is obvious that the mass of the optimized main girder is significantly less than that before the optimization, i.e., the mass of the main girder is optimized from 7203.03 kg in the initial design to 6018.82 kg, the reduction range is 16.44%, and the deflection of the main girder is optimized from the initial design 3.07 × 10^–3 m optimized to 3.07 × 10^–3 m, with a reduction of 7.49%.

Conclusion and Future Work

In this study, a COVID-19 prevention-inspired optimization algorithm, named Coronavirus Mask Protection Algorithm (CMPA), is proposed. The CMPA consists of three phases, i.e., the process of infection, virus transmission, and the immune process. Each phase is characterized by mathematical equations. The performance of the CMPA is evaluated by 23 famous benchmark functions, CEC2020 suite problems, and 3 truss design problems. Results indicate that the CMPA can provide competitive solutions compared to some state-of-the-art optimization algorithms. In addition, the CMPA is applied to optimize the structure parameters of a gantry crane. The results indicate that the CMPA is a promising optimization algorithm and can obtain optimal solutions. Based on the advantages of CMPA, it can be extended to multidisciplinary design optimization and multi-objective design optimization problems. Notably, the CMPA still has the potential to improve its performance on ultra-high-dimensional optimization problems, e.g., problems with more than 500-dimensional design variables. To deal with this problem, the CMPA performance can be improved by considering the transmission ability of the virus and the resistance of different populations.

Funding

This study was supported by Henan Natural Science Foundation, No. 222300420168, Yongliang Yuan, Science and Technology Plan Project of Henan Province, No. 222102210182, Jianji Ren, National Natural Science Foundation of China, No. 52005081, Xiaokai Mu, Natural Science Foundation of Henan Polytechnic University, B2021-31, Yongliang Yuan, Nonlinear equipment dynamics team of Henan Polytechnic University, T2019-5, Junkai Fan, Fundamental Research Funds for the Universities of Henan Province, NSFRF220415, Yongliang Yuan.

Data Availability

All data generated or analyzed during this study are included in this published article.

Declarations

Conflict of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.