Utilizing bee foraging behavior in mutational salp swarm for feature selection: a study on return-intentions of overseas Chinese after COVID-19

Abstract

We present a bee foraging behavior-driven mutational salp swarm algorithm (BMSSA) based on an improved bee foraging strategy and an unscented mutation strategy. The improved bee foraging strategy is leveraged in the follower location update phase to break the fixed range search of salp swarm algorithm, while the unscented mutation strategy on the optimal solution is employed to enhance the quality of the optimal solution. Extensive experimental results on public CEC 2014 benchmark functions validate that the proposed BMSSA performs better than nine well-known metaheuristic methods and seven state-of-the-art algorithms. The binary BMSSA (bBMSSA) algorithm is further proposed for feature selection by using BMSSA as the selection strategy and support vector machine as the classifier. Experimental comparisons on 12 UCI datasets demonstrate the superiority of bBMSSA. Finally, we collected a dataset on the return-intentions of overseas Chinese after coronavirus disease (COVID-19) through an anonymous online questionnaire and performed a case study by setting up a bBMSSA-based feature selection optimization model. The outcomes manifest that the bBMSSA-based feature selection model exhibits a conspicuous prowess, attaining an accuracy exceeding 93%. The case study shows that the development prospects, the family and job in the place of residence, seeking opportunities in China, and the possible time to return to China are the critical factors influencing the willingness to return to China after COVID-19.

Graphical Abstract

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1. Introduction

COVID-19 appeared in 2019 and is spreading rapidly in countries across the globe, causing severe impacts on different sectors (Q. Li et al., 2020; Xie et al., 2023). One of the major impacts of COVID-19 on global population migration has also been observed. Studying the behavioral intentions of individuals is a well-established research area, not a novel subject (G. Chen et al., 2022). Research has shown that intention is a good predictor of future migration behavior (Van Dalen & Henkens, 2008). The time migrants stay in the host countries affects the host and home country economies (Steiner, 2019). If immigrants intend to return, they are likely to invest their human and physical capital in their home countries, which positively affects the home countries’ economic development. This leads more and more researchers (Abarcar, 2017; Czaika, 2015; Tiwari, 2021; Toma & Villares-Varela, 2019) to investigate the important factors influencing immigrants’ intentions to return to their home countries. The return-intention is divided into two classes, return and not return, and the influencing factors can be treated as features. In this work, we collected a dataset on the return-intentions of overseas Chinese after COVID-19 through an anonymous online questionnaire. Consequently, the important factors influencing the return-intentions of overseas Chinese after COVID-19 can be mined by selecting the most effective features from the dataset, which is actually an optimization problem.

In recent years, optimization methods have garnered significant attention (B. Li et al., 2022). It is worth noting that while optimization methods are often categorized into deterministic and approximated approaches, such classifications may oversimplify the complexities of modern problem-solving (Bai et al., 2021; Ma et al., 2023). To fully appreciate the landscape of optimization, we must recognize that robustness and efficiency alone may not be sufficient benchmarks for evaluating their applicability, especially in combinatorial optimization cases (Cao et al., 2020; Ni et al., 2022, 2023). However, complex optimization problems exist in all areas of literature, such as feature selection (Qaraad et al., 2022b), neural network optimization (Hussein et al., 2023; Q. F. Luo et al., 2021), natural language processing (Shrestha et al., 2020), time scheduling problems (Sun et al., 2023), energy problems (Kumar et al., 2021), path planning (Wang et al., 2022), image segmentation (C. Chen et al., 2021; Xing et al., 2023b), and engineering design (J. Song et al., 2022; X. Zhou et al., 2022). However, traditional gradient-based optimization methods are often inefficient for solving high-dimensional problems, and gradient-based methods also suffer from local optimality problems (J. Luo et al., 2019). Swarm and evolutionary methods such as ant colony optimization and genetic algorithms, represent prominent categories of optimization algorithms grounded in iterative processes such as mutation and crossover (Qian et al., 2022; Zheng et al., 2022). These methods facilitate the exploration of solutions without reliance on gradient information, as long as the resulting solutions align with optimality criteria (Heidari, Akhoondzadeh, et al., 2022). Metaheuristic algorithms (MASs) can extract useful information from complex data and solve optimization problems more effectively than traditional optimization methods. Therefore, differential evolution (DE, Storn & Price, 1997), slime mold algorithm (SMA, S. M. Li et al., 2020), fruit fly optimization algorithm (FOA, Pan, 2012), moth-flame optimizer (MFO, Mirjalili, 2015), Harris hawks optimization (HHO, Heidari, Mirjalili, et al., 2019), firefly algorithm (FA, Yang & He, 2013), particle swarm optimization (PSO, Kennedy & Eberhart, 1995), bat algorithm (Yang, 2010), Runge Kutta optimizer (Ahmadianfar et al., 2021), colony predation algorithm (Tu et al., 2021), and a large number of metaheuristics have been developed to replace optimization methods and better solve practical optimization problems. However, the original MAS also has some shortcomings. Therefore, researchers have improved the MAS from exploration and exploitation (the two main aspects that affect the performance of algorithms) and proposed algorithms with great performance such as PSO with an aging leader and challengers (ALCPSO, W.-N. Chen et al., 2013), a hybrid approach based on chameleon swarm algorithm and mayfly optimization (Rizk-Allah et al., 2022), the ensemble of mutation strategies and control parameters values for DE (EPSDE, Mallipeddi et al., 2011), sine cosine algorithm (SCA) based on orthogonal parallel information (Rizk-Allah, 2019), PSO with a chaotic and Gaussian local search (CGPSO, Jia et al., 2011b), A-C parametric whale optimization algorithm (ACWOA, Elhosseini et al., 2019), DE based on chaos local search (DECLS, Jia et al., 2011a), SCA with quantum local search (Rizk-Allah, 2021), and generalized oppositional teaching learning based optimization (GOTLBO, X. Chen et al., 2016), etc.

The salp swarm algorithm (SSA, Mirjalili, Gandomi, et al., 2017) was proposed by simulating the group behavior of the salp swarm in 2017. There are two categories in the population: leaders and followers. In each evaluation, the leaders guide the followers and move towards the food in a chain-like behavior. The leader performs global exploration while the followers thoroughly explore locally during the movement. Although the SSA algorithm has the characteristics of a simple search mechanism and few processing parameters, the convergence speed of SSA is slow in solving complex optimization problems, and there is a tendency to fall into local optimal solutions. The research on the SSA algorithm is still in its initial stage, and to better optimize the shortcomings of the algorithm, many researchers have studied it. The optimization of SSA is mainly divided into two categories: the incorporation of various strategies and the combination with other metaheuristic algorithms.

Based on the characteristics of different MAS, the researchers merged them in different phases of SSA. Z. Zhou et al. (2022) took advantage of the strong exploration capabilities of ant lion optimizer (ALO) to improve the global exploration of SSA. Xiang et al. (2022) introduced the regularity in the individual position update rule of SCA into SSA to expand the search range of solutions. Singh et al. (2023) used the position update approach of the exploitation phase in HHO to guide the forward direction and speed of individuals in SSA. Since the unique hierarchy feature of grey wolf optimization (GWO) makes the global exploration capability of GWO superior, Qaraad et al. (2022a) and Zhao et al. (2022) took advantage of this character to improve SSA. To improve the performance of the photovoltaic system, Dagal et al. (2022) and Rizk-Allah and Hassanien (2021) embedded the PSO into the SSA. Saafan and El-Gendy (2021) proposed a hybrid algorithm by selectively performing an improved whale optimization algorithm or SSA, depending on the condition.

In addition to MAS, different strategies can also be employed to boost the efficiency of SSA. Sayed et al. (2018) proposed an improved chaos SSA (CSSA) based on chaos theory to address the slow convergence of SSA. In a similar study, Zhang, Chen, Heidari, et al. (2019) upgraded SSA using chaotic mechanisms, Gaussian mutation, and Cauchy mutation, leading to a chaos mutation-driven SSA (CMSSA) that aims to strike a balance between global and local search capabilities. Qais et al. (2019) recognized the need for improved equilibrium between global and local search capabilities in SSA and introduced an enhanced SSA (ESSA) based on Gaussian theory. Xia et al. (2022) embedded a quasi-opposition-based learning strategy into SSA (QBSSA). Kang et al. (2023) believed that combining the golden sine operator and opposition-based learning could mitigate the shortcomings of the original SSA. Hence, they proposed an improved SSA (ISSA) based on these strategies, and experimental results confirmed the effectiveness of this combination. Ren et al. (2021) introduced an upgraded SSA based on adaptive weights and the Levy flight mechanism (WLSSA), aiming to address the issue of SSA frequently getting trapped in local optima. Lin et al. (2023) perceived a lack of information exchange among individuals in SSA, leading to insufficient exploration capability. Consequently, they proposed a double mutation SSA (DMSSA) based on the cuckoo mutation strategy and an adaptive DE mutation strategy.

As mentioned earlier, numerous researchers have tried to enhance the SSA’s performance. However, some fundamental challenges persist, including local optima stagnation, suboptimal exploitation, and an imbalance in exploration and exploitation operations. These challenges prompted us to revisit SSA to address its limitations. Specifically, there are two key areas where SSA could benefit from improvement. Firstly, in the basic SSA, the swarm does not effectively share information between leaders and followers. This limitation impacts the algorithm’s exploration capability, potentially causing it to overlook critical search regions. Besides, as the algorithm progresses through iterations, the diversity of the population tends to decrease, particularly in the late stages of evaluation. This reduction in diversity can lead the algorithm towards local optima, hindering its ability to find globally optimal solutions.

To alleviate the shortcomings of SSA, an improved variant of SSA, the bee foraging behavior-driven mutational salp swarm algorithm (BMSSA), based on the improved food foraging strategy and unscented mutation mechanism, is proposed in this paper. For the problem of lacking exploration capability, the improved food foraging strategy based on bee foraging behavior (BF) of artificial bee colony (ABC) algorithm (Karaboga & Basturk, 2008) is introduced into SSA. BF expands the search range of the solution and effectively improves the global exploration capability of the algorithm through the exchange of location information between individuals. For the problem of decreasing population diversity, the unscented mutation strategy (MUS) is considered to be embedded in the SSA. MUS can generate multiple candidate solutions in the vicinity of a solution, thus enhancing the diversity of the algorithm. In this way, the algorithm’s exploration and exploitation capabilities can be effectively balanced. Moreover, a classification model with BMSSA as the search strategy and support vector machine (SVM) as the classifier is built to perform feature selection experiments to verify the ability of BMSSA to solve real-world problems. Finally, BMSSA is employed to explore the factors affecting the willingness of overseas Chinese to return to China after COVID-19.

In summary, the main contributions of this paper are as follows:

• By taking advantage of BF and MUS, a new BMSSA is proposed, enabling performance gains in both convergence rate and accuracy.

• A binary BMSSA (bBMSSA) algorithm for feature selection is proposed, achieving higher classification accuracy and fewer features.

• The return-intentions of overseas Chinese after COVID-19 and the critical influencing factors are studied using the proposed bBMSSA.

2. Preliminaries

2.1. Evolutionary feature selection methods overview

Feature selection is a way of dimensionality reduction that selects a subset from the input feature dataset without changing the original feature space. There are three types of feature selection methods: filter, wrapper, and embedding. Compared with the other two approaches, the wrapper method can obtain better results because it considers the relationship between features (Zhang, Liu, Wang, et al., 2021; X. Zhou et al., 2023). However, exhaustive enumeration of combinations of feature subsets is difficult, especially when dealing with high-dimensional data. Recently, several evolutionary algorithm-based methods have been used for choosing a subset of candidate features.

X.-F. Song et al. (2022) observed that even after dimensionality reduction using existing algorithms, the number of features remained substantial. Consequently, they innovatively proposed a novel three-stage hybrid feature selection method. The experimental results indicate that the proposed algorithm achieves optimal classification accuracy with relatively short execution times on most datasets. Eslami et al. (2023) introduced a novel binary aphid–ant mutualism algorithm. This algorithm effectively mitigates the risk of getting trapped in local optima by facilitating information exchange among individual members and employing a random crossover operator. Experimental results demonstrate that the proposed algorithm can identify optimal feature subsets on most datasets, improving classification accuracy. As Hussein et al. (2023) observed, the population diversity of MFO gradually decreased in the search process, potentially leading to the algorithm’s inability to escape local optima. Therefore, they proposed an enhancement approach that incorporates the Gaussian mutation strategy and scaling mechanism strategy to improve the performance of MFO. The performance of the proposed algorithm in solving the feature selection problem was evaluated on 20 different datasets. Zhang, Wang, Gong, et al. (2022) proposed an innovative feature selection method for high-dimensional unbalanced data with missing values. Their approach incorporates a filling risk index, an improved F-measure, and a PSO variant based on fuzzy clustering and pruning theory. Experimental results demonstrate that the proposed algorithm exhibits advantages in both classification accuracy and algorithm runtime when compared with the benchmark algorithms. Karimi et al. (2023) proposed a semi-supervised feature selection method based on ant colony optimization. This approach aims to find the optimal feature subset by considering the minimum redundancy among features and the maximum relevance between features and classes. Additionally, they enhance the ant colony algorithm using reinforcement learning principles. Altarabchi et al. (2023) proposed a two-stage surrogate-assisted evolutionary approach to reduce the computational cost of evolutionary algorithms when dealing with feature selection problems. Experimental results show that the proposed model converges faster and more accurately recognizes a subset of features.

The previous work illustrates that evolutionary algorithm-based feature selection methods can solve the feature selection problem well (Abualigah & Diabat, 2022; Ahmed et al., 2021; Arora & Anand, 2019; Devi et al., 2022; Dhindsa et al., 2022).

2.2. The basic SSA revisited

SSA (Mirjalili, Gandomi, et al., 2017) is an excellent metaheuristic algorithm that simulates the swarming behavior of bottle sea squirts while navigating and foraging in the ocean. The leaders in the population guide the population to find the optimal region in the search space, while the followers follow each other and use the region around the leaders to explore the optimal solution. The SSA is described as follows:

Step 1: Let N be the population size and Dim be the search space dimension, X_{i, j} is the location of the ith individual in jth dimension, where i = 1, 2, …, N and j = 1, 2, …, Dim. The initial population X_{i, j} is randomly generated in the search space using equation (1):

where r₁ is a random number within [0, 1], ub and lb are the upper bound and lower bound of search space, respectively.

Step 2: Calculate each individual’s fitness according to the objective function. Then, the fitness value is ranked, and the current food source is the position of the individual with the smallest fitness value, which is the optimal solution for the objective function. The population is divided into two subpopulations of leaders and followers, and the number of each subpopulation accounts for half of the population.

Step 3: If |$i\le \frac{N}{2}$|⁠, update the position of leaders according to equation (2):

where X_{i, j} is the position of ith individual in jth dimension, X_{best, j} denotes the position of the food source (best solution) on jth dimension, c₁ is a coefficient that decreases adaptively as the number of evaluations increases, and c₂ and c₃ are two random numbers within [0, 1]. c₁ can be described as equation (3):

where FEs denotes the current number of function evaluations and E denotes the maximum number of function evaluations.

Step 4: If |$i\gt \frac{N}{2}$|⁠, update the position of followers according to equation (4):

Step 5: Calculate the fitness value of each individual in the population and update the position of the optimal solution in the current evaluation according to the greedy selection rule.

Step 6: Steps 3∼5 are repeated until the maximum evaluation number is reached.

3. The Proposed BMSSA

3.1. Pipeline of BMSSA

From equation (4), the position update of the followers can be seen to be related to the previous individual, which makes the position update of each individual confined to a small range. Since the population exists in a chain form, if an individual falls into a local optimum, it may cause the whole population to gather to a fixed position, which makes the algorithm fall into a local optimum. As the number of evaluations increases, the population richness decreases. These can make the algorithm have a high probability of dropping into a local optimum and converge prematurely.

This paper combines BF and MUS into the basic SSA for improvement. BF breaks the inherent pattern of follower position update by performing a random search of the neighbors of the follower. MUS improves the quality of the optimal solution by exploring the vicinity of the optimal solution and generating multiple candidate solutions.

The flowchart of BMSSA is shown in Fig. 1. Specifically, BMSSA is described as follows:

Step 1: Initialize the population. The population is divided into two subpopulations of leaders and followers, each accounting for half of the population size. Calculate each individual’s fitness and decide the optimal one based on the fitness.

Step 2: Update the position of each leader according to equation (2).

Step 3: Each follower generates a new follower based on the BF strategy using equation (5), and the follower’s position is updated by comparing the fitness of the current follower and the new follower through a greedy selection mechanism.

Step 4: Update the position of X_best based on the MUS strategy.

Step 5: Compare the fitness values of all individuals and update the position of the current best individual.

Step 6: If the evaluation termination condition is satisfied, the position of the optimal individual is output; otherwise, Steps 2∼5 are repeated.

The pseudo-code of the proposed BMSSA is described in Algorithm 1. The details of the two improved strategies will be described in the following subsections.

3.2. Improved food foraging strategy for followers

As mentioned earlier, the position change of a follower is closely related to its previous individual. As a result, for the basic SSA, the fixed character of such position change hardly helps the individual to deal with the local optimum situation. It is inspired by BF in ABC algorithm (Karaboga & Basturk, 2008), combining the randomness of BF in ABC with the position updates of followers in SSA.

A random search around the follower is first performed according to BF to generate a candidate individual X_candidate. Subsequently, the fitness value of the individual is used as a criterion to determine the position of the follower X_follower. If the fitness of X_candidate is better than X_best, the position of X_best is replaced by X_candidate. Meanwhile, the information about the position of X_follower is considered valuable. Therefore, the position of X_follower also remains unchanged. Otherwise, the fitness of X_follower and X_candidate are compared. If the fitness of X_candidate is better than X_follower, the position of X_follower is replaced by X_candidate.

The candidate individual X_candidate is generated as shown in equation (5). From the equation, it can be seen that the candidate individual’s position is not limited to the position of the follower’s previous individual but is related to the position of any individual in the search space. This randomness strengthens the information exchange between individuals and breaks the fixed pattern of follower position change in basic SSA. Hence, the global exploration capability of SSA is enhanced.

where X_{random, j} is a random follower near on the X_follower on jth dimension. ϕ ∈ [−1, 1] is a random number.

3.3. MUS for the best individual

The SSA simulates the chain-forming behavior of the salp swarm. This chain structure can lead to an insufficient search range. As the number of evaluations increases, the insufficient population richness decreases, making the algorithm fall into the local optimum and converge prematurely. Therefore, MUS (Guo et al., 2021) inspired by the unscented transform (Julier & Uhlmann, 2004; Julier et al., 1995) is utilized to update the optimal solution. Mutating the optimal solution makes introducing new solutions during the search possible to ensure that it does not fall into a local optimum too early and further improves the algorithm’s local exploitation ability and the solution’s quality.

MUS enables further exploitation of the location information of X_best by generating multiple individuals around X_best. 2 × Dim candidate solutions are randomly generated according to X_best and its covariance matrix P_x. Compared with other mutation strategies, such as Gaussian mutation and Cauchy mutation, the most important feature of MUS is that multiple candidate solutions can be generated in one mutation, and the number of candidate solutions increases as the dimensionality increases. The mathematical model of MUS is described as follows:

where ( · )_s denotes the sth column of the square root matrix, θ is a small number, and θ = 1 × 10⁻² in this paper. According to equation (6), it can be found that the greater the value of κ is, the farther the candidate solution is away from X_best. Since it is developed on a small range of the current solution, θ is taken to be a smaller number. r₂ is a linearly decreasing factor that ensures that the algorithm exploits the current solution in a small range as the number of evaluations increases.

Figure 2 shows the schematic diagram of the generation of individuals for Dim = 2. Assume |$X_0^\prime$| is the current solution, |$X_1^\prime ,X_2^\prime ,X_3^\prime$|⁠, and |$X_4^\prime$| are four points obtained by MUS. Since Dim = 2, the mutation produces four individuals.

Then, calculate the fitness of 2 × D individuals and the best individual X_new in population |$X_s^{\prime }$| is selected to update the position of the current optimal solution.

3.4. Time complexity analysis

The computational complexity of the BMSSA is built around five key components: initialization, fitness evaluation, population updating, MUS, and BF. Mainly influenced by maximum evaluation times (E), population size (N), and problem dimension (Dim), the overall time complexity is the sum of the following terms:

• Initializing all individuals: |$\mathcal {O}(N\times {Dim})$|⁠.

• Calculating the fitness of the population: |$\mathcal {O}(N\log N)$|⁠.

• Updating the positions of search agents: |$\mathcal {O}(E\times N/2\times (Dim+3))$|⁠.

• Implementing BF: |$\mathcal {O}(E\times {N/2}\times (4+Dim))$|⁠.

• Performing MUS: |$\mathcal {O}(E\times (4+(N+2)\times {Dim}^2+N\log N))$|⁠.

Therefore, the total time complexity of the proposed BMSSA is |$\mathcal {O}(N\times {Dim})+\mathcal {O}(N\log N)+\mathcal {O}(E\times {N/2}\times (Dim+3))+\mathcal {O}(E\times N/2\times (4+Dim))+\mathcal {O}(E\times (4+(N+2)\times {Dim}^2+N\log N))\iff \mathcal {O}(E\times N\times Dim)$|⁠.

4. Experimental Results

In this section, the performance of BMSSA is verified using IEEE CEC 2014 benchmark functions. The proposed BMSSA was compared with two types of algorithms separately, including basic MAS [HHO (Heidari, Mirjalili, et al., 2019), FA (Yang & He, 2013), GWO (Mirjalili, Mirjalili, et al., 2014), HGS (Yang et al., 2021), DE (Storn & Price, 1997), SMA (S. M. Li et al., 2020), FOA (Pan, 2012), MFO (Mirjalili, 2015), and SSA (Mirjalili, Gandomi, et al., 2017)] and state-of-the-art (SOTA) algorithms [ALCPSO (W.-N. Chen et al., 2013), EPSDE (Mallipeddi et al., 2011), CGPSO (Jia et al., 2011b), ACWOA (Elhosseini et al., 2019), EOBLSSA (R. Chen et al., 2019), DECLS (Jia et al., 2011a), and GOTLBO (X. Chen et al., 2016)]. For the equality of the experiment, the experiment was conducted under the same environment configuration, i.e., N = 30, E = 300 000, and times per algorithm tested independently (flod = 30). Moreover, the Wilcoxon signed-rank test (Derrac et al., 2011), the Friedman test (García et al., 2010), and the Bonferroni–Dunn test (Demšar, 2006) were used to analyze the experimental results.

4.1. Parameter tuning

This section aims to conduct a sensitivity analysis on the parameter θ to investigate its impact on the performance of BMSSA. The value of the parameter θ largely determines the extent to which the algorithm explores the vicinity of the optimal solution. When the value of θ is too large, it reduces the algorithm’s exploration range near the optimal solution. This implies that more potential and possibly superior solutions in the vicinity may be overlooked, thereby experiencing slow convergence. On the other hand, when the value of θ is too small, it significantly weakens the algorithm’s exploration capability. This can lead to the algorithm getting stuck in local optima. Consequently, the choice of the θ value is intricately linked to achieving a balance between exploration and exploitation for the algorithm.

In this experiment, five different BMSSA variants with θ values set to 0.01, 0.03, 0.05, 0.07, and 0.09 are tested using the CEC 2014 benchmark with Dim = 30. Subsequently, the experimental result is analyzed using the Friedman test. Table 1 presents the Friedman test ranking of algorithm performance corresponding to various values of θ, with lower values indicating superior performance. The results displayed in the table reveal that the algorithm performs best when θ = 0.01, achieving the first rank with the lowest average ranking. Consequently, the value of θ is determined to be 0.01.

4.2. Ablation study on BMSSA

To measure BF and MUS’s effectiveness in BMSSA and find the optimal strategy, we conducted an ablation experiment of BMSSA on CEC 2014 benchmark functions. Dim was set as 30 in this experiment. BSSA indicates that only BF is added, and MSSA indicates that only MUS is included. The average ranking (AVR) and final ranking (Rank) value of these four optimization methods are shown in Table 2. The mean value (AVG) for each algorithm on each benchmark function was obtained by averaging the results of 30 independent runs of each algorithm. Then, the ranking result of each algorithm on each benchmark function is obtained by comparing AVG. AVR of each algorithm was obtained by averaging the ranking values of each algorithm on 30 benchmark functions. The Rank values were obtained by comparing the AVR values of all algorithms.

Figure 3 shows the ablation results of the convergence curves for the 12 benchmark functions. The convergence speed of BSSA is better than that of SSA, indicating that embedding BF is good for improving the global exploration ability of the algorithm, boosting the convergence speed of the algorithm, and contributing to the algorithm tripping out of the local optimum. Although BMSSA and BSSA converge at about the same speed in the early evaluation, the solution quality of BMSSA is better than that of BSSA in the late evaluation, indicating that the introduction of MUS enhances the local exploitation ability of the algorithm and improves the solution quality. BMSSA finally ranked first with an average ranking value of 1.6.

4.3. Balance and diversity analysis on BMSSA

In this subsection, the performance of the proposed algorithm is further analyzed in five aspects: population balance, population diversity, search trajectory, average fitness value, and convergence of BMSSA.

Figure 4 shows the population balance analysis plots. The algorithm performs global exploration at the beginning of the evaluation and local exploitation of the solution at the later evaluation stage. In the balanced analysis plots of F2 and F7, the exploitation proportion of BMSSA is higher than that of the basic SSA. The exploration proportion of BMSSA in F13 and F15 is higher than that of the basic SSA. It indicates that the proposed algorithm BMSSA effectively enhances the exploration and exploitation capability of the basic SSA.

Figure 5 shows the population diversity analysis plots. Since each search individual is distributed at random positions in the search space at the beginning of the evaluation, the initial distance between search individuals is far. However, the distance between individuals gradually decreases as the search region is continuously reduced to find the optimal solution. From the plots of F2, F7, F13, and F15, it can be concluded that the BMSSA curve smoothes out earlier than the SSA curve, which indicates that BMSSA can find the optimal solution earlier than SSA, verifying that the addition of BF speeds up the convergence rate.

As shown in Fig. 6, six functions F1, F2, F4, F11, F23, and F27, are selected from the CEC 2014 benchmark functions to analyze three aspects of the algorithm: the trajectory of the search, the average fitness value, and the convergence of the method. F1 and F2 are unimodal functions, F4 and F11 are multimodal functions, and F23 and F27 are composition functions. The first column displays 3D location distribution corresponding to various functions. The second column illustrates the 2D distribution of BMSSA on these functions, where the red dots represent the location of the optimal solution found and the black dots represent the search individuals. The third, fourth, and fifth columns respectively depict the search trajectory of BMSSA, average fitness values of BMSSA, and the convergence curves of BMSSA and SSA. From the convergence curves in Fig. 6 (rightmost), we can see that BMSSA has a more powerful seeking ability in finding the best solution than SSA, and the convergence speed of BMSSA is also outstanding in the composition functions F23 and F27.

4.4. Comparison with basic MAS

To study the performance of the proposed algorithm, experiments on CEC 2014 benchmark functions along with nine MAS, namely HHO (Heidari, Mirjalili, et al., 2019), FA (Yang & He, 2013), GWO (Mirjalili, Mirjalili, et al., 2014), HGS (Yang et al., 2021), DE (Storn & Price, 1997), SMA (S. M. Li et al., 2020), FOA (Pan, 2012), MFO (Mirjalili, 2015), and SSA (Mirjalili, Gandomi, et al., 2017), were conducted. The parameter setting of these algorithms is shown in Table 3.

The convergence results on 12 of 30 functions with Dim = 30 are plotted in Fig. 7. The convergence rate of BMSSA is faster than comparison algorithms on most functions because of the BF strategy. Since the population exists in a chain structure, the leaders direct the way forward, and the followers seek the optimal solution. Embedding the BF strategy into the followers’ position update can expand the search range of the solution, which can boost the global exploration ability and increase the chance of finding the optimal solution. Besides, the introduction of MUS can improve the solution’s quality. Improving the local exploitation capability of the algorithm by searching for solutions near the current optimal solution helps the algorithm to skip the local optimum. The convergence plots of F5, F11, F16, F24, and F30 show that BMSSA has a strong ability to jump out of the local optimum and thus is able to further improve the solution accuracy. FOA, FA, and MFO algorithms fall into a local optimum in the early iterative stage. Though the convergence speed of BMSSA is not as quick as that of HGS, the ability of BMSSA to find the optimal solution is stronger than that of HGS. The superior performance of BMSSA over the compared algorithms illustrates the effectiveness of MUS and BF.

Table 4 shows the ranking results of BMSSA and basic MAS. The overall mean values of BMSSA are ranked the first among these algorithms. BMSSA outperforms FA, FOA, and MFO for all 30 benchmark functions; outperforms DE for 24 functions and with similar performance on 1 function; outperforms HHO on 24 functions, similar performance on 6 functions; better than GWO on 27 functions, with similar performance on 2 functions; better than HGS on 21 functions, similar performance on 9 functions; and better than SSA on 20 functions, similar performance on 9 functions. The average ranking value of BMSSA is 1.433 with Dim = 50, which is 2% better than 1.467 that in Dim = 30. Compared with FA, FOA, and MFO, BMSSA outperforms them on all 30 benchmark functions in three dimensions. BMSSA outperforms DE on 26 functions, which is 2 more than 24 functions with Dim = 30, and outperforms SSA on 18 functions with Dim = 30. The average rank of BMSSA with Dim = 100 is 1.533, which is 51% better than the rank second algorithm, HGS. Moreover, it can be noticed that the average rank of HGS is the second in three dimensions.

4.5. Comparison with SOTA algorithms

To further verify the effectiveness of BMSSA, we experimentally compared BMSSA with SOTA algorithms, including ALCPSO (W.-N. Chen et al., 2013), EPSDE (Mallipeddi et al., 2011), CGPSO (Jia et al., 2011b), ACWOA (Elhosseini et al., 2019), EOBLSSA (Hussien, 2022), DECLS (Jia et al., 2011a), and GOTLBO(X. Chen et al., 2016). Table 5 is the detailed parameter setting of the comparison algorithms.

Figure 8 reports the convergence curves of 12 functions with Dim = 30. Although BMSSA converges more slowly in some functions, other algorithms have fallen into local optimum to different degrees. Moreover, BMSSA avoids this problem well and achieves better convergence accuracy. This also proves that adding MUS helps enhance the local exploration capability of the BMSSA and improves the probability of the algorithm jumping out of the local optimum and convergence accuracy.

The ranking results of BMSSA and SOTA algorithms are shown in Table 6. BMSSA has the first rank on average in all three dimensions, and the ranking value becomes smaller as the dimension increases. BMSSA outperforms EOBLSSA, CGPSO, ALCPSO, ACWOA, DECLS, EPSDE, and GOTLBO in 24, 27, 22, 29, 24, 15, and 16 of the 30 tested functions with Dim = 30, respectively. Although BMSSA is inferior to EPSDE and GOTLBO in 13 and 7 functions, the overall average ranking of BMSSA is the first, indicating that the overall performance of BMSSA is better than its competitors. The average ranking value of BMSSA is 1.7667 with Dim = 50, which is about 17% higher than that of 2.1333 with Dim = 30. The average ranking value of BMSSA is 1.7333 with Dim = 100, an improvement of about 18% from 2.1333 with Dim = 30, and about 1.8% from 1.7667 with Dim = 50. In addition, BMSSA outperforms CGPSO for all 30 functions with Dim = 100.

4.6. Comparison with SSA variants

In this section, seven SSA variants, including ISSA (Kang et al., 2023), ESSA (Qais et al., 2019), DMSSA (Lin et al., 2023), QBSSA (Xia et al., 2022), WLSSA (Ren et al., 2021), CSSA (Sayed et al., 2018), and CMSSA (Zhang, Chen, Heidari, et al., 2019) are chosen to be compared with BMSSA. The detailed parameter setting of these algorithms is depicted in Table 7.

Figure 9 provides detailed convergence results between BMSSA and other SSA variants across 12 test functions. Within these results, BMSSA demonstrates exceptional optimization performance on most of the test functions. Notably, in the test functions F5, F8, F10, F11, and F12, most comparative algorithms rapidly converge to local optima due to the local optima characteristics of these functions. However, BMSSA maintains a favorable convergence rate in these functions and can escape local optima, thereby enhancing the quality of the current solution. These observations validate the effectiveness and competitiveness of the BMSSA strategy. For complex functions, such as F23, F24, F25, and F29, BMSSA exhibits a notably slower convergence rate than CMSSA, WLSSA, and ESSA. Nevertheless, it ultimately achieves relatively superior convergence results.

Table 8 presents the ranking results and statistical significance of BMSSA and its variants compared with other algorithms at different dimensions (Dim = 30, 50, 100). When Dim = 30, ISSA, ESSA, DMSSA, QBSSA, WLSSA, CSSA, and CMSSA exhibit inferior performance compared with BMSSA in 28, 30, 13, 17, 15, 27, and 22 test functions. This indicates that BMSSA outperforms these algorithms significantly in these scenarios. Furthermore, at Dim = 50, BMSSA’s performance is superior to the aforementioned algorithms in 26, 30, 19, 17, 9, 30, and 22 problems, demonstrating its excellence and statistical significance across different dimensions. Similarly, when Dim = 100, BMSSA outperforms other algorithms in 26, 30, 19, 17, 9, 30, and 22 problems, reaffirming its stability and outstanding performance across various dimensions. It is noteworthy that WLSSA demonstrates comparable performance to BMSSA in half of the test functions, which is also a notable result.

4.7. Statistical analysis on BMSSA

4.7.1. Wilcoxon signed-rank test

The P-value results of BMSSA and basic MAS using the Wilcoxon signed-rank test with Dim = 30 are reported in Table 9, where the P-value less than 0.05 indicates that BMSSA has statistically significant differences from the compared algorithm. Only a few values greater than 0.05 can be seen, indicating that the experimental analysis is significant.

Table 10 reports the P-value results of BMSSA and SOTA algorithms for the Wilcoxon signed-rank test with Dim = 30. Most P-values are less than 0.05, indicating that BMSSA outperforms its competitors and has a high search exploitation capability. The embedded MUS helps BMSSA to jump out of local minima. MUS enhances the solution quality by randomly generating multiple candidate solutions near the current optimum, and the number of candidate solutions increases as the dimensionality increases. Therefore, the exploitation capability of the algorithm is also improved.

Table 11 displays the P-values obtained from the Wilcoxon signed-rank test for BMSSA and SSA variants in 30-Dim problems. The table results indicate that, except for WLSSA and CMSSA, BMSSA exhibits statistically significant differences when compared with other algorithms across these 30 test functions. However, it should be noted that BMSSA does not demonstrate significant differences when compared with WLSSA and CMSSA across the majority of composition functions.

4.7.2. Friedman test

Figure 10 presents algorithm rankings obtained through the Friedman test across different dimensions. Figure 10 (top) reports the Friedman test results of BMSSA with basic MAS for the experimental results in Subsection 4.4. As shown in Fig. 10 (top), it is clear that BMSSA achieves the best Friedman rank result on all three dimensions. Figure 10 (bottom) reports the Friedman test results of BMSSA with SOTA algorithms for the experimental results in Subsection 4.5. As shown in Fig. 10 (middle), BMSSA obtained the first rank on the Friedman test compared with the SOTA algorithms. Figure 10 (bottom) illustrates the Friedman test results for BMSSA and SSA variants. The experimental outcomes depicted in the figure indicate that BMSSA outperforms its variant counterparts across different dimensions. Therefore, the Friedman test also confirms the superiority of BMSSA.

4.7.3. Bonferroni–Dunn test

The Friedman test can test whether there are differences in performance between algorithms. However, the Bonferroni–Dunn test is required to draw statistical conclusions as a post-hoc test.

Critical difference (CD) is used to judge whether there is a statistical difference in performance between the comparison algorithm and BMSSA. CD is calculated as follows:

where α represents the significant level, q_α is a critical value which can be found in the lookup table (Demšar, 2006), k is the number of comparison algorithms, and Num is the total number of tests functions.

Figure 11 (top) shows the Bonferroni–Dunn test results of BMSSA with basic MAS for the experimental results in Subsection 4.4. In the comparison, experiment of BMSSA with basic MAS in Subsection 4.4, nine algorithms, including HHO, FA, GWO, HGS, DE, SMA, FOA, MFO, and SSA were compared with BMSSA. Therefore, k = 10 and Num = 30. The significant level α was selected as 0.05 and 0.1. According to equation (9), CD = 2.17 and CD = 1.98 with α = 0.05 and α = 0.1, respectively. The average rank of BMSSA is Avg_BMSSA = 1.47. If the difference in the average rank between the two algorithms is greater than CD+Avg_BMSSA, it is considered that there is a significant difference in the performance of the two algorithms. As shown in Fig. 11 (top), the average ranks (1.47, 1.43, and 1.53) of BMSSA are the lowest in three dimensions, respectively. BMSSA outperforms DE, HHO, SMA, FA, FOA, GWO, MFO, and SSA in both significant levels on three dimensions. There was no significant difference in the performance of BMSSA and HGS in the Bonferroni–Dunn test, except for the significant level of α = 0.1 and Dim = 30.

Figure 11 (middle) shows the Bonferroni–Dunn test results of BMSSA with SOTA algorithms for the experimental results in Subsection 4.5. Similarly, in the comparison experiment in Subsection 4.5, k = 8 and Num = 30, respectively. In addition, CD = 1.70 and CD = 1.55 with α = 0.05 and α = 0.1, respectively. The average ranks of BMSSA are 2.13, 1.77, and 1.73 on three dimensions, respectively. As shown in Fig. 11 (middle), BMSSA is more excellent than ALCPSO, CGPSO, ACWOA, EOBLSSA, and DECLS in three dimensions. For both significant levels with Dim = 30 and Dim = 50, EPSDE outperforms BMSSA. There is no significant difference in performance between BMSSA and EPSDE with Dim = 30. There was no significant difference in the performance of BMSSA and GOTLBO for the significant level of α = 0.05 and Dim = 100. However, the performance of BMSSA is better than GOTLBO at both significant levels in Dim = 50.

Figure 11 (bottom) displays the results of the Bonferroni–Dunn test comparing BMSSA with SSA variants. In this test, with k = 8 and Num = 30, the CD values are calculated as CD = 1.70 for α = 0.05 and CD = 1.55 for α = 0.1. BMSSA achieved average rank values of 2.49, 2.55, and 2.36 across three dimensions. As illustrated in Fig. 11 (bottom), BMSSA outperforms ISSA, ESSA, CSSA, and CMSSA across these three dimensions. However, there are no statistically significant differences between BMSSA, QGBSSA, and WLSSA at these two significance levels in these three dimensions.

5. Feature Selection with BMSSA

5.1. The proposed bBMSSA

Figure 12 shows the flowchart of the proposed feature selection algorithm. The feature selection problem is a discrete optimization problem whose solution is limited to binary space (Ahmadianfar et al., 2022; Xing et al., 2023a). Therefore, before applying the continuous BMSSA to the feature selection problem, it must be converted to the bBMSSA. The transfer function converted the continuous BMSSA into binary form (Zhang , Zhang, Peng, et al., 2019). The transfer function T( · ) and the position update are described in equations (10) and (11), respectively.

where r₃ is a random number in [0, 1].

Mapping continuous values to a range of 0 to 1 is achieved by utilizing an S-shaped curve. This smooth curve property gradually changes the transformed values when approaching 0 or 1, with the most rapid change occurring around 0.5. The denominator value in the function, in this case, 3, is responsible for controlling the sensitivity of the binary transformation. A larger denominator value makes the transition smoother, requiring larger random numbers to approach an output of 1, whereas a smaller denominator value increases sensitivity, necessitating smaller random numbers to approach an output of 1. This probabilistic form of binary transformation allows for several advantages. Firstly, it introduces diversity into the algorithm, as different iterations may select different features, thereby expanding the search space exploration. This aids in preventing the algorithm from getting stuck in local optima, particularly in high-dimensional feature spaces.

Classification accuracy and the number of selected feature subsets are the two main conflicting objectives of feature selection. When the classification accuracy in the classification result is higher, and the number of selected features is lower, better classification is obtained (Zhang, Cheng, Shi, et al., 2019). Therefore, a fitness function with two variables, classification accuracy and the number of features, is used to evaluate the quality of each solution. The fitness function is defined as follows:

where Er denotes the classification error rate of SVM, F_sub is the number of feature subsets, λ ∈ [0, 1], and β = 1 − λ represent the weight coefficient of the classifier error rate and the importance degree of the number of selected features, respectively.

In addition, after each feature subset is selected, a 10-fold cross-validation method is employed to measure the classification accuracy of the selected feature subset on the SVM classifier (Vapnik, 1995). In the training phase, a portion of the data is used to train the SVM classifier. After the training, another portion of data not used during training is used to evaluate the classifier’s performance. A 10-fold cross-validation method is used to divide the data into 10 subsets. This process consisted of iteratively using nine of these subsets for classifier training and the remaining one subset for testing. This process is repeated 10 times, each time using a different test subset, to obtain a comprehensive assessment of the performance of the feature subset across a variety of data samples.

5.2. Experimental set-up

Our bBMSSA was compared with feature selection methods based on metaheuristic algorithms on 12 datasets which are from the UCI data repository. The selected feature selection methods include binary SMA (bSMA, Abdel-Basset et al., 2021), binary SSA (BSSA, Faris et al., 2018), binary WOA (bWOA, Mafarja & Mirjalili, 2018), binary DE (BDE, Y. Chen et al., 2015), binary HHO (bHHO, Thaher et al., 2020), and binary snake optimization (BSO, Hashim & Hussien, 2022). The parameter setting of these algorithms is shown in Table 12.

Table 13 shows the number of samples, features, and categories for each dataset. In addition, all methods were tested under the same experimental environment: the maximum number of iterations was 50, and the population size was 20. To avoid the randomness of the feature selection methods based on swarm intelligence algorithms from generating errors in the final experimental results, all algorithms were run 10 times independently on each dataset. Besides, to evaluate the efficiency of each method, four data analysis metrics (average fitness, average error rate, average number of attributes selected, and average run time) (Mafarja & Mirjalili, 2018) were employed.

5.3. Experimental results and analysis

Table 14 shows the average fitness results of the compared algorithms on 12 datasets. Besides two datasets (Segment and Semeion), bBMSSA outperforms the compared methods on the other 10 datasets. bBMSSA finally ranks first with an average ranking (AVR) of 1.1667.

From the experimental results of the average error rate of feature selection between bBMSSA and other methods in Table 15, bBMSSA ranks first with an AVR value of 1.0833, which indicates that the classification accuracy of bBMSSA is generally better than the other six compared methods. bBMSSA achieves the first rank in classification accuracy performance on 11 datasets. In the Segment dataset, the classification accuracy of bBMSSA is not the best, but the mean values of bBMSSA differ very little from those of bHHO, which is the first in classification accuracy in this dataset. In general, bBMSSA shows good classification ability and achieves very high classification accuracy on datasets with large penglung and Semeion feature spaces. bBMSSA achieves very high classification accuracy on Dermatology, Hepatitisfulldata, Semeion, Wine, and Zoo, all of which have STD values of 0, indicating that bBMSSA is highly robust on these datasets.

The experimental results for the number of selected features are shown in Table 16. bBMSSA achieves a smaller number of features selected on most of the datasets. In the Bupa_liver, Hepatitisfulldata, Segment, and Semeion datasets, the number of features selected by bBMSSA was not the lowest, but it ranks in the top three in both datasets. Finally, the average ranking of bBMSSA is the first.

To better analyze the performance of bBMSSA, convergence curves were plotted for the fitness values of bBMSSA and the compared methods, as shown in Fig. 13. bBMSSA shows better convergence accuracy on 10 datasets than the other six algorithms in Fig. 13. bBMSSA has the most outstanding convergence speed performance on the five datasets, Bupa_liver, Congredd, Dermatology, Hepatitisfulldata, and Thyroid_2class. This indicates that the improved food foraging strategy can improve the global exploration ability of the algorithm, improving the solution quality and accelerating the convergence speed. Observing the convergence plots of the three datasets, Cleveland_heart, Segment, and Wine, it can be found that bBMSSA does not fall into local optimum but converges further when the number of iterations is 20, indicating that MUS can well balance the algorithm’s global exploration ability and local exploitation ability. In general, bBMSSA has good search ability and accurate search accuracy in solving the feature selection problem.

6. Case Study of Return-Intentions of Overseas Chinese after COVID-19

In this section, we collected a dataset on the return-intentions of overseas Chinese after COVID-19 through an anonymous online questionnaire and performed a case study by setting up a bBMSSA-based feature selection optimization on the dataset.

6.1. Questionnaire

In order to study the return-intentions of overseas Chinese after COVID-19, we conducted an online anonymous survey. The participating overseas Chinese were asked to fill out the online questionnaire. The questionnaire has sixteen multiple-choice questions, five of which allow participants to select one or more answer choices. Table 17 lists the details of the online questionnaire.

In this survey, a total of 4325 valid questionnaires were collected, of which 4057 questionnaires had the IP addresses of the respondent overseas, accounting for 93.8% of the total. Among the people who filled out the questionnaire in this survey, 89.7% were overseas Chinese and 10.3% were ethnic Chinese. In addition, people mainly gathered in Italy, Spain, Brazil, and other countries, with 1419 people in Italy, 1116 people in Spain, and 426 people in Brazil.

In this paper, we would like to explore the factors most affecting the willingness of overseas Chinese to return to China after COVID-19. Because of the good representativeness of these 4325 questionnaires can well reflect the different impacts of overseas Chinese suffering from COVID-19. Among the 4325 data collected, 62.6% chose not to return to China, 17.7% chose to return to China, and the remaining 19.7% did not have a clear choice.

6.2. Feature selection optimization model set-up based on bBMSSA

In this subsection, a feature selection optimization model is set up for mathematical data mining the factors most affecting the return-intentions of overseas Chinese after COVID-19 from the collected return-intentions dataset.

To avoid errors in the feature selection optimization model, we do not use the data that do not have a clear choice for returning to China. There are 3475 data for those who chose to return and those who did not, 764 data for those who chose to return, and 2711 data for those who chose not to return. There is a data imbalance between these two classes. However, if the classes of a dataset are imbalanced, the model training is biased back to the class with the more significant percentage, which may affect the accuracy of the classification model (Zhang, Wang, Heidari, et al., 2021).

Data sampling (Van Hulse et al., 2007) is one of the most commonly used methods to solve the issue of class imbalance. The balanced dataset is achieved by adjusting the sample size of the majority class, which accounts for the majority of the imbalanced dataset, and the minority class, which accounts for a smaller percentage. Data sampling methods can be classified as under-sampling or over-sampling schemes, where under-sampling removes samples from the majority class and oversampling adds more data to the minority class. In this paper, the balanced dataset was created with data under-sampling.

Our feature selection optimization model was set up with the following steps. Firstly, we processed the data by randomly removing 1183 data that people chose not to return to China. Consequently, the final return-intentions dataset used for the feature selection optimization model contains 764 data that people chose to return to China and 1528 data that people chose not to return to China, respectively. The two classes for classification are with a ratio of 1: 2. Secondly, the willingness to return to China (D38) in the dataset was selected as a label in the feature selection optimization model. Thirdly, the proposed bBMSSA algorithm with the SVM classifier introduced in Section 5 was employed for classification. In addition, 10-fold cross-validation was employed to measure the classification accuracy of the selected feature subset on the classifier. 10-fold cross-validation divides the dataset into ten parts, nine of which are used as training data and the remaining as test data. The proposed feature selection optimization model was experimented independently on the return-intentions dataset with 10 times of 10-fold cross-validation to guarantee accuracy. Therefore, a total of 100 times of model training and validation were performed.

6.3. Analysis on return-intentions

Our bBMSSA-based feature selection optimization model for the return-intentions dataset was compared with SOTA feature selection algorithms, including BDE (Y. Chen et al., 2015), BSSA (Faris et al., 2018), bWOA (Mafarja & Mirjalili, 2018), bSMA (Abdel-Basset et al., 2021), bHHO (Thaher et al., 2020), and BSO (Hashim & Hussien, 2022).

Figure 14 illustrates the convergence of these compared methods on our return-intentions dataset. Although bBMSSA’s convergence rate is not the fastest, our bBMSSA achieves higher convergence accuracy. The figure shows that the compared six algorithms are likely to fall into local optima to different degrees. In comparison, our bBMSSA demonstrates the capability of stepping out of the local optimal solution well after about 5, 32, and 44 iterations.

Figure 15 provides the boxplot of our bBMSSA against the compared feature selection methods and an additional SVM classifier without feature selection in terms of the classification error rate. The boxplot shows that our bBMSSA achieves a lower classification error rate compared with the compared algorithm. More specifically, the average classification error rates of bBMSSA and SVM are 6.98% and 9.51%, respectively. Therefore, the proposed bBMSSA with a smaller feature subset can improve the average classification accuracy of SVM with the full feature dataset by 2.8%.

Figure 16 counts the number of times each feature was selected during the experiment. According to the line graph, it can be observed that the four features, i.e., development prospects in the country of residence (D7), family or job reason to hold (D8), seeking opportunities in China (D12), and possible time to return to China (D33) were selected the most. Moreover, there are 10 features (D7, D8, D10, D12, D14, D17, D18, D19, D30, and D33) were selected more than 50 times. These features play an important role in deciding whether to return to China for overseas Chinese after COVID-19. About one-fourth of the 3475 questionnaire participants considered the development prospect of their current residence countries as a criterion for whether they would return to China. About one-half of the participants’ intentions to return to China were limited by external factors, such as family, job, etc. COVID-19 has led to economic downturns in some countries, and overseas Chinese are facing difficulties in schooling and employment, which may prompt overseas Chinese to consider returning to China. About 96% of those who have the intention to return to China plan to return to China in three years. About 45% of those who have no intention to return to China have no plans to return to China in recent years. On the other side, the three features, i.e., the country of residence (D1), industry engaged in (D2), and industries of interest in China (D23), were selected the least often, indicating that these three features are not the main factors influencing the willingness to return to China after COVID-19.

7. Conclusions and Future Works

This study proposed a new SSA variant named BMSSA based on BF and MUS. BF was employed in the search range of solutions. MUS was used to improve the optimal solution’s quality further, reducing the probability of the algorithm trapping into local optimum situations. The different dimensional tests performed with nine MAS and seven SOTA algorithms on 30 benchmark functions demonstrated the superiority of BMSSA, while statistical tests used the Friedman, Wilcoxon signed-rank, and post-hoc tests. Moreover, the experiment results of feature selection on 12 UCI datasets embody the ability of the proposed bBMSSA method to solve real-world problems. Finally, bBMSSA was employed to investigate the factors influencing the willingness of overseas Chinese to return to China after COVID-19. It is concluded from the analysis results that the development prospects in the country of residence, family or job reasons, seeking opportunities in China, and the possible time to return to China are the main factors influencing the return-intentions.

The incorporation of the MUS and the improved food foraging strategy has increased the algorithm’s time complexity. Therefore, in the subsequent work, one may consider employing parallel techniques to enhance the algorithm’s efficiency or devising alternative strategies to assist the algorithm in converging more rapidly while avoiding a significant increase in algorithm complexity. Furthermore, applications of the algorithm in other fields such as image segmentation, neural network learning, and parameter estimation can be investigated. We also would like to discover more useful knowledge from our return-intentions dataset soon.

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments on improving this paper. Many thanks also to Ali Asghar Heidari for his helpful proofreading. This work was supported by the Wenzhou Philosophy and Social Science Planning Project (Grant No. 22wsk201), the Basic Scientific Research Program of Wenzhou (Grant No. R20220098), Wenzhou Association for Science and Technology (2023 Soft Science Research Project with No. 10), and the Project of Wenzhou Key Laboratory Foundation (Grant No. 2021HZSY0071). J.X. was partially supported by the Graduate Scientific Research Foundation of Wenzhou University (Grant No. 3162023003072).

Declaration of AI and AI-assisted Technologies in the Writing Process

During the preparation of this work, the author(s) used chatGPT for grammar enhancement, proofreading, and paraphrasing for a few sentences. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication.

Conflict of interest statement

None declared.

References