2.1. Literature review

The research shows that the internal parameters in machine learning models and the feature space and sample space in big data have an important influence on the performance of classifiers. In the last decade or so, swarm intelligence algorithms have shown good benefits in solving such problems [73–76]. Furthermore, swarm intelligence algorithms exhibit notable characteristics of randomness and efficiency, encompassing a range of deterministic and stochastic techniques that find wide application in diverse optimization problems and practical scenarios, effectively harnessing the power of collective intelligence. Examples include optimization problems [77–80], traveler problems [81], complex network problems [82,83], path planning problems [84,85], and real-time detection problems [86]. Within the scope of this investigation, a refined variation of the Harris Hawks algorithm is introduced, leveraged to forecast forthcoming patterns in the advancement of the sharing economy. Originating from the pioneering work of AliAsghar Heidari and SeyedaliMirjalili in 2019, the Harris Hawks Optimization (HHO) algorithm represents a biomimetic intelligent optimization technique. The algorithm mimics the Harris Hawk predation characteristics and combines Levy Flights to achieve solutions to complex multidimensional problems. The algorithm is similar to common meta-heuristic algorithms, inspired by the habits of nature’s animals. It achieves global optimization algorithms by imitating the group hunting, raiding and siege strategy of the Harris Hawk. Since its introduction, the Harris Hawk algorithm has been used to solve optimization problems in a variety of fields and has achieved good results. Literature [87] introduced the HHO algorithm with chaotic search and opposition learning to perform parameter searches for PV cells and modules. The literature [88] used the HHO algorithm to analyze the stability of mass slope problems and to improve the accuracy of predictions. Literature [89] applied the HHO to in-vehicle location services for intelligent transportation systems to reduce the delay in the delivery of emergency data in in-vehicle networks to enhance the location rate. The literature [90] used the improved adaptive Harris Hawk algorithm for obtaining the optimal queue for the 2D grey scale gradient method and used this gradient method for image processing, possessing better results compared to other multi-stage min-valorization methods. The literature [91] used HHO for industrial safety. Using HHO for augmented artificial neural networks (ANN), a hybrid model ANN-HHO model was proposed for predicting the scour depth of spillways by dam outflow water and compared with two other hybrid models ANN-GA and ANN-PSO, showing the superiority and effectiveness of the hybrid model.

2.2. K-nearest neighbor classifier

The KNN algorithm is a simple and effective classification algorithm. It is based on the idea of a nearest neighbor, i.e. the class of a new sample is determined by the nearest samples of a known class. The fundamental concept underlying the K-nearest neighbors (KNN) algorithm involves the computation of the distance between the sample under classification and all the training samples, utilizing a distance metric. By identifying the k closest neighbors to the sample in question [92], a majority vote is subsequently employed to ascertain the category to which the sample belongs, based on the categories assigned to these nearest neighbors. Many research works have been done on this algorithm by scholars at home and abroad. For example, Wai Lam et al. [93] from the Chinese University of Hong Kong combined the KNN method with a linear classifier and achieved better classification results, with an accuracy of over 80% at a recall rate close to 90%. The literature [94] investigated the asymptotic nature of the K-nearest neighbor estimate of the regression function and obtained the asymptotic normality of the K-nearest neighbor estimate of the regression function and the coincidence of its Bootstrap statistic. The literature [95] establishes an efficient search tree for the nearest neighbor algorithm and improves the query rate. In the literature [96], an iterative nearest neighbor method is proposed to solve the problem of poor classification of KNN algorithms in the environment of small sample pools. In the case where insufficient classed samples are available, the local thematic features of the samples to be classified are amplified by retrieval to obtain similar samples with sufficiently fixed classes. The literature [97] gives parallel algorithms on reconfigurable mesh machines (RMESH) for K nearest neighbors in Euclidean space, among others. Li et al. proposed a two-layer hierarchical combination of text classification [98]. It was demonstrated that the combination of support vector machine and KNN methods could make SVM noise less sensitive and improve the efficiency of KNN methods, showing better classification performance. Shi et al. [99] propose a semi-supervised classification algorithm based on rough set error, using rough set error theory to extract negative case samples. A new classifier is constructed by combining SVM, Rocchio and Naive Bayes to improve the classification accuracy.

3.1. Exploration phase

Harris Hawks in the exploration phase roam many locations waiting for prey to arrive. Harris’s Hawks usually consider the position of their companions and prey when selecting their perches. Two main strategies are followed during this phase, as shown in Eq (2.1).

Eq (2.1)

Eq (2.2)

In Eq (2.1)., X(t+1) is the next updated position of the population iteration of Harris’s hawk. X_rand(t) refers to a random position in the current population. X_rabbit(t) refers to the prey’s location, i.e., the location of the current optimal solution. r₁, r₂, r₃, r₄ and q are random numbers between [0,1], respectively. UB and LB are used to control the upper boundary and lower bound, independently. In Eq (2.2), X_m(t) is the average position of individuals in the current Harris’s hawk population. where N is the population size and X_i(t) is the position of the i-th individual.

3.2. Transition factor

The escape energy E of the target is designed during the exploitation phase, which determines the exploitation behavior of the Harris Hawk, as shown in Eq (2.3). Eq (2.3) Eq (2.4) where E represents the escape energy of the target prey. E₀ denotes the escape energy of target in its initial state. T represents the maximum number of iterations. t denotes the current number of iterations. Harris Hawk chooses different exploitation behaviors depending on the value of E. When E≥1, the Harris Hawk searches for the target in a region far from the current solution. When E<1, Harris Hawk searches in the neighborhood of the current solution.

3.3. Exploitation phase

In the process of the exploitation phase, Harris’s Hawks choose the most appropriate way to attack their prey, depending on each situation. Four strategies describe the hunting behaviors of Harris’s hawk. The probability of prey escape is set to r. r<0.5 represents the state in which the prey escapes capture. r≥0.5 represents the state in which the prey is captured. Regardless of the state, the Harris hawk always surrounds the prey based on escape energy. When E≥0.5, the Harris Hawk performs a soft besiege. when E<0.5, the Harris Hawk performs a hard besiege.

3.3.2 Hard besiege.

When E<0.5 and r≥0.5, the energy of the prey target is at a low value. At this point, Harris’s hawk uses a surprise attack to approach the prey, a process that can be expressed by Eq (2.8).

Eq (2.8)

3.3.3 Soft besiege with progressive rapid dives.

When E≥0.5 and r<0.5, the energy of the prey target is at a high value. The probability of prey avoiding capture by Harris’s hawk was high at this time. Harris’s hawks performed soft besiege between prey captures. The subsequent random wandering of Levy-flight was used to simulate prey escape and Harris Hawk pursuit, a process that can be expressed by Eq (2.9). Eq (2.10) and Eq (2.11) determine the next updated position of the Harris Hawk. Eq (2.9) Eq (2.10) Eq (2.11) where u and v are both random numbers between [0,1], β takes 1.5. D is the dimension of the problem. S vector is randomly generated. Greedy selection is then used to determine which positional update is more favorable to the optimal solution, as shown in Eq (2.12). F() calculates the target fitness value.

Eq (2.12)

3.3.4 Hard besiege with progressive rapid dives.

When E<0.5 and r<0.5, Harris’s Hawk surrounds the low-energy prey target by forming an envelope. In the process, Harris’s hawk approaches the prey by narrowing the envelope. This process is described as Eq (2.13)–Eq (2.15).

Eq (2.13)Eq (2.14)Eq (2.15)

4.1. Simulated annealing based HHO

Simulated annealing (SA) algorithm is a random search algorithm first proposed by N. Metropolis et al [100]. Metropolis perceived a strong similarity between the solid annealing processes in the physical world and the general engineering combinatorial optimization situation. SA is a generally probability-based optimization algorithm where the solid annealing process can be summarized in three parts: the heating stage, the isothermal procedure and the cooling process:

1. Heating stage. This process is designed to increase the energy of the molecules of a solid so that the molecules within it have a more intense thermal movement, thus departing from the equilibrium state of the solid molecules. As the temperature reaches a sufficient value, the solid becomes a liquid, resulting in the disappearance of the non-uniform state in the solid system.
2. Isothermal processes. The states in the system always proceed in the direction of decreasing energy, and as the energy decreases, the system eventually enters equilibrium.
3. The cooling processes. The thermal movement of molecules is reduced, the energy of the system decreases, and a crystalline structure is formed.

The main idea of the simulated annealing algorithm in a combinatorial optimization problem is then shown as follows:

• Initialize the temperature values and solve the optimization problem.
• Execute the algorithm according to the Metropolis criterion.
• Output optimal solution.

In the SA algorithm, the Metropolis criterion is the assumption that the position of the previous agent is X(n). After further iterative updates of the population, the agent position is updated to X(n+1). At the same time, the energy of the search agent changes from E(n) to E(n+1). The probability of receiving a search agent from X(n) to X(n+1) is p. Eq (3.1) where T denotes the relative temperature; when moving to the next iteration of the update, if the energy has become smaller, then this change is accepted. If the energy has increased, the search agent has moved further away from the global optimum. In this case, the algorithm does not immediately discard it but judges it by probability: a random number ε is generated on a fixed interval [0,1]. If ε<p, the transfer in this case will also be accepted, otherwise the transfer fails to proceed to the next step of the cooling process, and so on. The cooling equation for the above cooling process is: Eq (3.2) where q denotes the cooling factor, the value of which is usually set to 0.99 [28]; T denotes the current temperature. When the temperature drops to the termination temperature Tend, the optimal value is output.

In this study, chaos mapping is introduced for the simulated annealing algorithm to further enhance the performance of SA. The operation of chaos mapping is shown in detail as follows: Eq (3.3) where μ denotes the control parameter, whose value is usually set to 4 [101]; β_i represents a random value ranging in [0,1]; and s stands for the number of the entire population. The chaotic strategy is strongly influenced by the initial conditions and can be understood as a search movement with a combination of overall ergodicity and randomness [102,103]. By effectively circumventing premature convergence of the population, this approach proficiently mitigates the occurrence thereof, expedites the convergence rate, and enhances the precision of solutions attained through the HHO algorithm.

4.2. Gaussian bare-bones

Gaussian Bare Bones is widely used as a variational strategy. Wang et al [104] introduced Gaussian skeleton variation into DE by intersecting the feasible solution obtained from Gaussian perturbation with the solution of the original DE. In this study, enhanced Gaussian Bare Bones were designed to be applied to HHO, and the new update procedure is shown in Eq (3.4). Eq (3.4) Eq (3.5) Eq (3.6) where N(μ, σ) is a Gaussian function with mean μ and variance σ. CR is the crossover factor, and its value is usually set to 0.3 [104–106]. are three random individuals in dimension j, while i≠i₁≠i₂≠i₃.

In the modified Gaussian Bare Bones, the new update position is generated by a Gaussian distribution. The method calculates the average of the intermediate position between the present capture and the optimal solution, utilizing the difference between the current solution and the optimal solution as a measure of variance. During the initial stages of evolution, when the disparity between the current solution and the global optimum is substantial, the population’s individuals encompass the entire search space. This stage primarily emphasizes global exploration. As the number of offspring increases, the deviation decreases, while the use of a linear decreasing factor b ensures that individuals in the later population will cover a smaller search area. At this point the algorithm can move more quickly into the behaviors of local exploitation, ensuring convergence at a later stage.

4.3. Framework of SGHHO

The entire framework of SGHHO combines both of these strategies and the flow graph of its overall framework is detailed in Fig 1. First, the population needs to be set up initially and the optimal position selected as the rabbit’s position in the algorithm. After the SGHHO algorithm is updated using the core formulation of HHO, a simulated annealing strategy is introduced. Guided by the SA strategy, the iterative process of SGHHO can accept points with larger energy values, i.e., poorer solutions. As a result, the strategy possesses a better vertical search capability with a larger search range, thus effectively improving the probability of HHO finding a global optimum. Next, Gaussian bare bones strategy is implemented, and the process of fine-tuning the Gaussian sampling allows SGHHO to ensure diversity exploration in the first stage while focusing on exploitation in the next phase. The switch in search behavior from exploration to exploitation allows SGHHO to converge to a higher level of accuracy.

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Fig 1. Flowchart of SGHHO.

https://doi.org/10.1371/journal.pone.0291626.g001

Eventually, the optimal solution in the entire population is updated until it reaches the maximum allowed number of iterations. Additionally, Algorithm 1 showcases the pseudo-code for the SGHHO method employed in this research.

Algorithm1. The pseudocode of SGHHO.

Initialize the population size N and maximum number of evaluation MaxFES;

Randomly initialize the location information of population X_i(i = 1,2,…,N);

While (termination condition is not satisfied) do

  Calculate the fitness value for each individual;

  Set X_rabbit to the current optimal solution;

  for (Each individual) do

    Updating the initial escape energy E₀ and escape force J;

    Update E by Eq (2.3);

    if (|E| ≥ 1) then

         if (q < 0.5) then

             Updating the position according to Eq (2.1);

         elseif (q ≥ 0.5) then

             Update the location according to the random selection in Eq (2.1);

         end

         if (|E| < 1) then

           if (r ≥ 0.5 and |E| ≥ 0.5) then

             Use Eq (2.5) to update the position;

           elseif (r ≥ 0.5 and |E| < 0.5) then

             Use Eq (2.8) to update the position;

           elseif (r < 0.5 and |E| ≥ 0.5) then

             Use Eq (2.12) to update the position;

           elseif (r < 0.5 and |E| < 0.5) then

             Use Eq (2.15) to update the position;

end if

    end for

    Calculate the fitness value for all individuals;

    Calculate the mutation position of the modified Gaussian bare bone strategy by Eq (2.15);

    Updating search agent locations by means of the modified simulated annealing strategy;

t = t + 1;

end while

return X_rabbit;

4.4. Computational complexity

The evaluation of algorithmic models involves considering the time complexity of an optimizer, which is an important metric. In this research, the time complexity of SGHHO consists of various steps, including population initialization, updating fitness values, modifying the positions of individuals in the population, executing the simulated annealing strategy, and implementing the Gaussian bare bone strategy. In the standard SGHHO algorithm, the time complexity of these first three components is calculated as O (population initialization) = O(N); O (fitness value update) = O(T*N) during T update iterations; and O (population individual position update) = O(T*N*D) during T iterations. where N denotes the assumed population size; T denotes the maximum number of iterative updates during the experiment; and D denotes the problem dimensions. The SA strategy and GB strategy are also included in the SGHHO algorithm to help SGHHO improve its performance. In particular, the time complexity of the SA strategy is O (SA strategy) = O(N); the time complexity of GB strategy is O (GB strategy) = O(N*D). So that the overall time complexity consumed by the SGHHO algorithm is O(SGHHO) = O(N*(T+TD+2+D)).

6.1. CEC 2014 function validation

In the section, IEEE CEC 2014 function set was chosen to prove each algorithm’s performance in optimization precision and convergence speed. In particular, algorithms are divided into four main types: singlet functions (F₁- F₃); multimodal functions (F4—F₈); hybrid modal functions (F₉- F₂₀) and composite modal functions (F₂₁- F₃₀). Table 1 shows the specific data for these 30 functions. In the table, the last two columns show the type of each algorithm and the corresponding global optimum value. Furthermore, to provide a more comprehensive assessment of each algorithm’s performance, two metrics, namely the average value (AVG) and the standard deviation value (STD), are utilized. The experimental tables highlight the optimal values for each problem in bold black font, ensuring their prominence. Additionally, non-parametric statistical tests such as the Wilcoxon signed-rank test [107] and the Friedman test [108] are employed to quantitatively evaluate the performance of SGHHO.

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Table 1. Description of the 30 IEEE CEC 2014 benchmark functions.

https://doi.org/10.1371/journal.pone.0291626.t001

6.2. Scalability analysis of SGHHO

The problem’s dimensionality corresponds to the number of factors to be optimized within the given optimization problem. Consequently, optimizing high-dimensional data allows for a more comprehensive evaluation of the algorithm’s overall performance and stability. Thus, the performance of the SGHHO algorithm can be thoroughly assessed. We use the original HHO and Farmland Fertility Algorithm (FFA) [109], which have better performance in the variable dimensionality case, as the reference objects for the experiments in this subsection. Within the experimental framework, scalability experiments are conducted using a test function set consisting of 30 functions from IEEE CEC 2014. The dimensions are varied, specifically set to 100, 500, and 1000, respectively. All experimental parameters remain constant, with the exception of the dimension setting (D). The metrics utilized to analyze the experimental results include AVG (average) and STD (standard deviation). A comprehensive overview of the results can be found in Table 2.

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Table 2. Experimental results of scalability tests on SGHHO.

https://doi.org/10.1371/journal.pone.0291626.t002

The table provides a clear visual representation, demonstrating that the SGHHO algorithm consistently outperforms the original HHO algorithm in unimodal functions (F1-F3), irrespective of the dimensional complexity. The AVG values achieved by the SGHHO algorithm exhibit a significant improvement compared to those of the original HHO algorithm. This superiority is evident in both low and high-dimensional problem settings. Also, SGHHO outperforms the FFA algorithm for F1 and F3 functions. And with the increasing complexity of the functions, the SGHHO algorithm has better optimization performance compared with the original HHO algorithm in multimodal functions (F4-F9). The implementation of the GB strategy successfully directs the HHO algorithm towards exploring regions that contain more optimal solutions, thereby enhancing the algorithm’s precision in generating solutions. The FFA, with its excessive local search capability, slightly outperforms the SGHHO on F4, F7 and F8. Further, in the mixed modal functions (F9-F20), the SGHHO algorithm still maintains its obvious superiority over FFA and HHO. Despite the variations in dimensional settings, the SGHHO algorithm demonstrates consistent performance in attaining optimal solutions. Among the composite functions (F21-F30), although there is a slight difference in the AVG value between the SGHHO algorithm and the original algorithm on the F29 function, it remains relatively small. Conversely, for all other functions, the SGHHO algorithm outperforms the HHO algorithm by a significant margin. The results of scalability experiments indicate that the SGHHO algorithm exhibits notable enhancements in optimization performance and stability across 93.3% of the functions in the IEEE CEC 2014 function set when considering variable dimensionality scenarios. This signifies the considerable improvements achieved by the SGHHO algorithm through the integration of SA and GB strategies.

6.3. The impact of two mechanisms

This subsection investigates the effect of the randomly introduced simulated annealing strategy and the Gaussian bare bone strategy on HHO. To enable effective verification of the role of the added mechanisms, this experiment performs all sequential sequencing of the SA and GB strategies, thus avoiding interactions between the mechanisms. The details are displayed in Table 3, where "S" and "G" denote "simulated annealing" and "Gaussian skeleton", respectively. 1 and 0 indicate that the strategy is selected and unselected, respectively. The four algorithms were compared on 30 test functions. Table 4 details the Avg and STD values obtained by the algorithms during the experiments. Similarly, the optimal values obtained under each function are marked in bold. Analysis of the table shows that SGHHO has the best solution accuracy out of the 30 functions and ranks first out of 19 functions. In addition, SGHHO is more stable than SHHO and GHHO in terms of STD values. This means that the SGHHO algorithm maximizes the performance benefits of both strategies.

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Table 3. Experimental design of mechanism combinations.

https://doi.org/10.1371/journal.pone.0291626.t003

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Table 4. Optimization results of each HHO variants on IEEE CEC 2014 functions.

https://doi.org/10.1371/journal.pone.0291626.t004

The p-values for the Wilcoxon signed rank test at a 5% confidence level are presented in Table 5. The symbols "+/-/ = " are utilized to signify the outcomes of the comparison between SGHHO and the other methods. From the table, SGHHO outperforms HHO on 24 test functions; it outperforms the GHHO algorithm on 13 functions; and it outperforms the SHHO algorithm on 10 functions. The integration of the two strategies demonstrates a substantial enhancement in the performance of HHO, attaining its maximum potential in the SGHHO algorithm. Additionally, the Friedman test was employed to quantitatively assess the algorithm’s performance. According to the average ranking of the algorithms, the SGHHO algorithm secures the top position among all combinations, signifying a significant advancement in achieving optimal performance. Not only is there a significant improvement in global search, but there is also a significant breakthrough in local exploitation. Therefore, the experimental results in this section serve as evidence for the efficacy of the SA and GB strategies in enhancing the performance of the original HHO algorithm.

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Table 5. The p-values of Wilcoxon test for HHO variants on the benchmark function.

https://doi.org/10.1371/journal.pone.0291626.t005

6.4. Comparison with other reported well-known optimizers

To verify the optimization performance of the SGHHO algorithm, 10 high-quality algorithms including BMWOA [110], CBA [111], EM [112], OBSCA [113], IGWO [114], MFO [115], ALPSO [116], CGPSO [117], SCADE [118] and HGWO [119] were selected for comparison. Furthermore, the IEEE CEC 2014 test set was utilized in the experiments to comprehensively evaluate the algorithms’ exploration and exploitation capabilities. To provide further insight into the performance comparisons, statistical tests such as the Wilcoxon rank sum test and the Friedman test were employed to assess the significance differences between SGHHO and the other algorithms. Furthermore, to allow experimental fairness, the whole algorithms were compared under the same settings, as well as each algorithm’s parameters were shown in Table 6.

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Table 6. Parameter setting for 10 metaheuristics.

https://doi.org/10.1371/journal.pone.0291626.t006

Table 7 shows AVG and STD values gained for all competitors on the set of functions tested. Where the optimal values obtained on each set of functions are bolded in order to allow more visual analysis of the gaps between the algorithms. The results in Table 7 demonstrate that SGHHO can reach the optimal solution of the function on 67% of the tested functions. In particular, on the unimodal function, SGHHO is only not optimal on the F2 function, second only to the ALSPSO and CBA algorithms. Compared to algorithms such as BMWOA, OBSCA, CGPSO and SCADE, the SGHHO showed strong competitiveness on the single-peak function. In addition, although the AVG values of SGHHO on the multimodal functions are not fully optimal, there is no significant difference compared to the optimal solution for each function and the average AVG ranking is better. On the hybrid functions, SGHHO can search for optimal solutions for most functions compared to other advanced algorithms. A comparison to this can be clearly seen in the F10-F20 functions. Also, using the Friedman test, this study compares the combined strength of SGHHO and these competing algorithms.

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Table 7. Comparative results for SGHHO and other reported methods on the benchmark function.

https://doi.org/10.1371/journal.pone.0291626.t007

As the data at the end of Table 8 indicates, SGHHO achieves the first overall average ranking in processing the CEC 2014 function set, followed by algorithms including ALCPSO, IGWO and CGPSO. Besides, according to the Wilcoxon test, the p-value of SGHHO is mostly less than 0.05 compared to other methods, which demonstrates the significant difference between the SGHHO algorithm compared to other competitors. It validates that the SGHHO algorithm, which incorporates the Gaussian Bare-Bones strategy and the simulated annealing strategy, is a more promising algorithm for different types of optimization problems, and that SGHHO significantly improves HHO’s optimization power.

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Table 8. P -values for SGHHO and other reported algorithms by Wilcoxon test on the benchmark function.

https://doi.org/10.1371/journal.pone.0291626.t008

Fig 3 illustrates the convergence curves of SGHHO in comparison to other state-of-the-art algorithms on the CEC 2014 test suite. The convergence curves provide a more intuitive picture of the difference in convergence quality and optimization capability between SGHHO and the competitors. The figure clearly demonstrates that SGHHO exhibits faster convergence rates for the majority of functions. For example, F1, F10, F16, F29 and F30. This is because SGHHO utilizes the Gaussian Bare Bones strategy for individual variation and can expand the global search early in the evolutionary process. Furthermore, an observation can be made from the results of F3, F5, F18, and F21, which indicate that even though SGHHO may not exhibit the fastest early convergence, it manages to sustain its progress towards the optimal value after ALPSO, CBA, and CGPSO have reached convergence. This signifies the ability of SGHHO to effectively prevent being trapped in local optima. Also, SGHHO converges rapidly in the later evolutionary stage through SA mechanism, and eventually achieves higher precision and better solutions in the search process when compared with other comparative algorithms. This makes algorithms such as OBSCA, ALCPSO, BMWOA and SCADE fall into premature convergence, while SGHHO can maintain the convergence trend until the global optimum. SGHHO demonstrates a remarkable capability to effectively maintain a balance between exploration and exploitation throughout the search process, thereby rejuvenating the evolution of the population. This ability proves instrumental in mitigating the issue of premature convergence that is commonly encountered in HHO.

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Fig 3. Convergence curves for SGHHO and advanced algorithms on 9 selected benchmark functions.

https://doi.org/10.1371/journal.pone.0291626.g003

A comparison of the mean, standard deviation values, Wilcoxon test and convergence figures leads to the conclusion that the proposed SGHHO significantly outperforms other competing algorithms. It can consistently maintain a leading position in function optimization problems, and the advantage of SGHHO becomes more obvious as the dimensionality of the problem increases.

Hence, in subsequent investigations, this approach can be extended to a broader range of scenarios, such as optimization of machine learning models [120], computer-aided medical diagnosis [121,122], pathology image segmentation [123–125], image denoising [126,127], fine-grained alignment [128], cancer diagnosis [129–131], medical signals [132,133], and structured sparsity optimization [134].

7.1. Methods & data collection

The sharing economy dataset is collected for this study and deals with data mainly from participants in the general public who have participated in the sharing economy. From these actual participants, 671 participants were selected for the study. The questionnaire was developed with reference to numerous scholars’ questionnaires on the sharing economy and predicted to have good reliability and validity. The questionnaire analyses 43 aspects of the respondents by examining their gender, education level, political status, age, profession, location, income level, consumption level, various forms of sharing economy and their perception of sharing economy (see Table 9). This study examines the basic situation of the public’s participation in sharing economy, explores the importance of these attributes and their inherent linkages, and builds a predictive model of future trends in the sharing economy on this basis.

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Table 9. Descriptions of each attribute.

https://doi.org/10.1371/journal.pone.0291626.t009

Since the above-mentioned questionnaires did not involve ethical issues, the review committee/ethics committee of Wenzhou University granted an exemption from ethical review. All participants in the questionnaires signed a consent form.

7.2. Condition configuration

The experiments involved in this work were completed on a host computer with 16GB of RAM, 3.6GHz main frequency, Windows 10 and coded on MATLAB R2016b software. In addition, the experimental parameters are extremely important to the simulation and subtle changes in the parameters can affect the final experimental outcome. To ensure the fairness of the study, the proposed models were quantitatively evaluated using statistical techniques including the calculation of average values (AVG) and standard deviations (STD). The AVG represents the average predictive performance of each model, while the STD indicates the degree of variation across 10 independent runs. To further assess the effectiveness of the proposed models, four commonly used classification metrics were employed, namely Accuracy (ACC), Sensitivity, Specificity, and Matthew’s correlation coefficient (MCC).

7.3. Experimental results and analysis of SGHHO-KNN

To investigate the key factors influencing the future development of sharing economy, this experiment uses a variety of machine learning models to make predictions from a real dataset collected, including BSGHHO-KNN, BSHHO-KNN, BGHHO-KNN, BP, RF, ELM and other classification models. Meanwhile, the classification results of each model will be statistically analyzed using four indicators, namely ACC, Sensitivity, Specificity and MCC. Furthermore, the comprehensive experimental findings are disclosed in Table 10, providing detailed insights into the results obtained. And it can be intuitively found that the BSGHHO-KNN model obtained 99.70%—the highest value of accuracy in predicting the future development trend of sharing economy, while the other five models were 98.66%, 93.45%, 51.56%, 99.11% and 65.73% respectively. While the BSHHO-KNN and RF models demonstrated satisfactory accuracy, it is worth noting that the BSGHHO-KNN model exhibited superior performance in terms of the Specificity and MCC metrics, surpassing both of them and attaining the highest classification outcomes. Based on the above four evaluation metrics it can be fully demonstrated that the BSGHHO-KNN model is feasible to be applied to the problem of predicting future trends in the sharing economy.

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Table 10. Average results of BSGHHO-KNN and other models on four indicators.

https://doi.org/10.1371/journal.pone.0291626.t010

Table 11 shows the standard deviation values derived from the BSGHHO-KNN model and the other five models after 10 independent experiments. The optimal values in the table are bolded for more visual analysis of the experimental data. After analyzing the results, it can be deduced that the BSGHHO-KNN model demonstrates exceptional stability in terms of ACC, sensitivity, and MCC, surpassing other models in these aspects. In terms of specificity indicators, RF model gains the best value.

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Table 11. Standard deviation values for BSGHHO-KNN and the other five models on the four indicators.

https://doi.org/10.1371/journal.pone.0291626.t011

To better compare the performance gap between the models, Fig 4 presents the histograms of AVG and STD values for the five models mentioned above. The examination of Fig 4 reveals that the BSGHHO-KNN model exhibits superior performance across all four indicators, namely ACC, Sensitivity, Specificity, and MCC, resulting in the most effective classification outcome. Following closely behind are the RF model and the BSHHO-KNN model. The figure shows that BSHHO-KNN and BGHHO-KNN both have been effective compared to the other classifiers. The experiments show that the HHO algorithm has excellent adaptability in combination with the KNN classifier. In addition, the BP model performed the worst on the sharing economy dataset. The ACC value was only 51.56% and the Sensitivity was 16.29%. This suggests that the BP model needs to be tuned with appropriate parameters for the specific problem. In addition, the BSHHO-KNN model and the BGHHO-KNN model do not perform the best on the dataset, which means that the BSGHHO algorithm can enable the KNN classifier to maximize its classification performance and thus achieve better classification results. In summary, the BSGHHO-KNN model outperforms other similar methods and can be used to investigate the key factors affecting future trends in the sharing economy.

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Fig 4. Comparison of SGHHO_KNN with well-known classifiers.

https://doi.org/10.1371/journal.pone.0291626.g004

In this experimental study, the proposed model successfully accomplishes the task of selecting the optimal subset of features during the entire process. Fig 5 counts the selected times of each feature in each experiment in the form of a line graph. From the figure: "form of sharing economy in the region" (F7), "attitude towards the sharing economy" (F30), "products in the sharing domain that have been used" (F22), "Factors of greatest concern when using shared product services" (F23), "Main aspects of distress caused by the sharing economy" (F28), "Negative effects of the sharing economy on society " (F39), "Main aspects of concerns about the sharing economy" (F35) and "Any suggestions for the development of the sharing economy" (F43) were selected most frequently. These six most common attributes appeared 10, 10, 5, 5, 5, 4, 4 and 4 times respectively. Therefore, this study concludes that these types of attributes may make a valuable contribution to predicting future trends in the sharing economy.

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Fig 5. The times each feature was selected by SGHHO_KNN during the 10-fold CV process.

https://doi.org/10.1371/journal.pone.0291626.g005

7.4. Discussion

The emergence of the sharing economy as a novel consumption model has introduced certain challenges, yet the public’s expectations for its future development remain high. Through the analysis of the questionnaire experiment results, it becomes evident that numerous factors influence the future trajectory of the sharing economy. Among the 43 attributes considered, attributes F7, F22, F23, F28, F30, F35, F39, and F43 stand out as the most significant in shaping the sharing economy’s future development trend. As a model intended to cater to the entire population, it is crucial to select the appropriate type of sharing economy that effectively serves the public, thereby playing a pivotal role in its overall advancement.

Attributes F22 and F23 pertain to consumer feedback regarding their experience with the sharing economy post-usage. In this context, the success of a particular type of sharing economy hinges on its capacity to satisfy diverse consumer needs at a reduced cost, while also garnering positive experiential feedback. This attribute plays a pivotal role in driving the development of the sharing economy. When consumers have a greater choice of products and services, providers are unable to price them effectively and consumer surplus is significantly increased. On the other hand, when the price of products and services is lower than normal, market demand increases as the price decreases, so the application of sharing economy model will help to increase market demand and thus contribute to the creation of greater value for money.

The F28, F35 and F39 belong to the troubles caused by sharing economy. Under this new model of sharing economy, the existing regulations and systems, etc. do not fit in with its development, thus problems such as taxation, labor security and information security arise one after another. Meanwhile, some of the current regulations are not yet effective in regulating the implementation of sharing economy model, and there is even the phenomenon of sharing economy model being bound by the traditional model legal system, which affects the model’s value due to the lack of laws and regulations. Therefore, under the premise of clarifying the development mechanism, solving the problem of fitting the environment of sharing economy model is a key factor in developing the sharing economy.

As per the attributes F30 and F43, participants hold a crucial position in regulating the activities of the sharing economy through SGHHO. However, the current process of gathering participant feedback and opinions is inadequate, and there is a lack of focus on establishing targeted channels for collecting and incorporating their input. Add to this the fact that sharing platforms have yet to be effectively improved and perfected, leaving a lack of comprehensive, real-time regulation of participants. Therefore, encouraging public participation in the regulation of the sharing sector is a vital measure for promoting the development of sharing economy.

In summary, the sharing economy exhibits positive momentum, with a wider scope of development and an increasing variety of sharing economy products. The future growth of the sharing economy relies on the support of national policies and the continuous enhancement of the management system, which serves as a political guarantee for its stable progress. Additionally, the improvement of social infrastructure is a prerequisite for establishing and maintaining the sharing economy, while the enhancement of citizens’ quality and their active and respectful involvement in the sharing economy are essential for its sustainable long-term development. Despite current challenges in the sharing economy’s development, the outlook for the future remains promising.