In classification problems, datasets often contain a large amount of features, but not all of them are relevant for accurate classification. In fact, irrelevant features may even hinder classification accuracy. Feature selection aims to alleviate this issue by minimizing the number of features in the subset while simultaneously minimizing the classification error rate. Single-objective optimization approaches employ an evaluation function designed as an aggregate function with a parameter, but the results obtained depend on the value of the parameter. To eliminate this parameter’s influence, the problem can be reformulated as a multi-objective optimization problem. The Whale Optimization Algorithm (WOA) is widely used in optimization problems because of its simplicity and easy implementation. In this paper, we propose a multi-strategy assisted multi-objective WOA (MSMOWOA) to address feature selection. To enhance the algorithm’s search ability, we integrate multiple strategies such as Levy flight, Grey Wolf Optimizer, and adaptive mutation into it. Additionally, we utilize an external repository to store non-dominant solution sets and grid technology is used to maintain diversity. Results on fourteen University of California Irvine (UCI) datasets demonstrate that our proposed method effectively removes redundant features and improves classification performance. The source code can be accessed from the website:

Classification challenges encompass a wide range of real-world issues, including image analysis [

These techniques are not without their drawbacks and restrictions, though. These include an effective amount of features in the subset and a relatively low classification accuracy rate. The implementation of an effective global search technology is necessary to address these shortcomings. The capabilities of evolutionary computation (EC) technology in global search is well known [

Evolutionary algorithms can be divided into single-objective and multi-objective categories. The single evaluation function in the single-objective approach takes into account both the chosen feature subset and the classification accuracy rate, which is controlled by a parameter. By contrasting the evaluation function values, the evolutionary algorithm seeks to identify the best solution. The multi-objective approach, on the other hand, incorporates at least two optimization goals into the evaluation process. Because the optimal answer cannot be found by simple comparison but rather by using a dominance relationship, the solutions acquired by a multi-objective evolutionary algorithm constitute a set. Minimizing the number of selected features and lowering the classification error rate are often the two key goals in the context of feature selection (FS) difficulties. As a result, FS issues can be categorized as multi-objective optimization problems (MOPs). Due to their population-based search process, which can produce numerous trade-off solutions in a single run, EC technologies are especially good at tackling multi-objective optimization issues. Multi-objective feature selection issues have been addressed using a number of EC methods, including GA, PSO, GWO, and ABC. The No-Free-Lunch (NFL) Theorem, which asserts that there is not one universally superior optimization technique that can resolve all optimization issues, should be noted. This suggests that certain problems may see performance degradation while using existing feature selection methods. The Whale Optimization Algorithm (WOA) [

A Multi-Strategy assisted Multi-Objective Whale Optimization Algorithm (MSMOWOA) is therefore suggested for feature selection in order to alleviate these limitations. We present a multi-strategy solution that combines the Grey Optimization technique, Levy flight, and adaptive mutation in order to improve the performance of MOWOA. In addition, An external repository in which non-dominated solutions are stored is employed to address multi-objective optimization problems. We use grid technology, splitting the grid depending on the non-dominated solutions, to maintain diversity within the repository. An elaborate experiment is created to assess the performance of the suggested algorithm. The outcomes show that MSMOWOA performs admirably in handling feature selection issues.

The main contributions of the paper are listed as follows:

A multi-strategy assisted multi-objective whale optimization algorithm is proposed to solve the feature selection problem.

The effectiveness of three strategies including grey wolf optimization, levy flight, and adaptive mutation is evaluated.

The performance of the proposed MSMOWOA is evaluated on 14 UCI datasets to research its efficiency for the multi-objective feature selection.

The remaining sections of the paper are organized as follows. The typical Whale Optimization Algorithm is described in

The Whale Optimization Algorithm (WOA) [

where

To simulate the bubble-net foraging behavior of humpback whales, two mechanisms of encircling prey and spiral updating position are described. The spiral updating position is modeled mathematically as follows:

where

To model the bubble-net foraging behavior of humpback whales, a probability of 50% is assumed to select between either the encircling prey mechanism or the spiral model to update the position of whales during the optimization process as in

where

For the search for prey, the humpback whales search for prey randomly. The mathematical model of the search for prey is as follows:

where

Multiple objectives that compete with one another make up multi-objective optimization issues. To give an example, the following formulation of the problem can be used without losing generality:

where the number of variables in the problem is represented by

Finding the best solution in a single-objective optimization issue is very simple because there is only one objective function. However, in multi-objective optimization situations, incomparability and conflicting objectives make it more difficult to find the best solution. If a solution exhibits better or equal values across all objective functions and has better values on at least one objective function, it can be said to be superior to another solution in such circumstances. The ensuing ideas are introduced to address this.

The challenges of feature selection include choosing pertinent features, removing superfluous and duplicate information, and decreasing the dimensionality of the data. Three different types of feature selection methods are currently in use: classical methods, single-objective based evolutionary algorithms, and multi-objective based evolutionary methods.

Sequential forward selection (SFS) and sequential backward selection (SBS) are two often used techniques in traditional feature selection methodologies [

Researchers have used evolutionary computation methods to get beyond the drawbacks of conventional feature selection methods in single-objective based evolutionary algorithms. This includes applying techniques like the Dragonfly Algorithm (DA) [

In multi-objective based evolutionary algorithms, the feature selection problem involves the conflicting objectives of maximizing classification accuracy and minimizing the number of selected features. Evolutionary algorithms based on multi-objective optimization have been widely adopted to address this challenge. Hamdani et al. [

The feature selection challenge in multi-objective based evolutionary algorithms combines the competing goals of maximizing classification accuracy and decreasing the number of selected features. Multi-objective optimization-based evolutionary methods have been widely used to address this problem. For feature selection, Hamdani et al. [

A natural optimization method that imitates the bubble-net hunting tactic is called the Whale Optimization Algorithm (WOA). In the feature selection industry, it has become more well-liked. Two hybridization models of WOA and simulated annealing (SA) were presented for various feature selection strategies in a study by Mafarja et al. [

The whole workflow of the proposed multi-strategy assisted multi-objective whale optimization algorithm is shown in the section. In order to improve the efficiency of WOA for feature selection, a multi-strategy which includes Grey Optimization strategy, levy flight, and adaptive mutation is introduced to it. Due to easily falling into the local optimal solution of the original WOA, the stage of WOA is determined by the value of the

Grey Wolf Optimization Algorithm (GWO) has a stronger convergence performance than WOA. Like Whale Optimization Algorithm, GWO is also proposed by Seyedali Mirjalili. And in the grey Wolf algorithm, the global optimal solution, the global suboptimal solution and the global third best solution of the Grey Wolf Optimization algorithm surround the prey in the same way as the contraction surrounding position update mechanism in the exploitation stage of the whale algorithm. In order to further improve the convergence speed of the whale algorithm and effectively reduce the feature dimension, under the condition of retaining the position update mechanism of spiral motion in the exploitation stage of WOA, the position update mechanism of GWO, as shown in

In the improved position update formula, the three best solutions which are the optimal solution, the suboptimal solution, and the third best solution are included. The improved whale algorithm not only combines the Grey Wolf mechanism to enhance the local exploitation ability, but also retains the global search ability of WOA.

Named after French mathematician Paul Levy, Levy Flight is isotropic with a step size that conforms to a heavy-tailed distribution.

Since the probability distribution of the step size of the “Levy flight” conforms to the heavy-tailed distribution, and in the study of heavy-tailed distributions, a probability distribution, called

The

Some MATLAB functions related to

1) Gaussian distribution

2) The Cauchy distribution is given by stblrnd

3) When the input parameters

The function stblrnd

Here,

The random number obtained from the

This paper suggests an adaptive mutation strategy that dynamically modifies the current individual mutation probability based on the last individual evaluation value, where the evaluation value is the classification error rate attained by the KNN classifier model, in order to further enhance the convergence of whale optimization algorithm. The classic mutation operator’s mutation probability is often kept constant during the iteration process or altered based on the number of iterations the person has received overall and the current iteration number. The adaptive mutation probability utilized in this study differs from the two more widely used traditional mutation techniques in that it is based on the magnitude of the individual’s evaluation value at the end of the algorithm’s operation. The present individual’s mutation probability is dynamically altered. The mutation probability is set to 0 for solutions that are superior to the previous individual but have a smaller evaluation value, meaning that they are retained rather than mutated. Contrarily, the mutation chance is suitably enhanced for the solutions with the higher evaluation values that are worse than the last individual, allowing the solutions with superior effect to be kept to the greatest extent.

However, in the iterative process of the standard multi-objective optimization methods, the solutions with bad fitness values are frequently directly discarded, despite the fact that these solutions with poor fitness values occasionally contain individuals with exceptional features in a particular aspect. Therefore, using these subpar solutions in their direct discard cannot fully utilize the particular resources of the whale population. This technique differs from the adaptive mutation probability suggested in this research. In order to effectively exploit the individual resources of the whale population and speed up population convergence, the mutation chance of these individuals with low fitness values is appropriately boosted.

The Pareto dominance in lines 4 and 29 and the generation of new solutions in lines 13 to 25 determine the suggested Algorithm 1’s complexity, respectively. The computing complexity of determining Pareto dominance is

All experimental programs are performed in Windows version of Windows 7 Ultimate edition, Intel(R) Core(TM) i5-3230M CPU @ 2.60 GHz 2.60 GHz, installed memory (RAM) 4.00 GB, Windows 7 Ultimate Edition. The system type is a 64-bit operating system. The programming language used in the experiments is Matlab, the version is Matlab R2020a.

The training set and test set each make up 70% and 30% of the total dataset, respectively. The classification error of a feature subset is calculated using the kNN algorithm. The test set’s prediction results are used as input when calculating additional assessment indicators. The performance of the nondominated subsets generated in the repository is assessed using the test set. The test set will be used to evaluate Pareto fronts in the repository, and the evaluation metrics’ average will be reported.

To evaluate the performance of the multi-objective optimization algorithm used in this paper, 14 benchmark datasets are employed. Fourteen datasets are extracted from the UCI Machine Learning repository. These datasets differ in the number of features (attributes) and samples (instances), where three high-dimensional datasets (datasets with far more features than samples) are included. These are arcene1 (Cancer discrimination dataset), Asian Religions (Asian Religious Scriptures study dataset), and gastroenterology1 (Gastroenterology dataset).

# | Dataset | # Features | # Samples | # Classes |
---|---|---|---|---|

1 | Audit risk | 26 | 776 | 2 |

2 | German | 20 | 1000 | 2 |

3 | Hill Valley with noise | 100 | 1212 | 2 |

4 | Ionosphere | 34 | 351 | 2 |

5 | Iris | 4 | 150 | 3 |

6 | Page blocks | 10 | 5473 | 5 |

7 | SPECT | 22 | 267 | 2 |

8 | Urban Land Cover1 | 147 | 168 | 9 |

9 | Vehicle Silhouettes | 18 | 846 | 4 |

10 | Yeast | 8 | 1484 | 10 |

11 | Zoo | 16 | 101 | 7 |

12 | Arcene1 | 10000 | 100 | 2 |

13 | Asian Religions | 8266 | 590 | 7 |

14 | Gastroenterology1 | 698 | 76 | 2 |

Two objective functions, namely the function to calculate the classification error rate and the function to calculate the ratio of the number of features in the feature subset to the total number of features, also known as the feature proportion, must be generally optimized when solving feature selection problems using the multi-objective optimization algorithm. The minimization objective functions stated in

The feature proportion is used as the first objective function, denoted as _{1}

where

For all algorithms in the experiment in this paper, the population size is set to 20, the archive size is set to 20, the number of iterations is set to 30 and the number of runs is set to 10. Other parameter settings of all algorithms are listed in

Parameter | MOWOA | MOPSO | MOGWO | MMOWOA | NSGA-II | |
---|---|---|---|---|---|---|

Archive parameters of adaptive grid search | nGrid | 7 | 7 | 10 | 10 | NA |

Alpha | 0.1 | 0.1 | 0.1 | 0.1 | NA | |

Beta | 2 | 2 | 4 | 4 | NA | |

Gamma | 2 | 2 | 2 | 2 | NA | |

Mutation parameter | Mu | 0.1 | 0.1 | NA | 0.1 | 0.1 |

pMutation | NA | NA | NA | NA | 0.4 | |

Crossover parameter | nCrossover | NA | NA | NA | NA | 0.7 |

Various indicators are used in this paper, mainly including second-level measures: error rate (Er), precision, recall, specificity, Jaccard coefficient (Jaccard_index), and third-level indicators: F-score and G-mean.

1) Error rate: It is determined by dividing the total number of erroneously classified samples by the total number of dataset samples, which is the percentage of incorrect predictions or classifications made by the KNN classifier. In general, the multiobjective optimization technique performs better the lower the classification error rate is. In

2) Precision: It focuses mostly on the outcomes of predictions. The performance of the multi-objective optimization technique improves with increasing accuracy. The following is a list of the calculation equation:

3) Recall: Its primary goal is to assess the actual samples. The multi-objective optimization approach performs better in general the greater the recall rate.

4) Specificity: The basic goal of specificity is to evaluate actual samples. How many of these samples that are genuinely negative are accurately predicted by the KNN classifier is referred to as specificity. The multi-objective optimization approach performs better in general the higher the specificity. In

5) Jaccard coefficient: It assesses both the actual sample and the anticipated outcome. When comparing all samples that are actually positive or where the projected result is positive, the Jaccard coefficient shows the likelihood that both the actual and predicted results will be positive. The multi-objective optimization approach generally performs better the higher the Jaccard coefficient index is.

6) F1-score: The harmonic mean of recall and precision is referred to as the F1-score, a third-level measurement. The precision and recall metrics are fully taken into account by the F1-score measurement. In general, the multi-objective optimization technique performs better the higher the F1-score is. In

7) G-mean: The third-level measure known as “G-mean” represents the geometric mean of recall and specificity. In this article, it serves as a trade-off between the classification performance of a majority class and a minor class and is frequently used to gauge the degree of data imbalance. The performance of the multi-objective optimization technique generally increases with the size of the G-mean. In

8) Hypervolume (HV): The HV denotes the area’s volume in the objective space that is bounded by the non-dominated solution set and the predetermined reference point that was produced by the multi-objective optimization technique. Without knowing the true Pareto Front, it can be used to assess the search outcomes of multi-objective optimization problems as well as the convergence and variety of multi-objective optimization techniques. Generally speaking, the multi-objective optimization technique performs better the higher the HV value is.

where

9) Feature rate (Fr): The term “Feature rate” (Fr) describes the proportion of features in a feature subset to all features. The multi-objective optimization approach performs better in general the lower the feature rate.

Four multi-objective evolutionary algorithms including MOWOA, MOGWO [

The section shows the results obtained by the proposed algorithm. First, the proposed algorithm MSMOWOA is compared with the original MOWOA in terms of various indicators. Then, in order to verify the performance of MSMOWOA, it is compared with other multi-objective evolutionary algorithms. Finally, the Wilcoxon rank sum test and Friedman test are employed to test the significant difference of MSMOWOA according to HV.

Dataset | Algorithm | Error rate | Precision | Recall | Specificity | Jaccard | F1-score | G-mean | HV | Fr |
---|---|---|---|---|---|---|---|---|---|---|

AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | ||

Audit risk | MOWOA | 0.076 (0.043) | 0.998 (0.003) | 0.984 (0.011) | 0.999 (0.002) | 0.982 (0.013) | 0.991 (0.006) | 0.991 (0.006) | 1.129 (0.024) | 0.201 (0.100) |

MSMOWOA | 0.004 (0.004) | 1.000 (0.000) | 0.989 (0.011) | 1.000 (0.000) | 0.989 (0.011) | 0.995 (0.006) | 0.995 (0.006) | 1.163 (0.004) | 0.041 (0.008) | |

German | MOWOA | 0.311 (0.015) | 0.765 (0.025) | 0.859 (0.042) | 0.387 (0.108) | 0.678 (0.013) | 0.808 (0.009) | 0.566 (0.096) | 0.809 (0.030) | 0.272 (0.152) |

MSMOWOA | 0.275 (0.012) | 0.777 (0.033) | 0.884 (0.055) | 0.370 (0.177) | 0.703 (0.014) | 0.825 (0.010) | 0.528 (0.212) | 0.872 (0.012) | 0.236 (0.118) | |

Hill Valley with noise | MOWOA | 0.480 (0.015) | 0.546 (0.023) | 0.575 (0.039) | 0.519 (0.049) | 0.388 (0.017) | 0.559 (0.018) | 0.545 (0.018) | 0.684 (0.027) | 0.137 |

MSMOWOA | 0.443 (0.010) | 0.574 (0.010) | 0.601 (0.030) | 0.555 (0.023) | 0.416 (0.012) | 0.587 (0.012) | 0.577 (0.012) | 0.737 (0.012) | 0.043 (0.026) | |

Ionosphere | MOWOA | 0.144 (0.031) | 0.964 (0.023) | 0.713 (0.060) | 0.985 (0.010) | 0.694 (0.055) | 0.818 (0.039) | 0.837 (0.034) | 0.999 (0.067) | 0.177 (0.103) |

MSMOWOA | 0.112 (0.020) | 0.942 (0.042) | 0.825 (0.060) | 0.968 (0.030) | 0.784 (0.059) | 0.878 (0.039) | 0.893 (0.035) | 1.086 (0.032) | 0.068 (0.018) | |

Iris | MOWOA | 0.043 (0.014) | 0.910 (0.036) | 1.000 (0.000) | 0.950 (0.022) | 0.910 (0.036) | 0.953 (0.020) | 0.975 (0.011) | 0.900 (0.011) | 0.361 (0.117) |

MSMOWOA | 0.027 (0.013) | 0.937 (0.042) | 1.000 (0.000) | 0.969 (0.021) | 0.937 (0.042) | 0.967 (0.022) | 0.984 (0.010) | 0.912 (0.011) | 0.420 (0.140) | |

Page blocks | MOWOA | 0.061 (0.013) | 0.966 (0.004) | 0.989 (0.002) | 0.695 (0.031) | 0.956 (0.003) | 0.978 (0.002) | 0.829 (0.018) | 1.039 (0.033) | 0.334 (0.097) |

MSMOWOA | 0.050 (0.004) | 0.967 (0.004) | 0.989 (0.002) | 0.707 (0.035) | 0.957 (0.003) | 0.978 (0.001) | 0.836 (0.020) | 1.055 (0.003) | 0.306 (0.064) | |

SPECT | MOWOA | 0.331 (0.060) | 0.725 (0.053) | 0.580 (0.065) | 0.839 (0.042) | 0.472 (0.037) | 0.640 (0.034) | 0.695 (0.030) | 0.808 (0.076) | 0.211 (0.101) |

MSMOWOA | 0.259 (0.028) | 0.752 (0.063) | 0.637 (0.062) | 0.840 (0.066) | 0.522 (0.029) | 0.686 (0.025) | 0.729 (0.019) | 0.897 (0.024) | 0.124 (0.056) | |

Urban Land Cover1 | MOWOA | 0.658 (0.065) | 0.091 (0.040) | 0.555 (0.223) | 0.461 (0.078) | 0.085 (0.037) | 0.155 (0.064) | 0.494 (0.119) | 0.597 (0.078) | 0.101 (0.099) |

MSMOWOA | 0.373 (0.036) | 0.204 (0.071) | 0.685 (0.099) | 0.713 (0.062) | 0.183 (0.058) | 0.306 (0.082) | 0.696 (0.044) | 0.882 (0.055) | 0.025 (0.011) | |

Vehicle Silhouettes | MOWOA | 0.419 (0.055) | 0.400 (0.040) | 0.473 (0.093) | 0.765 (0.041) | 0.274 (0.039) | 0.429 (0.049) | 0.597 (0.050) | 0.768 (0.020) | 0.283 (0.102) |

MSMOWOA | 0.354 (0.009) | 0.438 (0.030) | 0.511 (0.056) | 0.786 (0.023) | 0.307 (0.023) | 0.470 (0.027) | 0.633 (0.028) | 0.824 (0.011) | 0.268 (0.055) | |

Yeast | MOWOA | 0.553 (0.026) | 0.307 (0.020) | 0.517 (0.054) | 0.537 (0.018) | 0.239 (0.020) | 0.385 (0.026) | 0.526 (0.026) | 0.545 (0.006) | 0.504 (0.096) |

MSMOWOA | 0.519 (0.020) | 0.316 (0.017) | 0.514 (0.034) | 0.544 (0.023) | 0.243 (0.014) | 0.391 (0.018) | 0.528 (0.019) | 0.563 (0.012) | 0.525 (0.075) | |

Zoo | MOWOA | 0.289 (0.086) | 0.794 (0.093) | 1.000 (0.000) | 0.824 (0.073) | 0.794 (0.093) | 0.883 (0.060) | 0.907 (0.041) | 0.967 (0.035) | 0.216 (0.089) |

MSMOWOA | 0.162 (0.084) | 0.805 (0.186) | 1.000 (0.000) | 0.803 (0.290) | 0.805 (0.186) | 0.879 (0.139) | 0.850 (0.301) | 1.047 (0.059) | 0.177 (0.049) | |

Arcene1 | MOWOA | 0.275 (0.056) | 0.738 (0.089) | 0.742 (0.119) | 0.804 (0.068) | 0.584 (0.096) | 0.733 (0.076) | 0.769 (0.062) | 0.918 (0.099) | 0.093 (0.108) |

MSMOWOA | 0.230 (0.036) | 0.806 (0.089) | 0.880 (0.083) | 0.869 (0.059) | 0.726 (0.098) | 0.838 (0.067) | 0.873 (0.051) | 1.069 (0.055) | 0.002 (0.002) | |

Asian Religions | MOWOA | 0.619 (0.070) | 0.052 (0.016) | 0.330 (0.090) | 0.531 (0.044) | 0.047 (0.014) | 0.090 (0.026) | 0.415 (0.072) | 0.655 (0.050) | 0.085 (0.057) |

MSMOWOA | 0.570 (0.041) | 0.054 (0.020) | 0.300 (0.108) | 0.604 (0.035) | 0.048 (0.018) | 0.091 (0.032) | 0.416 (0.077) | 0.745 (0.030) | 0.022 (0.010) | |

Gastroentero-logy1 | MOWOA | 0.449 (0.048) | 0.624 (0.138) | 0.678 (0.147) | 0.615 (0.174) | 0.469 (0.100) | 0.469 (0.100) | 0.630 (0.076) | 0.758 (0.117) | 0.160 (0.184) |

MSMOWOA | 0.167 (0.065) | 0.904 (0.066) | 0.862 (0.072) | 0.895 (0.089) | 0.790 (0.084) | 0.880 (0.053) | 0.876 (0.061) | 1.072 (0.068) | 0.003 (0.003) |

In terms of the average error rate, compared with the original multi-objective whale algorithm, the proposed multi-strategy multi-objective whale algorithm MSMOWOA has achieved a lower classification error rate on all 14 data sets (100% of the data sets), and the performance of MSMOWOA is improved significantly. In terms of the average feature rate, MSMOWOA obtains a lower feature rate on 86% of the datasets, which also means that the feature number reduction rate (RR) of MSMOWOA algorithm is higher on 86% of the datasets. In

In terms of average precision, MSMOWOA outperforms MOWOA in 93% of datasets. In terms of average Recall, MSMOWOA also outperforms MOWOA in 71% of the datasets and achieves the same recall with MOWOA in the iris, page blocks, and zoo datasets. As for average Specificity, MSMOWOA is superior to MOWOA in 79% of datasets. From the perspective of Jaccard coefficient, MSMOWOA obtains higher Jaccard coefficient on all 14 datasets, namely 100% of the datasets.

For the third-level indicator F1-score, MSMOWOA gets a higher F-score on 86% of the datasets, and both MSMOWOA and MOWOA obtain the same F1-score on the page blocks dataset. For another third-level index G-mean, MSMOWOA also obtains a higher G-mean on 86% of the datasets.

From the HV, MSMOWOA achieves higher HV value on all 14 datasets, namely 100% of the datasets, and the convergence and diversity of the proposed multi-objective whale algorithm are significantly improved. However, due to the mixed Grey Wolf algorithm which is more complex than the original whale algorithm and the adaptive mutation strategy, the complexity of MSMOWOA is increased. As a result, the average running time of the proposed multi-objective whale algorithm on all data sets is higher than that of the original MOWOA algorithm to a different extent.

The average Pareto Front comparing MSMOWOA and MOWOA across all datasets is shown in

Dataset | Algorithm | Error rate | Precision | Recall | Specificity | Jaccard | F1-score | G-mean | HV | Fr |
---|---|---|---|---|---|---|---|---|---|---|

AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | AVG (STD) | ||

Audit risk | MSMOWOA | 0.004 (0.004) | 1.000 (0.000) | 0.989 (0.011) | 1.000 (0.000) | 0.989 (0.011) | 0.995 (0.006) | 0.995 (0.006) | 1.163 (0.004) | 0.041 (0.008) |

MOGWO | 0.010 (0.011) | 0.999 (0.002) | 0.991 (0.009) | 1.000 (0.001) | 0.991 (0.009) | 0.995 (0.004) | 0.996 (0.004) | 1.163 (0.004) | 0.048 (0.014) | |

MOPSO | 0.014 (0.014) | 0.999 (0.002) | 0.988 (0.012) | 1.000 (0.001) | 0.988 (0.014) | 0.994 (0.007) | 0.994 (0.007) | 0.993 (0.073) | 0.221 (0.070) | |

NSGA-II | 0.011 (0.014) | 1.000 (0.000) | 0.995 (0.012) | 1.000 (0.000) | 0.995 (0.012) | 0.998 (0.006) | 0.998 (0.006) | 1.161 (0.019) | 0.064 (0.024) | |

German | MSMOWOA | 0.275 (0.012) | 0.777 (0.033) | 0.884 (0.055) | 0.370 (0.177) | 0.703 (0.014) | 0.825 (0.010) | 0.528 (0.212) | 0.872 (0.012) | 0.236 (0.118) |

MOGWO | 0.279 (0.013) | 0.769 (0.032) | 0.899 (0.048) | 0.353 (0.166) | 0.706 (0.017) | 0.827 (0.012) | 0.523 (0.203) | 0.873 (0.016) | 0.164 (0.100) | |

MOPSO | 0.293 (0.015) | 0.777 (0.029) | 0.856 (0.024) | 0.426 (0.079) | 0.686 (0.013) | 0.813 (0.009) | 0.601 (0.050) | 0.745 (0.070) | 0.331 (0.089) | |

NSGA-II | 0.283 (0.013) | 0.785 (0.027) | 0.872 (0.052) | 0.441 (0.163) | 0.702 (0.015) | 0.825 (0.011) | 0.583 (0.206) | 0.870 (0.028) | 0.163 (0.074) | |

Hill Valley with noise | MSMOWOA | 0.443 (0.010) | 0.574 (0.010) | 0.601 (0.030) | 0.555 (0.023) | 0.416 (0.012) | 0.587 (0.012) | 0.577 (0.012) | 0.737 (0.012) | 0.043 (0.026) |

MOGWO | 0.444 (0.013) | 0.579 (0.026) | 0.581 (0.041) | 0.572 (0.042) | 0.407 (0.019) | 0.579 (0.019) | 0.575 (0.014) | 0.735 (0.014) | 0.033 (0.016) | |

MOPSO | 0.455 (0.027) | 0.556 (0.032) | 0.566 (0.050) | 0.545 (0.046) | 0.390 (0.033) | 0.560 (0.035) | 0.554 (0.028) | 0.514 (0.054) | 0.339 (0.059) | |

NSGA-II | 0.430 (0.011) | 0.583 (0.018) | 0.605 (0.047) | 0.561 (0.040) | 0.421 (0.022) | 0.593 (0.022) | 0.581 (0.016) | 0.627 (0.026) | 0.221 (0.037) | |

Ionosphere | MSMOWOA | 0.112 (0.020) | 0.942 (0.042) | 0.825 (0.060) | 0.968 (0.030) | 0.784 (0.059) | 0.878 (0.039) | 0.893 (0.035) | 1.086 (0.032) | 0.068 (0.018) |

MOGWO | 0.129 (0.015) | 0.904 (0.037) | 0.840 (0.053) | 0.948 (0.021) | 0.770 (0.038) | 0.869 (0.024) | 0.892 (0.024) | 1.077 (0.018) | 0.052 (0.010) | |

MOPSO | 0.127 (0.030) | 0.949 (0.022) | 0.734 (0.071) | 0.977 (0.012) | 0.705 (0.063) | 0.826 (0.043) | 0.846 (0.039) | 0.874 (0.077) | 0.247 (0.071) | |

NSGA-II | 0.092 (0.020) | 0.963 (0.022) | 0.853 (0.043) | 0.981 (0.012) | 0.825 (0.041) | 0.904 (0.025) | 0.915 (0.023) | 1.072 (0.038) | 0.110 (0.027) | |

Iris | MSMOWOA | 0.027 (0.013) | 0.937 (0.042) | 1.000 (0.000) | 0.969 (0.021) | 0.937 (0.042) | 0.967 (0.022) | 0.984 (0.010) | 0.912 (0.011) | 0.420 (0.140) |

MOGWO | 0.033 (0.010) | 0.931 (0.025) | 1.000 (0.000) | 0.961 (0.019) | 0.931 (0.025) | 0.964 (0.014) | 0.980 (0.010) | 0.907 (0.009) | 0.409 (0.128) | |

MOPSO | 0.038 (0.013) | 0.919 (0.033) | 1.000 (0.000) | 0.956 (0.018) | 0.919 (0.033) | 0.958 (0.018) | 0.978 (0.009) | 0.905 (0.010) | 0.389 (0.071) | |

NSGA-II | 0.037 (0.008) | 0.914 (0.025) | 1.000 (0.000) | 0.954 (0.013) | 0.914 (0.025) | 0.955 (0.014) | 0.977 (0.007) | 0.905 (0.007) | 0.358 (0.093) | |

Page blocks | MSMOWOA | 0.050 (0.004) | 0.967 (0.004) | 0.989 (0.002) | 0.707 (0.035) | 0.957 (0.003) | 0.978 (0.001) | 0.836 (0.020) | 1.055 (0.003) | 0.306 (0.064) |

MOGWO | 0.050 (0.002) | 0.967 (0.004) | 0.988 (0.003) | 0.703 (0.030) | 0.956 (0.003) | 0.977 (0.001) | 0.833 (0.017) | 1.055 (0.003) | 0.288 (0.065) | |

MOPSO | 0.051 (0.007) | 0.965 (0.003) | 0.990 (0.002) | 0.693 (0.033) | 0.956 (0.004) | 0.978 (0.002) | 0.828 (0.020) | 1.002 (0.072) | 0.301 (0.051) | |

NSGA-II | 0.049 (0.004) | 0.966 (0.003) | 0.988 (0.003) | 0.697 (0.022) | 0.956 (0.004) | 0.977 (0.002) | 0.830 (0.013) | 1.055 (0.004) | 0.327 (0.040) | |

SPECT | MSMOWOA | 0.259 (0.028) | 0.752 (0.063) | 0.637 (0.062) | 0.840 (0.066) | 0.522 (0.029) | 0.686 (0.025) | 0.729 (0.019) | 0.897 (0.024) | 0.124 (0.056) |

MOGWO | 0.263 (0.028) | 0.743 (0.041) | 0.625 (0.066) | 0.851 (0.034) | 0.513 (0.052) | 0.677 (0.046) | 0.728 (0.036) | 0.901 (0.025) | 0.126 (0.042) | |

MOPSO | 0.311 (0.041) | 0.703 (0.045) | 0.621 (0.077) | 0.814 (0.036) | 0.490 (0.055) | 0.656 (0.050) | 0.709 (0.041) | 0.791 (0.050) | 0.230 (0.062) | |

NSGA-II | 0.253 (0.029) | 0.763 (0.052) | 0.671 (0.061) | 0.854 (0.050) | 0.552 (0.033) | 0.711 (0.028) | 0.756 (0.021) | 0.908 (0.028) | 0.169 (0.053) | |

Urban Land Cover1 | MSMOWOA | 0.373 (0.036) | 0.204 (0.071) | 0.685 (0.099) | 0.713 (0.062) | 0.183 (0.058) | 0.306 (0.082) | 0.696 (0.044) | 0.882 (0.055) | 0.025 (0.011) |

MOGWO | 0.383 (0.057) | 0.229 (0.046) | 0.689 (0.146) | 0.719 (0.048) | 0.208 (0.050) | 0.342 (0.064) | 0.699 (0.061) | 0.889 (0.042) | 0.024 (0.010) | |

MOPSO | 0.514 (0.072) | 0.111 (0.037) | 0.617 (0.192) | 0.508 (0.084) | 0.102 (0.027) | 0.184 (0.043) | 0.551 (0.080) | 0.451 (0.071) | 0.382 (0.044) | |

NSGA-II | 0.468 (0.119) | 0.152 (0.074) | 0.571 (0.209) | 0.580 (0.129) | 0.136 (0.063) | 0.234 (0.099) | 0.562 (0.120) | 0.593 (0.109) | 0.267 (0.023) | |

Vehicle Silhouettes | MSMOWOA | 0.354 (0.009) | 0.438 (0.030) | 0.511 (0.056) | 0.786 (0.023) | 0.307 (0.023) | 0.470 (0.027) | 0.633 (0.028) | 0.824 (0.011) | 0.268 (0.055) |

MOGWO | 0.368 (0.017) | 0.429 (0.020) | 0.539 (0.033) | 0.762 (0.015) | 0.314 (0.014) | 0.477 (0.017) | 0.641 (0.018) | 0.812 (0.009) | 0.245 (0.035) | |

MOPSO | 0.342 (0.042) | 0.424 (0.036) | 0.536 (0.056) | 0.765 (0.028) | 0.310 (0.030) | 0.472 (0.035) | 0.639 (0.026) | 0.741 (0.060) | 0.281 (0.052) | |

NSGA-II | 0.359 (0.022) | 0.449 (0.027) | 0.565 (0.057) | 0.770 (0.026) | 0.333 (0.024) | 0.499 (0.027) | 0.659 (0.026) | 0.820 (0.018) | 0.244 (0.042) | |

Yeast | MSMOWOA | 0.519 (0.020) | 0.316 (0.017) | 0.514 (0.034) | 0.544 (0.023) | 0.243 (0.014) | 0.391 (0.018) | 0.528 (0.019) | 0.563 (0.012) | 0.525 (0.075) |

MOGWO | 0.523 (0.022) | 0.310 (0.024) | 0.527 (0.047) | 0.527 (0.027) | 0.243 (0.021) | 0.390 (0.027) | 0.526 (0.019) | 0.544 (0.024) | 0.483 (0.074) | |

MOPSO | 0.543 (0.014) | 0.307 (0.013) | 0.525 (0.046) | 0.530 (0.027) | 0.239 (0.010) | 0.386 (0.014) | 0.526 (0.016) | 0.558 (0.008) | 0.455 (0.045) | |

NSGA-II | 0.534 (0.012) | 0.305 (0.019) | 0.523 (0.036) | 0.531 (0.016) | 0.239 (0.016) | 0.385 (0.021) | 0.526 (0.016) | 0.561 (0.009) | 0.481 (0.051) | |

Zoo | MSMOWOA | 0.162 (0.084) | 0.805 (0.186) | 1.000 (0.000) | 0.803 (0.290) | 0.805 (0.186) | 0.879 (0.139) | 0.850 (0.301) | 1.047 (0.059) | 0.177 (0.049) |

MOGWO | 0.178 (0.077) | 0.691 (0.257) | 0.896 (0.315) | 0.710 (0.215) | 0.689 (0.258) | 0.864 (0.064) | 0.780 (0.289) | 1.031 (0.063) | 0.170 (0.064) | |

MOPSO | 0.171 (0.070) | 0.866 (0.054) | 1.000 (0.000) | 0.879 (0.050) | 0.866 (0.054) | 0.927 (0.031) | 0.937 (0.027) | 0.981 (0.020) | 0.244 (0.058) | |

NSGA-II | 0.175 (0.037) | 0.836 (0.075) | 0.995 (0.016) | 0.871 (0.063) | 0.831 (0.068) | 0.906 (0.040) | 0.930 (0.030) | 1.035 (0.037) | 0.187 (0.019) | |

Arcene1 | MSMOWOA | 0.230 (0.036) | 0.806 (0.089) | 0.880 (0.083) | 0.869 (0.059) | 0.726 (0.098) | 0.838 (0.067) | 0.873 (0.051) | 1.069 (0.055) | 0.002 (0.002) |

MOGWO | 0.269 (0.047) | 0.877 (0.065) | 0.810 (0.094) | 0.899 (0.060) | 0.727 (0.092) | 0.839 (0.061) | 0.851 (0.053) | 1.053 (0.053) | 0.001 (0.002) | |

MOPSO | 0.236 (0.055) | 0.779 (0.100) | 0.726 (0.162) | 0.833 (0.084) | 0.588 (0.100) | 0.736 (0.082) | 0.770 (0.072) | 0.539 (0.035) | 0.488 (0.005) | |

NSGA-II | 0.203 (0.051) | 0.788 (0.087) | 0.799 (0.119) | 0.837 (0.074) | 0.655 (0.107) | 0.788 (0.073) | 0.814 (0.059) | 0.586 (0.034) | 0.467 (0.005) | |

Asian Religions | MSMOWOA | 0.570 (0.041) | 0.054 (0.020) | 0.300 (0.108) | 0.604 (0.035) | 0.048 (0.018) | 0.091 (0.032) | 0.416 (0.077) | 0.745 (0.030) | 0.022 (0.010) |

MOGWO | 0.540 (0.042) | 0.063 (0.029) | 0.307 (0.127) | 0.614 (0.052) | 0.055 (0.026) | 0.104 (0.047) | 0.422 (0.117) | 0.753 (0.057) | 0.017 (0.012) | |

MOPSO | 0.524 (0.035) | 0.076 (0.022) | 0.471 (0.094) | 0.508 (0.031) | 0.070 (0.020) | 0.130 (0.036) | 0.487 (0.057) | 0.376 (0.019) | 0.483 (0.006) | |

NSGA-II | 0.469 (0.035) | 0.092 (0.028) | 0.459 (0.111) | 0.567 (0.047) | 0.083 (0.026) | 0.153 (0.044) | 0.506 (0.064) | 0.419 (0.026) | 0.468 (0.004) | |

Gastroenterology1 | MSMOWOA | 0.167 (0.065) | 0.904 (0.066) | 0.862 (0.072) | 0.895 (0.089) | 0.790 (0.084) | 0.880 (0.053) | 0.876 (0.061) | 1.072 (0.068) | 0.003 (0.003) |

MOGWO | 0.202 (0.104) | 0.833 (0.085) | 0.819 (0.182) | 0.842 (0.086) | 0.698 (0.147) | 0.814 (0.105) | 0.824 (0.094) | 1.017 (0.104) | 0.003 (0.002) | |

MOPSO | 0.377 (0.073) | 0.674 (0.119) | 0.618 (0.102) | 0.679 (0.139) | 0.465 (0.079) | 0.631 (0.075) | 0.639 (0.068) | 0.496 (0.043) | 0.432 (0.022) | |

NSGA-II | 0.367 (0.035) | 0.691 (0.105) | 0.672 (0.115) | 0.687 (0.125) | 0.501 (0.038) | 0.667 (0.034) | 0.670 (0.023) | 0.576 (0.018) | 0.359 (0.013) |

In terms of average error rate, compared with the other three multi-objective optimization algorithms, the proposed MSMOWOA achieves the lowest classification error rate among the four algorithms on 7 data sets (50% data sets), and the top two lowest classification error rate on 13 data sets (93% data sets). It is proved that the proposed MSMOWOA has a remarkable effect even compared with several popular multi-objective optimization algorithms. In terms of average feature rate, MSMOWOA can get a lower feature proportion on 14% of the datasets, but obtain the top two feature rate on 64% of the datasets, which also means that the feature reduction rate (RR) of MSMOWOA ranks the second position among the four multi-objective optimization algorithms on 64% of the datasets. In

For the third-level measure F1-score, MSMOWOA obtains the highest F1-score value in 21% of the datasets, and the top two average F1-score value in 79% of the datasets. For another third-level indicator G-mean, MSMOWOA obtains the highest G-mean in 36% of the datasets, and the top two average G-means in 64% of the datasets.

With respect to HV, MSMOWOA achieves higher HV value at 10 datasets, namely 71% of the datasets, which fully indicates that MSMOWOA still achieves very good convergence and diversity compared with other multi-objective optimization algorithms. In terms of the average running time, MSMOWOA combines the grey Wolf algorithm with high time complexity and utilizes adaptive mutation strategy, which greatly increases the algorithm complexity. On all the 14 datasets, the lowest average running time of the algorithm is not achieved. The second lowest average running time of the algorithm is obtained in 21% of the datasets, and the top three lowest average running time of the algorithm is obtained in 79% of the datasets.

Average Pareto Front plots of MSMOWOA, MOGWO, MOPSO, and NSGA-II are compared on the all datasets, as shown in

In order to more clearly show the performance of the proposed MSMOWOA, the Wilcoxon rank sum test and Friedman test are used to test statistical experimental results. The Wilcoxon rank sum test is performed with respect to HV indicator, which can simultaneously evaluate the convergence and diversity of the multi-objective optimization algorithm. Original hypothesis H0: there is no difference in the multi-objective whale algorithm before and after the improvement; New hypothesis H1: there is a significant difference in the multi-objective whale algorithm before and after the improvement. The significance level alpha is set at 0.05, and the results of Wilcoxon’s rank sum test are shown in

Dataset | MOWOA | MOGWO | MOPSO | NSGA-II |
---|---|---|---|---|

Audit risk | 1.083E−04 | 9.522E−01 | 1.083E−05 | 8.297E−02 |

German | 1.083E−05 | 8.387E−01 | 3.248E−04 | 4.023E−01 |

Hill Valley with noise | 1.083E−05 | 8.972E−01 | 1.083E−05 | 1.083E−05 |

Ionosphere | 1.000E−03 | 1.714E−01 | 7.578E−05 | 2.555E−01 |

Iris | 4.100E−02 | 3.217E−01 | 1.942E−01 | 1.684E−01 |

Page blocks | 7.000E−03 | 6.749E−01 | 6.062E−04 | 6.688E−01 |

SPECT | 1.949E−04 | 8.951E−01 | 2.165E−05 | 3.519E−01 |

Urban Land Cover1 | 1.083E−05 | 7.552E−01 | 1.083E−05 | 2.165E−05 |

Vehicle Silhouettes | 1.083E−05 | 2.443E−02 | 1.083E−05 | 7.817E−01 |

Yeast | 8.119E−04 | 5.470E−02 | 3.826E−01 | 4.242E−01 |

Zoo | 2.000E−03 | 3.630E−01 | 4.168E−03 | 4.480E−01 |

Arcene1 | 2.057E−04 | 5.903E−01 | 1.083E−05 | 1.083E−05 |

Asian Religions | 7.036E−04 | 7.817E−01 | 1.083E−05 | 1.083E−05 |

Gastroenterology1 | 1.083E−05 | 2.390E−01 | 1.083E−05 | 1.083E−05 |

+/−/≈ | 10/4/0 | 1/0/13 | 12/0/2 | 6/0/8 |

From

Friedman test is employed to verify the performance of MSMOWOA. The Friedman test is a nonparametric hypothesis test that can be used when comparing the performance of multiple sets of algorithms. The algorithm ranks the performance of each algorithm on various data sets and then determines whether each group of algorithms is truly significantly different by statistical ranking differences. Error rate and HV indicators both are used to verify the performance of MSMOWOA. The results are listed in

Algorithm | Ranking |
---|---|

MSMOWOA | 1.4643 |

MOGWO | 2.1071 |

MOPSO | 4.6786 |

NSGA-II | 2.9643 |

MOWOA | 3.7857 |

Algorithm | Ranking |
---|---|

MSMOWOA | 1.9643 |

MOGWO | 2.9286 |

MOPSO | 3.2143 |

NSGA-II | 2.1786 |

MOWOA | 4.7143 |

In order to clearly illustrate the overall performance of MSMOWOA, three key indicators including error rate, feature rate, and HV are selected to compute their average rank on each dataset. The result is shown in

In this paper, a multi-strategy assisted multi-objective whale optimization algorithm (MSMOWOA) is proposed to solve the feature selection problem. The MOWOA algorithm is improved by adjusting the balance point between the exploration and exploitation stages, which reduces the proportion of whales searching for local optima during the development stage and improves the ability to find global optima. Additionally, a hybrid grey wolf optimization strategy is integrated to enhance local development capability while retaining global search ability. The use of Levy distribution random numbers further enables mutation of poorly performing individuals in the whale population, improving its ability to escape from local optima and enhancing its global search ability. Finally, an adaptive mutation strategy is added to increase the mutation probability of low-fitness individuals and accelerate convergence speed. MSMOWOA is compared with other multi-objective optimization algorithms such as MOWOA and MOPSO on 14 benchmark datasets, demonstrating significant improvements in evaluation metrics such as average error rate and hypervolume (HV).

However, Compared with the original MOWOA, MSMOWOA has a significantly higher time complexity due to the integration of the computationally expensive grey wolf algorithm and the addition of an adaptive mutation strategy. However, MSMOWOA still outperforms NSGA-II on most datasets and MOGWO on some datasets. To further improve the proposed MSMOWOA algorithm, reducing its time complexity is a feasible idea. Other spiral curves such as hyperbolic spiral [

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

This work was supported in part by the Natural Science Youth Foundation of Hebei Province under Grant F2019403207, in part by the PhD Research Startup Foundation of Hebei GEO University under Grant BQ2019055, in part by the Open Research Project of the Hubei Key Laboratory of Intelligent Geo-Information Processing under Grant KLIGIP-2021A06, in part by the Fundamental Research Funds for the Universities in Hebei Province under Grant QN202220, in part by the Science and Technology Research Project for Universities of Hebei under Grant ZD2020344, in part by the Guangxi Natural Science Fund General Project under Grant 2021GXNSFAA075029.

Methodology: Chong Zhou; Formal analysis and investigation: Deng Yang; Writing-original draft preparation: Deng Yang; Writing-review and editing: Xuemeng Wei; Funding acquisition: Chong Zhou; Resources: Chong Zhou, Zhikun Chen, Zheng Zhang; Supervision: Chong Zhou.

All data supporting the findings of this study are available within the paper. All datasets used in the paper can be accessed from

The authors declare that they have no conflicts of interest to report regarding the present study.