To solve the problem of slow convergence and easy to get into the local optimum of the spider monkey optimization algorithm, this paper presents a new algorithm based on multi-strategy (ISMO). First, the initial population is generated by a refracted opposition-based learning strategy to enhance diversity and ergodicity. Second, this paper introduces a non-linear adaptive dynamic weight factor to improve convergence efficiency. Then, using the crisscross strategy, using the horizontal crossover to enhance the global search and vertical crossover to keep the diversity of the population to avoid being trapped in the local optimum. At last, we adopt a Gauss-Cauchy mutation strategy to improve the stability of the algorithm by mutation of the optimal individuals. Therefore, the application of ISMO is validated by ten benchmark functions and feature selection. It is proved that the proposed method can resolve the problem of feature selection.

Due to the increasing complexity of the object of the optimization problem, traditional algorithms are difficult to meet the requirements. Therefore, the design of new algorithms is a good way to solve them. Recently, there have been a lot of scholars who prefer metaheuristics due to their simplicity and high efficiency in solving problems. Many metaheuristics are of great importance in several areas [

Feature selection, as a kind of data pre-processing technique in machine learning, can get rid of unnecessary noise and redundancy features from the data set and extract essential components at the same time so that it can decrease the dimension of data and accelerate the performance of machine learning algorithms. Feature selection is one of the most critical problems in classification tasks. The search space can generate 2^{n} results for data with

A lot of researchers have used a variety of approaches to enhance the SMO’s performance. First, in optimization of the control parameters, Sharma et al. [

This paper presents a new algorithm based on multi-strategy (ISMO) by introducing four different strategies: refracted opposition-based learning strategy, non-linear adaptive dynamic weight factor strategy, and crisscross and Gauss-Cauchy mutation strategy. The application of ISMO is validated by ten benchmark functions and feature selection. The results indicate that ISMO is superior to other competitors.

The rest of this study is organized as follows. Part “Spider Monkey Optimization (SMO)” describes the mathematical model of the SMO. Part “Improved Spider Monkey Optimization (ISMO)” contains the proposed ISMO. Part “Experiment Results and Discussion” presents the ISMO’s performance evaluation and statistical analysis. In part “Feature Selection Optimization Comparison Experiment,” the applicability of ISMO in feature selection is evaluated. Finally, part “Conclusions and Future Research” summarizes the conclusions and future work.

The SMO comprises seven phases, which are addressed in the following subsection.

(1) Initialization phase: the SMO generates a uniformly distributed initial population of

(2) Local leader phase (LLP): each spider monkey adjusts its current location based on the experience of local group members. The location update formula is calculated as follows:

(3) Global leader phase (GLP): all the spider monkeys update their locations using the experience of global leaders. The location update equation is calculated as follows:

The locations of spider monkeys are updated based on a probability

(4) Global leader learning phase (GLL): the location of the global leader is updated by using the greedy selection in the population. In addition, check that the location of the global leader is being updated, and if not, add 1 to the Global Limit Count.

(5) Local leader learning phase (LLL): the location of the local leader is updated by using the greedy selection in the population. In addition, check that the location of the local leader is being updated, and if not, add 1 to the Local Limit Count.

(6) Local leader decision phase (LLD): if the location of any local leader is not updated up to a predetermined threshold, known as the Local Leader Limit, then all the members of the group will update their locations either through random initialization or using a combination of information from the global and local leader by

(7) Global leader decision phase (GLD): the location of the global leader is monitored, and if it is not updated up to a predetermined number of iterations called Global Leader Limit, then global leaders will divide the population into smaller groups.

The complete pseudo-code of the SMO is given in reference [

Based on the above analysis, the improvement of SMO is made in four aspects. First, the initial population is generated by a refracted opposition-based learning strategy to enhance diversity and ergodicity. Second, this paper introduces a non-linear adaptive dynamic weight factor to improve convergence efficiency. Then, using the crisscross strategy, using the horizontal crossover to enhance the global search and vertical crossover to keep the diversity of the population to avoid being trapped in the local optimum. At last, we adopt a Gauss-Cauchy mutation strategy to improve the stability of the algorithm by mutation of the optimal individuals. Therefore, the collaboration of the four search strategies can enhance diversification, exploration, and exploitation.

Opposition-based learning is a widely used approach for the estimation of population initialization [

Where the

Let

Shi et al. [

In

Taking the minimization problem as an example, the success value

In

In

The crisscross strategy [

The horizontal crossover strategy is to perform crossover operations in the same dimension of different populations. The formula is as follows:

The vertical crossover operation is performed in all dimensions of the newborn individual, and the crossover operation occurs with less probability than the horizontal crossover. The formula is as follows:

In

In the late iteration of the SMO, the rapid assimilation of the spider monkeys is prone to local optimal stagnation. The Gauss-Cauchy mutation strategy [

In

The steps of ISMO are illustrated in Algorithm 1.

The main steps of the proposed ISMO are illustrated in

To prove the validity and robustness of ISMO, we choose ten classical unimodal and multimodal functions, in which F1–F6 aims to check the convergence speed and precision, and F7–F10 aims to measure the ability to overcome local optimum, as illustrated in

Class | Function | Dimension | Range | |
---|---|---|---|---|

30 | [−100, 100] | 0 | ||

30 | [−10, 10] | 0 | ||

Unimodal functions | 30 | [−100, 100] | 0 | |

30 | [−100, 100] | 0 | ||

30 | [−5.12, 5.12] | 0 | ||

30 | [−100, 100] | 0 | ||

30 | [−5.12, 5.12] | 0 | ||

Multimodal functions | 30 | [−50, 50] | 0 | |

30 | [−32, 32] | 0 | ||

30 | [−600, 600] | 0 |

This study selected the performance of the SMO, ISMO, whale optimization algorithm (WOA), grey wolf optimizer (GWO), sine cosine algorithm (SCA), slime mould algorithm (SMA), sparrow search algorithm (SSA), chimp optimization algorithm (ChOA), and gaining-sharing knowledge based algorithm (GSK) for comparison, the parameters of algorithms are described in

Algorithm | Specifications |
---|---|

SMO | |

ISMO | |

WOA | |

GWO | |

SCA | |

SMA | |

SSA | _{2} |

ChOA | |

GSK |

As shown in

Function | Evaluation | SMO | WOA | GWO | SCA | SMA | SSA | ChOA | GSK | ISMO |
---|---|---|---|---|---|---|---|---|---|---|

F1 | Ave | 5.87E−17 | 2.98E−58 | 1.08E−27 | 4.35E−18 | 8.62E−268 | 9.17E−09 | 1.00E−48 | 1.01E−03 | |

SD | 1.02E−17 | 1.52E−57 | 1.34E−27 | 1.78E−17 | 0 | 1.80E−08 | 3.29E−48 | 2.98E−03 | ||

F2 | Ave | 2.52E−16 | 4.94E−26 | 8.74E−17 | 1.90E−11 | 8.89E−157 | 1.76E−05 | 1.38E−28 | 2.19E−03 | |

SD | 1.19E−17 | 2.66E−25 | 2.96E−17 | 8.13E−11 | 1.78E−156 | 1.45E−05 | 2.24E−28 | 2.78E−03 | ||

F3 | Ave | 9.72E−03 | 4.82E−10 | 2.69E−06 | 2.65E−10 | 1.40E−266 | 4.48E−07 | 1.40E−17 | 7.14E+01 | |

SD | 0 | 1.37E−09 | 3.31E−06 | 1.23E−09 | 0 | 8.82E−07 | 1.55E−17 | 1.66E+02 | ||

F4 | Ave | 2.60E−18 | 1.07E−14 | 1.65E−202 | 1.69E−52 | 2.16E−271 | 1.27E−11 | 7.87E−240 | 6.70E−72 | |

SD | 2.61E−18 | 5.02E−14 | 0 | 7.17E−52 | 0 | 2.53E−11 | 0 | 1.68E−71 | ||

F5 | Ave | 5.77E−17 | 1.64E−71 | 2.70E−29 | 1.00E−18 | 3.59E−259 | 8.10E−07 | 4.28E−50 | 8.09E−06 | |

SD | 8.50E−18 | 8.68E−71 | 3.88E−29 | 5.06E−18 | 0 | 1.57E−06 | 9.09E−50 | 1.61E−05 | ||

F6 | Ave | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.27E+01 | |

SD | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.61E+01 | ||

F7 | Ave | 4.84E−18 | 2.97E−84 | 3.78E+00 | 6.24E−21 | 0 | 8.40E−05 | 2.91E−44 | 1.21E+02 | |

SD | 8.70E−19 | 1.60E−83 | 3.83E+00 | 2.52E−20 | 0 | 3.24E−04 | 1.53E−43 | 5.68E+01 | ||

F8 | Ave | 3.59E−01 | 8.31E−02 | 4.36E−02 | 9.14E−01 | 3.84E−02 | 2.88E−04 | 6.37E−02 | 1.23E−01 | |

SD | 3.58E−02 | 1.48E−01 | 6.59E−03 | 4.95E−02 | 2.20E−02 | 1.58E−04 | 1.39E−02 | 2.19E−01 | ||

F9 | Ave | 2.57E−16 | 8.70E−24 | 8.07E−15 | 1.81E−10 | 3.00E−98 | 5.13E−04 | 1.07E−25 | 1.39E+00 | |

SD | 2.62E−17 | 4.35E−23 | 6.15E−15 | 6.33E−10 | 6.50E−114 | 5.11E−04 | 1.53E−25 | 9.24E−01 | ||

F10 | Ave | 6.38E−19 | 0 | 1.49E−02 | 1.80E−15 | 0 | 5.16E−09 | 0 | 6.32E−02 | |

SD | 1.48E−20 | 0 | 1.53E−02 | 9.64E−15 | 0 | 5.22E−09 | 0 | 5.33E−02 |

In this section, we compare the performance of ISMO with four different search strategies, for example, spider monkey optimization with refracted opposition-based learning strategy (ISMO-1), spider monkey optimization with non-linear adaptive dynamic weight factor (ISMO-2), spider monkey optimization with crisscross strategy (ISMO-3), and spider monkey optimization with Gauss-Cauchy mutation strategy (ISMO-4). From

Function | Evaluation | ISMO-1 | ISMO-2 | ISMO-3 | ISMO-4 | ISMO |
---|---|---|---|---|---|---|

F1 | Ave | 3.90E−19 | 1.46E−34 | 2.45E−61 | 0 | |

SD | 3.88E−19 | 1.30E−34 | 1.16E−61 | 0 | ||

F2 | Ave | 7.44E−13 | 1.42E−17 | 7.22E−34 | 0 | |

SD | 2.61E−13 | 5.68E−18 | 4.34E−35 | 0 | ||

F3 | Ave | 4.81E−02 | 1.77E+03 | 1.58E+00 | 0 | |

SD | 1.18E−02 | 8.79E+02 | 6.83E−01 | 0 | ||

F4 | Ave | 4.29E−21 | 1.43E−18 | 2.64E−31 | 0 | |

SD | 3.53E−21 | 8.47E−19 | 2.62E−31 | 0 | ||

F5 | Ave | 4.03E−21 | 4.96E−22 | 1.38E−61 | 0 | |

SD | 3.99E−21 | 4.86E−22 | 1.01E−61 | 0 | ||

F6 | Ave | 0 | 0 | 0 | 0 | |

SD | 0 | 0 | 0 | 0 | ||

F7 | Ave | 6.61E−21 | 4.63E−09 | 9.54E−30 | 0 | |

SD | 8.95E−21 | 6.55E−09 | 1.35E−29 | 0 | ||

F8 | Ave | 6.24E−06 | 3.32E−01 | 1.40E−11 | 9.04E−03 | |

SD | 5.09E−06 | 2.51E−02 | 7.75E−12 | 3.42E−03 | ||

F9 | Ave | 5.52E−12 | 2.23E−17 | 5.88E−31 | 3.00E−98 | |

SD | 5.25E−12 | 1.67E−17 | 1.85E−31 | 6.49E−114 | ||

F10 | Ave | 7.73E−22 | 9.05E−23 | 2.47E−61 | 0 | |

SD | 4.73E−22 | 7.39E−23 | 2.41E−61 | 0 |

The Wilcoxon signed-rank test and Friedman test [

Function | ISMO |
ISMO |
ISMO |
ISMO |
ISMO |
ISMO |
ISMO |
ISMO |
||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

R | R | R | R | R | R | R | R | |||||||||

F_{1} |
*** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + |

F_{2} |
*** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + |

F_{3} |
*** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + |

F_{4} |
*** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + |

F_{5} |
*** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + |

F_{6} |
1.70e− 08 | + | *** | + | 1.71e− 08 | + | 4.19e− |
+ | *** | + | 4.44e− 11 | + | 3.48e− 08 | + | *** | + |

F_{7} |
*** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + |

F_{8} |
*** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + |

F_{9} |
*** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + |

F_{10} |
*** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + | *** | + |

+/=/− | 10/0/0 | 10/0/0 | 10/0/0 | 10/0/0 | 10/0/0 | 10/0/0 | 10/0/0 | 10/0/0 |

^{***}denotes

Algorithm | Rank mean | Rank |
---|---|---|

SMO | 6.35 | 7 |

ISMO | 1.60 | 1 |

WOA | 4.30 | 4 |

GWO | 5.65 | 5 |

SCA | 6.05 | 6 |

SMA | 2.30 | 2 |

SSA | 6.75 | 8 |

ChOA | 3.60 | 3 |

GSK | 8.40 | 9 |

Chi-square | 56.91 | |

1.87E−09 |

In

This study applies ISMO to solve the feature selection. Each individual in the overall ISMO represents a combination of features, also called a feature subset. Each dimension is determined by the number of original features in the dataset. Each vector is made up of 0 and 1, where 0 indicates the unselected feature property, and 1 indicates the selection of the corresponding feature property. In the ISMO population initialization, the individual dimension is randomly [0, 1], so the individual vectors in the population consist of 0 and 1. In this study, the values of each dimension above 0.65 are set to 1, and the rest are set to 0 to obtain the individual vectors of 0 and 1.

To balance the minimization of selection features and the maximization of classification accuracy, the fitness function is given by:

To evaluate the performance of ISMO, the mean fitness (Mean), average classification accuracy (

To prove the validity of the ISMO feature selection, we choose seven datasets from the University of California, which are all common test cases for machine learning. A description of the dataset is given in

Number | Name | Characteristics | Samples | Categories |
---|---|---|---|---|

D1 | Avila | 10 | 20867 | 12 |

D2 | Fertility | 10 | 100 | 2 |

D3 | Wine | 13 | 178 | 3 |

D4 | Lymphography | 18 | 148 | 4 |

D5 | Dermatology | 33 | 366 | 6 |

D6 | Ionosphere | 34 | 351 | 2 |

D7 | Sonar | 60 | 208 | 2 |

To compare the overall performance of ISMO, it is compared with other methods, such as grey wolf optimizer (GWO), whale optimization algorithm (WOA), sine cosine algorithm (SCA), and K-nearest neighbor (KNN).

Dataset | ISMO | WOA | GWO | SCA | KNN |
---|---|---|---|---|---|

Avila | 82.3762 | 83.1227 | 82.5407 | 72.4228 | |

Fertility | 92.4700 | 97.4200 | 87.4000 | 79.2000 | |

Wine | 94.3334 | 94.7210 | 95.0562 | 71.5000 | |

Lymphography | 83.4237 | 84.2486 | 82.8378 | 72.6000 | |

Dermatology | 97.8879 | 98.3962 | 98.4916 | 85.2500 | |

Ionosphere | 90.4610 | 91.0776 | 90.9091 | 88.0000 | |

Sonar | 80.6644 | 82.6801 | 81.4423 | 80.1429 | |

Mean | 88.8024 | 90.2380 | 88.3825 | 78.4451 |

Dataset | ISMO | WOA | GWO | SCA | KNN |
---|---|---|---|---|---|

Avila | 82.5618 | 83.3461 | 82.5407 | 73.1544 | |

Fertility | 93.0000 | 98.0000 | 87.4000 | 80.0000 | |

Wine | 94.7191 | 95.0562 | 95.0562 | 72.2222 | |

Lymphography | 83.7838 | 84.5946 | 82.8378 | 73.3333 | |

Dermatology | 98.6034 | 98.9385 | 98.4916 | 86.1111 | |

Ionosphere | 90.7955 | 91.5341 | 90.9091 | 88.8889 | |

Sonar | 81.3462 | 83.3654 | 81.4423 | 80.9524 | |

Mean | 89.2585 | 90.6907 | 88.3825 | 79.2375 |

Dataset | Features | Number of samples | ISMO | WOA | GWO | SCA | KNN |
---|---|---|---|---|---|---|---|

Avila | 9 | 286 | 3.6000 | 3.9000 | 3.7000 | 10.0000 | |

Fertility | 10 | 20867 | 6.4444 | 6.0000 | 7.0000 | 10.0000 | |

Wine | 10 | 100 | 5.7000 | 5.0000 | 6.1000 | 13.0000 | |

Lymphography | 13 | 178 | 9.4000 | 9.0000 | 12.1000 | 18.0000 | |

Dermatology | 17 | 101 | 24.0706 | 18.2471 | 23.7794 | 33.0000 | |

Ionosphere | 18 | 148 | 14.5000 | 18.4000 | 22.0000 | 34.0000 | |

Sonar | 33 | 366 | 52.1000 | 51.1000 | 56.7000 | 60.0000 | |

Mean | 16.5450 | 15.9496 | 18.7685 | 25.4286 |

From

Aiming at the weakness of SMO, this study presents a new algorithm based on multi-strategy (ISMO). It can draw the following conclusions from the simulation test.

Firstly, inspired by the ideas of refracted opposition-based learning strategy, crisscross strategy, and Gauss-Cauchy strategy, a new position update is proposed to increase the population diversity and enhance global exploration. Moreover, according to the crisscross and Gauss-Cauchy strategies, the target value is updated with mutation to prevent it from getting into the local optimum. Furthermore, to verify the validity of ISMO, we simulated ten benchmark functions and compared them with different methods. Compared with other methods, ISMO has better performance than other methods. Finally, the Wilcoxon signed-rank test and the Friedman test prove that there is a more remarkable difference in ISMO.

Then, the ISMO is validated by the feature selection, and it is proved that ISMO is effective in choosing the best feature subset, reducing the feature dimensionality, and increasing the precision of classification.

Thus, ISMO is likely an excellent solution to numerical optimization problems. In the future, it will be considered how to improve the performance of this algorithm, even in the case of more complicated optimization problems.

The authors thank the anonymous reviewers for their valuable comments and suggestions.

The authors received no specific funding for this study.

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