1. Introduction
Swarm intelligence optimization algorithms (SIOAs) have consistently been one of the popular investigation fields in computer science, artificial intelligence, and machine learning [
1,
2]. They are many complex and challenging optimization problems existing in these research fields, and the traditional mathematical methods find it difficult to deal with these problems because of non-linearity and multimodality, while SIOAs make use of the stochastic components and are considerably efficient in resolving such issues [
3,
4], which has caused the research and application boom of SIOAs. Recent years have witnessed a group of SIOAs for mimicking the behaviors of animal groups in nature, such as the artificial fish-swarm algorithm (AFSA, 2002) [
5], artificial bee colony algorithm (ABC, 2006) [
6], firefly algorithm (FA, 2009) [
7], krill herd algorithm (KH, 2012) [
8], Drosophila food-search optimization algorithm (DFO, 2014) [
9], grey wolf optimizer (GWO, 2014) [
10], moth-flame optimization algorithm (MFO, 2015) [
11] and so on. These SIOAs have exhibited great potential to address engineering problems, with no exception to the field of analog circuit fault diagnosis.
As electronic systems have found their way into military, medical, aerospace, and other fields, individuals put forward stricter requirements on electronic systems in terms of stability, safety, and maintainability [
12,
13,
14,
15]. Research on fault diagnosis methods of circuit systems is an effective way to increase their reliability. When an electronic system fails, it should be able to immediately and effectively identify faults to avoid more severe situations. Therefore, circuit fault diagnosis evaluation has become the focus of intense research in the field of circuit design [
16,
17]. Recently, numerous circuit diagnosis methods have been proposed. Among them, support vector machine (SVM) is an excellent method on account of its small sample learning ability and short training time [
18,
19,
20]. Thus, SVM is frequently applied to analog circuit fault diagnosis. However, an emerging problem is how to select the parameters of SVM, which have high impacts on classification performance. Against this background, many researchers turned their eyes to SIOAs.
In [
21], a new mutation enhanced binary particle swarm optimizer (PSO)was proposed to find the optimal SVM parameters. In [
22], to improve the prediction accuracy, the ABC technique was employed to optimize the internal parameters of SVM. In [
23], Li et al. applied GWO to adjust the kernel parameter and regularization parameter of SVM. In [
24], a novel chaos embedded gravitational search algorithm (GSA) with SVM hybrid method was introduced, which hybridized the chaotic search and GSA with SVM. In [
25], the cuckoo search (CS)algorithm was adopted to tune the key parameters of SVM. Despite the success of the above-mentioned SIOAs, they still struggle in escaping from local minimums when the optimization task becomes more challenging, which makes it worth exploring new approaches.
The well-designed metaheuristic algorithm—whale optimization algorithm (WOA) [
26]—was proposed by Mirjalili and Lewis in 2016. Advantages abound, one of which is its strong optimization performance [
27,
28]. Therefore, WOA has been extensively applied to various fields, including optimal control problem [
29], feature selection [
30,
31], image segmentation [
32], reactive power scheduling problem [
33], parameter extraction of solar photovoltaic models [
34], etc. Petrović et al. [
35] presented a new approach for optimal single mobile robot scheduling based on WOA. Peng et al. [
36] described a new task scheduling optimization method for mobile equipment using WOA, which has high-grade performance on both efficiency and operational cost. Li et al. [
37] employed WOA to tune the parameters of extreme learning machines, which were applied to evaluate the aging degree of an electronic component.
Although WOA has been successfully adopted in numerous areas, several works figured out that it has the drawbacks of premature convergence and local optima stagnation [
38,
39]. For this consideration, many scholars have taken actions to enhance the performance of WOA. As an example, Ling et al. [
40] utilized Lévy Flight to promote WOA and obtained a better tradeoff between global and local search. Sun et al. [
41] put forward a cosine-based dynamic parameter updating method, which facilitated the performances of WOA. Yousri et al. [
42] introduced some chaotic variants where the parameters of the standard WOA were integrated with chaos maps. In [
43], a novel hybrid algorithm that integrated the Tabu search with WOA was developed to improve the convergence speed and local searchability. In fact, many WOA variants can facilitate the performance of the original WOA to a certain extent. But, deficiencies still remain as most of them only improve the ability of a single aspect, e.g., exploration, or maintain diversity, etc. This motivates us to provide novel modifications for enhancing the general performance of WOA.
In this paper, a multi-strategy ensemble whale optimization algorithm (MSWOA) is proposed to compensate for the shortcomings of WOA, and it mainly contains four highlights. First, whales are initialized by chaos theory so that they are more evenly distributed in the search domain, and the possibility of obtaining the global optimum is increased. Second, the random search strategy results in a poor convergence rate and stability since whales are searching around a random individual. Thus, an improved random searching strategy was developed to resolve the inefficiency of the previous scheme. Third, WOA has a chronic deficiency in losing diversity, because the original spiral updating position strategy drives the whole whale swarm to the current global best individual, and the search diversity would be hampered, especially in the early stage. Hence, the Lévy flight strategy was adopted to make a good balance between global and local search. Finally, the enhanced position revising mechanism was employed in MSWOA to strengthen exploration further.
Extensive comparative experiments were conducted to investigate the effectiveness of the proposed algorithm. For a start, numerical optimization experiments were carried out on nineteen widely applied benchmark functions (seven unimodal functions, six multimodal functions, and six fixed-dimensional multimodal functions), and the proposed MSWOA was compared with one promising GWO variants and five other types of well-designed SI algorithms. Experimental results revealed that for most functions, our proposal had considerable advantages in both search accuracy and convergence speed. Moreover, we applied MSWOA to tune the penalty parameter C and the kernel parameter γ of SVM, which was adopted for the fault diagnosis of analog circuits. Comparison studies showed that, with the comparison to the PSO and WOA methods, SVM optimized by MSWOA got an extraordinary average diagnostic accuracy. Namely, MSWOA has better practicability in circuit fault diagnosis.
This paper is organized as follows:
Section 2 introduces the WOA algorithm briefly;
Section 3 describes the proposed MSWOA algorithm in detail;
Section 4 illustrates the experimental setup of the benchmark functions and the comparison results, which prove that the proposed MSWOA has better search performance;
Section 5 applies the MSWOA to two diagnostic instances and performs a detailed analysis of results; finally,
Section 6 provides conclusions and future works.
6. Conclusions
Faced with complex optimization problems, the original WOA algorithm may drive the search to premature convergence or be stuck at the local optimal solutions, etc. To overcome those disadvantages, a multi-strategy ensemble whale optimization algorithm (MSWOA) was developed in this paper. The proposed MSWOA first adopted the chaotic initialization strategy so that the whale swarm could be evenly distributed in the search area. Then, two strategies of the conventional WOA was modified to facilitate its search performance. More precisely, by introducing the best search agent as a reference target, the random searching strategy was enhanced to reduce blindness and accelerate the convergence during optimization. Furthermore, the spiral updating position strategy was improved by the Levy flight, which has a high ability to balance exploration and exploitation. Finally, the enhanced position revising mechanism was designed to strengthen exploration ability further and yields a better MSWOA.
A large number of comparative experiments were conducted to validate the effectiveness of the MSWOA algorithm. For a start, we performed extensive numerical optimization simulation experiments based on nineteen classical benchmark problems, which were extensively used to test the performance of metaheuristics. Moreover, MSWOA was thoroughly compared with six well-established optimization methods. Comparison results indicated that the proposed MSWOA could achieve good results for the majority of the nineteen benchmark problems. Additionally, to further testify the performances of the MSWOA algorithm, we applied the MSWOA to the fault diagnosis of analog circuits. Empirical studies indicated that the diagnostic accuracy of the SVM optimized by the proposed MSWOA was the highest, which reflects that MSWOA possesses excellent practicability in the circuit fault diagnosis field. Therefore, the proposed MSWOA can be considered as a novel and efficient tool for addressing the numerical optimization problems and analog circuit fault diagnosis tasks.
However, for a few fixed-dimension multimodal functions, MSWOA had no obvious superiorities compared with other algorithms. Therefore, how to facilitate the MSWOA accuracy and speed for fixed-dimension multimodal functions still deserves further study.