A new model for maintenance prediction using altruistic dragonfly algorithm and support vector machine

3.1 ADA

Metaheuristics are a family of approximate optimization techniques that have become popular in the last two decades [21].

They have been inspired by animals’ behaviors, physical phenomena, or evolutionary concepts. The term “heuristic” originates from the ancient Greek word “heuriskein,” denoting the art of discovering new strategies or rules for problem-solving. The Greek suffix “meta,” meaning “upper-level methodology,” was combined with “heuristic” to coin the term “metaheuristic,” introduced by Glover [22]. Metaheuristics explore the search space and provide “acceptable” solutions in a reasonable time for solving hard and complex problems in science and engineering [21]. They are simple and flexible and can be applicable to different problems. They have better performance to avoid local optima. Metaheuristics find an acceptable solution but does not guarantee determining the optimal solution in each run [23]. The reason behind the robust searching mechanism of metaheuristic is the harmonization between two search schemas: exploration and exploitation [24].

While our work primarily explores the practical application and evaluation of metaheuristic algorithms in optimization scenarios, it is crucial to acknowledge and engage with alternative viewpoints that challenge the conventional understanding of their theoretical underpinnings. Camacho-Villalón et al. [25] offer a dissenting opinion regarding the efficacy of metaphorical inspirations driving the design of metaheuristic algorithms. This discourse enables a holistic examination of the strengths and limitations inherent in these methodologies, fostering a nuanced understanding of their role in computational optimization [26].

The dragonfly algorithm (DA), created by Mirjalili [27], is a nature-inspired metaheuristic optimization algorithm. Dragonflies, being small carnivorous insects, exhibit hunting behavior, preying on various smaller insects such as butterflies, bees, ants, and mosquitoes, while avoiding potential predators like spiders and birds [28]. The DA is designed based on the natural dynamics, involving migratory phases, and static behaviors, such as feeding swarms, observed in dragonflies [28]. These swarming behaviors align with the two primary phases of optimization in metaheuristics: exploration and exploitation. Five parameters are used to describe the swarming behavior of dragonflies.

Separation: The swarm exhibits a tendency for preventing static collisions with neighboring individuals. The mathematical calculation for the separation between two contiguous dragonflies is expressed as follows:

where X presents the position of the current individual and N is the total number of neighboring individuals.

Alignment: This value provides an indication of the current dragonfly’s relative velocity to its nearby neighbors. This is mathematically formalized by the following equation:

where V j stands for the velocity of the jth neighboring dragonfly.

Cohesion: reflects the tendency of the dragonflies to move toward the center of the swarms group. It is represented as follows:

Attraction: Based on the current location of the food, this parameter indicates how close the dragonflies should fly to it. It is calculated as follows:

where X is the position of the current dragonfly and F p is the position of the food source.

Distraction: This parameter regulates the distance at which dragonflies should navigate to evade potential enemy. This is modeled mathematically by

where E p is the enemy position and E i denotes the position of the enemy of the ith individual. In order to calculate the next position of the dragonflies, the step vector parameter is introduced and given as

where s , a , c , f , and e are random weight coefficients related, respectively, to separation, alignment, cohesion, attraction, and distraction, w presents the inertia weight, and t is the iteration counter [24].

As shown in the equation below, the new position of the dragonflies is determined with the help of the step vector:

To facilitate exploitation, low alignment and high cohesion weights are employed, whereas for exploration, high alignment and low cohesion weights are utilized [27]. In the absence of neighboring solutions, a random walk, specifically a Levy fly, is employed to enhance exploration, introduce randomness, and contribute to the exploitation of dragonflies. The Levy is calculated as follows:

where β is a constant, r 1 and r 2 are two random numbers in the range [0, 1] and σ is calculated as

Altruism refers to selfless concern and driven by a genuine desire to benefit others and promote their welfare. It is a simple technique that serves as a local search for the optimizer and reduces the likelihood of getting stuck in a local optimum [27]. It takes very less time to execute, and it is easy to conceptualize and implement [27]. Altruism is used to enhance the searchability of the DA.

The dragonfly that performs altruism is known as the altruist, and the dragonfly that receives the benefits is called the beneficiary [27]. A random number is used to vary the positions of the altruist and beneficiary, but they are always guaranteed to be between the lower and upper bounds of the problem according to Hamilton’s rule [29] in the following equation:

where c represents the fitness cost to the altruist, b denotes the fitness benefit to the beneficiary, and r signifies their genetic relatedness [29]. Its advantages over other optimization techniques lie in its enhanced exploration, global optima search, and its capacity to navigate complex solution spaces.

3.2 SVM

The SVM is a supervised ML algorithm introduced by Vapnick [30] capable of both classification and regression tasks. SVMs are often referred to as maximum margin classifiers, as they aim to identify the optimal separating hyperplane between two classes [31]. Let us establish labeled training examples as [ x i , y i ], an input vector x i ∈ R n , a class value y i ∈ { − 1 , 1 } , i = 1 , … , l . Linear separable means, for the training data a discriminant function w t x + b exists which defines a classification function sign ( w t x + b ) , for which:

where Y is the outcome, w ∈ R n is a weight vector, and the bias b is a scalar.

The margin is defined as the minimal distance of the boundary function w t x + b = 0 to all data points. The points located nearest to the dividing hyperplane are known as the support vectors. In general, in the case of linear separability, the decision rules are dictated by an optimal hyperplane, which separates the binary decision classes, can be expressed through the following equation in terms of the support vectors [32]:

where Y is the outcome and y i is the class value of the training example x i . The vector x = ( x 1 , x 2 , … , x n ) corresponds to an input and the vectors x i , i = 1 , … , N , are the support vectors. i are parameters that determine the hyper-plane. For the non-linearly separable case, a high-dimensional version of equation (9) is given as follows:

where K ( x , x i ) is defined as the kernel function, with the constraint 0 ≤ α i ≤ C and C is a trades off margin parameter.

The well-known kernel functions for SVM are

1. Linear kernels with the forms: K ( x i , x i ′ ) = ∑ j = 1 P x i x i j .

2. Polynomial kernels: K ( x i , x i ′ ) = 1 + ∑ j = 1 P x i x i j ′ d .

3. Radial basis function kernels: K ( x i , x i ′ ) = exp − γ ∑ j = 1 P ( x i j − x i j ) 2 .

4. Gaussian kernels: K ( x i , x i ′ ) = exp − ∑ j = 1 P ( x i j − x i j ) 2 2 σ .

In this work, we have used the RBF kernel and below are some advantages and disadvantages of utilizing the SVM algorithm:

Pros: Low generalization error, computationally inexpensive, handles non-linear data efficiently by using the kernel trick.

Cons: Prone to sensitivity regarding the selection of tuning parameters and choice of kernel. The SVM model is difficult to understand and interpret [33].

3.3 The ADA-SVM model

SVMs’ performance is greatly influenced by its parameters, particularly the regularization parameter C and the kernel parameter σ . The random choice of these parameters can lead to a poor generalization and suboptimal performance [34].

Metaheuristic algorithms, with their capacity to navigate complex parameter spaces and optimize objective functions, stand poised as potentially valuable contenders for enhancing the parameter selection process in SVM. The ADA lies in its capacity to maintain a balance between exploration (searching diverse areas of the solution space) and exploitation (concentrating on promising areas with high fitness values). This balance is attained through the altruistic behavior of dragonflies, where they collaborate to find better solutions. Figure 2 describes the proposed model using SVM optimized by ADA.

Figure 2

Flowchart of the suggested ADA-SVM model. Source: Created by the authors.

The process begins by initializing a population of dragonflies, these solutions correspond to different combinations of SVM’s parameters ( C and σ ). The ADA-SVM method incorporates the altruistic behavior of dragonflies. Dragonflies share information about their fitness values with each other and make adjustments to their positions accordingly. This encourages exploration of promising regions in the parameter space. Dragonflies in the population continue to explore the solution space while exploiting regions with higher fitness. This balance between exploration and exploitation is a key feature of ADA. The algorithm performs multiple iterations, allowing dragonflies to iteratively adjust their positions based on their interactions and information sharing. This process gradually converges toward optimal or near-optimal parameter values for SVM. Once the iterations are complete, the dragonfly with the best fitness value (representing the optimal or near-optimal parameters) is selected as the final solution.

The SVM pseudocode after applying the ADA algorithm:

Interpretable insights from the ADA-SVM model can aid in making maintenance decisions by ensuring an early detection of failures, and the allocation of maintenance resources to the assets or equipment with the highest predicted maintenance needs. Through the monitoring and continuous evaluation of model performance, organizations can refine and enhance their maintenance strategies over time.

3.4 Model complexity

The time complexity of an algorithm is the amount of time required for an algorithm to be executed. Since ADA-SVM combines both ADA and SVM, the overall complexity is influenced by both components. In the case of the proposed approach, solving the SVM optimization problem can have a time complexity of O ( n 2 ∗ d ) , where n is the number of training samples and d is the number of features. However, the proposed approach presents an additive time complexity induced by the search for optimum values of the parameters ( C and sigma) found by the metaheuristics. Indeed, the time complexity of the used metaheuristics is O ( p ) , where p represent the population size.

4.1 GAs with support vector machine model (GA-SVM)

GAs are stochastic search techniques inspired by natural genetics and evolutionary principles [35,36] applied in searching for the global optimum for many problems. The evolutionary process typically commences with a population of randomly generated individuals and progresses through generations. In each iteration, the fitness of each individual in the population is assessed; several individuals are chosen from the current population based on their fitness, and then modified to generate a new population. This new population is employed in the subsequent iteration of the algorithm. The algorithm concludes when either a maximum number of generations has been generated, or a satisfactory fitness level has been attained for the population.

GAs perform the search process according to the following phases: initialization, selection, crossover, and mutation. Initialization: In this stage, a population of genetic structures, called chromosomes, that are randomly distributed in the solution space. This configuration serves as the starting point for the search process [37].

Selection: The main purpose of the selection operator is to accentuate favorable solutions and discard unfavorable ones within a population based on their fitness values. The fitness value, determined by a fitness function, quantifies the optimality of a solution, providing a numerical representation of the chromosome’s performance.

This value is employed to rank a specific solution relative to all other solutions. Various techniques can be employed to implement selection in GAs, including tournament selection, Roulette wheel selection, rank selection, and others.

Crossover: It generates a new offspring by combining two randomly selected “best parents.” This operation involves swapping corresponding segments of a string representation of the parents, extending the search for new solutions in a broader direction [37].

Mutation: It is a mechanism used to explore a wider solution space in order to avoid converging in non-adequate local minima. Mutation is a rather random operation.

Termination: The algorithm terminates if the population has converged.

Pseudocode:

1. Initialize population: Generate p solutions randomly.

2. Evaluate fitness: For each p , compute its fitness.

   1. while ( Max h fitness ( h ) < Threshold)

   2. Selection: Probabilistically select a fraction of the best individuals in p .

   3. Crossover: Probabilistically form pairs of the selected individuals and produce two offsprings by applying the crossover operator. Add all offsprings to P new .

   4. Mutation: Choose m% of P New with uniform probability. For each selected solution, invert one randomly selected bit in its representation.

   5. Update population: Set P to P new .

   6. Evaluate fitness: For each solution p in P , compute its fitness ( p )

   7. end while

   8. Return the individual from P having the best fitness.

The SVM parameters targeted for optimization include the regularization parameter C , which determines the tradeoff cost between minimizing the training error and minimizing the model’s complexity, and the parameter σ , associated with the Gaussian kernel. σ defines the non-linear mapping from the input space to a high-dimensional feature space [38]. The chromosome X is represented as X = { x , y } , where x and y indicates, respectively, the regularization parameter C and σ . In order to search optimal values of SVM parameters, we start by defining the fitness function. In our study, the mean squared error (MSE) was used:

where n is the number of all data points, T is the value to be predicted (target), and O is the model output (prediction).

The tournament selection method is adopted to decide whether an individual can survive to the next generation so that they can be placed in a matting pool for crossover and mutation. By combining the information contained in the parent individuals, the crossover produces new individuals.

We consider X old = { x 1 , x 2 , … , x n } , Y old = { y 1 , y 2 , … , y n } . X old and Y old represent the pair of populations before crossover, and X new and Y new represent the pair of new populations after crossover operation.

Offspring solutions are generated using

where β was a random number distributed in [0, 1]. Different mutation strategies are used; the uniform mutation method is applied in this study. The values are randomly mutated upward or downward.

4.2 Particle swarm optimization with support vector machine model (PSO-SVM)

Particle swarm optimization (PSO) is a nature-inspired algorithm that iteratively optimizes a problem to enhance a candidate solution based on a given measure of quality. Initially developed by Russell Eberhart and James Kennedy in 1995 [39], PSO was inspired by the social behavior of birds and fish. It is known for its ease of implementation and computational efficiency in terms of memory requirements and speed. PSO stands out as one of the most useful and well-known metaheuristics, having been successfully applied to a variety of optimization problems. The fundamental concept of the algorithm involves creating a swarm of particles that traverse the space, actively seeking their goal. Two optimization properties underlie the PSO algorithm:

• A particle determines its best position based on its individual experience, incorporating knowledge from the movements of other particles. The evaluation of positions is conducted through a fitness function, and the definition of which depends on the specific problem being optimized.

• To facilitate optimal exploration of the problem space, each particle’s velocity incorporates a stochastic factor, enabling them to traverse unknown regions within the problem space. Each particle is characterized by a position and a velocity calculated as follows:

where x i is the position of the particle i , v i is the velocity of the particle i , PBest is the personal best position for the particle i , GBest is the global best position, Rand(0,1) is the random value between 0 and 1, C 1 is the acceleration constant corresponding to the cognitive component, C 2 is the acceleration constant corresponding to the social component. An inertia weight ( w ) is employed to regulate the velocity, influencing the convergence, exploration, and exploitation processes in the algorithm [40].

The way positions and velocities are initialized, which can have an important impact on the performance of the algorithm.

The PSO pseudocode is presented as follows:

Pseudocode:

1. Initialize particle population

2. for t = 1 To maximum number of generations

3. for each particle i do

   • ∗ if ( x i < Pbest i ) then

   • Pbest i = x i

   • Gbest i = mini Pbest i

   • ∗ end if

   • ∗ for each dimension d do

   • v i , d ( t + 1 ) = v i , d ( t ) + C 1 * R 1 ( P B e s t i ( t ) − x i , d ( t ) + C 2 * R 2 ( G B e s t i ( t ) − x i , d ( t )

   • x i , d ( t + 1 ) = x i , d ( t ) + V i , d ( t + 1 )

   • ∗ end for

   • ∗ end for

   • ∗ end for

4. End.

4.3 Grey wolf optimizer with support vector machine model (GWO-SVM)

The GWO algorithm emulates the leadership hierarchy and hunting behavior of grey wolves in nature. To simulate the leadership hierarchy, four types of grey wolves are utilized: alpha, beta, delta, and omega. The hunting process involves three primary steps as documented in the literature: searching for prey, encircling prey, and attacking prey [41].

For the purpose of mathematically modeling the social hierarchy of wolves, apha ( α ) is considered as the fittest solution. The second and third best solutions are denoted as beta ( β ) and delta ( δ ), respectively. The remaining candidate solutions are considered omega ( ω ).

Pseudocode:

• – Initialize the grey wolf population X i ( i = 1 , 2 , … , n )

• – Initialize a , A , and C

• – Calculate the fitness of each search agent

• X α = the best search agent

• X β = the second best search agent

• X δ = the third best search agent

• – while ( t < Max number of iterations) do

  • ∗ for each search agent

  • Update the position of the current search agent

  • ∗ End for

  • Update a , A , and C

  • Calculate the fitness of all search agent

  • Update X α , X β and X δ

  • t = t + 1

• – end while

• return X α ,

The components of vector a → linearly decrease from 2 to 0 over iterations, aiming to accentuate both exploration and exploitation [42].

4.4 DA with support vector machine model (DA-SVM)

The DA is a swarm-based optimization algorithm inspired by the social behavior of dragonflies. The structure of the DA is introduced in Section 3.1.

In the DA-SVM algorithm, each dragonfly is a solution, representing a combination of parameters set (C, σ ). During the optimization process, the DA iteratively updates the positions of the dragonflies based on their fitness values and interactions with other dragonflies in the swarm. The main process of the DA algorithm is given as follows:

Pseudocode:

1. Initialize the swarm X i ( i = 1 , 2 , … , n )

2. Initialize Δ X i ( i = 1 , 2 , … , n )

3. while (not stopping-criteria) do

4. Evaluate all dragonflies

5. Update the source food and enemy

6. Update coefficients s , a , c , f , and e

7. Calculate S i , A i , C i , F i , and E i using equations (1) to (5)

8. Update the neighboring radius.

9. if A dragonfly has at least one neighboring dragonfly then

10. Update velocity vector using equation (6)

11. Update position vector using equation (7)

12. else

13. Update position vector using Levy flight equation (8)

14. end if

15. Return the best agents.

16. end while

Table 1 presents a comparison study that highlights both the strengths and limitations of the optimization algorithms used.

5.1 The dataset

For Airlines companies, predicting engine failures ahead of time is a key to reducing the costly delays on their flights. The examination of engine status and functioning data historically pulled by the sensors is meant to ease the failure prediction of the material and equipment in usage, so the maintenance can be planned in a timely manner. The ML algorithm aims to predict engine failures within a given period of time by scrutinizing engine sensor values over time, so it can learn the relation between these values and their changes to the historical failures.

The dataset comprises simulated R2F events of aircraft engines, including operational settings and measurements from 21 sensors. The assumption is that the degradation pattern of the engine is manifested in its sensor readings. The training data file encompasses 20,632 cycle records for a total of 100 engines. A detailed description is provided in Table 2.

The pie chart in Figure 3 represents the distribution of data types in the dataset. It reveals that 71.4% of the entries are float64 (shown in grey), and the other 28.6% are int64 (depicted in green). This suggests that the majority of the dataset comprises floating-point numbers, with a significant portion consisting of integer values.

Figure 3

Distribution of data types in the dataframe. Source: Created by the authors.

The scatter plot in Figure 4 is primarily used for data visualization by reducing the data to three principal components, and it demonstrates the separability of the classes in the dataset using the first three principal components obtained from PCA. The data points are color-coded based on their class labels, with two classes: 0 (purple) and 1 (yellow), they are not linearly separable.

Figure 4

3D PCA scatter plot of the dataset. Source: Created by the authors.

5.2 Results on the aircraft engine dataset

This section presents the experimental results concerning the employed ML models. The code was executed using Jupyter Notebook in Python and ran on a Core i5-7300U CPU processor. The evaluation metrics used for assessing the models are outlined in Table 3:

TP and TN are True Positive and True Negative values, and FP and FN are, respectively, False Positive and False Negative values. For the engine dataset, True Positive (TP) means that engines need maintenance and selected by the model.

True Negative (TN): engines that are fine and not selected by the model.

False Positive (FP): engines that are not fine but selected by the model.

False Negative (FN): engines that need maintenance but are not selected by the model.

The parameter ranges for experiments are presented in Table 4.

For the engine dataset and to better understand what the best parameters for tuning the GAs are, several tests were made. In our study, the parameters for GA were as follows: population size 100, crossover rate 0.5, mutation rate 0.1. We applied different tournament sizes and different percentages of new individuals in each generation. The GA-SVM model achieved 95% accuracy, and the best values of C and σ , respectively are 0.48785306578063 and 0.105606222983.

As mentioned above, to enhance performance, the SVM with an RBF kernel function necessitates the identification of optimal values for parameters C and σ . For that, we use the PSO algorithm and the optimal parameters that yield to the highest accuracy are the ones that will be used in the SVM model. After running the algorithm for 20 iterations, setting parameters of PSO on the selected benchmark dataset are as follow: the number of particles in the swarm N = 64 , c 1 = 0.5 , c 2 = 0.3 , w = 0.9 .

The parameter ranges for experiments are mentioned above in Table 4.

For the engine dataset, the PSO-SVM model achieved 95% accuracy, and the optimal values of C and σ , respectively, are 6.03996867 and 0.27701551.

Now, for the GWO-SVM model, the performance of the algorithm is evaluated by 100 iterations. The GWO-SVM model achieved 97% accuracy, and the best values of C and σ , respectively, are 2.64530006074 and 0.07709845. The parameters used of DA are the total number of dragonflies is 20, which gives the best results, and the attribute of Levy is 1.5 within 100 iterations. The ADA-SVM model reached 98% of accuracy with 17.62821022 and 0.06507381 as best values related to C and σ . Table 5 summarizes the values found according to the performance metrics used.

In SVM, C controls the flexibility of the model and model’s ability to generalize well in unseen data. If a large value is assigned to constant c , then the margin tends to be smaller. In that case, as the tolerance of misclassification is very limited, the model tends to be less flexible. In contrast, if a very small value is assigned to constant c , then the margin tends to be larger. The tolerance level of misclassification is high, and this can result in a high misclassification rate and poor performance.

The hyperparameter σ controls the level of non-linearity introduced in the model. A small “ σ ” value leads to a highly non-linear decision boundary, making the model better suited for complex, non-linear data. Conversely, a large “ σ ” value tends to produce a more linear decision boundary, which is suitable for linearly separable data or cases where linearity is preferred. For the ADA-SVM model, the best accuracy is achieved with C = 17.63 and σ = 0.06 . When compared to the other models, these values represent, respectively, the higher value for C and smaller for σ (Figure 5).

Figure 5

ROC curve. Source: Created by the authors.

Looking at the results of Table 5, we can deduce that all the models reached significant accuracy results. The best performance was attained by the ADA-SVM and GWO-SVM models. The ADA-SVM outperforms in terms of accuracy and other metrics.

To statistically evaluate the performance differences among the algorithmic approaches employed in this study, we utilized Student’s t-test. The use of this test enables the rigorous comparison of mean performances between pairs of algorithms, allowing for an objective assessment of their relative efficacy in optimizing the SVM model for PdM tasks.

Through Student’s t -test, we aim to ascertain whether observed differences in the accuracy. First, we calculate the p-value and test statistic value. After conducting independent sample t-tests between ADA-SVM, GA-SVM, GWO-SVM, and DA-SVM configurations, the obtained p-value was found to be less than the chosen significance level of 0.05. The results are presented in Figure 6 and Table 6.

Figure 6

Plot of classification accuracy scores. Source: Created by the authors.

In the context of GA-SVM, the p-value is 0.002, which is lower than 0.05, indicating strong evidence against the null hypothesis. (“The difference between mean performance is probably real”.) The results obtained suggest a statistically significant difference in performance between the GA-SVM and ADA-SVM models. ADA-SVM is likely performing better based on the magnitude of the t-value.

Similar to GA-SVM vs ADA-SVM, PSO-SVM also shows a significant difference with a lower p-value (0.002) and a considerable t-value ( − 5.741 ).

There is a strong evidence of a significant difference between PSO-SVM and ADA-SVM, with ADA-SVM likely outperforming PSO-SVM based on the t-value.

For GWO-SVM, the p-value is greater than 0.05. The T -value (0.97) suggests a relatively smaller difference between GWO-SVM and ADA-SVM. This suggests that these models might perform somewhat similarly based on the measured performance metric.

5.3 Limitations

First, although SVM is a robust tool in diverse ML domains, it can have limitations such as the selection of appropriate hyperparameters, susceptibility to be prone to overfitting, and also its sensitivity to noisy data as outliers or misclassified data points. In addition, ADA itself can have limitations such as the choice of parameters tuning (the number of dragonflies, iterations, etc.). Now, once the integration of ADA with SVM is performed, this can lead to

1. higher processing time,

2. increased computational complexity,

3. higher impact on the effectiveness of the model due to the measurement errors or variations that could occur in real-world maintenance datasets.

5.4 Discussion

Recent studies have highlighted the effectiveness of soft computing, ML techniques, and metaheuristics across various domains, showcasing their adaptability and robustness. Singh and Shrivastava [43] introduced an advanced feature selection methodology by integrating three soft computing-based optimization algorithms: teaching learning-based optimization, elephant herding optimization and a novel hybrid algorithm combining these two algorithms. Their method achieves a remarkable accuracy of 97.96%. MunishKhanna et al. [44] explored the use of ant-lion optimization to select a subset of features. These selected features are then utilized to train and evaluate four ML classifiers and their ensemble. The strategy proposed can reduce the initial feature set by up to 50% without compromising performance in terms of accuracy. In their work [45], one of the objectives is the application of the artificial bee colony algorithm for the regression test case prioritization problem and comparing its performance with other algorithms. Moreover, this study also presents two greedy-based tie-breaking algorithms designed to efficiently rank test cases based on their importance and likelihood of success. Previous study presented an improved version of SVM for PdM in industrial applications. They incorporate an entropy-based feature selection method to improve the accuracy of the model. The results show significant improvement over standard SVM methods.

Buabeng et al. [46] combined multiple ML techniques to enhance fault diagnosis accuracy and efficiency in complex industrial systems. By integrating methods such as improved complete ensemble empirical mode decomposition with adaptive noise, principal component analysis and least squares support vector machine optimized by the combination of coupled simulated annealing and Nelder-Mead simplex optimisation algorithms (ICEEMDAN-PCA-LSSVM). The analysis results indicated that the proposed ICEEMDAN-PCA-LSSVM technique is highly versatile and surpassed all the compared classifiers in terms of accuracy, error rate, and other evaluation metrics considered. Khana et al. [47] introduced a multi-objective approach for the test suite reduction problem in which one objective is to be minimized and the remaining two maximized. In this study, experiments were performed on diverse versions of four web applications. This article introduces a multi-objective approach to address the test suite reduction problem, aiming to minimize one objective while maximizing the other two. The study conducts experiments on various versions of four web applications to validate the effectiveness of the proposed method. Through these experiments, the approach demonstrates its capability to optimize the test suite by significantly reducing the number of test cases while ensuring maximum test coverage and fault detection.

The ADA-SVM model stands out by incorporating the ADA for parameter optimization and SVM for classification. To thoroughly assess the proposed ADA-SVM model, it will be compared with other ML algorithms from the literature. Table 7 presents the results of six efficient classifiers from previous studies on aircraft engine data.

Table 7 compares the performance metrics of various ML models used for PdM of aircraft engines. The models listed include LSTM, decision tree (DT), artificial neural network (ANN), SVM, convolutional neural network (CNN), and Gaussian Naive Bayes (GNB).

This indicates that ADA-SVM outperforms all the other algorithms listed in Table 7 in terms of classification metrics. The high accuracy of ADA-SVM implies better predictive capabilities and reliability, which are crucial for minimizing downtime and maintenance costs in industrial applications.