In the current article, the lung cancer gene expression microarray dataset was chosen for the current study, We examined the known and publicly accessible microarray dataset, and Table 1 shows the data information used in this article.

The details of each dataset are as follows:

Lung_Adenocarcinoma: One is a pre-processed dataset of 7129 genes that includes 86 patients with primary lung adenocarcinoma, 62 of whom were still alive and 24 of whom had passed away [23,39].

Lung_Michigan: Ten non-neoplastic lung samples are also included along with 86 primary lung adenocarcinoma samples. 7129 genes characterize each sample [40].

Lung_Harvard1: A total of 203 snap-frozen lung cancers and healthy lungs were examined. The 203 specimens consist of 139 lung adenocarcinoma samples (labeled as ADEN), 17 samples of normal lung tissue (labeled as NORMAL), 6 small-cell lung carcinoma samples (labeled as SCLC), 20 pulmonary carcinoids samples (labeled as COID), and 21 squamous cell lung carcinoma samples (labeled as SQUA). 12600 genes are used to characterize each sample [23].

Lung_Harvard2: MPM (“Malignant Pleural Mesothelioma”) and lung ADCA (“Adenocarcinoma”) are separated into several categories. There are 181 samples of tissues (150 ADCA and 31 MPM). 32 of them, including 16 ADCA & and 16 MPM, are in the training set. For testing, the remaining 149 samples were utilized. 12533 genes are used to characterize each sample [52].

Lung_Ontario: Data on gene expression from 39 different NSCLC samples’ tumor samples. 24 of the patients in these samples showed distant metastasis (labeled as “relapse”) or local relapses of their tumor. According to radiological and clinical testing (labeled as “non-relapse”), the remaining 15 individuals are disease-free. 2880 genes are used to characterize the processed data [53].

Fig. 2 presents the class distribution of the previously explained, which clearly shows an imbalance in the data and the difference in varieties.

Class distribution imbalance is a common issue in lung cancer datasets, impacting classification model performance. Medical datasets, including lung cancer data, often suffer from skewed class distributions where the number of instances belonging to different classes is significantly imbalanced. To address this issue, it is required to apply specialized techniques such as resampling, where the dataset is adjusted to balance class proportions. Methods like oversampling the minority class, undersampling the majority class, and synthetic data generation using, e.g., AOA can help balance the class distribution. Thus, it enhances the model’s ability to learn from both classes and improve its predictive accuracy.

It is a newly proposed population-based algorithm, AOA utilizes the Arithmetic operators (Addition, Subtraction, Division, and Multiplication) for searching in search space to discover the best solution for a specific issue. One of the main advantages of AOA is the simplicity and considering both diversification (exploration) and intensification (exploitation) of the population [30].

AOA begins the search process with a random initial population with a specific number of solutions (population size). In the feature selection problem, the solution is represented as a vector of “binary values” (0 or 1) with length dataset diminution (total number of features in the dataset), where 0 denotes the not selected feature and 1 for the selected feature. Then the algorithm starts improving those solutions until the specified number of iterations is obtained. The improvement process contains two primary phases: The exploration phase aims to study the search space and determine the promising search regions, and the exploitation phase aims to deeply search in the explored search regions for the optimal solution. AOA determines whether to explore or exploit the search space by using the accelerated function, which is denoted as (AF); see Eq. (1) [30]. (1)AFCrnt_Iter=(min+CrntIter)(max−minMaxIter)where AFCiter indicates the accelerated function calculated for the current iteration (Crnt_Iter), and Max_Iter signifies the total number of iterations, min and max are parameters to determine the value of AF that increased linearly from min to max to perform exploration and exploitation. where if a random number between 0 and 1 is greater than AF, then the exploration phase will be performed; otherwise, the exploitation phase will be performed. Algorithm 1 presents the AOA pseudo-code [30].

AOA exhibits exploratory activity throughout the exploration phase. As per arithmetic operators, the mathematical computations employing either the D (division) or even the M (multiplication) operator produced highly scattered decisions or values (refer to several reigns), which commit to the exploratory search process. Nevertheless, owing to their significant dispersion compared to other operators (A and S), these operators (M and D) find it difficult to approach the objective. The impact of the distribution values of the various operators is shown using a function built on the foundation of four mathematical operations. Therefore, the scout research discovers a near-optimal solution that could be derived after multiple iterations. Moreover, the exploration operators (M and D) were used in this optimization phase to help the other phase (exploitation) in the research process via better mutual communication.

AOA’s exploration operators randomly determine the search area in different areas and get closer to discovering a better solution based on two primary search strategies (Multiplication and Division search strategy), which are expressed in Eq. (2). This search phase (exploratory search by performing M or D) is determined by the “accelerated function” (AF) presented in Eq. (1) for the condition that any number is greater than AF. The first operator (D) is determined by a random number (r1) less than 0.5 and 2^nd operator (M) is neglected until this operator ends its present task. Otherwise, the 2^nd operator (M) is activated to conduct the present task rather than D. A stochastic scale coefficient for the element is considered to create additional diversification paths and investigate different areas of the space of research. AOA envisions the following position update equations being offered for exploration games: (2)xi,j(CrntIter+1)={best (xj)÷(PF+ϵ)×((UBj−LBj)×µ+LBj),r2<0.5best (xj)×PF×((UBj−LBj)×µ+LBj), otherwisewhere the solution within the next iteration is denoted by xi,j(Crnt_Iter +1) which signifies the i^th solution’s j^th position at the present iteration, and best(x_j) indicates the j^th position in the best-obtained solution so far. ϵ indicates a small integer number, UB_j & LB_j signify the upper and lower bound value of the j^th position, respectively. µ indicates the control parameter to modify the search process. (3)PF(CrntIter)=1−CrntIter1αMaxIter1αwhere probability function (PF) represents a coefficient, PF (Crnt_Iter) indicates the function value at the t^th iteration, Crnt_Iter signifies the current iteration, and (Max_Iter) signifies the highest number of iterations. α represents a sensitive parameter and describes the accuracy of the exploitation across iterations.

In the exploitation phase, mathematical computations that either used addition (A) or subtraction (S) obtained high-density findings related to the exploitation search process. Nevertheless, these traders (A and S) may quickly reach the target owing to their low spread, unlike other traders. Therefore, despite the low spread of these traders (A and S), they are adept at rapidly approaching the optimal solution, which may be derived after multiple iterations [30]. Moreover, the exploitation operators (A & S) were operated in this phase of research (exploitation research by performing S or A) conditioned by the value of the function AF for the condition that r1 is not higher than the current AF_{Crnt_Iter} value, as shown in Eq. (1). In AOA, the exploitation operators (addition (A) and subtraction (S)) of AOA examine the search area in different dense areas and converge to discover the best solution based on two primary search strategies (search by addition (A) and subtraction (S) strategy) modeled in Eq. (4). (4)xi,j(CrntIter+1)={best(xj)−(PF+ϵ)×((UBj−LBj)×μ+LBj),r3<0.5best(xj)+PF×((UBj−LBj)×μ+LBj),otherwise

This phase utilizes the search space by executing a depth-first search. In this phase, modeled in Eq. (4), a random number (r3) conditions the 1^st operator (S), less than 0.5, and the other operator (A) is neglected until this operator ends its current task. Otherwise, the 2^nd operator (A) oversees performing the present task rather than S. These methods in this phase are like the partitions in the previous phase. Nevertheless, exploitative search operators (A and S) frequently try to avoid being stuck inside the local search perimeter. This approach supports exploration search methods to discover the best solution and keep the variety of candidate solutions. The parameter μ is designed to create a “stochastic value” at every iteration to maintain the exploration not only in the 1^st iterations but also during the last iterations.

A dataset consists of instances (rows), entities (columns), and classes. With datasets, the main problem is classifying the data or putting the invisible data into its proper class. The feature selection problem is the name provided for issues in this area. The main goal of the developed problem is the selection of the best dataset which will improve the classification precision or minimize the error rate. Some characteristics in the original dataset are unnecessary, irrelevant, or redundant, which has a detrimental impact on the classifier’s performance [54]. Therefore, feature selection issues are created to enhance the classifier’s performance and condense the size of the data collection.

The solutions to feature selection are only available as binary values since it is a binary optimization problem. A binary version of AOA must be created so that it may be utilized for feature selection problems. The number of features within the dataset is used to determine the length of the vector used to describe a solution in this work, which is a one-dimensional matrix. “1” or “0” stand in for each value in the matrix. If the value is “1,” it means that the associated property is chosen; otherwise, it is “0”.

In AOA after calculating the x_ij using Eqs. (2) and (4) and limiting the UB and LB to 0 and 1, respectively, the transfer function is used to obtain the binary solutions. AOA was suggested over continuous search space; Thus, we proposed a transfer function for binary AOA which is called S-shaped transfer function [55] to resolve the issues into binary search space as in Eq. (5). (5)TF(xij(CrntIter))=11+exp(−xij(CrntIter))where xij(CrntIter) is the i^th element in the dimension j and CrntIter indicates the current iteration number. Eq. (6) is used in AOA to update a solution for S-shaped transfer functions to acquire the subset of features. (6)xij(CrntIter+1)={0,  if  rnd>TF(xij(CrntIter))1,  if  rnd≤TF(xij(CrntIter))

The minimization of the number of chosen features while reducing the classification error rate may be thought of as a “multi-objective optimization” problem for feature selection. Each solution is assessed in accordance with the suggested fitness function, which relies on three classifiers to determine the solution’s classification error rate and several features that were specifically chosen for the solution. Both the AOA and modified AOA algorithms assess search performance using the objective function in Eq. (7).

(7)OF=γe+(1−γ)SFTFwhere OF is the objective function to be minimized, γ ∈ [0, 1] represents the importance of classification quality (1−γ) the importance of the number of features chosen and e indicates the error rate of a particular classier. Also, SF indicates the number of “selected features” and TF denotes the total number of features within the dataset [56].

Working with a high-dimensional dataset requires a high search capability of the algorithm, where the complexity of the algorithm may grow exponentially with the dimensionality of the feature space. We present a modification of AOA to search effectively in high-dimensional feature spaces. The modified version of AOA is called MAOA.

In the initialization phase of MAOA, the features are ordered according to their weights and are utilized to initialize the population. The initialization process starts with ranking the features using one of the classifiers presented in Section 4.1 and saving them in the ranked list. Then, the solutions in the population are initialized based on the ranked list, and the features in each solution are set to 1 (selected) from the random features of the top 50% of the ranked list, and the rest are set to random 0 or 1.

The exploitation phase in the MAOA algorithm is supported by local search to enhance the performance by adding significant features and eliminating redundant ones from the features based on their feature ranking. As can be noted in the AOA, exploitation, as in Eq. (4) is based on calculating the subtraction and addition operators of the best position, then a transfer function is applied. We assume that using a local search with a neighborhood strategy around the best position will enhance the findings. Therefore, the neighborhood operators that are proposed to replace the subtraction and addition operators.

Let’s consider the solution is x = [1, 0, 1, 0, 1, 0].

NB Change neighborhood method is used instead of the subtraction operator, which chooses a random feature and modifies its value by the “Not operator”, For instance, assuming that the third feature in Sol, which was chosen at random and has a value of 1, will now be changed to 0 by the not operator. Therefore, the updated solution would be as follows: x^* = [1, 0, 0, 0, 1, 0],

NBMove neighborhood method is used instead of the addition operator, which chooses a random feature and moves its location to a new position, For instance, assuming that the randomly selected feature is positioned at the first position in vector x (which equals 1), then it is relocated to a new random position” (let’s suppose the 4^th position) as illustrated in Fig. 3.

The updated solution would be as follows: x^* = [0, 1, 0, 1, 1, 0].

These neighborhood strategies are applied during each iteration to a randomly selected set of RSN solutions, where RSN represents the number of solutions considered for the local search update. Then, these selected solutions are assessed using the classifier, and the best solution among the neighbors is chosen to progress to the next iteration.

Since we intended to explore different types of machine learning models for microarray datasets, there are two significant processes for the prediction: The first step is to get the data ready for estimation, and the second is to compare the predicted models. The criteria used to compare the models include the average accuracy of 4-fold cross-validation, precision, recall, and F1_Score. The predictive classifying models used in this study are KNN (“K-Nearest Neighbor”). The machine learning models used for comparison are DT (“Decision Trees”), SVM (“Support Vector Machine”), RF (“Random Forest”), and Naive Bayes (NB) classifiers. These classifiers have been selected based on the time taken to train the model and the performance of the classifiers [57].

As shown in Table 2, the optimum findings were achieved with the K-NN and RF algorithms, and the NB algorithm shows comparable performance and has obtained good and best results in the Lung_Michigan dataset, while the worst results were found with the SVM and DT algorithms. For a clearer view, Fig. 4 shows the column chart for each classifier. The performance measure metrics are ordered from left to right as Average Precision, Recall, Accuracy, and F1_Score.

Overall, Fig. 4 shows that KNN has obtained the highest accuracy for 3 datasets out of 5, followed closely in 3 datasets, and outperformed by RF in 2 datasets. In some cases, NB and DT have obtained competitive accuracy scores, while the SVM has shown inferior performance. However, across all 5 datasets, the KNN has obtained higher metrics across all datasets, meaning that it has proven its suitability for the lung cancer classification task, mainly presented by F1_Score and precision. The rest of the classification algorithms have obtained inferior metric scores.

Since we are targeting to perform feature selection and classification simultaneously, we have chosen KNN as the evaluator to assess the selected subset of features and to produce the error rate used in Eq. (7).

FS-wrapper method and the KNN classifier were both used in several studies, and evaluated using k-fold cross-validation (4 folds used) [58,59]. K-fold cross-validation uses various random subsets of the data set each time. The other parts are likewise randomly chosen for training, while one component is randomly chosen for testing. Each experiment was performed in 10 independent runs for every dataset and the mean of these 10 runs was noted in terms of precision (accuracy), the value of the objective function, number of chosen features, boxplot, the convergence of time, and curves to execution.

The performance of the suggested algorithms and other optimization algorithms are tested on 5 lung cancer datasets. AOA (MAOA) algorithm was evaluated to the original AOA and five well-recognized optimization processes [60], such as PSO (“Particle Swarm Optimization”) [61], GWO (“Grey Wolf Optimizer”) [8], MFO (“Moth-Flame Optimization”) [62], FA (“Firefly Algorithm”) [63] and WOA (“Whale optimization algorithm”) [64].

The parameters settings of AOA and MOA used are:

α (sensitivity of exploitation, Eq. (3)) = 5.

To obtain a fair comparison, the algorithms under consideration have been applied with the same number of iterations along with population sizes of 100, and 20, respectively, with Accuracy where Max, Average Accuracy, and standard deviation are presented the Table 3 where the best-obtained results are bolded.

As shown in Table 3, MAOA performs well compared to other population-based algorithms; MAOA surpassed all other optimizers in terms of maximum classification precision across all 5 datasets. The outperformance of MAOA results from its capability to escape local optima with local search using the neighborhood strategy and improvements in initial solution diversity using the AOA solution initialization strategy.

To demonstrate the consistency of the outcomes produced by the suggested MAOA in terms of feature reduction, the boxplot is presented in Fig. 5. The datasets that show differences are plotted, including the Lung_Harvard2, Lung_Adenocarcinoma, and Lung_Ontario datasets. Notably, the Lung_Harvard2 and Lung_Michigan datasets are not included due to their similar results observed across various runs.

As shown in Fig. 5, MAOA is superior to the other algorithms, as can be seen in the line in the middle of the box, along with the upper and lower bounds for the three data sets used in the comparisons. The Wilcoxon statistical test was utilized to assess the importance of the reported findings. This test makes it possible to evaluate the importance and robustness of the MAOA compared to the original AOA and other optimization methods. The average classification accuracy across 10 runs was used to define the significance threshold for this test, which was set at 5%.

Table 4 shows the values of p achieved from the Wilcoxon trial of all competing methods. If the p-value in this test is below 5%, it means that MAOA shows significant enhancement over competing algorithms. Otherwise, there is no significant enhancement. The p-values for Lung_Harvard2 and Lung_Michigan are indicated as ‘-’ due to the uniform identity of outcomes across all comparators, where accuracy reached 100% (Table 3). This uniformity makes a t-test impractical. The findings in Table 5 show that MAOA shows an important enhancement over other algorithms in terms of average classification accuracy in most datasets. Table 5 shows the minimum, maximum, and mean number of features chosen.

As shown in Table 5, MAOA outperformed AOA and other algorithms in all datasets because it chose significantly fewer features. The superiority of MAOA is validated by a clear variation in all data sets. These results obtained by MAOA emphasize that it perfectly initializes the solutions by choosing the most informative features using the ranking method. Moreover, MAOA highlights its stability to perform very well against high-dimensional datasets. Moreover, with each iteration, the MAOA improves the solutions by selecting the relevant functionalities while removing the irrelevant ones. This increased classification precision. It should be noted that the changes made to the AOA have enhanced its mining and exploration abilities and given it more ability to search on high-dimension datasets.

Table 6 provides a detailed breakdown of the proportion of features preserved under each approach, addressing the aspect of feature complexity reduction.

In Table 6, each cell represents the percentage of features reserved for each algorithm. These percentages show how different algorithms performed in the dimensionality reduction processes for each dataset. MAOA significantly reduces the number of features compared to other algorithms for most datasets.

Furthermore, the training accuracy and the number of selected features in each iteration are utilized to produce the outcomes of the objective functions, as shown in Table 7.

Table 7 summarizes the minimum, mean, and standard deviation. All competing algorithms’ values for the objective function show that the MAOA outperforms them all in terms of generating the minimum values for the objective function. As revealed in Table 7, MAOA outperforms other methods (the best findings are shown in bold). These findings demonstrate the repeatability and stability of the MAOA method on datasets with various degrees of dimension.

To further evaluate and scale the MAOA, Fig. 6 exhibits the mean convergence curves of the competing algorithms across the five data sets, offering a visual representation of how the optimization process evolves over iterations. MAOA continues to exhibit the best objective function values, having the lowest values after 100 iterations. These findings demonstrate the appropriateness of the incorporated amendment to the AIC. Therefore, from the experiments, MAOA always surpasses the original AOA algorithm. This outperformance of the MAOA compared to the AOA derives from the use of a local search algorithm based on two neighborhood operators to enhance the exploitation capacity of the MAOA and prevent it from stagnating in the local optima.

Table 8 displays the average execution times of all competing methods to validate the impact of the improvements on the MAOA’s execution time. It should be observed that the MAOA performed better than all other algorithms in 4 out of 5 datasets in terms of average “execution time”. The MAOA execution time proves to be lower than other algorithms, also, employing the local search algorithm to reduce the selected features, leads to a reduction in the training time of the classifier, which makes MAOA outperform other methods in terms of objective function values, feature reduction, accuracy, and the statistical findings mentioned earlier. This combination of factors highlights the advantageous balance achieved by MAOA.

Lastly, to highlight the significance of the feature selection phase by the MAOA, Table 9 presents a comparison between the MAOA accuracy with feature selection and the accuracy without utilizing the feature selection produced by the best classifier taken from Table 2, which uses the full set of features. The reported findings identify the significance of the feature selection phase, as it has led to improved classification accuracy by reducing the dimensionality of such datasets and eliminating redundant and irrelevant features. Considering all the above findings, MAOA has proven to be a competitor when used to solve feature selection problems on high-dimension datasets.