https://orcid.org/0000-0002-7283-1538
Figures
Abstract
This study provides a comprehensive evaluation and classification of 35 soft decision-making (SDM) algorithms based on fuzzy parameterized fuzzy soft matrices (fpfs-matrices). Although fpfs-matrices offer a strong mathematical framework for modeling uncertainty, there has been a lack of large-scale comparisons of their derivative SDM methods in machine learning. To address this, we used the Comparison Matrix-Based Fuzzy Parameterized Fuzzy Soft Classifier (FPFS-CMC) to benchmark these 35 algorithms across ten diverse datasets from the UCI Machine Learning Repository. The methods were thoroughly assessed utilizing metrics such as accuracy, precision, recall (sensitivity), specificity, and F1-score, and statistical significance was confirmed using the Friedman and Nemenyi tests. Our results show that SDM methods via fpfs-matrices perform competitively in classification tasks involving uncertainty. Notably, the best algorithms according to F1-scores were A19 (Rank 1), YHX14 (Rank 2), and a three-way tie for Rank 3 among VMH16, AKO18o, and A19/2. By identifying the most effective algorithms and offering a structured decision-support framework, this research provides both a theoretical reference and practical guidance for practitioners selecting SDM methods for complex machine learning challenges.
Citation: Karakoç Ö, Memiş S, Sennaroglu B (2026) Classification of soft decision-making methods via fuzzy parameterized fuzzy soft matrices and their performance-based statistical analysis in machine learning. PLoS One 21(5): e0348760. https://doi.org/10.1371/journal.pone.0348760
Editor: Rajesh Kumar, National Institute of Technology, India (Institute of National Importance), INDIA
Received: January 10, 2025; Accepted: April 20, 2026; Published: May 13, 2026
Copyright: © 2026 Karakoç et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The datasets used in this study are available on Figshare at: https://doi.org/10.6084/m9.figshare.29120426 These datasets were originally obtained from the UCI Machine Learning Repository and are provided here in the exact form used in this study. The datasets are shared under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
Traditional mathematical methods may not be sufficient when dealing with problems that involve uncertainties. To overcome this issue, fuzzy sets [1] and soft sets [2] have been introduced to address uncertainty and applied in various fields. Various extensions of fuzzy and soft sets have emerged, including fuzzy soft sets [3,4], fuzzy parameterized soft sets [5], and fuzzy parameterized fuzzy soft sets (fpfs-sets) [6]. The concept of fpfs-sets has been particularly prominent due to its modeling ability. However, when dealing with a problem that has several criteria and high uncertainty, computerizing the fpfs-sets becomes necessary. To that end, soft matrices [7], fuzzy soft matrices [8] and fuzzy parameterized fuzzy soft matrices (fpfs-matrices) [9] have been proposed. fpfs-matrices have a broad scope by combining matrices, fuzzy and soft sets, and their extension. These provide the ability to address various types of uncertainty. Therefore, some soft decision-making (SDM) algorithms constructed by soft sets, fuzzy soft sets, fuzzy parameterized soft sets, fpfs-sets, soft matrices, and fuzzy soft matrices have been configured via fpfs-matrices. Since fpfs-matrices possess a modular structure and fuzzy criteria, they can be implemented more efficiently in applications. Recently, fpfs-matrices have been used in various decision-making problems, such as recruitment scenarios [10–12] and performance-based value assignment (PVA) problems in image denoising [13–16] as well as classification problems in machine learning [17–20].
Compared to algebraic research on new operations of soft sets [21], the potential applications of soft sets have been discussed in detail in various studies, and it has been stated that integrating soft sets into decision-making processes improves uncertainty management thanks to their flexible and parametric structures [22, 23]. In addition, the increasing importance of soft set theory in multi-criteria decision problems has been emphasized in a systematic review [24]. In this context, fpfs-matrices offer more detailed and adaptable analyses by integrating fuzziness and parametric structure compared to classical matrix models, and produce meaningful and consistent results, especially in multi-criteria decision support systems. In addition, SDM methods based on fpfs-matrices and fpfs-matrices-based k-nearest neighbor classifiers, such as FPFS-kNN [25], have been applied to energy efficiency analysis, evaluating heavy commercial vehicles in the logistics sector, energy planning in Türkiye, and academic performance evaluation. Aside from the studies mentioned above, the optimal feature selection-based dental caries prediction study [26] and the research demonstrating that fpfs-based machine learning algorithms effectively predict NOx emissions in the maritime field [27] have significantly inspired the application of this method across various disciplines. Following these developments, ifpifs-matrices structures, which offer a higher degree of uncertainty modeling capacity, have been developed, enhancing intuitiveness in decision-making. The studies [28,29] have showcased the advantages of this structure in applications such as evaluation and filter selection using multiple ifpifs-matrices. One of the most recent contributions is pioneering the development of next-generation algorithms for artificial intelligence applications by integrating the ifpifs-matrices-based SDM approach into adaptive machine learning models [30]. All these studies demonstrate that SDM methods address a broad range of applications when utilized with fpfs-matrices and ifpifs-matrices, providing an effective tool for modeling high degrees of uncertainty. The scope of this work is confined to fpfs-matrices due to their ease of implementation.
Although a comparison is provided, no performance metrics are provided to rank the algorithms’ success. Five test cases have been defined to eliminate the unsuccessful ones. These include five scenarios where an expert can naturally rank alternatives as the basis for the test cases [31]. An SDM technique completes a test case by generating its specified ranking order. To evaluate and compare the performance of algorithms that pass the test, one can use performance metrics and assess them with different machine learning algorithms. Latterly, SDM methods via fpfs-matrices have been applied to machine learning in a very striking way. The machine learning algorithms called FPFS-CMC [32] and FPFS-AC [33] use two SDM approaches. Despite advancements and successful implementations of various SDM algorithms in specific areas, a major gap persists in the literature: there is no unified framework for comparing their modeling capabilities and performance limits. Current research mainly uses isolated SDM techniques on narrow case studies, making objective benchmarking under consistent conditions difficult. As a result, researchers and practitioners lack a systematic framework for determining which SDM approach is best for different decision-making scenarios or high-dimensional data environments.
To address this issue, this study introduces a modular evaluation framework that combines SDM methods within machine learning architecture. Using the Comparison Matrix-Based Fuzzy Parameterized Fuzzy Soft Classifier (FPFS-CMC) as a standardized benchmarking tool, we provide a quantitative and statistically supported assessment of SDM performance. The main goal of this paper is to systematically classify 35 prominent SDM algorithms and perform an extensive comparative analysis across 10 real-world datasets from the UCI Machine Learning Repository. The key contributions of this research are fourfold:
- This paper offers the first comprehensive taxonomy and comparison of 35 SDM algorithms from the literature.
- By embedding SDM components into the modular FPFS-CMC structure, it converts qualitative decision-making tools into measurable machine learning classifiers.
- This study provides an objective ranking of SDM methods based on multiple performance metrics (accuracy, precision, recall, specificity, and F1-score), validated through rigorous statistical tests.
- It presents a decision-support framework that connects the abstract theory of fpfs-matrices with real-world data classification problems.
Considering these points, this paper introduces a significant methodological shift from “single case” applications to a “comprehensive” performance evaluation. By benchmarking 35 different SDM algorithms across ten real-world datasets, we provide a statistically validated reference for uncertainty modeling in artificial intelligence. This study highlights the most effective algorithms, such as A19 and YHX14, and creates a modular framework for future advancements in soft decision-making.
The organization of this paper is as follows: The second section introduces the concepts of fpfs-sets and fpfs-matrices, along with some related properties. The third section presents five test cases for fpfs-matrices and compares SDM methods across them. The fourth section provides a brief introduction to structured fpfs-matrices, their application fields, decision models, and the mathematical operators used. The fifth section compares and classifies algorithms based on their F1-scores. The performance changes of algorithms are discussed through statistical analysis. The sixth section analyzes the computational complexities of the SDM methods. The last section outlines the results and suggests directions for future work.
2. Materials & Methods
This section presents the concepts of fpfs-sets [6] and fpfs-matrices [9]. From now on, let be a parameter set,
be the set of all the fuzzy sets over
, and
. Here, a fuzzy set is denoted by
.
Definition 1.[6] Let be a universal set,
, and
be a function from
to
. Then, the set
, being the graphic of
, is called a fuzzy parameterized fuzzy soft set (fpfs-set) parameterized via
over
(or briefly over
).
Across the present paper, the set of all the fpfs-sets over is denoted by
. In
, since the
and
generate each other uniquely, the notations are interchangeable. Thus, as long as it leads to no confusion,
stands for an fpfs-set graph
.
Example 1. Let and
. Then,
is an fpfs-set over .
Definition 2.[9] Let . Then,
is called fpfs-matrix of
and is defined by
such that for and
,
Here, if and
, then
has order
.
Hereinafter, the set of all the fpfs-matrices parameterized via over
is denoted by
.
Example 2.The fpfs-matrix of provided in Example 1 is as follows:
Definition 3.Let . Then, feature fuzzification of
is defined by
Definition 4.Let such that
. Then, normalization
of
is defined by
Table 1 presents a list of 35 SDM algorithms [31,34–36], along with their abbreviated versions. For more details about the algorithms, see their references in Table 1.
3. Test Cases for the Comparison of the SDM Methods
This section outlines five test cases from [31] to compare the decision-making performance of SDM methods that use single, double, or multiple fpfs-matrices. Therefore, each test case consists of fpfs-matrices
, which has order
and manifests the same ranking order of alternatives without employing SDM methods. If an SDM method utilizes a single fpfs-matrix, we only use
. Similarly, if double, we use
and
. If an SDM method produces the ranking order provided in a test case, it is said to accomplish the test case. In this section, let
,
,
,
be the set of alternatives, and
be the set of parameters.
3.1. Test Case 1
Test Case 1 constructs three fpfs-matrices ,
, and
such that for all
and
,
and
. Therefore,
, for all
and
. For each fpfs-matrix herein, the ranking order of alternatives is
. For example,
3.2. Test Case 2
Test Case 2 constructs three fpfs-matrices ,
, and
such that for all
and
,
and
. Therefore,
, for all
and
. For each fpfs-matrix herein, the ranking order of alternatives is
. For example,
3.3. Test Case 3
Test Case 3 constructs three fpfs-matrices ,
, and
such that for all
and
,
,
, and if
, then
. Therefore,
and if
, then
, for all
and
. For each fpfs-matrix herein, the ranking order of alternatives is
. For example,
3.4. Test Case 4
Test Case 4 constructs three fpfs-matrices ,
, and
such that for all
and
,
,
, and if
, then
. Therefore,
and if
, then
, for all
and
. For each fpfs-matrix herein, the ranking order of alternatives is
. For example,
3.5. Test Case 5
Test Case 5 constructs three fpfs-matrices ,
, and
such that for all
and
,
. For each fpfs-matrix herein, the ranking order of alternatives is
. Here,
denotes the same ranking order. For example,
According to Table 2, 35 algorithms passed all test cases. Among 35 SDM algorithms, 12 are based on single fpfs-matrices, nine are based on double fpfs-matrices, and 14 are based on multiple fpfs-matrices. Parameters are predetermined and used as ,
,
,
,
,
,
,
,
, and
.
4. Classification of Configured SDM Methods
The study [64], which classifies classical MCDM methods into value/benefit-based, simple problem-solving, superiority-based problem-solving, and interactive problem-solving, inspired us in classifying SDM methods. It is crucial to classify SDM methods using soft set theory, a newly explored area, based on problem type and structure, thereby enabling the classification of SDM algorithms. During the classification, we considered the empirical study in which it was first used, the mathematical operator utilized therein, its performance in the defined test scenarios, the decision-making model, and the parameters employed. Based on the fpfs-matrices, 74 algorithms were configured and tabulated by their matrix structures, as shown in the tables. It discusses the original concept in depth, including the fpfs-based matrices, the decision model, and the mathematical operations herein.
We classified the methods based on the following criteria:
Empirical study: Each method and its corresponding application areas have been examined.
Operation: It denotes the mathematical operation used by that algorithm in the background.
Number of tests passed: Five test scenarios have been defined to eliminate unsuccessful algorithms. These scenarios are designed so that an expert can naturally list alternatives that serve as the basis for test cases. The criteria specify the number of tests completed, thereby facilitating evaluation.
Originality of Concept: Some SDM algorithms based on soft sets, fuzzy soft sets, fuzzy parameterized soft sets, fpfs-sets, soft matrices, and fuzzy soft matrices have been implemented using fpfs-matrices.
Decision Model: The mathematics used by algorithms indicates the decision model.
Parameter: It denotes parameter structures used in the algorithm.
Tables 3–5 detail SDM methods, including their abbreviations, the concepts they are based on, the decision model and mathematical operators they employ, the first empirical study in which they were used, and their parameters.
5. Performance Comparison of SDM Methods Using Machine Learning
Table 6 presents the properties of the datasets used in the simulation herein: Parkinsons[sic], Wine, Sonar, Ecoli, Hayes-Roth, Libras Movement, Teaching, Ionosphere, Whosaler, and Glass. We subsequently present the mathematical notation for the performance metrics to compare the aforesaid methods. To ensure consistency and fair benchmarking, all ten UCI datasets underwent a standardized preprocessing step. This included feature fuzzification according to Definition 3 and, when needed, normalization as outlined in Definition 4, converting raw attributes into compatible fpfs-matrix structures before classification.
The FPFS-CMC algorithm is presented in Algorithm 1. It uses the Pearson correlation coefficient to estimate feature weights based on the impact of the parameters on classification. Then, it creates two fpfs-matrices, one for training and one for testing, by fuzzifying the features of the training and testing samples based on their weights. It then constructs a comparison matrix by calculating pseudo-similarities between the training and testing fpfs-matrices. Afterward, the standard deviation is calculated for each column of the comparison matrix to obtain parameter weights. The comparison fpfs-matrix is then constructed by hybridizing the parameter weights and the comparison matrix. After that, the sMBR01 algorithm is applied to the comparison fpfs-matrix to obtain the optimal training sample. Then, the class label of the optimal training sample is assigned to the testing sample. This process is repeated for all the testing samples.
Algorithm 1. FPFS-CMC Algorithm’s Steps [32]
Procedure FPFC-CMC(
1. Compute using
and
2. Compute feature fuzzification of and
namely
and
3. for from
to
do
4. Compute the testing fpfs-matrix using
and
5. for from
to
do
6. Compute the training fpfs-matrix using
and
7.
8.
9.
10.
11.
12. end for
13. for from
to
do
15. end for
16.
17. Compute comparison fpfs-matrix using
and
18.
19.
20. end for
21. return
end procedure
Commonly used performance metrics [75], such as accuracy (Acc) (Eq. 1 and 10), precision (Pre) (Eq. 2 and 11), recall (Rec) (Eq. 3 and 12), specificity (Spe) (Eq. 4 and 13) and F1-score (F1) (Eq. 5 and 14) are presented to compare the performance of the algorithms. The mathematical notations of these performance metrics are as follows:
samples to be classified
let it be denoted as.
the correct classes of these instances,
let
denote the predicted classes of these samples and
the total number of classes.
Considering datasets containing binary classes, the true positive (), true negative (
), false positive (
), and false negative (
) values:
such that
Considering datasets with multiple classes, . True positive for the class
, true negative
, false positive
and false negative
Values of:
such that
5.1. Illustrative Example of Algorithm 1: FPFS-CMC
A data matrix sampled from the “Teaching Assistant Evaluation” dataset is provided below to demonstrate the implementation of the proposed method. This subset contains
samples distributed across three distinct classes (
) as indicated in the final column: class one consists of five samples, class two includes four samples, and class three contains six samples. In the initial iteration of the five-fold cross-validation process, the resulting training matrix
, the class vector
, and the test matrix
are established. Additionally, the ground truth labels for the test samples are represented by the vector
to facilitate the final performance evaluation. These matrices and vectors are obtained as follows:
As part of the procedural demonstration for the “Teaching Assistant Evaluation” dataset, the following refined text describes the specific stages of the FPFS-CMC algorithm: The matrices ,
, and
are provided as inputs to the FPFS-CMC. Following the classification task, the ground truth class labels
are utilized to evaluate the performance of the integrated SDM methods. While several metrics, such as Acc, Pre, Rec, Spe, and F1-score, are commonly used for comprehensive performance analysis, this numerical example focuses specifically on Acc to demonstrate the method’s fundamental effectiveness.
Secondly, the feature weights () are computed by analyzing the relationship between
and
using the Pearson correlation coefficient. In the FPFS-CM, this correlation is utilized to quantify the statistical significance of each attribute relative to the class labels. The primary objective of obtaining these feature weights is to provide a weighted foundation for constructing the training and testing fpfs-matrices in the subsequent phases, ensuring that the decision-making process prioritizes the most influential features of the dataset.
Thirdly, feature fuzzifications of and
are computed as follows:
In the subsequent phases, the training and testing samples from and
undergo a feature fuzzification process to derive their fuzzy representations, denoted as
and
, respectively. This transformation is a critical prerequisite for constructing the train and test fpfs-matrices. By mapping raw attribute values to a fuzzy membership space, the FPFS-CMC framework effectively captures inherent data uncertainties, providing a robust foundation for the multi-criteria classification process.
To illustrate the specific classification procedure, the following computational steps are executed for the first test sample
Fourthly, for each training sample , the test fpfs-matrix
and the corresponding training fpfs-matrix
are constructed utilizing the fuzzy representations
and
, respectively. As a specific instance, for the first test sample (
) and the first training sample (
), the test fpfs-matrix
and the training fpfs-matrix
are generated as follows:
and
The first rows of the fpfs-matrices and
consist of the
calculated in the second step. The second rows of these matrices correspond to the fuzzy representations of the first test sample
and the first training sample
, respectively.
In the fifth step, the matrix is constructed for the
test sample. Each row of this matrix represents the relationship between the test sample and a specific training sample
, while each column corresponds to one of the five distinct pseudo-similarity measures. For the first test sample (
), the
row of the matrix
is computed by applying the following similarity functions between
and
: Hamming pseudo-similarity, Chebyshev pseudo-similarity, Euclidean pseudo-similarity, Hausdorff pseudo-similarity, and Minkowski pseudo-similarity of fpfs-matrices. This systematic process continues for all training samples (
resulting in a comprehensive matrix
that captures the multifaceted proximity between the test instance and the entire training set.
In the sixth step, a column-based standard deviation analysis is performed on the matrix to determine the parameter weights (
). For each of the five pseudo-similarity measures
, the standard deviation (
) is calculated to assess the dispersion of similarity scores across the training set.
Sevently, these values are normalized to generate a parameter weight vector (), which quantifies the reliability and discriminative power of each similarity measure.
In the eighth step, these weights are integrated with the matrix to construct the comparison fpfs-matrix
.
Finally, the comparison fpfs-matrix is processed through the sMBR01 soft decision-making algorithm. The sMBR01 identifies the optimal training sample
by evaluating score functions within the matrix. For the
test sample, sMBR01
produces
. Therefore, the class label of this selected training instance
is then assigned as the predicted label
for the
test sample. This process is iteratively executed for all test instances until the final prediction vector
is generated, completing the classification cycle. Consequently,
is obtained.
The performance comparisons of the aforesaid SDM methods on the datasets are presented in Tables 7–16.
In Table 7, A19 stood out for its exceptional performance with single fpfs-matrices. YHX14, CEC11, G17, and LQP17 also performed exceptionally well, with accuracy above 95% and high precision, recall, and F1-score values. In the double fpfs-matrices, HG13, MRB02, ZXZ15, and RH18 yielded the best results. The top-performing algorithms using more than two fpfs-matrices were ZCW19, S19/5, RS16, Z14/2, and MR13.
In Table 8, notable achievements were observed with A19, YHX14, CEC11, and WQ14, each yielding impressive F1-scores exceeding 96% when using single fpfs-matrices. For double fpfs-matrices, AKO18o and VMH16 exhibited strong performance. Furthermore, A19/2 emerged as the most effective algorithm when leveraging more than two fpfs-matrices.
The comparison in Table 9 manifests varied SDM algorithm performance across different numbers of fpfs-matrices. It is noted that, among single fpfs-matrices, YHX14, MBR01, and A19 show the best performance. Furthermore, when utilizing double fpfs-matrices, VMH16 and AKO18 appear to perform favorably. Lastly, the results suggest that SS19/4 yields the best outcome when working with more than two fpfs-matrices.
Table 10 shows a comprehensive overview of the performance comparisons of algorithms on the Hayes dataset. It is worth noting that YHX14 and A19 demonstrate excellent results from the single-fpfs-matrices. Moreover, the double fpfs-matrices AKO18o algorithm stands out as the top performer. When considering matrices with more than two fpfs-matrices, the A19/2 algorithm emerges as the most favorable choice.
Table 11 provides performance results for the Libras dataset. We can state that A19 is the best algorithm. The findings indicate that among the double fpfs-matrices, AKO18a yields the most favorable outcomes. Considering two or more fpfs-matrices, it is evident that A19/2 yields the best result for the Libras movement dataset.
In Table 12, which details the performance of the teaching dataset, Algorithm A19 produces the best results. Among the double fpfs-matrices, AKO18o seems to generate the most favorable outcomes. Moreover, when considering two or more fpfs-matrices, A19/2 delivers the best results for the teaching dataset.
Table 13 provides a detailed comparison of algorithm performance on the Ionosphere dataset. A19 shows excellent results with single fpfs-matrices, while the double fpfs-matrices AKO18o algorithm stands out as the top performer. When considering matrices with more than two fpfs-matrices, the A19/2 algorithm emerges as the most favorable choice.
Table 14 compares algorithm performance on the Wholesalers dataset. YHX14 excels with single fpfs-matrices, while AKO18o stands out with double fpfs-matrices. A19/2 is the top choice for matrices with more than two fpfs-matrices.
Table 15 presents a comprehensive comparison of algorithm performance on the Glass dataset. A19 demonstrates commendable results with single fpfs-matrices, while the double fpfs-matrices AKO18o algorithm emerges as the leading performer. When evaluating matrices with more than two fpfs-matrices, the A19/2 algorithm appears to be the most favorable choice.
In Table 16, from the single fpfs-matrices, A19 emerges as the top performer, with the highest values across most metrics, including an F1-score of 79.40474. YHX14 and CEC11 also perform well, securing second and third positions. Most algorithms maintain high specificity (above 96%) and accuracy of around 94%.
When we consider Table 17, which compares all datasets for single matrices, we observe that A19 usually yields the best result, followed by MBR01 and YHX14. In most datasets, the CXL13 algorithm produces the worst results.
When analyzing Table 18, we notice that AKO18o usually yields the best result, followed by VMH16 and RH18. In most datasets, the AKO18a algorithm produces the worst results.
In Table 19, A19/2 shows the best result. In three of 10 datasets, the NKY17 algorithm generates the worst results.
Our study considered datasets from real-world application domains, including Medicine, Chemistry, Acoustics, and Artificial Intelligence. Our analysis based on Tables 17–19 identified the best-performing SDM methods for each dataset as follows:
- Hayes-Roth (Artificial Intelligence): A19, AKO18o, A19/2
- Ionosphere (Radio-Frequency Analysis): A19, VMH16, A19/2
- Libras (Biomechanics): A19, AKO18o, A19/2
- Parkinsons[sic] (Medical): A19, MRB02, MR13
- Sonar (Acoustics): YHX14, AKO18o, SS19/4
- Teaching Assistant Evaluation (Educational Sciences): A19, AKO18o, A19/2
- Wine (Chemistry): CEC11, AKO18o, A19/2
- Wholesaler (Business Analytics): YHX14, AKO18o, A19/2
- Glass (Forensic Sciences): A19, AKO18o, A19/2
- Ecoli (Biology): A19, AKO18o, A19/2
The success of A19 on the Wine dataset is specific to this dataset; it cannot be directly generalized to all problems in the field of Chemistry. Each dataset has different dynamics and features tailored to its field. Therefore, the superior performance of A19 on the Wine dataset does not imply that it will be generally successful in such applications. Consequently, the performance of specific algorithms can be evaluated more accurately by considering the unique structure of each dataset.
6. Statistical Evaluation
This section employs the corrected Friedman test [76] and the Nemenyi post hoc test [77], following the recommendations of [78], to assess the statistical significance of overall differences in performance across five performance metrics. The Friedman nonparametric test is used for multiple-hypothesis testing and provides a performance-based ranking of algorithms for each dataset. It assigns average ranks when algorithms are tied, with 1 indicating the best-performing algorithm, 2 the second-best, and so on. A rank of 35 represents the worst algorithm. It evaluates the Friedman statistic according to the, distribution with
degrees of freedom, where k denotes the number of algorithms. If a statistically significant difference in performance is observed, a post hoc test is used to determine which difference corresponds to which algorithm. The Nemenyi test is a commonly used post hoc test for comparing different classifiers. In this test, if the average ranks of two algorithms differ by more than the critical distance, their performance is significantly different. Initially, we calculate the average rank of each algorithm considered in our experiments, where the total number of algorithms
and the number of datasets
.
If the Friedman test statistic values for the accuracy, precision, recall, specificity, and F1 score values are
,
, and
, respectively, with 35 (
) degrees of freedom and the critical value for the Friedman test (Friedman, 1940) given for
and
is 9.49 at a significance level of
, we can conclude that the accuracy (
), precision (
), recall (
), F1-score (
), and specificity (
) values of the studied methods are significantly different. We can proceed with a post hoc test after rejecting the null hypothesis. The Nemenyi test [77] can be used when comparing all classifiers [78]. The Nemenyi diagrams are presented in Figs 1–6.
Figs 1–5 depict the top five SDM methods, which exhibit varying rankings across Acc, Spe, F1-score, and Rec performance. These methods are Alg-12 (A19), Alg-3 (YHX14), Alg-17 (VMH16), Alg-20 (AKO18o), and Alg-30 (A19/2).
In machine learning, relying solely on accuracy can be misleading, especially when precision and recall are inversely related. To address this, we used 10 diverse UCI datasets to establish a more reliable performance baseline. The F1-score was chosen as the main ranking metric because it provides a harmonic mean of precision and recall, ensuring that neither false positives nor false negatives are overlooked. This is particularly important in real-world situations where datasets may be imbalanced and a balanced evaluation of predictive quality is necessary.
To assess the importance of these rankings, we conducted a thorough two-stage statistical analysis using the Friedman and Nemenyi tests. First, the Friedman test was used to assess significant differences across the 35 SDM algorithms across all datasets. After rejecting the null hypothesis (which suggested that the performance differences were due to chance), we conducted the Nemenyi post-hoc test. This comparison examined pairs of algorithms to identify specific performance groups and notable gaps between methods.
Based on the improved F1-score analysis and the statistical results, Alg-12 (A19) proved to be the most reliable algorithm, followed by Alg-3 (YHX14). Other top-performing methods, such as Alg-30 (A19/2), Alg-20 (AKO18o), Alg-17 (VMH16), and Alg-26 (MR13/2), shared third place. The close match between our simulation-based F1 rankings and the Nemenyi test outcomes strongly supports the validity and broad applicability of the FPFS-CMC framework across different data scenarios.
7. Computational Complexity Analysis
This section provides a formal analysis of the computational complexity of the evaluated SDM methods using Big O notation. To ensure a clear understanding of how these complexities are derived, we first present a step-by-step analytical breakdown of G17() as a representative case study, examining its performance in relation to the number of samples (
) and attributes (
).
The computational efficiency of the G17() algorithm is primarily determined by the dimensions of the input fpfs-matrix
, where
represents the number of samples and
represents the number of parameters. In Step 1, the construction or initialization of the
matrix requires processing every entry, resulting in a complexity of
(
. While Step 2 is a simple index identification process of
, the subsequent operations in Steps 3 and 4 involve summations over the index set
and the entire parameter set for each of the
samples. In the worst-case scenario, where
includes all parameters, these steps require
operations, maintaining the dominant complexity at
(
. The final stages (Steps 5–7), which include finding the maximum value in the
vector, constructing the score matrix, and attaining the decision set, are linear operations relative only to the number of samples
. Consequently, when summing all computational costs, the higher-order term
(
prevails, leading to a total time complexity of
(
.
The memory overhead of the G17(R) algorithm is governed by the data structures required to store membership degrees and intermediate results. The most significant memory consumption occurs in Step 1, where the fpfs-matrix is stored, requiring
(
space to hold the membership values for
samples across
parameters. The auxiliary vectors generated in the following steps, such as
,
, and the score matrix
, are
column vectors, thus requiring only
(
additional space. Furthermore, the index sets
,
, and
consume negligible memory proportional to
or
. Since the storage of the initial fpfs-matrix remains the most resource-intensive component, the overall space complexity of the algorithm is formally defined as
(
.
The formal derivation of the G17() algorithm serves as a template for analyzing the complexity of the other methods examined in this study. By following a similar matrix-based procedure, including fuzzification, parameter-based summation, and score function evaluation, the time and space complexities of the remaining 34 SDM algorithms can be easily determined. These findings are summarized in Table 20, which shows the Big O notation for each method and highlights their scalability across various real-world problem sizes.
Here, denotes the row number of the fpfs-matrices,
,
,
denote the column number of the fpfs-matrices,
denotes the number of fpfs-matrices.
- For the largest problems: Algorithms with
and
complexity (e.g., CXL13, P18, MR13) are the most efficient for large-scale data due to their linear scalability.
- For Medium-Scale Problems: Algorithms with
or
complexity (e.g., DC15, ZZ19) perform reasonably well, but time costs increase rapidly with higher dimensions.
- For Small-Scale Problems: High-complexity algorithms, such as
(WQ14) should be employed only for very small datasets to prevent excessive computation time.
- Caution for Multiple Matrices: Algorithms with
complexity (e.g., SS19/5) becomes inefficient when the number of matrices (
) is high and unsuitable for large problems.
- Special Cases: Algorithms with
complexity (e.g., AKO18a, VMH16) are only practical when
and
They are small; otherwise, their performance deteriorates.
Therefore, SDM algorithms should be selected based on problem size (,
,
) and available computational resources. Notably, large datasets should be chosen with linear or linear-logarithmic complexity.
8. Results and Discussion
The experimental results and statistical evaluations presented earlier provide a comprehensive benchmark of 35 SDM algorithms. To fully grasp the practical significance of these findings, it is important to combine the performance results with the algorithms’ structural characteristics, as classified in Table 3.
The analyses herein show a strong link between an algorithm’s “Decision Model” and its success in machine learning classification tasks. The top-performing algorithm, A19, and the highly ranked YHX14 both use aggregation-based decision models. According to Table 3, A19 is based on an “indices set-based weighted aggregation” model, while YHX14 uses “Grey relational analysis”. The effectiveness of these models across datasets such as Hayes-Roth and Glass suggests that aggregation operators are better at capturing multidimensional uncertainty in UCI benchmark datasets than models based on spatial distance or belief functions.
The “Operation” column in Table 3 offers key insights into why some algorithms perform better. Algorithms that use weighted aggregation and comparison matrices (such as A19, MBR01, and AKO18o) tend to achieve higher F1-scores. For example, A19 uses index-set-based weighted aggregation, enabling a more detailed treatment of fuzzy parameters. MBR01 focuses on a binary comparison of alternatives, yielding more robust results in datasets with clear feature distinctions. Conversely, algorithms that rely solely on distance-based aggregation (such as CXL13) usually show lower performance metrics (Tables 17–19), indicating that simple geometric distance may not be sufficient to model complex fuzzy relationships in high-dimensional data.
A key finding of this study is the shift from “single-application” backgrounds to what can be called “general-purpose” effectiveness. Many algorithms in Table 3 were initially developed for specific “Empirical Studies,” such as medical diagnosis (XWL14), car selection (DC15), or house selection (G17). Our results show that although these algorithms were designed for niche decision problems, integrating them into the FPFS-CMC framework allows them to serve as powerful general-purpose classifiers. For instance, YHX14, created for wine quality identification, performed exceptionally well on the Sonar and Wholesaler datasets. This demonstrates the high modeling ability and versatility of fpfs-matrix-based SDM methods.
Choosing an optimal algorithm requires considering its parameter structures and computational costs. As shown in Table 3, some algorithms depend on complex weight matrices or multiple index sets (such as A19 and SS19/5). While these parameters can improve accuracy and F1-scores, they also increase computational complexity, as analyzed in Table 20. Algorithms with linear complexity (), like A19 and CEC11, strike the best balance between high performance and scalability for large-scale applications. Conversely, high-complexity models such as WQ14 (
) are better suited to small datasets despite their intensive decision models.
Understanding the practical implications of these findings requires combining the performance results with the structural features and dataset characteristics. The varied structural properties of the ten UCI datasets, as shown in Table 6, are crucial in explaining the fluctuations in the performance of SDM algorithms. Three possible factors that can influence these outcomes are as follows:
- Concerning the ranking criteria and the F1-score, we used the F1-score as the main metric to evaluate the performance of SDM algorithms within the machine learning framework. Unlike accuracy, which can be misleading in certain data distributions, the F1-score offers a harmonic mean of precision and recall. Using this measure, we ensured that the rankings accurately reflect a balanced assessment of each algorithm’s ability to maintain high predictive performance for both majority and minority classes, providing a more dependable benchmark for decision-making.
- The number of features in the datasets, which varies widely from 5 to 90, significantly impacts performance. In high-dimensional datasets like Libras Movement and Sonar, algorithms that use feature fuzzification and precise parameter weighting within the FPFS-CMC framework, specifically A19 and YHX14, consistently outperform others. This indicates that these models are better at handling the increased complexity introduced by a larger number of parameters than traditional methods.
- The datasets cover a range of multi-class complexities, from binary classification to problems with up to 15 classes. While many SDM algorithms perform well on binary classification tasks, such as Parkinsons[sic] or Sonar, the performance gap widens in multi-class settings, such as Ecoli (7 classes) or Libras Movement (15 classes). In these more challenging scenarios, algorithms such as A19 and A19/2 prove to be the most effective, demonstrating greater scalability across diverse and complex decision-making environments.
In conclusion, the success of an SDM algorithm depends on the complex interaction among its mathematical operations, decision model, and the dataset’s structural dynamics. This study shows that while algorithms like A19 and YHX14 provide superior modeling capabilities, their effectiveness is confirmed by their resilience to the challenging features of real-world data. By combining qualitative classifications with quantitative benchmarks, this research provides a structured framework for selecting the most suitable SDM method based on both theoretical soundness and empirical reliability.
9. Conclusion
The primary goal is to comprehensively classify SDM methods, assess their effectiveness in machine learning modeling, and explore their potential applications. The classification of SDM algorithms relies on the decision model, empirical studies, mathematical operators, and the number of fpfs-matrices. The FPFS-CMC algorithm was employed to categorize SDM algorithms across ten real datasets from the UCI Machine Learning Repository. F1-scores ranked algorithms in machine learning, and the results were subjected to statistical analysis.
This process helped identify which SDM algorithms are most suitable for real-world applications, and which produce the best results. In this context, the proposed FPFS-CMC framework offers a systematic evaluation method that combines SDM methodologies with machine learning performance metrics, providing a practical tool for selecting appropriate algorithms based on empirical data.
The studies showed that SDM algorithms performed very well in analyzing machine learning techniques, confirmed by statistical tests. The FPFS-CMC classifier proved effective for SDM methods in machine learning. Additionally, Nemenyi post hoc testing revealed that Alg-12 (A19) performed the best, followed by Alg-3 (YHX14). Alg-30 (A19/2), Alg-20 (AKO18o), and Alg-17 (VMH16) shared the third position, with MR13/2 ranked fourth. These rankings align with the results of the simulation studies, which validated the algorithms’ performance across 10 datasets. The consistency between statistical analysis results and simulation findings further demonstrates the robustness and reliability of the proposed evaluation approach, highlighting its methodological significance for machine learning-based decision analysis.
This work focuses mainly on fpfs-matrices because of their ease of implementation. Since algorithmic performance varies across datasets, no single algorithm can be considered universally optimal. Evaluations must consider each dataset’s inherent features and complexity. The constraints and complexity of an algorithm may not suit every dataset. Understanding the compatibility between the algorithm and the dataset is crucial. While this can be seen as a disadvantage, it underscores the importance of dataset–algorithm compatibility and offers valuable guidance for practitioners applying SDM algorithms to real-world problems.
Furthermore, while this study establishes a comprehensive performance baseline for 35 SDM algorithms, future research will focus on conducting detailed sensitivity analyses to systematically investigate the impact of various internal parameters within the fpfs-matrices. Given the diverse parameter structures involved, a dedicated framework will be developed to evaluate how fluctuations in these variables influence the final decision-making outcomes, thereby adding further robustness and empirical depth to the FPFS-CMC. Future research may also extend this scope to more complex algebraic structures, such as interval-valued intuitionistic fuzzy parameterized matrices [79], which provide greater flexibility in modeling linguistic uncertainties in noise removal and image processing tasks. Incorporating advanced multi-criteria group decision-making (MCGDM) frameworks, especially those using N-soft set models and their parameter-reduction techniques, offers a promising path to improve the scalability of SDM methods [80]. Additionally, integrating hybrid expert knowledge systems, like Fermatean fuzzy soft sets [81] or Pythagorean fuzzy N-soft models combined with the PROMETHEE approach [82,83], could further enhance the robustness and ranking accuracy of the FPFS-CMC framework in complex group settings. Beyond purely mathematical extensions, exploring the behavioral and cognitive aspects of decision-making, such as the influence of emotional intelligence and proactive decision-making scales [84,85], could lead to more human-centered AI systems. Lastly, applying these combined machine learning and MCDM approaches to emerging fields, such as AI-driven personalization in smart city infrastructure and urban planning [86], remains an exciting direction for developing next-generation adaptive artificial intelligence systems.
Declaration of generative AI and AI-assisted technologies in the writing process
During the preparation of this manuscript, the authors used Grammarly to improve the English writing. The authors have reviewed and edited the output and take full responsibility for the content of this publication.
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