SAPEVO-PC: Integrating Multi-Criteria Decision-Making and Machine Learning to Evaluate Navy Ships

Abstract

The selection of a navy ship is essential to guarantee a country’s sovereignty, deterrence capabilities, and national security, especially in the face of possible conflicts and diplomatic instability. This paper proposes the integration of concepts related to multi-criteria decision making (MCDM) methodology and machine learning, creating the Simple Aggregation of Preferences Expressed by Ordinal Vectors—Principal Components (SAPEVO-PC) method. The proposed method proposes an evolution of the SAPEVO family, allowing the inclusion of qualitative preferences, and adds concepts from Principal Component Analysis (PCA), aiming to simplify the decision-making process, maintaining precision and reliability. We carried out a case study analyzing 32 warships and ten quantitative criteria, demonstrating the practical application and effectiveness of the method. The generated rankings reflected both subjective perceptions and the quantitative performance data of each ship. This innovative integration of qualitative data with a quantitative machine learning algorithm ensures comprehensive and robust analyses, facilitating informed and strategic decisions. The results showed a high degree of consistency and reliability, with the top and bottom rankings remaining stable across different decision-makers’ perspectives. This study highlights the potential of SAPEVO-PC to improve decision-making efficiency in complex, multi-criteria environments, contributing to the field of marine science.

1. Introduction

Military research and development in the modern world are not only two of the main factors in building the military power of a state, strengthening its strategic independence, and ensuring technological advantages, but they also represent essential factors in the scientific, technological, and economic progress of the state [1].

The relationship between the arms race and strategic stability has been meaningful for states to increase military power and the consequent effects on their rivalries with other states [2]. According to [3], armaments are fundamental to military power, emphasizing large naval, air, and land weapons, nuclear weapons, and ballistic missile capability. Also, Gilkova [1] has stated that the leading countries with regard to spending on military research and development not only dominate the creation and development of weapons, military equipment, and ammunition, but are also currently the most developed countries in the world in the field of robotics, information, and other types of civilian technology.

In this context, selecting a warship is crucial for any nation with significant maritime interests, as this choice directly reflects the country’s defense capabilities, power projection, and national security [4]. This process requires the meticulous analysis of strategic, budgetary, and technological demands.

Additionally, choosing a warship has implications beyond the purely military scope, influencing a country’s diplomatic and geopolitical relations [5], mainly due to the enormous growth in maritime traffic activities [6]. The decision may signal a commitment to maintaining regional peace and stability, or reflect a nation’s capability to deter potential threats. Thus, selecting a warship becomes a strategic and political issue of considerable magnitude, with long-lasting implications for national security and prestige, as discussed by [7], which emphasizes how naval capabilities can impact international perceptions and a nation’s status. Therefore, choosing a warship involves technical, economic, political, and strategic considerations that require a holistic and well-grounded approach.

Given the complexity surrounding most military problems, in the early days of World War II, operations research (OR) emerged as a scientific approach to decision-making in the military sphere [8]. Within the armed forces (AF), a routine OR technique employed is the multi-criteria decision making (MCDM) methodology, employed to refer to a collection of formal methodologies that aim to explicitly incorporate multiple criteria in assisting stakeholders and groups in exploring significant decisions [9]. These decisions typically involve conflicting objectives, ambiguous, non-repeatable uncertainties, and allocating costs and benefits among various individuals, businesses, groups, and organizations [10,11].

Greco et al. [12] have stated that the fundamental elements of MCDM consist of a set of alternatives (or observations), a minimum of two criteria (or variables), and the involvement of at least one decision-maker (DM). When considering a finite set of alternatives, the DM may encounter three primary types of problem: choice problems entail selecting a subset that contains the best alternatives, ranking problems involve arranging the alternatives from the best to the worst, and sorting problems entail categorizing the alternatives into pre-defined and ordered groups [13].

The complexity of the decision is enhanced by the need to balance multiple criteria, such as combat capability, operational range, and cost efficiency. It characterizes the problem as a typical multi-criteria decision scenario, which can be effectively addressed by implementing MCDM concepts. These methods are designed to structure and solve complex decision-making problems where conflicting criteria must be evaluated simultaneously [14].

Costa et al. [15] carried out an extensive review of the literature on MCDM in the military field, and highlighted several recent studies, such as the choice of a military hospital care vessel to support the fight against the COVID-19 pandemic [4], ranking policing strategies [16,17], the selection of drones to be acquired for use in naval warfare [18] and public security [19], and the classification of aircraft [20,21], military bases [22] and military personnel [23].

A complex factor in decision problems is usually the volume of data generated, which has grown exponentially with new communication technologies such as 5G and the rapid development of the Internet of things (IoT) [24]. Thus, according to [25], it is necessary to develop tools that provide a rapid analysis of a large number of data to support the decision-making process.

The explosion of data in today’s era, driven by technology such as 5G and the Internet of things, underscores the need for agile tools to analyze vast amounts of information and support complex decision-making processes [25]. In this sense, it is increasingly imperative that researchers develop and apply methodologies that facilitate the analysis of vast datasets. This quest requires integrating more advanced and efficient data analysis techniques to extract meaningful insights and support complex decision-making processes.

In this scenario, principal component analysis (PCA) gains much importance, as it allows the analysis of observations described by quantitative interdependent variables to extract meaningful information, representing them as a set of new orthogonal variables [26]. PCA is an unsupervised machine learning (ML) technique [27] that aims to identify a relatively small number of factors representing the behavior set of original interdependent variables [28].

Still, in this context, several authors have pointed out that the number of pairwise evaluations represents a significant challenge in multi-criteria methods. Saaty [29], known for developing the Analytic Hierarchy Process (AHP) method, recognizes that directly comparing criteria and alternatives in a large set can be costly and susceptible to inconsistencies. Likewise, Greco et al. [12], in a comprehensive analysis of multi-criteria approaches, highlighted the practical and cognitive difficulty of dealing with a large volume of comparisons, emphasizing the need for methods that effectively deal with this complexity.

As a contribution to mitigating this problem that is so relevant today, this paper proposes a new MCDM method called Simple Aggregation of Preferences Expressed by Ordinal Vectors—Principal Components (SAPEVO-PC), which integrates concepts from multi-criteria analysis and advanced ML techniques. The proposed method reduces the number of pairwise evaluations to obtain the criteria weights, using PCA concepts to evaluate the performance of the alternatives. The methodology above can be useful for dealing with vast datasets and extracting significant insights.

The SAPEVO-PC method stands out for its ability to handle large volumes of data efficiently and its capacity to provide robust and reliable decision support. Integrating advanced ML techniques into the MCDM framework permits an analysis that can uncover deeper insights and patterns within the data. The method enables the analysis of qualitative data (obtaining criteria weights) with less cognitive effort and quantitative data (analysis of the alternatives’ performance), using each approach’s advantages.

Given the above, this paper proposes a methodology for evaluating warships based on concepts related to MCDM and ML. We analyzed data from the military, economic, and strategic spheres of 32 ships from different countries’ navies and evaluated them using 10 quantitative criteria. The SAPEVO-PC method stands out for the following characteristics related to the decision-making context:

• Criteria weights through expert opinion: Allow the determination of criteria weights considering the opinions of multiple decision-makers, with a reduced number of pairwise comparisons required;
• Identification of Correlations: By identifying correlations between the original variables, this verifies correlated military capabilities, providing insights into how different attributes of warships report to each other;
• Data Reduction: Creating factors that represent the original criteria provides structural data reduction, simplifying complex datasets into more manageable and interpretable components without losing essential information;
• Performance Indicators: Developing consolidated rankings by creating performance indicators based on generated factors guarantees a comprehensive and robust evaluation framework. It allows for a more accurate assessment of warship capabilities.

By proposing and applying the SAPEVO-PC method, this research offers a scientifically robust and replicable methodology that can be applied to various tactical, operational, and strategic challenges in the maritime domain and broader scientific fields. The ability to incorporate the opinions of multiple decision-makers and reduce the number of assessments required increases the efficiency and effectiveness of the decision-making process, making it a valuable tool for both academic research and practical applications in defense strategy and naval engineering.

SAPEVO-PC provides an innovative approach to warship evaluation by combining MCDM and ML techniques, and improving the selection process’s accuracy and reliability. This method stands out, especially in the context of military problems, as it can deal with complex criteria and strategic considerations involved in choosing warships.

This work can contribute to academia and society by presenting a model that integrates MCDM and ML concepts. The proposed framework has a robust dual nature and can be replicated in tactical, operational, and strategic applications for military and civil environments.

2. Materials and Methods

MCDM methodologies constitute a comprehensive set of formal approaches designed to explicitly integrate multi-criteria consideration into problem-solving, supporting stakeholders and groups in exploring important decisions [9]. These methodologies involve essential elements, such as actions (alternatives), at least two criteria, and a DM. The main issues addressed include the alternatives’ selection, ordering, or orderly ranking [12].

Over the years, several MCDM models have been proposed, concentrated in two major groups of methodologies: aggregation methods using a single synthesis/compensatory criterion (American school), and outranking/non-compensatory methods (French/European school) [30].

Based on utility theory, the first group considers two types of relationship between variables and indifference [31]. In this specific scenario, the relations of incomparability are not considered, and the transitivity between preferences is assumed [5]. It means that if alternative A is preferred over alternative B and B is preferred over C among the main methodologies belonging to this group, we highlight:

• AHP, proposed by [29]: method which structures complex problems into a hierarchy of objectives, criteria, and sub-criteria, allowing pairwise comparisons of alternatives and the synthesis of priorities;
• MACBETH (Measuring Attractiveness by a Categorical-Based Evaluation Technique) [32]: this uses qualitative judgments to construct an attractiveness scale for different alternatives;
• MAUT (Multiple Attribute Utility Theory), developed by [10]: this method uses utility functions to model the decision-makers’ preferences across multiple criteria, helping to identify the best alternative based on the maximization of expected utility;
• SMART (Simple Multi-Criteria Attribute Rating Technique), proposed by [33]: this employs rating scales to score and weigh alternatives against criteria, facilitating decision-making;
• TODIM (Multi-criteria Interactive Decision-making), presented by [34]: this interactive method considers the prospective behavior of decision-makers, utilizing prospect theory;
• TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution), implemented by [35]: this method identifies the best alternative based on its proximity to the ideal solution; and
• UTA (Additive Utility Theory) [36]: this method uses additive utility functions to model decision-makers’ preferences.

The second group of MCDM methods is characterized by the outranking relationship between alternatives, marked by non-transitivity relationships between preferences. The methods of this group extend a basic set of situations of preference relations based on four types: indifference relationship, weak preference, strict preference, and incomparability relationship [37]. The two main approaches that make up this class are considered method families:

• ELECTRE (Elimination and Choice Translating Reality for Enrichment Evaluation), proposed by [38]: it is used to solve complex decision problems by constructing outranking matrices, which help compare alternatives based on multiple criteria. Various versions of the ELECTRE method, such as ELECTRE I, II, III, IV, and TRI are adapted to different types of problems and contexts; and
• PROMETHEE (Preference Ranking Method for Enrichment Evaluation), developed by [39]: the method is based on constructing preference flows, allowing direct comparison between alternatives. PROMETHEE I and II are the best-known versions, with the former used for partial ranking and the latter for complete ranking.

2.1. SAPEVO Method

The Simple Aggregation of Preferences Expressed by Ordinal Vectors (SAPEVO), introduced by [40], takes an approach that handles input data in an ordinal manner when evaluating variables in a specific context. This method facilitates the processing of data that reflects the subjectivity of the decision-maker. To deal with multiple decision-makers, the methodology called Simple Aggregation of Preferences Expressed by Ordinal Vectors—Multi Decision-makers (SAPEVO-M), proposed by [41], offers a scenario-focused assessment with the participation of multiple decision-makers. In addition to improving the previous model, this approach reinforces its consistency.

The distinguishing feature of this model lies in its ability to transform preferences in an ordinal manner. This process is used to derive preference relationships between the alternatives in each criterion and to determine the criteria weights, resulting in the definition of the corresponding degrees of priority [41]. SAPEVO-M procedures are conducted in two stages:

• Transform ordinal preferences of criteria into a vector of criteria weights; and
• Integrate the vector criteria of different decision-makers.

Due to the relevance of the method, several variants have emerged over time, configuring a family of MCDM methods.

Table 1. Methods of the SAPEVO family and their characteristics.

Method	Compensatory	No Compensatory	Ordinal	Cardinal	Ordination	Classification	Group	Reduction of Pairwise Comparisons	ML
SAPEVO [40]	X		X		X
SAPEVO-M [41]	X		X		X		X
PROMETHEE-SAPEVO-M1 [18]	X		X	X	X
ELECTRE-MOr [20]	X		X	X		X	X
SAPEVO-M-NC [42]		X	X		X		X
SAPEVO-H2 [43]	X		X	X	X		X
SAPEVO-PC, proposed in this paper	X		X	X	X		X	X	X

Source: authors.

presents the SAPEVO family’s methods and their main characteristics.

Compensatory methods allow compensation between criteria, which means that a high performance in one criterion can compensate for the low performance of an alternative in another criterion. In contrast, non-compensatory methods, such as ELECTRE and PROMETHEE, do not allow such compensations. Non-compensatory logic establishes outranking relationships between pairs of alternatives, in which alternative A outranks an alternative B if it is at least as good as B in the criterion under consideration, and this is the idea of central expression expressed by the concept of outranking [44].

Ordination methods rank alternatives from best to worst; classification methodologies distribute alternatives into pre-defined and ordered categories [13]. Data used in MCDM can also be classified as cardinal or ordinal. The first are those that have information about the magnitude of the differences between alternatives, allowing arithmetic operations such as addition, subtraction, and multiplication to be carried out. Ordinal data only indicate the order or ranking of alternatives without providing information about the differences’ magnitude.

Analyzing the data in Table 1, the methods belonging to the SAPEVO family constitute a comprehensive set of approaches encompassing the main typologies of MCDM problems, such as sorting, selection, and classification. However, despite the diversity of applications and variations within the SAPEVO family, all these methods share a fundamental characteristic: the performance of evaluations alongside the criteria and alternatives.

Equation (1) shows the number of pairwise evaluations required for the application of the original SAPEVO methods:

$$P=n·\left({\displaystyle \frac{n-1}{2}}\right)$$

(1)

P represents the number of pairwise evaluations required to obtain the weights of each criterion and the scores of each alternative based on the qualitative criteria; n corresponds to the number of criteria or alternatives analyzed, with P being calculated for each analysis (of the criteria and alternatives, separately).

There is a notable increase in pairwise evaluations in decision scenarios involving many criteria and alternatives. This increase can result in considerable cognitive strain on the decision-makers, making the decision-making process more challenging and subject to judgment inconsistencies. The accumulation of peer evaluations can overwhelm decision-makers, making it difficult to maintain consistency and accuracy in comparisons. In this sense, an important innovation of SAPEVO-PC is the significant reduction in pairwise comparisons required during the assessment process.

By incorporating the SAPEVO-PC method, the evaluation process becomes more efficient, reducing the cognitive load of decision-makers and increasing the overall reliability of the results. This efficiency is achieved without compromising the comprehensiveness and accuracy of assessments.

Another significant innovation is the integration of ML algorithms with MCDM techniques. This combination improves analytical capabilities and provides a more robust framework for decision support in complex problems.

Another innovative feature of SAPEVO-PC is the integration of ML algorithms into MCDM analysis. This integration helps reduce the complexity of high-dimensional data while preserving essential patterns and relationships. Furthermore, the methodology proposed in this article ensures that both qualitative expert opinions and quantitative data-based insights are used effectively in the decision-making process. The result is a more reliable and accurate assessment of alternatives, providing a comprehensive evaluation framework.

The SAPEVO-PC method stands out for its ability to effectively deal with complex and multidimensional data sets. This feature is crucial in the maritime domain, where decision-making often involves multiple criteria and diverse stakeholder perspectives. Combining qualitative and quantitative analyses, the method’s two-phase approach guarantees a balanced and complete assessment. These innovations fill existing gaps in the literature and offer a useful tool for decision support in complex problems.

2.2. SAPEVO-PC Method

This topic describes the concepts and steps for applying the SAPEVO-PC method. The method has two distinct phases: predominantly qualitative and quantitative. The first phase consists of obtaining the criteria weights, in which the qualitative opinions of multiple decision-makers are considered to determine the relative importance of each criterion. This phase ensures that expert opinions and preferences are effectively integrated into decision-making. The second phase involves evaluating the performance of alternatives using ML algorithms, specifically PCA. This quantitative phase provides data-driven insights to create performance indicators, providing a robust and objective assessment of each alternative. By combining these two phases, SAPEVO-PC offers a comprehensive and balanced approach to multi-criteria decision-making, integrating qualitative input from experts with quantitative analytical rigor.

In the qualitative phase, we must define the criteria called Positive Ideal Element (PIE) and Negative Ideal Element (NIE) to determine the criteria weights. We emphasize that the PIE is the criterion considered the most important, while the NIE is the least important, according to the decision-maker’s point of view.

The pairwise assessments are carried out only by comparing the PIE with each of the other criteria and, similarly, comparing the NIE with each other. The SAPEVO-PC method is based on a seven-point ordinal scale, as shown in

Table 2. Conceptual scale of the SAPEVO-PC method.

Degree	Description
1	The PIE and the other criterion are equally important
2	The PIE is slightly more important than the other criterion
3	The PIE has moderate importance when compared with the other criterion
4	The PIE has considerable importance when compared with the other criterion
5	The PIE has great importance when compared with the other criterion
6	The PIE has very great importance when compared with the other criterion
7	The PIE has absolute importance when compared with the other criterion

Source: authors.

. This scale ranges from 1 (Equality) to 7 (Absolutely best/worst), allowing the decision-maker to establish preferences qualitatively, according to their subjective evaluations.

To make the pairwise comparisons of the criteria about the NIE, symmetrical values will be assigned to those indicated in Table 2, considering that this parameter is the worst among all the criteria under analysis (all grades attributed to the NIE about the others will be negative).

A significant reduction is observed when it comes to the number of parity evaluations in SAPEVO-PC compared to other methods of the SAPEVO family and AHP. Equation (2) represents the number of pairwise evaluations (K) required to obtain the criteria weights and evaluate the alternatives in qualitative criteria using the SAPEVO-PC method.

$$K=2n-3$$

(2)

where n is the number of criteria or alternatives being evaluated.

For comparison purposes,

Table 3. Comparison between the number of peer-to-peer evaluations.

Number of Criteria or Alternatives (n)	SAPEVO Classic (P)	SAPEVO-PC (K)
2	1	1
3	3	3
4	6	5
5	10	7
6	15	9
7	21	11
8	28	13
9	36	15
10	45	17
11	55	19
12	66	21
13	78	23
14	91	25
15	105	27

Source: authors.

shows the number of comparisons required for the application of the classic SAPEVO and SAPEVO-PC methods, considering the number n of criteria:

It is observed that SAPEVO-PC becomes advantageous from 4 evaluation criteria, and this advantage increases as the number of criteria and alternatives increases.

2.2.1. Phase 1: Qualitative Evaluation for Obtaining Criteria Weights

Step 1. Define a set of decision criteria. In this step, the criteria C = {C₁; C₂;...; C_n} will be assessed in the decision-making process.

Step 2. Identify the criteria considered the best (most desirable, most important) and the worst (least desirable, least important). At this stage, the decision-maker determines which criterion is best or worst without pairwise comparisons.

Step 3. To evaluate the preference of the best general criterion concerning the other criteria (A⁺) using the 7-point scale presented in Table 2. The result of the evaluation of the precedence of the best criterion (PIE) about the others is presented through the vector represented in Equation (3):

$$A^{+}=\left\{a_{P1},a_{P2},\dots ,a_{Pn}\right\}$$

(3)

where P_j represents the preference of the best PIE criterion over a given criterion j, and n represents the criterion number. It is observed that the element a_PP = 1 because it represents the comparison of the element PIE with itself.

Step 4. Establish the preference of all criteria about NIE (A⁻) using the symmetric values (multiplied by −1) in Table 2. The result of the evaluations of precedence of the other criteria about the NIE forms the vector represented in Equation (4):

$$A^{-}=\left\{a_{N1},a_{N2},\dots ,a_{Nn}\right\}$$

(4)

where N_j indicates the preference of criterion j over the worst NIE criterion. We emphasize that the element a_NN = 1 because it corresponds to the comparison of NIE with itself.

Step 5. Calculate the element A^*, represented by the global sum of A⁻ and A⁺ for each criterion, according to Equation (5):

$${\mathrm{A}}^{\mathrm{*}}=\left\{({\mathrm{a}}_{\mathrm{N}1}+{\mathrm{a}}_{\mathrm{P}1}),({\mathrm{a}}_{\mathrm{N}2}+{\mathrm{a}}_{\mathrm{P}2}),\dots ,({\mathrm{a}}_{\mathrm{N}\mathrm{n}}+{\mathrm{a}}_{\mathrm{P}\mathrm{n}})\right\}$$

(5)

This term represents the overall level of precedence for each of the criteria.

Step 6. Calculate the normalized A_N precedence value for each criterion according to the vector represented in Equation (6):

$${\mathrm{A}}_{\mathrm{N}}=\left\{\left({\displaystyle \frac{{\mathrm{a}}_1^{\mathrm{*}}-\max {\mathrm{a}}_{\mathrm{j}}^{\mathrm{*}}}{\min {\mathrm{a}}_{\mathrm{j}}^{\mathrm{*}}-\max {\mathrm{a}}_{\mathrm{j}}^{\mathrm{*}}}}\right),\left({\displaystyle \frac{{\mathrm{a}}_2^{\mathrm{*}}-\max {\mathrm{a}}_{\mathrm{j}}^{\mathrm{*}}}{\min {\mathrm{a}}_{\mathrm{j}}^{\mathrm{*}}-\max {\mathrm{a}}_{\mathrm{j}}^{\mathrm{*}}}}\right),\dots ,\left({\displaystyle \frac{{\mathrm{a}}_{\mathrm{n}}^{\mathrm{*}}-\max {\mathrm{a}}_{\mathrm{j}}^{\mathrm{*}}}{\min {\mathrm{a}}_{\mathrm{j}}^{\mathrm{*}}-\max {\mathrm{a}}_{\mathrm{j}}^{\mathrm{*}}}}\right)\right\}$$

(6)

where a^*_j indicates the sum of the elements N_j and P_j. It should be noted that the normalized values obtained range from 0 to 1, where the weight of the NIE equals 0 and the PIE receives a maximum weight of 1. However, as in the classic SAPEVO models, this normalization can result in criteria with weights equal to 0, which conceptually makes no sense, as such a criterion would not be considered in the analysis.

Gomes et al. [41] suggested assigning the weight of 1% of the lowest subsequent value to circumvent this limitation. However, these values remain very close to 0 in practice. In this paper, we propose to assign the weight of 20% of the lowest subsequent value to the criterion with the least importance. The results represent cardinal scores of the criteria and alternatives to the qualitative criteria, clarifying the preference relationships for each variable in each evaluation context.

Step 7. Calculate the values of the criteria weights, W_S, represented on a percentage scale, according to the vector presented by Equation (7):

$${\mathrm{W}}_{\mathrm{S}}=\left\{\left({\displaystyle \frac{A_{N_1}}{{\sum }_{j=1}^N{A}_{N_j}}}\right),\left({\displaystyle \frac{A_{N_2}}{{\sum }_{j=1}^N{A}_{N_j}}}\right),\dots ,\left({\displaystyle \frac{A_{N_n}}{{\sum }_{j=1}^N{A}_{N_j}}}\right)\right\}$$

(7)

Steps 1 to 7 are performed for each DM involved in decision-making.

Step 8. Calculate the Group Decision Value, W_G, represented on a percentage scale, according to Equation (8):

$${\mathrm{W}}_{\mathrm{G}}=\left\{\left({\displaystyle \frac{{\sum }_{j=1}^D{W}_{S_1}}{\mathrm{D}}}\right),\left({\displaystyle \frac{{\sum }_{j=1}^D{W}_2}{\mathrm{D}}}\right),\dots ,\left({\displaystyle \frac{{\sum }_{j=1}^D{W}_{S_n}}{\mathrm{D}}}\right)\right\}$$

(8)

WG is the vector of group decision values for the criteria; D is the total number of decision-makers; WS is the weight of the i-th criterion the d-th decision-maker assigns.

In this step, we compute the average of the decision-makers’ evaluations to derive the group decision value. It involves taking the mean of the individual weights assigned by each decision-maker to each criterion. The group decision value provides a collective assessment that incorporates the perspectives of all decision-makers, ensuring a balanced and representative evaluation.

2.2.2. Phase 2: Quantitative Evaluation Integrating MCDM and ML

The second phase of the SAPEVO-PC method is conducted using PCA principles. The concepts and details of PCA will be explained and detailed in this section, as well as the integration of this ML technique with the algorithm proposed in this article.

Initially, we have to address factor analysis, which is useful when seeking to analyze variables that present relatively high correlation coefficients, with the establishment of new variables that capture the collective behavior of the original variables [28], summarizing the data and generating hypotheses [45]. Fávero and Belfiore [46] define each new variable as a factor grouping variables based on established criteria. Therefore, factor analysis is a multivariate technique that seeks to identify several factors representing the behavior set of interdependent original variables [46].

Among the techniques for determining factors, PCA is the most widely used since it assumes that uncorrelated factors can be extracted from linear combinations of the original variables. In other words, PCA allows, based on a set of original variables correlated with each other, another set of variables (factors) to be determined, resulting from the linear combination of the original set [28].

In a historical context, Pearson [47] and Spearman [48] are the founders of factor analysis. While the first author developed the mathematical model of what is conventionally called correlation, the second proposed a methodology to evaluate the inter-relationships between variables. Decades later, Hotelling [49] proposed the term Principal Component Analysis as the analysis that establishes components from the maximization of the variance of original data.

In order to mathematically implement PCA, Fávero and Belfiore [28] initially suggested a database with several observations, n, and, for each observation, i (i = 1,..., n), values corresponding to each of the k metric variables X. To extract factors from the k variables, it is necessary to define the matrix of ρ correlations with the values of Pearson’s linear correlation between each pair of variables, as illustrated in the matrix (9).

$$\mathsf{\rho }=\left(\begin{array}{cc}\begin{array}{cc}1& {\mathsf{\rho }}_{12}\\ {\mathsf{\rho }}_{21}& 1\end{array}& \begin{array}{cc}\dots & {\mathsf{\rho }}_{1\mathrm{k}}\\ \dots & {\mathsf{\rho }}_{2\mathrm{k}}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ {\mathsf{\rho }}_{\mathrm{k}1}& {\mathsf{\rho }}_{\mathrm{k}2}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \dots & 1\end{array}\end{array}\right)$$

(9)

The correlation matrix ρ is symmetric concerning the main diagonal, with values equal to 1. ${\mathsf{\rho }}_{12}$ represents the Pearson correlation between X₁ and X₂, calculated using Equation (10) [28]:

$${\mathsf{\rho }}_{12}={\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{X}}_{1\mathrm{i}}-\overline{{\mathrm{X}}_1}\right).\left({\mathrm{X}}_{2\mathrm{i}}-\overline{{\mathrm{X}}_2}\right)}{\sqrt{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{X}}_{1\mathrm{i}}-\overline{{\mathrm{X}}_1}\right)}^2}.\sqrt{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{X}}_{2\mathrm{i}}-\overline{{\mathrm{X}}_2}\right)}^2}}}$$

(10)

$\overline{{\mathrm{X}}_1}$ and $\overline{{\mathrm{X}}_2}$ correspond to the means of the variables X₁ and X₂.

Pearson’s correlation consists of a measure of the degree of the linear relationship between two metric variables, ranging from −1 to 1 [50]. Values close to one of these limits correspond to the existence of a linear relationship between the two variables under analysis. Thus, they can significantly contribute to extracting a single factor [46]. In contrast, a value close to 0 indicates that the linear relationship between the two variables is practically non-existent [51].

For a correct extraction of factors based on original variables, the matrix of ρ correlations must present relatively high and statistically significant values [46,52]. According to Hair et al. [53], if a considerable amount of values is below 0.30, there is a preliminary indication that factor analysis may be inappropriate. In this sense, Fávero and Belfiore [46] have stated that the researcher must use the Kaiser–Meyer–Olkin (KMO) statistic and Bartlett’s sphericity test to verify the overall adequacy of factor extraction.

The KMO statistic ranges from 0 to 1, providing the proportion of variance common to all variables in the sample, possibly attributed to a common factor [54]. According to [28], values closer to 1 indicate that the variables share a very high percentage of variance with high Pearson correlations. On the other hand, values close to 0 result from low Pearson correlation indices, which may indicate that factor analysis will not be adequate. Obtaining the KMO statistic proposed by Kaiser [55] is possible using Equation (11):

$$\mathrm{K}\mathrm{M}\mathrm{O}={\displaystyle \frac{{\sum }_{\mathrm{l}=1}^{\mathrm{k}}{\sum }_{\mathrm{c}=1}^{\mathrm{k}}{\mathsf{\rho }}_{\mathrm{l}\mathrm{c}}^2}{{\sum }_{\mathrm{l}=1}^{\mathrm{k}}{\sum }_{\mathrm{c}=1}^{\mathrm{k}}{\mathsf{\rho }}_{\mathrm{l}\mathrm{c}}^2+{\sum }_{\mathrm{l}=1}^{\mathrm{k}}{\sum }_{\mathrm{c}=1}^{\mathrm{k}}{\mathsf{\phi }}_{\mathrm{l}\mathrm{c}}^2}},\mathrm{l}\ne \mathrm{c}$$

(11)

where l and c correspond to the rows and columns of the correlation matrix $\mathsf{\rho }$, respectively. Φ represents the partial correlation coefficients between two variables [28]. Pearson’s correlation coefficients ρ are also known as zero-order correlation coefficients, while φ coefficients are also called higher-order correlation coefficients. Three variables are called first-order correlation coefficients, four second-order correlation coefficients, and so on [46].

When we want to know the correlation between two variables, we use the partial correlation coefficients, disregarding the effects of the other variables present in the sample under analysis [56]. For example, considering three variables X₁, X_2, and X₃, the first-order correlation coefficients can be defined by Equation (12) [28]:

$${\mathsf{\phi }}_{\mathrm{a}\mathrm{b},\mathrm{c}}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{a}\mathrm{b}}-{\mathsf{\rho }}_{\mathrm{a}\mathrm{c}}·{\mathsf{\rho }}_{\mathrm{b}\mathrm{c}}}{\sqrt{\left(1-{\mathsf{\rho }}_{\mathrm{a}\mathrm{c}}^2\right)·(1-{\mathsf{\rho }}_{\mathrm{b}\mathrm{c}}^2)}}}$$

(12)

where φ_ab,c represents the correlation between X_a and X_b, keeping X_c constant, where a, b, and c can assume values 1, 2, or 3, corresponding to the three variables X₁, X_2, and X₃ under analysis. Equation (13) represents the general expression of a given partial correlation coefficient (second order) for a situation with four variables [28]:

$${\mathsf{\phi }}_{\mathrm{a}\mathrm{b},\mathrm{c}\mathrm{d}}={\displaystyle \frac{{\mathsf{\phi }}_{\mathrm{a}\mathrm{b},\mathrm{c}}-{\mathsf{\phi }}_{\mathrm{a}\mathrm{d},\mathrm{c}}·{\mathsf{\phi }}_{\mathrm{b}\mathrm{d},\mathrm{c}}}{\sqrt{\left(1-{\mathsf{\phi }}_{\mathrm{a}\mathrm{d},\mathrm{c}}^2\right)·(1-{\mathsf{\phi }}_{\mathrm{b}\mathrm{d},\mathrm{c}}^2)}}}$$

(13)

where φ_ab,cd corresponds to the correlation between X_a and X_b, with X_c and X_d constant, where a, b, c, and d can take the values 1, 2, 3, or 4, corresponding to the four variables under analysis. According to [28], obtaining correlation coefficients of higher orders (five or more variables) will be based on determining partial correlation coefficients of lower orders. The authors also point out that even if Pearson’s correlation coefficient between two variables is 0, the partial correlation coefficient between them may not be equal to 0 since this depends on the values of Pearson’s correlation coefficients between each of these variables and the others present in the database.

Factor analysis is appropriate if the partial correlation coefficients between the variables are low [46], which means that such variables share a high percentage of variance. Disregarding one or more of them can impair the quality of the extraction of the analyzed factors. In this sense,

Table 4. Relationship between KMO statistics and the global adequacy of factor analysis.

KMO Statistics	Global Adequacy of Factor Analysis
Between 1.00 and 0.90	Very good
Between 0.90 and 0.80	Good
Between 0.80 and 0.70	Average
Between 0.70 and 0.70	Reasonable
Between 0.60 e 0.50	Bad
Less than 0.50	Unacceptable

Source: [28].

presents, according to [28], a relationship between KMO statistics and the global adequacy of factor analysis.

The sphericity test proposed by Bartlett [57] compares Pearson’s correlation matrix with an identity matrix I of equal dimension. If the values of the principal diagonals of each matrix present differences that are not statistically different from 0 at a given significance level, the extraction of the factors will be inadequate. According to [28], Pearson’s correlations between each pair of variables are statistically equal to 0, making extracting factors from the original variables impossible. Therefore, expression (14) defines the null (H₀) and alternative (H₁) hypotheses of Bartlett’s sphericity test:

$${\mathrm{H}}_0:\mathsf{\rho }=\left(\begin{array}{cc}\begin{array}{cc}1& {\mathsf{\rho }}_{12}\\ {\mathsf{\rho }}_{21}& 1\end{array}& \begin{array}{cc}\dots & {\mathsf{\rho }}_{1\mathrm{k}}\\ \dots & {\mathsf{\rho }}_{2\mathrm{k}}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ {\mathsf{\rho }}_{\mathrm{k}1}& {\mathsf{\rho }}_{\mathrm{k}2}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \dots & 1\end{array}\end{array}\right)=\mathrm{I}=\left(\begin{array}{cc}\begin{array}{cc}1& 0\\ 0& 1\end{array}& \begin{array}{cc}\dots & 0\\ \dots & 0\end{array}\\ \begin{array}{cc}⋮& ⋮\\ 0& 0\end{array}& \begin{array}{cc}\ddots & ⋮\\ \dots & 1\end{array}\end{array}\right)$$

(14)

$${\mathrm{H}}_1:\mathsf{\rho }=\left(\begin{array}{cc}\begin{array}{cc}1& {\mathsf{\rho }}_{12}\\ {\mathsf{\rho }}_{21}& 1\end{array}& \begin{array}{cc}\dots & {\mathsf{\rho }}_{1\mathrm{k}}\\ \dots & {\mathsf{\rho }}_{2\mathrm{k}}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ {\mathsf{\rho }}_{\mathrm{k}1}& {\mathsf{\rho }}_{\mathrm{k}2}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \dots & 1\end{array}\end{array}\right)\ne \mathrm{I}=\left(\begin{array}{cc}\begin{array}{cc}1& 0\\ 0& 1\end{array}& \begin{array}{cc}\dots & 0\\ \dots & 0\end{array}\\ \begin{array}{cc}⋮& ⋮\\ 0& 0\end{array}& \begin{array}{cc}\ddots & ⋮\\ \dots & 1\end{array}\end{array}\right)$$

The statistic corresponding to this test is a χ² statistic, represented by Equation (15):

$${\mathsf{\chi }}_{\mathrm{B}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{t}}^2=-⌈\left(\mathrm{n}-1\right)-\left({\displaystyle \frac{2.\mathrm{k}+5}{6}}\right)⌉·\mathrm{l}\mathrm{n}\left|\mathrm{D}\right|$$

(15)

With ${\displaystyle \frac{\mathrm{k}.(\mathrm{k}-1)}{2}}$ degrees of freedom, where n is the sample size, and k is the number of variables. Moreover, D is the determinant of the matrix of ρ correlations [46].

According to [28], Bartlett’s sphericity test, therefore, allows the verification for a given level of statistical significance and some degrees of freedom, whether the total value of the χ²_Bartlett statistic is greater than the critical value of the statistic. If this happens, we state that the Pearson correlations between the pairs of variables are statistically different from 0, which means that factors can be extracted based on the original variables.

After verifying the global adequacy of the factor analysis, we must start by defining the factors by principal components, determining the eigenvalues and eigenvectors of the matrix of ρ correlations, and calculating the factor scores [46].

A factor represents the linear combination of original variables. In this sense, for k variables, a maximum number of k factors (F₁, F₂,..., F_k) is analogous to the maximum number of clusters that can be defined based on a sample with n observations. Thus, for k variables, we have Equation (16):

$$\begin{array}{c}\begin{array}{c}{\mathrm{F}}_{1\mathrm{i}}={\mathrm{s}}_{11}·{\mathrm{X}}_{1\mathrm{i}}+{\mathrm{s}}_{21}·{\mathrm{X}}_{2\mathrm{i}}+\cdots +{\mathrm{s}}_{\mathrm{k}1}·{\mathrm{X}}_{\mathrm{k}\mathrm{i}}\\ {\mathrm{F}}_{2\mathrm{i}}={\mathrm{s}}_{12}·{\mathrm{X}}_{1\mathrm{i}}+{\mathrm{s}}_{22}·{\mathrm{X}}_{2\mathrm{i}}+\cdots +{\mathrm{s}}_{\mathrm{k}2}·{\mathrm{X}}_{\mathrm{k}\mathrm{i}}\end{array}\\ \begin{array}{c}⋮\\ {\mathrm{F}}_{\mathrm{k}\mathrm{i}}={\mathrm{s}}_{1\mathrm{k}}·{\mathrm{X}}_{1\mathrm{i}}+{\mathrm{s}}_{2\mathrm{k}}·{\mathrm{X}}_{2\mathrm{i}}+\cdots +{\mathrm{s}}_{\mathrm{k}\mathrm{k}}·{\mathrm{X}}_{\mathrm{k}\mathrm{i}}\end{array}\end{array}$$

(16)

The terms are called factorial scores, representing a linear model’s parameters that relate a given factor to the original variables [28]. According to the expression, it is possible to obtain these scores by determining the eigenvalues and eigenvectors of the matrix of $\mathsf{\rho }$ correlations (17).

$$\mathsf{\rho }=\left(\begin{array}{cc}\begin{array}{cc}1& {\mathsf{\rho }}_{12}\\ {\mathsf{\rho }}_{21}& 1\end{array}& \begin{array}{cc}\dots & {\mathsf{\rho }}_{1\mathrm{k}}\\ \dots & {\mathsf{\rho }}_{2\mathrm{k}}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ {\mathsf{\rho }}_{\mathrm{k}1}& {\mathsf{\rho }}_{\mathrm{k}2}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \dots & 1\end{array}\end{array}\right)$$

(17)

In which the matrix of correlations k × k, according to Fávero and Belfiore [28], presents k eigenvalues λ₂ (λ₂₁ ≥ λ₂₂ ≥... ≥ λ_2k), obtained through Equation (18):

$$\det \left({\mathsf{\lambda }}^2-\mathrm{I}-\mathsf{\rho }\right)=0,\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{\mathsf{\lambda }}_1^2+{\mathsf{\lambda }}_2^2+\dots +{\mathsf{\lambda }}_{\mathrm{k}}^2=\mathrm{k}$$

(18)

where I is the identity matrix, also with dimensions k × k.

Thus, Fávero and Belfiore [46] define the matrix of eigenvalues L2 according to the expression (19):

$${\mathsf{\Lambda }}^2=\left(\begin{array}{cc}\begin{array}{cc}{\mathsf{\lambda }}_1^2& 0\\ 0& {\mathsf{\lambda }}_2^2\end{array}& \begin{array}{cc}\dots & 0\\ \dots & 0\end{array}\\ \begin{array}{cc}⋮& ⋮\\ 0& 0\end{array}& \begin{array}{cc}\ddots & ⋮\\ \dots & {\mathsf{\lambda }}_{\mathrm{k}}^2\end{array}\end{array}\right)$$

(19)

To define the eigenvectors of the matrix ρ based on the eigenvalues, we must solve the systems of Equation (20) for each eigenvalue λ₂ (λ₂₁ ≥ λ₂₂ ≥... ≥ λ_2k):

$$\left(\begin{array}{cc}\begin{array}{cc}{\mathsf{\lambda }}_{\mathrm{k}}^2-1& -{\mathsf{\rho }}_{12}\\ -{\mathsf{\rho }}_{21}& {\mathsf{\lambda }}_{\mathrm{k}}^2-1\end{array}& \begin{array}{cc}\dots & -{\mathsf{\rho }}_{1\mathrm{k}}\\ \dots & -{\mathsf{\rho }}_{2\mathrm{k}}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ -{\mathsf{\rho }}_{\mathrm{k}1}& -{\mathsf{\rho }}_{\mathrm{k}2}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \dots & {\mathsf{\lambda }}_{\mathrm{k}}^2-1\end{array}\end{array}\right).\left(\begin{array}{c}\begin{array}{c}{\mathrm{v}}_{1\mathrm{k}}\\ {\mathrm{v}}_{2\mathrm{k}}\end{array}\\ \begin{array}{c}⋮\\ {\mathrm{v}}_{\mathrm{k}\mathrm{k}}\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}0\\ 0\end{array}\\ \begin{array}{c}⋮\\ 0\end{array}\end{array}\right)\to \left\{\begin{array}{c}\begin{array}{c}{(\mathsf{\lambda }}_{\mathrm{k}}^2-1)·{\mathrm{v}}_{1\mathrm{k}}-{\mathsf{\rho }}_{12}·{\mathrm{v}}_{2\mathrm{k}}\dots -{\mathsf{\rho }}_{1\mathrm{k}}·{\mathrm{v}}_{\mathrm{k}\mathrm{k}}=0\\ -{\mathsf{\rho }}_{21}·{\mathrm{v}}_{1\mathrm{k}}+{(\mathsf{\lambda }}_2^2-1)·{\mathrm{v}}_{2\mathrm{k}}\dots -{\mathsf{\rho }}_{2\mathrm{k}}·{\mathrm{v}}_{\mathrm{k}\mathrm{k}}=0\end{array}\\ \begin{array}{c}⋮\\ -{\mathsf{\rho }}_{\mathrm{k}1}·{\mathrm{v}}_{1\mathrm{k}}-{\mathsf{\rho }}_{\mathrm{k}2}·{\mathrm{v}}_{2\mathrm{k}}\dots +{(\mathsf{\lambda }}_{\mathrm{k}}^2-1)·{\mathrm{v}}_{\mathrm{k}\mathrm{k}}=0\end{array}\end{array}\right.$$

(20)

Thus, Favero and Belfiore [28] state that the factorial scores of each factor are calculated by determining the eigenvalues and eigenvectors of the matrix ρ. It is possible to define the vectors of the factorial scores by the expression (21) for the k-th factor [46].

$${\mathrm{S}}_{\mathrm{k}}=\left(\begin{array}{c}\begin{array}{c}{\mathrm{S}}_{1\mathrm{k}}\\ {\mathrm{S}}_{2\mathrm{k}}\end{array}\\ \begin{array}{c}⋮\\ {\mathrm{S}}_{\mathrm{k}\mathrm{k}}\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}{\displaystyle \frac{{\mathrm{v}}_{1\mathrm{k}}}{\sqrt{{\mathsf{\lambda }}_{\mathrm{k}}^2}}}\\ {\displaystyle \frac{{\mathrm{v}}_{2\mathrm{k}}}{\sqrt{{\mathsf{\lambda }}_{\mathrm{k}}^2}}}\end{array}\\ \begin{array}{c}⋮\\ {\displaystyle \frac{{\mathrm{v}}_{\mathrm{k}\mathrm{k}}}{\sqrt{{\mathsf{\lambda }}_{\mathrm{k}}^2}}}\end{array}\end{array}\right)$$

(21)

According to [28], the factorial scores of each factor are standardized by the respective eigenvalues. Therefore, the factors calculated by Equation (16) should be obtained by multiplying each factorial score by the corresponding original variable, standardized using the Z-scores procedure. Thus, it is possible to calculate the k-th factor by Equation (22):

$${\mathrm{F}}_{\mathrm{k}\mathrm{i}}={\displaystyle \frac{{\mathrm{v}}_{1\mathrm{k}}}{\sqrt{{\mathsf{\lambda }}_{\mathrm{k}}^2}}}·\mathrm{Z}{\mathrm{X}}_{1\mathrm{i}}+{\displaystyle \frac{{\mathrm{v}}_{2\mathrm{k}}}{\sqrt{{\mathsf{\lambda }}_{\mathrm{k}}^2}}}·\mathrm{Z}{\mathrm{X}}_{2\mathrm{i}}+\cdots +{\displaystyle \frac{{\mathrm{v}}_{\mathrm{k}\mathrm{k}}}{\sqrt{{\mathsf{\lambda }}_{\mathrm{k}}^2}}}·\mathrm{Z}{\mathrm{X}}_{\mathrm{k}\mathrm{i}}$$

(22)

where ZX_i corresponds to the standardized value of each variable X for a given observation i. All the extracted factors present, among themselves, Pearson correlations equal to 0; that is, they are orthogonal to each other [28]

The innovative integration of the W_S weights, obtained in phase 1, with the PCA values occurs in calculating the factor scores. The original criteria weights (WS) are combined with the PCA-derived values in this process. Specifically, each weight of the original criteria arising from phase 1 is multiplied by the corresponding value, ensuring that qualitative expert opinions and strictly quantitative data-based values are incorporated into the final assessment.

This integration allows for a more comprehensive and accurate assessment of alternatives, taking advantage of the strengths of both methods. Considering qualitative and quantitative data, the factor equation can be represented by Equation (23):

$${\mathrm{F}}_{\mathrm{k}\mathrm{i}}={\displaystyle \frac{{\mathrm{v}}_{1\mathrm{k}}}{\sqrt{{\mathsf{\lambda }}_{\mathrm{k}}^2}}}·\mathrm{Z}{\mathrm{X}}_{1\mathrm{i}}·W_{S_1}+{\displaystyle \frac{{\mathrm{v}}_{2\mathrm{k}}}{\sqrt{{\mathsf{\lambda }}_{\mathrm{k}}^2}}}·\mathrm{Z}{\mathrm{X}}_{2\mathrm{i}}·W_{S_2}+\cdots +{\displaystyle \frac{{\mathrm{v}}_{\mathrm{k}\mathrm{k}}}{\sqrt{{\mathsf{\lambda }}_{\mathrm{k}}^2}}}·\mathrm{Z}{\mathrm{X}}_{\mathrm{k}\mathrm{i}}·W_{S_k}$$

(23)

where W_S represents the weight of each criterion obtained in phase 1; k represents the number of original criteria under analysis.

Fávero and Belfiore [46] point out that it is fundamental to establish a criterion to obtain the appropriate amount of factors representing the original variables. Although there are several ways to obtain such a quantity, an analysis must be made based on the magnitude of the eigenvalues calculated from the matrix of ρ correlations [28].

Considering that the eigenvalues represent the percentage of variance shared by the original variables for the formation of each factor, Fávero and Belfiore [28] point out that lower percentages of variance shared by the original variables form the factors extracted based on smaller eigenvalues. In this sense, factors extracted from eigenvalues of less than one cannot represent the behavior of even one original variable [46].

Among the ways of determining the number of factors, Ruscio and Roche [58] highlight the latent root criterion or Kaiser criterion, in which only the factors corresponding to eigenvalues more significant than one are considered. In this work, we used PCA factor analysis, in which the first factor F₁, formed by the highest percentage of variance shared by the original variables, is also called the main factor [28].

After establishing the factors, it is possible to define the factor loadings, represented by Pearson correlations between the original variables and each factor generated [59].

Table 5. Factor loadings between original variables and factors.

Variable Factor	F1	F2	$\dots$	Fk
X1	c11	c12	$\dots$	c1k
X2	c21	c22		c2k
$⋮$	$⋮$	$⋮$		$⋮$
Xk	ck1	ck2		ckk

Source: [28].

presents the factor loadings for each variable–factor pair.

Based on the latent root criterion (only eigenvalues greater than 1), we assume that the factor loadings between the factors corresponding to eigenvalues of less than one and all the original variables are low, as they will have already presented higher loads with factors previously extracted from higher eigenvalues [46]. Similarly, original variables that share a small portion of variance with the other variables will only present high factor loadings in a single factor. If this occurs for all original variables, there will be no significant differences between the correlation matrices ρ and identity I, making the χ²_Bartlett statistic very low and, therefore, the factor analysis will be inappropriate [28].

Since the factor loadings are the Pearson correlations between each variable and each factor, the sum of the squares of these loads in each row of Table 5 will always equal one because each variable shares part of its percentage of variance with all k factors. In addition, the sum of the variance percentages will be 100% [28].

However, if several factors less than k are extracted based on the latent root criterion, each row’s sum of squares of the factor loadings will not equal 1. In this case, the sum is called commonality, which corresponds to the total shared variance of each variable in all factors extracted from eigenvalues greater than 1 [26]. Therefore, Fávero and Belfiore [28] present the Equation (24):

$${\mathrm{c}}_{\mathrm{k}1}^2+{\mathrm{c}}_{\mathrm{k}2}^2+\dots =\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}{\mathrm{X}}_{\mathrm{k}}$$

(24)

The analysis of commonalities aims to verify if any variable does not share a significant percentage of variance with the extracted factors [46]. Fávero and Belfiore [28] state that, although no cutoff point defines a given commonality as being high or low because the sample size can interfere with this judgment, the verification of considerably low commonalities compared to the others may suggest the reconsideration of the inclusion of the respective variable in the factor analysis.

Thus, after defining the factors based on the scores, the factor loadings will equal the estimated parameters of a multiple linear regression model, having as dependent variables a standardized ZX and, as explanatory variables, the factors themselves. The adjustment coefficient R² of each model will equal the commonality of the original variable [28].

On the other hand, the sum of the squares of the factor loadings in each column of Table 2 will be equal to the respective eigenvalue. That happens due to the ratio between each eigenvalue and the total number of variables corresponding to the percentage of variance shared by all k original variables for forming each factor [28], as illustrated by Equation (25):

$${\mathrm{c}}_{1\mathrm{k}}^2+{\mathrm{c}}_{2\mathrm{k}}^2+\cdots +{\mathrm{c}}_{\mathrm{k}\mathrm{k}}^2={\mathsf{\lambda }}_{\mathrm{k}}^2$$

(25)

Finally, it is possible to create a performance ranking between observations. In this sense, a widely accepted procedure is the criterion of weighted sum and ordering, based on the sum, for each observation, of the values obtained from all factors that have eigenvalues greater than 1, weighted by the respective percentages of shared variance [28]. The equation for the final score (P_i) of an alternative i is given by Equation (26):

$${\sum }_{j=1}^k{F}_{ij}·{Var}_j$$

(26)

where: P_i is the final score of an alternative i; F_ij is the value of factor j for an alternative i; Var_j is the percentage of the shared variance of factor j; k is the total number of factors considered (those with eigenvalues greater than 1). This criterion considers the performance referring to all the original variables because the consideration only of the main factor may not consider, for example, the upbeat performance obtained in a given variable that eventually shares a considerable percentage of variance with the second factor [46].

The flowchart presented in Figure 1 summarizes the SAPEVO-PC methodology.

In summary, a set of data containing alternatives and criteria is initially used. Then, the qualitative criteria are evaluated by multiple decision-makers (represented by “N” DMs). The results of these assessments are integrated with the factor scores, allowing the extraction of factors using statistical analyses such as the KMO Test and the Bartlett Sphericity Test to ensure data adequacy. Finally, a performance assessment of the alternatives is carried out, considering the factors calculated by integrating qualitative and quantitative assessments.

2.3. Related Studies

The selection of warships is a complex process requiring advanced decision-making tools. Recent studies have extensively utilized MCDM methodologies to address this complexity [60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76]. Vukić et al. [60] used the AHP method to analyze and prioritize the selection criteria and types of vehicle to monitor navigation safety. Tenório et al. [62] introduced the THOR method to aid the Brazilian Navy in selecting warships amidst budget constraints, effectively quantifying uncertainties. Similarly, Manap and Voulvoulis [63] developed a risk-based framework for selecting sediment-dredging strategies, integrating environmental, economic, and social risks.

The importance of sustainability in naval decisions is emphasized by Pesce et al. [64], who applied MCDM to select sustainable alternatives for cruise ships and assess environmental sustainability in maritime engineering. Gumusay et al. [65] used AHP and GIS to evaluate site suitability for marina construction, combining environmental, economic, and social factors. Maceiras et al. [61] applied a multi-criteria methodology to identify the most promising alternative fuel among those under development for possible implementation on military ships.

Hybrid MCDM models have been extensively studied by several people. Balin et al. [66] combined fuzzy AHP and TOPSIS to select critical gas turbine components, demonstrating the robustness of hybrid approaches. Similarly, Monprapussorn et al. [67] integrated MCDM and GIS to optimize hazardous waste-transport routes. Bayraktar and Nuran [68] applied the TOPSIS method to improve marine energy efficiency, emphasizing considering multiple criteria such as fuel consumption and emissions.

The application of MCDM to maritime search and rescue operations was explored by Wielgosz and Malyszko [69], and Malyszko [70], highlighting criteria such as effectiveness and safety. Santos et al. [71] employed AHP-Gaussian and Pearson’s correlation to rank warships for the Brazilian navy, aligning with strategic defense programs.

Considering environmental and safety criteria, Stavrou et al. [64] applied MCDM to locate ship-to-ship transfer operations. Malyszko [73] proposed a multi-criteria assessment for search and rescue units, emphasizing the role of decision-support systems. Hansson et al. [74] evaluated the need for alternative marine fuels to reduce greenhouse gas emissions from maritime transport. They employed life cycle analysis (LCA) to assess the environmental impact of alternative fuels, highlighting the importance of sustainable fuel options but not specifically applying MCDM for ship selection. Similarly, Ren and Lützen [75] described a methodology for applying LCA in the maritime sector to evaluate the environmental performance of shipping operations, focusing on environmental impacts without directly addressing MCDM for ship selection. Baesens et al. [76] applied deep learning convolutional neural networks to classify a set of warship images automatically.

The reviewed studies consistently underscore the efficacy of MCDM methods for warship selection and other strategic decisions in naval engineering. Integrating hybrid techniques, such as AHP, TOPSIS, and fuzzy logic is widely recognized as robust for handling complexity and uncertainties in decision-making processes [62,63,66].

The literature review reveals that MCDM approaches significantly enhance the decision-making processes for warship selection and other strategic defense decisions by systematically integrating multiple criteria. Adopting hybrid techniques and fuzzy logic increases the robustness and precision of decision models.

The literature review reveals that previous applications predominantly utilize classical MCDM methods such as AHP, THOR, and various hybrid MCDM approaches. However, these studies lack the integration of deeper mathematical and statistical analyses through ML algorithms to provide more reliability and robustness to the decision-making process. In this context, the methodology proposed in this paper is innovative, as it proposes integrating ML techniques and MCDM models into a real and complex problem related to maritime science and engineering.

Our methodology uniquely combines the qualitative strengths of the MCDM approach with the quantitative rigor of ML techniques, especially PCA. Analyzing real data from 32 military ships provides a robust framework for comprehensive and accurate assessments, filling a gap in the literature.

Studies by [59,61] and others have demonstrated the utility of MCDM methods in various maritime and defense-related applications. However, integrating ML principles, particularly PCA, adds a new dimension to our approach. PCA reduces data complexity while preserving essential standards, facilitating more accurate performance assessments of warships. This integration addresses the limitations of traditional MCDM methods by increasing analytical depth and providing an understanding of the factors that influence ship performance.

The SAPEVO-PC method is aligned with the literature trend of developing decision-support methods that integrate PCA and MCDM techniques, as cited in [77,78]. However, SAPEVO-PC offers innovative advantages, such as integrating state-of-the-art MCDM and ML techniques, while allowing the DM’s opinion to be considered in the decision-making process. This feature promotes more accurate and coherent decision-making aligned with the institutions’ strategic objectives.

Our proposed method stands out for its ability to effectively deal with complex and multidimensional datasets, which is crucial in the maritime domain. This innovative approach increases the accuracy of decision-making and provides a replicable model for addressing tactical, operational, and strategic challenges in marine science and engineering.

3. Methodology

According to the classification proposed by [79], this research can be quantitative, combining case studies and mathematical modelling [80]. This study focuses on evaluating and establishing a ranking of warships, considering the primary vessels available in the world’s navies. This economic–military evaluation incorporates tactical, operational, and strategic aspects to ensure a comprehensive assessment. By utilizing robust mathematical and statistical methods, this research aims to provide a replicable and objective framework for analyzing the capabilities of different warships.

The warship dataset analyzed in this study is highly relevant as it has been meticulously compiled from reliable sources that provide detailed and accurate data on military ships, such as [81,82], ensuring the accuracy and relevance of information. These sources were selected for their credibility and the comprehensiveness of the data offered.

Quantitative data on warships were manually extracted from the identified sources. This process involved reading reports in detail and carefully transcribing relevant information into a central spreadsheet. After initial collection, the data were validated by military experts to ensure accuracy and reliability. This validation process included cross-checking information with multiple sources and correcting any inconsistencies identified.

The dataset includes a comprehensive range of criteria such as displacement, speed, armament, and technological capabilities, ensuring a robust and complete assessment of each warship. By leveraging these reliable sources, the dataset reflects the current state of naval capabilities and provides a solid basis for analyzing and ranking the world’s most advanced warships.

Table 6. Criteria used in the analysis.

Criterion	Trigram	Unit of Measurement	Description	Monotonicity
Displacement	DSP	Tons (t)	Refers to the total weight of the ship, measured in tons. Displacement includes the weight of the ship’s structure, fuel, cargo, crew, and other components. A higher displacement generally indicates a larger and more robust ship.	Maximization
Speed	SPD	Knots	The maximum speed the ship can achieve is measured in knots (nautical miles per hour). Speed is crucial for quick maneuvers, threat evasion, and strategic positioning in combat situations.	Maximization
Cargo Capacity	CAP	Tons (t)	The amount of useful cargo the ship can carry is measured in tons. It includes armaments, ammunition, vehicles, supplies, and other equipment. A higher cargo capacity allows the ship to be more versatile and self-sufficient during extended operations.	Maximization
Armament	ARM	Number of weapons	The number of weapon systems onboard the ship, including guns, missiles, torpedoes, and other armaments. A ship’s firepower is critical to its defense and attack capabilities, directly influencing combat effectiveness.	Maximization
Cost	CST	US Dollars (USD)	The total construction cost of the ship is measured in US dollars. Cost is a significant factor in military planning and budgeting, affecting the economic feasibility of building and maintaining a fleet.	Minimization
Range	RNG	Nautical miles	The maximum distance the ship can travel without refueling is measured in nautical miles. A greater range allows the ship to operate for longer periods in open waters without relying on frequent logistical support.	Maximization
Crew	CRE	Number of personnel	The total number of personnel required to operate the ship, including officers, sailors, and technical staff. The size of the crew directly impacts operational efficiency and the ship’s capacity to perform various tasks.	Minimization
Year of Commissioning	YOC	Year	The year the ship was completed and commissioned. Newer ships tend to incorporate more advanced technology and designs, offering better performance and capabilities.	Maximization
Missile Range	MSL	Miles	The maximum distance the missiles onboard the ship can reach is measured in miles. A longer missile range allows the ship to strike targets at greater distances, providing a strategic advantage.	Maximization
Radar Range	RAD	Miles	The maximum range of the ship’s radar system is measured in miles. A long-range radar improves the ship’s ability to detect and track threats at greater distances, enhancing both defensive and offensive effectiveness.	Maximization

Source: authors.

presents the criteria analyzed in our study, as well as a description of each of them.

These 10 criteria for evaluating warships were carefully considered to cover a wide range of capabilities and operational characteristics, allowing for a robust and diverse analysis. The criteria presented in Table 6 were chosen based on their relevance and direct impact on ship operations and capabilities.

The proposed methodology aims to provide a comprehensive and flexible vision, as each navy may have different objectives when employing a ship, such as operations against natural disasters, social assistance, patrolling, combating piracy, or other tactical, operational, and strategic jobs. For example, payload capacity and displacement are critical for humanitarian assistance operations, while missile range and weapons classification are essential for combat missions. Including diverse criteria allows the method to be adapted to different scenarios and specific needs.

Furthermore, the choice of criteria was limited by the availability of reliable quantitative data in the analyzed databases. We recognize that a complete assessment of a ship’s military strength requires additional, more specific, and confidential information, both quantitative and qualitative. However, the selected criteria provide a solid basis for the comparative analysis proposed in this study.

In this paper, we did not present the specific names of the ships in the analysis based on several important considerations, especially related to the sensitive nature of military data. Military data, including detailed specifications of warships, are highly sensitive and confidential. Disclosing specific information can compromise the operational security of armed forces and expose strategic vulnerabilities. The ships’ explicit identification can reveal details about the capabilities and disposition of naval fleets from different countries.

In this sense, many nations have strict confidentiality policies regarding their military capabilities. Using generic names for the ships respects these policies, ensuring no confidential information is inadvertently disclosed. While the specific names of the ships are not disclosed, the data used in the analysis are real and based on verified information from reliable sources. They include technical and operational specifications such as displacement, speed, cargo capacity, armament, cost, range, crew, year of commissioning, missile range, and radar range.

In addition, this study uses the Soft Systems Methodology (SSM) [83], with the CATWOE analysis depicted in Figure 2, encapsulating the key elements of evaluating Navy Ships using ML and MCDM techniques.

The CATWOE structured critical analysis around the study, focusing on Customers (C), Actors (A), Transformations (T), Worldview (W), Owners (O), and Environmental constraints (E).

The primary objective of this analysis is to present a scientific methodology that can be replicated in this case and various tactical, operational, and strategic problems in the maritime sphere and scientific fields. Using generic names allows the focus to remain on the evaluation criteria and the methodology employed without compromising sensitive information.

Table 7. The decision matrix presenting the 32 alternatives evaluated on 10 criteria.

Ships	DSP	SPD	CAP	ARM	CST	RNG	CRE	YOC	MSL	RAD
Ship 1	14,564	30	1000	80	−4500	9000	−158	2016	100	300
Ship 2	100,000	31.5	9000	90	−8000	12,000	−5500	1975	800	350
Ship 3	65,000	25	5000	50	−4100	10,000	−1600	2017	600	320
Ship 4	45,400	28	4000	60	−2300	7000	−1600	2013	450	280
Ship 5	58,600	29	4500	70	−3200	8000	−2000	1990	700	300
Ship 6	28,000	32	4200	80	−5000	11,000	−1500	1988	900	350
Ship 7	12,500	34	4100	80	−4500	10,000	−1200	1982	800	320
Ship 8	12,000	30	3800	112	−1500	9000	−310	2019	600	340
Ship 9	3500	40	1200	50	−600	6000	−75	2008	400	250
Ship 10	10,000	30	1300	96	−1400	8300	−300	2007	500	270
Ship 11	5520	30	1400	80	−1000	4500	−300	2002	250	260
Ship 12	6000	30	2000	68	−800	6000	−200	2012	500	270
Ship 13	5690	29	1800	60	−900	4000	−230	2003	400	280
Ship 14	7000	30	2000	68	−1000	5000	−230	2017	500	300
Ship 15	5800	29	1800	68	−900	4000	−230	2004	400	280
Ship 16	7200	30	1800	64	−950	4000	−190	2013	300	310
Ship 17	6600	30	1900	70	−700	8000	−300	2011	500	290
Ship 18	3200	28	1500	70	−500	7000	−180	2008	400	270
Ship 19	5290	26	1600	72	−600	5000	−146	2006	500	290
Ship 20	3100	44	1200	60	−700	6000	−80	2010	300	240
Ship 21	5400	30	1800	70	−1000	8000	−200	2013	600	280
Ship 22	8000	32	2200	48	−1200	7000	−190	2010	450	320
Ship 23	6050	29	1600	40	−1200	5000	−200	2002	500	290
Ship 24	5800	28	1700	48	−1000	4500	−200	2002	500	290
Ship 25	4200	29	1200	30	−800	4000	−180	1990	300	250
Ship 26	9500	30	2000	90	−1800	4000	−350	1989	700	310
Ship 27	4100	29	1200	40	−600	4000	−180	1977	400	260
Ship 28	3600	25	1000	40	−300	5000	−150	1992	300	230
Ship 29	21,300	18.8	3000	68	−1800	10,800	−177	2006	700	340
Ship 30	12,490	32	2500	80	−2000	8000	−485	1983	800	330
Ship 31	2600	25	760	30	−370	4500	−110	2020	280	230
Ship 32	7350	30	1600	48	−1200	7000	−190	2009	400	320

Source: authors.

presents 32 real warships from navies worldwide, evaluated according to the 10 quantitative criteria previously described and presented.

Although a comparison of ships of the same class is more direct and specific, we chose to evaluate a variety of ships of different sizes and characteristics, providing a more comprehensive and generalizable analysis. This study aims to provide a methodology that can be applied to different ships and missions, from large-scale ocean operations to coastal patrols or in difficult-to-access areas. The analysis presented here can be adapted to different scenarios, with certain mission requirements and expanding the criteria to cover additional aspects of ship operation.

The complexity of decisions within the military context necessitates a comprehensive analysis incorporating various facets of military operations, including strategy, operations, logistics, technology, and intelligence. In this regard, integrating opinions from advisors with diverse military profiles is crucial to addressing the complex nature of military operations and high-level decision-making. With this objective, we consulted five distinct profiles of military decision-makers, each tailored to specific specializations within the military framework, as depicted in

Table 8. Description of the profiles of the five experts consulted.

ID	Profile	Background	Specialization	Function
DM 1	Operational Tactics Advisor	Experience in terrestrial or naval combat operations	Specialized in the tactical deployment of troops and equipment in various combat scenarios	Provides insights into effective combat strategies, enemy engagement tactics, and real-time decision-making in battlefield conditions
DM 2	Logistics and Supply Chain Advisor	A senior officer with extensive experience in military logistics	Specialist in supply chain management, logistics planning, and resource allocation	Ensures efficient distribution of resources, from munitions to medical supplies, essential for sustained military operations
DM 3	Intelligence and Security Advisor	Former intelligence officer with field experience in intelligence gathering and analysis	Specialized in cybersecurity, counter-intelligence, and the acquisition of actionable intelligence	Advises on threat assessment, enemy capabilities, and security protocols to protect sensitive information and operations
DM 4	Technology and Innovations Advisor	Military engineer focused on emerging technologies	Expert in integrating cutting-edge technology into military operations	Guides the adaptation of new technologies to enhance operational effectiveness and strategic advantage
DM 5	Strategic Planning and Policy Advisor	Senior military strategist with experience in defense policy formulation	Skilled in long-term strategic planning, policy development, and military diplomacy	Provides high-level strategic guidance policy recommendations and develops frameworks for international co-operation

Source: authors.

:

The aggregation of experts from distinct domains offers a breadth of perspectives and experiences that are pivotal for a thorough understanding of the potential impacts of each decision. Each expert, whether specialized in operational tactics, logistics, intelligence, technology, or strategic planning provides invaluable insights that aid in mitigating the risks associated with individual biases in the decision-making process.

Figure 3 illustrates the proposed methodology in this paper for assessing warships.

Phase 1 begins with a dataset of warships from navies around the world, followed by qualitative criteria evaluation by DMs across various domains such as Operational Tactics, Logistics and Supply Chain, Intelligence and Security, Technology and Innovations, and Strategic Planning and Policy, culminating in group consensus. Phase 2 involves adapting PCA concepts, conducting validation tests, integrating qualitative and quantitative data, performing a performance evaluation of ships, ranking the ships, and conducting a sensitivity analysis considering the DMs’ individual and consensus opinions, leading to the final decision.

We used the R programming language to enable the implementation of the SAPEVO-PC method, as illustrated in the next section.

4. Results and Discussion

This section will detail the results according to the proposed methodology, which follows the SAPEVO-PC steps presented previously. Initially, we have the qualitative phase of the analysis.

4.1. Obtaining Criteria Weights Using a Qualitative Scale

The qualitative analysis began by determining the criteria to be considered in the decision in Step 1, as shown in Table 2. All preference assignments presented in this section were obtained through interviews with military experts. Initially, the Positive Ideal Element (PIE) and Negative Ideal Element (NIE) were identified through a strategic analysis of the criteria. DM 1, an Operational Tactics Advisor, defined the armament as PIE and the cost as NIE.

Table 9. Establishment of preferences for obtaining weights by DM 1.

Criteria	A+ (ARM)	A− (CST)	A*	AN	WS
Displacement	3	−3	0	0.57	9.55%
Speed	4	−5	−1	0.64	10.74%
Cargo Capacity	2	−4	−2	0.71	11.93%
Armament	1	−7	−6	1.00	16.71%
Cost	7	1	8	0.06	0.95%
Range	5	−4	1	0.50	8.35%
Crew	6	−2	4	0.29	4.77%
Year of Commissioning	3	−5	−2	0.71	11.93%
Missile Range	3	−6	−3	0.79	13.13%
Radar Range	3	−5	−2	0.71	11.93%

Source: authors.

summarizes his preferences. The A⁺ column represents the preference values assigned by DM 1 for the criterion Armament concerning the others. For this decision-maker, Armament was considered moderately more important (degree 3) compared to displacement, considerably more important (degree 4) than the speed criterion, slightly more important (degree 2) than cargo capacity, equally important (degree 1) to itself, absolutely more important (degree 7) than cost, very important (degree 5) in terms of range, presents very great importance (degree 6) for the crew criterion, and is moderately more important (degree 3) than the criteria year of commissioning, missile range, and radar range.

On the other hand, the A⁻ column represents how much the NIE, the cost criterion, is less important than the others, following the same scale of preferences but with symmetric values.

The values A*, A_N, and W_S were obtained respectively by Equations (4) to (6). In the analysis of the weights assigned to the criteria (W_S) by the SAPEVO-PC method, it was observed that the Operational Tactics Advisor (DM 1) significantly prioritized aspects crucial for effectiveness and readiness on the battlefield. Armament, with the highest weight of 16.71%, stood out as the criterion of greatest importance. This focus on armament reflects the advisor’s priority to maximize offensive and defensive capabilities, which is essential for military operations that rely on technological superiority and firepower to achieve tactical objectives effectively. The emphasis on this criterion suggests an approach favoring investment in advanced technologies and weapon systems that can offer a decisive advantage during confrontations.

Besides armament, the Missile Range and Radar Range criteria, with weights of 13.13% and 11.93%, respectively, also receive considerable attention. These criteria are fundamental for the capability of force projection over long distances and advanced surveillance, allowing the tactical advisor to plan and execute operations with a comprehensive understanding of the theatre of operations. Missile Range ensures that forces can precisely strike strategic targets, while Radar Range secures early threat detection, which is essential for maintaining operational security and rapid response capability in conflict environments.

On the other hand, the Cost criterion, with the lowest weight of just 0.95%, reflects the advisor’s stance that cost considerations, while important, are secondary to maintaining superior military capability. This low weight indicates a willingness to invest in cutting-edge equipment and technology, regardless of costs, to ensure that the units under his command have the best tools available. This approach is typical in contexts where operational effectiveness and tactical supremacy are valued more than resource economization, aligning defense investments with long-term strategic goals and readiness requirements.

The results obtained with the SAPEVO-PC method corroborate its reliability in generating weights that effectively represent and quantify the decision-maker’s perceptions. The percentage weights assigned align with the decision-maker’s strategic priorities and tactical considerations, particularly in areas critical to military operations, such as armament, missile range, and radar capabilities. By accurately reflecting these preferences in a quantifiable form, the SAPEVO-PC method increases reliability and adds significant relevance to the decision-making process. This alignment ensures that the derived weights are not arbitrary but are based on the decision-maker’s strategic knowledge and operational realities, thus validating the usefulness and accuracy of the method in complex decision environments.

Continuing with the analysis using the SAPEVO-PC method, the same procedures were carried out for the other DM, with the A⁺, A⁻, and W_s value_s consolidated in

Table 10. Weights obtained by SAPEVO-PC, considering individual and group opinions.

	DM 2			DM 3			DM 4			DM 5			Group
Criteria	A+	A−	Ws	A+	A−	Ws	A+	A−	Ws	A+	A−	Ws	WG
Displacement	2	−5	13.29%	3	−3	8.77%	4	−3	8.56%	7	1	0.95%	8.07%
Speed	2	−6	14.49%	2	−6	13.78%	4	−6	12.22%	5	−2	5.97%	12.05%
Cargo Capacity	1	−7	16.91%	3	−4	10.03%	6	−4	7.33%	3	−4	10.74%	11.01%
Armament	3	−5	12.08%	2	−5	12.53%	1	−7	17.11%	2	−6	14.32%	13.71%
Cost	2	1	6.04%	7	−4	5.01%	7	1	0.98%	1	−7	16.71%	6.75%
Range	3	−4	10.87%	2	−6	13.78%	3	−4	11.00%	5	−3	7.16%	11.32%
Crew	6	−2	4.83%	6	1	1.00%	6	−2	4.89%	2	−5	13.13%	4.97%
Year of Commissioning	4	−5	10.87%	3	−5	11.28%	2	−6	14.67%	4	−3	8.35%	11.29%
Missile Range	7	1	0.97%	4	−3	7.52%	2	−4	12.22%	5	−1	4.77%	6.60%
Radar Range	5	−5	9.66%	1	−7	16.29%	2	−3	11.00%	6	−2	4.77%	11.60%

Source: authors.

.

Analyzing the weights obtained with the application of SAPEVO-PC for the other decision-maker’s values is consistent with the profile of each one, and these were also verified. The W_S results for DM 2 reflect a clear prioritization of carrying capacity (16.91%), indicative of a logistics and supply chain consultant’s focus on resource management and distribution efficiency. This consultant also gave significant weight to speed (14.49%) and displacement (13.29%), suggesting an appreciation of logistic vehicles or vessels’ rapid deployment and physical capabilities. Lower weights for costs (6.04%) and crew (4.83%) imply a strategic focus on operational capacity over economic or labor considerations, aligning with a logistics-oriented approach where ensuring material availability and logistical readiness are key criteria.

For DM 3, the radar range received the most weight (16.29%), highlighting the intelligence and security consultant’s emphasis on surveillance and early threat-detection capabilities. It was closely followed by speed and range (13.78%), reflecting the importance of rapid response and broad operational reach in intelligence and security operations. DM 4, with a solid background in technology, attributed the greatest weight to weapons (17.11%), illustrating a significant focus on integrating advanced weapons systems, probably reflecting the value attributed to technological superiority. The year of commissioning also receives a high weight (14.67%), indicating a preference for newer and possibly more technologically advanced platforms. The relatively balanced weights in other criteria, such as missile range (12.22%) and radar range (11.00%), demonstrate a comprehensive approach that values offensive capabilities and surveillance technology.

The strategic planning and policy consultant (DM 5) emphasized costs (16.71%), suggesting a strategic priority on fiscal efficiency and budget management. It was followed by the crew (13.13%), focusing on human resource management and perhaps the importance of well-trained personnel in implementing strategic plans. Lower weights for critical operational capabilities such as missile range (4.77%) and radar range (4.77%) may indicate a broader, more policy-oriented approach where specific technical details are less prioritized against overarching strategic and political considerations.

Adding the individual weights together, we arrive at the weight obtained by the group. In general, weapons have the highest average weight (13.71%), indicating a collective recognition of the importance of firepower and defense capabilities in operational scenarios. This is closely followed by the year of commissioning (11.29%) and range (11.32%), suggesting a shared value placed on modernity and operational range. Consistency in weighting between different consultants for criteria such as radar range (11.60%) and payload capacity (11.01%) further highlights common strategic considerations that transcend individual specializations, highlighting a balanced approach to military personnel capabilities and readiness for group decision-making.

To finalize the analysis of the weights obtained through the SAPEVO-PC method, it is essential to highlight the multi-decision-maker characteristic that the method facilitates. This feature allows for individualized analysis and group aggregation, significantly enhancing the decision-making process’s reliability, transparency, and flexibility. By employing this method, individual decision-makers can input their specific preferences and priorities, ensuring that each one’s perspective is thoroughly represented and considered. Subsequently, these individual assessments are aggregated to form a group consensus, which reflects a collective decision.

This dual approach ensures that each decision-maker’s views are considered, thus increasing the personal relevance and acceptance of the decision outcomes, but also leverages the diverse expertise and insights of the group to form a more balanced and comprehensive decision. Such a methodology is particularly beneficial in complex decision environments where multiple stakeholders are involved and integrating diverse viewpoints can significantly enhance strategic outcomes.

Overall, the SAPEVO-PC method’s capacity to accommodate multiple decision-makers strengthens the decision-making process and fosters a collaborative environment where decisions are not only shared but are also more likely to be supported and implemented effectively due to the inclusive process. This approach inherently boosts the decision’s legitimacy and efficacy, making it a robust tool in strategic decision-making.

4.2. Quantitative Analysis

Before analyzing the alternatives, evaluating the correlation matrix between the original variables is important. Correlation analysis allows us to identify which variables are highly correlated and we can combine them into a single factor [84]. In this sense, the heat map of correlations is a valuable tool for visualizing these correlations and identifying which variables are highly correlated. Figure 4 shows the heat map of the correlations of the 10 variables evaluated in this study.

The correlation matrix uses a color scale to represent the magnitude of correlations, with lighter colors indicating stronger correlations (positive or negative). We highlight the high correlation (0.916) between displacement and load capacity by analyzing the main patterns observed. This correlation makes sense, as ships with larger displacement tend to have greater carrying capacity for weapons, supplies, and other equipment needed for prolonged operations. Other correlations occur between the crew criterion and the displacement (0.93) and load capacity (0.93) criteria, as larger ships generally require more crew for operation and maintenance, justifying the correlation.

The cost is highly correlated with the load capacity. Ships with greater cargo capacity tend to be more expensive due to the need for more materials, advanced technology and infrastructure to support the greater weight and volume. Furthermore, autonomy and load capacity have a high and positive brightness. This is justified because ships with greater cargo capacity can carry more fuel and supplies, allowing greater autonomy on long-range missions. The speed shows a negative shine with the gearbox, although weak. It may indicate that heavier ships tend to be slower due to their weight and inertia.

The insights generated by the heatmap make much sense when evaluating the characteristics of warships, which gives relevance and reliability to the dataset analyzed in this article. The strong positive correlation between variables such as displacement, cargo capacity, armament, and crew aligns with the expectation that larger ships are better equipped and require more personnel for operation and maintenance. Likewise, the correlations observed between cost, autonomy, and cargo capacity reflect an intuitive understanding that more robust ships with greater autonomy require significant investments. These coherent correlation patterns provide a solid basis for applying SAPEVO-PC, ensuring that the factors formed are representative and meaningful. This overview validates data quality and improves confidence in subsequent analysis, resulting in robust and applicable insights for strategic decision-making in naval planning.

This diagnosis is a preliminary indication of how factor analysis can group variables into a given factor without substantial loss of their variances. Next, the general adequacy of the factor analysis is verified by applying KMO statistics and Bartlett’s sphericity test. These statistics are essential in determining whether factor analysis is appropriate for the data and whether the variables are highly correlated enough to form significant factors.

Based on the result of the KMO statistic obtained (0.77), the overall adequacy of the factor analysis can be considered satisfactory [54]. The next step is to evaluate the result of Bartlett’s sphericity test, which verifies the null hypothesis that the correlation matrix is an identity matrix.

From the result of Bartlett’s statistic, with a significance level of 0.05 and 45 degrees of freedom, we can say that the Pearson correlation matrix is statistically different from the identity matrix with the same dimension because the p-value of Bartlett’s test (2.47 × 10⁻³⁷) was lower than the established significance level. Thus, we rejected the null hypothesis, meaning a significant correlation exists between the observed variables. Figure 5 shows how much each original variable contributes to forming the generated main components (PCs).

The graph is divided into facets, each representing a main component (PC1 to PC10). Each facet bar represents an original variable’s contribution to that principal component.

Analyzing PC1, we observed that variables such as Crew, Autonomy, Displacement, and Capacity had significant factor loadings, indicating that PC1 can be interpreted as a component that captures the dimension of size and operational capacity of ships. Furthermore, we verified that, except for speed and year of commissioning, the other variables are represented by PC1.

On the other hand, when evaluating PC2, Armament and Year of Commissioning had positive and significant loadings, suggesting that PC2 is capturing aspects related to the ships’ military power and technological level. PC3 presents highlighted loads for Speed, a criterion that does not present a significant load with PC1 or PC2. In our study, components beyond the third (PC3) were considered less significant in explained variance, although they may still contain specific insights into less evident variations in the data.

The first principal components (PC1, PC2, and PC3) explain most of the variance in the data (around 80%), being the most important for understanding the main dimensions of variation between ships.

The factor-loading graph clearly shows how each variable contributes to the main components. This visualization makes identifying key dimensions of variation in ship data easier and highlights important relationships between variables. Analysis of factor loadings helps to interpret the main components and provides a solid basis for comparative assessment of ships based on multiple criteria.

The next step is to define how many significant components to consider in the analysis. In this work, we used the Kaiser criterion to select eigenvalues greater than 1. In this context,

Table 11. Eigenvalues and variances of each Principal Component.

Main Components	PC1	PC2	PC3	PC4	PC5	PC6	PC7	PC8	PC9	PC10
Eigenvalues	5.55	1.23	1.12	0.91	0.44	0.32	0.28	0.08	0.05	0.02
Proportion of Variance	0.56	0.12	0.11	0.09	0.04	0.03	0.03	0.01	0.00	0.00
The Proportion of Cumulative Variance	0.56	0.68	0.79	0.88	0.93	0.96	0.99	0.99	1.00	1.00

Source: authors.

presents the eigenvalues and variances associated with each Principal Component.

Evaluating the eigenvalues and considering the Kaiser criterion, we observed that only the first three components (PC1, PC2, and PC3) have eigenvalues greater than or equal to 1. From this result, we conclude that the first three components are relevant to explain the variability of the data. Therefore, we can proceed with extracting the factors from these components. Next, we obtained the factorial scores, as shown in

Table 12. Factorial scores.

Criteria	PC1	PC2	PC3
Displacement	0.155423	−0.24365	−0.20246
Speed	−0.0036	−0.0486	0.805706
Cargo Capacity	0.171826	−0.10539	−0.07351
Armament	0.093602	0.511918	0.267572
Cost	−0.16459	0.093697	−0.02715
Range	0.145684	0.240196	−0.05238
Crew	−0.15912	0.305348	0.01627
Year of Commissioning	−0.07837	0.411382	−0.31889
Missile Range	0.137907	0.151027	0.101244
Radar Range	0.137278	0.351931	−0.08195

Source: authors.

.

The factorial scores generate the factors in a linear model from the original variables. The scores are the eigenvectors divided by the square root of the respective eigenvalues. The correlation between the original variables and the factors would be the factor loadings. In general, the values of the factorial scores corroborate and quantify the graphical analyses performed in Figure 5.

At this point, the qualitative data (weights obtained in phase 1) are integrated with the factor scores calculated in the quantitative phase, according to Equation (23). To illustrate the multi-decision nature, let us consider the WG weights presented in Table 10. With this, we can calculate the values corresponding to the three factors of each observation in the sequence, already adding the weights obtained by SAPEVO-PC in the qualitative phase and the factor scores from the quantitative phase.

Table 13. Factors extracted for each observation.

Ships	PC1	PC2	PC3
Ship 1	0.021127	0.099273	0.004603
Ship 2	0.355221	−0.20351	0.088186
Ship 3	0.096416	−0.05724	−0.13233
Ship 4	0.037867	−0.05422	−0.06278
Ship 5	0.102727	−0.08085	0.017531
Ship 6	0.153825	0.012533	0.080186
Ship 7	0.126845	−0.01277	0.119095
Ship 8	0.053404	0.271553	0.049097
Ship 9	−0.071	−0.0403	0.083082
Ship 10	−0.00152	0.15239	0.060968
Ship 11	−0.04307	0.039131	0.03912
Ship 12	−0.0378	0.047597	−0.00613
Ship 13	−0.04902	−0.02064	−0.01387
Ship 14	−0.03501	0.065949	−0.01822
Ship 15	−0.04395	0.012582	−4.1 × 10−5
Ship 16	−0.04903	0.033232	−0.02167
Ship 17	−0.02388	0.074605	−0.00556
Ship 18	−0.04433	0.056257	−0.01864
Ship 19	−0.04093	0.058816	−0.03
Ship 20	−0.06909	−0.0094	0.14294
Ship 21	−0.02175	0.080846	−0.00443
Ship 22	−0.02988	−0.00997	−0.02792
Ship 23	−0.05059	−0.08381	−0.05002
Ship 24	−0.04925	−0.05504	−0.04558
Ship 25	−0.08199	−0.18225	−0.04257
Ship 26	0.014868	0.067119	0.091879
Ship 27	−0.06599	−0.16873	0.005674
Ship 28	−0.08625	−0.13163	−0.0716
Ship 29	0.035279	0.099727	−0.13419
Ship 30	0.047499	0.04325	0.098805
Ship 31	−0.11303	−0.09539	−0.14878
Ship 32	−0.03773	−0.00911	−0.04684

Source: authors.

shows these results.

We emphasize that all the extracted factors present, among themselves, Pearson correlations equal to 0, that is, they are orthogonal to each other [28]. After establishing the factors, it is possible to define the factor loadings, which correspond to the Pearson correlation coefficients between the original variables and each factor.

Table 14. Factor loadings of variables.

Criteria	PC1	PC2	PC3
Displacement	0.86302136	−0.30061206	−0.22759255
Speed	−0.01998727	−0.05995903	0.90574198
Cargo Capacity	0.95410681	−0.13003317	−0.08263133
Armament	0.51974595	0.63160448	0.30079325
Cost	−0.91390286	0.11560292	−0.03051812
Range	0.80894478	0.29635372	−0.05888141
Crew	−0.88355602	0.37673853	0.01828981
Year of Commissioning	−0.43514459	0.50756273	−0.35848847
Missile Range	0.76576306	0.18633711	0.11381451
Radar Range	0.76226830	0.43421274	−0.09212555

Source: authors.

presents the factor loadings of the variables analyzed in the present study.

It is possible to observe that 6 of the 10 variables present higher correlations with the first factor. In comparison, the criteria Armament and Speed present higher correlations with the second and third factors. This higher correlation proves the need for three factors for all variables to share significant percentages of variance. With the factor loadings obtained, we can calculate the commonalities, which correspond to the percentage of variance shared by the 10 original variables for the formation of each factor.

Table 15. Commonalities of each original variable.

Criteria	Commonalities
Displacement	0.8869719
Speed	0.8243631
Cargo Capacity	0.9340564
Armament	0.7595367
Cost	0.8495138
Range	0.7456842
Crew	0.9229377
Year of Commissioning	0.5754847
Missile Range	0.6340683
Radar Range	0.7780808

Source: authors.

presents the commonalities of each original variable of the dataset:

Evaluating the commonality values referring to the three extracted factors, we observed relatively high values for 8 of the 10 criteria (above 74%). If we consider the factor loadings of all factors, all commonalities would be equal to 1. We can draw a graph for the first three factors, known as a loading plot. We plotted the factor loadings of each variable on each of the orthogonal axes. Thus, Figure 6 shows the factors PC1 and PC2, represented respectively on the x– and y–axes.

Through the analysis of the loading plot, the behavior of the correlations is apparent. Except for the Speed and Year of Commissioning, we noticed that all other variables presented a correlation above 0.5 with PC1 (x–axis) and PC2 (y–axis). On the other hand, the number of Helicopter Carriers presents the highest correlation with the PC2 (axis of the ordinates).

To obtain the final ranking of alternatives, the SAPEVO-PC method uses the criterion of weighted sum and order, according to Equation (26).

Table 16. Ranking of ships considering the opinion of five military experts with different profiles.

Ships	DSP	SPD	CAP	ARM	CST	RNG	CRE	YOC	MSL	RAD	p	Ranking
Ship 1	14,564	30	1000	80	−4500	9000	−158	2016	100	300	0.024	10
Ship 2	100,000	31.5	9000	90	−8000	12,000	−5500	1975	800	350	0.182	1
Ship 3	65,000	25	5000	50	−4100	10,000	−1600	2017	600	320	0.032	7
Ship 4	45,400	28	4000	60	−2300	7000	−1600	2013	450	280	0.007	12
Ship 5	58,600	29	4500	70	−3200	8000	−2000	1990	700	300	0.049	5
Ship 6	28,000	32	4200	80	−5000	11,000	−1500	1988	900	350	0.096	2
Ship 7	12,500	34	4100	80	−4500	10,000	−1200	1982	800	320	0.082	3
Ship 8	12,000	30	3800	112	−1500	9000	−310	2019	600	340	0.069	4
Ship 9	3500	40	1200	50	−600	6000	−75	2008	400	250	−0.035	26
Ship 10	10,000	30	1300	96	−1400	8300	−300	2007	500	270	0.025	9
Ship 11	5520	30	1400	80	−1000	4500	−300	2002	250	260	−0.015	16
Ship 12	6000	30	2000	68	−800	6000	−200	2012	500	270	−0.016	17
Ship 13	5690	29	1800	60	−900	4000	−230	2003	400	280	−0.031	25
Ship 14	7000	30	2000	68	−1000	5000	−230	2017	500	300	−0.013	15
Ship 15	5800	29	1800	68	−900	4000	−230	2004	400	280	−0.023	21
Ship 16	7200	30	1800	64	−950	4000	−190	2013	300	310	−0.026	23
Ship 17	6600	30	1900	70	−700	8000	−300	2011	500	290	−0.005	14
Ship 18	3200	28	1500	70	−500	7000	−180	2008	400	270	−0.020	19
Ship 19	5290	26	1600	72	−600	5000	−146	2006	500	290	−0.019	18
Ship 20	3100	44	1200	60	−700	6000	−80	2010	300	240	−0.023	22
Ship 21	5400	30	1800	70	−1000	8000	−200	2013	600	280	−0.003	13
Ship 22	8000	32	2200	48	−1200	7000	−190	2010	450	320	−0.021	20
Ship 23	6050	29	1600	40	−1200	5000	−200	2002	500	290	−0.044	28
Ship 24	5800	28	1700	48	−1000	4500	−200	2002	500	290	−0.039	27
Ship 25	4200	29	1200	30	−800	4000	−180	1990	300	250	−0.073	31
Ship 26	9500	30	2000	90	−1800	4000	−350	1989	700	310	0.027	8
Ship 27	4100	29	1200	40	−600	4000	−180	1977	400	260	−0.057	29
Ship 28	3600	25	1000	40	−300	5000	−150	1992	300	230	−0.072	30
Ship 29	21,300	18.8	3000	68	−1800	10,800	−177	2006	700	340	0.017	11
Ship 30	12490	32	2500	80	−2000	8000	−485	1983	800	330	0.043	6
Ship 31	2600	25	760	30	−370	4500	−110	2020	280	230	−0.091	32
Ship 32	7350	30	1600	48	−1200	7000	−190	2009	400	320	−0.027	24
WG	8.07%	12.05%	11.01%	13.71%	6.75%	11.32%	4.97%	11.29%	6.60%	11.60%

Source: authors.

presents the ranking of the 32 ships analyzed in this study, considering 10 quantitative criteria and the opinion of five military experts with different profiles.

Evaluating the results obtained, we verified that Ship 2, which occupies the first position, presents an excellent and exceptional performance across several key criteria. It is extremely large, with a displacement of 100,000 tons, providing a substantial payload capacity of 9000 tons and a good armament rating of 90. These characteristics contribute significantly to its high performance score. Furthermore, Ship 2 has a remarkable speed of 31.5 knots and an excellent range of 12,000 nautical miles, making it highly versatile and capable of long-duration missions. Although it has a relatively high cost and crew, the ship’s capabilities justify the investment.

Ship 6 ranks second, mainly due to its high speed of 32 knots and excellent range of 11,000 nautical miles. Its weapon rating of 80 is also strong, contributing to its defensive and offensive potential. Ship 7, which occupies third position, stands out for its speed of 34 knots, one of the highest among major ships, combined with a moderate displacement of 12,500 tons and a carrying capacity of 4100 tons. Its armament rating of 80 is robust, complementing its combat capabilities. Despite the high cost, the balance between speed, good autonomy and range, and missile range guarantees its final ranking.

Ship 8, in fourth position, is notable for its very strong armament of 112, the highest among the capital ships. It maintains good operational flexibility with a moderate displacement of 12,000 tons and a speed of 30 knots. Her 3800-ton payload and range of 9000 nautical miles are adequate, and the relatively low cost and a small crew are advantageous. Being a modern ship built in 2019, it likely incorporates advanced technology. Ship 5, in fifth place, combines a high displacement with good speed and load capacity. Additionally, it has a range of 8000 nautical miles. Despite the high cost and large crew, the general capabilities, including a good missile range, justify its high position in the ranking.

Regarding the worst place in the ranking, we highlight Ship 31, which occupies the last position as it presents a poor performance in terms of displacement, speed, load capacity, and armament. Despite being a new ship, it has low relative autonomy, and low missile and radar range, which are reasons that justify its poor performance score. Ship 25 is in the 31st position and shows limited displacement and cargo capacity capabilities. Despite good speed, its armament rating and range are reportedly low. Ship 28, in the 30th position, has low displacement, armament, load capacity, and speed, despite an advantageous cost.

Analysis of the final ranking reveals that the flagships exhibit strong performance across several criteria, especially in displacement, carrying capacity, armament, speed, and range. These ships justify their high costs and large crews through their robust capabilities and advanced features. On the other hand, ships in the last positions demonstrate limited capabilities in key areas such as displacement, cargo capacity, armament, and range. The PCA-based methodology, incorporating the weighted sum and ordering criteria, provides a comprehensive and reliable ship performance assessment. This approach ensures that the assessment is grounded in quantifiable data and robust statistical analysis, making it a valuable tool for strategic decision-making in naval engineering and military planning.

A sensitivity analysis was carried out to evaluate the robustness and reliability of the results obtained, considering the weights obtained for each decision-maker, detailed in Table 9 and Table 10. This procedure allows the checking of the consistency and stability of the final ranking in the face of variations in the criteria weights, ensuring that decisions are well founded and reflect multiple perspectives.

Table 17. Individual and consensus rankings, considering the weights obtained by SAPEVO-PC.

Ships	DM 1	DM 2	DM 3	DM 4	DM 5	Consensus
Ship 1	11	8	11	13	10	11
Ship 2	1	1	1	1	1	1
Ship 3	8	7	8	10	7	7
Ship 4	15	12	15	14	12	14
Ship 5	6	5	6	6	5	6
Ship 6	2	2	2	2	2	2
Ship 7	3	3	3	4	3	3
Ship 8	4	4	4	3	4	4
Ship 9	23	20	23	25	26	25
Ship 10	10	9	10	8	9	10
Ship 11	25	21	25	22	16	23
Ship 12	19	19	19	17	17	18
Ship 13	26	26	26	26	25	26
Ship 14	17	17	17	15	15	16
Ship 15	24	25	24	23	21	24
Ship 16	21	22	21	24	23	22
Ship 17	12	13	12	12	14	12
Ship 18	22	23	22	20	19	21
Ship 19	20	24	20	18	18	20
Ship 20	18	14	18	19	22	17
Ship 21	13	15	13	11	13	13
Ship 22	14	16	14	16	20	15
Ship 23	28	28	28	28	28	28
Ship 24	27	27	27	27	27	27
Ship 25	30	30	30	30	31	30
Ship 26	9	10	9	7	8	8
Ship 27	29	29	29	29	29	29
Ship 28	31	31	31	31	30	31
Ship 29	7	11	7	9	11	9
Ship 30	5	6	5	5	6	5
Ship 31	32	32	32	32	32	32
Ship 32	16	18	16	21	24	19

Source: authors.

presents the individual and consensus rankings, considering the weights obtained by SAPEVO-PC.

The analysis results show high coherence between the decision-makers and consensus rankings. It is observed that Ship 2 consistently maintained the first position in all rankings, indicating its superior performance in critical criteria, regardless of variations in the weights assigned by decision-makers. Likewise, Ship 6 remained second in all rankings, standing out as another high-performance alternative. Ships 7, 8, and 5 also showed small variations in their positions but consistently remained among the best classified, reinforcing the robustness of the evaluation criteria used.

On the other hand, the ships in the last positions also showed remarkable stability. Ship 31 maintained the last position in all rankings, followed closely by ships 28, 27, and 25, which were also consistently in the last positions. This stability at the ends of the ranking suggests that, regardless of variations in criteria weights, there is clear stability about which ships outperform and underperform. These observations give relevance and reliability to the proposed methodology, showing that the assessments are robust and well founded, even from different decision-makers’ perspectives.

The individual and group rankings were presented to military experts and validated by them. According to experts, the methodology proposed in this article effectively represented the ranking by merging their subjective perceptions with quantitative performance data for each ship. This validation reinforces the reliability and relevance of the SAPEVO-PC method, highlighting its practical usefulness in complex decision-making scenarios in marine science and engineering.

SAPEVO-PC provides a solid framework for warship selection, although it is most useful in scenarios with a significant amount of data and requiring expert input, which may not always be feasible. Compared to classical methods such as AHP or TOPSIS, SAPEVO-PC provides a more integrated and reliable approach by combining MCDM and ML, which increases decision accuracy but at a higher cost in terms of complexity and resource requirements.

5. Conclusions

This paper explored the complexity inherent in problems related to marine science and engineering, highlighting the need for effective approaches to support decision-making in challenging environments. The SAPEVO-PC methodology, presented as an evolution in the SAPEVO family, demonstrated its significant usefulness in evaluating military ships, providing an innovative way of assigning weights to criteria from the decision-maker’s perspective. Introducing Ideal Positive Elements (PIE) and Ideal Negative Elements (NIE) was crucial to this approach, allowing the consideration of qualitative preferences in complex scenarios.

The application of the SAPEVO-PC method, focusing on simplifying the comparison process and substantially reducing the number of peer assessments, proved valuable in dealing with the wide range of criteria and alternatives involved in the proposed case study, consisting of 32 alternatives and 10 criteria. The efficiency and effectiveness of these methodologies offer a notable contribution to the field of decision-making in challenging contexts where precision and agility are imperative. This approach addresses the limitations inherent in traditional multi-criteria methods, such as the high number of peer assessments required, which can be costly and prone to inconsistencies.

Furthermore, integrating MCDM with ML algorithms has significantly advanced analysis and decision-making in complex marine science and engineering contexts. PCA concepts made it possible to reduce the dimensionality of the data, transforming ten original variables into three main factors without losing the essence of the information. This combined approach simplifies the assessment process and increases the accuracy and objectivity of results, facilitating interpretation and practical application. Integrating qualitative data with quantitative ML algorithms offers a more robust and comprehensive analysis, allowing military decision-makers to evaluate the various criteria involved in ship selection with greater confidence and efficiency.

However, it is essential to emphasize that no single metric can fully encompass ship performance, as a comprehensive assessment requires the analysis of multiple factors that influence the quality and tactical employment of these naval assets. We also highlight that only quantitative performance data were considered by applying the SAPEVO-PC method, excluding aspects such as the technological level of these means and the qualification of the crew, for example. Future research could expand the analyzed criteria to include operability in adverse weather conditions, robustness and redundancy of vital systems, damage control capabilities, and a more detailed analysis of each ship’s armaments.

Based on the results obtained, it can be stated that the SAPEVO-PC method has vast potential that can be explored in several other areas besides naval science and engineering. Future studies could focus on adapting and optimizing SAPEVO-PC for different contexts and types of data, exploring its applicability in emerging areas such as Artificial Intelligence, smart cities, and IoT. Furthermore, studies can investigate the combination of SAPEVO-PC with other ML and Big Data analytic techniques to further improve the accuracy and efficiency of MCDM decisions.

In short, this study makes a significant contribution through the methodology developed, which has the potential to be applied in different situations, going beyond the exclusive emphasis on the resulting ranking. The methodology represents a significant advance in promoting more informed, faster, and reliable decisions, contributing to the optimization of resources and the effectiveness of operations in decision-making environments.