SAPEVO-H² Multi-Criteria Modelling to Connect Decision-Makers at Different Levels of Responsibility: Evaluating Sustainability Projects in the Automobile Industry

Abstract

Decision-making in complex environments, especially sustainable ones, requires flexible methodologies to handle multiple criteria and stakeholder perspectives. This study introduces the SAPEVO-H² method (Simple Aggregation of Preferences Expressed by Ordinal Vectors—Hybrid and Hierarchical), an extensive model from the SAPEVO family, which offers multi-criteria analysis through a hierarchical structure of variables evaluated by groups partitioned into levels concerning their respective responsibilities. The proposal allows flexible analysis, considering inputs through ordinal and cardinal information. The validation of the methodology is demonstrated through a case study involving an automobile manufacturing company, which focuses on prioritizing sustainability projects based on multiple objectives aimed at minimizing polluting gas emissions. Within a hierarchical structure of five levels, the individual level results are presented. In addition, a sensitivity analysis is applied, exposing the most sensitive variables to changes concerning the highest levels. Then, we discuss the main contributions and limitations concerning the mathematical proposal and the conclusions and proposals for future work.

1. Introduction

Deciding on the sustainable sphere is characterized by high and unique complexity resulting from the interdependence between environmental, social, legislative, and economic variables [1]. As exposed in [2], the understanding and structuring of this complexity is essential for the development of models for decision support, which effectively capture the nuances of the perceptions and preferences of the stakeholders involved in the decision-making processes from strategic to operational levels [3].

When it comes to organizational development regarding the sustainability variable as the main point of strategic advancement in a globalized world, the business direction in prioritizing scientific and technological projects represents a challenging task that unfolds at multiple organizational levels [4]. However, considering the complexity of the alignment between variables and perceptions of preference in the face of these unknowns, the correct structuring and evaluation in sustainable scenarios require robust technical support [5]. The complexity of the interactions between variables requires advanced analytical tools for modeling and assessment, highlighting the need for innovative methods through mathematical, statistical, and computational support [6].

In this scenario, Multi-Criteria Decision Analysis (MCDA), an operational research area, provides methodologies that enable the decision-making process to be more rational and efficient, making possible the structuring, analysis, and decision-making in the spheres of public and private organizations [7], favoring transparency and information in the analytical process [8], especially for cases in which sustainability is the main focus of decision-making, as this study seeks to address [9]. Whenever there is a need for strategic direction, optimization, acquisition, and feasibility of technologies, MCDA models are great allies of decision-making, treating multiple variables in favor of transparent and assertive decision-making in complex environments [10].

Emerging as a central problem to validate the axiomatic model, this study addresses a central research question: how to enable, through mathematical modelling, the integration of multiple decision-makers’ perspectives across distinct hierarchical levels of leadership? The objective is to structure variables systematically, evaluate alternatives rigorously, and prioritize sustainability projects. Such integration requires robust analytical support to ensure coherence and effectiveness throughout the organizational decision-making chain.

In response to the question posed, this study is based on the exploration of a multi-criteria model for decision aid based on the evaluation of complex variables at different organizational levels, enabling a group evaluation in which the stakeholders are distributed in a hierarchical structure of objectives, criteria, and alternatives, in this scenario projects, to be analyzed and compared in favor of transparent and substantial decision-making. In this context, as an axiomatic model, a hierarchical approach, Simple Aggregation of Preferences Expressed by Ordinal Vectors—Hybrid and Hierarchical (SAPEVO-H²), will be explored and detailed in the following sections of this study. The SAPEVO-H² method, an evolution of the SAPEVO family of methods, was first presented in [11]. In this paper, we explore the final version of the mathematical modelling, presenting some advancements compared with previous versions, including improvements in the axiomatic structure, resulting in the final version of the model.

Due to increasing regulatory pressures, consumer awareness, and environmental concerns, sustainability has become a central strategic priority in the automobile industry. Evaluating sustainability projects is critical for balancing economic performance with environmental and social responsibilities [12]. Given the multifaceted criteria, studies underscore the complexity of prioritizing initiatives to reduce emissions, optimize resource use, and advance clean technologies. Despite various multi-criteria approaches for sustainability assessment, there remains a gap in methodologies that integrate hierarchical decision structures with multiple evaluators of varying expertise [13]. Addressing this gap, the study uses the SAPEVO-H² method to evaluate sustainability projects, thereby advancing decision-making practices in this evolving industry.

This article is organized into six parts. The introduction is presented in this section. Section 2 presents the literature review on sustainability from the perspective of multi-criteria decision models. Section 3 exposes the axiomatic basis of the model in exploration, detailing its main technical aspects of evaluating the variables. To validate the SAPEVO-H² method, Section 4 presents a case study in the field of sustainability. Section 5 presents the discussion of the survey, reflecting the feasibility of the model, the differentials of this new approach, and the limitations of the model. Finally, Section 6 presents the study conclusions and proposals for future work.

2. Literature Review

Decision-making in complex environments involves multidimensional considerations beyond technical, organizational, and operational performance. Socioeconomic, environmental, and regulatory factors must be integrated to drive strategic choices [14]. Thus, complex environments require robust and modular methodologies that can deal with the diversity and interdependence of these factors [15].

In a world where the pressure for natural resources is increasing, and environmental concerns are at the center of global agendas, sustainable decision-making emerges as a guiding principle to ensure that present choices do not compromise the ability of future generations to meet their own needs [16]. Defining the scenario of sustainable development in the automotive industry, it is understood that automobile manufacturers face significant challenges when seeking the transition to zero-emission vehicles or at least operate on the minimization of polluting gases [17]. Incorporating scientific and technical approaches allows companies to identify and prioritize technological solutions and business strategies that best align with market needs and sustainability imperatives [18].

In this context, the incorporation of multi-criteria methodologies into the decision-making process allows a holistic evaluation of alternatives [19], facilitating the identification of more effective solutions to the challenges faced by the automotive industry in the transition to more sustainable operations and the construction of products that are more appropriate to the sustainable parameter [20]. In the automotive industry, incorporating the multi-criteria approach in decision-making processes is favorable in structuring and evaluating criteria from a sustainability perspective, such as energy efficiency, environmental impact, safety, costs, and market acceptance, among other parameters of great relevance [21].

2.1. Multi-Criteria Decision-Making and Industrial Sustainability

Decision-making is a complex task that involves multiple perspectives, constraints, and variables [22]. The MCDA is a process used for several decades to support decision-making in different contexts [23], involving methodologies ranging from simple techniques to robust modeling, many of which require computational support for data processing and interaction of decision-makers in the treatment of variables [24].

The MCDA has gained significant traction in industrial contexts as a powerful approach to tackle complex decision problems involving multiple conflicting criteria [25]. In manufacturing, energy, and chemical processing industries, MCDA methods support managers in balancing economic, environmental, and social factors to optimize operational and strategic decisions [26]. These methods enable systematic evaluation and prioritization of alternatives, from technology adoption to resource allocation, by integrating data under uncertainty and stakeholder diversity [27].

In the face of sustainability, incorporating sustainable criteria in decision-making drives the search for innovative solutions [28], considering the immediate interests of the parties involved and the collective well-being and preservation of natural resources [29]. This approach emphasizes the importance of considering the broader context when evaluating alternatives. It drives the search for decision models that align with short-term objectives and long-term goals for a sustainable future [30,31].

Applications of MCDA methods span several industrial sectors beyond automotive manufacturing, including chemical processing, energy production, manufacturing, and waste management [32]. For instance, MCDA has been employed to evaluate renewable energy technologies considering cost, environmental impact, social acceptance, supply chain sustainability to balance efficiency and ecological footprint, and waste treatment to optimize technology selection based on multiple performance metrics [33,34].

As presented by [26], the literature on the analysis of variables in the field of sustainability is broad and constantly growing, with different interpretations and implementations of this concept available to date. In addition, the authors point out that assessments under sustainability aspects can vary considerably, from a micro to a macro scale, which means that the inclusion of various processes cannot always be considered with the same approaches.

Considering the grid of complexity attributed to the problems of the sustainable sphere and the need to provide consensus among the entities participating in the analysis and decision-making process, a given action must be evaluated and directed in a structured, transparent, and reliable way [35]. In this context, the models belonging to the MCDA can meet these needs, supporting the decision-making under discussion [36].

The use of data and information of both qualitative and quantitative nature in sustainability assessments is fundamental, considering that a wide variety of data typologies must be considered [37]. However, the explicit inclusion of qualitative data or mixed information for utility and outranking in the methods should be treated with caution due to the need to manipulate the information at the input stage, so the axiomatic framework for considering the treatment of perceptions should be objective and transparent as to the mathematical logic attributed to the decision model [38].

As noted in [39], among the multi-criteria methods frequently used in sustainability assessments, the Analytic Hierarchy Process (AHP), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE), and Simple Additive Weighting (SAW) stand out. Each method has specific advantages and limitations, but all seek to integrate different evaluation criteria in a structured way. There are notable cases in the automotive industry, the context of validation of this study, where MCDA methods have been applied to improve sustainability [34]. For example, studies have shown how the applications of AHP and PROMETHEE and their combination for the implementation of multi-criteria methodology hybrid methods have been favorable for evaluating the introduction of electric vehicle technologies in urban transport fleets, considering factors such as cost, range, charging infrastructure, and emission reduction [40,41,42].

Classical MCDA methods, as described above, have been widely applied in diverse industrial sectors to support complex decision-making problems. Their applications include supplier selection, production planning, quality assessment, maintenance scheduling, and risk management, where multiple conflicting criteria must be balanced systematically. For example, AHP is frequently used to evaluate suppliers based on cost, quality, and delivery time; TOPSIS and PROMETHEE have been applied to optimize production scheduling and maintenance prioritization, proving their effectiveness in enhancing operational efficiency and strategic decision-making across manufacturing and service industries [43,44].

2.2. MCDA Group Approach

Decision-making problems in real, complex environments are rarely analyzed based on a single variable. Therefore, especially in high-level decisions, multiple conflicting variables are expected to be considered [45]. In addition, the presence of multiple stakeholders, or decision-making agents, is also expected due to the need to integrate different perspectives and preferences in a given decision-making scenario, and the difference between these is a crucial complexity factor that impacts the final decision to be directed [46]. Group evaluation involves capturing diverse opinions and integrating them cohesively to reach a consensus [47]. Methods with these characteristics aim to identify and incorporate stakeholders’ different views and preferences, from decision-makers to community members affected by the decisions [48].

Within the scope of the MCDA, the methods for group analysis follow a structured process, which begins with the identification of stakeholders and their preferences, the definition of evaluation criteria, the collection and processing of data, the construction of the preference model, and the generation of decision recommendations [49]. During each stage, interaction and communication between decision-makers are crucial to ensuring the representativeness of different perspectives in the decision-making process [50]. While MCDA models for group assessment offer a comprehensive and inclusive approach to decision-making, there are challenges to consider [51].

The application of MCDA often challenges the selective allocation of specialized evaluators for specific variables [23]. Conventional MCDA approaches often require all participants to evaluate all variables, regardless of their specialization or specific knowledge [52]. This uniformity in participation can limit the accuracy of assessments, especially when group members have distinct expertise [53]. This can result in a less in-depth or less accurate assessment of certain variables, as not all evaluators have the same knowledge or experience in all areas relevant to the decision-making process [54].

However, the evolution of the MCDA has highlighted the need for more flexible and adaptable approaches. New perspectives are emerging, focusing on the more targeted division of labor among evaluators [22]. These approaches selectively allocate evaluators based on their areas of expertise, allowing them to focus on the variables where they have specialized knowledge. This adds greater accuracy to assessments and contributes to a more efficient decision-making process, where specialized knowledge is applied more strategically and in-depth, increasing the relevance of individual contributions to the decision-making process. Therefore, part of the methodological proposal presented in this study seeks to fill this gap, offering a model that builds a hierarchy of variables and enables the distribution of decision-makers to hierarchical levels of knowledge to promote transparent decision-making processes adjusted to the objectives of the problem.

2.3. The SAPEVO Family

The search for consensus among decision-makers in a process is complex, and in most cases, it is necessary to integrate a stakeholder who moderates the decision-making process to guide the flow in capturing preferences, building consensus, and indicating the final decision. In this scenario, the Simple Aggregation of Preferences method Expressed by Ordinal Vectors (SAPEVO method) [44] operates under an ordinal input approach for evaluating variables in a given scenario. Concerning the multi-criteria methodology, a specific method provides the processing of subjective and tacit data to the decision maker, enabling the conversion of these subjective points into cardinal scores, numerically expressing a relative degree of importance or performance of the decision variables.

A few years after the validation of the original model in 1997 [55], some extensions and proposals were presented to the academic community, as exposed by [56], such as the SAPEVO-M method [57], an extension of the original model intended for group assessment. In addition to a group evaluation model, there was an axiomatic improvement of the previously developed model, thus increasing its consistency.

The main technical characteristic of the model concerns the process of ordinal transformation of preferences, which is used to obtain the degrees of preference relations between the alternatives in each criterion and to obtain the weights of the criteria, thus generating their respective degrees of priority. The SAPEVO method is currently presented as a family of methods, just like other classic MCDA models. The SAPEVO family of methods currently consists of five methodologies with different technical capabilities. Some approaches use the entire basis of the SAPEVO method in its axiomatic structure of aggregation, and others consider only the use of its ordinal basis in a partial format to support the transcription of ordinal inputs in cardinal values [58,59].

3. The SAPEVO-H² Method

The SAPEVO-H² method, proposed in [11], offers a versatile and adaptable methodological framework to support decision-making processes within complex and hierarchical environments. Such contexts are typical in large corporations and public, military, and governmental organizations, where decisions involve multiple layers of responsibility and diverse stakeholder expertise. Within this study, particularly in sustainability-related evaluations, the method facilitates the simultaneous participation of multiple decision-makers. Each evaluator contributes partial or complete assessments on hierarchical subsets of variables aligned with their specific domain knowledge and operational roles, ensuring that expertise is efficiently leveraged at the appropriate levels of the decision hierarchy.

The results generated by the model can indirectly reflect the preferences expressed by decision-makers in evaluations carried out at different levels of the hierarchy. These individual or group assessments are structured within the hierarchical architecture of the model, allowing each local evaluation to contribute meaningfully to the overall decision-making framework. When systematically aggregated, these localized judgments are transformed into a coherent global prioritization of variables, thus offering a comprehensive understanding of how specific inputs influence broader strategic outcomes.

This paper presents the finalized version of the SAPEVO-H² method, which incorporates substantial axiomatic refinements that significantly enhance the model, bringing theoretical coherence and scalability. One of the principal improvements is the removal of the former dependence on sub-criteria levels, a change that simplifies the hierarchical structure and consequently improves interpretability and ease of application. Incorporating the hyperbolic tangent function as a smoothing mechanism for ordinal-to-cardinal data transformation is a significant advancement. This feature promotes more stable and continuous aggregation of preferences, reducing abrupt transitions in scoring and enhancing robustness, particularly in scenarios involving individual decision-makers. Furthermore, the consistency verification process and the aggregation logic have been refined to ensure tighter alignment between local evaluations and the resultant global prioritization, improving the method’s overall fidelity and practical applicability.

Figure 1 illustrates the axiomatic structure that guides the applicability of the SAPEVO-H² method.

The axiomatic foundation was designed to enable multiple decision-makers’ simultaneous and distributed participation, each contributing according to their domain-specific expertise within a pre-defined hierarchical structure. This framework supports the decomposition of the decision problem into multiple tiers, where each level encompasses a specific set of variables that progressively contribute to achieving a strategic objective at the top of the hierarchy. At each tier, assessments may be conducted either by individual experts or collaborative groups, thereby acknowledging the multidisciplinary nature and complexity of real-world decision contexts. This method enhances applicability by supporting multiple elicitation formats, including both qualitative assessments and quantitative data, making it adaptable to environments with varying levels of analytical maturity and data availability.

The framework models the decision environment as a structured hierarchy, involving a set of decision-makers D = {d₁, d₂, …, d_n}, each of whom may contribute to evaluating the problem according to their respective roles. The decision structure is organized into hierarchical levels N = {n₁, n₂, …, n_l}, where each level represents a progressive layer of abstraction or specificity within the decision-making process.

At each level, a distinct group of variables is defined, representing goals, criteria, or alternatives that collectively contribute to fulfilling a higher-level strategic objective located at the top of the hierarchy, where:

▪

$V$: Variables set, where $v_r\in V,r=1,\dots s.$

▪

$C$: Criteria set, where $c_j\in C,j=1,\dots k.$

▪

$A$: Alternatives set, where $a_i\in A,i=1,\dots b.$

Concerning the variables, $m$ represents each decision-maker within the set of decision-makers D, represented by $n$ as the total number of decision-makers. The symbol $h$ exposes each hierarchical level within the set of hierarchical levels N, and the total number of levels is represented by $l$.

Looking for the variables, $r$ is the index representing each variable within the set of variables $V$, and the total number of variables is $s$, which can be approached as goals. The criteria and alternatives are represented by $j$ and $i$, respectively, considering the total number for each exposed set as $k$ and $b$.

Figure 2 illustrates a representative configuration of the hierarchical evaluation structure adopted by the SAPEVO-H² method. To ensure meaningful comparative analysis, each evaluation set within a given hierarchical level must contain at least two distinct variables. This requirement allows for establishing preference relations among variables, which are fundamental for deriving relative importance scores at each stage of the decision-making process.

An important aspect to consider within the hierarchical evaluation framework is the possibility of multiple evaluators assessing the same criteria or variables at a given level. This overlapping evaluation scenario introduces an additional layer of complexity in the decision-making process, as it requires mechanisms to handle potential conflicts, divergences, and negotiation among evaluators. Such interactions can enrich the analysis by incorporating diverse perspectives but also demand robust aggregation and consensus-building procedures to ensure coherent outcomes. The SAPEVO-H² model is designed with sufficient flexibility to accommodate these cases, enabling the aggregation of multiple ordinal or cardinal inputs for the same criterion while preserving transparency in the synthesis of preferences. Furthermore, the model allows exclusive evaluation situations where a single expert is responsible for a specific criterion, facilitating tailored applications according to organizational structures and decision governance.

3.1. Variables Evaluation

At each hierarchical level, the evaluation of variable sets may be conducted individually or collectively, depending on the structure and scope of the analysis. The method of exploration accommodates three distinct evaluation modalities:

▪

group-based ordinal assessments;

▪

individual ordinal assessments; and

▪

cardinal evaluations (single or group of decision-makers).

This methodological flexibility ensures that the method can adapt to various organizational contexts and levels of stakeholder involvement.

Starting with the group-based evaluation using ordinal data, the method employs a qualitative assessment approach grounded in a seven-point ordinal scale, as

Table 1. Ordinal Scale for Qualitative Variable and Criteria.

Preference Descriptions	Ordinal Values
Less important	−3
Much less important	−2
Less important	−1
Equivalent	0
More important	1
Much more important	2
More important	3

presents. This scale expresses relative preferences or importance levels among the variables or criteria under analysis, enabling comparative judgments in scenarios where quantitative data may be unavailable or inappropriate.

When multiple decision-makers evaluate variables, the method employs a pairwise comparison approach based on the established ordinal scale (Table 1). Each decision-maker ($d_m$) constructs a preference matrix that captures their comparative judgments between all variables within the set. These judgments are then aggregated to derive the relative importance of each variable. Equation (1) formalizes this process, where $x_r$ represents the pairwise input and $v_r$ denotes the computed utility for each variable, reflecting its perceived significance within the group context. The resulting scores are subsequently normalized to ensure comparability, and the ordinal preferences are transcribed into cardinal values, enabling their use in the following aggregation and ranking procedures.

$$e_r={\displaystyle \frac{\sum v_r-\mathit{min}{v}_r}{{maxv}_r-\mathit{min}{v}_r}}$$

(1)

Once the individual utility scores $e_r$ are derived from the pairwise comparisons, the model aggregates these values across all decision-makers. Equation (2) computes the total utility for each variable by summing up the individual contributions from all m decision-makers. This aggregated score reflects the collective judgment regarding the relative importance of each variable within the decision set.

$$e_r={\displaystyle \frac{{\sum }_{m=1}^n{e}_{rm}}{m}}$$

(2)

A normalization step is applied to standardize these values and facilitate their use in further hierarchical aggregation. Equation (3) ensures that the total importance values are converted into a proportional scale, typically summing to 1, resulting in normalized weights ${\upsilon }_r$ for each variable.

$${\upsilon }_r={\displaystyle \frac{e_r}{\sum e_r}}$$

(3)

However, a variable may sometimes receive zero weight due to unanimous deference or neutral evaluation. The model incorporates a corrective mechanism to preserve numerical stability and ensure that no variable is entirely excluded from the analysis, especially in iterative or multi-level aggregations. Following the principle introduced in [46], if a variable obtains a zero normalized weight, it is assigned a small non-zero value equal to 1% of the smallest positive weight among the other variables. This adjustment is formalized in Equation (4). This process ensures continuity in the aggregation process and prevents discontinuities that could compromise the hierarchical synthesis of preferences.

$${\upsilon }_r=\left\{\begin{array}{c}{\upsilon }_{r}if{\upsilon }_r>0\\ 0.01\min \left({\upsilon }_r>0\right)if{\upsilon }_r=0\end{array}\right.$$

(4)

Following the process of evaluation of variables, in decision scenarios where a single expert is responsible for evaluating a given set of qualitative variables, SAPEVO-H² incorporates a mathematical smoothing mechanism to mitigate the subjectivity inherent in isolated judgments. Specifically, the model applies the hyperbolic tangent function as a transformation tool to convert discrete ordinal preferences, originally derived from the verbal scale in Table 1, into continuous cardinal values.

The ordinal scale spans integer values from −3 to +3, representing relative importance or performance differences between variables. When these inputs are processed through the hyperbolic tangent, extreme values are compressed within a bounded interval, effectively narrowing the influence spread among strongly rated variables. This transformation reduces overemphasis on outlier judgments and ensures a smoother gradient in the resulting weight distribution [60]. Figure 3 and

Table 2. Ordinal Scale Transcribed by Hyperbolic Tangent.

Ordinal Values	Hyperbolic Tangent Values
−3	−0.995
−2	−0.964
−1	−0.762
0	0.000
1	0.762
2	0.964
3	0.995

illustrate how this function translates pairwise ordinal scores into normalized cardinal measures, as expressed in Equation (5).

$$a_r=tgh\left(a_r\right)$$

(5)

The model introduces a normalization procedure grounded in predefined boundary conditions to ensure that the smoothed scores derived from individual ordinal assessments are proportionally scaled for subsequent aggregation. Following the approach outlined in [8], two reference points are established: the maximum theoretical sum of preference values, which corresponds to a scenario in which one variable is consistently preferred over all others, and the minimum adequate sum, representing the lowest possible aggregation of preference values under rational consistency assumptions. These bounds are Equations (6) and (7), where n denotes the number of variables evaluated within the given set.

$$maximumsum=(n-1)$$

(6)

$$minimumsum=\left(n-1\right)-1$$

(7)

Once the transformed scores $a_r$ are obtained via the hyperbolic tangent function for each comparison, the relative importance of each variable is determined through a normalization step, formalized in Equation (8). This equation ensures that the resulting weights are constrained within a [0, 1] interval, providing a consistent and interpretable scale across decision contexts.

$${\upsilon }_r={\displaystyle \frac{\sum a_r-minimumsum}{maximumsum-minimumsum}}$$

(8)

In specific decision scenarios, particularly those involving performance metrics or measurable outcomes, variables may be evaluated using quantitative data rather than subjective judgments. SAPEVO-H² supports cardinal analysis to accommodate these cases, allowing preferences to be derived from directly observed numerical values. This extension is particularly useful when alternatives are assessed based on criteria such as cost, efficiency, or emissions, in contexts in which objective measurements are available and meaningful.

Being the third format of variable evaluations, the cardinal analysis method incorporates threshold-based preference functions to translate quantitative inputs into normalized importance scores. For each criterion, analysts define two parameters: a minimum threshold ($L_{min}$), below which no preference is assigned, and a maximum threshold ($L_{max}$), above which full preference is granted. Depending on the structure of the criterion, one of three preference functions may be applied: a quasi-criterion, a linear preference up to $L_{max}$, or a criterion model with $L_{min}$ to $L_{min}$. These functions transform raw performance values $v_r$ into degrees of importance $e_r$, aggregating and normalizing using the same principles applied in the ordinal case, ensuring consistency across the hybrid decision-making process. Figure 4 presents the function design.

In the context of cardinal evaluation, the method provides three distinct preference functions to translate quantitative performance values into degrees of importance. The quasi-criterion function is the most restrictive, suitable for binary decisions in which any value below a minimum threshold $L_{min}$ is entirely disregarded, and any value above it is fully preferred as expressed in Equation (9). This model is ideal for scenarios where a performance criterion represents a strict cutoff, such as compliance, safety limits, or eligibility.

The linear preference function introduces a gradual increase in preference up to a defined maximum threshold, denoted as $L_{max}$. As presented in Equation (10), this approach is suitable for criteria in which higher performance progressively enhances utility, but beyond a certain point, further improvements no longer yield additional preference. In contrast, the third function is more flexible, incorporating a region of indifference below $L_{min}$, a linear transition between $L_{min}$ and $L_{max}$, and full preference once the performance surpasses $L_{max}$, as described in Equation (11).

$$e_r=\left\{\begin{array}{c}0if{v}_r\le L_{min}\\ 1if{v}_r>L_{min}\end{array}\right.$$

(9)

$$e_r=\left\{\begin{array}{c}{\displaystyle \frac{v_r}{L_{max}}}if{v}_r\le L_{max}\\ 1if{v}_r>L_{max}\end{array}\right.$$

(10)

$$e_r=\left\{\begin{array}{c}0if{v}_r\le L_{min}\\ {\displaystyle \frac{v_r-L_{min}}{L_{max}-L_{min}}}if{L}_{min}<v_r<L_{max}\\ 1if{v}_r\ge L_{max}\end{array}\right.$$

(11)

The selection of the appropriate preference function, as well as the specification of threshold values ($L_{min}$ and $L_{max}$), must be carried out jointly by the analyst and the decision-makers. These parameters should reflect both the technical characteristics of the criterion under consideration and the strategic intent of the decision process. This collaborative definition ensures that the mathematical treatment of the data aligns with real-world expectations and preserves the interpretability of the results.

It is also important to account for the orientation of the criterion—whether higher values are desirable or undesirable. In cases where the objective is minimization (e.g., emissions, costs, or risk), the normalized performance scores must be inverted to reflect proper preference logic. This is achieved by computing the complement of the value, where $e_r=1-e_r$. Once all variables have been evaluated and appropriately transformed, the aggregated importance of each alternative $a_{ij}$ is calculated using the standard summation and normalization procedures described in Equations (2) and (3), thus integrating the results into the broader decision-making hierarchy.

3.2. Outranking and Aggregation Analysis

After the evaluation phase, which establishes the relative importance of variables within each hierarchical level, the SAPEVO-H² method proceeds with the synthesis of results through two key procedures: outranking analysis and additive aggregation. These mechanisms allow for the progressive integration of preferences across the hierarchy, translating localized judgments into global performance indicators. The outranking index reflects the comparative advantage of each alternative or variable within its set, while the aggregation procedure consolidates these indices throughout the hierarchical structure, culminating in a final prioritization of other options aligned with the strategic objectives of the decision problem.

For each hierarchical level under analysis, the model constructs an outranking index that quantifies the relative dominance of each variable compared to its peers at the same level. This local index is computed by weighing the importance of each variable $e_{rh}$ by the preference scores inherited from the immediately superior level ${\upsilon }_{j(h-1)}$, and then averaging across all connections, as formalized in Equation (12):

$${\varphi }^{+}{e}_{rh}={\displaystyle \frac{{\sum }_{j=1}^n{e}_{rh}{\upsilon }_{j\left(h-1\right)}}{n}}$$

(12)

Next, to assess the net dominance of a variable over others, the model calculates the difference between its outranking power and that of all competing variables at the same level. This results in a comprehensive outranking index $\varphi e_{rh}$, as defined in Equation (13):

$$\varphi e_{rh}=\sum _{r=1}^s{\varphi }^{+}{e}_{rh}-{{\varphi }^{+}e}_{sh}$$

(13)

After calculating the local indices, the model initiates the additive aggregation process to propagate the importance of each variable upward through the hierarchy. The relative importance of a variable $e_{rh}$ concerning a higher-level construct $s_{r(h-1)}$ is computed by multiplying its local weight by the weight of the corresponding parent variable. For deeper hierarchies, where a variable is linked to multiple upper levels, the global importance is obtained via the summation defined in Equation (14).

$$e_{rhl}=\sum _{l=1}^n{e}_{rh}{k}_l$$

(14)

This recursive aggregation process allows for a global ranking of alternatives while preserving local performance insights at each criterion or attribute level. Such dual-level visibility ensures that decision-makers can identify the best alternatives and understand the specific dimensions in which each alternative excels or underperforms. Moreover, the approach supports individual and group-based decision contexts, maintaining the transparency and traceability of stakeholder preferences throughout the model.

The integration of outranking and additive aggregation in SAPEVO-H² enables the generation of a comprehensive ranking of alternatives and enhances the explanatory power. By preserving the hierarchical traceability of preferences, decision-makers are empowered to perform multi-level diagnostics, identifying which variables or criteria most significantly influenced the final positioning of each alternative. This capability is particularly valuable in strategic decision environments, where transparency, justification of choices, and stakeholder alignment are essential. Additionally, the ability to perform local sensitivity analyses at each hierarchical node contributes to the robustness of the decision process, allowing analysts to simulate the impact of changes in priorities or stakeholder inputs across different levels of abstraction.

3.3. Consistency Analysis When Performed Pairwise Evaluations

The consistency analysis for the pairwise evaluation approach used in SAPEVO-H² was initially proposed in [56]. This technique was designed to enhance the reliability of ordinal inputs by identifying and quantifying inconsistencies in pairwise comparisons, ensuring that the axiomatic basis of the method is preserved even in complex judgment matrices. Its incorporation reinforces the robustness of the model in group decision-making scenarios, where divergent or contradictory opinions may arise during the elicitation process.

In this procedure, the consistency of the judgments provided by the decision-makers is assessed by comparing the original input matrix to a derived reference matrix known as the ideal transitive matrix (Figure 5). The process begins with the original pairwise comparison matrix, represented as an $n$ × $n$ matrix for $n$ variables, where only the upper diagonal is considered, since comparisons are reciprocal. From this, the model constructs the ideal transitive matrix by applying logical transitivity rules: for each element $a_{ij}$, the expected value is calculated based on the sum $a_{1j}+a_{i1}$, and the result is clipped to remain within the bounds of the original scale, as shown in Equation (14):

$$a_{ij}=\left\{\begin{array}{c}-3if{(a}_{1j}+a_{i1})\le -3\\ a_{1j}+a_{i1}if-3<(a_{1j}+a_{i1})<3\\ 3if{(a}_{1j}+a_{i1})\ge 3\end{array}\right.$$

(15)

Next, a comparison matrix is generated by comparing each entry of the original input matrix to its corresponding value in the ideal transitive matrix. This yields a binary consistency matrix, where each entry is set to 1 if the original and ideal values match, and 0 otherwise. This matrix reflects the number of consistent judgments across the set of comparisons. Figure 6 illustrates an example of such a binary matrix structure.

The total number of consistent judgments, denoted bp, is calculated by summing all values in the consistency matrix. This value is then divided by the total number of unique comparisons ${\displaystyle \frac{n(n-1)}{2}}$ to produce the consistency index $\mathsf{\lambda }$, as shown in Equation (15).

$$\mathsf{\lambda }={\displaystyle \frac{bp}{\left(\frac{n(n-1)}{2}\right)}}$$

(16)

To assist in interpreting this index, the model classifies the level of consistency into five categories: high (0–10%), average (10–20%), low (20–30%), inconsistent (30–40%), and very inconsistent (above 40%). This classification helps analysts identify when a set of judgments may require review or re-elicitation to ensure the integrity of the decision-making process.

The inclusion of a consistency mechanism within the SAPEVO-H² method represents a significant advancement in ensuring the methodological rigor of MCDA models based on ordinal judgments. By systematically identifying incoherencies in pairwise comparisons, the approach provides the logical structure of preference data and fosters greater transparency and accountability in group decision contexts. This is particularly relevant in complex environments, where stakeholder alignment and confidence in the decision process are critical.

4. Case Study

As part of exposing the numerical implementation and feasibility of the mathematical modeling analyses in the proposal, this study considers a decision-making scenario within the automotive industry. To this end, Energy Car, a vehicle manufacturer with a fictitious name for study purposes, seeks guidance in prioritizing its sustainable projects.

Currently, the organization has a set of six projects for sustainable development purposes, with the primary focus on minimizing polluting gas emissions in the manufacturing operations that the company operates. However, to prioritize these projects, it is favorable to analyze the technical profiles of the variables and recognize the alignment of each alternative with the organizational objectives, especially those aligned with the strategic objective of zero emissions.

To collect data and information related to the problematic situation in question, a series of interviews was conducted to provide a better understanding of the scenario studied from the point of view of the experience of people directly linked to the evaluation. The interviews were conducted with two administrative directors, three administrative managers, and three project coordinators. In the sequence, we present the expertise profile of decision-makers:

• D-A: Chief Executive Officer of Energy Car, with over 20 years of experience in the automotive industry, focusing on strategic management and sustainable business practices;
• D-B: Chief Operating Officer, responsible for integrating operations into the company’s core strategies and overseeing the implementation of industrial and technological initiatives;
• D-C: Head of Environmental Compliance, with extensive knowledge of environmental regulations and sustainability standards;
• D-D: Director of R&D, specializing in the development of innovative and sustainable automotive technologies;
• D-E: Plant Manager, overseeing daily operations and implementing sustainable manufacturing processes;
• D-F: Supply Chain Manager, focusing on sustainable sourcing and logistics optimization;
• D-G: Senior Engineer, expert in eco-design and materials science;
• D-H: Project Manager, responsible for on-the-ground implementation and monitoring.

During the interviews, objectives were identified, fully or partially, from a zero-emission perspective in manufacturing operations. For the interview process, a scenario was built in which the organization aims to achieve zero emissions as a long-term development goal. To define the objectives and align the influences, a set of 12 objectives was listed, all aligned with the strategic objective of minimizing polluting emissions. In addition, five more criteria were established for evaluating the set of projects to be prioritized. Figure 7 presents the objectives, criteria, and alternatives already structured according to the perspective of the hierarchical method proposed in this study.

With the respective structure shown in Figure 7, the five levels of evaluation were constructed, and the newly proposed SAPEVO-H² method is favorable to the decision-making process, enabling the alignment and exposure of the relative importance of the variables at each connected level. In this context, the five levels of assessment are:

• Level 1: Strategic objectives to minimize gas emissions in manufacturing operations;
• Level 2: Tactical objectives are fundamental to the feasibility of strategic objectives.
• Level 3: Tactical objectives: means to achieve all or part of the fundamental objectives;
• Level 4: Evaluation criteria for projects concerning the tactical objectives of projects;
• Level 5: Projects to boost the sustainable aspect of the organization under evaluation.

The six projects focus on various aspects of environmental sustainability within the company. Each project addresses sustainability goals, from reducing water consumption to promoting environmental education and optimizing manufacturing processes. The following topics provide detailed descriptions of these initiatives, highlighting their objectives and implementation strategies:

• Clean Water (CW): Aims to reduce water consumption and improve water recycling processes within the manufacturing plants. The project focuses on implementing advanced water treatment systems to recycle wastewater, reducing the overall water footprint of the manufacturing process. This includes installing filtration units, reverse osmosis systems, and developing rainwater harvesting infrastructure.
• Green Education (GE): Seeks to promote environmental awareness and sustainable practices among employees and the local community, developing educational programs and workshops on sustainability topics such as energy efficiency, waste reduction, and the importance of biodiversity. It also includes creating an online learning platform with resources and courses on environmental stewardship.
• Eco Materials (EM): Incorporates environmentally friendly materials into the production process, involving researching and utilizing sustainable materials such as recycled plastics, biodegradable composites, and low-impact metals. The main goal is to reduce the environmental impact of raw material extraction and processing while maintaining product quality and performance.
• Eco Design (ED): Focuses on designing products with minimal environmental impact throughout their lifecycle. The project focuses on incorporating eco-design principles into the development of new automotive models. This includes designing for disassembly, using modular components, and selecting materials that are easier to recycle. The project aims to create vehicles that are both sustainable and innovative.
• Zero Waste (ZW): Aims to achieve zero waste to landfill status by enhancing waste management practices. The strategy aims to implement comprehensive waste management strategies, including recycling, composting, and waste-to-energy technologies. It focuses on reducing waste generation at the source, improving sorting and recycling processes, and finding sustainable ways to repurpose waste materials.
• Optimal Manufacturing (OM): Aims to optimize manufacturing processes to enhance efficiency and reduce environmental impact, adopting advanced manufacturing technologies such as automation, precision machining, and real-time data analytics to improve process efficiency. The project seeks to reduce energy consumption, minimize resource waste, and enhance overall production sustainability.

To apply the model to the stakeholders, after the interviews, an exposition of the proposed model was made for the common understanding of all, understanding the structure of inputs necessary for processing the information and constructing the preferences in a quantitative character in the analysis by the new SAPEVO-H² method. The numerical implementation is exposed in the following section.

4.1. Numerical Application

For a structural application, the analysis of the first three levels of the decision-making process will be considered, limiting itself to the objectives and analysis of strategic and tactical variables of the central objective of this case study. Subsequently, the evaluation of an operational nature will be considered, thus promoting the analysis of the projects to be prioritized and their respective evaluation criteria.

4.1.1. Strategic and Tactical Levels Evaluation

To start the evaluation process, this section will evaluate and analyze levels 1, 2, and 3 by stakeholders D-A, D-B, D-C, D-D, and D-E. Starting with the evaluation of the first level, there are three objectives: Enable the energy transition (EET), Optimize the supply chain (OSC), and Boost research and development in design, operation, and technology (BRD).

At level 1, the assignments of the decision-makers D-A and D-B were evaluated.

Table 3. Level 1 Evaluation by D-A.

	EET	OSC	BRD	Punctuation	Individual Utility
EET	0	2	0	2	1
OSC	−2	0	3	1	0.8
BRD	0	−3	0	−3	0

and

Table 4. Level 1 Evaluation by D-B.

	EET	OSC	BRD	Punctuation	Individual Utility
EET	0	2	−1	1	0.8
OSC	−2	0	−1	−3	0
BRD	1	1	0	2	1

show the individual attributions of the two decision-makers and the aggregate result in

Table 5. Level 1 Final Utilities.

	Punctuation	Individual Utility
EET	1.8	0.5
OSC	0.8	0.222
BRD	1	0.278

. The two matrix evaluations had high consistency (0%). The values of Punctuation and Individual Utility are the representation of $v_r$ and $e_r$, respectively, based on Equation (1), from Section 3.

Based on the attributions and partial results obtained, it is understood that the essential objective is to promote the energy transition, with the variable of highest preference being cardinal, with the aggregation of the two decision-makers. Subsequently, the development of design, operations, and technology research was established in order of importance, preceded by the optimization of supply chain processes.

Once the first level of the hierarchy is analyzed, the process is sequenced in the evaluation of the variables present in the second level: Prioritize the use of renewable energies (PRE), Keep partnerships with suppliers who are committed to sustainable practices (KPS), Promote sustainable disposal policies (PSP), and Optimize manufacturing processes (OMP).

At level 2, only stakeholder D-A will act as an evaluator, and then a single-decision process will be operated at this level. After assigning D-A concerning the variables at this level. As the ordinal transformation is the same in all model implementations,

Table 6. Level 2 Aggregation in Level 1.

	EET	OSC	BRD	Final Utility
PRE	0.4767	0.1671	0.3938	0.3849
KPS	0.1697	0.1671	0.1062	0.1515
PSP	0.3303	0.1671	0.1231	0.2365
Weights	0.5	0.222	0.278

details the individual utilities concerning the three objectives of the higher level (EET, PSC, and BRD), and by the aggregation modeling, the final utility is presented as well. The three matrix evaluations obtained high consistency (0%).

Maintaining a trend logic and realizing the preference in the variables with higher sustainable inclinations under direct alignment is possible. In this scenario, the preference for using renewable energies proved to be the primary objective of level two, preceded by promoting sustainable policies in the organization. The strategy of maintaining partners with sustainable characteristics proved to be the one with the least impact and preference within the process.

Next, it is time to evaluate the variables of the third level, namely: Review and improve existing processes (RIP), Promote automation of manufacturing activities (PAM), Prioritize the use of sustainable materials (PSM), Mitigate waste and waste of natural resources (MWN), and Promote sustainable ideas among employees (PSI). The decision-makers D-C, D-D, and D-E evaluate the level.

In the process, the five variables will be analyzed against the four objectives of the higher level (PRE, KPS, PSP, and OMP). The decision-maker D-C will evaluate against the four objectives, the decision-maker D-D will evaluate against three, and the decision-maker D-E will evaluate against only one objective.

Table 7. Level 3 Partial Utilities by D-C, D-D, and D-E.

	D-C				D-D			D-E
	PRE	KPS	PSP	OMP	PRE	KPS	PSP	PSP
PRE	0.5	1	0.67	1	0	1	0.67	1
PAM	0	0	0	0.67	0	0.562	0	0.33
PSM	1	1	1	0	1	0.937	0.67	0
MWN	1	0	0.67	0.33	0.5	0.625	0.33	0
PSI	0.5	0	0.67	0	0	0	1	0
Inconsistency	0%	0%	0%	0%	0%	0%	0%	0%

shows the partial utilities obtained.

Considering the attributions made and the partial utilities, it is then possible to aggregate the preferences and obtain utilities at level 3; in this way,

Table 8. Level 3 Aggregation in Level 2.

	FOR	KPS	PSP	OMP	Final Utility
RIP	0.1110	0.3898	0.2349	0.5989	0.2934
PAM	0.0011	0.1096	0.0018	0.2995	0.0855
PSM	0.4440	0.3776	0.2936	0.0009	0.2977
MWN	0.3330	0.1218	0.1762	0.0998	0.2109
PSI	0.1110	0.0011	0.2936	0.0009	0.1125
Weights	0.3849	0.1515	0.2365	0.2272

exposes the preferences of the variables concerning the objectives of level 2.

From a sustainability perspective, from the point of view of the stakeholders of the evaluation process, the most prominent variables regarding preferences are the prioritization of sustainable materials and the review of production processes. Then, in order of preference, there is the mitigation of waste, the promotion of sustainable ideas to the company’s employees, and finally, the automation of manufacturing activities.

As it is a hierarchical model, the level 3 variables can also be measured in terms of their respective importance in the other higher levels; in this case, they are also evaluated, through an aggregation factor, for level 1, concerning the objectives of EET, CSO, and BRD. It is noteworthy that when additional levels to the higher level are evaluated in sequence, the final utility will always be the same, given the distribution of the weights of the higher level. However, the analysis becomes richer by clarifying the preferences about the elements of the higher level individually, that is, how each variable was preferred for an individual objective of the higher level.

Table 9. Level 3 Aggregation in Level 1.

	EET	CSO	BRD
RIP	0.2106	0.4216	0.3398
PAM	0.0267	0.1682	0.1252
PSM	0.3727	0.1868	0.2514
MWN	0.2399	0.1552	0.2034
PSI	0.1501	0.0682	0.0803

exposes the performances.

Analyzing the variable of level 3 in character to each level 1 objective, the preference in the PSM variable is recognized when analyzing the energy transition objective, as well as the MWN and RIP variables. On the other hand, when analyzing the optimization of the processes, the highlight remains with the OSC variable. In the third objective, aimed at research and development, the emphasis is on the RIP and PSM variables, both aligned with issues of processes, materials, and technologies.

With the analysis of the third level, it is understood that the variables are evaluated at strategic and tactical levels. In this scenario, the next section will be dedicated to evaluating levels 4 and 5, dealing with criteria and alternatives (projects) for evaluating and prioritizing actions.

4.1.2. Operational Levels Evaluation

To start the evaluation process, this section will evaluate and analyze levels 1, 2, and 3 by stakeholders D-A, D-B, D-C, D-D, and D-E.

Considering the continuity of the process of assignments and evaluations at the hierarchical levels, it is time to analyze the criteria and projects in each criterion. As initially exposed in the hierarchical structure (Figure 7), there are five criteria representing aspects of importance for the projects to be analyzed, namely: Environmental impact (EI), Operational efficiency (OE), Waste reduction (WR), Energy Efficiency (EE), and Project Budget (PB).

As projects, six alternatives will be analyzed from the perspective of the five evaluation criteria, with the upper level being to be built. The projects are: Clean Water (CW), Green Education (GE), Eco Materials (EM), Eco Design (ED), Zero Waste (ZW), and Optimal Manufacturing (OM).

Considering a set of five criteria and six alternatives, at levels 4 and 5, three decision-makers (D-F, D-G, and D-H) will be considered. Regarding the evaluations carried out at level 4,

Table 10. Level 4 Partial Utilities by D-F, D-G, and D-H.

	D-F					D-G					D-H
	RIP	PAM	PSM	MWN	PSI	RIP	PAM	PSM	MWN	PSI	RIP	PAM	PSM	MWN	PSI
EI	0	0	0.7895	0.6667	0	0.5	0	1	1	0	0.3333	0	1	0.75	1
OE	1	1	0	0	1	1	1	0	0	1	1	1	0	0	1
WR	0	0	1	1	0	0	0	1	1	0	0	0	1	1	1
EE	1	0.6667	0.4211	0.6667	1	1	1	1	1	0.5	1	0	0	0.75	0
PB	0	0	0.4211	0.3333	1	0.5	0.5	1	0.5	0.5	0.3333	0	1	0	0
Inconsistency	0%	0%	20%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	20%	0%

presents the partial utilities concerning the third level (RIP, PAM, PSM, MWN, and PSI).

The preferences of the stakeholders are then aggregated, and the weights obtained at the higher level are influenced, aggregating and constructing the weights of the criteria at the fourth level of the hierarchy.

Table 11. Level 4 Aggregation in Level 3.

	RIP	PAM	PSM	MWN	PSI	Final Utility
EI	0.1086	0.0010	0.2892	0.2786	0.1250	0.1909
OE	0.3909	0.5795	0.0015	0.0010	0.3750	0.2070
WR	0.0011	0.0010	0.3110	0.3458	0.1250	0.1800
EE	0.3909	0.3220	0.1473	0.2786	0.1875	0.2659
PB	0.1086	0.0966	0.2510	0.0961	0.1875	0.1562
Weights	0.2934	0.0855	0.2977	0.2109	0.1125

presents the respective information.

Based on the information constructed, it is understood that energy efficiency is one of the main evaluation criteria for project selection, preceded by the operational efficiency of manufacturing processes. Subsequently, however, exposing a certain similarity between the performances, the environmental impact proved relatively essential regarding waste reduction in manufacturing processes. Finally, exposing 15% of the amount distributed, the budgetary aspects of the feasibility of each project were presented.

It should be noted that even though there were variations between the weights of the criteria, which is expected, there was a certain proximity and even a leveling between the weights obtained, not exposing any variable with more than half of the proportional importance, which shows that all the criteria listed for evaluation have a relative degree of relevance in the decision-making process.

Table 12. Level 4 Aggregation in Levels 2 and 1.

	Level 2				Level 1
	FOR	KPS	PSP	OMP	EAT	CSOs	FRG
EI	0.2471	0.1857	0.1962	0.0935	0.2163	0.1517	0.1764
OE	0.0866	0.2170	0.2035	0.4081	0.1548	0.2883	0.2360
WR	0.2672	0.1602	0.1892	0.0359	0.2179	0.1209	0.1590
EE	0.2227	0.2774	0.2398	0.3586	0.2408	0.3025	0.2819
PB	0.1764	0.1596	0.1713	0.1039	0.1702	0.1366	0.1467

shows the individual importance of the level 2 and level 1 objectives for understanding the performance of the criteria for each top-level objective.

With the analysis of the performances at the higher levels, the construction of the relevance of the EE criterion, which gave good individual performances in each objective, but not the maximum in all, proves to be an attractive characteristic in the evaluation. This is one of the factors of interest in multi-criteria models, where it does not necessarily need to be the best in all perspectives, but rather to present median preferences in most of the evaluation criteria, being a compensatory factor.

If there were any variable with relevant performance in only one factor and low performance in the others, it would be a matter of investigation to understand why a given variable was relevant only in one and not in the others. This aspect is of great value in terms of the transparency of the model, which will reflect the transparency of the decision-making process.

Once the weights of the criteria have been obtained, it is time to evaluate the variables of the last level, these being the projects in the scope of sustainability to be prioritized in the management of the organization under discussion. For this part of the process, six alternatives will be considered. The variables will be analyzed from the perspective of the five criteria previously evaluated (EI, OE, WR, EE, and PB).

Table 13. Project Budget Evaluation by D-F, D-G, and D-H.

	Cardinal Values	Thresholds (D-F)	Thresholds (D-G)	Thresholds (D-H)
CW	USD 250,000.00	Lmin USD 300,000.00	Lmin USD 250,000.00	Lmin USD 0
GE	USD 120,000.00	Lmax USD 1,000,000.00	Lmax USD 900,000.00	Lmax USD 800,000.00
EM	USD 360,000.00
ED	USD 720,000.00
ZW	USD 90,000.00
OM	USD 1,200,000.00

presents the cardinal values from the project budget, along with the different thresholds exposed by each decision-maker.

Table 14. Level 5 Partial Utilities by D-F, D-G, and D-H.

	D-F						D-G				D-H
	EI	OE	WR	EE	PB	EI	OE	WR	EE	PB	EI	OE	WR	EE	PB
CW	1	0	0.333	1	1	1	0	0.333	1	1	1	0.238	0.333	1	0.687
GE	1	0	0.667	0.333	1	0.667	0	0.667	0.333	1	0.500	0	0.667	0.294	0.850
EM	0.895	0.846	0.667	0.333	0.914	1	0.333	0.667	0.333	0.831	1	0.524	0.667	0.294	0.550
ED	0.579	0.461	0.667	0	0.400	0.667	0.333	0.667	0	0.277	0.500	0.524	0.667	0	0.100
ZW	0.316	0	1	0.333	1	0.333	0	1	0.333	1	0	0.286	1	0.294	0.887
OM	0	1	0	1	0	0	1	0	1	0	0	1	0	0.941	0
Inc.	13.3%	6.7%	0%	0%	0%	0%	0%	0%	0%	0%	0%		0%	0%	0%

provides the partial utilities.

Once the individual assignments and partial utilities were constructed, it was possible to transcribe the different perspectives of the stakeholders in the project evaluations. With the respective attributions, the process of aggregating the amounts and constructing the aggregated preferences for level 5 is carried out.

Table 15. Level 5 Aggregation in Level 4.

	RIP	PAM	PSM	MWN	PSI	Final Utility
CW	0.2867	0.0364	0.0999	0.3396	0.2336	0.2070
GE	0.2071	0.0004	0.1998	0.1088	0.2477	0.1432
EM	0.2767	0.2601	0.1998	0.1088	0.1995	0.2027
ED	0.1668	0.2014	0.1998	0.0011	0.0675	0.1203
ZW	0.0620	0.0436	0.2997	0.1088	0.2510	0.1429
OM	0.0006	0.4581	0.0010	0.3330	0.0007	0.1838
Weights	0.1909	0.2070	0.1800	0.2659	0.1562

shows the respective information.

Considering the assignments performed, there is a perception of projects of greater relevance to the organization. In this case, priority was given to the expenditure and reuse of water, which is one of the leading manufacturing inputs in many cases. The Eco Materials project also proved very relevant, having a relative approximation in terms of its importance to the first place. Finally, the preference was preceded by the Optimal Manufacturing, Green Education, and Zero Waste projects, which were practically tied, and last place was the Eco Design project.

Considering that the weights of the criteria are of great relevance, even projects of high monetary cost stand out among the main ones, as is the case with OM, which presented the highest budget but good results in the others. Aspects such as this reflect the importance of understanding the influences of other variables on alternatives or actions to the problem.

As previously performed, in addition to the evaluation,

Table 16. Level 5 Aggregation in Levels 3 and 2.

	Level 1					Level 2
	RIP	PAM	PSM	MWN	PSI	FOR	KPS	PSP	OMP
CW	0.2036	0.1534	0.2227	0.2315	0.1694	0.2175	0.2087	0.2040	0.1913
GE	0.0923	0.0595	0.2002	0.1809	0.1178	0.1725	0.1403	0.1470	0.0914
EM	0.1961	0.2055	0.2086	0.1959	0.2149	0.2037	0.2019	0.2053	0.1989
ED	0.1048	0.1239	0.1278	0.1226	0.1342	0.1242	0.1178	0.1234	0.1124
ZW	0.0939	0.0849	0.1902	0.1754	0.1290	0.1677	0.1393	0.1468	0.0994
OM	0.3094	0.3728	0.0504	0.0938	0.2346	0.1144	0.1922	0.1735	0.3065

and

Table 17. Level 5 Aggregation in Level 1.

	EET	CSOs	BRD
CW	0.2109	0.2007	0.2050
GE	0.1567	0.1224	0.1354
EM	0.2038	0.2013	0.2019
ED	0.1226	0.1171	0.1190
ZW	0.1544	0.1254	0.1364
OM	0.1516	0.2331	0.2023

present the respective performance of the projects in the objectives that make up the hierarchical structure.

Having detailed performances at each level makes it possible to observe how favorable a project can be in each objective, presenting their respective performances. This feature of detailing, exposing the interconnections present between the variables of the decision-making process, is one of the main gains of the mathematical model in the proposal, corroborating the tracking and better understanding of the construction of cardinal preferences carried out by inputs in part of the process, which bring a reflection at the macro level of analysis.

4.2. Sensitivity Analysis

To recognize the robustness of the model being implemented and proposed in this study, a sensitivity analysis is then performed in the decision-making process. To vary the values, let us consider the change in preferences at level 1 and realize how significant the prioritization of projects can be, as previously obtained; that is, the analysis will show how impactful the variation of preferences at level 1 can be concerning the variables at level 5.

Five scenarios will be considered for the sensitivity analysis process: S1, S2, S3, S4, and S5. The weights obtained in the evaluation will be considered in the first scenario, while in the second scenario, weights equal to the three variables will be considered. Finally, the other three scenarios will present a variation with one dominant variable and two equivalent levels of importance.

Table 18. Different Weights in Each Scenario.

	S1	S2	S3	S4	S5
EET	0.5	0.3333	0.5	0.25	0.25
CSO	0.2222	0.3333	0.25	0.5	0.25
BRD	0.2778	0.3333	0.25	0.25	0.5

shows the different weights for the five scenarios.

Performing the model aggregation calculations through the new weights obtained at level 1, we have five different degrees of importance for the projects, with the variables being established at level 5.

Table 19. Different Scores in Each Scenario.

S1		S2		S3		S4		S5
CW	0.2070	CW	0.2054	CW	0.2069	OM	0.2050	CW	0.2054
EM	0.2027	EM	0.2021	EM	0.2027	CW	0.2044	EM	0.2022
OM	0.1838	OM	0.1955	OM	0.1847	EM	0.2021	OM	0.1973
GE	0.1432	ZW	0.1386	GE	0.1428	ZW	0.1354	ZW	0.1381
ZW	0.1429	GE	0.1380	ZW	0.1426	GE	0.1342	GE	0.1375
ED	0.1203	ED	0.1194	ED	0.1203	ED	0.1189	ED	0.1194

and Figure 8 present the cardinalities obtained in each of the five scenarios in detail to understand the results generated.

Based on the results obtained, it can be concluded that changing preferences at more distant levels in the hierarchical structure does not cause significant impacts on the cardinalities obtained. In other words, changes in preferences at the strategic levels cause only minor variations in the importance and priorities at the operational levels.

Observing the performances generated, it is understood that in scenarios S1, S2, and S3, the order of importance among the alternatives was maintained, keeping the CW, EM, OM, GE, ZW, and DE projects as priority orders. It is also noticeable that the GE and ZW projects have a respective equivalence in almost all scenarios. A significant variation is only observed when arriving at the S4 scenario, where the OM alternative benefits and leads the ordering. Scenario S5 is relatively similar to the ordering of the first three scenarios.

With the sensitivity analysis of the model, it is recognized that the OM project may be more sensitive to changes in preferences at higher levels of the hierarchy. On the other hand, the other alternatives are relatively stable in the decision-making process. For significant changes in preferences and respective priority orders, it would be necessary to change preferences at the levels closest to the variables in which it is sought to have a better detail of the analysis and its respective robustness in the decision-making process.

The sensitivity analysis conducted in this study follows a systematic approach, considering possible variation of utilities, where the values of selected criteria were varied within predefined plausible ranges reflecting hypothetical fluctuations observed or expected in the decision environment. This scenario-based approach enables the identification of variables with the most significant influence on the final prioritization outcomes. The sensitivity analysis was also performed to demonstrate the variability of preferences under different scenarios, highlighting how minimal or substantial changes at higher hierarchical levels can significantly alter global prioritization. By quantifying how changes in specific criteria affect the aggregated rankings, the analysis provides meaningful insights into model robustness and the stability of decision recommendations.

5. Discussion

The feasibility of the proposed model highlights the ability to hierarchize multiple evaluation criteria, offering a clear representation of the relationships between variables at different levels. The applied multi-criteria hierarchical analysis provided a systematic approach to complex decision-making, enabling the weighted consideration of several variables and their relationships. Notably, one of the points of adequacy of the proposed modeling stands out in the possibility of having a group evaluation, partitioning the stakeholders so that it is necessary to evaluate only the scenarios in which they have previous expertise. The practice of distributing decision-makers at different levels makes it possible to scale the levels of responsibility without the need to give weight to decision-makers.

The modeling seeks to fill gaps in the decision-making models when considering a group of evaluators. At first, the feasibility of constructing a structure of variables for analysis, in which each decision-maker does not need to evaluate variables of which he or she is not aware, is favorable in the composition of divergent preferences and characteristics in organizations and public sectors, with hierarchies of responsibilities in the face of a common objective. From the methodological point of evaluation, modeling the outranking relationships between variables at each level offers a more refined view of the interactions, allowing the identification of dominances and overclassifications that are not readily perceptible from the human point of view and in more traditional models.

One of the differentiating points lies in its ability to capture nuances in the relationships between variables, revealing patterns of dominance and outranking at different levels of the hierarchy. This provides a deeper and more granular understanding of preferences by accurately highlighting the most relevant variables and their interrelationships. Considering this respective importance, when operating an organizational system, it is necessary to observe and have a clear exposure to the impacts of specific preferences and decisions on other decisions or relationships in the organizational context, whether in public or private entities.

Regarding the aspect of transparency in the decision-making process, the model explores the construction of all preferences and traceability. This factor becomes relevant, exposing the interconnections between the variables, established preferences, and associations between the perceptions exposed among the stakeholders of the decision-making process. With the SAPEVO-H² method, decision-making processes in environments that are not only organizational but also public policy can prove to be a great ally when considering the transparency in the attributions and what the aggregation between the factors corroborates to lead to the effectiveness of a decision action.

The practical applicability of this model is varied, especially in complex decision-making situations, such as project selection, investment evaluation, or business strategies. Its ability to provide a detailed hierarchical view of preferences can be valuable to managers, helping them prioritize actions and resources more informedly. In the case of an interactive model, the development of an open computational model with online access favoring society, the implementation of the model in different scenarios of decision-making, and the exposition of all mathematical processes performed in the conversion of ordinal inputs into cardinal punctuations, as well as the construction of the final preferences between the variables at all established levels. Being an online access software, the computational model will bring a format of asynchronous access and evaluation among the decision-makers of the decision-making process, with each one being able to participate and evaluate the process through their device.

Considering that the assignment of preferences between variables primarily influences the construction of the final decision, the pairwise comparison between variables is favorable, considering that it is not always simple to perform a global preference assignment of a variable. In this context, preferences are clarified through a sequence of preference assignments between only two variables. In addition, the possibility of evaluating the consistency of attributions clarifies how committed each decision-maker can be to the decision-making process.

The quantitative increment provided by the normalization through the hyperbolic tangent function introduces a new perspective, particularly favorable in environments characterized by uncertainty. In the context of quantitative input analysis, the model offers three different functions, facilitating the application of modelling to real-world problems where using qualitative and quantitative data for decision-making is common. However, a limitation of the current proposal is that the axiomatic structure does not yet incorporate a treatment for uncertainty. Addressing this gap will be a focus of future research efforts.

An additional consideration for extending the applicability of the SAPEVO-H² model lies in addressing scenarios where multiple evaluators assess identical criteria or variables within the hierarchy. While the current model accommodates the aggregation of multiple inputs, it does not explicitly incorporate advanced negotiation or conflict resolution mechanisms that naturally arise in such contexts. This limitation suggests an opportunity for future work to integrate consensus-building algorithms or interactive decision-support tools.

To illustrate the viability of the SAPEVO-H² method, we compare it with other well-known multi-criteria methods, such as SAPEVO-M, AHP, ELECTRE, and PROMETHEE. Each method has unique characteristics, and the following discussion highlights the differences and advantages of SAPEVO-H² based on the findings. SAPEVO-H², like SAPEVO-M and PROMETHEE, is designed for outranking problems, where alternatives are compared to determine their relative ranking. In contrast, AHP is primarily used for selection problems, and ELECTRE is used for classification. The SAPEVO-H² method focuses on outranking, allows it to prioritize variables effectively, and clarifies how much better a variable is than the other in a cardinal aspect.

The method employs a hierarchical model with multiple levels, including objectives, criteria, and alternatives. This structure is more detailed than the two-level structures used by SAPEVO-M, AHP, ELECTRE, and PROMETHEE, which only consider criteria and alternatives. The hierarchical approach of SAPEVO-H² enables more nuanced decision-making by incorporating various levels of decision criteria, making it particularly suitable for complex decisions that require comprehensive analysis. Based on this hierarchical structure, the model features allow multiple decision-makers to evaluate the same process according to their responsibilities and expertise. This differs from SAPEVO-M, where all decision-makers evaluate all variables, and AHP performs individual assessments followed by aggregation using the geometric mean. ELECTRE and PROMETHEE typically require consensus for group analysis, making them less flexible in accommodating diverse opinions. The asynchronous structure of the SAPEVO-H² method can be favored for more specialized evaluations, leveraging the specific knowledge of each decision-maker.

The SAPEVO-H² method is flexible in accepting both ordinal and cardinal data inputs, different from SAPEVO-M, which is limited to ordinal inputs and scenarios with fewer criteria and alternatives, AHP, which is constrained by the Saaty scale for input data, and ELECTRE and PROMETHEE that are limited to cardinal data. This specificity enables the methodology to integrate a broader range of information, making it adaptable to various decision-making contexts and data types. In this case, the model stands out considering qualitative and quantitative variables, making it adaptable to complex and straightforward decisions with varying hierarchies.

Some limitations of SAPEVO-H² are that all variables need to be evaluated to construct the aggregation, and if more than ten variables are considered at a single level, the use of cardinal inputs for preferences is suggested. AHP in the traditional format is limited to fifteen criteria or alternatives and only considers the Saaty scale for preferences, making it unsuitable for inconsistent data. ELECTRE and PROMETHEE do not handle qualitative inputs and rely heavily on direct attribution for weight representation.

In addition, we acknowledge that AHP effectively addresses hierarchical decision problems, although the literature examples rarely demonstrate applications with more than three levels. AHP remains a foundational tool for hierarchical decision analysis, and the proposed model offers several advantages in addressing complex decision scenarios. Notably, SAPEVO-H² accommodates deeper hierarchical structures beyond the common three-level frameworks and supports distributed evaluation by multiple experts across different decision levels. This flexibility enables the model to capture specialized knowledge and diverse stakeholder perspectives effectively. Moreover, integrating ordinal-to-cardinal smoothing transformations enhances stability and interpretability in preference aggregation.

Regarding the presented model, future work will focus on extending the SAPEVO-H² framework to incorporate non-compensatory aggregation methods, including outranking approaches such as ELECTRE and PROMETHEE. These methodologies can address the limitations of compensatory models by reducing or eliminating the compensability between criteria, thus providing decision-makers with more nuanced prioritization that respects critical thresholds and veto effects. Combining hierarchical evaluation with non-compensatory aggregation can improve robustness and stakeholder satisfaction, particularly in complex and sensitive decision environments.

Through the mathematical feasibility of implementing the case study related to the prioritization of projects in the scope of sustainability in the automotive industry, it was possible to understand the objectives to be achieved and have these as a reflection in the construction of the importance among the variables that led to the final prioritization of the projects. The implementation of SAPEVO-H² modeling clarified the variables at their evaluation levels, thus enabling the aggregation between them at different levels and presenting the aggregate performance of the alternatives, criteria, and objectives. Regarding the consistency of the pairwise evaluations, high and medium consistency indices were found in all parts analyzed. In addition to the aggregation model, the analyses of outranking and sensitivity proved relevant because of the recognition of the perception of the robustness of the mathematical proposal.

Implementing the SAPEVO-H² method in the case study demonstrates its viability in connection with organizational development. By focusing on sustainable solutions within the automotive industry, the model aligns closely with the Sustainable Development Goals (SDGs), particularly in minimizing polluting emissions [61]. This case study, involving an automobile manufacturing company, serves as a practical validation of the method. The case study emphasized prioritizing projects to reduce emissions and improve environmental sustainability, showcasing the model’s effectiveness in real-world applications.

This approach not only reinforces the commitment to environmental sustainability but also highlights the versatility of the SAPEVO-H² model in adapting to different sustainability variables. The flexibility of the model allows for incorporating a broad spectrum of criteria, whether environmental, social, or economic, thereby reflecting the interconnected nature of the SDGs. By providing a tool to assess and prioritize projects, SAPEVO-H² ensures that initiatives making significant contributions to global sustainable development goals are given due consideration. The successful application of this model in the Energy Car case study underscores its potential to guide decision-making in diverse organizational contexts.

Within the Contributions to Sustainable Policies framework, the proposed method emerges as an enabler for the design and evaluation of policies that promote the SDGs. By enabling detailed analysis of different sustainability scenarios and their repercussions on sustainable development goals, the model provides a solid basis for informed decision-making. This is particularly valuable for policymakers, who can utilize the insights generated by model implementations to develop more effective SDG strategies. The ability of the model to integrate diverse perspectives and provide a multi-criteria assessment means that it can serve as an instrument in promoting innovative and sustainable solutions, encouraging a more holistic and integrated approach to sustainability across different sectors and organizational and government levels.

It is understood that all the amounts listed through the attributions made between the problem variables reflect the point of view of the organization’s stakeholders used as a basis for validating the proposed model, not necessarily representing a general opinion of the automotive industry in terms of sustainability. In line with some limitations of the study, we highlight that the study brings the perception of restricted decision-making to the prioritization of projects in the scope of sustainability in an automobile manufacturer, where the integration of new decision-makers or the inclusion of new variables, whether objectives, criteria or alternatives, may reflect a new scenario with different results from those exposed in this study.

6. Conclusions

This study introduces a methodological approach called SAPEVO-H², based on the principles of MCDA models. This model offers an innovative perspective structured in a hierarchical framework of multilevel evaluation. It skillfully incorporates the participation of multiple decision-makers in an evaluation process capable of simultaneously dealing with qualitative and quantitative data, considering asynchronous evaluations, and partitioning stakeholders for evaluations of the variables aligned with their respective responsibilities.

The proposed approach fluidly integrates quantitative information through cardinal evaluations into an ordinal evaluation model, a hallmark of the SAPEVO family methodologies. The model establishes a consortium of decision-makers, outlining the assessments according to the elements of their expertise, which are segmented into strategic, tactical, and operational levels, thus addressing the various levels of impact within the situation under analysis.

The robustness of the model is highlighted by its implementation approach, validating the preferences inserted into the axiomatic processing process and enhancing the consistency and reliability of decision-making analyses. In addition, this model allows for a comprehensive exploration of hierarchical elements at all levels of evaluation, employing an additive aggregation approach. Additionally, it enables an overrating analysis, discerning the relative superiority of elements within each assessed level based on the overrating relationship between them.

To validate the proposed approach, the study uses the analysis and prioritization of projects in the scope of sustainability in a car manufacturer as a case study, exposing the feasibility and flexibility of application in a real scenario of decision-making. It begins by elucidating the understanding of the problem and its structure of objectives to be reflected in the criteria and projects listed. This analysis reveals the preferences between the variables of each level, elucidating their influences on the operational levels and subsequently outlining the performance by preferences of the projects in all the variables at the other levels. In addition, the outranking assessment examines the relationships between the variables at each level of assessment, clarifying their degrees of superiority over the others in detail.

From the findings presented throughout this study, we found that the adaptability and flexibility of this model make it applicable in several high-level decision-making domains. Future studies will focus on the evolution of this framework, seeking its application and testing in other decision-making environments, identifying areas for improvement, and strengthening its robustness. The imminent development of an online computing platform will integrate multiple decision-makers, fostering a dynamic real-time evaluation environment leveraging numerical and graphical resources to support improved decision-making processes.