Develop a Hybrid Neutrosophic Fuzzy Approach with Score Matrix Methods for Vague Attribute Weight Determination and Multi-Attribute Group Decision Making Problems

In Multiple Attribute Group Decision-Making (MAGDM) scenarios, increasing the number of decision-makers introduces difficulty, inconsistency in conditions, and ambiguity in information. During this process, clear details about characteristic measurements are often unavailable, leading to the use of interval numbers instead of actual integers [1]. Decision-makers use interval numbers, which lack clear preference data, as the evaluation language in uncertain MAGDM. Thus, handling interval variables becomes a primary challenge. When a decision-making issue involves numerous attributes, it is referred to as a multi-attribute decision-making MADM issue [2]. This type of issue involves ranking options and choosing the best one. Since the 1960s, MADM has significantly advanced and found widespread use as a crucial component of contemporary decision-making technology. Its approach and concept have been utilized in numerous domains, including investment choices, research analysis, personnel administration, supplier selection, healthcare equipment choice, and sustainable urban development [3]. The two fundamental components of a comprehensive MADM process are the combination and interpretation of selection data. The accurate expression of decision data through language usage can be referred to as the description of decision data [4]. In real-world settings, decision-makers' analysis can be ambiguous and complicated, making it challenging to provide appropriate analysis values for MADM situations when faced with insufficient, erroneous, or incorrect data. Fuzzy language is considered the most effective tool for expressing such fuzzy facts [5].

As a result, various sets describing unclear decision data have emerged, such as fuzzy sets, soft sets, pathogenic hypersoft sets, Neutrosophic Sets, and their variations. Issues involving collaborative decision-making can be broadly characterized as scenarios where several specialists or decision-makers attempt to reach a common solution to a decision issue, comprising a collection of potential answers or alternatives expressed in terms of intuitionistic fuzzy sets [6]. Experts are required to provide their unique perspectives on each option. Issues often arise when some specialists feel their perspectives have not been fully considered, leading to disputes over the outcome. This can result in actions that go against the chosen course of action or a lack of commitment in future group decision-making scenarios. Therefore, there is a growing need in various societal contexts for decisions to be made by consensus. Certain crucial scenarios in MAGDM issues are supported and handled using the Gaussian distribution approach [7].

A Neutrosophic Set explicitly characterizes the indefiniteness, in contrast with intuitionistic fuzzy collections along with gap value intuitionistic sets of fuzzy values. Three fundamental elements make up a Neutrosophic collection: truthfulness membership (T), indeterminacy membership (I), and falsity membership (F), all of which are specified separately from each other [8]. Using a Neutrosophic Set in actual intellectual and technical domains would be a greater challenge. It also supplied set-theoretic procedures and various features of Single-Valued Neutrosophic Sets (SVNSs) and Interval Neutrosophic Sets (INSs). SVNSs convey data that is ambiguous, vague, incompatible, and lacking in the actual world. Handling such ambiguous and inconsistent data would be more appropriate [9].

A novel approach to multicriteria fuzzy decision-making issues was introduced, where the options' attributes are expressed by interval-valued fuzzy sets. Methods for determining the level of resemblance between interval-valued fuzzy sets were also established [10]. Since actual intervals, as opposed to clear numerical ranges between 0 and 1, may be used to indicate the criterion values of the alternatives, the proposed technique is more adaptable than the one previously described [11].

An interval-valued intuitionistic analysis matrix method for organizing and making decisions is presented, based on the mathematical aggregating generator and hybrid aggregating operation. Real-world example is provided to confirm the efficacy of the created method. It has been shown that Vague Sets constitute intuitively fuzzy sets after several scholars examined the ambiguity of information [12]. The vague theory of sets has drawn greater interest since it was first mentioned in the scientific literature. Due to the inherently biased nature of mental processes and the increasing complexities of social and economic circumstances, numerous everyday situations involve data in the form of imprecise valuations. In a vague set, interval-based participation is employed rather than point-based membership, as it is in fuzzy sets [13]. A thorough analysis of the differences between fuzzy sets with intuition and Vague Sets was provided, considering algebraic characteristics, graphic depictions, and real-world applications. Investigators have attempted to explore the vagueness of information through the use of intuitionistic fuzzy sets andVague Sets. Distinctions between intuitionistic fuzzy sets and Vague Sets were found in both graphic and algebraic terms [14].

A new approach for calculating the relationship coefficient of period insufficiency, ranging from [-1, 1], using alpha-cuts over vague studies through statistical confidence intervals is demonstrated by example. An attempt was made to establish the connection coefficients of interval unclear sets within the range [0, 1] [15]. This study introduces a novel approach that yields an interval Vague Correlate Coefficient of Intervals Vague Sets, which serves as a connection coefficient. Ambiguous datasets were kept consistent by employing dependencies on information, addressing the use of fuzzy logic in relational database systems to extract more significance from the information [16]. It is demonstrated that ambiguity in information values and imprecision in their relationships can be represented by a fuzzy relationship information structure with appropriate interpretation for the fuzzy membership functions. Research has been conducted on connection operators for fuzzy relationships and the suitability of fuzzy logic for expressing integrity requirements. Fuzzy Functional Dependence (FFD) is a generalization of classical functional dependency defined by incorporating a fuzzy similarity measurement equivalent for domain value comparison [17]. A set of beneficial and comprehensive reasoning axioms has been provided, and the resulting issue of FFDs has been studied. The issue of efficient join reduction of fuzzy relationships for a given collection of fuzzy functional relationships is examined [18]. It is demonstrated that the conceptual framework of existing relational databases with functional dependency can be expanded to fuzzy relationships meeting fuzzy functional requirements by imposing an appropriate limit on equals [19].

1.1 Problem statement

Accurately establishing the weights of characteristics is crucial in the world of decision-making, especially in complicated circumstances combining many qualities and group inputs. Nevertheless, these circumstances frequently involve ambiguity and uncertainty makes determining weight precisely difficult. In real-world circumstances, there is often inherent ambiguity and unclear data that is difficult to handle using existing techniques.

1.2 Challenges

• Vagueness and Ambiguity: The process of determining the weight of a characteristic can be complicated by its ambiguous borders and inherent vagueness, which are common in decision-making scenarios.
• Group Decision-Making: Combining the choices of multiple decision-makers introduces additional complexity, necessitating techniques capable of integrating diverse inputs.
• Indeterminate Information: Existing approaches often fail to sufficiently handle the partial, incompatible, or ambiguous data present in real-world decision-making situations.

1.3 Motivation

Robust MAGDM techniques are becoming increasingly important in the modern complex decision-making environment, especially in industries such as financial services, healthcare, and technology. When faced with the complexities of everyday situations, marked by confusion, unpredictability, and the need for consensus among multiple decision-makers, existing decision-making approaches often falter. The development of a Hybrid Neutrosophic Fuzzy Approach with Score Matrix Methods (HNFA-SMM) for MAGDM issues and vague component weight estimation are driven by several key factors:

Handling Vagueness and Uncertainty: Real-world decision-making situations frequently involve ill-defined characteristics that display vagueness and ambiguity. This ambiguity might not be sufficiently captured by existing fuzzy sets or traditional crisp sets. Incorporating degrees of reality, unpredictability, and falsehood into Neutrosophic Fuzzy Sets (NFSs) allows for a more sophisticated treatment of this ambiguity and a more realistic portrayal of actual circumstances.

Complexity of Group Decision-Making: Decision-making processes often require input from a variety of specialists or stakeholders, each with different preferences and viewpoints. Combining these diverse contributions fairly and meaningfully presents several challenges. A robust MAGDM model must skillfully integrate these divergent perspectives to reach a consensus that accurately represents the collective viewpoints of the group.

Need for Accurate Attribute Weight Determination: The weights assigned to each attribute significantly impact the overall decision-making process. Accurate determination of these weights is crucial for the effectiveness of the decision-making model, ensuring that each attribute's importance is appropriately reflected in the final decision.

Precisely assigning weights to characteristics is essential in the field of MAGDM, especially when dealing with imprecise and ambiguous data. Although both existing fuzzy and Neutrosophic techniques have their drawbacks, they provide structures for addressing uncertainty and ambiguity. A hybrid model that combines these strategies, strengthened by score matrix techniques, may offer a more reliable and adaptable solution.

Neutrosophic logic, which includes three elements—Truth (T), Unpredictability (I), and Falsity (F)—expands classical logic to account for indeterminacy. Fuzzy logic uses degrees of participation to describe uncertainty in information, addressing approximate thinking rather than predetermined and accurate thinking. By combining fuzzy and Neutrosophic logic, the hybrid Neutrosophic fuzzy modelling approach can manage various forms of ambiguity and unpredictability, capturing more details about characteristics in MAGDM. Score matrix techniques provide a systematic approach to evaluating and ranking options according to a set of standards, making it possible to measure and compare the efficacy of different options effectively.

The proposed technique is divided into several steps: First, the selection issue and criteria are determined, with each criterion precisely described and assigned a predetermined measuring scale. Then, decision-makers provide information used to build NFS based on all criteria, employing fuzzy membership functions to address vagueness and defining membership functions for truth, unpredictability, and falsehood. Next, for each decision-maker, a score matrix is generated, where each cell contains a Neutrosophic fuzzy number indicating how well an option performs relative to a criterion. Rows represent options, while columns represent criteria. This comprehensive technique efficiently handles unclear characteristic weight calculation in MAGDM, ensuring a more thorough and accurate analysis.

3.1 Preliminaries

Definition 3.1.1: Let E is collection of variables and X starting point global collection. Iff $E$ maps into $F$ of all fuzzy subsets of X, then the pair ($\mathrm{F}, \mathrm{E}$) - fuzzy soft set over X.

Let $O=\left\{O_1, O_2, \ldots, O_n\right\}$ - set of n alternatives; $\mathrm{D}= \left\{d_1, d_2, \ldots, d_n\right\}$ - set of Decision Makers (DMs); $\lambda= \left\{\lambda_1, \lambda_2, \ldots, \lambda_n\right\}$ - weight vector of DMs $\lambda_k \geq 0, k=1,2, \ldots, l$ and $\sum_{k=1}^l \lambda_k=1 ; U=\left\{u_1, u_2, \ldots, u_n\right\}$ - set of attributes m.

Definition 3.1.2: Fuzzy Sets

Let fuzzy set $\bar{A}$ in a universe of discourse $X$ is characterized by a membership function $\mu_{\bar{A}}: X \rightarrow[0,1]$, where, $\mu_{\bar{A}}(x)$ the degree of membership of an element x in the fuzzy set $\bar{A}$.

$\bar{A}=\left\{\left(x, \mu_{\bar{A}}(x)\right) \mid x \in X\right\}$              (1)

Definition 3.1.3: Neutrosophic Set

Let $\bar{N}$ in a universe of discourse Neutrosophic Set $X$ is characterized by three membership functions: falsity membership $F_{\bar{N}}$; truth membership $T_{\bar{N}}$ and indeterminacy membership $I_{\bar{N}}$. Each function maps X to the interval $[0,1]$.

$\bar{N}=\left\{\left(x, T_{\bar{N}}(x), F_{\bar{N}}(x), I_{\bar{N}}(x)\right) \mid x \in X\right\}$                   (2)

Definition 3.1.4: Score Matrix Method

The Score Matrix Method involves constructing a matrix to evaluate alternatives based on criteria. Let $A= \left\{A_1, A_2, \ldots, A_m\right\}$-set of alternatives and $C=\left\{C_1, C_2, \ldots, C_n\right\}$-set of criteria. The score matrix $S$ is defined as follows:

$S=\left[S_{i j}\right]_{m \times n}$         (3)

where, $s_{i j}$ is alternative score $A_i$ based on criterion $C_j$. In a hybrid Neutrosophic fuzzy context, $s_{i j}$ is represented by a Neutrosophic fuzzy number.

$s_{i j}=\left(\mu T_{i j}, T_{i j}, I_{i j}, F_{i j}\right)$           (4)

Definition 3.1.5: Aggregation of Scores

To aggregate the scores, define an aggregation operator. One common approach is the weighted average, where each criterion $C_j$ is assigned a weight $W_j$.

$S_i=\sum_{j=1}^n w_j \cdot s_{i j}$            (5)

For hybrid NFS, the aggregation is applied to each component separately.

$S_i=\left(\sum_{j=1}^n w_j \cdot \mu T_{i j}, \sum_{j=1}^n w_j \cdot T_{i j}, \sum_{j=1}^n w_j \cdot I_{i j}, \sum_{j=1}^n w_j \cdot F_{i j}\right)$       (6)

where, $\mu_{i j}, T_{i j}, I_{i j}, F_{i j}$ are the components of the Neutrosophic fuzzy number $s_{i j}$.

Definition 3.1.6: Defuzzification and Decision Making

To make a final decision, the aggregated Neutrosophic fuzzy scores need to be defuzzified. One common defuzzification method is the centroid method. For a hybrid Neutrosophic fuzzy number $S_i=\left(\mu T_i, T_i, I_i, F_i\right)$ the defuzzified score $D_i$ can be calculated as:

$D_i=\frac{\mu T_i, T_i+\left(1-I_i\right)+\left(1-F_i\right)}{4}$        (7)

where, $S_i=\left(\mu T_i, T_i, I_i, F_i\right)$ is the aggregated Neutrosophic fuzzy score of alternative A_i.

Definition 3.1.7: Final Decision

Alternative defuzzified score

$A^*=arg \min _i D_i$            (8)

These preliminaries provide the foundational concepts and mathematical formulations necessary for developing an HNFA-SMM for vague attribute weight determination and MADGM problems.

Definition 3.1.8: Vague Sets defined operator Vague Weighted Averaging (VWA)

Rules for VWA

$w_i^*=0\ i f\ 1 \leq i \leq r$,              (9)

$w_r^*=\frac{6(n-1) \alpha-2(n-r-1)}{(n-r+1)(n-r+2)}$           （10）

$w_n^*=\frac{2(2 n-2 r+1)-6(n-1) \alpha}{(n-r+1)(n-r+2)}$           （11）

$w_i^*=\frac{n-i}{n-r} w_r^*+\frac{i-r}{n-r} w_n^*$ if $r<i<n$           （12）

With every component being identified by a Neutrosophic fuzzy number (NFN) with partial knowledge of characteristic lifting weights. The score operation is then utilized to determine the rating of every attribute appreciated to generate the achieved matrices of the collectively Neutrosophic fuzzy choice matrix.

Everyone will provide an approach consisting of formulas and figures to construct a Hybrid Neutrosophic Fuzzy Approach (HNFA) with Score Matrix Methods for VWA calculation. Fuzzy logic and Neutrosophic reasoning are used in this method to successfully handle imprecise property values.

3.2 Method 1: HNFA-SMMs

Input

$A=\left\{A_1, A_2, \ldots, A_m\right\}$: Set of alternative

$C=\left\{C_1, C_2, \ldots, C_n\right\}$: Set of criteria

$W=\left\{w_1, w_2, \ldots, w_n\right\}$: Weights assigned to each criterion $C_j$

$S=\left[s_{i j}\right]$: score matrix where $s_{i j}$ represents the Neutrosophic fuzzy score of alternative $A_i$ with respect to criterion $C_j$.

Output

Final Weight Vector $W^*$; Determined weights for each criterion

Step 1: Initialization

Initialize an empty hybrid NFSs $\widetilde{H}$.

Step 2: Score Matrix Construction

Construct the score matrix $S=\left[s_{i j}\right]$ where each $s_{i j}$ is a Neutrosophic fuzzy number representing the score of alternative $A_i$ on criterion $C_j$.

Step 3: Weighted Aggregation

For each alternative $A_i$ compute the aggregated Neutrosophic fuzzy score $\mathrm{S}_{\mathrm{i}}$ using the weighted average method using Eq. (6)

Step 4: Defuzzification

Defuzzify the aggregated Neutrosophic fuzzy scores $S_i$ using the centroid method using  Eq. (7).

Step 5: Determine final weights

Normalize the defuzzied scores $D_i$ to obtain the final weights $W^*$:

$w_j^*=\frac{D_j}{\sum_{i=1}^n D_i}, j=1,2, \ldots, n$           (13)

Pseudocode

Input:

    $\mathrm{A}=$ set of m alternatives

    $\mathrm{C}=$ set of p criteria

    $D=$ set of $n$ experts

    $\lambda=$ indeterminacy penalty

    NFN[i][j][k] = (T, I, F) triplet from expert k

Output:

    $\mathrm{X}=\mathrm{m} \times \mathrm{p}$ score matrix

Begin

    for $i=1$ to $m$ : \# Loop over alternatives

        for $\mathrm{j}=1$ to p : \# Loop over criteria

        sum ascore $=0$

        for $\mathrm{k}=1$ to n : \# Loop over experts

            $(\mathrm{T}, \mathrm{I}, \mathrm{F})=\mathrm{NFN}[\mathrm{i}][\mathrm{j}][\mathrm{k}]$

            score $\mathrm{k}=\mathrm{T}-\mathrm{F}-\lambda * \mathrm{I}$

            sum score $+=$ scoreach

        $\mathrm{X}[\mathrm{i}][\mathrm{j}]=$ sum and score $/ \mathrm{n}$

    return X

End

Example 1: Let's illustrate this algorithm with a hypothetical scenario involving three alternatives $A_1, A_2, A_3$ and four criteria $C_1, C_2, C_3, C_4$.

Score Matrix Construction: The score matrix S is constructed as follows:

$S=\left[\begin{array}{llll}(0.6,0.7,0.1,0.2) & (0.4,0.6,0.3,0.4) & (0.5,0.8,0.2,0.1) & (0.7,0.5,0.4,0.3) \\ (0.3,0.4,0.5,0.6) & (0.8,0.6,0.4,0.2) & (0.2,0.3,0.7,0.5) & (0.6,0.4,0.6,0.3) \\ (0.4,0.6,0.3,0.4) & (0.7,0.5,0.2,0.1) & (0.6,0.7,0.4,0.2) & (0.5,0.8,0.3,0.1)\end{array}\right]$

Assume initial weights W = [0.3, 0.2, 0.3, 0.2].

Weighted Aggregation: Compute the aggregated NFS Score $S_i$ for each alternative A_i:

$\begin{gathered}S_1=(0.5,0.7,0.3,0.25) ; S_2=(0.5,0.5,0.4,0.375) \\ S_3=(0.5,0.6,0.3,0.25)\end{gathered}$

Defuzzification: Defuzzify the aggregated scores S_i:

$\begin{gathered}D_1=\frac{0.5+0.7+(1-0.3)+(1-0.25)}{4}=0.6125 \\ D_2=\frac{0.5+0.5+(1-0.4)+(1-0.375)}{4}=0.55 \\ D_3=\frac{0.5+0.6+(1-0.3)+(1-0.25)}{4}=0.6125\end{gathered}$

Determine final weights: Normalize the defuzzified scores $D_i$ to obtain $W^*$:

$\begin{gathered}w_1^*=\frac{0.6125}{0.6125+0.55+0.6125} \approx 0.378 \\ w_2^*=\frac{0.55}{0.6125+0.55+0.6125} \approx 0.343 \\ w_3^*=\frac{0.6125}{0.6125+0.55+0.6125} \approx 0.378 \\ W^*=[0.378,0.343,0.378]\end{gathered}$

This algorithm effectively combines fuzzy logic and Neutrosophic logic to handle vague attribute weight determination in MAGDM scenarios. Adjustments can be made in practice to suit specific contexts and requirements.

Table 1. Score matrix for alternatives A_i and Criteria $C_j$

Each cell $(i, j)$ in Table 1 represents the NFS score of alternative $\mathrm{A}_i$ based on criterion $C_j$. The score $(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d})$ corresponds to $\mu T_{i j}=a, T_{i j}=b, I_{i j}=c, F_{i j}=d$. These Scores are used in the weighted aggregation step of the algorithm to compute the aggregated NFS score $S_i$ for each alternative $A_i$. These scores are used in the weighted aggregation step of the algorithm to compute the aggregated NFS score $S_i$ for each alternative $A_i$. Table 1 helps visualize how each alternative performs across different criteria in a NFS facilitating decision-making processes in contexts where attribute weights are vague or uncertain. Adjustments and expansions can be made based on specific needs and additional criteria. To successfully address the imprecise attributes weight analysis and decision-making difficulties, an HNFA with score matrix (SM) for MAGDM method must be developed in several phases. This is a description of organized algorithms for this method:

Algorithm 2: Hybrid Neutrosophic fuzzy method for MAGDM with score matrix

Input

n: No. of decision-makers (experts); m: No. of alternatives; p: No. of criteria.

$X=\left\{X_{i j}\right\}$         (14)

where, $X_{i j}$ represents the Neutrosophic fuzzy scores for alternative A on criterion $C_j$.

$W=\left\{w_j\right\}$            (15)

Initial attribute weights (may be vague or uncertain)

$\varepsilon$: Convergence threshold; $\lambda$: Learning rate for weight adjustment.

Output

$S=\left\{S_i\right\}$                 （16）

Neutrosophic fuzzy aggregated scores for each alternative $A_{i .}$

Step 1: Initialize the attribute weights

$W=\left\{w_j^0\right\}$           (17)

with initial values, set iteration counter k = 0.

Step 2: Iterative weight adjustment loop

Repeat until convergence:

Step 2.1: Weight normalization

Normalize weights to ensure consistent scaling:

$w_j^k=\frac{w_j^k}{\sum_{j=1}^P w_j^k}$            (18)

Step 2.2: Neutrosophic fuzzy score aggregation

Calculate aggregated score $S_i$ for each alternative $A_i$:

$S_i=\frac{\sum_{j=1}^P w_j^k \cdot X_{i j}}{\sum_{j=1}^P w_j^k}$               (19)

Here, $X_{i j}$ can be a composite Neutrosophic fuzzy value, e.g., based on:

$X_{i j}=T_{i j}-F_{i j}-\lambda \cdot I_{i j}$               (20)

where, T, I, F denote the truth, indeterminacy, and falsity memberships respectively.

Step 2.3: Weight update using deviation from mean

Adjust weights to reflect their influence relative to the average across all alternatives:

$w_j^{k+1}=w_j^k+\lambda\left(X_{i j}-\frac{1}{m} \sum_{i=1}^m X_{i j}\right)$                  (21)

Step 3: Convergence check

Check if weight changes are minimal:

$\max \_$diff $=\max \left(\left|w_j^{k+1}-w_j^k\right|\right)$                 (22)

If max_diff < $\varepsilon$, stop the iteration, otherwise, increment k and repeat step 2.

Step 4: Output aggregated scores

Return the final aggregated scores:

$S=\left\{S_i\right\}$               (23)

for i = 1, 2, ..., m

Computational complexity: For K iterations:

• Score aggregation: O(m⋅p)
• Weight update: O(p⋅m)
• Convergence check: O(p)

Overall complexity: O(K⋅m⋅p)

Scalability: Efficient for small to moderate mmm and p; for large-scale problems, parallelizing score and weight updates is recommended.

Establishing the attribute's values and constructing the score matrix are the first steps in the method. It calculates the aggregated NFS scores for every option and repeatedly modifies the weights depending on the results. Neutrosophic fuzzy logic is utilized in this technique to manage ambiguity in processes of decision-making, and it supports ambiguous or unclear characteristic weights. The convergent criteria guarantee that the procedure finishes when the weights stabilize or converge.

Modify variables like $\epsilon$ (integration criterion) and $\lambda$ (learning rate) by specific industry needs and empirical results. The approach can be modified to include extra restrictions or complications unique to the existing MAGDM situation.

Example 2: After applying the HNFA-SMMs, the following results were obtained:

Attribute Weights (Final)

Criterion 1 (w₁): 0.3

Criterion 2 (w₂): 0.2

Criterion 3 (w₃): 0.5

These weights represent the final adjusted values after the iterative process.

3.3 Neutrosophic fuzzy aggregated scores

Table 2 interprets as follows:

$S_i^{\text {Membership}}$: Indicates the degree of fulfillment or acceptance of each alternative.

$S_i^{\text {Indeterminary}}$: Shows the level of uncertainty or ambiguity associated with each alternative.

$S_i^{\text {Non-membership}}$: Reflects the degree of rejection or non-fulfillment.

Table 2. NFS aggregated scores S_i for each alternative A_i

Interpretation: Attribute weights: The final attribute weights w₁ = 0.3, w₂ = 0.2, and w₃ = 0.5 indicate the relative importance of each criterion in the decision-making process. To demonstrate how well the proposed HNFA-SMM works for unclear characteristic weight identification and MAGDM issues, the developers compare it to an established technique called Technique X, which is frequently employed in situations involving similar making choices.

Example 3: Problem Description: Explore an issue of decision-making in which a committee must assess and choose, based on many criteria, the best supplier for a manufacturing contract: Price, Excellence, Delivered Time, and Services. Every attribute is rated from 1 to 10, with higher numbers denoting better results.

Data

Dealer A: Quality (8), Price (7), Service (9), Delivery Time (6)

Dealer B: Quality (7), Price (6), Service (8), Delivery Time (8)

Dealer C: Quality (6), Price (8), Service (7), Delivery Time (7)

3.4 Existing method (NFA) results

Using method X, the attribute weights were determined as follows:

Price: 0.3

Quality: 0.2

Delivery Time: 0.3

Service: 0.2

Score Calculation (NFA):

For Dealer A:

Score $_{A, N F A}=7 \times 0.3+8 \times 0.2+6 \times 0.3+9 \times 0.2=7.5$

For Dealer B:

Score $_{B, N F A}=6 \times 0.3+7 \times 0.2+8 \times 0.3+8 \times 0.2=7.0$

For Dealer C:

Score $_{C, N F A}=8 \times 0.3+6 \times 0.2+7 \times 0.3+7 \times 0.2=7.1$

3.5 Proposed HNFA-SMM results

Using HNFA-SMM, the attribute weights were determined through the HNFA detailed earlier.

Price: 0.25

Quality: 0.15

Delivery Time: 0.3

Service: 0.3

Score Calculation (HNFA-SMM):

For Dealer A:

$\begin{gathered} {Score }_{A, H N F A-S M M} =(7 \times 0.25+8 \times 0.15+6 \times 0.3+9 \times 0.3)=7.45\end{gathered}$

For Dealer B:

$\begin{gathered} { Score }_{B, H N F A-S M M} =(6 \times 0.25+7 \times 0.15+8 \times 0.3+8 \times 0.3)=6.95\end{gathered}$

For Dealer C:

$\begin{gathered}\text { Score }_{C, H N F A-S M M} =(8 \times 0.25+6 \times 0.15+7 \times 0.3+7 \times 0.3)=7.0\end{gathered}$

Table 3. Comparison results of proposed and existing systems

Table 3 explains that HNFA-SMM yields a similar result. HNFA-SMM scores for Dealer B are marginally lower than for NFA. Dealer C compared to HNFA-SMM, NFA yields a little higher result. Suppose that 4 DMs have been assigned to pick suppliers in a production industry. Assessing five supplier choices $(\rho ; 1,2,3,4,5)$ based on five performance criteria: shipment, quality, adaptability, Service, and pricing.