Functional dependency-based group decision-making with incomplete information under social media influence: An application to automobile

Abstract

In today’s complex decision-making environment, accounting for attribute interdependencies and expert relationships is crucial. Traditional models often assume attribute independence and overlook the significant impact of expert relationships on decision outcomes. Also, amidst the dynamic and ever-changing decision-making landscape, the effect of news and real-time updates on alternative rankings is significant. In complex decision-making environments, information is constantly evolving, and staying up-to-date with the latest developments is paramount. To overcome these limitations, this study aims to develop a novel model that effectively captures attribute dependencies and incorporates the influence of social media on alternative ordering. To establish the model, the Decision-making trial and evaluation laboratory (DEMATEL) method and regression analysis are integrated to capture attribute dependencies. Furthermore, social network analysis (SNA) is employed to develop a trust propagation model for determining experts’ weights. Additionally, we present a two-stage multi-skilled and high potential multi-criteria decision-making (MCDM) framework, where the base-criterion method (BCM) is adopted to evaluate attribute weights and the well-known traditional Vlekriterijumsko KOmpromisno Rangiranje (VIKOR) method is redefined using Heronian mean (HM) operator to capture the relationships between arguments. Despite uncertainties, the proposed fuzzy-BCM-VIKOR-Heronian (F-BCM-VIKOR-H) approach enhances flexibility by addressing inconsistent data in complex decision-making problems. Similarly, certain news or future updates about any alternative or attribute can significantly affect the ranking. Acknowledging the significance of timely information, the proposed approach actively considers the effect of such news through the formation of an updated matrix. By factoring in the latest developments, we ensure that the proposed decision-making model remains relevant and adaptable, capturing the most current insights into alternative performance. To demonstrate the model’s effectiveness, we apply the proposed approach to a numerical illustration in the electronics industry, specifically for ranking cars. Sensitivity analysis evaluates the model’s stability, and comparing the results with existing approaches showcases its advantage and superiority.

Keywords

Group decision making VIKOR SNA attribute dependencies news influence

1 Introduction

In the dynamic environment of decision-making, the integration of real-world factors is of paramount importance for achieving accurate and informed choices. News, as a potent driver of public opinion and sentiment, has the potential to significantly impact the reputation and performance assessment of alternatives. While the influence of news on consumer perceptions and decision-making has been widely recognized across domains like business and social media, its specific impact within decision-making frameworks and ranking methodologies remains relatively unexplored. This study aims to fill these research gaps by investigating the influence of news on rankings within a decision-making framework. Despite the acknowledged significance of news in shaping opinions and sentiments, limited research has been dedicated to comprehending how news directly affects the relative rankings of alternatives. Motivated by the imperative of informed decision-making across diverse domains such as business, finance, marketing, and public policy, this research seeks to enhance the effectiveness of ranking methodologies by capturing the influence of real-world factors. By examining the impact of news on rankings, decision-makers can be provided with more accurate, context-aware, and up-to-date insights, thereby empowering them to make well-informed choices.

Multi-criteria decision-making (MCDM) methods traditionally focus on optimizing criteria weights and alternative rankings. Over the past decades, several weight determination and ranking methods have been given by researchers [11 , 36]. Recent advancements such as the Base-Criterion method (BCM) [16] and fuzzy extensions [17] have offered promising avenues for refining decision-making processes. One of the most popular MCDM techniques, VIKOR [52], offers several compromise alternatives. The VIKOR method stands out not only for its ability to address group utility maximization and individual regret minimization but also for its capacity to accurately capture the subjective preferences of decision-makers. In order to accommodate uncertainty in decision-making scenarios, Wang and Chang [51] introduced a fuzzy extension to the VIKOR approach. The versatility and capabilities of the VIKOR method have made it applicable to a wide array of real-world decision-making problems [2 , 42]. However, despite these developments, traditional MCDM methods rely on the assumption of attribute independence, which often doesn’t hold in practical scenarios. This impracticality becomes evident when considering vital aggregation operators such as the Bonferroni mean (BM), Choquet integral (CI), Sugeno integral (SI), and Heronian Mean (HM). The HM operator, in particular, excels in measuring correlations among combined arguments, setting itself apart from other operators [15 , 49]. Consequently, the conventional MCDM methods, can introduce distortions in aggregate results, leading to inaccurate rankings of alternatives. Thus, the study combines F-BCM and VIKOR-H. The main idea for integrating F-BCM and VIKOR-H is due to the following points:

In multidimensional decision analysis systems, it may make complicated decision-making issues considerably more efficient and simpler, notwithstanding uncertainty.

When taking the decision-making process into account, F-BCM employs subjective data that mirrors the judgment of the decision-makers. The F-BCM approach provides for an entirely consistent pairwise comparison of evaluation criteria, unlike AHP and F-BWM.

The traditional VIKOR technique is enhanced by the use of innovative aggregation operators that may capture the linkages among the aggregated arguments and eliminate the outcomes of the results of the inconsistent data.

Group decision-making (GDM) has drawn a lot of attention in recent years for resolving multi-criteria decision-making issues because it integrates the skills and expertise of several specialists, resulting in a more trustworthy outcome [7, 9]. In GDM different experts have different opinions, and perceptions of the same set of alternatives, hence group consensus is important in GDM. Hard consensus is another name for traditional consensus, which solely takes into account the numbers 0 (no consensus obtained) or 1 (consensus established). However, finding harmony could be challenging. Additionally, changing the agreement level in a genuine decision-making dilemma from 0 to 1 is impracticable [20, 21]. Therefore, the concept of soft consensus [12, 56] came into existence and is extensively used in group decision problems. Significantly, in the conventional GDM procedure for obtaining agreement, the decision-making experts are assumed to be independent, with the objective trust ties between them being disregarded [27]. However, this assumption no longer seems to be accurate given the rise of online social networks. Entities have connections of trust in social network environments. Social network analysis (SNA) has been used in studies to demonstrate how decision-making complexity may be reduced while decision-making quality is improved [23, 33]. The trust ties between experts based on SNA were taken into consideration by Wu [26], who also developed a trust-based consensus model. A conflict relationship analysis approach based on SNA was used by Ding [45] to identify conflict relationships among experts. Hence, it is worth exploring and developing a novel GDM framework to facilitate decision-making in social networks.

Expert weights play a crucial role in GDM, yet traditional approaches tend to treat all experts equally, overlooking the intricate dynamics of social networks. According to SNA, strong social ties suggest a higher level of trust and collaboration between members, making them more inclined to openly share opinions and expertise. Conversely, members with more connections, such as numerous friendships or interactions within the social network, hold greater importance and influence within the group. Based on this concept, the paper introduces a trust propagation model to determine experts’ weights in GDM scenarios. The model utilizes information on social ties and connections among experts to calculate their respective weights. It considers both the trust-based collaboration potential and the influence-based importance of experts enhancing the accuracy and reliability of expert weight assignments. Additionally, it is frequently challenging for experts to provide accurate decision-making preferences due to the inherent ambiguity and uncertainty in human judgments as well as the persistence of decision-making. To convey expert preferences, TFNs are used in this paper, due to their clear interpretability, computational efficiency, broad applicability and ease of implementation. This research suggests a consensus-based approach that derives expert weights by taking into account the relationships of trust between the experts.

Furthermore, experts may not always be able to have sufficient previous knowledge of all the characteristics or options. The problem of missing information has been addressed using a variety of cutting-edge strategies, such as granular computing technology [18]. A social impact network (SIN) is utilized by Capuano [37] to approximate the missing values while taking into account the relative relevance of the experts to one another. In [24] social network community analysis is explored as a solution to the problem of imperfect fuzzy preference relations. A sophisticated approach to handle imperfect interval-valued intuitionistic multiplicative preference relations was put out by Meng [14]. When dealing with imperfect additive preference relations, Zhang [19] established a personalized individual semantics-based failure mode and impacts analysis technique. In practice, when an expert maintains strong ties with another member, they tend to seek decision support and advice more willingly from that member. Building upon this concept, the paper adopts social trust relationships between experts to estimate their incomplete information. By utilizing these trust relationships, the model aims to enhance the accuracy of expert information, providing a valuable approach for dealing with incomplete data in decision-making processes.

The study delves into a crucial aspect that is often overlooked in existing research –the inter-relationship between attributes in real-life decision-making scenarios. It highlights that attributes are not always independent of each other; rather, they exhibit either favorable or unfavorable connections. When one attribute changes, it affects another, ultimately influencing decision outcomes. Recognizing and extracting these attribute dependencies is of paramount importance for accurately describing group decision-making processes in real-world contexts. By addressing this research gap, the study brings novel insights to the forefront, contributing to a deeper understanding of decision-making dynamics in complex environments.

Considering all the aforementioned points in mind the study develops a novel group decision-making model having the following contributions:

By ingeniously integrating the F-BCM and VIKOR-H methods, the study presents a distinct viewpoint for ranking alternatives, enhancing the decision-making process.

The development of fuzzy preference relations for Triangular Fuzzy Numbers significantly reduces the computational complexity compared to earlier studies, making the model more efficient and practical.

To calculate experts’ weights in social networks, a trust propagation model is employed, surpassing the conventional practice of assigning identical weights. The study also addresses the common problem of incomplete or insufficient information by leveraging trust relationships among experts, increasing the model’s reliability.

The study focuses on determining attribute dependencies, a crucial factor often overlooked in decision-making models. By combining DEMATEL and regression analysis, the independence among attributes is explored, improving the model’s accuracy in real-world scenarios.

This study investigates the effect of social media platforms on the decision-making process. Recognizing the significance of media updates and news, the model incorporates attribute dependencies to more accurately evaluate future decisions.

In summary, this study aims to contribute to the existing body of knowledge by shedding light on the impact of news on rankings within a decision-making framework. The paper presents a fuzzy group decision-making approach capable of handling incomplete and uncertain information, while considering attribute dependencies. These findings hold tremendous potential to enhance decision-making processes across various fields. The integration of news into ranking methodologies has the potential to revolutionize how we evaluate alternatives and entities in today’s fast-paced, information-driven world.

The paper’s reminder is organized as follows: Some fundamental terms and study backgrounds are defined in Section 2. The suggested technique is described in Section 3 and is comprised of four steps. Section 4 presents a simulation experiment for resolving supplier selection challenges to demonstrate the viability and validity of the suggested strategy. Section 5 presents comparisons and debates, followed by discussions in Section 6. Section 7 concludes with the findings.

2 Preliminaries

2.1 Fuzzy measure

In recent years [30, 44] fuzzy measure showed as an effective framework to describe the vague connection.

2.1.1 Regular fuzzy measure

Definition 1 [44]. Let X represents a finite set and P(X) represent the power set of X, then a set value function μ : P (X) → [0, 1] is referred to as a regular fuzzy measure if it meets the following criteria.

μ (φ) = 0

μ (X) = 1

IfE ⊂ P (X) , F ⊂ P (X) , E ⊂ F, thenμ (E) ⩽ μ (F)

2.1.2 2-order additive fuzzy measure

Experts may not always be able to provide the values of the complete fuzzy measure in real-world circumstances. In the event that there are n qualities, experts must offer a fuzzy measure for 2ⁿ values. Thus, to address such an issue and to reduce the complexity of the computation [34] introduced the concept of 2-order fuzzy measure which also effectively considers the interaction effects.

2.2 Shapley function

In recent years the cooperative game theory has received extensive consideration [14]. The interaction between criteria is critically considered in cooperative game theory (CGT). According to some academics, measuring a singleton set is insufficient to assess the overall significance of a collection of criteria. It is also important to take into account the measure of all the subsets that contain that condition. As a result, the idea of the Shapley function developed, which is connected to the collection of fuzzy measurements to represent the significance of each unique criterion [34].

Definition 2 [34]. The Shapley value of each member x_i is defined as follows, assuming that is the fuzzy measure μ specified on a set X having n elements. $ν_{i} = \sum_{k = 0}^{n - 1} γ_{k} \sum_{K \subset X \ x_{i}, | K | = k} (μ_{i K} - μ_{K})$ (1) here $γ_{k} = \frac{(n - k - 1)! k!}{n!}$ , |K| is the cardinality of K and 0 ! =1 .

Theorem 1 [34]. For a given fuzzy measure μ defined on set X, the Shapley values for each element in X meet the requirement that $\sum_{i = 1}^{n} ν_{i} = 1 .$

2.3 Social network analysis

The SNA is a useful tool for examining the connections between initiatives and other social entities [30]. It may be used to research structural balance and location characteristics including centrality, prestige, and trust [32]. In the traditional consensus-building process of GDM, the objective trust relationships between experts are frequently disregarded. However, when taking into account real-life situations, any change in an expert’s individual opinions depend on their trust relationships with others. As a result, the final conclusions in GDM challenges might be dramatically changed by the trust relationships among specialists. Essentially, a social network ia social structure made up of a set of edges (L) and a set of nodes (E). The edges reflect the connections of trust between the nodes, or expert nodes.

A social network algorithm based on the idea of modularity, the Louvain algorithm is used by [53] and assesses the quality of partitions created. The approach can find hierarchical community structures and is computationally efficient. The Louvain algorithm’s optimization objective is to increase the modularity of the whole community network. Social network clustering makes heavy use of the Louvain method.

Definition 3 [53]. An indicator called Modularity Q is used to assess the importance of community network detection. It defines the community’s proximity and is defined as follows: $Q = \frac{1}{2 m} \sum_{i} \sum_{j} [w_{ij} - \frac{k_{i} k_{j}}{2 m}] δ (c_{i}, c_{j})$ (2) where w_ij denotes the weights of the edges between nodes i and j and m denotes the total weight of all edges. In the event that the network is not a weighted graph, the weights of all edges are assumed to be 1. The node i’s cluster is represented by c_i, and its weights are represented by k_i, respectively. We have δ (c_i, c_j) = 1 if both i and j are members of the same cluster c. If not, δ (c_i, c_j) = 0.

Definition 4 [53]. ΔQ, the measure of modularity change, is defined as follows: $\begin{matrix} Δ Q = [\frac{w_{in} + 2 k_{i, in}}{2 m} - {[\frac{w_{tot} + k_{i}}{2 m}]}^{2}] \\ - [\frac{w_{in}}{2 m} - {[\frac{w_{tot}}{2 m}]}^{2} - {[\frac{k_{i}}{2 m}]}^{2}] \end{matrix}$ (3) where w_in stands for the total weight of all the edges in cluster c, w_tot for the total weight of all the edges connected to cluster c’s points, k_i for the total weight of all the edges connected to point i, k_i,in for the total weight of all the edges connected to point i’s connections to cluster c’s points and m denotes the total weight of all edges.

2.4 Louvain algorithm

The following two steps make up the Louvain algorithm:

The expert social network relationship is produced by converting the directed graph with nodes into an undirected graph. Each individual point is treated as a cluster, its neighbor nodes are added, and ΔQ is then calculated using Equation (3). The loop is iterated until the ΔQ of each cluster is stable.

A node-like view is taken of the obtained cluster. The total of the weights of the edges between the nodes in the corresponding two clusters determines the weight of the edge between nodes. Step 1 is then performed once more till the overall modularity stays the same. The experts are grouped into K clusters using the aforementioned two stages.

3 The proposed measure based GDM model

This section describes the proposed MAGDM process in detail. The necessary computations are carried out by making the use of fuzzy measure, Shapley function, Markov stationary chain etc. The steps required to get the desired results are described into four phases:

To compile the necessary ranking of alternatives.

To acquire experts’ weights.

To obtain the membership value of attributes.

To obtain the updated ranking considering the social media influence.

The next subsections go into great depth on the four stages. Figure 1 depicts the suggested methodology’s flowchart.

Fig. 1

Flowchart of measure based GDM model.

A new approach based on Functional dependency among attributes:

The criteria are not always independent of each other, in most of the real-life circumstances there exists dependency among criteria. Hence, the consideration of dependency becomes an essential part of decision making. A change in the independent criteria can bring a change in all the dependent criteria. The dependency can be classified considering three aspects.

Positive dependence: Two or more attributes are said to be positively dependent if intensification in the independent criteria, causes the dependent criteria to increase.

Negative dependence: Two or more attributes are said to be negatively dependent if the increase in the independent criteria, results in the decreases in dependent criteria.

Independence: Two or more attributes are said to be independent if the attributes have no effect on each other.

Example: A car company decides to improves the engine capacity of the car then the company in turn increases the price of the car. Similarly, a change in the look and mileage of the car also brings a change in the price. Hence price is dependent criteria depending on all other factors. Maruti Suzuki Brezza increases its engine capacity from 1200 cc to 1462 cc which in turn increases its price from 7.82 lakh to 8.91 lakh. Thus, any news or future update about any criteria like C_i for a particular alternative, this affect all the other criteria that are related to or dependent on criteria C_i for that alternative.

For determining the dependency among attributes, we used the combination of DEMATEL and regression analysis, as described in section 3.4.1 in detail.

3.1 Proposed ranking method

This section includes a ranking approach for multi-criteria decision-making problems.

3.1.1 Determining attribute weights through F-BCM

Haseli [17] proposed Fuzzy Base-Criterion Method as an extension of BCM method. In F-BCM method the experts can represent their opinions linguistically, to capture the uncertainty and vagueness present in the human decisions. F-BCM is proficient to acquire completely reliable outcomes and compute crisp weights by means of fewer pairwise comparisons than the prevailing MCDM approaches named as BWM [25] and AHP [31]. Since secondary comparisons are not required in F-BCM, it is more accurate and has less computing cost. Hence, in this work, the Fuzzy BCM approach is favored for computing the criterion weights. The TFNs for pairwise comparisons is shown in Table 1.

Table 1
TFNs for pairwise comparison

Linguistic Set TFNs

Equally Important (1,1,1)

Moderately Important (0,0,0.25)

Essentially Important (0,0.25,0.5)

Very Strongly Important (0.25,0.5,0.75)

Extremely Important (0.5,0.75,1)

Absolutely Important (0.75,1,1)

Linguistic Set	TFNs
Equally Important	(1,1,1)
Moderately Important	(0,0,0.25)
Essentially Important	(0,0.25,0.5)
Very Strongly Important	(0.25,0.5,0.75)
Extremely Important	(0.5,0.75,1)
Absolutely Important	(0.75,1,1)

The detailed procedure of F-BCM is described as follows:

Govern the set of evaluation criteria {C₁, C₂, C₃, …, C_n} in agreement with the opinion of experts.

From the given criteria choose one criterion as the base criteria.

This phase involves the computation of pairwise comparisons. The comparative fuzzy preference of the base criteria over the remaining criteria is shown in Table 1. The resultant vector from the fuzzy base comparisons is represented as shown. ${\tilde{T}}_{B} = ({\tilde{T}}_{B 1}, {\tilde{T}}_{B 2}, {\tilde{T}}_{B 3}, \dots, {\tilde{T}}_{Bn})$

$T_{B}^{⌣}$ characterizes the fuzzy base criterion over the remaining criteria. The ambiguous importance of the base criteria relative to the other criteria is represented by $T_{Bj}^{⌣}$ .

A non-linear programming model is created to calculate the optimal fuzzy weights as follows-

\begin{matrix} \min \tilde{ξ} \\ s . t . \\ {\begin{matrix} | \frac{{\tilde{w}}_{B}}{{\tilde{w}}_{j}} - {\tilde{T}}_{Bj} | ⩽ \tilde{ξ} \\ \sum_{j = 1}^{m} C ({\tilde{w}}_{j}) = 1 \\ \begin{matrix} a_{j}^{w} ⩽ b_{j}^{w} ⩽ c_{j}^{w} \\ a_{j}^{w} ⩾ 0 \end{matrix} \end{matrix} \end{matrix}

(4) where

{\tilde{w}}_{B} = (a_{B}^{w}, b_{B}^{w}, c_{B}^{w})

represents the base criteria,

{\tilde{w}}_{j} = (a_{j}^{w}, b_{j}^{w}, c_{j}^{w})

represents the other criteria’s,

\tilde{ξ} = (a^{ξ}, b^{ξ}, c^{ξ})

represents epsilon to be minimized,

{\tilde{T}}_{Bj}

represents ambiguous importance of the base criteria relative to the other criteria,

\sum_{j = 1}^{m} C ({\tilde{w}}_{j}) = 1

presents the constraints that the sum of weights ariteria is equal to 1 and

C ({\tilde{w}}_{j})

represents the crisp value of criteria’s.

3.1.2 Ranking oalternatives through F-VIKOR-H

The stepwise procedure of F-VIKOR-H is as follows:

Evaluate the alternatives w.r.t the established attributes for the construction of the fuzzy decision matrix according for the given problem. If there are “m” criteria and “n” alternatives then the decision matrix is expressed as-

A = [\begin{matrix} (x_{11}^{a}, x_{11}^{b}, x_{11}^{c}) & (x_{12}^{a}, x_{12}^{b}, x_{12}^{c}) & \dots & (x_{1 m}^{a}, x_{1 m}^{b}, x_{1 m}^{c}) \\ (x_{21}^{a}, x_{21}^{b}, x_{21}^{c}) & (x_{22}^{a}, x_{22}^{b}, x_{22}^{c}) & \dots & (x_{2 m}^{a}, x_{2 m}^{b}, x_{2 m}^{c}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (x_{n 1}^{a}, x_{n 1}^{b}, x_{n 1}^{c}) & (x_{n 2}^{a}, x_{n 2}^{b}, x_{n 2}^{c}) & \dots & (x_{mn}^{a}, x_{mn}^{b}, x_{mn}^{c}) \end{matrix}]

Convert the non-beneficiary attributes into beneficiary attributes. If C₁ is the non-beneficiary attribute then its values are transformed into $(\frac{1}{x_{k 1}^{a}}, \frac{1}{x_{k 1}^{b}}, \frac{1}{x_{k 1}^{c}}),$ where, k denotes the number of alternatives.

Normalize the given decision matrix A. $N = [\begin{matrix} (γ_{11}^{a}, γ_{11}^{b}, γ_{11}^{c}) & (γ_{12}^{a}, γ_{12}^{b}, γ_{12}^{c}) & \dots & (γ_{1 m}^{a}, γ_{1 m}^{b}, γ_{1 m}^{c}) \\ (γ_{21}^{a}, γ_{21}^{b}, γ_{21}^{c}) & (γ_{22}^{a}, γ_{22}^{b}, γ_{22}^{c}) & \dots & (γ_{2 m}^{a}, γ_{2 m}^{b}, γ_{2 m}^{c}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (γ_{n 1}^{a}, γ_{n 1}^{b}, γ_{n 1}^{c}) & (γ_{n 2}^{a}, γ_{n 2}^{b}, γ_{n 2}^{c}) & \dots & (γ_{mn}^{a}, γ_{mn}^{b}, γ_{mn}^{c}) \end{matrix}]$

where normalized values are given as $(γ_{11}^{a} = \frac{γ_{11}^{a}}{γ_{j}^{m}}, γ_{11}^{b} = \frac{γ_{11}^{b}}{γ_{j}^{m}}, γ_{11}^{c} = \frac{γ_{11}^{b}}{γ_{j}^{m}})$

where $γ_{j}^{m} = \max_{j} (x_{ij}^{c})$ .

The individual regret and group utility used in the traditional VIKOR approach are computed via the fuzzy IGWHM function [41]. $S_{i}^{p, q} = \frac{{(\sum_{i = 1}^{n} \sum_{j = i}^{n} {(w_{i} γ_{i})}^{p} {(w_{j} γ_{j})}^{q})}^{\frac{1}{p + q}}}{{(\sum_{i = 1}^{n} \sum_{j = i}^{n} {(w_{i})}^{p} {(w_{j})}^{q})}^{\frac{1}{p + q}}}$ (5) $= (\begin{matrix} \frac{{(\sum_{i = 1}^{n} \sum_{j = i}^{n} {(w_{i} γ_{i}^{a})}^{p} {(w_{j} γ_{j}^{a})}^{q})}^{\frac{1}{p + q}}}{{(\sum_{i = 1}^{n} \sum_{j = i}^{n} {(w_{i})}^{p} {(w_{j})}^{q})}^{\frac{1}{p + q}}}, \\ \frac{{(\sum_{i = 1}^{n} \sum_{j = i}^{n} {(w_{i} γ_{i}^{b})}^{p} {(w_{j} γ_{j}^{b})}^{q})}^{\frac{1}{p + q}}}{{(\sum_{i = 1}^{n} \sum_{j = i}^{n} {(w_{i})}^{p} {(w_{j})}^{q})}^{\frac{1}{p + q}}}, \\ \frac{{(\sum_{i = 1}^{n} \sum_{j = i}^{n} {(w_{i} γ_{i}^{c})}^{p} {(w_{j} γ_{j}^{c})}^{q})}^{\frac{1}{p + q}}}{{(\sum_{i = 1}^{n} \sum_{j = i}^{n} {(w_{i})}^{p} {(w_{j})}^{q})}^{\frac{1}{p + q}}} \end{matrix})$ where p, q≥0, w_j describe the weights of criteria, w_j >0 and ∑w_j = 1. $R_{i}^{p, q} = \frac{{(\max_{i} \sum_{j = i}^{n} {(w_{i} γ_{i})}^{p} {(w_{j} γ_{j})}^{q})}^{\frac{1}{p + q}}}{{(\max_{i} \sum_{j = i}^{n} {(w_{i})}^{p} {(w_{j})}^{q})}^{\frac{1}{p + q}}}$ (6) $= (\begin{matrix} \frac{{(\max_{i} \sum_{j = i}^{n} {(w_{i} γ_{i}^{a})}^{p} {(w_{j} γ_{j}^{a})}^{q})}^{\frac{1}{p + q}}}{{(\max_{i} \sum_{j = i}^{n} {(w_{i})}^{p} {(w_{j})}^{q})}^{\frac{1}{p + q}}}, \\ \frac{{(\max_{i} \sum_{j = i}^{n} {(w_{i} γ_{i}^{b})}^{p} {(w_{j} γ_{j}^{b})}^{q})}^{\frac{1}{p + q}}}{{(\max_{i} \sum_{j = i}^{n} {(w_{i})}^{p} {(w_{j})}^{q})}^{\frac{1}{p + q}}}, \\ \frac{{(\max_{i} \sum_{j = i}^{n} {(w_{i} γ_{i}^{c})}^{p} {(w_{j} γ_{j}^{c})}^{q})}^{\frac{1}{p + q}}}{{(\max_{i} \sum_{j = i}^{n} {(w_{i})}^{p} {(w_{j})}^{q})}^{\frac{1}{p + q}}} \end{matrix})$

Now, recognize the maximum and minimum values of the $S_{i}^{p, q}$ and $R_{i}^{p, q}$ as- $S^{*} = min S_{i}^{p, q}, S^{-} = max S_{i}^{p, q},$ $R^{*} = min R_{i}^{p, q}, R^{-} = max R_{i}^{p, q}$

The results attained by S^* and R^* have the supreme group utility and lowest individual regret of the opponent, correspondingly.

Lastly, we combine the features of the group utility and individual regret in the form of $Q_{i}^{p, q}$ to estimate the positions of the alternative as shown in Equation (7)- $Q_{i}^{p, q} = k \frac{S_{i}^{p, q} - S^{*}}{S^{-} - S^{*}} + (1 - k) \frac{R_{i}^{p, q} - R^{*}}{R^{-} - R^{*}}$ (7)

where $S_{i}^{p, q}$ and $R_{i}^{p, q}$ denotes the utility and regret measure of alternative i and S^-, S^*, R^-, R^* denotes the maximum and minimum values of $S_{i}^{p, q}$ and $R_{i}^{p, q}$ respectively. The parameter k plays a significant part in the evaluation of the compromise solution and indicates the weight of strategy for the majority of the criteria. The parameter can have any value from the range [0, 1] but is often assigned a value of 0.5 in order for the compromise solution to both have the most group utility and the least amount of individual regret for the opponent.

With regard to $S_{i}^{p, q}, R_{i}^{p, q}$ and $Q_{i}^{p, q}$ the alternatives are listed in the decreasing order. Here, we find three rating positions that aid in assessing the compromise solution. The options that were rated first and second, respectively, in terms of $Q_{i}^{p, q}$ are designated as W⁽¹⁾ and W⁽²⁾, respectively, for the following evaluations. If the following criteria are met, the compromise solution includes the alternative W⁽¹⁾:

C1: Acceptable advantage

$Q (W^{(1)}) - Q (W^{(2)}) ⩾ DQ, where DQ = \frac{1}{n - 1}$ and n denotes the total number of alternatives.

C2: Acceptable stability in decision-making. The alternative W⁽¹⁾ attains the initial position in the ranking list considering the values of S or R. Within a decision-making process, this compromise solution is stable.

A set of compromise solutions can be created if one, or both, of these requirements are not met. The set of compromise solution contains both W⁽¹⁾and W⁽²⁾ if only criteria C2 does not met. The compromise solution set which is comprised of W⁽¹⁾, W⁽²⁾, …, W^(k), if criteria C1 does not hold, where W^(k) is evaluated from the inequality Q (W⁽¹⁾) -- Q (W^(k))<DQ.

3.2 To obtain weight of experts

The expert weights can be attained by performing the following steps:

Social network matrix: Consider a group of experts E ={ e₁, e₂ … e_m } under the social network, the trust degree of expert e_i to expert e_jcan be signified with a TFNs to represent the ambiguous nature of experts. Consequently, a trust relationship matrix, W_m×m = (w_ij)can be constructed where w_ijmentions the trust degree of expert e_i to expert e_i. Therefore, W_m×m = (w_ij) can be observed as a directed matrix.

W = [\begin{matrix} w_{11} & \dots & w_{1 m} \\ ⋮ & ⋱ & ⋮ \\ w_{m 1} & \dots & w_{mm} \end{matrix}]

where w_ij represents the TFNs.

Considering the everyday problems, it is not at all times true that the trust degree of expert e_i to expert e_j is equivalent to the trust degree of expert e_j to expert e_i. Therefore, for a W_m×m = (w_ij) , w_ij = w_ji cannot generally be recognized. Moreover, experts might not recurrently connect with each other. Consequently, a absence of trust between experts might cause experts not to have a direct trust relationship, which specifies that the elements in W_m×m are partially unknown. Therefore, the trust matrix is typically an incomplete asymmetric matrix. For the evaluation of the missing elements in the trust matrix, we propose a trust propagation model.

Trust propagation model: Suppose that w_ij is unknown which means that expert e_i to expert e_j do not have a direct trust relationship. For expert e_i to expert e_j, if we create a connection over k experts, i.e., there is a directed relationship e_i ⟶ e₁ ⟶ . . . ⟶ e_k ⟶ e_j, then we can represent the trust relationship as $a_{ij}^{1} = w_{i 1} + w_{12} + \dots w_{kj}$ , similarly suppose there exists another path connecting expert e_i to expert e_j, through m experts, then we can represent the trust relationship as $r = {[r_{i}]}_{m \times 1} = {[\sum_{j = 1}^{n} t_{ij}]}_{m \times 1}$ and so on. Suppose there exists a total of p paths between expert e_i and expert e_j then for the computation of w_ij the trust propagation model is developed as:

{\begin{matrix} MaxZ = \sum_{p} a_{ij}^{p} x_{p} \\ s . t . \\ \begin{matrix} \sum_{k = 1}^{p} x_{k} = 1 \\ 0 ⩽ Z ⩽ 1 \end{matrix} \end{matrix}

(8) where

a_{ij}^{p}

trust relationship represents the between expert e_i and expert e_j through path p and x_p represents the coefficient of objective function. Complete the trust relationship matrix using the results of the optimization model.

Clustering: This step is used to cluster the experts based on the trust relationships amongst experts. In the framework of a social network, some experts might have the similar viewpoint because of certain trust or connections with each other’s, and thus clustering based on expert trust relationships is more objective. Therefore, the paper applies the Louvain algorithm based on the idea of modularity in a social network and the connection between decision experts to cluster experts.

Fuzzy preference relation: A Fuzzy Preference Relation (FPR) [43] P on a set of alternatives X is a fuzzy set on the product set X×X, i.e., a relation on X characterized by a membership function $μ_{P} : X \times X \to [0, 1] .$

Definition 5. Consider two triangular fuzzy numbers A = (m1, n1, p1) and B = (m2, n2, p2) then the preference of A over B is defined as given in Equation (9)- $P_{AB} = \frac{1}{2} [\frac{(m_{1} - m_{2}) + (n_{1} - n_{2}) + (p_{1} - p_{2})}{\sqrt{m_{1}^{2} + n_{1}^{2}} \bar{+ p_{1}^{2}} \cdot \sqrt{m_{2}^{2} + n_{2}^{2}} \bar{+ p_{2}^{2}}} + 1]$ (9)

Property 1. The fuzzy preference relation P is additive reciprocal i.e., P_AB + P_BA = 1.

Proof: Let A = (m1, n1, p1) and B = (m2, n2, p2) be the two triangular fuzzy numbers, since $P_{AB} = \frac{1}{2} [\frac{(m_{1} - m_{2}) + (n_{1} - n_{2}) + (p_{1} - p_{2})}{\sqrt{m_{1}^{2} + n_{1}^{2} + p_{1}^{2}} \cdot \sqrt{m_{2}^{2} + n_{2}^{2} + p_{2}^{2}}} + 1]$

and $P_{BA} = \frac{1}{2} [\frac{(m_{2} - m_{1}) + (n_{2} - n_{1}) + (p_{2} - p_{1})}{\sqrt{m_{1}^{2} + n_{1}^{2} + p_{1}^{2}} \cdot \sqrt{m_{2}^{2} + n_{2}^{2} + p_{2}^{2}}} + 1]$

$P_{AB} + P_{BA} = \frac{1}{2} [\frac{(m_{1} - m_{2}) + (n_{1} - n_{2}) + (p_{1} - p_{2})}{\sqrt{m_{1}^{2} + n_{1}^{2}} \bar{+ p_{1}^{2}} \cdot \sqrt{m_{2}^{2} + n_{2}^{2}} \bar{+ p_{2}^{2}}} + 1 + \frac{(m_{2} - m_{1}) + (n_{2} - n_{1}) + (p_{2} - p_{1})}{\sqrt{m_{1}^{2} + n_{1}^{2}} \bar{+ p_{1}^{2}} \cdot \sqrt{m_{2}^{2} + n_{2}^{2}} \bar{+ p_{2}^{2}}} + 1]$

$P_{AB} + P_{BA} = 1 .$

Property 2. The preference relation P is transitive i.e., $P_{AB} ⩾ \frac{1}{2}$ and $P_{BC} ⩾ \frac{1}{2} \Rightarrow P_{AC} ⩾ \frac{1}{2}$ .

Proof: Let A = (m1, n1, p1) and B = (m2, n2, p2) and C = (m3, n3, p3) be the three triangular fuzzy numbers, since $P_{AB} ⩾ \frac{1}{2}$ and $P_{BC} ⩾ \frac{1}{2}$ , we have $\frac{(m_{1} - m_{2}) + (n_{1} - n_{2}) + (p_{1} - p_{2})}{\sqrt{m_{1}^{2} + n_{1}^{2} + p_{1}^{2}} \cdot \sqrt{m_{2}^{2} + n_{2}^{2} + p_{2}^{2}}} ⩾ 0$ and $\frac{(m_{2} - m_{3}) + (n_{2} - n_{3}) + (p_{2} - p_{3})}{\sqrt{m_{3}^{2} + n_{3}^{2} + p_{3}^{2}} \cdot \sqrt{m_{2}^{2} + n_{2}^{2} + p_{2}^{2}}} ⩾ 0$

now, $\frac{(m_{1} - m_{3}) + (n_{1} - n_{3}) + (p_{1} - p_{3})}{\sqrt{m_{3}^{2} + n_{3}^{2} + p_{3}^{2}} \cdot \sqrt{m_{1}^{2} + n_{1}^{2} + p_{1}^{2}}} = \frac{(m_{1} - m_{2} + m_{2} - m_{3}) + (n_{1} - n_{2} + n_{2} - n_{3}) + (p_{1} - p_{2} + p_{2} - p_{3})}{\sqrt{m_{3}^{2} + n_{3}^{2} + p_{3}^{2}} \cdot \sqrt{m_{1}^{2} + n_{1}^{2} + p_{1}^{2}}}$ $= \frac{(m_{1} - m_{2}) + (n_{1} - n_{2}) + (p_{1} - p_{2})}{\sqrt{m_{3}^{2} + n_{3}^{2} + p_{3}^{2}} \cdot \sqrt{m_{1}^{2} + n_{1}^{2} + p_{1}^{2}}} + \frac{(m_{2} - m_{3}) + (n_{2} - n_{3}) + (p_{2} - p_{3})}{\sqrt{m_{3}^{2} + n_{3}^{2} + p_{3}^{2}} \cdot \sqrt{m_{1}^{2} + n_{1}^{2} + p_{1}^{2}}} ⩾ 0$ ⇒ $P_{AC} ⩾ \frac{1}{2}$

Thus, from property 2.1 and 2.2, the preference relation P is a total ordering relation.

Lemma 1. By the preference relation P if $P_{AB} > \frac{1}{2}$ we say that A is preferred over B.

Lemma 2. By the preference relation P if $P_{AB} = \frac{1}{2}$ we say that A is indifferent to B.

Expert weights: The weight of expert is determined combining the trust relationships of experts in the similar cluster and the preference of clusters as written in Equation (10).

ω_{i} = \sum_{k = 1, k \neq i}^{n} P (C_{i}, C_{k}) + \sum_{j = 1, j \neq i}^{p} w_{ij}

(10)

where p denotes the number of experts, w_ij represents the trust relationship of expert i to the expert j of the same cluster, P (C_i, C_k) signifies the preference of the cluster to which expert i belongs with the other clusters.

The final weights exp_i are determined by normalizing the values obtained in the previous step as shown in Equation (11). $\exp_{i} = \frac{ω_{i}}{\sum_{i = 1}^{p} ω_{i}}$ (11)

3.3 To obtain membership value of attributes

This section uses the fuzzy measure to collect the judgment of experts, due to their ability to deliver the highest performance in expressing the interaction effect of attributes. The steps required to attain the membership value of attributes are:

Given an expert collection E ={ e₁, e₂ … e_m } and the set of attributes C ={ c₁, c₂ … c_n }.

Fuzzy measure provided by experts: The experts may use various forms to represent their initial opinions. In the present study the experts are required tooffer their opinion for the attributes in terms of fuzzy measure μ_i. Since, it in real life scenario it is impractical for the decision makers to provide the measure values for the entire subsets of attributes, the 2- order additive fuzzy measure is used to gather the preliminary opinions of experts. The experts provide the fuzzy measure for only those attributes, for which they are having any prior knowledge.

Combination of fuzzy measure by Shapley function: Since the attribute importance cannot be determined by considering the singleton fuzzy measure, it becomes essential to consider fuzzy measure of subsets containing the attribute also. Hence, the Shapley function is used to obtain the Shapley value of each attribute $s_{k}^{i}$ for a particular expert e_i, in the form of attribute global importance matrix. $[\begin{matrix} s_{1}^{1} & \begin{matrix} s_{1}^{2} & \dots \end{matrix} & s_{1}^{n} \\ ⋮ & \begin{matrix} ⋮ & ⋱ \end{matrix} & ⋮ \\ s_{m}^{1} & \begin{matrix} s_{m}^{2} & \dots \end{matrix} & s_{m}^{n} \end{matrix}]$

Estimation of incomplete information: In many real-life situations the experts cannot have an understanding for every attribute, hence the obtained global importance matrix in step 2 will have some empty places. The incomplete information of expert about any particular attribute is determined by considering the influence of experts on each other. For instance, expert 1 doesn’t have prior knowledge about attribute 1 then the Shapley value of attribute 1 for expert 1 is calculated as shown in Equation (12): $s_{1}^{1} = w_{12} s_{1}^{2} + w_{13} s_{1}^{3} \dots . . w_{1 m} s_{1}^{m}$ (12)

Membership value of attributes: Obtain the membership values α_k of each attribute by using Equation (13), where $s_{k}^{j}$ represents the Shapley value of attribute k for expert j and exp_j represents the weight of the experts: $α_{k} = \frac{\sum_{j = 1}^{m} \exp_{j} s_{k}^{j}}{\sum_{k = 1}^{m} \sum_{j = 1}^{m} \exp_{j} s_{k}^{j}}$ (13)

3.4 To obtain the updated ranking considering the social media influence

Many times, certain news, either positively or negatively affect the ranking of alternative. This section describes the effect of the news on the decision matrix and ranking order of alternatives. The following steps are required to update the information.

3.4.1 Functional dependency between attributes

In this section steps involved to capture the dependency among attributes are presented in terms of DEMATEL and regression analysis.

3.4.1.1. DEMATEAL: The DEMATEL method was introduced by Gabus and Fontela [1]. The DEMATEL method is useful for the construction of a network relation design for the examination the internal relationships among attributes. The steps involved in the procedure are:

Determine the preliminary influence matrix: Considering the degree of influence of criteria i on criteria j, the preliminary influence matrix A = [a_ij] _m×m is formed. The experts are asked to determine the elements of matrix a_ij based on the 5-point scale as shown in Table 2.

Normalize the initial influence matrix: The initial influence matrix is normalized using N = k × A where: $k = \max_{i, j} {\frac{1}{\max_{i} \sum_{j = 1}^{n} | a_{ij} |}, \frac{1}{\max_{j} \sum_{j = 1}^{n} | a_{ij} |}}$

Determine the total influence matrix: The representation of the total influence matrix T = [t_ij] _m×m is shown below: $T = D {(I - D)}^{- 1}$

The cause-and-effect values: Determine the r and c values which represents the summation of row and column values: $r = {[r_{i}]}_{m \times 1} = {[\sum_{j = 1}^{n} t_{ij}]}_{m \times 1}$ and $c = {[c_{j}]}_{1 \times m}^{'} = {[\sum_{j = 1}^{n} t_{ij}]}_{1 \times m}^{'}$ . Determine r_i + c_i values, if the positive value of r_i + c_i represents that the criteria belongs to the cause group otherwise it belongs to the effect group.

3.4.1.2. Regression analysis: After clustering the attributes into cause-and-effect collections, the next step is to perform statistical analysis on the attributes set. For this regression analysis is performed on attributes to find the relationship between cause-and-effect attributes. Statistically dependency among attributes can be determined by curve fitting method. Since, linear regression is an effective statistical tool to determine dependency, it has been utilized in the paper.

Table 2
5-point scale for determining influence matrix

Influence of criteria on other criteria 5-point scale

No impact 0

Low impact 1

Medium impact 2

High impact 3

Very high impact 4

Influence of criteria on other criteria	5-point scale
No impact	0
Low impact	1
Medium impact	2
High impact	3
Very high impact	4

3.4.2 Effect on other attributes

Consider that any future update announces β_i percentage increment/decrement in any criteria C_i hence it effects all attributes that are dependent on C_i. The effect on the dependent criteria C_j can be determined using Equation (14)- $β_{j} = \frac{α_{j}}{α_{i}} \times β_{i}$ (14)

here, α_i, α_j denotes the membership value of attributes as obtained in third phase.

3.4.3 Update the decision matrix

The decision matrix is updated considering the updated value of attributes. The attributes are updated by adding/subtracting the value of β_i.

4 An application

This section validates the reliability of the proposed approach through a simulation test concerning a case study of a car selection problem MCDM problem.

4.1 Problem description

Car selection is one of the most common problems considering the invention of new companies with many features. We choose the data set from the database archive.ics.uci.edu consisting of 1728 objects and 6 attributes, out of which we selected 10 cars (Cr₁, Cr₂, … Cr₁₀) and 6 attributes (Att₁, Att₂, … Att₆) to validate the proposed framework. The attributes are buying price, maintenance price, number of doors, number of persons, luggage boot, and safety. The given decision matrix is represented in Table 3. To capture the uncertainty, and to deal with linguistic data, Table 3 is fuzzified using triangular membership functions to have fuzzy decision matrix.

Table 3
Initial decision-making matrix

Att ₁ Att ₂ Att ₃ Att ₄ Att ₅ Att ₆

Cr ₁ VH VH 2 2 S L

Cr ₂ VH VH 2 2 M L

Cr ₃ VH VH 2 2 M H

Cr ₄ VH VH 2 2 B L

Cr ₅ VH VH 2 3 M L

Cr ₆ VH VH 2 4 S L

Cr ₇ H H 4 2 S M

Cr ₈ H H 4 4 S M

Cr ₉ H VH 2 4 M H

Cr ₁₀ H VH 3 2 B H

	Att ₁	Att ₂	Att ₃	Att ₄	Att ₅	Att ₆
Cr ₁	VH	VH	2	2	S	L
Cr ₂	VH	VH	2	2	M	L
Cr ₃	VH	VH	2	2	M	H
Cr ₄	VH	VH	2	2	B	L
Cr ₅	VH	VH	2	3	M	L
Cr ₆	VH	VH	2	4	S	L
Cr ₇	H	H	4	2	S	M
Cr ₈	H	H	4	4	S	M
Cr ₉	H	VH	2	4	M	H
Cr ₁₀	H	VH	3	2	B	H

4.2 Implementation

4.2.1 To determine attribute weights via F-BCM

The steps required to attain criteria weights are:

At first, decision maker selects buying cost P₁ as a base-criteria from the given collection of criteria.

The fuzzy base-comparison vector based on TFNs revealed by the experts for pairwise comparisons of the fuzzy base-criterion with remaining criteria as shown. ${\tilde{T}}_{B} = {(0.75, 1, 1), (0.25, 0.5, 0.75), (0.25, 0.5, 0.75),$ (0.5, 0.75, 1) , (0.5, 0.75, 1)}.

A non-linear constrained optimization problem is formed using the Equation (4) as shown below:

\begin{matrix} Min \tilde{ξ} \\ s . t . \\ {\begin{matrix} | \frac{(a_{1}, b_{1}, c_{1})}{(a_{2}, b_{2}, c_{2})} - (0.75, 1, 1) | ⩽ (k, k, k) \\ | \frac{(a_{1}, b_{1}, c_{1})}{(a_{3}, b_{3}, c_{3})} - (0.25, 0.5, 0.75) | ⩽ (k, k, k) \\ | \frac{(a_{1}, b_{1}, c_{1})}{(a_{4}, b_{4}, c_{4})} - (0.25, 0.5, 0.75) | ⩽ (k, k, k) \\ | \frac{(a_{1}, b_{1}, c_{1})}{(a_{5}, b_{5}, c_{5})} - (0.5, 0.75, 1) | ⩽ (k, k, k) \\ | \frac{(a_{1}, b_{1}, c_{1})}{(a_{6}, b_{6}, c_{6})} - (0.5, 0.75, 1) | ⩽ (k, k, k) \\ \frac{1}{6} (\begin{matrix} a_{1} + 4 b_{1} + c_{1} + a_{2} + 4 b_{2} + c_{2} + a_{3} + 4 b_{3} + c_{3} + \\ a_{4} + 4 b_{4} + c_{4} + a_{5} + 4 b_{5} + c_{5} + a_{6} + 4 b_{6} + c_{6} \end{matrix}) = 1 \\ \begin{matrix} a_{1} ⩽ b_{1} ⩽ c_{1} \\ a_{2} ⩽ b_{2} ⩽ c_{2} \\ \begin{matrix} a_{3} ⩽ b_{3} ⩽ c_{3} \\ a_{4} ⩽ b_{4} ⩽ c_{4} \\ \begin{matrix} a_{5} ⩽ b_{5} ⩽ c_{5} \\ a_{6} ⩽ b_{6} ⩽ c_{6} \\ \begin{matrix} a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6} ⩾ 0 \\ k ⩾ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}

by solving the above optimization model, the fuzzy weights of criteria are obtained as:

${\bar{w}}_{1} = (0.32567, 0.40731, 0.80358)$ ,

${\bar{w}}_{2} = (0.03137, 0.0784, 0.27465)$ ,

${\bar{w}}_{3} = (0.03369, 0.08381, 0.29848)$ ,

${\bar{w}}_{4} = (0.03369, 0.08381, 0.29848)$ ,

${\bar{w}}_{5} = (0.03211, 0.08087, 0.28797)$ ,

${\bar{w}}_{6} = (0.03211, 0.08087, 0.28797) .$

the fuzzy weights are converted in crisp weights as:

w₁ = 0.45975,

w₂ = 0.10327,

w₃ = 0.11123,

w₄ = 0.11123,

w₅ = 0.10726,

w₆ = 0.10726

These weights are further used in the F-VIKOR-H model.

4.2.2 Ranking of alternatives via F-VIKOR-H

The first and the foremost step is the construction of a 10 × 6 decision matrix considering the evaluations of ten cars based on the given set of attributes. The decision matrix is displayed in Table 3. Here, attributes buying price, maintenance price, luggage boot, safety are expressed in terms of linguistic terms.

Buying price: very high, high, med, low

Maintenance price: very high, high, med, low

Luggage boot: small, med, big

Safety: low, med, high

Considering the given decision matrix, a 10 × 6 fuzzy decision matrix is computed as displayed in Table 5 with the use of the (Triangular fuzzy numbers) TFNs as shown in Table 4.

The fuzzy decision matrix is normalized using as displayed in Table 6.

Table 4
TFNs for conversion of linguistic attributes

TFNs Linguistic Set for Linguistic

M1, M2 and M3 Set for Mv5

(0,0,0.25) Very low Very small

(0,0.25,0.5) Low Small

(0.25,0.5,0.75) Medium Medium

(0.5,0.75,1) High Big

(0.75,1,1) Very high Very big

TFNs	Linguistic Set for	Linguistic
(0,0,0.25)	Very low	Very small
(0,0.25,0.5)	Low	Small
(0.25,0.5,0.75)	Medium	Medium
(0.5,0.75,1)	High	Big
(0.75,1,1)	Very high	Very big

Table 5

Fuzzy decision-making matrix

	Att ₁	Att ₂	Att ₃	Att ₄	Att ₅	Att ₆
Cr ₁	(0.75,1,1)	(0.75,1,1)	(0,0,0.25)	(0,0,0.25)	(0,0.25,0.5)	(0,0.25,0.5)
Cr ₂	(0.75,1,1)	(0.75,1,1)	(0,0,0.25)	(0,0,0.25)	(0.25,0.5,0.75)	(0,0.25,0.5)
Cr ₃	(0.75,1,1)	(0.75,1,1)	(0,0,0.25)	(0,0,0.25)	(0.25,0.5,0.75)	(0.5,0.75,1)
Cr ₄	(0.75,1,1)	(0.75,1,1)	(0,0,0.25)	(0,0,0.25)	(0.5,0.75,1)	(0,0.25,0.5)
Cr ₅	(0.75,1,1)	(0.75,1,1)	(0,0,0.25)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0,0.25,0.5)
Cr ₆	(0.75,1,1)	(0.75,1,1)	(0,0,0.25)	(0.25,0.5,0.75)	(0,0.25,0.5)	(0,0.25,0.5)
Cr ₇	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0.25)	(0,0.25,0.5)	(0.25,0.5,0.75)
Cr ₈	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0.25,0.5)	(0.25,0.5,0.75)
Cr ₉	(0.5,0.75,1)	(0.75,1,1)	(0,0,0.25)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)
Cr ₁₀	(0.5,0.75,1)	(0.75,1,1)	(0,0.25,0.5)	(0,0,0.25)	(0.5,0.75,1)	(0.5,0.75,1)

Table 6

Normalized decision-making matrix

	Att ₁	Att ₂	Att ₃	Att ₄	Att ₅	Att ₆
Cr ₁	(0.75,1,1)	(0.75,1,1)	(0,0,0.333)	(0,0,0.333)	(0,0.25,0.5)	(0,0.25,0.5)
Cr ₂	(0.75,1,1)	(0.75,1,1)	(0,0,0.333)	(0,0,0.333)	(0.25,0.5,0.75)	(0,0.25,0.5)
Cr ₃	(0.75,1,1)	(0.75,1,1)	(0,0,0.333)	(0,0,0.333)	(0.25,0.5,0.75)	(0.5,0.75,1)
Cr ₄	(0.75,1,1)	(0.75,1,1)	(0,0,0.333)	(0,0,0.333)	(0.5,0.75,1)	(0,0.25,0.5)
Cr ₅	(0.75,1,1)	(0.75,1,1)	(0,0,0.333)	(0,0.333,0.666)	(0.25,0.5,0.75)	(0,0.25,0.5)
Cr ₆	(0.75,1,1)	(0.75,1,1)	(0,0,0.333)	(0.333,0.666,1)	(0,0.25,0.5)	(0,0.25,0.5)
Cr ₇	(0.5,0.75,1)	(0.5,0.75,1)	(0.333,0.666,1)	(0,0,0.333)	(0,0.25,0.5)	(0.25,0.5,0.75)
Cr ₈	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.333,0.666,1)	(0,0.25,0.5)	(0.25,0.5,0.75)
Cr ₉	(0.5,0.75,1)	(0.75,1,1)	(0,0,0.333)	(0.333,0.666,1)	(0.25,0.5,0.75)	(0.5,0.75,1)
Cr ₁₀	(0.5,0.75,1)	(0.75,1,1)	(0,0.333,0.666)	(0,0,0.333)	(0.5,0.75,1)	(0.5,0.75,1)

Next, we calculate the value of $S_{i}^{p, q}$ and $R_{i}^{p, q}$ using IGWHM and taking p = q = 1. Further they are converted into crisp set to obtain the value of $Q_{i}^{p, q}$ . Finally ranking is obtained using $Q_{i}^{p, q}$ . The results are shown in Table 7.

Table 7

Ranking of alternatives

	$S_{i}^{p, q}$	$R_{i}^{p, q}$	Crisp $S_{i}^{p, q}$	Crisp $R_{i}^{p, q}$	$Q_{i}^{p, q}$	Ranking
Cr ₁	(0.32577,0.36355,0.56921)	(0.14075,0.16757,0.37117)	0.39153	0.197034	0	10
Cr ₂	(0.34518,0.38547,0.59238)	(0.15416,0.18098,0.38905)	0.413241	0.211186	0.07353	9
Cr ₃	(0.38104,0.42059,0.62903)	(0.18098,0.20779,0.4248)	0.448739	0.239491	0.203821	5
Cr ₄	(0.36665,0.40899,0.61651)	(0.16757,0.19438,0.40693)	0.436521	0.225339	0.150383	8
Cr ₅	(0.34518,0.4146,0.62396)	(0.15416,0.19952,0.41377)	0.437921	0.227665	0.15788	7
Cr ₆	(0.35317,0.42423,0.63405)	(0.15929,0.20465,0.42061)	0.447359	0.233082	0.188411	6
Cr ₇	(0.37173,0.54911,0.86354)	(0.1727,0.3533,0.8454)	0.571951	0.405215	0.787549	4
Cr ₈	(0.40065,0.60998,0.92598)	(0.19124,0.40273,0.91956)	0.627759	0.453621	1	1
Cr ₉	(0.41048,0.57463,0.87508)	(0.19952,0.37757,0.86461)	0.597345	0.42907	0.887783	2
Cr ₁₀	(0.40341,0.56631,0.86627)	(0.19438,0.37073,0.85434)	0.589154	0.421943	0.85656	3

Thus, car 8 has been ranked as the optimal car.

4.3 To obtain weight of experts

Five experts are asked for the evaluation of the criteria. The trust relationship between the experts are expressed in the form of matrix as demonstrated in Table 8.

The incomplete elements of the trust relationship matrix are determined using the trust propagation model and are shown in Table 9.

The experts are grouped into K clusters using social network Louvain algorithm based on their level of trust with one another as shown in Table 10. After that, each cluster’s preference and trust relationship are determined.

Table 8
Trust relationship matrix

– (0.25,0.5,0.75) (0.5,0.75,1) (0.5,0.75,1)

(0.5,0.75,1) – (0,0.25,0.5) (0.25,0.5,0.75)

(0.75,1,1) – (0.5,0.75,1) (25,0.5,0.75)

(0.25,0.5,0.75) (0.5,0.75,1) (0.75,1,1) – (0.25,0.5,0.75)

(0.75,1,1) (0.25,0.5,0.75) (0,0.25,0.5) –

–	(0.25,0.5,0.75)	(0.5,0.75,1)		(0.5,0.75,1)
(0.5,0.75,1)	–	(0,0.25,0.5)	(0.25,0.5,0.75)
(0.75,1,1)		–	(0.5,0.75,1)	(25,0.5,0.75)
(0.25,0.5,0.75)	(0.5,0.75,1)	(0.75,1,1)	–	(0.25,0.5,0.75)
	(0.75,1,1)	(0.25,0.5,0.75)	(0,0.25,0.5)	–

Table 9

Complete trust relationship matrix

–	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)
(0.5,0.75,1)	–	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.3825,0.6325,0.8825)
(0.75,1,1)	(0.3005,0.5505,0.8005)	–	(0.5,0.75,1)	(0.25,0.5,0.75)
(0.25,0.5,0.75)	(0.5,0.75,1)	(0.75,1,1)	–	(0.25,0.5,0.75)
(6,0.617,0.867)	(0.75,1,1)	(0.25,0.5,0.75)	(0,0.25,0.5)	–

Table 10

Clustering results

Clusters		Trust relationship
C ¹	{1, 3}	(0.625,0.875,1)
C ²	{2, 5}	(0.566,0.816,0.941)
C ³	{4}	(1,1,1)

The weights of experts are determined using Equation (11).

e₁ = 0.210069, e₂ = 0.195539, e₃ = 0.187078,

e₄ = 0.247733, e₅ = 0.15958

4.4 To obtain membership values of attributes

The experts provide their initial opinion in the form of fuzzy measure as displayed in Table 11.

Table 11
Initial evaluations of attributes with fuzzy measure

μ₁(Expert e₁) {1} 0.25 {2} 0.1 {3} 0.13 {4} 0.21

{1,2} 0.3 {1,3} 0.26 {1,4} 0.29 {2,3} 0.2

{2,4} 0.25 {3,4} 0.22 {1,2,3} 0.35 {2,3,4} 0.36

{1,3,4} 0.45 {1,2,4} 0.4 {1,2,3,4} 1.00

μ₂(Expert e₂) {3} 0.27 {4} 0.3 {5} 0.16 {6} 0.29

{3,4} 0.4 {3,5} 0.45 {3,6} 0.3 {4,5} 0.31

{4,6} 0.46 {5,6} 0.59 {3,4,5} 0.5 {4,5,6} 0.65

{3,5,6} 0.67 {3,4,6} 0.52 {3,4,5,6} 1.00

μ₃(Expert e₃) {2} 0.31 {3} 0.27 {5} 0.45 {6} 0.12

{2,3} 0.35 {2,5} 0.53 {2,6} 0.39 {3,5} 0.49

{3,6} 0.29 {5,6} 0.6 {2,3,5} 0.56 {3,5,6} 0.67

{2,5,6} 0.64 {2,3,6} 0.44 {2,3,5,6} 1.00

μ₄(Expert e₄) {1} 0.21 {2} 0.09 {3} 0.54 {5} 0.2

{1,2} 0.25 {1,3} 0.55 {1,5} 0.26 {2,3} 0.6

{2,5} 0.22 {3,5} 0.56 {1,2,3} 0.61 {2,3,5} 0.7

{1,3,5} 0.69 {1,2,5} 0.3 {1,2,3,5} 1.00

μ₅(Expert e₅) {1} 0.14 {2} 0.26 {5} 0.31 {6} 0.15

{1,2} 0.4 {1,5} 0.35 {1,6} 0.23 {2,5} 0.45

{2,6} 0.5 {5,6} 0.33 {1,2,5} 0.51 {2,5,6} 0.62

{1,5,6} 0.39 {1,2,6} 0.54 {1,2,5,6} 1.00

μ₁(Expert e₁)	{1} 0.25	{2} 0.1	{3} 0.13	{4} 0.21
	{1,2} 0.3	{1,3} 0.26	{1,4} 0.29	{2,3} 0.2
	{2,4} 0.25	{3,4} 0.22	{1,2,3} 0.35	{2,3,4} 0.36
	{1,3,4} 0.45	{1,2,4} 0.4	{1,2,3,4} 1.00
μ₂(Expert e₂)	{3} 0.27	{4} 0.3	{5} 0.16	{6} 0.29
	{3,4} 0.4	{3,5} 0.45	{3,6} 0.3	{4,5} 0.31
	{4,6} 0.46	{5,6} 0.59	{3,4,5} 0.5	{4,5,6} 0.65
	{3,5,6} 0.67	{3,4,6} 0.52	{3,4,5,6} 1.00
μ₃(Expert e₃)	{2} 0.31	{3} 0.27	{5} 0.45	{6} 0.12
	{2,3} 0.35	{2,5} 0.53	{2,6} 0.39	{3,5} 0.49
	{3,6} 0.29	{5,6} 0.6	{2,3,5} 0.56	{3,5,6} 0.67
	{2,5,6} 0.64	{2,3,6} 0.44	{2,3,5,6} 1.00
μ₄(Expert e₄)	{1} 0.21	{2} 0.09	{3} 0.54	{5} 0.2
	{1,2} 0.25	{1,3} 0.55	{1,5} 0.26	{2,3} 0.6
	{2,5} 0.22	{3,5} 0.56	{1,2,3} 0.61	{2,3,5} 0.7
	{1,3,5} 0.69	{1,2,5} 0.3	{1,2,3,5} 1.00
μ₅(Expert e₅)	{1} 0.14	{2} 0.26	{5} 0.31	{6} 0.15
	{1,2} 0.4	{1,5} 0.35	{1,6} 0.23	{2,5} 0.45
	{2,6} 0.5	{5,6} 0.33	{1,2,5} 0.51	{2,5,6} 0.62
	{1,5,6} 0.39	{1,2,6} 0.54	{1,2,5,6} 1.00

The Shapley function is utilized to aggregate the information provided by the experts in terms of fuzzy measure. The Shapley value of each attribute for a particular expert e_i is shown in Table 12.

Table 12

Attribute global importance matrix

	Att ₁	Att ₂	Att ₃	Att ₄	Att ₅	Att ₆
e ₁	0.30083	0.20417	0.21917	0.27583	–	–
e ₂	–	–	0.21583	0.2225	0.25583	0.30583
e ₃	–	0.2175	0.19083	–	0.39917	0.1925
e ₄	0.165	0.13	0.51667	–	0.18833	–
e ₅	0.165	0.34333	–	–	0.27333	0.21833

From Table 12 it is clear that expert 1 have prior knowledge only about the criteria M₁, M₂, M₃ and M₄ and not about the remaining criteria. The incomplete information in the table can be determined by Equation (12) considering the influence of experts on each other as shown in Table 13.

Table 13

Incomplete information estimation

	Att ₁	Att ₂	Att ₃	Att ₄	Att ₅	Att ₆
e ₁	0.30083	0.20417	0.21917	0.27583	0.18161	0.15368
e ₂	0.1375	0.12241	0.21583	0.2225	0.25583	0.30583
e ₃	0.16485	0.2175	0.19083	0.19341	0.39917	0.1925
e ₄	0.165	0.13	0.51667	0.1524	0.18833	0.17434
e ₅	0.165	0.34333	0.14167	0.19173	0.27333	0.21833

Finally, Equation (13) is utilized to evaluate the membership values of criteria as demonstrated in Table 14.

Table 14

Membership values of criteria

	Att ₁	Att ₂	Att ₃	Att ₄	Att ₅	Att ₆
α_i	0.14226	0.14709	0.20761	0.15576	0.19141	0.15587

4.5 Functional dependency

Next step is to determine the functional dependency between attributes. The DEMATEL approach is applied to cluster attributes into cause-and-effect group.

The experts are asked to form the initial influence matrix based on the 5-point scale from Table 2. The initial influence matrix is shown in Table 15.

Table 15
Initial influence matrix

Att ₁ Att ₂ Att ₃ Att ₄ Att ₅ Att ₆

Att ₁ 0 3 2 2 2 3

Att ₂ 2 0 1 1 1 2

Att ₃ 3 2 0 1 1 1

Att ₄ 3 2 1 0 1 1

Att ₅ 3 2 2 2 0 1

Att ₆ 4 3 2 2 1 0

	Att ₁	Att ₂	Att ₃	Att ₄	Att ₅	Att ₆
Att ₁	0	3	2	2	2	3
Att ₂	2	0	1	1	1	2
Att ₃	3	2	0	1	1	1
Att ₄	3	2	1	0	1	1
Att ₅	3	2	2	2	0	1
Att ₆	4	3	2	2	1	0

The total influence matrix is shown in Table 16.

Table 16

Total influence matrix

	Att ₁	Att ₂	Att ₃	Att ₄	Att ₅	Att ₆
Att ₁	0.9501	1.0295	0.7071	0.7071	0.5976	0.8268
Att ₂	0.7561	0.5376	0.4454	0.4454	0.3743	0.5507
Att ₃	0.871	0.7318	0.4024	0.4793	0.4072	0.5305
Att ₄	0.871	0.7318	0.4793	0.4024	0.4072	0.5305
Att ₅	1.005	0.8444	0.624	0.624	0.3929	0.6121
Att ₆	1.2132	1.0419	0.7127	0.7127	0.5446	0.6411

The r and c values for the attributes are shown in Table 17.

Table 17

r and c values

Attributes	r_i + c_i	r_i - c_i
Att ₁	–0.8482	10.4846
Att ₂	–1.8075	8.0265
Att ₃	0.0513	6.7931
Att ₄	0.0513	6.7931
Att ₅	1.3786	6.8262
Att ₆	1.1745	8.5579

From the Table 17 it is observed that first two attributes are classified as cause group and the last two as effect group. Cause group = {Att₁, Att₂}, Effect group = {Att₃, Att₄, Att₅, Att₆}.

Further, the regression analysis is performed on the decision matrix, to determine the dependency between cause-and-effect attributes, and the outcomes are represented in Table 18.

Table 18

Regression analysis results

	Att ₁	Att ₂	Att ₃	Att ₄	Att ₅	Att ₆
Att ₁	–		0.417*			0.307*
Att ₂		–	0.427**
Att ₃			–
Att ₄				–
Att ₅					–
Att ₆						–

^*Signifies p-value < 0.05; ^**signifies p-value < 0.01.

It is observed from Table 18 that the buying price is dependent on the number of doors and safety and the maintenance price is dependent on the number of doors.

Att₁→ { Att₃, Att₆ }

Att₂→ { Att₃ }

Recently a news came by the safety minister of India that is to have a minimum of 6 safety bags compulsory in all the vehicles that carry at least 8 passengers. As a consequence, all the vehicles updated their safety by increasing the amount of safety bags. Due to the increase in the attribute safety, there is a change in the dependent attribute buying price. The change in the dependent and independent attribute is determined using Equation (14). The updated matrix is displayed in Table 19.

Table 19

Updated decision matrix

	Att ₁	Att ₂	Att ₃	Att ₄	Att ₅	Att ₆
Cr ₁	(0.814,0.907,1)	(0.75,1,1)	(0,0,0.333)	(0,0,0.333)	(0,0.25,0.5)	(0.5,0.75,1)
Cr ₂	(0.814,0.907,1)	(0.75,1,1)	(0,0,0.333)	(0,0,0.333)	(0.25,0.5,0.75)	(0.5,0.75,1)
Cr ₃	(0.262,0.355,0.448)	(0.75,1,1)	(0,0,0.333)	(0,0,0.333)	(0.25,0.5,0.75)	(0.5,0.75,1)
Cr ₄	(0.814,0.907,1)	(0.75,1,1)	(0,0,0.333)	(0,0,0.333)	(0.5,0.75,1)	(0.5,0.75,1)
Cr ₅	(0.814,0.907,1)	(0.75,1,1)	(0,0,0.333)	(0,0.333,0.666)	(0.25,0.5,0.75)	(0.5,0.75,1)
Cr ₆	(0.814,0.907,1)	(0.75,1,1)	(0,0,0.333)	(0.333,0.666,1)	(0,0.25,0.5)	(0.5,0.75,1)
Cr ₇	(0.307,0.399,0.492)	(0.5,0.75,1)	(0.333,0.666,1)	(0,0,0.333)	(0,0.25,0.5)	(0.5,0.75,1)
Cr ₈	(0.307,0.399,0.492)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.333,0.666,1)	(0,0.25,0.5)	(0.5,0.75,1)
Cr ₉	(0.185,0.278,0.371)	(0.75,1,1)	(0,0,0.333)	(0.333,0.666,1)	5,0.5,0.75)	(0.5,0.75,1)
Cr ₁₀	(0.185,0.278,0.371)	(0.75,1,1)	(0,0.333,0.666)	(0,0,0.333)	(0.5,0.75,1)	(0.5,0.75,1)

Applying the steps of ranking procedure as described in section, to the updated decision matrix, the outcomes obtained are shown in Table 20.

Table 20

Ranking of alternatives

	$S_{i}^{p, q}$	$R_{i}^{p, q}$	Crisp $S_{i}^{p, q}$	Crisp $R_{i}^{p, q}$	$Q_{i}^{p, q}$	Ranking
Cr ₁	(0.18488,0.23437,0.3732)	(0.03538,0.05176,0.09314)	0.249263	0.055925	0	10
Cr ₂	(0.20824,0.26054,0.39992)	(0.04036,0.05724,0.09926)	0.275052	0.06143	0.044422	9
Cr ₃	(0.33275,0.43269,0.65038)	(0.13337,0.22204,0.46252)	0.452311	0.247344	0.54816	5
Cr ₄	(0.23411,0.28826,0.42762)	(0.04533,0.06273,0.10537)	0.302463	0.066936	0.091173	8
Cr ₅	(0.20824,0.29391,0.43555)	(0.04036,0.06483,0.10771)	0.303238	0.067896	0.093574	7
Cr ₆	(0.21789,0.30518,0.44722)	(0.04226,0.06693,0.11005)	0.314308	0.070003	0.112298	6
Cr ₇	(0.32084,0.45679,0.66969)	(0.11885,0.21516,0.41715)	0.469616	0.232771	0.553492	4
Cr ₈	(0.35146,0.52103,0.73628)	(0.13281,0.24957,0.46196)	0.528642	0.265509	0.682148	3
Cr ₉	(0.41048,0.57463,0.87508)	(0.19952,0.37757,0.86461)	0.597345	0.42907	1	1
Cr ₁₀	(0.40341,0.56631,0.86627)	(0.19438,0.37073,0.85434)	0.589154	0.421943	0.978685	2

As compared with the previous ranking Car8 which was ranked first earlier, experienced an impact of news and now occupies third position. Also, other alternatives experienced change in the ranking.

5 Comparative and sensitivity analysis

Comparative analysis with existing models is demonstrated in the present section to illustrates the validity of the proposed approach, along with the corresponding discussions.

Also, to demonstrate the stability of proposed approach sensitivity analysis is performed with respect to the different parameters involved.

5.1 Comparative analysis

This section compares the proposed ranking method with other existing well-known approaches by taking the numerical example as illustrated in section 4. The comparison involves two traditional methods presented by Hwang and Yoon [8] TOPSIS method and Opricovic [47] VIKOR method. Also, the recent extensions of VIKOR method including Ashwani [4] VIKOR method, Ahmet [5] TFN-VIKOR method, Jiang [41] IF-VIKOR method, Guangzheng [29] TrFN-VIKOR method. The results are demonstrated through Table 21 and Fig. 2.

Table 21
Ranking of alternatives by different methods

Methods Ranking

Proposed method Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁

TOPSIS [8] Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₃ > Cr₄ > Cr₅ > Cr₂ > Cr₁

VIKOR [47] Cr₈ > Cr₉ > Cr₇ > Cr₁₀ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁

VIKOR [4] Cr₈ > Cr₉ > Cr₇ > Cr₁₀ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁

TFN-VIKOR [5] Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁

IF-VIKOR [41] Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁

TrFN-VIKOR [29] Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₅ > Cr₆ > Cr₄ > Cr₂ > Cr₁

Methods	Ranking
Proposed method	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁
TOPSIS [8]	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₃ > Cr₄ > Cr₅ > Cr₂ > Cr₁
VIKOR [47]	Cr₈ > Cr₉ > Cr₇ > Cr₁₀ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁
VIKOR [4]	Cr₈ > Cr₉ > Cr₇ > Cr₁₀ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁
TFN-VIKOR [5]	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁
IF-VIKOR [41]	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁
TrFN-VIKOR [29]	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₅ > Cr₆ > Cr₄ > Cr₂ > Cr₁

Fig. 2

Ranking by different approaches.

From the comparison results demonstrated in Table 21 and Fig. 2 the following conclusions are recorded.

The outcomes by the proposed method are compatible with those of the existing approaches that is, the car X₈. As a result, the proposed method is trustworthy and efficient.

The method TOPSIS [8] does not include any parameter whereas other methods VIKOR [47], VIKOR [4], TFN-VIKOR [5], IF-VIKOR [41], TrFN-VIKOR [29] all involve a single parameter but the present approach involves two more additional parameters which makes the method more flexible than the existing methods. Subjective preferences of the experts is effectively expressed by the use of parameters.

The arguments are simply integrated in the existing methods but the present approach efficiently captures the interrelationships among the arguments to be integrated.

Furthermore, the effectiveness of the proposed approach is demonstrated through the Spearman correlation coefficients of the ranking results with different methods is displayed in Table 22. Since r ⩾ 0.9, it establishes a strong similarity among the ranking obtained by the proposed and other prevailing approaches and hence validates the proposed approach.

Table 22

SSCs of the ranking results by different methods

	Proposed method	TOPSIS	VIKOR	VIKOR (2020)	TFN-VIKOR	IF-VIKOR	TrFN-VIKOR
Prop. method	1	0.97576	0.97576	0.97576	0.98788	0.98788	0.98788
TOPSIS	–	1	0.97576	0.97576	0.98788	0.98788	0.98788
VIKOR	–	–	1	1	0.98788	0.98788	0.98788
VIKOR (2020)	–	–	–	1	0.98788	0.98788	0.98788
TFN-VIKOR	–	–	–	–	1	1	0.96364
IF-VIKOR	–	–	–	–	–	1	0.96364
TrFN-VIKOR	–	–	–	–	–	–	1

To further elaborate the benefits of the presented approach we analyzed the comparison one by one.

Two well-known utility theory-based approaches, TOPSIS [8] and VIKOR [47], leverage the proximity of ideal points to identify the best options. VIKOR [47] is more adaptable than TOPSIS [8] and is utilized in many different decision-making sectors since it may employ the compromise coefficient to obtain a compromise answer. Only real-valued data may be used with these approaches, and they struggle to solve MCDM issues in complicated fuzzy contexts. It is clear that our proposed approach may be utilized to determine both real numbers and fuzzy numbers for making decisions. Therefore, compared to the other two approach [8, 47] our proposed method offers a larger range of applications.

Ashwani [4] developed a hybrid MCDM approach integrating BWM and VIKOR method. Although BWM method performs better than analytic hierarchy process (AHP), it has a high computational complexity, which limits its use.

TFN-VIKOR is a hybrid VIKOR method given by Ahmet [5] which calculates the weights using fuzzy Shannon entropy and employes VIKOR to attain the optimal alternative. The major disadvantage of Shannon entropy to attain attribute weights is the hypersensitivity or high sensitivity of significance to the entropy values of different attributes [22], additionally, it disregards the preferences of the decision-makers.

IF-VIKOR is a hybrid VIKOR method proposed by Jiang [41] which estimates the attribute weights using CRITIC method and employes intuitionistic VIKOR to rank the alternatives. CRITIC method has a shortcoming in properly capturing the conflicting relationships among attributes, since it simply employs the Pearson correlation for this purpose [4].

TrFN-VIKOR is a hybrid VIKOR method proposed by Guangzheng [29] which estimates the attribute weights using maximum deviation method and the highest group utility and lowest individual loss values using distance measure of TrFN. The major disadvantage of maximum deviation method to compute attribute weights lies in the fact that it is highly affected by extreme values, also it has high computational complexity as compared to other weight determination methods. Also, the distance measure of TrFN produces unacceptable results under many situations.

Through the above discussions we can conclude the following:

The proposed model can handle complex decision-making problems involving uncertainty as compared to TOPSIS [8] and VIKOR [47].

The decision makers can effectively express their subjective preference with the help of parameters involved in the proposed method. This produces more adaptable results as compared to the existing ones.

Due to the use of high generalized heronian mean operators the proposed approach is more effective as compared to previous ones.

The dependency among attributes is the one of the vital issues is decision-making which is not considered in any of the above methods. The proposed approach effectively establishes dependency among attributes.

The proposed approach is a decision-making method which considers the effect of social media, new or any future update on ranking order.

5.2 Validation of news influence on ranking: Comparing updated ranking with original ranking

This section is aimed to examine the impact of news on rankings within a framework. The influence of news, whether positive or negative, on consumer perceptions and decision-making is widely recognized. It is crucial to assess how effectively incorporating news into ranking methodologies captures its influence. In this context, a comparative analysis is conducted between the updated rankings, which account for the effect of news, and the original rankings.

The purpose of this evaluation is to determine the extent to which the rankings of alternatives are influenced by news related to specific criteria. Specifically, we aim to validate whether alternatives receiving positive news exhibit higher rankings, while those associated with negative news experience lower rankings. This analysis helps assess the influence of news on the ranking of alternatives.

Original ranking: Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁

Now, suppose news emerges highlighting the significance of safety in the automobile industry. Consequently, all suppliers are expected to enhance their safety measures. Suppliers 9 and 10 already have high safety ratings, while Supplier 8 needs to improve its safety performance. However, it is important to note that increasing safety measures often lead to higher prices, which can impact consumer buying decisions. As a result, Supplier 8 experiences a negative effect from the news, leading to a decline in its ranking, which can be validated through the updated ranking shown below.

Updated ranking: Cr₉ > Cr₁₀ > Cr₈ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁

By comparing the updated rankings with the original rankings and analyzing the alignment with the actual impact of news, we can effectively validate the influence of news on specific criteria within the ranking. This analysis provides valuable insights into the capability of the proposed method to accurately reflect the impact of news on decision-making processes.

5.3 Sensitivity analysis

The three parameters p, q, and k have an obvious impact on the decision outcomes, as shown in section 3. Therefore, a sensitivity analysis is carried out in the current section to explore the impact of these parameters on the ranking findings. Table 23 lists the ranking outcomes for various parameter values.

Table 23
Ranking of results for different values of p, q, k

Value of p, q Value of k Ranking results

0 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₅ > Cr₃ > Cr₄ > Cr₂ > Cr₁

0.25 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₅ > Cr₃ > Cr₄ > Cr₂ > Cr₁

p = q = 0.5 0.5 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₅ > Cr₃ > Cr₄ > Cr₂ > Cr₁

0.75 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₅ > Cr₃ > Cr₄ > Cr₂ > Cr₁

1 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₅ > Cr₃ > Cr₄ > Cr₂ > Cr₁

0 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁

0.25 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁

p = q = 1 0.5 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁

0.75 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁

1 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁

0 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁

0.25 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁

p = q = 2 0.5 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁

0.75 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁

1 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁

0 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁

0.25 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁

p = q = 3 0.5 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁

0.75 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁

1 Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁

Value of p, q	Value of k	Ranking results
	0	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₅ > Cr₃ > Cr₄ > Cr₂ > Cr₁
	0.25	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₅ > Cr₃ > Cr₄ > Cr₂ > Cr₁
p = q = 0.5	0.5	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₅ > Cr₃ > Cr₄ > Cr₂ > Cr₁
	0.75	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₅ > Cr₃ > Cr₄ > Cr₂ > Cr₁
	1	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₆ > Cr₅ > Cr₃ > Cr₄ > Cr₂ > Cr₁
	0	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁
	0.25	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁
p = q = 1	0.5	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁
	0.75	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁
	1	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₅ > Cr₄ > Cr₂ > Cr₁
	0	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁
	0.25	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁
p = q = 2	0.5	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁
	0.75	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁
	1	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁
	0	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₆ > Cr₄ > Cr₅ > Cr₂ > Cr₁
	0.25	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁
p = q = 3	0.5	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁
	0.75	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁
	1	Cr₈ > Cr₉ > Cr₁₀ > Cr₇ > Cr₃ > Cr₄ > Cr₆ > Cr₅ > Cr₂ > Cr₁

The following conclusions are drawn from Table 23.

There are 20 possible ranking outcomes for various parameter settings, but the best option remains the same, i.e., the automobile Cr₈. This suggests that the parameters p, q and k for choosing the best alternative, have no significant impact on the decision-making results.

Only a small difference is seen between Cr₄ and Cr₆, and the ranking results stay constant when the values of parameters p and q are fixed while the value of parameter v is modified. This shows that the decision-making outcomes are only marginally dependent on the parameter k.

The ranking results slightly change with the change in value of parameters p and q. This indicates that the decision-making results are marginally sensitive with respect to the parameters p and q.

When the values of the parameters p and q are changed, the ranking results somewhat alter. This shows that the decision-making outcomes a only marginally sensitive to the p and q parameters.

Our suggested strategy is only somewhat sensitive to the parameters p and q, as well as the compromise coefficient k, as a result of the aforementioned observation. The suggested strategy is stable in relation to the ideal alternative since the optimal alternative is consistent in all of the circumstances.

This demonstrates the potency of our suggested approach. Additionally, our suggested approach is adaptable since the decision-maker’s subjective preferences may be expressed through the adjustment of the parameters.

6 Discussions

This section provides a comparative analysis from different perspective to clearly differentiate the proposed model from existing models.

Ranking method: The present study presents a different perspective for the ranking of alternatives by developing a novel ranking method integrating the F-BCM and VIKOR- H method. In contrast to AHP and F-BWM, the F-BCM method has the advantage of allowing fully consistent pairwise comparisons of evaluation criteria, and the conventional VIKOR method is enhanced by the use of new aggregation operators that can capture the correlations of the aggregated arguments. This innovative approach effectively removes the impact of anomalous data, yielding reliable results.

Initial opinions: The measure of importance regarding the attributes is not always additive. To model the uncertainty in the opinion collection many methods have been widely applied. The utilization of fuzzy measure to consider the initial evaluations of experts in the present study is an effective tool to consider initial opinions as it reflects the relationship between attributes, in contrast to ANP which might introduce additional complexity.

Weight of experts: In most of the GDM models’ experts are considered to be totally independent of each other. However, in social network environments, experts develop connections of trust with other experts. In order to construct the weights of the experts in a social network context, a trust propagation model is developed while taking into account the process of attaining agreement in group decision-making. The proposed model’s decision-making quality is increased while the complexity is reduced by the application of social network analysis (SNA).

Incomplete information estimation: Acknowledging that real world experts possess incomplete knowledge, the proposed approach incorporates social network analysis to determine the global importance of experts and estimate incomplete information. By effectively handling trust relationships among experts, the model gains robustness and adaptability in decision-making scenarios. This allows to account for varying degrees of expertise, ensuring a more comprehensive and reliable decision-making process. Opting for social network analysis is advantageous due to its ability to capture a holistic view of expert importance, which ultimately offers a more robust solution than approaches like Dempster-Shafer theory, Bayesian networks.

Attributes dependency: In real-life decision-making scenarios, attributes often exhibit subtle associations rather than complete independence. The proposed approach acknowledges the crucial role of attribute dependency in influencing alternative rankings. To address this vital aspect, we integrate the well-known DEMATEL method with regression analysis. In contrast to AHP, which introduces complexity and is sensitive to small changes, the integrated approach enables us to effectively capture and account for attribute dependencies. Consideration of these interrelationships, enhances the accuracy and relevance of the decision-making process, providing valuable insights in complex decision scenarios.

News or future update: In complex real-life decision-making problems, information is constantly evolving, and external factors, such as news and updates from various sources like television or social media, significantly influence the performance of alternatives. The proposed approach takes a proactive stance in analyzing how these updates affect the decision-making process. By thoroughly examining attribute dependency and the dynamics of update processes, we ensure that proposed model provides reliable estimations considering the effect of news or future updates.

7 Conclusion

This paper introduces a novel decision-making approach that centers on the crucial influence of news dynamics. In complex decision-making environments, where information constantly evolves, understanding the impact of news and real-time updates is paramount. Traditional models often overlook this dynamic factor. Proposed approach bridges this gap by seamlessly integrating the influence of news into the decision-making process. Within this complex environment, where interdependencies among attributes govern the decision outcomes, the proposed approach adeptly establish attribute dependencies while recognizing their intricate interrelationships. Through a trust propagation model, expert weights within social networks are calculated, taking into account the interdependence that experts often share in evaluating alternatives. The strength of the proposed approach lies in the way we uniquely incorporate news influence on ranking orders. By integrating an updated decision matrix, we enable the model to adapt in real time to the ever-changing news environment, providing a more accurate reflection of decision dynamics. Furthermore, the integrated F-BCM-VIKOR-H approach addresses uncertain data and provides robust decisions, considering both attribute interdependencies and the impact of real-time events. The MCDM model is explored using the problem of car selection. The authenticity and stability of the results are validated using comparative and sensitivity analysis. This comprehensive approach ensures a more holistic decision-making process, relevant in the face of evolving digital scenarios and social media dynamics.

The proposed approach considers the influence of social media and updates, the impact of external factors and changing dynamics on decision-making outcomes can be complex and difficult to capture comprehensively. Future research could explore more sophisticated methods to address these challenges and improve the robustness of decision-making models. Also, the following issues can be considered in future. First, the proposed method can be adapted to resolve MAGDM problems in incomplete environment. Further it can be combined with other theories such as game theory [40], regret theory [51] and rough set theory [57]. The proposed model can be extended to different information systems like intuitionistic fuzzy numbers, neutrosophic fuzzy numbers, hesitant fuzzy information systems [28] etc. Also, the proposed model can be expanded to include information systems in dynamic decision environment. The reduction of the decision rules is also a worth exploring topic.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Functional dependency-based group decision-making with incomplete information under social media influence: An application to automobile

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Fuzzy measure

2.1.1 Regular fuzzy measure

2.1.2 2-order additive fuzzy measure

2.2 Shapley function

3 The proposed measure based GDM model

3.1.1 Determining attribute weights through F-BCM

Table 1 TFNs for pairwise comparison Linguistic Set TFNs Equally Important (1,1,1) Moderately Important (0,0,0.25) Essentially Important (0,0.25,0.5) Very Strongly Important (0.25,0.5,0.75) Extremely Important (0.5,0.75,1) Absolutely Important (0.75,1,1)

3.4.1 Functional dependency between attributes

Table 2 5-point scale for determining influence matrix Influence of criteria on other criteria 5-point scale No impact 0 Low impact 1 Medium impact 2 High impact 3 Very high impact 4

4 An application

4.1 Problem description

Table 3 Initial decision-making matrix Att 1 Att 2 Att 3 Att 4 Att 5 Att 6 Cr 1 VH VH 2 2 S L Cr 2 VH VH 2 2 M L Cr 3 VH VH 2 2 M H Cr 4 VH VH 2 2 B L Cr 5 VH VH 2 3 M L Cr 6 VH VH 2 4 S L Cr 7 H H 4 2 S M Cr 8 H H 4 4 S M Cr 9 H VH 2 4 M H Cr 10 H VH 3 2 B H

4.2.1 To determine attribute weights via F-BCM

4.2.2 Ranking of alternatives via F-VIKOR-H

Table 4 TFNs for conversion of linguistic attributes TFNs Linguistic Set for Linguistic M1, M2 and M3 Set for Mv5 (0,0,0.25) Very low Very small (0,0.25,0.5) Low Small (0.25,0.5,0.75) Medium Medium (0.5,0.75,1) High Big (0.75,1,1) Very high Very big

Table 8 Trust relationship matrix – (0.25,0.5,0.75) (0.5,0.75,1) (0.5,0.75,1) (0.5,0.75,1) – (0,0.25,0.5) (0.25,0.5,0.75) (0.75,1,1) – (0.5,0.75,1) (25,0.5,0.75) (0.25,0.5,0.75) (0.5,0.75,1) (0.75,1,1) – (0.25,0.5,0.75) (0.75,1,1) (0.25,0.5,0.75) (0,0.25,0.5) –

Table 15 Initial influence matrix Att 1 Att 2 Att 3 Att 4 Att 5 Att 6 Att 1 0 3 2 2 2 3 Att 2 2 0 1 1 1 2 Att 3 3 2 0 1 1 1 Att 4 3 2 1 0 1 1 Att 5 3 2 2 2 0 1 Att 6 4 3 2 2 1 0

5.1 Comparative analysis

5.3 Sensitivity analysis

7 Conclusion

Declaration of competing interest

References

Table 1
TFNs for pairwise comparison

Linguistic Set TFNs

Equally Important (1,1,1)

Moderately Important (0,0,0.25)

Essentially Important (0,0.25,0.5)

Very Strongly Important (0.25,0.5,0.75)

Extremely Important (0.5,0.75,1)

Absolutely Important (0.75,1,1)

Table 2
5-point scale for determining influence matrix

Influence of criteria on other criteria 5-point scale

No impact 0

Low impact 1

Medium impact 2

High impact 3

Very high impact 4

Table 4
TFNs for conversion of linguistic attributes

TFNs Linguistic Set for Linguistic

M1, M2 and M3 Set for Mv5

(0,0,0.25) Very low Very small

(0,0.25,0.5) Low Small

(0.25,0.5,0.75) Medium Medium

(0.5,0.75,1) High Big

(0.75,1,1) Very high Very big

Table 8
Trust relationship matrix

– (0.25,0.5,0.75) (0.5,0.75,1) (0.5,0.75,1)

(0.5,0.75,1) – (0,0.25,0.5) (0.25,0.5,0.75)

(0.75,1,1) – (0.5,0.75,1) (25,0.5,0.75)

(0.25,0.5,0.75) (0.5,0.75,1) (0.75,1,1) – (0.25,0.5,0.75)

(0.75,1,1) (0.25,0.5,0.75) (0,0.25,0.5) –

Table 15
Initial influence matrix

Att ₁ Att ₂ Att ₃ Att ₄ Att ₅ Att ₆

Att ₁ 0 3 2 2 2 3

Att ₂ 2 0 1 1 1 2

Att ₃ 3 2 0 1 1 1

Att ₄ 3 2 1 0 1 1

Att ₅ 3 2 2 2 0 1

Att ₆ 4 3 2 2 1 0