Emergency supplier evaluation framework fusing probabilistic hesitant fuzzy set group decision-making and clustering-based methods

Abstract

To address the issues of inaccurate expert weight allocation in existing large-group decision-making methods and information loss in multi-attribute group emergency decision-making processes, we propose a clustering-based method using the probabilistic hesitant fuzzy set. Recognizing the diverse contributions of the decision makers within each cluster-to-cluster consistency and the varied impacts of preferences in different clusters on overall group preference, we introduce a two-layer weight model. Specifically, from a two-dimensional viewpoint, we determine the weights of decision members within each cluster using an expert evaluation distance formula and the weights of each cluster using fuzzy entropy. Subsequently, by incorporating the Maclaurin symmetric mean operator (MSMO), we establish the ranking of decision alternatives. Finally, we assess the effectiveness and applicability of the proposed method by applying it to analyze the case of the Tonga volcanic eruption.

Keywords

Multi-attribute group decision-making weight model cluster analysis Maclaurin symmetric mean operator

1 Introduction

In recent years, there has been a significant increase in major natural disasters such as floods, earthquakes, fires, volcanic eruptions, and other catastrophic events. Among these, the eruption of Mount Taal in Batangas Province, Philippines, affected over 42,000 people [1]. The coal mine fire accident in Chongqing trapped 357 people [2]. Typhoon “Dussehra” has caused 1.4545 million people in the entire province to be affected, with an emergency evacuation of 363,000 people and resettlement of 150,000 people [3]. Influenced by factors such as climate change and certain human activities, the global incidence of disasters is rapidly rising. It is projected that by 2030, the frequency of large and medium-sized disasters worldwide will reach 560 occurrences annually, averaging 1.5 occurrences per day [4]. Probabilistic hesitant fuzzy set(PHFS) [5], as one of the effective representations of fuzzy information, expresses the membership degree of elements in a probabilistic form. This effectively reduces the problem of information loss in the decision-making process. Due to containing more information than ordinary hesitant fuzzy sets, research on probabilistic hesitant fuzzy sets is generally applied to large-group decision-making. To address the above challenges, the probability hesitant fuzzy set (PHFS) provides a more intuitive representation of experts’ risk preferences. PHFS incorporates probability distributions, introducing an additional layer of complexity to the modelling process. Specifically, significant advancements have also been made in the realm of distance metrics for PHFS [6 –8]. As of now there is little research on expert weights, which is the first research gap in the extant literature.

The decision-making process involves an increasing number of DMs, each possessing unique areas of expertise and solutions. By classifying them accordingly, the resulting decisions can be more grounded in reality [9]. Kristina et al. [10] introduced the widely adopted K-means algorithm. Extending this algorithm, Zhou et al. [11] introduced the fuzzy C-means algorithm based on fuzzy membership degrees. Mengüç et al. [12] proposed a hybrid approach consisting of modelling and a K-means mathematical algorithm. Wang et al. [13] proposed a group decision consensus model based on two-dimensional binary linguistic terms. The weighting is based on the consistency level of judgment matrices provided by experts. The better the consistency of the judgment matrices, the higher the corresponding expert weights. Such methods only consider the quality of evaluations reflected in the logical consistency of experts during assessment, without accounting for the quality of evaluations based on the degree of alignment with objective reality. Therefore, when judgment matrices provided by various experts are entirely consistent, the weight coefficients for each expert are the same, making it impossible to differentiate the quality of individual judgment matrices. The above research has demonstrated the positive impact of PHFS in addressing decision problems. Nevertheless, PHFS solely consider DMs’ hesitations and fuzziness during the decision process, overlooking the urgency of unexpected events and the credibility of the information provided by decision experts in high-pressure situations. The proposed method in this paper includes the introduction of a weight model, which helps mitigate the impact of subjective weights, making the decision results closer to real situations. Moreover, this paper presents a decision model based on the PHFS environment, considering decision-makers’ hesitations, a factor that cannot be disregarded in emergency response scenarios. This is the second research gap in the extant literature.

The issue of expert weights is of utmost importance in group decision-making, Pang et al. [14] determined expert weights by comparing DMs’ assessments with the relative rankings of alternative solutions. In large-scale group decision-making. Methods for determining expert weights [15 –17] often assume that the greater the difference between an individual and group opinions, the lower is the weight assigned to the individual. The above expert weight determination method is too simple and has certain limitations, which affects the credibility of the evaluation results. Therefore, the first research objective of this paper is to develop a weight determination method based on DMs within a set that is directly linked to their contributions to enhancing the consistency of the set.

The literature review indicates that Multiple Attribute Group Decision Making (MAGDM) problems are applied in various aspects of life [18 –26], and the integration of attribute information [27 –31] has become a significant topic in MAGDM research.A review of the literature indicates that property information integration has emerged as a crucial topic of research on multi-attribute group decision-making (MAGDM).In real-world decision-making, the evaluation parameters of solution attributes are not mutually independent. Due to differences in decision-makers’ subjective consciousness, such as risk preferences, different decision results may arise, making it challenging to demonstrate the rationality of the outcomes. In such circumstances, the Maclaurin Symmetric Mean (MSM) operator is undoubtedly a favorable choice. The Maclaurin symmetric mean (MSM), initially proposed by Maclaurin [32], has undergone extensions and development [33] to capture the relationships among multiple input parameters. Furthermore, some researchers have integrated MSM into the fuzzy linguistic environment [34 –36] to demonstrate the practicality and efficacy of property information integration. However, it is worth noting that in the context of emergency response, the third research gap in the existing literature lies in considering how DMs can effectively cluster to select appropriate emergency response suppliers under prevailing conditions within the PHFS environment. Therefore, the second research objective of this paper is to effectively address the challenge of accurately classifying experts in the group decision-making process.

In this paper we categorize DM weights into two parts, namely aggregation weights and internal member weights. These weights should differ for different aggregation preferences, primarily due to the varying number of internal DMs within the aggregation and the distinctiveness of the information each aggregation provides. Furthermore, in real-life scenarios, DM weights may fluctuate due to differences in knowledge structure, socio-cultural background, and familiarity with decision solutions. Therefore, the third research objective of this paper is to propose an emergency supplier decision-making framework suitable for addressing PHFS based on meeting the first two research objectives.

This paper proposes a clustering method based on PHFS by achieving three established research objectives. The effectiveness and applicability of the method are demonstrated through case analysis and comparative experiments with existing approaches.

We organize the rest of the paper as follows: Section 2 introduces the fundamental concepts of PHFS and the Maclaurin symmetric mean operator (MSMO). Section 3 presents the expert clustering model and the method for calculating expert composite weights. Section 4 delineates the specific steps for addressing emergency group decision-making problems. Subsequently, in Section 5, we conducted a case study to illustrate the application of the proposed method and provided a comparative analysis. Finally, Section 6 summarizes the paper.

The subsequent chapter will provide a comprehensive introduction to several pertinent concepts essential for the present study, building upon the aforementioned review of existing research.

2 Preliminaries

This section introduces the fundamental concepts of PHFS and the MSMO sorting approach, which will be used in the following sections.

2.1 Probabilistic hesitant fuzzy set

Definition 1 [5]. Let X be a non-empty set, the probabilistic hesitant fuzzy set (PHFS) is defined as

$H_{p} = {< x, h (p_{x}) > | x \in X}$ (1) where h (p_x) = {γ^λ (P^λ) |λ = 1, 2, 3 . . . . . , k} is called the probabilistic hesitant fuzzy element (PHFE).

Definition 2 [5]. Let two PHFEs be h₁ (p) = {γ₁^λ (P₁^λ) |λ = 1, 2, 3 . . . . . , k₁} and h₂ (p) = {γ₂^λ (P₂^λ) |λ = 1, 2, 3 . . . . . , k₂}. The adjustment steps are summarized in Algorithm 1.

Algorithm 1 The adjustment steps of two PHFEs
Required.h₁ (p), h₂ (p).
Output.h₁ (p) and h₂ (p) are exactly the same.
Step 1: Standardize h₁ (p) and h₂ (p) to get $h_{1} (\bar{\bar{p}})$ and $h_{2} (\bar{\bar{p}})$ .
Step 2: ${\bar{\bar{p}}}^{1} = \min {{\tilde{p}}_{1}^{1}, {\tilde{p}}_{2}^{1}}$ .
Step 3: If $m_{θ_{n}, i}, {\bar{m}}_{p (Φ), i}$ and ${\tilde{m}}_{p (Φ), i}$ , then ${\bar{\bar{p}}}^{2} = min {\tilde{p_{1}^{2}}, \tilde{p_{2}^{1}} - {\bar{\bar{p}}}^{1}}$ ;
If ${\bar{\bar{p}}}^{1} = \tilde{p_{2}^{1}}$ , then ${\bar{\bar{p}}}^{2} = min {\tilde{p_{2}^{2}}, \tilde{p_{1}^{1}} - {\bar{\bar{p}}}^{1}}$ .
Step 4: If ${\bar{\bar{p}}}^{1} = \tilde{p_{1}^{1}}$ and ${\bar{\bar{p}}}^{2} = \tilde{p_{1}^{2}}$ , then ${\bar{\bar{p}}}^{3} = min {\tilde{p_{1}^{3}}, \tilde{p_{2}^{1}} - {\bar{\bar{p}}}^{1} - {\bar{\bar{p}}}^{2}}$ ;
If ${\bar{\bar{p}}}^{1} = \tilde{p_{1}^{1}}$ and ${\bar{\bar{p}}}^{2} = \tilde{p_{2}^{1}}$ , then ${\bar{\bar{p}}}^{3} = min {\tilde{p_{2}^{2}}, \tilde{p_{1}^{2}} - {\bar{\bar{p}}}^{2}}$ ;
If ${\bar{\bar{p}}}^{1} = \tilde{p_{2}^{1}}$ and ${\bar{\bar{p}}}^{2} = \tilde{p_{2}^{2}}$ , then ${\bar{\bar{p}}}^{3} = min {\tilde{p_{2}^{3}}, \tilde{p_{2}^{2}} - {\bar{\bar{p}}}^{1} - {\bar{\bar{p}}}^{2}}$ ;
If ${\bar{\bar{p}}}^{1} = \tilde{p_{2}^{1}}$ and ${\bar{\bar{p}}}^{2} = \tilde{p_{1}^{1}} - {\bar{\bar{p}}}^{1}$ , then ${\bar{\bar{p}}}^{3} = min {\tilde{p_{1}^{2}}, \tilde{p_{2}^{2}} - {\bar{\bar{p}}}^{2}}$ ... , so on.
Return. Until the probability information of the membership degree in $h_{1} (\bar{\bar{p}})$ and $h_{2} (\bar{\bar{p}})$ is exactly the same.

Definition 3. [5] In MAGDM problems, the evaluated scheme set is denoted as A ={ A₁, A₂, ⋯ , A_i, ⋯ , A_M }, the attribute set as U ={ u₁, u₂, ⋯ , u_i, ⋯ , u_M }, and the decision expert group’s evaluation data information for the j-th attribute of the i-th solution is represented by PHFE h (p). Here, PHFE is defined as: $\begin{matrix} h_{ij} (p_{ij}) = {(γ_{ij}^{(k)}, p_{ij}^{(k)}) | γ_{ij}^{(k)} \in [0, 1], \sum_{k = 1}^{h_{ij} (p_{ij})} p_{ij}^{(k)} \\ \leq 1, k = 1, 2, \dots, h_{ij} (p_{ij})} \end{matrix}$

2.2 The Maclaurin symmetric mean operator sorting approach

Definition 3.: Let a_i (i = 1, 2, ... , n) be a set of non-negative real numbers and r = 1, 2, ... , n. Define

$MS M^{(r)} (\begin{matrix} a_{1}, a_{2}, \dots, a_{n} \end{matrix}) = {(\frac{\sum_{1 \leq i_{1} < \dots < i_{r} \leq n} \prod_{j = 1}^{r} a_{i_{j}}}{C_{n}^{r}})}^{\frac{1}{r}}$ (2)

where i = 1, 2, ... , n, are all r-tuples in the ergodic combination 1, 2,..., n and $C_{n}^{r} = \frac{n!}{r! (n - r)!}$ is the binomial coefficient.

In this paper, the weight model is constructed for decision-making problems in PHFS environment, recognizing the diverse contributions of the decision makers within each cluster-to-cluster consistency and the varied impacts of preferences in different clusters on overall group preference, we introduce a two-layer weight model. Subsequently, incorporating the Maclaurin symmetric mean operator, we establish the rankings of decision alternatives. Therefore, in the following sections, how the model of this article is constructed will be shown in detail.

3 A group decision framework for emergency supplier selection

3.1 Compatibility-based clustering method

Given k experts evaluating m attributes of the solutions, the preference matrix R_λ provided by the λ-th expert for the m attributes of the solutions is considered. First, use the Kendall’s intra-group data consistency analysis method, which involves conducting a consistency test on the judgment matrix. Subsequently, the normalized individual ranking vector T_λ = (T_λ1, T_λ2, T_λ3, . . . , T_λm) ^S is obtained using R_λ, where λ = (1, 2, . . . , k).

Definition 4. Let R_a = (r^a_ij) m × m and R_b = (r^b_ij) m × m, if for all i, j ∈ Z, r^a_ij = r^b_ij, then R_a and R_b are called completely compatible.

Definition 5. Let R_a = (r^a_ij) m × m, R_b = (r^b_ij) m × m, T_a = (T_a1, T_a2, . . . , T_am) ^S, and T_b = (T_b1, T_b2, . . . , T_bm) ^S be the normalized ranking vectors obtained by the eigenvalue method for T_a and T_b. Let L (a, b) represent the similarity coefficient between individual ranking vectors T_a and T_b, which generally satisfies the following properties:

L (a, b) = 1 exhibits reflexivity;

L (a, b) = L (b, a) shows symmetry;

L (a, b) = 1 indicates that the individual ranking vectors of the a-th and b-th experts are completely compatible;

For all a, b ≤ k, 0 ≤ L (a, b) ≤ 1. The closer L (a, b) is to 1, the more similar are the vectors T_a and T_b, vice versa.

In this paper we use the cosine value of the angle between vectors to represent the similarity between individual ranking vectors T_a and T_b as follows:

$L (a, b) = \frac{\sum_{k = 1}^{m} T_{ak} \times T_{bk}}{\sqrt{\sum_{k = 1}^{m} T_{ak}^{2} \times \sum_{k = 1}^{m} T_{bk}^{2}}}$ (3)

We summarize in Algorithm 2 the clustering of experts based on the similarity of compatibility in the expert vector matrix.

Algorithm 2 Clustering of experts based on the similarity of compatibility in the expert vector matrix
Required. B = (L (a, b)) k × k
Output. All the experts complete the classification.
Step 1: Based on the compatibility matrix B = (L (a, b)) k × k, identify the maximum element in the off-diagonal elements as $L_{max}^{(1)} = max L (a, b)_{a, b = 1, \dots, k, a \neq b}$ . Then, assign the two experts c and d with the highest compatibility to a new group denoted as G₁ = { c, d }.
Step 2: Remove the compatibility related to the expert sorting vectors of c and d. Then, add the compatibility between the new group G₁ and other expert sorting vectors as L (e, f) = max{ L (c, f) , L (d, f) }, thereby reconstructing the compatibility matrix.
Step 3: Find the expert corresponding to the maximum compatibility $L_{max}^{(2)} =$ max{ L (c, f) , L (d, f) } among the off-diagonal elements of the new matrix. Then, form a group G₂ with the previous new group.
Step 4: Remove the expert corresponding to the maximum compatibility in this iteration and add it to the new group G₂, thereby reconstructing the compatibility matrix.
Step 5: Repeat Steps 3 and 4 until all the experts belong to one large cluster.
Step 6: Record the maximum compatibility value during the classification and merging process.
Step 7:Through data comparison, a reasonable threshold T is calculated, which is classified according to the distance between the given threshold T and the maximum compatibility and the maximum compatibility.
Return. Expert clustering grouping.

3.2 The method of expert weight acquisition

3.2.1 Expert clustering group weights model

Cluster the k decision experts participating in the decision-making process into M clusters and let G = { G¹, G², . . . , G^M } (M > 1) be the cluster set, where cluster G^M contains n_M DMs and satisfies $\sum_{M = 1}^{M} n_{M} = K .$

Definition 6. Let the PHFE be $h_{ij} (p_{ij}) = {({γ_{ij}}^{λ}, {P_{ij}}^{λ}) | λ = 1, 2, 3 . . . . ., k | \sum_{λ = 1}^{k} {P_{ij}}^{λ} = 1}$ , then its geometric distance d = d (h (p)) is defined as

$\begin{matrix} d (h_{ij} (p_{ij})) = \sum_{λ = 1}^{k} \\ \times \frac{min {γ_{ij}, p_{ij}} \sqrt{γ_{ij}^{2} + p_{ij}^{2}}}{\sqrt{γ_{ij}^{2} + p_{ij}^{2}} + \sqrt{{(γ_{ij} - 1)}^{2} + {(p_{ij} - 1)}^{2}}} . \end{matrix}$ (4)

Shannon [37] proposed the concept of information entropy in 1948, i.e.,

$\begin{matrix} H (X) = H (P_{1}, P_{2}, \dots, P_{n}) = - \sum_{i = 1}^{n} (p_{i} ln p_{i}) \end{matrix}$ (5)

Higher entropy values indicate lower information content and result in smaller weight allocations, reflecting increased ambiguity and decreased impact on the decision outcomes. The formula for calculating the weights between groups is as follows:

$\begin{matrix} {\overset{´}{ω}}_{M} = \frac{1 - B_{M}}{M - \sum_{M = 1}^{M} B_{M}} \end{matrix}$ (6) Where

$B_{M} = - \frac{\sum_{M = 1}^{M} θ K ln θ_{K}}{ln K}, M = 1, 2, 3 . . . . ., k$ (7) and

$θ_{K} = \frac{d (h_{i j} (p_{i j}))}{\sum_{M = 1}^{K} d (h_{i j} (p_{i j}))} = \frac{\sum_{λ = 1}^{k} \times \frac{\min {γ_{i j}, p_{i j}} \sqrt{γ_{_{i j}}^{2} + p_{_{i j}}^{2}}}{\sqrt{γ_{_{i j}}^{2} + p_{_{i j}}^{2}} + \sqrt{{(γ_{_{i j}} - 1)}^{2} + {(p_{_{i j}} - 1)}^{2}}}}{\sum_{M = 1}^{K} \times \sum_{λ = 1}^{k} \times \frac{\min {γ_{_{i j}}, p_{_{i j}}} \sqrt{γ_{_{i j}}^{2} + p_{_{i j}}^{2}}}{\sqrt{γ_{_{i j}}^{2} + p_{_{i j}}^{2}} + \sqrt{{(γ_{_{i j}} - 1)}^{2} + {(p_{_{i j}} - 1)}^{2}}}}$ (8)

3.2.2 Expert weight model

In the context of MAGDM, there are K decision experts D_l (l = 1, 2, . . . , K), Z attributes u_i (i = 1, 2, . . . , Z), and expert weights ω_{D
_l} (l = 1, 2, . . . , K). There are A candidate solutions P_j (j = 1, 2, . . . , A). The scoring provided by the first expert for the j-th solution is expressed as D_{lp
_j}, while D_{lu
_i} is used for scoring the i-th attribute.

For experts, the discordance in the decision matrix indicates the level of uncertainty associated with their assessments of the available options. As discordance increases, trust diminishes, necessitating a reduction in their objective weights. Therefore, expert discordance functions O_Z (D_l) and S_A (D_l) are defined, along with objective weights ω_{D
_l} as follows:

Definition 7. In the MAGDM problem, the separation rate of expert ratings of attributes is expressed as

$O_{Z} (D_{l}) = \sum_{i = 1}^{Z} \frac{| D_{l u_{i}} - \frac{1}{K} \sum_{l = 1}^{K} D_{l u_{i}} |}{\frac{1}{K} \sum_{l = 1}^{K} D_{l u_{i}}}$ (9)

Expert D_l gives attribute u_i a score that affects the average separation rate of attribute u_i scores.

Definition 8. In the MAGDM problem, the separation rate of experts’ ratings of solutions is expressed as $S_{A} (D_{l}) = \sum_{j = 1}^{A} \frac{| D_{l p_{j}} - \frac{1}{K} \sum_{l = 1}^{K} D_{l p_{j}} |}{\frac{1}{K} \sum_{l = 1}^{K} D_{l p_{j}}}$ (10)

For expert D_l, the separation rate between the score of plan P_j and the average score of plan P_j is given.

Definition 9. Expert D_l’s weight ω_{D
_l} is calculated as follows:

$\begin{matrix} ω_{D_{l}} = 1 - \\ \frac{\sum_{i = 1}^{Z} \frac{| D_{l u_{i}} - \frac{1}{K} \sum_{l = 1}^{K} D_{l u_{i}} |}{\frac{1}{K} \sum_{l = 1}^{K} D_{l u_{i}}} + \sum_{j = 1}^{A} \frac{| D_{l p_{j}} - \frac{1}{K} \sum_{l = 1}^{K} D_{l p_{j}} |}{\frac{1}{K} \sum_{l = 1}^{K} D_{l p_{j}}}}{\sum_{l = 1}^{K} \sum_{i = 1}^{Z} \frac{| D_{l u_{i}} - \frac{1}{K} \sum_{l = 1}^{K} D_{l u_{i}} |}{\frac{1}{K} \sum_{l = 1}^{K} D_{l u_{i}}} + \sum_{l = 1}^{K} \sum_{j = 1}^{A} \frac{| D_{l p_{j}} - \frac{1}{K} \sum_{l = 1}^{K} D_{l p_{j}} |}{\frac{1}{K} \sum_{l = 1}^{K} D_{l p_{j}}}} \end{matrix}$ (11)

Experts are known to have subjective weights of

$ω_{D_{l}} = (ω_{D_{1}}, ω_{D_{2}}, ω_{D_{3}}, . . . . . . ω_{D_{K}})$ (12)

The combined empowerment method is employed to determine the overall weight of the expert as follows:

$ω_{k} = {\overset{´}{ω}}_{M} \times ω_{D_{l}}$ (13)

3.3 Weighted composite value

By Definition 6, the PHFE geometric distance is

$\begin{matrix} d (h_{ij} (p_{ij})) = \sum_{λ = 1}^{k} \times \\ \frac{min {γ_{ij}, p_{ij}} \sqrt{γ_{ij}^{2} + p_{ij}^{2}}}{\sqrt{γ_{ij}^{2} + p_{ij}^{2}} + \sqrt{{(γ_{ij} - 1)}^{2} + {(p_{ij} - 1)}^{2}}} . \end{matrix}$ (14)

Definition 10. In the MAGDM problem, the dispersion coefficient model of PHFE h_ij (p_ij) is

$\begin{matrix} e (h_{ij} (p_{ij})) = \frac{1}{2} \sum_{D_{l} = 1}^{K} \sum_{D'_{l} = 1}^{K} \times \\ \sqrt{{(γ_{ij}^{(D_{l})} - γ_{ij}^{(D'_{l})})}^{2} + {(p_{ij}^{(D_{l})} - p_{ij}^{(D'_{l})})}^{2}} \end{matrix}$ (15)

Definition 11. In the MAGDM problem, the weighted comprehensive value of PHFE h_ij (p_ij) is definedas

$\begin{matrix} PHFE h_{ij} (p_{ij})^{ω} = {\tilde{h}}_{ij} (p_{ij}) = \\ {[d (h_{ij} (p_{ij}))]}^{ω_{j}} + {[e (h_{ij} (p_{ij}))]}^{ω_{j} - 1} \end{matrix}$ (16)

Among them, ω_j represents the attribute weight.

The decision framework should be presented in Section 4, based on the newly proposed weight model and distance model in this section.

4 Solution procedure of the proposed decision-making framework

We summarize in Algorithm 3 the decision-making steps based on PHFS and the MSMO sorting approach.

Algorithm 3 The steps of the proposed method
Required.R^λ (λ = 1, 2, . . . . , k), ω_{D _l}, and ${\tilde{h}}_{ij} (p_{ij})$ , H_i.
Output.h₁ (p) and h₂ (p) are exactly the same.
Step 1: Evaluate the performance of suppliers on m attributes by k decision experts and obtain q preference matrices R^λ (λ = 1, 2, . . . . , k);
Step 2: Obtain the normalized ranking vector T_λ of R^λ using the eigenvalue method and cluster the experts through compatibility clustering;
Step 3: Calculate the inter-cluster weights of the experts using the entropy method, i.e., Equations (6)–(8);
Step 4: Calculate the trustworthiness of the preference matrices and construct the attribute weight determination model based on trustworthiness, obtaining the objective weights ω_{D _l} through Equations (9)–(12);
Step 5: Calculate the comprehensive weight of the experts ω_k by using the weights between expert groups and the individual weights of experts ω_{D _l} through Equation (13);
Step 6: Draw the final expert decision table based on the expert comprehensive weights and expert initial rating data;
Step 7: Calculate the weighted comprehensive value ${\tilde{h}}_{ij} (p_{ij})$ of each attribute according to Definition 11
Step 8: Compare each plan on a single attribute and establish a comparison matrix. Here the comparison matrix $H_{j} = {[H_{{ii}^{'}}]}_{M \times M} = {[{(\frac{{\tilde{h}}_{ij} (p_{ij})}{{\tilde{h}}_{ij} (p_{ij})})}_{i \times i^{'}}]}_{M \times M}$ , i, i′ = 1, 2, … . Z; j = 1, 2, … A;
Step 9: Calculate the aggregate value $H_{j}^{i}$ of each scheme on the j-th attribute through Equation (3) combined with the comparison matrix H_ii′;
Step 10: Calculate the collective value $H_{j}^{i}$ of each solution on all the attributes, where $H_{i} = \sum_{i = 1}^{Z} H_{j}^{i}$ ;
Step 11: Compare the size of H and rank the alternatives by the properties of MSMO.
End for
Return. The optimal emergency supplier.

In order to validate the effectiveness and superiority of the decision framework proposed in this paper, subsequent section will conduct case analysis and comparative analysis, accompanied by comprehensive discussions.

5 Case study

5.1 Problem description

Due to the complexity and variability of emergencies, it is challenging for any department or individual to make decisions independently. Therefore, emergency decision-making requires the simultaneous participation of multiple experts from different disciplines. Additionally, during the initial stages of emergencies, information is incomplete, decision-makers face time constraints, data scarcity, and their thinking tends to be ambiguous, preferring to express preferences using fuzzy information. Therefore, in a fuzzy environment, considering the challenges in emergency decision-making, methods such as fuzzy theory, multi-attribute decision-making, and group decision-making are chosen for solution selection. The volcanic eruption disaster in Tonga in January 2022 is an example of a complex and dynamic emergency event, requiring the simultaneous involvement of experts from various departments. Hence, using this case is suitable for validating the effectiveness and feasibility of the proposed method in this paper.

In order to safeguard people’s lives and basic livelihoods, the Tonga Prime Minister’s Office declared a state of emergency nationwide and immediately established the Accident Emergency Response Team. Relevant departments quickly set up the Emergency Decision and Rescue Command Center, convening experts from fields such as firefighting, medical, power supply, and environmental protection. Based on the actual situation, we select six emergency suppliers A ={ A_i, i = 1, . . . , 6 }. To select the most reasonable plan, the government invited 50 emergency decision-making experts from different fields, i.e., K ={ k₁, . . . , k₅₀ }, to simulate the real scenario of a volcanic eruption. The decision-making process as follows:

Step 1: Six attribute categories, denoted as U ={ u₁, . . . , u₆ }, are identified. These categories encompass “u₁: material cost”, “u₂: human casualties”, “u₃ speed of assistance”, “u₄: post-disaster recovery”, “u₅: economic losses”, and “u₆: social impact.” The initial preference matrix of the DM, as illustrated in Table 1.

Table 1
DMs’ initial probability preference matrix

u ₁ u ₂ u ₃ u ₄ u ₅ u ₆

A1 (0.9,0.533) (0.1,0.585) (0.6,0.314) (0.8,0.703) (1.0,0.706) (0.8,0.687)

(0.3,0.342) (0.6,0.145) (1.0,0.124) (1.0,0.106) (0.4,0.161) (0.1,0.184)

(0.6,0.125) (0.9,0.270) (0.4,0.562) (0.9,0.191) (0.7,0.133) (0.5,0.129)

A2 (0.1,0.531) (0.2,0.222) (0.8,0.850) (0.9,0.601) (0.6,0.786) (0.7,0.912)

(1.0,0.028) (0.4,0.569) (0.5,0.022) (1.0,0.062) (0.5,0.138) (0.4,0.061)

(0.4,0.441) (0.5,0.209) (0.6,0.128) (0.4,0.337) (0.7,0.076) (0.9,0.027)

A3 (0.9,0.026) (0.5,0.662) (0.8,0.234) (0.4,0.266) (0.5,0.291) (0.2,0.651)

(0.5,0.416) (0.1,0.181) (0.3,0.112) (0.9,0.557) (0.8,0.084) (1.0,0.179)

(0.7,0.558) (0.2,0.157) (0.9,0.654) (0.7,0.177) (0.6,0.625) (0.6,0.170)

A4 (0.4,0.673) (0.6,0.768) (1.0,0.431) (0.3,0.410) (0.6,0.708) (0.1,0.651)

(0.6,0.034) (0.8,0.165) (0.4,0.485) (0.8,0.370) (1.0,0.044) (0.9,0.131)

(0.3,0.293) (0.7,0.067) (0.2,0.084) (0.9,0.220) (0.1,0.248) (0.8,0.218)

A5 (0.2,0.118) (0.2,0.345) (0.2,0.407) (0.5,0.746) (0.2,0.224) (0.1,0.730)

(0.8,0.088) (0.9,0.359) (1.0,0.160) (0.1,0.167) (0.4,0.441) (0.2,0.024)

(0.9,0.794) (1.0,0.296) (0.7,0.433) (1.0,0.087) (0.8,0.335) (0.3,0.246)

A6 (0.2,0.947) (0.7,0.904) (1.0,0.222) (0.4,0.235) (0.1,0.721) (0.5,0.042)

(0.9,0.012) (0.5,0.051) (0.1,0.240) (0.3,0.536) (0.7,0.043) (0.9,0.652)

(0.3,0.041) (0.8,0.045) (0.7,0.538) (0.1,0.229) (0.6,0.236) (0.8,0.306)

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
A1	(0.9,0.533)	(0.1,0.585)	(0.6,0.314)	(0.8,0.703)	(1.0,0.706)	(0.8,0.687)
	(0.3,0.342)	(0.6,0.145)	(1.0,0.124)	(1.0,0.106)	(0.4,0.161)	(0.1,0.184)
	(0.6,0.125)	(0.9,0.270)	(0.4,0.562)	(0.9,0.191)	(0.7,0.133)	(0.5,0.129)
A2	(0.1,0.531)	(0.2,0.222)	(0.8,0.850)	(0.9,0.601)	(0.6,0.786)	(0.7,0.912)
	(1.0,0.028)	(0.4,0.569)	(0.5,0.022)	(1.0,0.062)	(0.5,0.138)	(0.4,0.061)
	(0.4,0.441)	(0.5,0.209)	(0.6,0.128)	(0.4,0.337)	(0.7,0.076)	(0.9,0.027)
A3	(0.9,0.026)	(0.5,0.662)	(0.8,0.234)	(0.4,0.266)	(0.5,0.291)	(0.2,0.651)
	(0.5,0.416)	(0.1,0.181)	(0.3,0.112)	(0.9,0.557)	(0.8,0.084)	(1.0,0.179)
	(0.7,0.558)	(0.2,0.157)	(0.9,0.654)	(0.7,0.177)	(0.6,0.625)	(0.6,0.170)
A4	(0.4,0.673)	(0.6,0.768)	(1.0,0.431)	(0.3,0.410)	(0.6,0.708)	(0.1,0.651)
	(0.6,0.034)	(0.8,0.165)	(0.4,0.485)	(0.8,0.370)	(1.0,0.044)	(0.9,0.131)
	(0.3,0.293)	(0.7,0.067)	(0.2,0.084)	(0.9,0.220)	(0.1,0.248)	(0.8,0.218)
A5	(0.2,0.118)	(0.2,0.345)	(0.2,0.407)	(0.5,0.746)	(0.2,0.224)	(0.1,0.730)
	(0.8,0.088)	(0.9,0.359)	(1.0,0.160)	(0.1,0.167)	(0.4,0.441)	(0.2,0.024)
	(0.9,0.794)	(1.0,0.296)	(0.7,0.433)	(1.0,0.087)	(0.8,0.335)	(0.3,0.246)
A6	(0.2,0.947)	(0.7,0.904)	(1.0,0.222)	(0.4,0.235)	(0.1,0.721)	(0.5,0.042)
	(0.9,0.012)	(0.5,0.051)	(0.1,0.240)	(0.3,0.536)	(0.7,0.043)	(0.9,0.652)
	(0.3,0.041)	(0.8,0.045)	(0.7,0.538)	(0.1,0.229)	(0.6,0.236)	(0.8,0.306)

Step 2: Following Definition 4, we standardize the initial expert rating matrix R^λ and obtain the normalized ranking vector T^λ of R^λ using the characteristic root method. Tables 2–5 display this process.

Table 2

Initial rating matrix for DMs

Expert k₁’s ratings of the plans
	u	u	u	u	u	u
A1	0.9	0.1	0.4	0.9	1.0	0.8
A2	0.4	0.4	0.8	0.9	0.6	0.7
A3	0.5	0.1	0.9	0.9	0.6	0.6
A4	0.3	0.7	0.4	0.3	0.6	0.1
A5	0.9	0.9	1.0	0.5	0.4	0.1
A6	0.2	0.7	1.0	0.3	0.6	0.9

Table 3

Initial rating matrix for DMs

Expert k₂’s ratings of the plans
	u	u	u	u	u	u
A1	0.3	0.1	0.4	0.9	1.0	0.5
A2	0.1	0.4	0.8	0.9	0.6	0.7
A3	0.5	0.2	0.3	0.9	0.6	0.6
A4	0.4	0.6	1.0	0.9	0.1	0.1
A5	0.9	1.0	0.7	0.5	0.4	0.1
A6	0.2	0.7	0.7	0.3	0.6	0.8

Table 4

Initial rating matrix for DMs

Expert k₄₉’s ratings of the plans
	u	u	u	u	u	u
A1	0.3	0.1	0.4	0.8	1.0	0.8
A2	0.1	0.4	0.8	0.9	0.6	0.7
A3	0.7	0.2	0.9	0.7	0.6	0.2
A4	0.4	0.8	0.4	0.3	0.6	0.1
A5	0.9	0.9	0.7	0.5	0.4	0.1
A6	0.2	0.7	0.7	0.1	0.6	0.9

Table 5

Initial rating matrix for DMs

Expert k₅₀’s ratings of the plans
	u	u	u	u	u	u
A1	0.9	0.9	0.6	1.0	1.0	0.1
A2	0.4	0.2	0.8	0.9	0.6	0.7
A3	0.7	0.2	0.9	0.9	0.5	0.6
A4	0.4	0.8	0.2	0.8	0.6	0.8
A5	0.9	1.0	0.7	1.0	0.4	0.3
A6	0.2	0.7	0.7	0.4	0.6	0.9

According to the compatibility matrix, the experts are clustered according to the expert clustering steps, through data comparison, the calculated threshold T = 1.0000-0.9890 = 0.011 and the results are shown in Table 6.

Table 6

Expert clustering results

Gather	Number of members	Expert k_l
G1	29	K1,K5,K11,K13,K14,K18,K19,K21,K22,K23,K24,K25, K26, K27,K28,K29,K30,K31,K33,K35,K36,K38,K39,K42,K43,K45,K47,K48, K49
G2	16	K2,K3,K4,K6,K7,K8,K9,K12,K15,K17,K20,K32,K40,K41,K46,K50
G3	1	K34
G4	3	K16,K37,K44
G5	1	K10

Step 3: Through the entropy method, i.e., Equations (6)–(8), the weights between expert clustering groups are derived, yielding the aggregation weights ${\overset{´}{ω}}_{1} = 0.5182$ , ${\overset{´}{ω}}_{2} = 0.2856$ , ${\overset{´}{ω}}_{3} = 0.0545$ , ${\overset{´}{ω}}_{4} = 0.0905$ , and ${\overset{´}{ω}}_{5} = 0.0513$ .

Step 4: The aggregated internal expert weights ω_{D
_l} are obtained through Equation (9)–(12) and the expert comprehensive weights ω _k are obtained through Equation (13)

Step 5: After calculating the weight of each decision-making member within the aggregation, the final decision-making fuzzy matrix is constructed based on the individual weights of the experts, as shown in Table 9.

Step 6: Calculate the weighted comprehensive value of each attribute ${\tilde{h}}_{ij} (p_{ij})$ , as shown in Table 10.

Step 7: Compare each plan on a single attribute and establish a comparison matrix. $\begin{matrix} H_{1} = {[H_{{ii}^{'}}]}_{6 \times 6} = \\ [\begin{matrix} 1.0000 1.1460 1.0663 0.8173 0.9877 0.6403 \\ 0.8726 1.0000 0.9305 0.7132 0.8619 0.5587 \\ 0.9378 1.0747 1.0000 0.7665 0.9263 0.6005 \\ 1.2236 1.4021 1.3046 1.0000 1.2085 0.7834 \\ 1.0124 1.1602 1.0795 0.8274 1.0000 0.6483 \\ 1.5618 1.7897 1.6653 1.2764 1.5426 1.0000 \end{matrix}] \end{matrix}$ $\begin{matrix} H_{2} = {[H_{{ii}^{'}}]}_{6 \times 6} = \\ [\begin{matrix} 1.0000 0.8879 0.8645 0.8969 0.9442 0.8583 \\ 1.1262 1.0000 0.9736 1.0100 1.0633 0.9666 \\ 1.1568 1.0271 1.0000 1.0375 1.0922 0.9929 \\ 1.1150 0.9901 0.9639 1.0000 1.0528 0.9570 \\ 1.0591 0.9404 0.9156 0.9499 1.0000 0.9091 \\ 1.1651 1.0345 1.0072 1.2764 1.0449 1.0000 \end{matrix}] \end{matrix}$ $\begin{matrix} H_{3} = {[H_{{ii}^{'}}]}_{6 \times 6} = \\ [\begin{matrix} 1.0000 0.9099 0.9846 0.9994 1.0608 1.0123 \\ 1.0990 1.0000 1.0821 1.0983 1.1658 1.1125 \\ 1.0156 0.9241 1.0000 1.0150 1.0774 1.0281 \\ 1.0006 0.9105 0.9852 1.0000 1.0615 1.0129 \\ 0.9427 0.8577 0.9282 0.9421 1.0000 0.9543 \\ 0.9878 0.8989 0.9727 0.9872 1.0479 1.0000 \end{matrix}] \end{matrix}$

Table 7

Gathering the weights of internal decision-making experts

Gather	Aggregate internal DM weight
G1	ω₁ = 0.02011 ω₅ = 0.02007 ω₁₁ = 0.02008 ω₁₃ = 0.02007 ω₁₄ = 0.01994 ω₁₈ = 0.02013 ω₁₉ = 0.01997 ω₂₁ = 0.02005 ω₂₂ = 0.02007 ω₂₃ = 0.01990 ω₂₄ = 0.01990 ω₂₅ = 0.02002 ω₂₆ = 0.01997 ω₂₇ = 0.01994 ω₂₈ = 0.01999 ω₂₉ = 0.02007 ω₃₀ = 0.02021 ω₃₁ = 0.01997 ω₃₃ = 0.02017 ω₃₅ = 0.02000 ω₃₆ = 0.02019 ω₃₈ = 0.01993 ω₃₉ = 0.01978 ω₄₂ = 0.01986 ω₄₃ = 0.01998 ω₄₅ = 0.02007 ω₄₇ = 0.02018 ω₄₈ = 0.01979 ω₄₉ = 0.02014
G2	ω₂ = 0.02010 ω₃ = 0.02009 ω₄ = 0.01996 ω₆ = 0.01991 ω₇ = 0.02004 ω₈ = 0.01989 ω₉ = 0.01983 ω₁₂ = 0.01982 ω₁₅ = 0.02010 ω₁₇ = 0.01992 ω₂₀ = 0.02004 ω₃₂ = 0.02020 ω₄₀ = 0.01987 ω₄₁ = 0.02002 ω₄₆ = 0.01996 ω₅₀ = 0.01992
G3	ω₃₄ = 0.01996
G4	ω₁₆ = 0.02002 ω₃₇ = 0.01992 ω₄₄ = 0.01985
G5	ω₁₀ = 0.01999

Table 8

Comprehensive weight of each decision-making expert

Expert	K1	K2	K3	K4	K5	K6	K7	K8	K9
Weight	0.026	0.014	0.014	0.014	0.026	0.014	0.014	0.014	0.014
Expert	K10	K11	K12	K13	K14	K15	K16	K17	K18
Weight	0.003	0.026	0.014	0.026	0.026	0.014	0.005	0.014	0.026
Expert	K19	K20	K21	K22	K23	K24	K25	K26	K27
Weight	0.026	0.014	0.026	0.026	0.026	0.026	0.026	0.026	0.026
Expert	K28	K29	K30	K31	K32	K33	K34	K35	K36
Weight	0.026	0.026	0.026	0.026	0.014	0.026	0.003	0.026	0.026
Expert	K37	K38	K39	K40	K41	K42	K43	K44	K45
Weight	0.005	0.026	0.026	0.014	0.014	0.026	0.0266	0.004	0.026
Expert	K46	K47	K48	K49	K50
Weight	0.014	0.026	0.026	0.026	0.014

Table 9

DMs’ final probability fuzzy preference matrix

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
A1	[1.2, 0.34], [0.2, 0.14], [0.6, 0.16], [0.8, 0.10], [0.4, 0.18], [0.1, 0.08]	[0.1, 0.54], [0.4, 0.06], [0.0, 0.06], [0.6, 0.08], [1.2, 0.16], [0.8, 0.06], [0.2, 0.04]	[0.5, 0.4], [0.3, 0.14], [0.4, 0.14], [0.1, 0.10], [0.8, 0.10], [0.7, 0.04], [1.3, 0.08]	[1.2, 0.14], [0.6, 0.24], [1.0, 0.40], [0.7, 0.08], [0.1, 0.04], [0.2, 0.06], [1.3, 0.04]	[1.3, 0.50], [0.7, 0.28], [0.9, 0.02], [0.3, 0.02], [0.1, 0.06], [0.5, 0.08], [0.2, 0.04]	[1.0, 0.34], [0.4, 0.10], [0.6, 0.18], [0.1, 0.36], [0.2, 0.02]
A2	[0.5, 0.16], [0.1, 0.62], [0.3, 0.14], [0.0, 0.06], [1.3, 0.02]	[0.5, 0.36], [0.3, 0.30], [0.4, 0.04], [0.0, 0.02], [0.1, 0.14], [0.7, 0.04], [0.6, 0.10]	[1.0, 0.48], [0.6, 0.26], [0.4, 0.06], [0.1, 0.04], [0.2, 0.06], [0.8, 0.10]	[1.2, 0.36], [0.6, 0.22], [0.3, 0.10], [0.5, 0.20], [0.1, 0.04], [0.2, 0.06], [1.3, 0.02]	[0.8, 0.46], [0.4, 0.26], [0.5, 0.06], [0.1, 0.10], [0.9, 0.06], [0.7, 0.02], [0.6, 0.04]	[0.9, 0.56], [0.5, 0.30], [0.1, 0.06], [1.2, 0.02], [0.2, 0.04], [0.3, 0.02]
A3	[0.7, 0.10], [0.4, 0.20], [0.5, 0.12], [0.9, 0.30], [0.1, 0.06], [0.6, 0.12], [0.2, 0.04], [1.2, 0.06]	[0.1, 0.34], [0.4, 0.14], [0.7, 0.22], [0.6, 0.18], [0.0, 0.04], [0.3, 0.08]	[1.2, 0.34], [0.2, 0.08], [0.6, 0.28], [0.1, 0.04], [0.4, 0.12], [1.0, 0.12], [0.0, 0.02]	[1.2, 0.3], [0.6, 0.28], [0.9, 0.14], [0.1, 0.08], [0.2, 0.02], [0.5, 0.16], [0.3, 0.02]	[0.8, 0.42], [0.4, 0.28], [0.7, 0.06], [0.6, 0.12], [0.1, 0.08], [0.2, 0.02], [1.0, 0.02]	[0.8, 0.04], [0.4, 0.10], [0.7, 0.08], [0.1, 0.14], [0.0, 0.10], [0.3, 0.44], [1.3, 0.10]
A4	[0.4, 0.18], [0.3, 0.20], [0.2, 0.10], [0.1, 0.10], [0.5, 0.42]	[0.9, 0.10], [0.4, 0.18], [0.6, 0.12], [0.8, 0.34], [0.1, 0.06], [0.5, 0.02], [1.0, 0.14], [0.2, 0.04]	[0.5, 0.34], [0.7, 0.18], [0.3, 0.16], [1.3, 0.20], [0.0, 0.02], [0.2, 0.04], [0.1, 0.06]	[0.4, 0.20], [0.6, 0.26], [0.2, 0.08], [1.0, 0.28], [0.1, 0.06], [1.2, 0.10], [0.0, 0.02]	[0.8, 0.34], [0.1, 0.36], [0.4, 0.22], [1.3, 0.08]	[0.1, 0.58], [0.6, 0.12], [0.0, 0.06], [1.0, 0.18], [1.2, 0.04], [0.2, 0.02]
A5	[1.2, 0.50], [0.6, 0.32], [0.1, 0.04], [0.3, 0.06], [0.2, 0.06], [1.0, 0.02]	[1.2, 0.24], [0.7, 0.12], [0.6, 0.08], [1.3, 0.14], [0.1, 0.12], [0.0, 0.06], [0.2, 0.04], [0.3, 0.20]	[1.3, 0.04], [0.5, 0.20], [0.9, 0.20], [0.1, 0.10], [0.0, 0.06], [0.7, 0.04], [0.3, 0.34], [0.2, 0.02]	[0.7, 0.26], [0.4, 0.22], [0.1, 0.20], [0.6, 0.26], [0.2, 0.04], [1.3, 0.02]	[0.5, 0.28], [0.3, 0.32], [0.6, 0.16], [0.1, 0.02], [0.2, 0.04], [1.0, 0.14], [0.0, 0.04]	[0.1, 0.72], [0.2, 0.08], [0.0, 0.08], [0.4, 0.12]
A6	[0.3, 0.58], [0.1, 0.32], [0.0, 0.10]	[0.9, 0.54], [0.5, 0.28], [0.4, 0.02], [0.1, 0.04], [0.6, 0.02], [0.2, 0.06], [1.0, 0.04]	[1.3, 0.08], [0.5, 0.22], [0.9, 0.38], [0.1, 0.20], [0.0, 0.06], [0.2, 0.02], [0.7, 0.04]	[0.4, 0.26], [0.2, 0.20], [0.3, 0.06], [0.1, 0.22], [0.5, 0.22], [0.0, 0.04]	[0.8, 0.16], [0.4, 0.14], [0.1, 0.60], [0.0, 0.08], [0.9, 0.02]	[1.2, 0.38], [0.6, 0.34], [1.0, 0.14], [0.1, 0.04], [0.2, 0.06], [0.4, 0.02], [0.7, 0.02]

Table 10

The weighted comprehensive value of each attribute ${\tilde{h}}_{ij} (p_{ij})$

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
A1	1.0458	0.8930	0.9870	1.0125	1.0298	1.0969
A2	0.9126	1.0057	1.0847	0.9958	1.0444	1.0612
A3	0.9808	1.0330	1.0024	1.0170	1.0213	0.9226
A4	1.2796	0.9957	0.9876	0.9892	1.1366	0.9109
A5	1.0588	0.9458	0.9304	1.0111	0.9929	1.0962
A6	1.6333	1.0404	0.9750	1.0968	0.9630	1.0250

$\begin{matrix} H_{4} = {[H_{{ii}^{'}}]}_{6 \times 6} = \\ [\begin{matrix} 1.0000 1.0168 0.9956 1.0236 1.0014 0.9231 \\ 0.9835 1.0000 0.9792 1.0067 0.9849 0.9079 \\ 1.0044 1.0213 1.0000 1.0281 1.0058 0.9272 \\ 0.9770 0.9934 0.9727 1.0000 0.9783 0.9019 \\ 0.9986 1.0154 0.9942 1.0221 1.0000 0.9219 \\ 1.0833 1.1014 1.0785 1.1088 1.0848 1.0000 \end{matrix}] \end{matrix}$ $\begin{matrix} H_{5} = {[H_{{ii}^{'}}]}_{6 \times 6} = \\ [\begin{matrix} 1.0000 0.9860 1.0083 0.9060 1.0372 1.0694 \\ 1.0142 1.0000 1.0226 0.9189 1.0519 1.0845 \\ 0.9917 0.9779 1.0000 0.8986 1.0286 1.0605 \\ 1.1037 1.0883 1.1129 1.0000 1.1447 1.1803 \\ 0.9642 0.9507 0.9722 0.8736 1.0000 1.0310 \\ 0.9351 0.9221 0.9429 0.8473 0.9699 1.0000 \end{matrix}] \end{matrix}$ $\begin{matrix} H_{6} = {[H_{{ii}^{'}}]}_{6 \times 6} = \\ [\begin{matrix} 1.0000 1.0336 1.1889 1.2042 1.0006 1.0701 \\ 0.9675 1.0000 1.1502 1.1650 0.9681 1.0353 \\ 0.8411 0.8694 1.0000 1.0128 0.8416 0.9001 \\ 0.8304 0.8584 0.9873 1.0000 0.8310 0.8887 \\ 0.9994 1.0330 1.1882 1.2034 1.0000 1.0695 \\ 0.9345 0.9659 1.1110 1.1253 0.9350 1.0000 \end{matrix}] \end{matrix}$

Step 8: Calculate the aggregate value $H_{j}^{i}$ of each scheme on the j-th attribute through Equation (3) combined with the comparison matrix H_ii′, as shown in Table 11.

Table 11

The aggregate value of each scheme $H_{j}^{i}$

r = 1		r = 2		r = 3		r = 4		r = 5		r = 6
$H_{1}^{1}$	0.9429	$H_{1}^{1}$	0.9399	$H_{1}^{1}$	0.9367	$H_{1}^{1}$	0.9334	$H_{1}^{1}$	0.9299	$H_{1}^{1}$	0.9262
$H_{1}^{2}$	0.8228	$H_{1}^{2}$	0.8202	$H_{1}^{2}$	0.8175	$H_{1}^{2}$	0.8146	$H_{1}^{2}$	0.8115	$H_{1}^{2}$	0.8083
$H_{1}^{3}$	0.8843	$H_{1}^{3}$	0.8815	$H_{1}^{3}$	0.8785	$H_{1}^{3}$	0.8754	$H_{1}^{3}$	0.8722	$H_{1}^{3}$	0.8687
$H_{1}^{4}$	1.1537	$H_{1}^{4}$	1.1500	$H_{1}^{4}$	1.1462	$H_{1}^{4}$	1.1421	$H_{1}^{4}$	1.1379	$H_{1}^{4}$	1.1333
$H_{1}^{5}$	0.9546	$H_{1}^{5}$	0.9516	$H_{1}^{5}$	0.9484	$H_{1}^{5}$	0.9451	$H_{1}^{5}$	0.9415	$H_{1}^{5}$	0.9378
$H_{1}^{6}$	1.4726	$H_{1}^{6}$	1.4680	$H_{1}^{6}$	1.4630	$H_{1}^{6}$	1.4578	$H_{1}^{6}$	1.4524	$H_{1}^{6}$	1.4466
$H_{2}^{1}$	0.9086	$H_{2}^{1}$	0.9084	$H_{2}^{1}$	0.9080	$H_{2}^{1}$	0.9078	$H_{2}^{1}$	0.9075	$H_{2}^{1}$	0.9073
$H_{2}^{2}$	1.0233	$H_{2}^{2}$	1.0230	$H_{2}^{2}$	1.0227	$H_{2}^{2}$	1.0224	$H_{2}^{2}$	1.0221	$H_{2}^{2}$	1.0218
$H_{2}^{3}$	1.0511	$H_{2}^{3}$	1.0508	$H_{2}^{3}$	1.0505	$H_{2}^{3}$	1.0502	$H_{2}^{3}$	1.0499	$H_{2}^{3}$	1.0496
$H_{2}^{4}$	1.0131	$H_{2}^{4}$	1.0128	$H_{2}^{4}$	1.0125	$H_{2}^{4}$	1.0122	$H_{2}^{4}$	1.0120	$H_{2}^{4}$	1.0117
$H_{2}^{5}$	0.9620	$H_{2}^{5}$	0.9621	$H_{2}^{5}$	0.9618	$H_{2}^{5}$	0.9615	$H_{2}^{5}$	0.9612	$H_{2}^{5}$	0.9610
$H_{2}^{6}$	1.0586	$H_{2}^{6}$	1.0583	$H_{2}^{6}$	1.0579	$H_{2}^{6}$	1.0576	$H_{2}^{6}$	1.0573	$H_{2}^{6}$	1.0570
$H_{3}^{1}$	0.9945	$H_{3}^{1}$	0.9943	$H_{3}^{1}$	0.9940	$H_{3}^{1}$	0.9938	$H_{3}^{1}$	0.9936	$H_{3}^{1}$	0.9934
$H_{3}^{2}$	1.0930	$H_{3}^{2}$	1.0927	$H_{3}^{2}$	1.0925	$H_{3}^{2}$	1.0923	$H_{3}^{2}$	1.0921	$H_{3}^{2}$	1.0918
$H_{3}^{3}$	1.0100	$H_{3}^{3}$	1.0098	$H_{3}^{3}$	1.0096	$H_{3}^{3}$	1.0094	$H_{3}^{3}$	1.0092	$H_{3}^{3}$	1.0090
$H_{3}^{4}$	0.9951	$H_{3}^{4}$	0.9949	$H_{3}^{4}$	0.9947	$H_{3}^{4}$	0.9945	$H_{3}^{4}$	0.9943	$H_{3}^{4}$	0.9941
$H_{3}^{5}$	0.9375	$H_{3}^{5}$	0.9373	$H_{3}^{5}$	0.9371	$H_{3}^{5}$	0.9369	$H_{3}^{5}$	0.9367	$H_{3}^{5}$	0.9365
$H_{3}^{6}$	0.9824	$H_{3}^{6}$	0.9822	$H_{3}^{6}$	0.9820	$H_{3}^{6}$	0.9818	$H_{3}^{6}$	0.9816	$H_{3}^{6}$	0.9814
$H_{4}^{1}$	0.9934	$H_{4}^{1}$	0.9933	$H_{4}^{1}$	0.9931	$H_{4}^{1}$	0.9930	$H_{4}^{1}$	0.9929	$H_{4}^{1}$	0.9928
$H_{4}^{2}$	0.9770	$H_{4}^{2}$	0.9769	$H_{4}^{2}$	0.9768	$H_{4}^{2}$	0.9767	$H_{4}^{2}$	0.9766	$H_{4}^{2}$	0.9765
$H_{4}^{3}$	0.9978	$H_{4}^{3}$	0.9977	$H_{4}^{3}$	0.9976	$H_{4}^{3}$	0.9975	$H_{4}^{3}$	0.9974	$H_{4}^{3}$	0.9972
$H_{4}^{4}$	0.9706	$H_{4}^{4}$	0.9704	$H_{4}^{4}$	0.9703	$H_{4}^{4}$	0.9702	$H_{4}^{4}$	0.9701	$H_{4}^{4}$	0.9700
$H_{4}^{5}$	0.9920	$H_{4}^{5}$	0.9919	$H_{4}^{5}$	0.9918	$H_{4}^{5}$	0.9917	$H_{4}^{5}$	0.9916	$H_{4}^{5}$	0.9915
$H_{4}^{6}$	1.0761	$H_{4}^{6}$	1.0760	$H_{4}^{6}$	1.0758	$H_{4}^{6}$	1.0757	$H_{4}^{6}$	1.0756	$H_{4}^{6}$	1.0755
$H_{5}^{1}$	1.0011	$H_{5}^{1}$	1.0008	$H_{5}^{1}$	1.0006	$H_{5}^{1}$	1.0003	$H_{5}^{1}$	1.0001	$H_{5}^{1}$	0.9998
$H_{5}^{2}$	1.0154	$H_{5}^{2}$	1.0151	$H_{5}^{2}$	1.0148	$H_{5}^{2}$	1.0146	$H_{5}^{2}$	1.0143	$H_{5}^{2}$	1.0140
$H_{5}^{3}$	0.9928	$H_{5}^{3}$	0.9926	$H_{5}^{3}$	0.9924	$H_{5}^{3}$	0.9921	$H_{5}^{3}$	0.9919	$H_{5}^{3}$	0.9916
$H_{5}^{4}$	1.1050	$H_{5}^{4}$	1.1047	$H_{5}^{4}$	1.1044	$H_{5}^{4}$	1.1041	$H_{5}^{4}$	1.1038	$H_{5}^{4}$	1.1035
$H_{5}^{5}$	0.9653	$H_{5}^{5}$	0.9650	$H_{5}^{5}$	0.9648	$H_{5}^{5}$	0.9645	$H_{5}^{5}$	0.9643	$H_{5}^{5}$	0.9640
$H_{5}^{6}$	0.9362	$H_{5}^{6}$	0.9359	$H_{5}^{6}$	0.9357	$H_{5}^{6}$	0.9354	$H_{5}^{6}$	0.9352	$H_{5}^{6}$	0.9349
$H_{6}^{1}$	1.0829	$H_{6}^{1}$	1.0822	$H_{6}^{1}$	1.0816	$H_{6}^{1}$	1.0809	$H_{6}^{1}$	1.0803	$H_{6}^{1}$	1.0797
$H_{6}^{2}$	1.0477	$H_{6}^{2}$	1.0471	$H_{6}^{2}$	1.0464	$H_{6}^{2}$	1.0458	$H_{6}^{2}$	1.0452	$H_{6}^{2}$	1.0446
$H_{6}^{3}$	0.9108	$H_{6}^{3}$	0.9103	$H_{6}^{3}$	0.9097	$H_{6}^{3}$	0.9092	$H_{6}^{3}$	0.9087	$H_{6}^{3}$	0.9082
$H_{6}^{4}$	0.8993	$H_{6}^{4}$	0.8988	$H_{6}^{4}$	0.8982	$H_{6}^{4}$	0.8977	$H_{6}^{4}$	0.8972	$H_{6}^{4}$	0.8967
$H_{6}^{5}$	1.0823	$H_{6}^{5}$	1.0816	$H_{6}^{5}$	1.0810	$H_{6}^{5}$	1.0803	$H_{6}^{5}$	1.0797	$H_{6}^{5}$	1.0791
$H_{6}^{6}$	1.0120	$H_{6}^{6}$	1.0113	$H_{6}^{6}$	1.0107	$H_{6}^{6}$	1.0101	$H_{6}^{6}$	1.009	$H_{6}^{6}$	1.0089

Step 9: Compute the set values $H_{j}^{i}$ for each solution on all attributes and conduct robustness analysis on this method by varying the parameter r. Here, $H_{i} = \sum_{i = 1}^{Z} H_{j}^{i}$ , as shown in Table 12.

Table 12

Comprehensive values at different r

r = 1		Rank	r = 2		Rank	r = 3		Rank
H ₁	5.9235	4	H ₁	5.9191	4	H ₁	5.9144	4
H ₂	5.9791	3	H ₂	5.9750	3	H ₂	5.9707	3
H ₃	5.8469	5	H ₃	5.8427	5	H ₃	5.8383	5
H ₄	6.1368	2	H ₄	6.1317	2	H ₄	6.1264	2
H ₅	5.8941	6	H ₅	5.8895	6	H ₅	5.8849	6
H ₆	6.5380	1	H ₆	6.5318	1	H ₆	6.5254	1
r = 4		Rank	r = 5		Rank	r = 6		Rank
H ₁	5.9096	4	H ₁	5.9047	4	H ₁	5.8995	4
H ₂	5.9663	3	H ₂	5.9618	3	H ₂	5.9570	3
H ₃	5.8338	5	H ₃	5.8291	5	H ₃	5.8242	5
H ₄	6.1209	2	H ₄	6.1152	2	H ₄	6.1093	2
H ₅	5.8801	6	H ₅	5.8751	6	H ₅	5.8699	6
H ₆	6.5188	1	H ₆	6.5119	1	H ₆	6.5046	1

From Table 12, it is evident that when not considering attribute weights, the ranking of solutions for the six logistics suppliers is A₆> A₄ > A₂ > A₁ > A₃ > A₅. Therefore, selecting the sixth logistics supplier is deemed the optimal solution. Meanwhile, due to the variation in the parameter r, there are subtle changes in the ranking of attribute values, indicating that the operator constructed in this paper is more flexible. This also demonstrates the stability of the proposed method. The numerical results validate the effectiveness of the proposed approach.

5.2 Comparative experiment

To showcase the adaptability and dependability of the proposed decision framework, we conduct a comparative experiment in this section. The two representative decision-making methods are selected for comparison based on the literature on multi-granularity linguistic large-scale group decision-making (LGDM) methods:

The multi-layer weight-solving multi-granularity linguistic LGDM method proposed by Wang et al. [38];

The attribute multi-granularity dual-layer weight LGDM method proposed by Xu et al. [39].

For ease of representation, we denote the proposed method and the three selected methods as Z₁, Z₂, and Z₃, respectively.

5.2.1 Analysis of experiment results

The experimental results for methods Z₁, Z₂, and Z₃ are presented in Tables 13 and 14.

Table 13
Different workstation methods for aggregating weights

Clustering weight solution method Clustering weight

Z ₁ ${\overset{´}{ω}}_{1} = 0.5182$ ${\overset{´}{ω}}_{2} = 0.2856$ ${\overset{´}{ω}}_{3} = 0.0545$ ${\overset{´}{ω}}_{4} = 0.0905$ ${\overset{´}{ω}}_{5} = 0.0513$

Z ₂ ${\overset{´}{ω}}_{1} = 0.4861$ ${\overset{´}{ω}}_{2} = 0.2973$ ${\overset{´}{ω}}_{3} = 0.0512$ ${\overset{´}{ω}}_{4} = 0.1338$ ${\overset{´}{ω}}_{5} = 0.0316$

Z ₃ ${\overset{´}{ω}}_{1} = 0.3976$ ${\overset{´}{ω}}_{2} = 0.3041$ ${\overset{´}{ω}}_{3} = 0.1073$ ${\overset{´}{ω}}_{4} = 0.1152$ ${\overset{´}{ω}}_{5} = 0.0758$

Clustering weight solution method	Clustering weight
Z ₁	${\overset{´}{ω}}_{1} = 0.5182$ ${\overset{´}{ω}}_{2} = 0.2856$ ${\overset{´}{ω}}_{3} = 0.0545$ ${\overset{´}{ω}}_{4} = 0.0905$ ${\overset{´}{ω}}_{5} = 0.0513$
Z ₂	${\overset{´}{ω}}_{1} = 0.4861$ ${\overset{´}{ω}}_{2} = 0.2973$ ${\overset{´}{ω}}_{3} = 0.0512$ ${\overset{´}{ω}}_{4} = 0.1338$ ${\overset{´}{ω}}_{5} = 0.0316$
Z ₃	${\overset{´}{ω}}_{1} = 0.3976$ ${\overset{´}{ω}}_{2} = 0.3041$ ${\overset{´}{ω}}_{3} = 0.1073$ ${\overset{´}{ω}}_{4} = 0.1152$ ${\overset{´}{ω}}_{5} = 0.0758$

Table 14

Comparison of different decision-making algorithms

Decision algorithm	Decision result score value	Rank
Z ₁	H₁ = 5.9235 H₂ = 5.9791 H₃ = 5.8469 H₄ = 6.1368 H₅ = 5.8941 H₆ = 6.5380	A₆ > A₄ > A₂ > A₁ > A₃ > A₅
Z ₂	${\hat{s}}_{1} = 4.6494$ ${\hat{s}}_{2} = 4.7891$ ${\hat{s}}_{3} = 4.3648$ ${\hat{s}}_{4} = 4.8165$ ${\hat{s}}_{5} = 4.4951$ ${\hat{s}}_{6} = 5.0473$	A₆ > A₄ > A₂ > A₁ > A₃ > A₅
Z ₃	C₁ = 1.0564 C₂ = 1.1096 C₃ = 1.0239 C₄ = 1.1472 C₅ = 1.0047 C₁ = 1.1769	A₆ > A₄ > A₂ > A₁ > A₅ > A₃

Through a comparison of the clustering weight calculation results in Table 13, it is evident that this study aligns with [38] and [39] in determining the most crucial rankings in the decision-making process. However, there are significant differences in the specific allocation of cluster weights. The reasons for these differences are as follows:

The approach in [38] involves solving expert weights by constructing a trust matrix among experts; after clustering the expert group, the weights of the experts within each sub-cluster are summed to obtain the weight of each cluster.

The method in [39] determines the cluster weights using the entropy method, leading to larger weights for clusters with fewer members, affecting the accuracy of the decision outcomes.

In contrast, we classify experts by first establishing an expert compatibility matrix, and then calculating clustering weights from the perspective of two-dimensional distance. We consider not only the comprehensive values of each PHFE but also the differences between their internal elements PHFN (both membership degree and occurrence probability influence the weights). Additionally, due to the variable parameter r, the decision results vary, making our approach more flexible, adaptable to different DMs’ needs, and thereby enhancing its generality. The comparative analysis above indicates that our proposed decision-making method is more flexible, reliable, and better suited for solving MAGDM problems with independent attribute values.

The effectiveness and superiority of the proposal in this paper will be demonstrated through case analysis and comparative analysis in this section. Subsequently, the subsequent section will provide a summary of the work conducted in this paper as well as future prospects.

6 Conclusions

The method proposed in this paper offers a new approach to solving MAGDM problems by considering the relationships between decision experts. In Section 2, we introduced the basic concepts of PHFS and the Maclaurin Symmetric Mean Operator (MSMO). Section 3 elaborated on the calculation methods for expert clustering models and composite weights of experts. Section 4 outlined the specific steps to address emergency team decision problems. Subsequently, in Section 5, we conducted a case study to illustrate the application of the proposed method and provided a comparative analysis. Finally, Section 6 summarized the paper. By constructing a compatibility matrix among the experts, we derive the decision experts’ weights, ensuring the rationality of decision expert weighting. In practical applications, our method can be used for emergency decision-making in sudden events, the selection of post-disaster management plans for natural disasters, assisting managers in decision-making, with significant practical significance and application value.

While advancing research on MAGDM, future work should address the following limitations: (1) We assume that the DMs are entirely rational and neglect their psychological behaviors; improving our method could involve accounting for DMs’ bounded rationality. (2) Limited datasets may hinder a comprehensive understanding of the superiority of our proposed method; future research should incorporate additional datasets and conduct statistical analyses to ascertain its advantages. Additionally, this paper focuses on the scenario where the weights of solution attributes are known. In the future, we will delve into the exploration and research of situations where the weights of solution attributes are unknown.

Funding

This research was supported in part by the National Natural Science Foundation of China (Grant No. 11971434), Zhejiang Provincial Natural Science Foundation of China (Grant No. LY21G010002), the Fundamental Research Funds for the Provincial Universities of Zhejiang (Grant No. XT202308), and the Modern Business Research Center of Zhejiang Gongshang University, which is a key Research Institute of Social Sciences and Humanities of the Ministry of Education of China.

References

China Emergency Information Website. Philippines volcano erupts, China provides timely assistance. (2023-7-10) [2023-7-10]. https://www.emerinfo.cn/2023-07/10/c_1212242638.htm

National Work Safety Emergency Rescue Center. Chongqing Energy Investment Chongqing New Energy Company Songzao Coal Mine “9·27” major fire accident rescue. (2021-06-25) [2021-06-25]. https://www.nwserc.cn/aqyj/jyal/202106/t20210625_389929.shtml

Ministry of EmergencyManagement of China. The National Defense Headquarters specially dispatched and deployed the defense work against Typhoon “Dusuri” to upgrade the flood prevention and typhoon emergency response to Level 2. (2023-7-27) [2023-7-27]. https://www.mem.gov.cn/xw/yjyw/202307/t20230727_457533.shtml

, The United Nations report shows that the number of disasters worldwide is rising rapidly [N].China Science Daily, 2022-04-29(1). https://news.sciencenet.cn/sbhtmlnews/2022/4/369229.shtm

Zhu

, Decision method for research and application based on preference relation, Working Paper, Nanjing: Southeast University, 2014.

and Xu

, A consensus model for large-scale group decision making with hesitant fuzzy information and changeable clusters, Information Fusion 41 (2018), 217–231.

Zhang

, Xu

and He

, Operations and integrations of probabilistic hesitant fuzzy information in decision making, Information Fusion 38 (2017), 1–11.

Ding

, Wang

Y.M.

and Goh

, TODIM dynamic emergency decision-making method based on hybrid weighted distance under probabilistic hesitant fuzzy information, International Journal of Fuzzy Systems 23 (2021), 474–491.

Song

G.X.

and Yang

, Decision behavior analysis in group decision making, Academic Exploration 13(3) (2000), 48–49.

10.

Sinaga

K.P.

and Yang

M.S.

, Unsupervised K-means clustering algorithm, IEEE Access 8 (2020), 80716–80727.

11.

Zhou

and Yang

, Effect of cluster size distribution on clustering: A comparative study of k-means and fuzzy c-means clustering, Pattern Analysis and Applications 23 (2020), 455–466.

12.

Mengüç

, Aydin

and Ulu

, Optimisation of COVID-19 vaccination process using GIS, machine learning, and the multi-layered transportation model, International Journal of Production Research (2023), 1–14.

13.

Wang

, Rodríguez

R.M.

, Wang

, et al. A two-stage minimum adjustment consensus model for large scale decision making based on reliability modeled by two-dimension 2-tuple linguistic information, Computers & Industrial Engineering 151 (2021), 106973.

14.

Pang

and Song

, Multi-attribute group decision making method for interval-valued intuitionistic uncertain language with completely unknown experts’ weights, Computer Science 45(1) (2018), 47–54.

15.

Wan

, Xing

and Zhang

, Algorithm of adjusting weights of decision-makers in multi-attribute group decision-making based on entropy theory, Control and Decision 25(6) (2010), 907–910.

16.

Zhu

, Determination of bidirectional objective weight of expert based on interval two-tuple linguistic group decision making, Computing Technology and Automation 37(3) (2018), 106–110.

17.

Liang

and Wang

, Generalized TODIM emergency group decision making method for basic uncertain information interval-valued hesitant fuzzy set based on reliability degree, Control and Decision 38(07) (2023), 1988–1996.

18.

Chen

Z.S.

, Zhou

, Zhu

C.Y.

, et al. Prioritizing real estate enterprises based on credit risk assessment: an integrated multi-criteria group decision support framework [J], Financial Innovation 9(1) (2023), 120.

19.

Chen

Z.S.

, Lu

J.Y.

, Wang

X.J.

, et al. Identifying digital transformation barriers in small and medium-sized construction enterprises: A multi-criteria perspective [J], Journal of the Knowledge Economy (2024), 1–37.

20.

Chen

Z.S.

, Zhu

, Wang

Z.J.

, et al. Fairness-aware large-scale collective opinion generation paradigm: A case study of evaluating blockchain adoption barriers in medical supply chain [J], Information Sciences 635 (2023), 257–278.

21.

Agrawal

, Seh

A.H.

, Baz

, et al. Software security estimation using the hybrid fuzzy ANP-TOPSIS approach: Design tactics perspective [J], Symmetry 12(4) (2020), 598.

22.

Sahu

and Srivastava

R.K.

, Needs and importance of reliability prediction: An industrial perspective [J], Information Sciences Letters 9(1) (2020), 33–37.

23.

Sahu

and Srivastava

R.K.

, Predicting software bugs of newly and large datasets through a unified neuro-fuzzy approach: Reliability perspective [J], Advances in Mathematics: Scientific Journal 10(1) (2021), 543–555.

24.

Kumar

, Khan

S.A.

, Agrawal

, et al. Measuring the security attributes through fuzzy analytic hierarchy process: Durability perspective [J], ICIC Express Letters 12(6) (2018), 615–620.

25.

Kumar

, Khan

S.A.

and Khan

R.A.

, Durable security in software development: Needs and importance [J], CSI Communication 39(7) (2015), 34–36.

26.

Kumar

, Alenezi

, Ansari

M.T.J.

, et al. Evaluating the impact of malware analysis techniques for securing web applications through a decision-making framework under fuzzy environment [J], Int. J. Intell. Eng. Syst 13(6) (2020), 94–109.

27.

Liu

, He

and Yu

, Generalized hybrid aggregation operators based on the 2-dimension uncertain linguistic information for multiple attribute group decision making, Group Decision and Negotiation 25 (2016), 103–126.

28.

Yin

, Wang

and Li

, The multi-attribute group decision-making method based on interval grey trapezoid fuzzy linguistic variables, International Journal of Environmental Research and Public Health 14(12) (2017), 1561.

29.

, Wang

, Zhou

, et al. Method of multiple attribute group decision making based on 2-dimension interval type-2 fuzzy aggregation operators with multi-granularity linguistic information, International Journal of Fuzzy Systems 19 (2017), 1880–1903.

30.

Sahu

, Srivastava

R.K.

, Kumar

, et al. Integrated hesitant fuzzy-based decision-making framework for evaluating sustainable and renewable energy [J], International Journal of Data Science and Analytics 16(3) (2023), 371–390.

31.

Kumar

, Baz

, Alhakami

, et al. A hybrid fuzzy rule-based multi-criteria framework for sustainable-security assessment of web application [J], Ain Shams Engineering Journal 12(2) (2021), 2227–2240.

32.

Wang

, Yang

and Li

, Multi-criteria decision-making method based on single-valued neutrosophic linguistic Maclaurin symmetric mean operators, Neural Computing and Applications 30 (2018), 1529–1547.

33.

DeTemple

D.W.

and Robertson

J.M.

, On generalized symmetric means of two variables, Publikacije Elektrotehničkog fakulteta. Serija Matematika i fizika (634/677) (1979), 236–238.

34.

Wang

, Yang

and Li

, Multi-criteria decision-making method based on single-valued neutrosophic linguistic Maclaurin symmetric mean operators, Neural Computing and Applications 30 (2018), 1529–1547.

35.

, Liu

and Ju

, Some new intuitionistic linguistic aggregation operators based on Maclaurin symmetric mean and their applications to multiple attribute group decision making, Soft Computing 20 (2016), 4521–4548.

36.

Liu

and Qin

, Maclaurin symmetric mean operators of linguistic intuitionistic fuzzy numbers and their application to multiple-attribute decision-making, Journal of Experimental & Theoretical Artificial Intelligence 29(6) (2017), 1173–1202.

37.

Shannon

C.E.

, A mathematical theory of communication, The Bell System Technical Journal 27(3) (1948), 379–423.

38.

Wang

, Zhang

and Zhang

, Multi-granularity linguistic large group decision-making based on cloud model and multi-layer weight determination, Control and Decision 36(09) (2021), 2257–2266.

39.

and Sun

, Two-layer weight large group decision-making method based on multigranularity attributes, Control and Decision 31(10) (2016), 1908–1914.