The large-scale group consensus multi-attribute decision-making method based on probabilistic dual hesitant fuzzy sets

: We proposed a novel decision-making method, the large-scale group consensus multi-attribute decision-making method based on probabilistic dual hesitant fuzzy sets, to address the challenge of large-scale group multi-attribute decision-making in fuzzy environments. This method concurrently accounted for the membership and non-membership degrees of decision-making experts in fuzzy environments and the corresponding probabilistic value to quantify expert decision information. Furthermore, it applied to complex scenarios involving groups of 20 or more decision-making experts. We delineated five major steps of the method, elaborating on the specific models and algorithms used in each phase. We began by constructing a probabilistic dual hesitant fuzzy information evaluation matrix and determining attribute weights. The following steps involved classifying large-scale decision-making expert groups and selecting the optimal classification scheme based on effectiveness assessment criteria. A global consensus degree threshold was established, followed by implementing a consensus-reaching model to synchronize opinions within the same class of expert groups. Decision information was integrated within and between classes using an information integration model, leading to a comprehensive decision matrix. Decision outcomes for the objects were then determined through a ranking method. The method’s effectiveness and superiority were validated through a case study on urban emergency capability assessment, and its advantages were further emphasized in comparative analyses with other methods.


Introduction
Multi-attribute decision-making (MADM) is a process in which multiple people make decisions and consider multiple attributes.It is widely used in the decision-making (DM) of complex problems, such as emergency management [1], watershed management [2], and investment selection [3].Building on decision-making (DM) research, scholars have developed a variety of models and algorithms for group decision-making (GDM) [4,5].With the increase in the number of people participating in GDM, the problems related to MADM gradually develop into large-scale group multiattribute decision-making (LSG-MADM) problems [6][7][8].Large-scale group decision-making, frequently encountered in today's society, is advancing as a prominent subject within decision science.Compared with traditional MADM methods, LSG-MADM methods are more applied to scenarios with multi-domain intersection and complex problems [9].A minimum of twenty decision-making experts is typically required for LSG-MADM processes [10].Incorporating a consensus mechanism into LSG-MADM issues and setting up the large-scale group consensus multi-attribute decision-making (LSGC-MADM) method can help different viewpoints better fit together [11].The major areas of study for LSGC-MADM are consensus mechanisms [12], group clusters [13], and cooperative behaviors [14].
Scholars have achieved some research results, such as Du introduced a decision support approach for tackling large-scale decision-making in social networks, merging constrained community detection with multi-stage multi-cost consensus models to address clustering and consensus complexities [15].Yu et al. enhanced group decision-making with the Enhanced Minimum Cost Consensus Model (EMCC), leveraging explicit adjustment paths and coordination elasticity to prevent over-adjustment and enhance consensus efficiency and adaptability [16].Chen et al. proposed an expertise-structure and risk-appetite-integrated two-tiered framework for collective opinion generation in large-scale group decision-making [17].Although scholars have made progress in researching large-scale decisionmaking, considering the fuzziness of decision-makers' thought processes and the complexity of decisionmaking events, we address decision information using the form of probabilistic dual hesitant fuzzy sets, thereby better capturing the authenticity and completeness of decision information.
To express the ambiguity and uncertainty of human thinking, since Zadeh proposed the concept of fuzzy sets (FSs) [18], the research theory of fuzzy DM has become more and more abundant.In order to deal with more complex DM problems, Atanasso and Yager defined intuitionistic fuzzy sets (IFSs) [19] and Pythagorean fuzzy sets (PFSs) [20], respectively.Torra proposed hesitant Fuzzy Sets (HFSs) [21], which can better characterize decision-making experts' indecision in evaluating information.Scholars are progressively incorporating probabilistic information into decision-making processes.For instance, Wang introduced a novel approach utilizing the Probabilistic Language Term Set to address the Probabilistic Language Preference Relationship (PLPR) in decision-making scenarios [22].Liu et al. introduced a group decision-making approach based on the incomplete probabilistic language term set (InPLTS), effectively managing uncertain decision information through specialized categorization, a mathematical programming model for consistency and consensus, and a reliability-induced operator [23].Xu and Zhou proposed probabilistic hesitant fuzzy sets (PHFSs) [24] based on HFSs, which can provide membership degrees and their corresponding probabilities.Scholars such as Hao et al. defined probabilistic dual hesitant fuzzy sets (PDHFSs) [25], which can collect membership and nonmembership evaluation information and their corresponding probability information.Consequently, PDHFSs offer a more nuanced capability for depicting evaluation information compared to fuzzy sets like PFSs and HFSs.
In recent years, there has been a significant increase in the development of diverse fuzzy sets of DM methods [26,27].However, it is essential to note that existing research has the following issues: 1) Most PDHFSs' DM methods are based on individual or small and medium-sized group DM.There needs to be more study or literature that discusses large-scale group decision-making based on PDHFSs.2) When decision-making experts use fuzzy preference relationships to express evaluation information, some existing studies ignore individual consensus levels, which may lead to conflicting DM results, resulting in low consensus in group preference information aggregation.
The literature review highlights a significant research gap in using probabilistic dual hesitant fuzzy information for decision-making in large groups, especially given the recent development of probabilistic dual hesitant fuzzy sets which have not been widely studied worldwide.This study aims to fill this gap by applying these fuzzy sets to make consensus decision-making more accurate and reliable for large groups, addressing the challenges of ambiguity and uncertainty.To more effectively tackle the issues above and the obstacles, this study presents the Large-Scale Group Consensus Multi-Attribute Decision-Making Method based on Probabilistic Dual Hesitant Fuzzy Sets (the LSGC-MADM Method based on PDHFSs).This method is important for advancing the theoretical basis of fuzzy decision-making and offers a practical, scalable solution for various fields where reaching consensus is key.Consequently, this research significantly contributes to the decision-making literature, presenting an effective instrument for navigating complex decision-making scenarios.The decisionmaking method is specifically designed for use by decision-making experts in complex scenarios involving 20 or more participants.Initially, an evaluation matrix with probabilistic dual hesitant fuzzy information (PDHFI) is formed, drawing on expert preference information.The entropy method is then applied to ascertain the weights of attributes.Following this, group similarity is measured using the equal probability distance metric, and the scheme for expert group classification is determined based on the net-making classification method and the classified test criteria.Next, the consensus-reaching model is adopted to achieve consensus in decision-making opinions within each expert class.Ultimately, the decision-making objects are ranked using the ranking method after integrating decision-making information within and between classes.In contrast to other methods, this method examines experts' probability information, membership degree, and non-membership degree in group decision-making.However, it also looks at the consensus degree of these experts when making largescale group decisions.Therefore, the method proposed in this study makes the DM results more objective, reasonable and reliable.
The main contributions of this paper are as follows: 1) A comprehensive large-scale group consensus decision-making method, named the Large-Scale Group Consensus Multi-Attribute Decision-Making Method based on Probabilistic Dual Hesitant Fuzzy Sets, is proposed, integrating multiple approaches.2) A group similarity measurement method is constructed based on probabilistic dual hesitant fuzzy evaluation information, utilizing the equal probability distance method, and a netmaking classification method is proposed to classify decision-making experts.3) A global consensus threshold is established to build the consensus-reaching model, which judges and adjusts the evaluation information of experts within each class, achieving consensus on the decision-making information of experts.4) A comprehensive expert weight, combining class weight and class deviation weight, is used to obtain a comprehensive information decision matrix, from which the final evaluation result of the decision object is derived.The LSGC-MADM Method based on PDHFSs advances the field of decision-making under uncertainty.This method facilitates the coordination of opinions across different expert categories and the integration of these insights to inform action, offering an effective, scalable solution for large-scale fuzzy decision-making challenges applicable in domains requiring sophisticated decision strategies.
This paper is structured as follows: Section 2 introduces PDHFSs and the related concepts of the LSGC-MADM problem.Section 3 proposes the research framework and this study's specific models and methods.Section 4 conducts a case study on applying the LSGC-MADM Method based on PDHFSs.This study is summarized and projected in Section 5.

Probabilistic dual hesitant fuzzy sets (PDHFSs)
Definition 1 [28]: Let X be the domain, then is called a probabilistic dual hesitant fuzzy set on X . ( ) ( )| ( )   respectively represent the degree of membership and non-membership and the corresponding probability distribution information, among which and # ( ) respectively represent the number of corresponding elements in the membership and non-membership degree, and satisfy # ( )  and *  represent the maximum value of the membership degree and non-membership degree, respectively, where Suppose two PDHFEs are 1 pd and 2 pd , respectively.The operation law is defined as follows: where 0 be a PDHFE, and the score function of it can be expressed as Eq (6).
Among them, i h   and i p p  represent the membership value and the corresponding probability of the membership part.j g   and j q q  represent the non-membership value and the corresponding probability of the non-membership part, respectively.Definition 4: The comparison between two PDHFEs 1 pd and 2 pd can be expressed as follows: pd is considered to be better than 2 pd , recorded as , it means that pd pd  .
Definition 5: Any PDHFE can be normalized.In the normalized PDHFE, the sum of all membership and non-membership probability values is 1, respectively.Let a PDHFE be , | , | pd h g h p g q     , and then its normalized form is as Eq (7) [30].

Description of the large-scale group consensus multi-attribute decision-making (LSGC-MADM) problem
The LSGC-MADM problem is an interactive activity among many individuals in a social environment [31].Scholars widely study group classification and consensus building as effective methods to solve LSGC-MADM problems.Consensus and selection are the two fundamental processes of the consensus-reaching model [32].The consensus process includes the measurement of group consensus degree, the identification of disagreements, and the regulation of opinions.The difficulties of large-scale consensus decision problems include the following points: 1) Effective classification of large-scale group members.2) Identify the individual with a low consensus contribution degree.3) Measure the consensus level of the population.4) Building an effective consensus guidance mechanism for the group can quickly reach a consensus.A consensus decisionmaking (CDM) solution can be obtained by DM members of a large-scale group (LSG) using the LSGC-MADM Method based on PDHFSs proposed in this study.

Main methods and models
This section primarily introduces the process and steps of the Large-Scale Group Consensus Multi-Attribute Decision-Making Method based on Probabilistic Dual Hesitant Fuzzy Sets (the LSGC-MADM Method based on PDHFSs).It also provides a detailed overview of the specific methods and models included in this study.The flowchart of the LSGC-MADM Method based on PDHFSs is illustrated in Figure 1.

The LSGC-MADM method based on PDHFSs
Suppose that in the LSGC-MADM Method based on PDHFSs process, the set of T decisionmaking experts is , the set of decision-making objects is , and the set of decision-making attributes is . The specific steps of the LSGC-MADM Method based on PDHFSs proposed in this study are summarized as follows.
Step 1: Construct the probabilistic dual hesitant fuzzy information (PDHFI) evaluation matrix and calculate the attribute weights.Decision-making experts provide PDHFI for each attribute of the decision-making object.This process results in the formation of a comprehensive PDHFI evaluation matrix ( ) k PD , encompassing inputs from all experts.The weights j  for each attribute C j are determined using the entropy method.Detailed algorithms and formulas related to Step 1 are presented in Section 3.2.1.
Step 2: Classify decision-making experts into group classes and establish optimal classification.Based on the PDHFI evaluation matrix from all experts, we employ a group classification model to categorize the decision-making experts.The optimal classification result is determined according to the criteria p I , which are used to test the effectiveness of the classification.Detailed steps of this group classification model are outlined in Section 3.2.2.
Step 3: Calculate and adjust expert opinions to achieve internal consensus within each class.By constructing the consensus-reaching model, the decision-making evaluation value of each class of experts are harmonized to reach internal consensus.For a detailed description of the steps involved in the consensus-reaching model, refer to Section 3.2.3.
Step 4: Utilize the decision-making information integration model to obtain the comprehensive decision matrix.Employ the decision-making information integration model to merge both intra-class and inter-class expert decision-making information, resulting in a comprehensive decision-making information matrix Z i R .The specific steps of this model are detailed in Section 3.2.4.
Step 5: Rank decision-making objects to identify the optimal decision.According to the comprehensive decision-making information matrix Z i R and the sum of squared deviations

Construct the PDHFI evaluation matrix of each decision expert (≥20 experts)
Step 1：Construct the PDHFI evaluation matrix and calculate the attribute weights.
(1) ( ) Find the location of the decision information that needs to be adjusted

Update decision information matrix
A decision information matrix that has reached the consensus goal Step 3: Calculate and adjust expert opinions to achieve internal consensus within each class.
Step 4: Obtain the comprehensive decision matrix.

Intra-class and inter-class decision information integration
Comprehensive decision-making information matrix Step 5: Rank decision-making objects.shown in Eq (8).

Ranking of decision-making objects Final decision result
, where (Ⅱ) The attribute weight In this study, we calculate the attribute weight j  using the entropy method [33].The entropy value of the decision-making attribute j C , denoted as j e , is used in the calculation of j  , which is detailed in Eq (9).(1 )

The group classification model
An expert group classification model is constructed to classify experts into groups according to their decision-making evaluation values.Experts within the same class possess relatively consistent decision-making evaluation value.Therefore, the weights of experts in the same class can be considered as equal values [34].

(Ⅰ) Group similarity measurement
This section designs a similarity measurement between experts based on the equal probability distance (EPD) to measure the consistency of experts.
The probability values continue to be compared according to the above method until 1 NH and 2 NH are empty sets, and finally, the value of YH is obtained.
3) The non-membership part of 1 pd and 2 pd is processed according to the above method.YG is used to replace YH , and YG value can be obtained, and pd is defined as Eq (12).
, k l

SM
represents the similarity between decision-making experts k E and l E , the calculation formula is shown in Eq (13).
, , , (Ⅱ) Classification of expert groups (ⅰ) The net-making classification method based on the similarity matrix.The similarity between decision-making experts constitutes the expert similarity matrix S SM .Since S SM is a symmetric matrix, only the upper triangular needs to be calculated, as Eq (14).
This section uses the similarity matrix-based net-making classification method to classify experts [35].The specific steps are as follows: 1) Set the cut level, that is, the similarity threshold as 3) In the upper triangular matrix r P , substitute the elements on the main diagonal with the corresponding numbers of the experts.Convert all "1" to "*" in the elements above the main diagonal within r P , and remove any elements that possess a value of "0" .Utilize "*" as the nodal point for constructing warp and weft lines, thereby weaving the network.Experts k E linked through this network are deemed to be in the same class.This approach effectively generates a preliminary classification scheme ( 1, 2,... ) c c C   for the expert groups, with C representing the total number of identified classes.(ⅱ) Set the classification effect test criteria p I .
In order to choose the best similarity threshold e  , it is necessary to set the criteria for checking the classification effect.The larger the proportion of the sum of squares of the inter-class decision information, the better the classification effect of experts [36].According to Eq (6), the score function of the expert k E on the j C of the i A can be obtained.The average information of all experts in the c  on the , and the average information value is . The definition of the criteria p I to test the expert classification effect is as Eq (15). ) Among them,     .The optimal expert group classification scheme can be obtained when the p I value is maximized.

The consensus-reaching model
In order to obtain a decision-making scheme that is satisfied by the experts in the same class, it is necessary to consider whether the decision-making opinions of the experts in the same class reach a certain level of consensus.The five steps that comprise the consensus-reaching model that is built in this section are as follows.
(ⅰ) Determine the global consensus degree threshold c C  .Set the initial value of the adjustment times t to " 0 " , and the initial decision information score matrix to be , where  , ; ) of the expert k E to the object i A can be obtained, as Eq (16).
2) Calculate the evaluation score value of the expert group.Since the weights of experts in the same class are treated as equal values, the group evaluation score c i   of the object i A by the expert group in class c  can be obtained by calculating the mean value, as Eq (17).
3) Calculate the consensus level of the expert class except the expert l E .According to k i  and CL is the consensus level of other experts except the expert l E on the object i A in the expert group of class c  , as Eq ( 18). ( \ ) represents the set of other experts except the expert l E in the expert group of class c  .
4) Calculate the consensus level of individual experts.

i CL is the consensus level of individual experts in class c
 on the object i A , as Eq (19).
5) Calculate the global consensus degree of the expert group.
c CL  is the global consensus degree of the expert group in class c  , as Eq (20).
( ) If the value of c CL  is greater than the value of c C  , proceed to step 5. Otherwise, proceed to Step 3.
(ⅲ) Determine which experts' opinions require modification and in which locations those opinions need to be modified.
1) Calculate the cumulative consensus degree l CD of expert l E , as Eq (21).
l CD reflects the contribution degree of expert l E to the consensus of the group in the consensus-reaching process.If 0 l CD  , it means that the expert l E plays a positive role in the process of group consensus-reaching.

( )
2) Adjust the decision-making information that experts need to modify.Find the expert with the smallest value of CD .At this time, ( , )  i j is the position where the decision information needs to be adjusted, let ( , )  p q equal ( , ) i j .(ⅳ) Adjust the decision information of the element in the position ( , )  p q to form a new decisionmaking information matrix.
1) Calculate the consensus contribution degree of individual experts.The consensus level ij CL of all experts in class c  to the i A on the j C is calculated as Eq (22).The consensus level of all CL , as Eq (23).The consensus contribution degree of l E in c  to i A on C j is l ij CD , as Eq (24).
Calculate the consensus contribution degree ( ) of all experts in c  to p A on the q C , and set the decision-making expert with the highest consensus contribution degree value as lh E .
2) Adjust the decision-making information of the expert ls E at the ( , ) p q position [37].Let  represent the tuning parameter, satisfying 0 1    .Modify the decision information of the ( , )   p q position while maintaining the information of the other position elements intact, resulting in a new decision-making information score matrix denoted as * ( ) , S pd  , as Eq (25).
Then, let (ⅴ) After the iterative steps, the final consensus decision-making information matrix ( ) ( ) and k c E  .At this juncture, a consensus has been reached by all experts.

3.2.4.
The decision-making information integration model (Ⅰ) Intra-class decision-making information integration Integrate the information of ( ) ( ) on the attribute dimension j  , and obtain the decisionmaking information matrix ( )' ( ) of each expert for the object i A , as Eq (26), where Since the weights of experts within the same class are equal, the decision-making information matrix  expert group for i A is calculated as shown in Eq (27).
(Ⅱ) Inter-class decision-making information integration In this section, the class weight and class deviation weight are comprehensively considered to obtain the comprehensive decision-making information matrix.
(ⅰ) Calculate the class weight c n  , as Eq (28).The class weight is determined by the ratio of the number of experts in this class to the total number of experts. 1) Calculate the distance c D between the mean value of each class's decision information and all decision information, as Eq (29).The calculation of 2) Calculate the class deviation weight c p  according to c D , as Eq (30).( Generally, when the decision-making result is focused on the opinions of the majority of experts, take 0.5

 
. Unless otherwise specified, take 0.5   .(ⅳ) Calculate the comprehensive decision-making information matrix Z i R , as Eq (32).
SR of decision-making information needs to be further compared, as Eq (33). ) ,( 1, 2,..., ) expertise in various fields of emergency management, including but not limited to natural disaster response, public safety, urban planning, health, and sanitation.These experts were asked to provide personalized assessment information based on their professional knowledge and practical experience.In this way, the evaluation aims to capture the unique insights of each expert, with the goal of developing a comprehensive and in-depth understanding of the cities' emergency management capabilities [38].This case study not only promises to provide valuable insights into the cities' emergency preparedness and response capabilities for the government but also, through comparative analysis, can reveal the strengths and weaknesses of each city in terms of emergency management.This, in turn, can guide future policy making and resource allocation, enhancing the resilience of cities and their ability to respond to sudden public health emergencies.

Decision-making process
In this section, the LSGC-MADM Method based on PDHFSs is applied to evaluate the emergency management capabilities of three cities.The specific steps are as follows.
Step 1: PDHFI for each attribute of three cities is provided by decision-making experts.The resulting PDHFI evaluation matrix is presented in Table 1 (showing partial information).The weights of each attribute are calculated using the entropy method, as detailed in Table 2.
Step 2: The group classification model in Section 3.2.2 is used to classify all experts and generate classification results.The expert similarity , , ( ) is calculated according to Eq (12).The decision-making experts are classified according to the net-making classification method.It can be calculated that the minimum similarity between the twenty decision-making experts is 0.8247, and the maximum similarity is 0.9155.Classification becomes meaningful when the similarity threshold e  falls within the interval of [0.8247, 0.9155).The relationship between the classification effect test criteria p I and e  value is shown in Figure 2. The experts' classification is optimal when the p I value peaks at 8.8599.At this point, the experts are divided into four classes, as shown in Table 3.
Step 3: Set the global consensus degree threshold .As an example, the experts in class 1  reached a consensus using the consensus-reaching model in Section 3.2.3.The consensus decisionmaking information matrix is obtained after class 1  reaches consensus, as shown in Table 4.
Step 4: Using the decision-making information integration model in Section 3.2.4,the decisionmaking information matrix c i R  of the four expert groups for i A can be obtained, as shown in Table 5.The calculation results of the weights for each class are shown in Table 6.The Eq (32) is used to get the comprehensive decision-making information matrix Z i R for the three evaluation cities, shown in Table 7.
Step 5: According to the decision-making score of i A , the evaluation cities are ranked to obtain the optimal decision scheme.Since 1 3 2 , the emergency management capability of the three cities is ranked as 1

Comparison of different methods
This section presents a comparative analysis between the method proposed in this study and other existing decision-making methods, demonstrating the efficacy and superiority of the proposed method.Table 8 presents a comprehensive comparison of the pertinent attributes associated with the four distinct decision-making methods.Zhang et al. [39] suggested a hesitant fuzzy language adaptive consensus model based on individual cumulative consensus contributions in the previous study as a way to find emergency medical facilities.This is called "Method 1"."Method 2" was proposed by Garg and Kaur [40], who proposed a PDHFSs method based on the MSM operator to quantify the gesture information of patients with cerebral hemorrhage.Wu and Xu [41] developed a large-scale consensus decision-making model with hesitant fuzzy information and variable clusters, which is documented as "Method 3".The LSGC-MADM Method based on PDHFSs is proposed in this study called "Method 4".The case problem presented in Section 4.1 is then solved using four different decision-making methods, and the final decision-making results are displayed in Table 9.
The ranking of decision-making methods varies slightly as a result of the distinct characteristics and focal points inherent in each method.According to the findings presented in Table 9, it is evident that Methods 1, 3 and 4 collectively assert that city A1 possesses the most effective emergency management capability.Conversely, Methods 2-4 collectively contend that city A2 exhibits the weakest emergency management capability.This observation highlights the reliability and validity of Method 4, which is the decision-making method proposed in this study.The adaptive consensus model in Method 1 requires that experts' weights and decision-making information be changed all the time.This could mean that the final decisions are different from what was known at the start.The absence of a consensus-building mechanism in Method 2 may lead to errors in resolving complex group decision-making problems.While Method 3 and Method 4 yield identical decision-making outcomes, Method 3 fails to incorporate the non-membership and probability information of decision-making experts, thereby limiting its ability to comprehensively depict decision-making information.The LSGC-MADM Method based on PDHFSs effectively addresses the issue of incomplete decisionmaking information collection and exhibits a wider range of applicability.
ranked to determine the optimal decision-making result.The specific steps of the ranking method are shown in Section 3.2.5.

and 0 YH
 .Then, compare the probability values of two elements in the first position in 1

1 :
Suppose the two probabilistic dual hesitant fuzzy elements are 1 | 0.5, 0.2 | 0.3, 0.3 | 0.2) .After the above calculation steps, we can get represent the evaluation values of the th k and th l experts on the decision-making attribute C j of the object i A , where 0 k l T 

2 )
Construct an upper triangular matrix r P , and the rules for the values of matrix r P

.
When the number of decision expert classes is 1 or each expert constitutes their own class, the classification becomes meaningless.This section calculates the criteria p I for testing the classification effect by continuously adjusting the similarity Calculate the global consensus degree c CL  of the expert group in class c  .1) Calculate the evaluation score of the individual expert.According to the weighted summation of the expert k E in the class c  in the dimension of the attribute C j , the individual evaluation score ( 1,...

l
CD and denote it as ls E .Calculate the consensus contribution degree ls ij CD of ls E to i A on the j C .Denote the smallest value of Calculate the class deviation weight c p  .This section further determines the class deviation weight c p  according to the deviation value between classes.
and the final comprehensive class weight is c z  .
result of the object i A is considered to be better than * To conduct an in-depth evaluation of the emergency management capabilities of three Chinese cities( 1,2,3) i A i  in response to sudden incidents, the government decision-making department has initiated a comprehensive assessment project.This evaluation focuses on three core decision-making attributes ( to fully understand the comprehensive strength of each city in crisis response.To ensure the scientific accuracy of the assessment, the government management department has taken into account the geographic location of the cities, historical disaster records, and existing emergency management facilities and resources, while setting preference coefficients to reflect the importance of different attributes.The government department carefully selected 20 decision-making experts ( 1, 2,..., 20) k E k  with extensive experience and

Figure 2 .
Figure 2. The relationship between Ip and αe value.

Table 1 .
The PDHFI evaluation matrix of 20 decision-making experts.

Table 2 .
The weight of each attribute.

Table 3 .
The classification results of the decision-making experts.

Table 4 .
The consensus decision-making information matrix.

Table 5 .
The decision-making information matrix.

Table 6 .
The class weight, the class deviation weight, and the comprehensive class weight.

Table 7 .
The comprehensive decision-making information matrix.

Table 8 .
Comparison of different decision-making methods.

Table 9 .
Decision-making results of four decision-making methods.Method 4" is the decision-making method proposed in this study.
* "Method 4" is the decision-making method proposed in this study.