A probabilistic linguistic opinion dynamics method based on the DeGroot model for emergency decision-making in response to COVID-19.

Emergency decision-making entails a multi-criteria problem with a short period and urgent events, which creates difficulties for decision makers to undertake an optimal decision. To ensure the validity and rationality of decision results, the probabilistic linguistic term set is adopted to represent the evaluation information of experts because it can assign different probabilities or importance to different linguistic terms, which is closely related to human cognition. In addition, to portray the dynamic changes in the emergency decision-making process, this study develops a new dynamics method based on the DeGroot model with probabilistic linguistic information. First, to simulate the transition matrix of probabilistic linguistic opinions, the basic operational rules are defined based on the transformation function and expectation function. Next, three forms of influence matrices incorporating similarity, self-persistence, and authority degrees are constructed, and the consensus conditions of the models are discussed. Then, considering the social networks and incomplete trust relationships between experts, a fourth trust-based influence matrix is devised. A case study of emergency decision-making for assessing response plans to COVID-19 is performed to verify the feasibility and effectiveness of the dynamic method. Furthermore, a sensitivity analysis is conducted. Finally, comparisons with classical methods are performed to illustrate the superiorities of the proposed algorithms.

Introduction

The emergence of the coronavirus disease 2019 (COVID-19) pandemic has been the worst crisis in this century, posing an unprecedented threat to global public health and the economy. After the epidemic broke out, China immediately took many effective actions to activate the public health emergency mechanism for preventing the spread of the epidemic, and thus has made outstanding achievements in controlling it. At the beginning of the outbreak, due to the lack of a specific medicine for treating the COVID-19 virus, the implementation of policy interventions guided by scientific evidence was crucial to fighting the pandemic. Accordingly, much research has been conducted to alleviate the adverse effects of COVID-19. For instance, Mardani, Saraji, Mishra, and Rani (2020) designed an extended hesitant fuzzy framework to analyze the key challenges of adopting digital health intervention during the COVID-19 outbreak. Wan, Yan, and Dong (2021) constructed a consensus model for large-scale group decision-making (LSGDM) with probabilistic linguistic preference relations based on personalized individual semantics (PIS) and applied it to COVID-19 surveillance. Ren, Liao, and Liu (2020) proposed a multi-criteria decision-making (MCDM) method with Dempster-Shafer (DS) theory and generalized Z-numbers to select an appropriate medicine for patients with COVID-19. However, selecting an appropriate response plan for a specific outbreak area is challenging, and it represents an emergency decision-making problem. This type of problem is characterized by suddenness, uncertainty, and disruption, with limited time and heavy tasks. To solve emergency decision-making problems, several proposals have been explored (Huang et al., 2022, Liu et al., 2022a, Zheng et al., 2022).

Due to the inherent complexity and uncertainty of decision-making problems, conflicting and non-cooperative behaviors of decision-makers are inevitable. Moreover, it should be noted that an inappropriate emergency decision can lead to significant economic losses and can have severe social consequences. Hence, the goal of emergency decision-making is for experts to achieve unanimous consensus results for the selection or ranking of currently feasible alternatives. Opinion dynamics is a useful tool to investigate consensus formation. It is related to the opinion evolution procedure of a group of decision makers sharing the same discussion topics. In recent years, many opinion dynamics models have been proposed, wherein decision makers can interact with each other and exchange their opinions (Bernardes, Stauffer, & Kertész, 2002). The DeGroot model is one of the most classical models, and it employs a stochastic matrix to represent the relationships and interactions among decision makers (Berger, 1980, DeGroot, 1974). There have been numerous studies on this model. The Friedkin-Johnsen (FJ) model (Friedkin and Johnsen, 1990, Friedkin et al., 2016) is the most representative variant of the DeGroot model. Both the FJ model and DeGroot model are linear and thus can be examined by linear methods. Another is the nonlinear model, which was originally proposed by  Hegselmann and Krause (2002), and since then, the bounded confidence (BC) model has been introduced. Among many variations of the BC model, the commonly used BC models are the Deffuant-Weisbuch (DW) model (Zha, Liang, Kou, Dong, & Yu, 2019) and Hegselmann–Krause (HK) model (Wu et al., 2022a, Yang et al., 2014). These studies contributed many valuable and practical theories to opinion dynamics. For instance, Ding, Chen, Dong, and Herrera (2019) explored the impact of experts’ self-confidence and node degree on the formation of consensus opinions and discussed the convergence speed of the DeGroot model. Li and Wei (2019) developed a two-stage dynamic consensus model for LSGDM based on a social network (SN), wherein the DeGroot model was used to present the opinion evolution process in each sub-group. Dong, Ding, Martínez, and Herrera (2017) analyzed the necessary conditions for the DeGroot model to reach a consensus based on the concept of leadership and then generalized the consensus strategy to solve the decision-making problem. Zhou, Wu, Abdulrahman, and Herrera (2020) proposed a two-step communication opinion dynamics model based on the DeGroot model, in which the self-persistence of experts and influence index of SNs were considered. Su, Xu, Zhao, and Liu (2020) extended the DeGroot model to describe the evolution process of a hesitant fuzzy (HF) evaluation matrix and applied the HF-DeGroot model to manage a public health event.

An essential part of the decision-making process is collecting evaluation information from experts. However, in real life, people may not evaluate the alternatives in an accurate numerical form, and instead, they tend to make assessments under the linguistic setting. Therefore, the fuzzy linguistic approach has received increasing attention, and several linguistic computational models have been proposed. For example, Rodriguez, Martinez, and Herrera (2012) introduced the idea of hesitant fuzzy linguistic term set (HFLTS), which enables experts to capture the uncertainty with multiple possible linguistic terms. The HFLTS originally worked under the assumption of linguistic terms with equal probabilities, which means that the importance of these linguistic terms is not differentiated. In fact, the HFLTS may not depict all information of experts when they have different preferences for linguistic terms. To address this problem, Zhang, Dong, and Xu (2014) initiated the concept of linguistic distribution assessment (LDA), which provides symbolic proportion information on linguistic terms. Following on the aforementioned pioneering work, Xiao, Wang, and Zhang (2020) introduced a two-stage consensus model to manage individual consistency and group consensus with linguistic distribution preference relations in the group decision making (GDM). Wu, Zhang, Kou, Zhang, Chao, Li, Dong, and Herrera (2021) reviewed the crucial elements and applications of distributed linguistic information processing in the decision-making process, discussing current challenges and future research directions.

Furthermore, considering the unknown probability distribution information of certain linguistic terms, Pang, Wang, and Xu (2016) proposed the probabilistic linguistic term set (PLTS). The evaluation information expressed in the forms of the LDA and PLTS includes not only multiple possible linguistic terms, but also relevant probabilistic information. However, these sets have notable differences. Namely, the LDA follows a completely known probability distribution, meaning that the sum of probabilities of all possible linguistic terms is one. Further, the sum of probabilities of potential linguistic terms in the PLTS is not allowed to exceed the value of one. However, in a practical problem, it can be difficult to attain complete information on the probability distribution. Hence, a PLTS-based model that represents multiple preferences and hesitancy of experts might be more appropriate for real-world scenarios to a certain extent. The PLTS is an effective linguistic model, and there are fruitful studies on this model. Mo (2020) developed a new method for emergency decision-making of gypsum mine collapse accidents based on PLTS and D number theory. Fang, Liao, Yang, and Xu (2019) proposed a probabilistic linguistic evidential reasoning approach to address MCDM problems with several uncertainties, and applied it to the screening of high-risk populations for lung cancer. Lin, Chen, Xu, Gou, and Herrera (2020) put forward two methods to solve MCDM problems, combining a multiplicative analytic hierarchy process and a score function based on the concentration degree. (Liu & Yang, 2022) designed an innovative feedback mechanism based on the particle swarm optimization (PSO) algorithm that can facilitate solving consensus problems under the probabilistic linguistic environment. Gou, Xu, Liao, and Herrera (2020) defined a generalized linguistic representation model named the probabilistic double hierarchy linguistic term set (PDHLTS), and extended the VIKOR (VIse Kriterijumska Optimizacija i Kompromisno Resenje) method to the probabilistic double hierarchy linguistic context. Subsequently, based on the PDHLTS, Gou, Xiao, Huang, and Deng (2021) conceived a MCDM method to evaluate real economic development from the perspective of economic financialization.

Although the aforementioned decision-making methods have substantially contributed to both scientific research and practical applications, there are still some limitations that need to be addressed:

The existing methods largely failed to consider the evaluations and opinions with fuzzy linguistic information in the context of opinion dynamics. Even less attention has been paid to constructing effective opinion dynamics models for emergency decision-making with PLTS. Therefore, it is promising to develop a fuzzy linguistic methodology to address this challenge.

Most decision-making methods are aimed at static information for a specific time point, which makes them potentially inappropriate for constantly changing data. For instance, the outbreak of COVID-19 was not particularly damaging or contagious in the early stages, but over time the virus spread rapidly, severely affecting other fields, such as politics, economy, and culture. In such a case, static decision-making approaches can neither handle dynamic changes of events well nor can reflect the corresponding changes in attribute values. However, dynamic decision-making methods are suitable for solving such problems. Therefore, for this type of problem, a dynamic model should be developed according to the specificity of emergency decision-making.

Opinion dynamics is a complex process with many interactions and relationships, which are commonly characterized by a SN and can be represented by a trust matrix. For example, Zhang, Gao, and Li (2020) considered a consensus reaching-based GDM problem that incorporated SNs between experts and interval fuzzy preference relations. Gao and Zhang (2021) focused on devising a consensus model based on noncooperative behavior management to deal with PIS-based GDM problems, where the trust degrees of experts were dynamically adjusted in the SN. Zhang and Li (2021) considered the individual consistency and developed a minimum adjustment-based optimization model to achieve consensus. Nevertheless, most related literature (Gao and Zhang, 2021, Zhang et al., 2020, Zhang and Li, 2021) fails to consider a real situation where a trust matrix is generally incomplete because some experts may not offer all direct trust values for each expert. Particularly, there have been few studies considering a probabilistic linguistic environment. To accurately model a practical decision process, it is necessary to develop a method that can evaluate the missing trust values.

According to the aforementioned analysis, this study exploits the PLTS to represent decision-makers’ assessments to express their uncertainty and hesitation information appropriately and accurately, and develops a dynamic method to solve emergency decision-making problems. The main innovations of this work can be summarized as follows.

A novel score function of PLTS is defined, which can remarkably distinguish different PLTSs, and its strong robustness is demonstrated with specific examples. The comparison results indicate that the proposed score function can overcome some drawbacks of the existing methods with desirable features. In addition, the operational rules of PLTSs are defined to make the probabilistic linguistic elements fit the matrix operations better, which lays the foundation for constructing opinion dynamics models.

Compared with other linguistic computational methods (Liu et al., 2022b, Wu et al., 2022b, Zhang et al., 2021), the proposed PLTS-based method is capable of conveying evaluation information more precisely and reflecting human perception habits better, which makes it valuable in practical applications. Furthermore, the corresponding decision-making method based on the DeGroot model are constructed, considering the similarity degree, self-persistence and authority degree of an expert, which can visualize the dynamic process of reaching consensus between experts. To the best of our knowledge, this is the first work that incorporates the opinion dynamics model and PLTS to illustrate the evolution of participants’ opinions.

In the context of a SN, an expert’s opinion is influenced by other experts’ opinions and consequently subject to evolve. Considering the external trust and internal self-confidence of experts, a trust-based opinion dynamics model based on the DeGroot model is developed. In addition, to estimate incomplete trust relationships between decision-makers, the trust propagation operator is defined. The proposed method fully mines the social relationships and decision information between experts.

The rest of this paper is arranged as follows: Section 2 displays a review of basic concepts about PLTS, social network analysis (SNA) and DeGroot model. Section 3 develops the probabilistic linguistic DeGroot (PL-DeGroot) opinion dynamics model and the corresponding algorithms are introduced in Section 4. Section 5 applies the proposed method to a numerical example for emergency decision-making of COVID-19 epidemic and conducts comparisons with other studies, and the sensitivity analysis is also discussed. Finally, Section 6 summarizes the contributions of this work and points out future work.

Preliminaries

This section briefly reviews some preliminaries of the PLTS, SNA, and DeGroot model.

PLTS

The linguistic term set (LTS) (Zadeh, 1975) can be regarded as a linguistic evaluation scale, which contains a finite number of ordered linguistic terms. The subscript-asymmetric LTS and subscript-symmetric LTS are two conventional LTSs, where is a positive integer. Based on the subscript-symmetric LTS, Pang et al. (2016) defined the PLTS, allowing experts to express qualitative information with uncertainty and hesitation in the linguistic setting.

Pang et al. (2016)

Let be a LTS, the PLTS can be defined as: where represents the th linguistic term associated with the corresponding probability , and is the number of different elements in .

Gou and Xu (2016)

Let be a LTS, the transformation function can be defined as: In addition, represents the equivalent information of linguistic term , which can be obtained by its inverse function :

Yu, Du, and Xu (2020)

Given a PLTS , the normalized PLTS can be obtained by the following steps:

Normalize the probability of PLTS. If , the corresponding PLTS is , where .

Normalize the granularity of PLTS. All PLTSs have the same linguistic terms. If the linguistic term does not appear in , then its corresponding probability is zero.

In this paper, if there is no special explanation, the PLTS is a normalized PLTS.

For two PLTSs and based on the LTS , the normalization forms of them are derived from two steps:

Normalize the probability of PLTS: and . Thus, the sums of probabilities in and are equal to 1.

Normalize the granularity of PLTS: and . For convenience, we denote and as and . Thus, they have the same length.

Many ranking methods of PLTSs have been proposed for practical applications, such as the score function (Lin et al., 2020), expectation function (Wu, Liao, Xu, Arian, & Francisco, 2018), and possibility degree-based ranking methods (Mao, Wu, Dong, Wan, & Jin, 2019). In general, it is suggested to utilize a ranking method based on the expectation function and deviation degree. Let be any PLTS based on the LTS , is the subscript of linguistic term , Pang et al. (2016) presented the definition of expectation function and deviation degree as:

Then the ranking rules between and are constructed as:

If , then ;

If , then

if , then ;

if , then .

Besides, Lin et al. (2020) proposed the score function as follows: where is the subscript of linguistic term . The larger the , the better the .

After reviewing the PLTS basics, the decision-making problem with probabilistic linguistic information can be analyzed. Let be a set of alternatives, be a set of criteria and be a set of experts. Each expert evaluates different alternatives regarding each criterion with PLTSs on the predefined LTS . Then all PLTSs form a probabilistic linguistic decision matrix for expert , which can be expressed as follows: where .

To obtain a collective evaluation of multiple experts, Wu and Liao (Wu et al., 2018) developed a group aggregation operator as follows:

Wu et al. (2018)

During the decision-making process, a group of experts provide their opinions as PLTSs . The weight of expert is , satisfying and , then the collective evaluation can be denoted as: where ,

SNA

The SN represents a platform that allows agents to exchange information and communicate with each other, focusing on the relationships between social entities, such as families, companies, and nations. Meanwhile, SNA has emerged as an important technique for analyzing the structure, relationships, and properties of SNs. The effectiveness of the existing SNA frameworks and techniques for modeling social trust relationships between a panel of agents has been verified using various decision-making methods. Wu et al., 2015, Wu et al., 2017, Zhang et al., 2018.

In general, a SN consists of a set of nodes and a set of edges , where nodes denote agents (e.g., experts or organizations) and edges denote all different relationships between nodes. Table 1 shows a SN among five experts .

Table 1

Different representation schemes of a SN.

Graph	Sociomatrix	Algebraic
	A=1100101100101101011001011	e_1Re_2, e_1Re_5, e_2Re_3, e_3Re_1, e_3Re_4, e_4Re_1, e_4Re_3, e_5Re_2, e_5Re_4

Liu, Xu, Montes, and Herrera (2019)

The sociometric is utilized to represent a SN such that where implies that there is a direct relationship between expert and .

It is obvious that the above sociomatrix denotes a binary relation between experts, and the elements of decision matrix indicate either complete trust or no trust at all. In some situations, experts may find it inconvenient to express their judgments in terms of a binary relation, but rather feel more comfortable characterizing the uncertainty of a trust relationship in the SN with linguistic terms. Hence, this study aims to construct a social trust network, where experts explicitly express their qualitative evaluations of other experts using trust degrees in the form of PLTSs. Thus, the sociomatrix can be named the probabilistic linguistic sociomatrix, and its mathematical expression is as follows. where is the probabilistic linguistic trust degree from expert to .

DeGroot model

In practice, experts’ opinions on a decision-making problem may change over time, which means that their opinions are a kind of dynamic information. The DeGroot model (DeGroot, 1974) can be regarded as a representative technique for estimating the evolution process of an expert, and it can be described as follows.

Let be a set of experts, and denote the opinions of all experts at time . In the DeGroot model, experts are mutually influenced, and matrix is called the influence matrix, where is the weight that expert assigns to expert . For each expert, it holds that . Then the evolution process for expert can be expressed as: where is a row stochastic matrix.

The opinion fusion process for all experts can be expressed in the matrix form as ; however, it can be expressed more compactly by .

Li and Wei (2019)

For any initial opinion , if there exists a constant , such that , then a consensus can be reached, and is the consensus opinion.

In the DeGroot model, consensus conditions have been demonstrated, as given by Lemma 1, Lemma 2.

(Li & Wei, 2019)

If and only if there is an integer , such that the matrix power contains at least one column of strictly positive elements, then a consensus among all experts can be achieved.

Lemma 1 represents a sufficient consensus condition that does not rely on the initial opinions of experts. Thus, it can be used to verify whether a consensus can be reached among experts after the influence matrix is determined.

Based on the DeGroot model, let be the stationary probability vector for the influence matrix , where and . Then, the ultimate consensus opinion can be expressed by:

Lemma 2 indicates that the ultimate consensus can be expressed as a linear combination of the initial opinions of all experts. Notably, the final consensus is derived from the influence matrix and experts’ initial opinions.

Probabilistic linguistic opinion dynamics method

In this section, an innovative score function and operational rules for PLTSs are presented, and the PL-DeGroot opinion dynamics model is proposed.

Operational laws of PLTSs

To enrich the calculation among PLTSs, an expectation function of PLTSs is defined, which can convert the PLTS to a crisp number in the range of . For any PLTS based on the LTS , the expectation function is defined as follows: where is the transformation function.

It can be observed that . In particular, , if and only if ; , if and only if .

Since and the transformation function is a monotonically increasing function, we have and . Thus, . If , then ; and , . Obviously, if , then ; and , . This completes the proof.

Given two PLTSs and , based on a LTS . Then, the score functions and the deviation degrees can be obtained by Eq. (3). Therefore, holds, which contradicts human cognition. It can be concluded that the difference between and cannot be distinguished by Eq. (3).

Given two PLTSs and based on a LTS . By applying Eq. (4), the score functions can be obtained, which means that and are equivalent. However, this result does not reflect the actual situation, and the score function proposed by Lin et al. (2020) cannot be adopted to compare these two PLTSs. Thus, to remove the above limitations of expectation (score) function and yield the precise ranking results, an improved score function is proposed.

Motivated by information energy (Liao et al., 2015, Zhong and Xu, 2020), a novel score function of PLTS is defined as follows: where is the hesitance degree function, is the number of different elements in and is a function of hesitance degree .

It should be noted that the higher the hesitance degree of a PLTS is, the less deterministic information it contains, and the smaller the PLTS will be. Hence, is a monotonically decreasing function, and in this study, it is expressed as follows:

The definition of the score function takes into account both the hesitance degree and expected value of a PLTS. Accordingly, a new comparison rule is that a larger value of the score function indicates a better PLTS, which can be extended to other decision-making methods, such as determining the best alternative, and deducing the ranking of feasible alternatives.

In particular, if , then . The function , if , then ; if , then .

Continued to Example 2

Based on Eq. (9), , , and can be obtained, then . If the expectation function of PLTS (Wu et al., 2018) is adopted, then , which demonstrates the availability of the proposed score function.

Continued to Example 3

By Eq. (9), we obtain , , and , then , which is the same as the result of applying expectation function in Wu et al. (2018). Hence, Eq. (9) can differentiate between and rationally and effectively.

Given three PLTSs , and based on a LTS . By Eq. (9), there is , , and , , , then and , which are the same results as applying the possibility degree in Mao et al. (2019). This proves the robustness of the proposed score function, because Eq. (9) can guarantee that the order between and remains invariant under a small disturbance in .

Considering the limitations of the existing PLTS operational rules in terms of the calculation accuracy and computational complexity (i.e., out-of-bounds problem (Mao et al., 2019) and unreasonable operational results (Yue, Xie, & Chen, 2020)), several basic operational rules for PLTSs based on the expectation function are given.

Let be a LTS, and be any two PLTSs on , and . Some operational rules of PLTSs are given below:

(1) ;

(2) ;

(3) ;

(4) ;

(5) .

The operational rules for PLTSs are similar to those for real numbers, and they satisfy the distributive law and associative law. Let and be two PLTSs, and , then operational rules for PLTSs satisfy the following properties:

(1) ;

(2) ;

(3) ;

(4) ;

(5) ;

(6) .

Let be a LTS, and be two PLTSs on , according to Definition 7, and . Let , then and .

Through normalization, the dimension of probabilistic linguistic information can be reduced. Then, it is necessary to consider only the difference in the probability distribution in the calculation process, which greatly preserves the original decision information.

Let be a LTS, and be any two PLTSs on . The distance measure between and can be defined as: where is a parameter. When , Eq. (11) is reduced to the Hamming distance measure; when , Eq. (11) is reduced to the Euclidean distance measure.

The distance measure satisfies the following properties:

(1) ;

(2) , if ;

(3) .

As described in Wu et al. (2018), the relationship between the distance and similarity measures is complementary; thus, the corresponding similarity measure between and is derived by .

PL-DeGroot opinion dynamics model

Let be a set of experts and be a vector representing the opinions of experts at time . During the decision-making process, experts interact with each other and exchange their opinions, and they may adjust their opinions at the next moment. Assume that is the influence matrix, which represents the influence weight assigned to expert by . Meanwhile, it should be noted that is a row stochastic matrix, so each element is nonnegative and the sum of elements in each row is one. Then, the PL-DeGroot opinion dynamics model can be developed, and the evolution process for expert at time can be expressed as: where is the opinion of expert expressed by a PLTS at time , and it can be denoted as .

Based on the theory of DeGroot model, we can get an important conclusion below.

Let be a row stochastic matrix, and be a probabilistic linguistic column matrix, then where is the initial opinion vector of experts expressed by PLTSs.

According to the associative law of PLTS operations, we can easily get .

For any initial opinion , if there exists a constant such that , then all experts can reach a consensus, and refers to the consensus opinion.

A real-world decision-making problem concerning the selection of a global supplier is offered to illustrate the evolution process of PL-DeGroot opinion dynamics model. Five experts are invited to participate in the performance evaluation of a given supplier. They provide their evaluation information by means of PLTSs, where the LTS is predefined. Suppose that is the original individual opinion for experts, where , , , , and . is the influence matrix, which is presented below. Assume that is a stationary probability vector of the influence matrix . Then, after several iterations, it holds that Thus, the performance score of the given supplier is 0.8343, which is the consensus opinion of experts. The opinion evolution process of the five experts is shown in Fig. 1.

Fig. 1

The opinion evolution process of five experts.

PL-DeGroot opinion dynamics model for solving decision-making problems

The construction of the influence matrix is the main part of the DeGroot model, and many methods for determining the influence matrix in the crisp number environment have been studied (Ding et al., 2019, Dong et al., 2017, Li and Wei, 2019, Zhou et al., 2020). However, in the fuzzy environment, especially in the probabilistic linguistic environment, many of these methods may not be applicable. Therefore, this study proposes a new method using four different algorithms to determine the influence matrix and discuss the consensus conditions in different situations.

Influence matrix construction

In most studies, the influence matrix is determined subjectively and with limited objectivity, ignoring hidden information in experts’ opinions. To compensate for this shortcoming, this study incorporates both objective and subjective factors to construct the influence matrix.

There are many interactions between experts at each stage of the decision-making process, which makes experts’ opinions more susceptible to the influence of others and evolve over time. The similarity can measure the closeness degree between two experts, and similar opinions are more simply conveyed to each other. Hence, this study uses the similarity measure between experts to construct the influence matrix.

Algorithm 1. The PL-DeGroot model based on similarity degree

Input: The initial opinions of experts , respectively.

Output: The influence matrix .

Step 1: Calculate the similarity measure between and by Eq. (11). Then, the influence matrix can be expressed as: where represents the opinion transition weight assigned by expert to .

Step 2: Output.

Step 3: End.

Assume that each expert is not completely stubborn, and has a certain degree of self-persistence in his/her own opinion, but can be influenced by others in a common group. Then, let be the self-persistence degree of expert , indicating the extent to which expert insists on maintaining the previous opinion in repeated iterations. A higher self-persistence degree suggests that an expert is less likely to accept others’ opinions, and the expert’s weight can be increased appropriately. Therefore, the self-persistence degree can be considered in the transition weights of opinions, and the corresponding algorithm is given below.

Algorithm 2. The PL-DeGroot model based on self-persistence degree

Input: The self-persistence degree of experts , respectively.

Output: The influence matrix .

Step 1: Calculate the similarity between and by Eq. (11).

Step 2: Derive the influence matrix , it is expressed as:

Step 3: Output.

Step 4: End.

Additionally, the authority figure of a decision group plays a critical role in the decision-making process, and this expert can be more likely to influence others, who will also be more inclined to trust. In fact, a high authority degree of an expert increases the possibility of influencing the evolution of other experts’ opinions when interacting with them, which consequently affects the final decision result. Therefore, the authority degree should be considered in the influence matrix construction process.

Algorithm 3. The PL-DeGroot model based on authority degree

Input: The authority figure and the authority degree .

Output: The influence matrix .

Step 1: Calculate the similarity between and by Eq. (11).

Step 2: Derive the influence matrix , it is expressed as: Step 3: Output.

Step 4: End.

In addition to the ease of exchanging information among decision makers with similar interests, decision makers within the same SN can readily interact with each other in terms of opinions and influence each other, leading to the evolution of opinions. Therefore, the SN can also be taken as a source for constructing the influence matrix. However, it has become increasingly common for experts to be involved in a SN of the actual decision-making problem, and the SNA has become an important tool for group relational analysis to coordinate various opinions so as to form a group’s agreement (Zhang et al., 2021). Nevertheless, currently, there has been no PLTS-based method that considers the impacts of the SN and opinion interaction and evolution on the efficiency of the decision implementation process. As a result, it is of high significance to take into account the SN of participants along with the evolution of opinions to provide an approximate framework for decision-making in practice.

In the following, the influence matrix is constructed based on the SNA to fully explore the impact of social relationships on opinion evolution. This study mainly focuses on trust relationships between experts. Due to the limited information and the ambiguity of human cognition, it is difficult for experts to provide trust values to the remaining experts directly. In view of the transitivity of trust propagation, indirect trust relationships between experts can be inferred. A t-norms-based trust propagation is implemented in this work to estimate missing trust values in the fuzzy sociomatrix.

Zhang et al. (2018)

Let be a path from and (its length is ), and be the set of all given trust values. Then, the trust propagation operator () can be used to estimate the trust value, and it is defined as follows: Specifically, if the length from to is two (i.e., ), then the trust value is calculated by:

When there are different trust paths between two experts, it is necessary to fuse the trust values derived from multiple sources. Without loss of generality, this study applies the order weighted averaging (OWA) operator to obtain the total trust value.

Yager (1996)

Suppose that there are trust paths from to , and denote the set of trust values, then the total trust value can be calculated by: where is a permutation of such that is the th largest value, and is the weight vector with and .

For obtaining the OWA weights, a quantifier-guided method (Zadeh, 1983) based on the linguistic quantifiers is adopted as follows: where can be expressed by with . Moreover, the values are (0,1), (0.3,0.8), (0,0.5), and (0.5,1), representing “all”, “most”, “at least half”, and “as many as possible” of the proportional quantifiers, respectively.

According to the above analysis, an algorithm for the PL-DeGroot model based on SNA is given below.

Algorithm 4. The PL-DeGroot model based on SNA

Input: The SN among experts and the confidence degree of expert .

Output: The influence matrix .

Step 1: Estimate the missing values of SN by Eq. (17).

Step 2: Aggregate multiple paths by Eqs. (18)–(20) and construct the complete probabilistic linguistic sociomatrix denoted by .

Step 3: Derive the influence matrix .

Let be a diagonal matrix of experts’ confidence on the decision-making problem, where represents the confidence degree of expert and the matrix is normalized with . Then, the influence matrix can be expressed as follows: where is a unit matrix, and .

Step 4. Output.

Step 5. End.

Evolution analysis of the PL-DeGroot opinion dynamics model

This section analyzes evolution processes in two special situations for the PL-DeGroot opinion dynamics model and discusses the consensus conditions.

In situation 1, let be a predefined LTS, and consider a situation where experts have only two types of opinions and . Suppose that there are experts, of which experts hold the opinion and the remaining experts hold the opinion .

In Algorithm 1, the experts’ initial opinions can be expressed as , based on Eq. (14), the opinion transition matrix is given as:

At time , there is , then . Thus, we have The result shows that the experts cannot reach a consensus and they insist on their previous opinions.

In Algorithm 2, all experts have a hesitation degree of zero and they are completely confident in their opinions. Based on Eq. (15), the opinion transition matrix is given as: Applying the theory of matrices, the matrix can be divided into two parts, and , which are denoted as:

The opinion transition matrix can be rewritten as where and are both stochastic matrices.

Then we have

Thus, , we have The result indicates that if all experts are completely self-persistent, their opinions cannot reach a consensus.

In Algorithm 3, according to Eq. (16), the opinion transition matrix is given as: In a similar way, the matrix can be denoted as where and

Since where and .

Thus, , we have Similarly, Algorithm 3 does not reach a consensus on the experts’ opinions in situation 1, where experts are reluctant to change their opinions.

Algorithms 1–3 have the same effect under situation 1. This phenomenon can be explained using the Markov Chain theory (Su et al., 2020). Let be the state space composed of all experts’ opinions, which can be decomposed into two irreducible closed sets denoted by and . Then, regardless of the transfer probability between the internal states of and , their states will remain unchanged, since there is no transition between them.

According to the simulation results shown above, when experts are confronted with only two diametrically opposed opinions (extreme opinions), they do not modify their initial opinions regardless of the influence of similarity, self-persistence, and authority. The underlying reason for unrealized consensus is the refusal of experts to transfer from one opinion to another, which is also demonstrated in real life. In view of that, the PL-DeGroot model is employed in this study to investigate the tendency in opinion transformation of incompletely opposite groups, which is a common scenario in opinion evolution.

In situation 2, let be a predefined LTS, and consider a situation where experts have opinions different from two types of opinions and . To simplify the analysis, suppose that there are experts, of which experts hold the opinion , the experts hold the opinion , and the last one holds any opinion in the form of PLTS. Then, the similarity measure between the expert and others can be given as:

For Algorithm 1, the opinion transition matrix is given as: where

According to matrix , . Then, the elements of the th column are all positive, satisfying the consensus condition in Lemma 1, which means that experts can reach a unanimous consensus. The consensus opinion consists of the stationary probability vector and the initial opinions of experts. In conclusion, for similarity-based dynamic models, experts will eventually reach a consensus regardless of the values of the third opinion that differs from two extreme values. Algorithms 2 and 3 can be examined in a similar way.

To analyze the evolution of various opinions in the probabilistic linguistic context, it is assumed that the PL-DeGroot model is time-invariant.

The consensus condition of Algorithm 4 relies on the social trust network structure rather than the influence matrix. It reaches consensus under the condition that there is at least one globally accessible point in the network. The detailed proof process can be found in Berger (1980) . For the sake of brevity, the proof is omitted in this work.

Decision-making method based on the PL-DeGroot model

The principal goal of emergency decision-making is to obtain a satisfying and reasonable decision result, and one of the evaluation rules is to judge whether the opinions of experts can reach a consensus. Based on the analysis presented in the previous Section 3.2, once the initial opinions of experts are provided, the prediction of their ultimate opinions can be obtained from a dynamic perspective. Thus, the PL-DeGroot model can be applied to the emergency decision-making.

The specific steps of the decision-making method based on the PL-DeGroot model are described as follows.

Step 1: Collect the initial opinions of all experts.

Step 2: Obtain the collective decision matrix. Aggregate the evaluation values under each attribute to obtain the global decision information of expert for alternative by Eq. (5).

Step 3: Construct the opinion transition matrix and apply Algorithm 1, 2, 3 or 4 to obtain the opinion transition matrix. For Algorithms 1–3, the influence matrix is presented as: where .

Step 4: Obtain experts’ weights. Calculate the limit of matrix , if is fully regular, then is determined, and this is the final weight of each expert for alternative . When and , then is the stationary probability vector for the influence matrix , and it holds that

Step 5: Calculate the final opinion of expert on alternative by: where is the weight of criterion , and is the evaluation value of expert for alternative with respect to criterion .

Step 6: Rank the alternative . In Step 4, the weight assignment matrices of different experts for each alternative are obtained; then apply the weighted averaging (WA) operator to obtain the comprehensive score as follows: The larger the value of is, the better the alternative is.

The flowchart of the proposed decision-making method based on the PL-DeGroot model is depicted in Fig. 2, which illustrates the decision process.

Fig. 2

The flowchart of proposed method based on opinion evolution.

Case study

The proposed PL-DeGroot model is verified by a numerical example (adapted from Su et al. (2020)) concerning public health emergency decision-making. In addition, the comparison analysis and sensitivity analysis are both furnished.

Background

The outbreak of COVID-19 was in Wuhan, China, and as the epidemic spread, numerous cases appeared in other parts of the country and abroad, causing severe economic losses worldwide. Since the epidemic outbreak, China has undertaken effective measures and activated a public health emergency response mechanism, realizing remarkable achievements in epidemic control. However, the selection of an appropriate response plan for a specific area is an emergency decision-making problem, and it is challenging to make an accurate and scientific decision. Thus, developing a reasonable decision-making method to handle emergency events is highly promising.

Due to the complexity and uncertainty of emergency events, decision makers can have difficulty providing a crisp value for evaluation but find it convenient to make judgments on the merits of alternatives in natural language. However, to achieve the objectivity and accuracy of decision results, many influencing factors, such as political and economic factors, should be considered. Experts may have multiple evaluation opinions at the same time and assign them with different weights. Therefore, the PLTS can serve as an effective tool for assessing the emergency event. In fact, the emergency decision-making problem is not always static, as it may change at any time, so it can be considered a dynamic evolution process. Thus, the proposed PL-DeGroot opinion dynamics model can be suitable for emergency decision-making and achieve full consensus among experts.

To better justify the applicability of the proposed method in practical decision-making problems, suppose that an authority assembles experts to select an optimal response plan for a regional outbreak of COVID-19. By consulting relevant experts in the epidemiology field and with the support of experts in the field of emergency management, the following criteria are considered: time pressure (), virus information and emergency supplies (), closeness to the public (), mortality (), global epidemic situation (), resumption of work (), and social impact (). is the response plan that needs to be evaluated and the epidemic control measures are defined as follows:

: Normal traffic flow between cities should be established, but epidemic prevention checkpoints should be set up at a reasonable distance. People with abnormal health conditions should be immediately quarantined. Heroic deeds in the fight against the epidemic should be publicized and honored via TV news, broadcasts, and the government should strengthen the social security system.

: The main epidemic areas should be “closed”, while the remaining areas should be normally accessible but with epidemic checkpoints set-up. People traveling between cities need to be quarantined at home, and those showing abnormal signs need to be quarantined immediately. In addition, the nucleic acid test is required. The government should enhance public mental health education by promoting epidemic prevention and control through Weibo.

: The transportation between different cities can work normally, but a 14-day quarantine is required for travelers. Schools and campuses need to pay great attention to the physical and mental health of students and strengthen efforts to prevent the epidemic. The government should promote epidemic prevention through the official WeChat account.

: Every household and city in the main epidemic area should be “closed”. People passing through cities in the epidemic area are required to offer the nucleic acid test report and be quarantined for 14 days. The government needs to be informed in real-time about the progress of the virus-related research and the distribution of necessary supplies. Furthermore, the epidemic status in other countries should be monitored, and the psychological counseling for the public should be enhanced.

: Vehicles and people are strictly prohibited from passing through main epidemic areas. The government should broadcast news on the progress of the virus spreading, issue new policies to ease the pressure on industries severely affected by the epidemic, and provide a free psychological counseling website.

Decision process

This section applies the proposed decision-making method based on the PL-DeGroot model with different algorithms to assess and select an optimal response plan during the COVID-19 outbreak.

Decision process based on Algorithm 1

First, the opinions of all experts are collected. The specific steps of the decision process are summarized as follows:

Step 1: Each expert allocates PLTSs to evaluate decision information on response plan concerning criterion , where is a predefined LTS. The evaluations are shown in Table A.1, Table A.2, Table A.3, Table A.4, Table A.5. The weight vector of criteria is given as .

Table A.1

The evaluation matrix of expert .

	c_1	c_2	c_3	c_4	c_5	c_6	c_7
x_1	{s_−3(0.1),s_−2(0.4), s_−1(0.2),s_0(0.1), s_1(0.2)}	{s_−1(0.5),s_0(0.3), s_1(0.2)}	{s_0(0.2),s_1(0.5), s_2(0.3)}	{s_−2(0.1),s_−1(0.8), s_1(0.1)}	{s_−1(0.3),s_0(0.4), s_1(0.3)}	{s_0(0.1),s_1(0.2), s_2(0.5),s_3(0.2)}	{s_−2(0.7),s_−1(0.1), s_0(0.2)}
x_2	{s_−3(0.2),s_−2(0.3), s_−1(0.1),s_0(0.1)s_1(0.1),s_2(0.2)}	{s_−1(0.6),s_0(0.2), s_1(0.2)}	{s_0(0.1),s_1(0.6), s_2(0.3)}	{s_−1(0.6),s_0(0.2), s_1(0.2)}	{s_−1(0.2),s_0(0.5), s_1(0.2),s_2(0.1)}	{s_0(0.3),s_1(0.4), s_2(0.3)}	{s_−3(0.1),s_−2(0.1), s_−1(0.1),s_0(0.4), s_1(0.2),s_20.1}
x_3	{s_−3(0.3),s_−2(0.2), s_−1(0.5)}	{s_−3(0.5),s_−2(0.3), s_−1(0.2)}	{s_−1(0.2),s_0(0.2), s_1(0.3),s_2(0.3)}	{s_−3(0.4),s_−2(0.3), s_−1(0.3)}	{s_−3(0.2),s_−2(0.3), s_−1(0.3),s_0(0.2)}	{s_−3(0.1),s_−2(0.1), s_−1(0.6),s_1(0.2)}	{s_−2(0.4),s_−1(0.2), s_1(0.2),s_2(0.1), s_3(0.1)}
x_4	{s_−3(0.2),s_−2(0.2), s_−1(0.2),s_0(0.4)}	{s_−3(0.3),s_−2(0.2), s_1(0.5)}	{s_1(0.4),s_2(0.5), s_3(0.1)}	{s_−1(0.7),s_0(0.3)}	{s_−1(0.3),s_0(0.35), s_1(0.35)}	{s_1(0.9),s_2(0.1)}	{s_−2(0.4),s_−1(0.2), s_1(0.2),s_2(0.1), s_3(0.1)}
x_5	{s_−3(0.1),s_−2(0.4), s_−1(0.2),s_0(0.1), s_1(0.2)}	{s_−1(0.2),s_0(0.2), s_1(0.3),s_2(0.2)}	{s_0(0.1),s_1(0.5), s_2(0.4)}	{s_0(0.1),s_1(0.5), s_2(0.2),s_3(0.2)}	{s_0(0.65),s_1(0.2), s_2(0.15)}	{s_−1(0.55),s_0(0.15), s_1(0.3)}	{s_−3(0.1),s_−2(0.3), s_0(0.4),s_2(0.1), s_3(0.1)}

Table A.2

The evaluation matrix of expert .

	c_1	c_2	c_3	c_4	c_5	c_6	c_7
x_1	{s_−3(0.3),s_−2(0.3), s_−1(0.3),s_0(0.1)}	{s_−3(0.2),s_−2(0.1), s_−1(0.2),s_0(0.3),s_1(0.2)}	{s_−2(0.2),s_−1(0.5), s_0(0.3)}	{s_−1(0.5),s_0(0.1), s_1(0.4)}	{s_−3(0.3),s_−2(0.2), s_−1(0.3),s_0(0.2)}	{s_−3(0.1),s_−2(0.1), s_−1(0.2),s_1(0.2),s_2(0.4)}	{s_−3(0.3),s_−2(0.4), s_−1(0.3)}
x_2	{s_−3(0.5),s_−2(0.15), s_−1(0.35)}	{s_−3(0.3),s_−2(0.2), s_−1(0.3),s_0(0.1), s_1(0.1)}	{s_−2(0.4),s_−1(0.4), s_0(0.2)}	{s_−2(0.3),s_−1(0.4), s_0(0.1),s_1(0.2)}	{s_−3(0.3),s_−2(0.3), s_−1(0.2),s_0(0.2)}	{s_−3(0.1),s_−2(0.2), s_−1(0.3),s_0(0.2), s_1(0.2)}	{s_−3(0.1),s_−2(0.5), s_−1(0.4)}
x_3	{s_−3(0.7),s_−2(0.2), s_−1(0.1)}	{s_−3(0.4),s_−2(0.3), s_−1(0.2),s_0(0.1)}	{s_−2(0.6),s_−1(0.3), s_0(0.1)}	{s_−2(0.5),s_−1(0.3), s_0(0.1),s_1(0.1)}	{s_−3(0.3),s_−2(0.4), s_−1(0.2),s_0(0.1)}	{s_−3(0.1),s_−2(0.4), s_−1(0.3),s_0(0.2)}	{s_−2(0.4),s_−1(0.5), s_0(0.1)}
x_4	{s_−1(0.1),s_0(0.1), s_1(0.4),s_2(0.4)}	{s_−2(0.1),s_−1(0.3), s_0(0.1),s_1(0.3), s_2(0.2)}	{s_−1(0.1),s_0(0.1), s_1(0.25),s_2(0.55)}	{s_0(0.1),s_1(0.35), s_2(0.35),s_3(0.2)}	{s_−3(0.1),s_1(0.2), s_2(0.3),s_3(0.4)}	{s_−3(0.3),s_−2(0.3), s_−1(0.2),s_0(0.2)}	{s_−3(0.1),s_−2(0.1), s_−1(0.1),s_1(0.2), s_2(0.3)}
x_5	{s_−2(0.35),s_−1(0.2), s_0(0.3),s_1(0.15)}	{s_−3(0.1),s_−2(0.2), s_−1(0.3),s_0(0.2), s_1(0.2)}	{s_0(0.45),s_1(0.25), s_2(0.3)}	{s_−1(0.2),s_0(0.5), s_1(0.15),s_2(0.15)}	{s_−3(0.1),s_1(0.3), s_2(0.4),s_3(0.2)}	{s_−3(0.4),s_−2(0.4), s_−1(0.2)}	{s_−3(0.1),s_−2(0.2), s_−1(0.2),s_1(0.3), s_2(0.2)}

Table A.3

The evaluation matrix of expert .

	c_1	c_2	c_3	c_4	c_5	c_6	c_7
x_1	{s_−3(0.5),s_0(0.3),s_1(0.2)}	{s_−3(0.2),s_−2(0.2),s_−1(0.3),s_1(0.3)}	{s_−1(0.6),s_1(0.3),s_2(0.1)}	{s_−3(0.3),s_−2(0.4),s_−1(0.1),s_0(0.2)}	{s_−3(0.3),s_−2(0.4),s_−1(0.2),s_0(0.1)}	{s_−1(0.1),s_0(0.2),s_1(0.3),s_2(0.4)}	{s_−2(0.5),s_−1(0.1),s_1(0.4)}
x_2	{s_−2(0.5),s_−1(0.2),s_0(0.2),s_1(0.1)}	{s_−3(0.3),s_−2(0.3),s_−1(0.2),s_0(0.2)}	{s_−1(0.5),s_0(0.4),s_1(0.1)}	{s_−2(0.4),s_−4(0.3),s_0(0.3)}	{s_−3(0.4),s_−2(0.3),s_−1(0.3)}	{s_−2(0.1),s_−1(0.2),s_0(0.2),s_1(0.25),s_2(0.25)}	{s_−3(0.4),s_−2(0.2),s_−1(0.3),s_1(0.1)}
x_3	{s_−2(0.65),s_−1(0.1),s_1(0.25)}	{s_−3(0.3),s_−2(0.4),s_−1(0.3)}	{s_0(0.1),s_1(0.3),s_2(0.3),s_3(0.3)}	{s_−3(0.5),s_−2(0.3),s_−1(0.2)}	{s_−3(0.4),s_−2(0.2),s_−1(0.4)}	{s_−2(0.2),s_−1(0.2),s_0(0.1),s_1(0.3),s_2(0.2)}	{s_−2(0.7),s_−1(0.2),s_1(0.1)}
x_4	{s_−1(0.1),s_1(0.45),s_2(0.35),s_3(0.1)}	{s_1(0.6),s_2(0.4)}	{s_−2(0.2),s_−1(0.4),s_0(0.3),s_1(0.1)}	{s_−1(0.1),s_1(0.4),s_2(0.4),s_3(0.1)}	{s_1(0.3),s_2(0.4),s_3(0.3)}	{s_−3(0.3),s_−2(0.2),s_−1(0.2),s_1(0.3)}	{s_1(0.2),s_2(0.3),s_3(0.5)}
x_5	{s_0(0.3),s_1(0.25),s_2(0.45)}	{s_−1(0.2),s_1(0.5),s_2(0.1),s_3(0.2)}	{s_−1(0.1),s_1(0.3),s_2(0.3),s_3(0.3)}	{s_−1(0.25),s_0(0.2),s_1(0.3),s_2(0.25)}	{s_−1(0.1),s_0(0.3),s_1(0.2),s_2(0.3),s_3(0.1)}	{s_−3(0.4),s_−2(0.3),s_−1(0.3)}	{s_1(0.3),s_2(0.4),s_3(0.3)}

Table A.4

The evaluation matrix of expert .

	c_1	c_2	c_3	c_4	c_5	c_6	c_7
x_1	{s_−3(0.1),s_−2(0.2),s_−1(0.2),s_0(0.5)}	{s_−3(0.3),s_−2(0.3),s_−1(0.3),s_0(0.1)}	{s_−2(0.1),s_−1(0.3),s_0(0.1),s_1(0.3), s_2(0.2)}	{s_−3(0.2),s_−2(0.2),s_−1(0.3),s_1(0.3)}	{s_−3(0.2),s_−2(0.3),s_−1(0.3),s_1(0.2)}	{s_0(0.1),s_1(0.5),s_2(0.1),s_3(0.3)}	{s_−2(0.1),s_−1(0.2),s_0(0.2),s_1(0.4),s_2(0.1)}
x_2	{s_−3(0.2),s_−2(0.2),s_−1(0.4),s_0(0.2)}	{s_−3(0.5),s_−2(0.3),s_−1(0.2)}	{s_−3(0.1),s_−2(0.4),s_−1(0.2),s_1(0.3)}	{s_−3(0.1),s_−2(0.2),s_−1(0.1),s_1(0.6)}	{s_−3(0.2),s_−2(0.2),s_−1(0.4),s_1(0.2)}	{s_−1(0.1),s_0(0.4),s_1(0.3),s_2(0.2)}	{s_0(0.5),s_1(0.3),s_2(0.2)}
x_3	{s_−3(0.3),s_−2(0.3),s_−1(0.3),s_0(0.1)}	{s_−3(0.7),s_−2(0.2),s_−1(0.1)}	{s_0(0.3),s_1(0.4),s_2(0.3)}	{s_−3(0.3),s_−2(0.3),s_−1(0.4)}	{s_−3(0.2),s_−2(0.2),s_−1(0.3),s_1(0.3)}	{s_−2(0.1),s_−1(0.3),s_0(0.35),s_1(0.25)}	{s_−2(0.1),s_−1(0.3),s_0(0.4),s_1(0.2)}
x_4	{s_−2(0.1),s_−1(0.2),s_1(0.4),s_2(0.3)}	{s_0(0.2),s_1(0.7),s_2(0.1)}	{s_−1(0.2),s_0(0.2),s_1(0.4),s_2(0.2)}	{s_1(0.6),s_2(0.3),s_3(0.1)}	{s_1(0.75),s_2(0.15),s_3(0.1)}	{s_−2(0.2),s_−1(0.3),s_0(0.3),s_1(0.2)}	{s_0(0.1),s_1(0.2),s_2(0.3),s_3(0.4)}
x_5	{s_−2(0.2),s_−1(0.2),s_0(0.2),s_1(0.4)}	{s_0(0.4),s_1(0.4),s_2(0.2)}	{s_0(0.2),s_1(0.5),s_2(0.3)}	{s_0(0.1),s_1(0.5),s_2(0.4)}	{s_0(0.5),s_1(0.4),s_2(0.1)}	{s_−3(0.2),s_−2(0.3),s_−1(0.4),s_1(0.1)}	{s_0(0.2),s_1(0.3),s_2(0.3),s_3(0.2)}

Table A.5

The evaluation matrix of expert .

	c_1	c_2	c_3	c_4	c_5	c_6	c_7
x_1	{s_−3(0.2),s_−2(0.3),s_−1(0.3),s_0(0.1),s_1(0.1)}	{s_−3(0.3),s_−2(0.3),s_−1(0.1),s_1(0.3)}	{s_−2(0.2),s_−1(0.2),s_0(0.2),s_1(0.1), s_2(0.3)}	{s_−3(0.2),s_−2(0.3),s_−1(0.4),s_1(0.1)}	{s_−3(0.5),s_−2(0.25),s_−1(0.25)}	{s_−3(0.2),s_−2(0.3),s_1(0.1),s_2(0.2)}	{s_−3(0.3),s_−2(0.1),s_0(0.3),s_1(0.3)}
x_2	{s_−3(0.3),s_−2(0.2),s_−1(0.1),s_0(0.4)}	{s_−3(0.4),s_−2(0.3),s_−1(0.2),s_0(0.1)}	{s_−2(0.2),s_−1(0.4),s_0(0.1),s_1(0.1),s_2(0.1),s_3(0.1)}	{s_−3(0.3),s_−2(0.3),s_−1(0.3),s_0(0.1)}	{s_−3(0.4),s_−2(0.2),s_−1(0.2),s_0(0.1)}	{s_−3(0.3),s_−2(0.4),s_−1(0.2),s_1(0.1)}	{s_−2(0.4),s_−1(0.3),s_1(0.1),s_2(0.2)}
x_3	{s_−3(0.4),s_−2(0.2),s_−1(0.2),s_1(0.2)}	{s_−3(0.5),s_−2(0.4),s_−1(0.1)}	{s_−1(0.1),s_0(0.2),s_1(0.5),s_2(0.2)}	{s_−3(0.3),s_−2(0.3),s_−1(0.2),s_1(0.2)}	{s_−1(0.3),s_0(0.2),s_1(0.3),s_2(0.2)}	{s_−3(0.4),s_−2(0.3),s_−1(0.2),s_0(0.1)}	{s_−3(0.4),s_−2(0.2),s_−1(0.1),s_0(0.4)}
x_4	{s_−2(0.1),s_−1(0.1),s_0(0.1),s_1(0.2),s_2(0.5)}	{s_−1(0.1),s_0(0.1),s_1(0.2),s_2(0.3),s_3(0.3)}	{s_−2(0.2),s_−1(0.3),s_1(0.5)}	{s_−2(0.1),s_−1(0.2),s_0(0.2),s_1(0.4),s_2(0.1)}	{s_−1(0.3),s_0(0.2),s_1(0.3),s_2(0.2)}	{s_−3(0.5),s_−2(0.2),s_−1(0.1),s_1(0.2)}	{s_−1(0.1),s_1(0.3),s_2(0.3),s_3(0.3)}
x_5	{s_−2(0.3),s_−1(0.2),s_0(0.1),s_1(0.4)}	{s_−2(0.1),s_−1(0.2),s_0(0.2),s_1(0.5)}	{s_−1(0.1),s_0(0.1),s_1(0.4),s_2(0.4)}	{s_−2(0.3),s_−1(0.3),s_0(0.2),s_1(0.2)}	{s_−2(0.3),s_−1(0.3),s_0(0.2),s_1(0.2)}	{s_−3(0.6),s_−2(0.3),s_−1(0.1)}	{s_−1(0.2),s_0(0.2),s_1(0.3),s_2(0.2),s_3(0.1)}

Step 2: Calculate the similarity degrees between experts’ opinions for the response plans by Eq. (11). Then, the opinion transition matrix is obtained by Eq. (14), and it is shown in Table 2.

Table 2

The opinion transition matrix for different based on Algorithm 1.

		e_1	e_2	e_3	e_4	e_5
x_1	e_1	0.2311	0.1911	0.1941	0.2011	0.1826
	e_2	0.1906	0.2306	0.1960	0.1891	0.1937
	e_3	0.1880	0.1902	0.2238	0.1988	0.1992
	e_4	0.1969	0.1856	0.2011	0.2263	0.1901
	e_5	0.1812	0.1927	0.2041	0.1927	0.2293
x_2	e_1	0.2392	0.1842	0.2010	0.1986	0.1770
	e_2	0.1762	0.2288	0.2037	0.1922	0.1991
	e_3	0.1867	0.1979	0.2223	0.1975	0.1956
	e_4	0.1900	0.1923	0.2034	0.2289	0.1854
	e_5	0.1721	0.2023	0.2047	0.1884	0.2325
x_3	e_1	0.2343	0.1804	0.1991	0.1988	0.1874
	e_2	0.1795	0.2331	0.1935	0.1935	0.2004
	e_3	0.1937	0.1891	0.2279	0.2024	0.1869
	e_4	0.1912	0.1871	0.2002	0.2254	0.1961
	e_5	0.1839	0.1977	0.1885	0.2000	0.2299
x_4	e_1	0.2309	0.1917	0.1940	0.2032	0.1802
	e_2	0.1886	0.2273	0.1955	0.2000	0.1886
	e_3	0.1871	0.1915	0.2227	0.2027	0.1960
	e_4	0.1947	0.1947	0.2013	0.2212	0.1881
	e_5	0.1797	0.1912	0.2028	0.1959	0.2304
x_5	e_1	0.2336	0.1916	0.1846	0.2079	0.1823
	e_2	0.1885	0.2299	0.1908	0.1908	0.2000
	e_3	0.1816	0.1908	0.2299	0.2069	0.1908
	e_4	0.1996	0.1861	0.2018	0.2242	0.1883
	e_5	0.1806	0.2014	0.1921	0.1944	0.2315

Step 3: Obtain experts’ weights for different response plans, and the results are displayed in Table 3.

Table 3

The experts’ weights for different response plans based on Algorithm 1.

wri	e_1	e_2	e_3	e_4	e_5
x_1	0.1975	0.1979	0.2039	0.2017	0.1990
x_2	0.1925	0.2012	0.2072	0.2011	0.1980
x_3	0.1964	0.1974	0.2019	0.2042	0.2001
x_4	0.1961	0.1993	0.2034	0.2047	0.1965
x_5	0.1967	0.1999	0.1999	0.2049	0.1986

Step 4: Calculate the final opinion matrix for experts by Eq. (25), which is listed in Table 4.

Table 4

The final opinion matrix for experts based on Algorithm 1.

	x_1	x_2	x_3	x_4	x_5
e_1	0.0978	0.0988	0.0899	0.1232	0.1108
e_2	0.0742	0.0572	0.0444	0.1154	0.0860
e_3	0.0795	0.0722	0.0631	0.1293	0.1164
e_4	0.0867	0.0781	0.0626	0.1308	0.1168
e_5	0.0605	0.0492	0.0547	0.1074	0.0814

Step 5: Rank the alternatives by comprehensive score . According to Eq. (26), and of response plans are obtained, so the ranking result is .

Decision process based on Algorithm 2

Algorithm 2 is applied to manage the case presented in Section 5.1. The implementation steps are as follows:

Step 1: Collect the information of experts.

Step 2: Assume that the self-persistence degree of expert is . Derive the opinion transition matrix, which is given in Table A.6.

Table A.6

The opinion transition matrix for different based on Algorithm 2.

		e_1	e_2	e_3	e_4	e_5
x_1	e_1	0.8000	0.0497	0.0505	0.0523	0.0475
	e_2	0.1239	0.5000	0.1274	0.1229	0.1258
	e_3	0.0969	0.0980	0.6000	0.1025	0.1026
	e_4	0.1018	0.0960	0.1039	0.6000	0.0983
	e_5	0.0705	0.0750	0.0795	0.0750	0.7000
x_2	e_1	0.8000	0.0484	0.0528	0.0522	0.0466
	e_2	0.1142	0.5000	0.1320	0.1246	0.1292
	e_3	0.0960	0.1018	0.6000	0.1016	0.1006
	e_4	0.0985	0.0998	0.1055	0.6000	0.0962
	e_5	0.0673	0.0791	0.0800	0.0736	0.7000
x_3	e_1	0.8000	0.0471	0.0520	0.0519	0.0490
	e_2	0.1170	0.5000	0.1261	0.1261	0.1308
	e_3	0.1003	0.0980	0.6000	0.1049	0.0968
	e_4	0.0987	0.0966	0.1034	0.6000	0.1013
	e_5	0.0716	0.0770	0.0735	0.0779	0.7000
x_4	e_1	0.8000	0.0498	0.0505	0.0529	0.0468
	e_2	0.1221	0.5000	0.1265	0.1294	0.1220
	e_3	0.0963	0.0986	0.6000	0.1043	0.1008
	e_4	0.1000	0.1000	0.1034	0.6000	0.0966
	e_5	0.0701	0.0746	0.0790	0.0763	0.7000
x_5	e_1	0.8000	0.0500	0.0482	0.0543	0.0475
	e_2	0.1223	0.5000	0.1239	0.1239	0.1299
	e_3	0.0943	0.0991	0.6000	0.1075	0.0991
	e_4	0.1029	0.0960	0.1040	0.6000	0.0971
	e_5	0.0705	0.0786	0.0750	0.0759	0.7000

Step 3: Obtain the final weight assignment of experts, which is given in Table 5.

Table 5

The experts’ weights for different response plans based on Algorithm 2.

wri	e_1	e_2	e_3	e_4	e_5
x_1	0.3216	0.1290	0.1676	0.1652	0.2166
x_2	0.3129	0.1327	0.1721	0.1657	0.2166
x_3	0.3194	0.1286	0.1656	0.1680	0.2184
x_4	0.3196	0.1305	0.1675	0.1689	0.2136
x_5	0.3203	0.1309	0.1636	0.1690	0.2162

Step 4: Calculate the final opinion matrix for experts and rank alternatives.

According to the weight assignment matrix of different experts for each alternative, the comprehensive scores of response plans are obtained as and ; thus, the ranking result is .

Decision process based on Algorithm 3

Next, Algorithm 3 is used to solve the case presented in Section 5.1, and the implementation steps are given in the following.

Step 1: Acquire the evaluations from experts.

Step 2: Assume that the authority parameter is set to , and expert is selected as an authority figure. The opinion transition matrix is obtained by Algorithm 3 for different is given in Table A.7.

Table A.7

The opinion transition matrix for different based on Algorithm 3.

		e_1	e_2	e_3	e_4	e_5
x_1	e_1	0.5460	0.1128	0.1147	0.1187	0.1078
	e_2	0.4851	0.1467	0.1247	0.1203	0.1232
	e_3	0.4808	0.1216	0.1431	0.1271	0.1274
	e_4	0.4951	0.1167	0.1264	0.1423	0.1195
	e_5	0.4695	0.1248	0.1322	0.1248	0.1487
x_2	e_1	0.5571	0.1072	0.1170	0.1156	0.1031
	e_2	0.4611	0.1497	0.1332	0.1257	0.1303
	e_3	0.4787	0.1268	0.1425	0.1266	0.1254
	e_4	0.4841	0.1225	0.1295	0.1458	0.1181
	e_5	0.4540	0.1334	0.1350	0.1242	0.1534
x_3	e_1	0.5503	0.1059	0.1169	0.1167	0.1102
	e_2	0.4667	0.1515	0.1258	0.1258	0.1302
	e_3	0.4900	0.1196	0.1441	0.1280	0.1183
	e_4	0.4860	0.1189	0.1272	0.1432	0.1247
	e_5	0.4741	0.1274	0.1215	0.1289	0.1481
x_4	e_1	0.5457	0.1132	0.1146	0.1201	0.1064
	e_2	0.4819	0.1451	0.1248	0.1277	0.1205
	e_3	0.4793	0.1227	0.1427	0.1298	0.1255
	e_4	0.4916	0.1229	0.1271	0.1397	0.1187
	e_5	0.4671	0.1243	0.1317	0.1272	0.1497
x_5	e_1	0.5495	0.1126	0.1085	0.1223	0.1071
	e_2	0.4816	0.1468	0.1219	0.1219	0.1278
	e_3	0.4702	0.1235	0.1488	0.1339	0.1235
	e_4	0.4993	0.1164	0.1262	0.1403	0.1178
	e_5	0.4685	0.1306	0.1246	0.1261	0.1502

Step 3: Obtain the final weight assignment of experts, which is given in Table 6.

Table 6

The experts’ weights for different response plans based on Algorithm 3.

wri	e_1	e_2	e_3	e_4	e_5
x_1	0.5153	0.1198	0.1229	0.1236	0.1184
x_2	0.5147	0.1197	0.1258	0.1229	0.1169
x_3	0.5160	0.1171	0.1231	0.1239	0.1199
x_4	0.5138	0.1207	0.1229	0.1255	0.1171
x_5	0.5160	0.1206	0.1191	0.1264	0.1179

Step 4: Calculate the final opinion matrix for experts and rank alternatives.

According to the weight assignment matrix of different experts for each alternative, the comprehensive scores of response plans are and , so the ranking result is .

Decision process based on Algorithm 4

Further, Algorithm 4 is adopted for solving the emergency decision-making problem concerning the COVID-19 epidemic. It should be noted that due to the limited knowledge of experts and the vagueness of the human mind, there can be a high possibility that incomplete trust information exists in the assessment process. Experts provide the probabilistic linguistic sociometric matrix based on the LT as follows:

Step 1: Collect the opinions of all experts.

Step 2: Estimate the missing values in the incomplete decision matrix by Eq. (17).

In Fig. 3, it can be seen that at least one path exists between any two experts who are not linked directly. Take as an example, based on Eq. (17), the trust values of three paths from to are obtained as and .

Fig. 3

The trust relationships among five experts.

Step 3: Obtain the complete probabilistic linguistic sociomatrix .

The weights of different paths are calculated by Eqs. (18)–(20), then has the values of and . In this work, the linguistic quantifier “most” is considered. Multiple paths are aggregated by Eq. (18), then . The other missing values of matrix can be computed in the same manner. The final trust matrix is given as:

Step 4: Derive the influence matrix.

Assume that the confidence degree of experts is , then the influence matrix can be derived as where is the normalized matrix of .

Step 5: Calculate the final collective evaluation matrix.

After the influence matrix reaches stability, the weight vector of experts is . The final collective evaluation is computed by (refer to Table 7) where is the evaluation value of expert for response plan with respect to criterion .

Table 7

The final collective evaluation based on Algorithm 4.

	c_1	c_2	c_3	c_4	c_5	c_6	c_7
x_1	0.3405	0.2985	0.5237	0.3115	0.2695	0.6729	0.3934
x_2	0.2802	0.2103	0.4273	0.3827	0.2833	0.5251	0.4728
x_3	0.2301	0.1256	0.6658	0.2210	0.3364	0.4494	0.3633
x_4	0.5936	0.6842	0.5537	0.7189	0.7478	0.3621	0.7592
x_5	0.4797	0.5929	0.7068	0.6323	0.5996	0.2265	0.6731

Step 6: Apply the WA operator to obtain the comprehensive score and the alternatives are ranked by

Then, we can get and . Thus, the ranking of response plans is .

Comparison and discussion

The decision-making method based on the PL-DeGroot model is compared with other traditional consensus models. Based on the comparison results, the advantages of the proposed method are discussed in this section.

Comparison of proposed algorithms

As can be seen from the results obtained using Algorithms 1–4, the ranking indexes of different algorithms are similar. The consistency of the results indicates the effectiveness of the proposed method. Because Algorithms 1–3 are all similarity-based models, they can be compared under the same conditions. First, based on the simulation results, Algorithms 1 and 2 produce almost identical results, indicating that the authority degree of an expert has a more significant impact on the final ranking than similarity and self-persistence indicators.

Next, the speed of reaching consensus for the three algorithms is analyzed, as exhibited in Fig. 4. For Algorithms 1 and 3, nine iterations are required for the collective opinion to reach a consensus. In this sense, neither similarity nor authority can affect the speed of reaching the consensus, but the self-persistence degree is a dominant factor in delaying the consensus. In terms of the calculation process, all three algorithms are relatively simple, but Algorithm 1 is the clearest and most intuitive.

Fig. 4

The speed of reaching consensus for three algorithms.

In what follows, Algorithm 4 is utilized to explore the time needed to reach a consensus on experts’ opinions. The opinion evolution process for different response plans is presented in Fig. 5, Fig. 6, where it can be seen that after six iterations, the experts can form a consensus opinion. Compared with other algorithms proposed in this study, Algorithm 4 takes less time to reach a consensus. Besides, the major superiority of Algorithm 4 is that it exploits implicit information among experts and takes into account the trust relationship, making the decision result more reasonable.

Fig. 5

The opinion evolution process for different response plan .

Fig. 6

The opinion evolution process for different response plans .

Finally, from the perspective of actual applications, decision makers need to choose an appropriate PL-DeGroot model based on the complexity of algorithms and the accessibility of collecting information. In terms of the influence matrix construction, Algorithms 1–3 are opinion dynamics models based on similarity and do not require additional information, while Algorithm 4 involves a network of trust relationships between experts and requires more prior information. In addition, both Algorithms 2 and 3 include parameters, whose determination is a highly important task. For the sake of space limitation, we do not discuss the selection of parameters at length. However, there are no additional parameters in Algorithm 4, which makes it easier to perform. As for the speed of reaching consensus, Algorithm 4 has a remarkable advantage over Algorithms 1–3. Moreover, in terms of algorithm complexity, the complexity of four algorithms is , while Algorithm 1 seems to be the easiest to understand. In summary, the selection of decision models is determined by the decision environment and participants’ preferences in reality.

Comparison with traditional decision-making models

To confirm the effectiveness of the proposed methodology, a comparison analysis with the existing baseline methods is conducted. The comparison method included the VIKOR method, the method based on the score function, the consensus model in Wu and Xu (2016), and the consensus model in Zhang et al. (2021).

The VIKOR method has been a widely used method that was developed to solve decision-making problems with both conflicting and non-commutable attributes (Wu, Xu, Jiang, & Zhong, 2019). It performs a compromise ranking of finite decision alternatives by maximizing the group utility and minimizing individual regret values. For simplicity, in this study, it is assumed that the weight parameter of balancing group utility and individual regret is 0.5. Then, the compromise decision indexes are calculated as and . Thus, the ranking of response plans is . The ranking obtained by the VIKOR method is obviously different from the results of other methods, but the optimal plan is the same. This may be explained from two aspects. On the one hand, this method is constrained by the parameter, and the ranking result varies with the value of the selected parameter. On the other hand, this method is susceptible to extreme values, so the ranking of an alternative is influenced by the worst evaluation under all criteria and thus becomes underperforming.

Referring to the method in Lin et al. (2020), the score function and the group aggregation operator are used to obtain the ranking of alternatives. The collective decision matrix is obtained by Eqs. (5), (9), as shown in Table A.8, and the score values of five response plans are presented in Table 8. Based on the weights of criteria, the final score values of the response plans are calculated by , then and are obtained. Thus, the ranking result is derived as .

Table A.8

The collective evaluation matrix for experts.

	c_1	c_2	c_3	c_4	c_5	c_6	c_7
x_1	{s_−3(0.14),s_−2(0.24),s_−1(0.3),s_0(0.22),s_1(0.1)}	{s_−3(0.2),s_−2(0.18),s_−1(0.28),s_0(0.14),s_0(0.16),s_1(0.24),s_2(0.18)}	{s_−2(0.1),s_−1(0.32),s_0(0.16),s_1(0.24),s_2(0.18)}	{s_−3(0.14),s_−2(0.2),s_−1(0.42),s_0(0.06),s_1(0.18)}	{s_−3(0.26),s_−2(0.23),s_−1(0.27),s_0(0.14),s_1(0.1)}	{s_−3(0.06),s_−2(0.08),s_−1(0.1),s_0(0.08),s_1(0.26),s_2(0.32),s_3(0.1)}	{s_−3(0.12),s_−2(0.36),s_−1(0.14),s_0(0.14),s_1(0.22),s_2(0.02)}
x_2	{s_−3(0.24),s_−2(0.37),s_−1(0.23),s_0(0.18),s_1(0.04),s_2(0.04)}	{s_−3(0.3),s_−2(0.22),s_−1(0.3),s_0(0.12),s_1(0.06)}	{s_−3(0.02),s_−2(0.2),s_−1(0.3),s_0(0.16),s_1(0.22),s_2(0.08),s_3(0.02)}	{s_−3(0.08),s_−2(0.24),s_−1(0.34),s_0(0.14),s_1(0.2)}	{s_−3(0.26),s_−2(0.2),s_−1(0.26),s_0(0.16),s_1(0.1),s_2(0.02)}	{s_−3(0.08),s_−2(0.14),s_−1(0.16),s_0(0.22),s_1(0.25),s_2(0.15)}	{s_−3(0.06),s_−2(0.24),s_−1(0.22),s_0(0.18),s_1(0.2),s_2(0.1)}
x_3	{s_−3(0.32),s_−2(0.31),s_−1(0.18),s_0(0.1),s_1(0.09)}	{s_−3(0.44),s_−2(0.3),s_−1(0.24),s_0(0.02)}	{s_−2(0.12),s_−1(0.08),s_0(0.14),s_1(0.32),s_2(0.26),s_3(0.08)}	{s_−3(0.22),s_−2(0.28),s_−1(0.36),s_0(0.08),s_1(0.06)}	{s_−3(0.18),s_−2(0.16),s_−1(0.3),s_0(0.13),s_1(0.19),s_2(0.04)}	{s_−3(0.1),s_−2(0.2),s_−1(0.2),s_0(0.15),s_1(0.29),s_2(0.06)}	{s_−3(0.06),s_−2(0.36),s_−1(0.26),s_0(0.18),s_1(0.1),s_2(0.02),s_3(0.02)}
x_4	{s_−3(0.04),s_−2(0.08),s_−1(0.18),s_0(0.06),s_1(0.31),s_2(0.31),s_3(0.02)}	{s_−2(0.22),s_−1(0.08),s_0(0.1),s_1(0.44),s_2(0.3),s_3(0.06)}	{s_−2(0.08),s_−1(0.2),s_0(0.2),s_1(0.31),s_2(0.21)}	{s_−2(0.02),s_−1(0.06),s_0(0.06),s_1(0.49),s_2(0.27),s_3(0.1)}	{s_−3(0.02),s_−1(0.06),s_0(0.04),s_1(0.38),s_2(0.31),s_3(0.19)}	{s_−3(0.22),s_−2(0.18),s_−1(0.22),s_0(0.14),s_1(0.22),s_2(0.02)}	{s_−3(0.04),s_−2(0.04),s_−1(0.4),s_0(0.1),s_1(0.26),s_2(0.28),s_3(0.24)}
x_5	{s_−3(0.02),s_−2(0.25),s_−1(0.16),s_0(0.2),s_1(0.28),s_2(0.09)}	{s_−3(0.02),s_−2(0.06),s_−1(0.18),s_0(0.22),s_1(0.44),s_2(0.3),s_3(0.06)}	{s_−1(0.04),s_0(0.17),s_1(0.39),s_2(0.34),s_3(0.06)}	{s_−2(0.06),s_−1(0.15),s_0(0.22),s_1(0.33),s_2(0.2),s_3(0.04)}	{s_−3(0.02),s_−2(0.06),s_−1(0.08),s_0(0.33),s_1(0.26),s_2(0.19),s_3(0.06)}	{s_−3(0.32),s_−2(0.26),s_−1(0.31),s_0(0.03),s_1(0.08)}	{s_−3(0.04),s_−2(0.1),s_−1(0.08),s_0(0.16),s_1(0.24),s_2(0.24),s_3(0.14)}

Table 8

The score values of five response plans.

F(Lcij(p))	c_1	c_2	c_3	c_4	c_5	c_6	c_7
x_1	0.3966	0.4022	0.6437	0.4211	0.3313	0.7135	0.5171
x_2	0.3322	0.3038	0.5297	0.4527	0.3428	0.5486	0.3959
x_3	0.2841	0.2003	0.7498	0.3225	0.4161	0.4917	0.8830
x_4	0.7155	0.9018	0.7051	0.9470	0.9718	0.4004	0.7399
x_5	0.5553	0.6761	0.9476	0.7672	0.7097	0.2853	0.6353

Furthermore, we compare the proposed methods with the consensus model in Chen et al., 2021, Wu and Xu, 2016. As described in Wu and Xu (2016) and Chen et al. (2021), the following steps are implemented.

Step 1: Calculate the consensus degree of evaluation information for three levels. The consensus degree between the collective opinion and individual opinion is defined as , where denotes the distance measure given in Eq. (11).

Step 1.1: For the criterion level: The consensus degree on alternative is defined as , where . By applying the above formula, the results are obtained, as shown in Table 9.

Table 9

The consensus degrees of five experts for each alternative.

CLri	x_1	x_2	x_3	x_4	x_5
e_1	0.7236	0.6886	0.6757	0.6600	0.6812
e_2	0.7433	0.7557	0.4150	0.5550	0.4652
e_3	0.7664	0.7793	0.8038	0.7017	0.6910
e_4	0.7560	0.7376	0.8229	0.7531	0.7519
e_5	0.7905	0.7871	0.7764	0.6548	0.7443

Step 1.2: For the alternative level: The overall degree of consensus on alternative is defined as , where . Then and are obtained for each alternative.

Step 1.3: For the decision matrix level: The overall consensus degree is obtained by and .

Step 2: If the consensus threshold is set to , and the feedback adjustment process is conducted to improve the consensus degree. The feedback mechanism utilizes the identification rules (IRs) to detect which expert’s evaluation should be revised, and then direction rules (DRs) are adopted to describe in which direction the evaluation should be revised.

Step 2.1: IRs are used to determine the evaluation whose consensus degree is less than the predefined threshold.

IRs for alternatives: . Then is obtained.

IRs for experts of identified alternative: . Then is obtained.

IRs for criteria: . According to IRs, the modified elements are .

Step 2.2: DRs: If , then expert does not need to modify evaluation; otherwise, is employed to adjust the expert’s evaluation, where is the adjustment parameter determined by decision makers. After one adjustment iteration, the overall consensus degree is .

Step 3: When the predetermined consensus threshold is reached, the selection process can proceed. For the sake of calculation, the weighted score method is applied directly. Assuming that the weights of experts are equal, the following results are obtained , and . The ranking of response plans is .

If a larger consensus threshold is set, more than one feedback adjustment will be required and the computational process will be more complex, especially for LSGDM problems. In addition, the adjustment parameter value also affects the speed of reaching consensus. In this study, is set to facilitate the consensus. If a smaller value is assigned to , experts will be less willing to change their opinions. Several iterations are needed to satisfy the consensus condition.

To illustrate the superiority of the PL-DeGroot model, a comparison with the consensus model in Zhang et al.’s method (Zhang et al., 2021) is conducted. In Zhang et al. (2021), a SN-based consensus reaching process (CRP) with minimum adjustment distance was performed to achieve consensus in the failure mode and effect analysis (FMEA) problem. In this work, Zhang et al.’s method is employed to fit the case context, and the consensus model with minimum adjustment distance is adopted. After calculation, the decision matrices of experts and are adjusted. Applying the optimization-based consensus model, the overall consensus degree is higher than the predefined threshold, and the ranking indexes are , and . Thus, the ranking of response plans is . Although Zhang et al.’s method (Zhang et al., 2021) could achieve consensus among the decision panel relatively fast, it ignored the weights of experts, which had considerable importance in the decision-making process.

The comparison results of the existing methods are presented in Table 10. It should be noted that the proposed methods generate the identical optimal alternative as the five conventional decision-making methods, illustrating that the proposed method is credible. These main differences between the methods can be explained as follows.

Table 10

Comparison between the existing decision-making methods.

Methods	Ranking results	Consensus decision	Opinion interaction and evolution	The weights of experts
VIKOR method	x_4≻x_1≻x_2≻x_5≻x_3	Not considered	Not considered	Not considered
The consensus model in Wu and Xu (2016)	x_4≻x_5≻x_1≻x_2≻x_3	Similarity-based consensus model	Not considered	Not considered
The consensus model in Zhang et al. (2021)	x_4≻x_5≻x_1≻x_2≻x_3	Optimization-based consensus model	Not considered	Not considered
Algorithm 1	x_4≻x_5≻x_1≻x_2≻x_3	Similarity-based PL-DeGroot opinion dynamics model	Considered	Weight assignment matrix based on similarity degree
Algorithm 2	x_4≻x_5≻x_1≻x_2≻x_3	Similarity-based PL-DeGroot opinion dynamics model	Considered	Weight assignment matrix based on self-persistence degree
Algorithm 3	x_4≻x_5≻x_1≻x_2≻x_3	Similarity-based PL-DeGroot opinion dynamics model	Considered	Weight assignment matrix based on authority degree
Algorithm 4	x_4≻x_5≻x_1≻x_2≻x_3	Based on the trust relationships among the experts	Considered	SNA
The method based on the proposed score function	x_4≻x_5≻x_1≻x_3≻x_2	Not considered	Not considered	Not considered

Consensus decision. The consensus models in Chen et al., 2021, Wu and Xu, 2016, Zhang et al., 2021 are based on the traditional feedback mechanism. Therefore, in each iteration of the CRP, experts whose individual opinions differ significantly from the collective opinion are required to revise their original opinions. An effective strategy of feedback mechanism is to offer adjustment suggestions to individuals. Such methods usually need a moderator to coordinate and modify the evaluation value of experts. Due to the moderator’s limited rationality, the obtained results may have reduced objectivity. Additionally, changes in evaluation are associated with the cost and resources, but most of the methods fail to consider the consensus budget. However, the proposed method based on the PL-DeGroot model includes an automatic adjustment mechanism that originates from the interaction and information exchange between experts, reducing the complexity of the CRP. This mechanism is convenient for emergency decision-making problems with time constraints.

The process of information fusion. The ranking results obtained by the VIKOR method and the method based on the proposed score function differ slightly from the other methods’ results. This may be because all methods, except for the two mentioned methods, use the overall decision matrix but ignore the compensating effect between the criteria in the data collection process. Thus, the negative information is neutralized during the phase of information fusion and cannot be preserved to the end, which results in the loss of information and affects the accuracy of the final decision results. In contrast, in the proposed method, the negative information representing defects is disseminated effectively, leading to a drop in the ranking of , which ensures the accuracy of the emergency decision-making result.

Expert weight vector determination. In most methods, an expert’s weight is determined directly or assigned by a moderator, lacking in both scientific and interpretability properties. However, in this work, an expert-weighting approach is developed where experts’ weights are determined objectively depending on the experts’ expertise. For instance, Algorithm 1 uses the similarity degrees between experts to weigh their importance; the self-persistence and authority degrees are considered in the PL-DeGroot opinion dynamics model; Algorithm 4 applies the trust degrees between experts in the SN to deduce the experts’ weights.

In summary, the comparison analysis results reveal that the proposed decision-making method, which considers practical features, can solve the consensus problem with the probabilistic linguistic information and the opinion dynamics model. The proposed method is especially suitable for emergency decisions where time is of high importance and information is dynamically changing. Moreover, the proposed method illustrates the evolution of decision-makers’ views, considering their changes from conflict to the agreement, thus making the decision result more reasonable. Although many significant achievements have been made in the field of consensus model research, most studies have not highlighted the importance of opinion evolution. Therefore, this study builds a bridge between opinion dynamics and consensus in emergency decision-making, thus facilitating the modeling of individual opinion evolution significantly by incorporating the algorithms developed based on opinion dynamics.

Sensitivity analysis

The consensus condition of the PL-DeGroot model has been discussed in Section 4.2. Therefore, this section focuses on the influence of parameters in Algorithms 2 and 3 on the final decision result.

Parameter sensitivity analysis of Algorithm 2

In Algorithm 2, the self-persistence degrees of five experts are fixed. For simplicity of illustration, the self-persistence of expert is selected to investigate the effect of parameter on the ranking results. When the self-persistence degree of expert changes from 0.1 to 0.9, the ranking results change as shown in Fig. 7. From Fig. 7, has a slight impact on the final ranking results, and is always the best alternative. This demonstrates the high stability of Algorithm 2 at different values of parameter . In view of the practical background, the quarantine policy of is the most rigorous and thus provides the best results, which conforms to the actual situation. In contrast, the images of the ranking values of all response plans never cross with the change of , namely, the ranking of response plans remains steady. Depending on the trend of the image, the gap between alternatives and can become smaller, but it still shows relatively good stability. Particularly, and are similar from the perspective of specific measures to prevent and control the epidemic, which may account for their narrowing gap.

Fig. 7

The influence of different on the comprehensive score .

Next, let is set and the same experiment is performed to further investigate the influence of on the ranking results. The corresponding changes are shown in Fig. 8. Considering that each expert has different levels of self-persistence, experts show different changing trends of parameter . In general, the proposed method has low sensitivity to parameter .

Fig. 8

The influence of different on the comprehensive score .

Parameter sensitivity analysis of Algorithm 3

In Algorithm 3, parameter is introduced to adjust the authority degree of an expert ( is assumed to be an authority figure). For , changes in final comprehensive score values of the response plans are depicted in Fig. 9, where it can be seen that the authority parameter has a negligible effect on the ranking order. The changes in results are almost stable, which proves that Algorithm 3 enjoys high robustness. The gap between response plans and decreases when , which implies that the degree of dominance of over is not remarkable. This is also consistent with the results in other methods.

Fig. 9

The influence of different on the comprehensive score .

Next, each of the remaining experts is set as an authority figure in turn and the same experiment is performed. The results indicate similar conclusions, as shown in Fig. 10. Although there is a slight fluctuation in , as shown in Fig. 10(b), it does not affect the final ranking results and obeys a steady trend. Therefore, the proposed method can be considered reliable and reasonable in terms of the comparative and sensitivity analyses that are performed in this study.

Fig. 10

The influence of different authority figures on the comprehensive score .

Experimental assessment

To further illustrate the effectiveness of the proposed method, the following evaluation indexes are adopted.

The number of iterations (), this parameter indicates the average number of iterations needed to satisfy the consensus reaching condition. The lower the number of iterations is, the faster the convergence speed to reach a consensus is. As described in Section 5.3.1, Algorithm 4 is the most efficient method among the proposed algorithms in terms of reaching the consensus.

The adjusted degree (), this parameter represents the average adjusted degree between the initial opinion and the final opinion meeting the reached- consensus condition. The smaller the adjusted degree is, the lower the loss of opinion is, and the more information is preserved. As presented in Table 11, the proposed algorithms have similar capability to preserve the initial information, but Algorithms 1 and 4 may have a slight advantage. The adjusted degree is calculated by where is presented in Eq. (26).

Table 11

The adjustment degree of experts for different algorithms.

AD_r	e_1	e_2	e_3	e_4	e_5	Average value
Algorithm 1	0.0938	0.0621	0.0199	0.0298	0.0730	0.0557
Algorithm 2	0.0809	0.0750	0.0271	0.0209	0.0859	0.0579
Algorithm 3	0.0563	0.0996	0.0400	0.0210	0.1105	0.0655
Algorithm 4	0.0809	0.0750	0.0197	0.0169	0.0859	0.0557

From the above comparison analyses, sensitivity analyses, and experimental analyses, it can be found that the proposed PL-DeGroot model with different algorithms is reasonable and effective and can solve practical emergency decision-making problems.

Conclusion

The PLTS is a powerful representation for capturing complex and uncertain information of decision-makers in the linguistic context. However, the inherent internal structure of the PLTS makes it challenging to portray changes in the probabilistic linguistic information. In this study, the PL-DeGroot opinion dynamics model is proposed to address the dynamic evolution problem of probabilistic linguistic information. First, operations based on the transformation function and expectation function for PLTSs are introduced, which lays a solid foundation for the application and extension of probabilistic linguistic information. Second, the DeGroot model based on PLTSs is presented. This model considers three parameters: similarity degree, self-persistence, and authority degree. In addition, this model not only can predict the tendency of opinions but can also improve decision-making efficiency. Moreover, considering the relationships between experts, the SN and incomplete trust relationships between experts are integrated into the PL-DeGroot model. Finally, the proposed algorithms are adopted to select the optimal response plan for COVID-19. Compared with the classical static decision-making methods, the proposed method is more adaptive in an emergency decision environment and ensures the consensus of experts.

Compared with traditional consensus models, the PL-DeGroot opinion dynamics model without a feedback mechanism modifies experts’ opinions in a self-adjusting manner, thus saving considerable effort and cost devoted to the CRP, which is appropriate for solving complex emergency decision-making problems.

The proposed method considers the importance of experts and adopts different opinion fusion mechanisms. Regarding the assignment of experts’ weights, the opinion evolution process is performed to obtain the importance of the experts and apply it in the opinion fusion process to obtain the final ranking values of the alternatives. The comparison and sensitivity analyses validate the effectiveness and feasibility of the proposed model for solving emergency decision-making problems.

In the decision-making problem, relationships between experts are not reflected only in the evaluation information but also in their SN. Therefore, it is significant to consider the SN in the CRP. Algorithm 4 considers the SN when constructing the influence matrix, making the best use of decision information.

Considering the evaluation evolution process, an information propagation method is proposed to estimate the missing values in the SN between experts and then these values are used in the DeGroot model, allowing uncertainty and incomplete information in a decision environment.

To the best of the authors’ knowledge, the proposed method is the first emergency decision-making method with probabilistic linguistic information that considers an incomplete SN between experts and opinion evolution problems. The results obtained in the case study of selecting optimal COVID-19 response plans can serve as a guide for developing similar methods.

For clarifying the proposed method, the data utilized in the case of selecting COVID-19 response plans serve as an example of the proposed framework. In future work, real data can be used in the proposed method to examine its performance further. In addition, the DeGroot model could be applied to different types of fuzzy environments, such as a hesitant fuzzy set and a probabilistic hesitant fuzzy set. Furthermore, dynamic and complex SNs can be considered and the corresponding improvements can be made in the proposed method to be applicable to many different situations.

CRediT authorship contribution statement

Yuanyuan Liu: Methodology, Writing – original draft, Software. Youlong Yang: Conceptualization, Data curation, Writing – review & editing.

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.