Introduction

Background

In the global scope, landslide disasters seriously threaten the safety of human life and property because of their sudden, multiple, mass and other characteristics [1]. Among them, China is one of the countries with the most frequent landslide disasters in the world. China has a vast territory, numerous mountains and plateaus, complex geological structures, and serious slope geological disasters. Whether it is the collapse and sliding of natural slopes or the slope instability caused by human industrial and agricultural activities, it will bring huge losses to the safety of people’s lives and property. According to the statistical results of the geological disasters report issued by the China Geological Survey from 2010 to 2020, there were 129,000 geological disasters in China in the past 11 years, including 92,000 landslide disasters, accounting for 71.3% of the total disasters, which accounted for the largest proportion of various geological disasters. At the same time, in the past decade, geological disasters in China have caused nearly 50 billion yuan of direct economic losses and a large number of casualties [2, 3]. With the need of economic development, more and more artificial slopes have been formed in people’s production and life, followed by the deformation and destruction of artificial slopes. For example, the open pit mining of Fushun Coal Mine and Daye Iron Mine have all had slope instability accidents. The huge disasters caused by landslides should be paid enough attention to by human beings.

The disaster caused by landslide is huge. To prevent the spread of the disaster to the maximum extent, it is particularly important to carry out landslide disaster control work. The effect of landslide treatment is closely related to the selected treatment alternatives. In the decision-making of landslide disaster treatment alternatives, there are usually multiple evaluation factors and multiple alternatives. In landslide disaster, experts come from different departments and the numbers may be exceed 20. Among the complex indicators to be considered, the accurate values of these indicators cannot be provided. In addition, different experts have preferences for different alternatives, so the selection of landslide disaster treatment alternatives is an LSGDM problem.

Literature review and motivation

Because many practical problems are uncertain and complex, the application of uncertainty theory in LSGDM is more and more extensive, such as rough sets (RS), and so on [4,5,6,7]. Because experts may come from different professional fields, their evaluation opinions may have great conflicting. Since LSGDM problem composed of more than 20 experts has become a research hotpot [8], among which consensus reaching process (CRP), clustering analysis of experts and aggregation method of LSGDM information have become three hot research directions. The purpose of CRP is to promote experts to quickly reach an agreement on the evaluation of LSGDM issues [9,10,11,12,13]. At present, there are many CRP models. Many clustering methods are applied to the process of expert clustering, which can save the iteration time of CRP [14,15,16].

However, these researches on LSGDM only consider the participation of experts in decision-making, and do not consider the participation of the public. In the era of rapid growth of social media, the public can express their feelings and preferences on decision-making problems. Mining and analyzing the public social media big data can provide important basis for decision-making behavior of the public events, such as emergency decision-making, environmental protection, etc. [17,18,19]. Therefore, we conduct sentiment analysis (SA) on the public’s evaluation opinions and participate in decision-making as another decision-making subject. In LSGDM issues involving the participation of experts and the public, it is necessary to consider how to obtain and handle the public opinions. Applying social media data in decision-making has become an important research direction [20,21,22].

Yao pointed out that positive, negative and boundary regions correspond to three-way decision-making behaviors, namely, acceptance, rejection and non commitment [23, 24]. At present, TWD has been widely used to solve multi-attribute decision making (MADM) problems [5, 6, 25,26,27]. TWD can be used as a granular computing method, can also provide a reasonable explanation for classification and ranking results, and can provide an effective tool for solving uncertain MADM problems [28, 29]. The three-way MADM method objectively divides all objects into three domains. However, in some cases, it is necessary to consider not only the classification of objects, but also the final ranking results of objects. In recent years, three-way MADM method for ranking candidate objects has received extensive attention [5, 30,31,32]. We mainly focus on using TWD method to solve the LSGDM problems, including the acquisition methods of conditional probability and loss function. At present, in some TWD models, the loss functions are mainly based on the experience of experts [26, 31,32,33]. The loss function obtained in this way has a strong subjectivity, to make the loss function more objective, according to the research [24, 34]. Jia et al. [5] proposed a method for calculating the relative loss function. As is known to all, the evaluation value expressed by fuzzy number can only describe the membership degree of the alternative to a certain attribute, but can not describe the degree of hesitation. Xu [35] proposed IFN. Based on Xu’s research results, Liu et al. [6] put forward a method for calculating the relative loss function obtained by IFNs. Furthermore, it is very important to study the method of obtaining conditional probability. Jia et al. [5] assume that conditional probabilities of different alternatives are the same. In actual LSGDM problems, different alternatives should have different conditional probabilities. Nevertheless, in the LSGDM problem, it is difficult to obtain an objective conditional probability. Therefore, some scholars have proposed solutions from different perspectives [6, 26, 30, 31, 36].

The motivations of the current work are as follows:

  1. (1)

    The selection of landslide disaster treatment alternatives is a LSGDM problem. In LSGDM, a large number of experts participate in decision-making, and each expert may have different domain knowledge. Hence, evaluation opinions of experts are highly uncertain. Experts can easily express the evaluation opinions of each alternative under each attribute through fuzzy numbers, and the public’s evaluation opinions can also get the corresponding fuzzy numbers through sentiment analysis. However, fuzzy number can only describe the membership degree. IFS expresses the evaluation opinions on alternatives through membership degree and non-membership degree. It can describe fuzziness and uncertainty in the evaluation opinions. So that, we propose a method to automatically convert fuzzy numbers into IFNs in LSGDM, which can make good use of the advantages of IFS in dealing with uncertainty problems to solve LSGDM problems.

  2. (2)

    On LSGDM, some experts’ opinions may differ greatly from collective opinions. However, they are not willing to revise their evaluation opinions. To reach a consensus on opinions of experts as soon as possible, it is necessary to manage the non-cooperative behaviors of experts reasonably and effectively. Therefore, we adopt intuitionistic fuzzy C-means to cluster experts, which can improve the efficiency of CRP. Then, aggregate evaluation information of expert subgroups, identify the most uncooperative expert subgroup, propose modification suggestions to the subgroup, and give reasonable weight punishment to the most uncooperative expert subgroup. Thus, the CRP of expert evaluation opinions can be carried out quickly and effectively.

  3. (3)

    On many practical LSGDM problems, the public participation can make decisions more reasonable and effective. Therefore, we not only considered the experts’ evaluation of LSGDM event, but also considered the public’s participation in the event, gave the method for obtaining and handling the public evaluation information, and gave the aggregation method of the public evaluation information and experts’ evaluation information.

  4. (4)

    TWDs can reduce mistakes in decision-making and make decision-making more reasonable. It can not only give ranking results of alternatives, but also give classification results of alternatives. We propose a method based on TWD to classify and rank alternatives. This method is applied to the selection of landslide treatment alternatives.

Contributions and structure

The main contributions of this paper are as follows:

  1. (1)

    We use the fuzzy neighborhood operator proposed in Refs. [4, 37] to convert evaluation values represented by fuzzy numbers into IFNs, and obtain the intuitionistic fuzzy evaluation decision matrix described by IFNs. Therefore, we can use the advantages of IFS to solve the LSGDM problem.

  2. (2)

    Taking into account the participation of experts and the public in the evaluation of LSGDM events, respectively, the method of obtaining and handling the public evaluation information is given, and the aggregation method of the public evaluation information and expert evaluation information is given. It expands the number and types of decision makers in LSGDM, and provides a new idea for the study of decision makers in LSGDM.

  3. (3)

    We propose a novel method based on TWD to classify and rank alternatives, and give the corresponding algorithm description. The method is applied to the decision making of landslide treatment alternatives selection. In this process, we first cluster the experts, and then consider the identification and management of non-cooperative behavior. It supplies a new idea for classification and ranking of alternatives in LSGDM.

  4. (4)

    An effective LSGDM method based on TWD and IFS (IF-TW-LSGDM) is provided to study the selection of alternatives for landslide treatment.

The rest of this paper is arranged as follows: in “Preliminaries” section, some basic concepts of SA, LSGDM, fuzzy neighborhood operators (FNOs) and IFS are introduced. In “Evaluation aggregation in LSGDM with the participation of the public and experts” section, an evaluation information aggregation method in LSGDM with the participation of the public and experts is proposed. In “IFNs-based TWD” section, a new TWD model based on IFS is proposed, which classifies alternatives into positive, boundary and negative domains, and ranks them. Then, calculation steps and algorithm description are given. In “An illustrative example” section, IF-TW-LSGDM model is established for landslide treatment alternatives selection. The model is applied to a specific example. In “Comparisons and analyses” section, effectiveness and feasibility of IF-TW-LSGDM method are verified, and the influence of parameter on ranking and classification results is analyzed. In “Conclusions” section, main work and contributions are summarized, and future research directions are proposed.

Preliminaries

Some basic concepts of SA, LSGDM, IFS and FNOs are introduced in this section.

LSGDM

Normally, LSGDM problems are described as follows: attribute set \(C=\left\{ c_1,c_2,\ldots ,c_n\right\} \) with weight vector \(w=\left( w_1,w_2,\ldots ,w_n\right) \) satisfy \(w_j\in \left[ 0,1\right] \) and \(\sum _{j=1}^nw_j=1\), alternative set \(Y=\left\{ y_1,y_2,\ldots ,y_m\right\} \), expert set \(P=\left\{ p_1,p_2,\ldots ,p_s\right\} \left( s\ge 20\right) \). An expert \(p_k\left( k=1,2,\ldots ,s\right) \) builds a decision table as shown in Table 1. In Table 1, \(e_{ij}^k\) represents the preference level given by the kth expert \(p_k\) for alternative \(y_i\) w.r.t. attribute \(c_j\). Here, \(j=1,2,\ldots ,n;i=1,2,\ldots ,m\).

Table 1 Decision table of expert \(p_k\)
Full size table

By gathering the evaluation opinions of experts, we obtained decision matrix \(E=\left( E_1,E_2,\ldots ,E_s\right) \) to represent all decision evaluation opinions of experts in LSGDM problems. The decision matrix of experts is shown in Table 2.

Table 2 Decision matrix of experts
Full size table

Covering-based FRS model

Ye et al. [4] improved the existing FNOs and put forward a reflexive fuzzy \(\beta \)-neighborhood operator in fuzzy \(\beta \)-covering approximation space (F\(\beta \)CAS).

Definition 1

[4, 38] Let \(E_k\) be an expert’s decision table, for \(\forall y_i,y_l\in Y\), the improved FNO \({\mathcal {N}}\) is defined as follows:

$$\begin{aligned} {\mathcal {N}}\left( y_i\right) \left( y_l\right) =\bigwedge _{c_j\in C} {\mathcal {I}}\left( y_{ij},y_{lj}\right) , \end{aligned}$$
(1)

where \({\mathcal {I}}\) is a R-implicator: \(\forall x,y\in \left[ 0,1\right] \), \({\mathcal {I}}_{{\mathcal {T}}}(x,y)=\textrm{sup}\{z\in \left[ 0,1\right] \mid {\mathcal {T}}\left( x,y\right) \le y\}\), \({\mathcal {T}}\) is a t-norm.

Definition 2

[4, 37] Let \(E_k\) be an expert’s decision table. \({\mathcal {I}}\) is a R-implicator, \({\mathcal {T}}\) is a t-norm. \(P\left( Y\right) \) is expressed as the family of all fuzzy subsets of Y. Through improved FNO \({\mathcal {N}}\), \(\forall y_i,y_l\in Y\) and \(A\in P\left( Y\right) \), the lower and upper fuzzy rough approximation operators of A are defined as

$$\begin{aligned}{} & {} {\underline{N}}\left( A\right) (y_i)=\textrm{inf}_{y_l\in Y}{\mathcal {I}}\left( {\mathcal {N}}\left( y_i\right) \left( y_l\right) ,A\left( y_{l}\right) \right) , \end{aligned}$$
(2)
$$\begin{aligned}{} & {} {\overline{N}}\left( A\right) \left( y_i\right) =\textrm{sup}_{y_l\in Y}{\mathcal {T}}\left( {\mathcal {N}}\left( y_i\right) \left( y_l\right) ,A\left( y_{l}\right) \right) . \end{aligned}$$
(3)

IFSs

Definition 3

[39] An IFS T on a finite universe U can be represented as follows:

$$\begin{aligned} T=\left\{ \langle y,\mu _T\left( y\right) , \nu _T\left( y\right) \rangle \mid y\in U\right\} . \end{aligned}$$
(4)

where \(\mu _T\left( y\right) \in \left[ 0,1\right] \) is called membership degree, and \(\nu _T\left( y\right) \in \left[ 0,1\right] \) is called non-membership degree, and \(0\le \mu _T\left( y\right) +\nu _T\left( y\right) \le 1\), for all \(y\in U\). The indeterminacy degree of y to T is expressed as \(\pi _T\left( y\right) =1-\mu _T\left( y\right) -\nu _T\left( y\right) \).

Definition 4

[35] Suppose \(T_1=\left( \mu _1,\nu _1\right) \) and \(T_2=\left( \mu _2,\nu _2\right) \) be two IFNs, \(k>0\), then operations of \(T_1\) and \(T_2\) are defined as:

(1) \(T_1\oplus T_2=\left( \mu _1+\mu _2-\mu _1\mu _2, \nu _1\nu _2\right) \);

(2) \(T_1\otimes T_2=\left( \mu _1\mu _2, \nu _1+\nu _2-\nu _1\nu _2\right) \);

(3) \(kT_1=\left( 1-\left( 1-\mu _1\right) ^k,\nu _1^k\right) \);

Definition 5

[40] Let \(T_1=\left( \mu _1,\nu _1\right) \) and \(T_2=\left( \mu _2,\nu _2\right) \) be two IFNs, Euclidean distance between \(T_1\) and \(T_2\) is defined as follows:

$$\begin{aligned} d\left( T_1,T_2\right) =\sqrt{\frac{1}{2}\left( \left( \mu _1-\mu _2\right) ^2+\left( \nu _1-\nu _2\right) ^2 +\left( \pi _1-\pi _2\right) ^2\right) }, \end{aligned}$$
(5)

where \(\pi _1 =1-\mu _1-\nu _1\) and \(\pi _2 =1-\mu _2-\nu _2\) are the degrees of hesitancy of \(T_1=\left( \mu _1,\nu _1\right) \) and \(T_2=\left( \mu _2,\nu _2\right) \), respectively.

Suppose \(\Omega =\left\{ T_i\right\} \) is a set of IFNs, where \(T_i=\left( \mu _i,\nu _i\right) \), the weight vector of \(T_i\) is \(w=\left( w_1,\ldots ,w_n\right) \), where \(\sum _{i=1}^n w_i=1\) and \(0\le w_i\le 1\), \(i=1,2,\ldots ,n\), Xu [35] proposed the intuitionistic fuzzy weighted averaging (IFWA) operator as

$$\begin{aligned} IFWA\left( T_1,T_2,\ldots ,T_n\right) =\left( 1-\prod _{i=1}^n\left( 1-\mu _i\right) ^{w_i},\prod _{i=1}^n \nu _i^{w_i}\right) . \end{aligned}$$
(6)

Sentiment analysis (SA)

Sentiment analysis refers to extracting people’s feelings or opinions from written words. At present, SA methods mainly include methods based on sentiment lexicon, methods based on machine learning and methods based on deep learning [41, 42]. Generally speaking, sentiments can be classified into positive, negative and neutral sentiments [43].

SA based on the sentiment lexicon only needs to match pre processed words with words in the sentiment lexicon, and then calculate sentiment score and judge sentiment polarity according to the word matching degree. This method is simple in calculation and does not require additional resources. This paper uses this method to analyze the sentiment of the public opinion. The core of SA based on sentiment lexicon is the construction of sentiment lexicon. An sentimental lexicon is a collection of nouns, verbs, adjectives, adverbs and other words with fixed sentimental tendencies, such as the words "happy" and "hero" with positive sentiments, and the words "sad" and "traitor" with negative sentimental tendencies. At present, Chinese sentimental lexicons that are widely used include HowNet, NTUSD of Taiwan University and Dalian University of Technology’s sentiment lexicon database. Main steps of SA are as follows [44].

(1) Data collection: Obtain various forms of information related to the target affairs released by the public. For example: posts and texts, etc.

(2) Data preprocessing: Preprocess text data, including word segmentation, word deletion and part-of-speech tagging. This study uses Jieba Chinese word segmentation database for word segmentation. Then, part-of-speech categories are assigned to tokens.

(3) Edit an sentiment lexicon: Sentimental words are marked to represent positive, hesitant and negative sentiment orientations.

(4) Determine sentiment orientation: Calculate the sentiment score of text to determine polarity of each alternative in the sentence.

Evaluation aggregation in LSGDM with the participation of the public and experts

The public is the experiencer of the target affairs, and usually they can express more valuable evaluation information about the target affairs. With the continuous development of information technology, the public has gradually become an important part of decision-making bodies. As we know in “LSGDM” section, \(E=\left( E_1,E_2,\ldots ,E_s\right) \) be the decision matrix given by experts. Here, we suppose \(E_T=\left( t_{ij}\right) _{m\times n}\) be the public decision matrix, where \(t_{ij}\) is obtained by SA.

Generation of the public decision evaluation information

To obtain the public decision data, the final word segmentation results are first obtained from the Chinese word segmentation database of Jieba, and the part of speech tagging is performed. Stop words were then deleted because they could not express any opinion.

Then, the term frequency-inverse dense frequency method (TF-IDF) [45] is applied to form a common feature word set. Furthermore, on the basis of Dalian University of Technology’s sentiment lexicon database, a sentiment lexicon dictionary is established. Next, analyze whether each sentimental word is in the sentimental lexicon. If yes, judge whether there are adverbs and negative words to modify the sentimental word; If yes, process the sentimental word. An sentimental word may have different sentimental polarities (negative or positive), which are commendatory in some sentences and derogatory in others. To judge the polarity of the word more accurately, we use the KNN algorithm [46] to calculate: the polarity of sentiment word A is determined by the sum of the polarity values of the four sentiment words in the front and the back. Distance between the two polarity values of A and the sum is calculated, respectively, and the polarity of A is represented by the smaller one. By averaging the polarity of all sentimental words, SA value \(T_s\) of the whole text can be obtained. \(-1\le t\le 1\). Through the above method, we can get SA value \(T_{s_{ij}}\) of each alternative with respect to each attribute.‘ Then, the initial public decision matrix is obtained as \(E_{T_s}=\left( T_{s_{ij}}\right) _{m\times n}\), where \(T_{s_{ij}}\in \left[ -1,1\right] \). The evaluation value \(e_{ij}^k\) of the decision matrix given by experts is between 0 and 1. To be consistent with the evaluation value standard of experts, we need to standardize the value of the initial public decision matrix as follows:

$$\begin{aligned} t_{ij}=\frac{T_{s_{ij}}-\textrm{min}_iT_{s_{ij}}}{\textrm{max}_iT_{s_{ij}}-\textrm{min}_iT_{s_{ij}}}. \end{aligned}$$
(7)

Convert LSGDM to IF-LSGDM

As is known to all, the evaluation value expressed by fuzzy number can only describe the membership degree of the alternative to a certain attribute, but cannot describe the degree of hesitation. Xu [35] proposed IFN. To describe fuzzy data better, we convert the evaluation values of experts decision matrices and the public decision matrix into IFNs.

Based on the idea of Çoker [47], fuzzy information system is changed to intuitionistic fuzzy information system by using fuzzy rough set (FRS) model. There are likely operations in References [4, 48]. For the decision matrix of expert \(p_k\), an attribute fuzzy set \(C_j^k\) can be constituted of fuzzy evaluation values of all objects under attribute \(c_j^k\), namely, \(C_j^k=\frac{e_{1j}^k}{y_1}+\frac{e_{2j}^k}{y_2}+\cdots +\frac{e_{mj}^k}{y_m}\). According to the Reference [49], FNOs can effectively deal with fuzzy continuous data, Ye [4] uses the FRS model given in “Covering-based FRS model” section to compute lower and upper approximations of \(C_j^k\). Therefore, IFS can be obtained as \(\widetilde{C_j^k}=\left\{ \left<y_i,{\underline{N}}\left( C_j\right) \left( y_i\right) ,1-{\overline{N}}\left( C_j\right) \left( y_i\right) \right>\mid y_i\in Y\right\} \).

For brevity, we convert the fuzzy evaluation value \(e_{ij}^k\) to an intuitionistic fuzzy evaluation value \(\widetilde{e_{ij}^k}\), which is represented by \(\widetilde{e_{ij}^k}=\left( \mu _{e_{ij}^k},\nu _{e_{ij}^k}\right) \). Next, let \(\mu _{e_{ij}^k}={\underline{N}}\left( C_j\right) \left( y_i\right) \) and \(\nu _{e_{ij}^k}=1-{\overline{N}}\left( C_j\right) \left( y_i\right) \). Therefore, we can obtain an intuitionistic fuzzy information system. In the same way, we will convert the decision matrices obtained by experts and the public into decision matrices expressed by IFNs, that is, \(E_k=\left( \widetilde{e_{ij}^k}=\left( \mu _{e_{ij}^k},\nu _{e_{ij}^k}\right) \right) _{m\times n}\), \(k=1,2,\ldots ,s\) and \(E_T=\left( \widetilde{t_{ij}}=\left( \mu _{t_{ij}},\nu _{t_{ij}}\right) \right) _{m\times n}\) Therefore, an LSGDM problem becomes an IF-LSGDM problem.

CRP based on IFSs

For LSGDM, the main goal of CRP is to make the evaluation information of experts have certain consistency, thereby reduce conflicts. In general, opinions of experts are somewhat different at the beginning. Hence, CRP is an iterative process that gradually reduces the non-cooperative behavior among experts. For non-cooperative behavior, Xu [50] proposed two solutions: (1) Opinion adjustment. Adjust the opinions of experts to make them closer to the collective opinions; (2) Weight adjustment. Reduce the influence of uncooperative experts to reach greater consensus. Wu et al. [51] put forward a method to identify and manage non-cooperative behavior in IF-LSGDM. We will use this approach to achieve consensus reaching among experts in this section.

According to the description of LSGDM problem in “LSGDM” section, operation steps of IF-CRP are as follows.

Step 1. Standardize the initial decision matrix \(E=\left( E_1,E_2,\ldots ,E_s\right) \) to obtain normalized matrix \({\dot{E}}=\left( \dot{E_1},\dot{E_2},\right. \left. \ldots ,\dot{E_s}\right) \),

$$\begin{aligned} {\dot{e}}_{ij}^k=\left( {\dot{\mu }}_{e_{ij}^k},{\dot{\nu }}_{e_{ij}^k}\right) =\left\{ \begin{aligned}&\left( \mu _{e_{ij}^k},\nu _{e_{ij}^k}\right) , \ \text{ benefit } \text{ criterion }\\&\left( \nu _{e_{ij}^k},\mu _{e_{ij}^k}\right) , \ \text{ cost } \text{ criterion } \end{aligned} \right. \end{aligned}$$
(8)

where \(k=1,2,\ldots ,s\).

In the same way, we can get the normalized public decision matrix \({\dot{E}}_T=\left( {\dot{t}}_{ij}=\left( {\dot{\mu }}_{t_{ij}},{\dot{\nu }}_{t_{ij}}\right) \right) _{m\times n}\).

Step 2. Cluster decision makers. Clustering analysis based on experts’ evaluation opinions is the basis of LSGDM. In this paper, experts are classified into \(L(1\le L\le s)\) clusters using the intuitionistic fuzzy C-means proposed by Xu [52]. Suppose that weight vector of each cluster is denoted as \(\lambda =\left( \lambda _1,\lambda _2,\ldots ,\lambda _L\right) \), \(\lambda _l=\frac{\left( n_l\right) ^2}{\sum _{l=1}^L\left( n_l\right) ^2}\), where \(n_l\) is the number of experts in the lth cluster \(g_l\). Obviously, \(\lambda _l \in \left[ 0,1\right] \) and \(\sum _{l=1}^L\lambda _l=1\).

The cluster decision matrix \({\dot{E}}_l=\left( {\dot{e}}_{ij}^l\right) _{m\times n}\) can be obtained as \({\dot{e}}_{ij}^l= \left( {\dot{\mu }}_{e_{ij}^l},{\dot{\nu }}_{e_{ij}^l}\right) = \left( 1-\prod _{\varrho =1}^{n_l}\left( 1-{\dot{\mu }}_{e_{ij}^{l_\varrho }}\right) ^{\frac{1}{n_l}},\right. \left. \prod _{\varrho =1}^{n_l} \left( {\dot{\nu }}_{e_{ij}^{l_\varrho }}\right) ^{\frac{1}{n_l}}\right) \), where \(\frac{1}{n_l}\) is weight of experts in the cluster \(g_l\). Collect decision matrix is \({\ddot{E}}=\left( \ddot{e}_{ij}\right) _{m\times n}\), where \(\ddot{e}_{ij}=\left( \ddot{\mu }_{ij},\ddot{\nu }_{ij}\right) = \left( 1-\prod _{l=1}^L\left( 1-{\dot{\mu }}_{e_{ij}^l}\right) ^{\lambda _l},\prod _{l=1}^L \left( {\dot{\nu }}_{e_{ij}^l}\right) ^{\lambda _l}\right) \).

Step 3. Consensus measures. Individual consensus level is calculated as \(CI_l^t=1-d\left( {\dot{E}}_l^t,{\ddot{E}}^t\right) =1-\frac{1}{m\times n}\sum _{i=1}^m\sum _{j=1}^nd\left( {\dot{e}}_{ij}^{l,t},\ddot{e}_{ij}^t\right) \), where \(d\left( {\dot{e}}_{ij}^{l,t},\ddot{e}_{ij}^t\right) \) is Euclidean distance of two IFNs, \(t\ge 0\) is the tth iteration of CRP. Collect consensus level can be obtained as \(GCI^t=\frac{1}{L}\sum _{l=1}^L CI_l^t\). Obviously, \(GCI^t\in \left[ 0,1\right] \). If the level of collective consensus is low, CRP will be used to manage the differences among experts. Otherwise, proceed to the next selection process.

Step 4. Manage non-cooperative behaviors. Usually, some subgroups are unwilling to adjust their initial evaluation opinions. Managing these non-cooperative behavior is an important and difficult task for CRP. We use Wu et al.’s [51] method to identify and manage non-cooperative behavior. First, identify non-cooperative behavior of L subgroups, get collective consensus level GCI and the least cooperative subgroup \(g_*\). Then \(g_*\) is punished with weight to obtain a new weight vector \(\lambda ^*=\left( \lambda _1^*,\lambda _2^*,\ldots ,\lambda _L^*\right) \), and the decision matrix of cluster \(g_*\) is recalculated to obtain the new cluster decision matrix \({\dot{E}}_l^*=\left( {\dot{e}}_{ij}^{l,*}\right) _{m\times n}\) and the new collect decision matrix \({\ddot{E}}^*=\left( \ddot{e}_{ij}^*\right) _{m\times n}\) of each expert subgroup after reaching a consensus.

Information fusion

In practical application, some events should fully consider the opinions of the public, while some highly professional events need to emphasize the opinions of expert groups. Therefore, the final decision matrix \(X=\left( \chi _{ij}\right) _{m\times n}\) can be calculated as follows:

$$\begin{aligned} \chi _{ij}{} & {} =W_{et}\ddot{e}_{ij}^*\oplus \left( 1-W_{et}\right) {\dot{t}}_{ij}\nonumber \\{} & {} =\left( 1-\left( 1-\ddot{\mu }_{e_{ij}^*}\right) ^{W_{et}}\left( 1-{\dot{\mu }}_{t_{ij}}\right) ^{1-W_{et}}, \right. \nonumber \\{} & {} \quad \left. \left( \ddot{\nu }_{e_{ij}^*}\right) ^{W_{et}}\left( {\dot{\nu }}_{t_{ij}}\right) ^{1-W_{et}}\right) , \end{aligned}$$
(9)

where \(W_{et}\in \left[ 0,1\right] \) is adjustment parameter between expert groups decision evaluation and the public decision evaluation. In particular, when \(W_{et}=1\), the method degenerates into an LSGDM model that only considers the evaluation opinions of expert groups. When \(W_{et}=0\), the method degenerates into a decision-making model that only considers the public evaluation opinions.

IFNs-based TWD

In this section, a new TWD model on the basis of IFS is proposed, which classifies objects into positive, boundary and negative domains, and ranks them.

The relative loss functions based on IFNs

Loss function in TWD theory consists of three actions and two states. These three actions and two states are denoted as \({\mathbb {B}}=\{b_P,b_B,b_N\}\) and \(\Gamma =\Gamma \left( D,\lnot D\right) \). Among them, \(b_P\), \(b_B\) and \(b_N\) represent three actions to classify the object y, in other words, respectively, determine \(y\in POS\left( D\right) \), \(y\in BND\left( D\right) \) and \(y\in NEG\left( D\right) \). The initial loss functions proposed by Yao [24] is shown in Table 3.

Table 3 Initial loss functions [24]
Full size table
Table 4 Relative loss functions in intuitionistic fuzzy environment [6]
Full size table

Through studies of references [6], the form of relative loss functions on the basis of IFNs is shown in Table 4, \(D_j\) and \(\lnot D_j\) represent two states related to attribute \(c_j\). If the object \(y_i\) is in \(D_j\), then \(y_i\) has the attribute \(c_j\). On the contrary, \(y_i\) does not have the attribute \(c_j\). \(\chi _{ij}=\left( \mu _{\chi _{ij}},\nu _{\chi _{ij}}\right) \) is the evaluation value of the final decision matrix, and \(\chi _{ij}^\prime \) is the complement of \(\chi _{ij}\). According to the research of Li et al. [34], the parameter \(\rho \left( \rho \in \left[ 0,1\right] \right) \) is applied to calculate the loss of taking \(b_N\).

For \(\forall y_i\in Y\), \(i=1,2,\ldots m\), the expected losses of taking actions of \(b_\blacklozenge \left( \blacklozenge =P,B,N\right) \) are expressed as follows:

$$\begin{aligned}{} & {} R\left( b_P\mid \left[ y_i\right] \right) =l_{PP}P_r\left( D\mid \left[ y_i\right] \right) + l_{PN}P_r\left( \lnot D\mid \left[ y_i\right] \right) \end{aligned}$$
(10)
$$\begin{aligned}{} & {} R\left( b_B\mid \left[ y_i\right] \right) =l_{BP}P_r\left( D\mid \left[ y_i\right] \right) + l_{BN}P_r\left( \lnot D\mid \left[ y_i\right] \right) \end{aligned}$$
(11)
$$\begin{aligned}{} & {} R\left( b_N\mid \left[ y_i\right] \right) =l_{NP}P_r\left( D\mid \left[ y_i\right] \right) + l_{NN}P_r\left( \lnot D\mid \left[ y_i\right] \right) , \end{aligned}$$
(12)

where \(P_r\left( D\mid \left[ y_i\right] \right) \) is the conditional probability of an alternative \(y_i\) belonging to D, \(P_r\left( \lnot D\mid \left[ y_i\right] \right) =1-P_r\left( D\mid \left[ y_i\right] \right) \) is the conditional probability of an alternative \(y_i\) belonging to \(\lnot D\).

According to Table 4, we can obtain expected loss as follows [6]:

$$\begin{aligned} R\left( b_P\mid \left[ y_i\right] \right)= & {} \left( 1-\left( 1-\nu _{\chi _{ij}}\right) ^{1-P_r}, \mu _{\chi _{ij}}^{1-P_r}\right) \end{aligned}$$
(13)
$$\begin{aligned} R\left( b_B\mid \left[ y_i\right] \right)= & {} \left( 1-\left( 1-\mu _{\chi _{ij}}\right) ^{\rho \times P_r}\left( 1-\nu _{\chi _{ij}}\right) ^{\rho \times \left( 1-P_r\right) }, \right. \nonumber \\{} & {} \left. \nu _{\chi _{ij}}^{\rho \times P_r}\mu _{\chi _{ij}}^{\rho \times \left( 1-P_r\right) }\right) \end{aligned}$$
(14)
$$\begin{aligned} R\left( b_N\mid \left[ y_i\right] \right)= & {} \left( 1-\left( 1-\mu _{\chi _{ij}}\right) ^{P_r}, \nu _{\chi _{ij}}^{P_r}\right) , \end{aligned}$$
(15)

where \(P_r\) is the abbreviation of \(P_r\left( D\mid \left[ y_i\right] \right) \).

According to Bayesian decision procedure, best rule is minimum cost decision rule [24]. Liu et al. [6] gave three different decision rules based on IFSs, namely, positive, negative and comprehensive viewpoint. We use the third decision rule based on comprehensive viewpoint. The decision rules (P), (B) and (N) can be given as follows:

(P) If \(I\left( R\left( b_P\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_B\mid \left[ y_i\right] \right) \right) \) and \(I\left( R\left( b_P\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_N\mid \left[ y_i\right] \right) \right) \), decide \(y_i\in POS\left( D\right) \);

(B) If \(I\left( R\left( b_B\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_P\mid \left[ y_i\right] \right) \right) \) and \(I\left( R\left( b_B\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_N\mid \left[ y_i\right] \right) \right) \), decide \(y_i\in BND\left( D\right) \);

(N) If \(I\left( R\left( b_N\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_P\mid \left[ y_i\right] \right) \right) \) and \(I\left( R\left( b_N\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_B\mid \left[ y_i\right] \right) \right) \), decide \(y_i\in NEG\left( D\right) \).

\(I\left( R\left( b_\blacklozenge \mid \left[ y_i\right] \right) \right) \) \(\left( \blacklozenge =P,B,N\right) \) is ideal positive degree of expected losses. Their calculation formulas are as follows [6]:

$$\begin{aligned}{} & {} I\left( R\left( b_P\mid \left[ y_i\right] \right) \right) =1-\sqrt{\frac{\left( \mu _{\chi _{ij}}^{1-P_r}\right) ^2+ \left( \left( 1-\nu _{\chi _{ij}}\right) ^{1-P_r}\right) ^2}{2}} \end{aligned}$$
(16)
$$\begin{aligned}{} & {} I\left( R\left( b_B\mid \left[ y_i\right] \right) \right) \nonumber \\{} & {} \quad =1-\sqrt{\frac{\left( \left( 1-\mu _{\chi _{ij}}\right) ^{\rho \times P_r}\left( 1-\nu _{\chi _{ij}}\right) ^{\rho \times \left( 1-P_r\right) }\right) ^2+ \left( \mu _{\chi _{ij}}^{\rho \times \left( 1-P_r\right) }\nu _{\chi _{ij}}^{\rho \times P_r}\right) ^2}{2}} \end{aligned}$$
(17)
$$\begin{aligned}{} & {} I\left( R\left( b_N\mid \left[ y_i\right] \right) \right) =1-\sqrt{\frac{\left( \left( 1-\mu _{\chi _{ij}}\right) ^{P_r}\right) ^2+ \left( \nu _{\chi _{ij}}^{P_r}\right) ^2}{2}} \end{aligned}$$
(18)

Calculation method of conditional probability based on IFNs

In this section, we use the grey relational degree proposed by Liu et al. [6] to calculate conditional probability. The calculation steps are as follows:

Step 1. Fix an intuitionistic fuzzy positive ideal state (IFPIS) and intuitionistic fuzzy negative ideal state (IFNIS).

In general, we select the maximum attribute value as IFPIS and the minimum attribute value as IFNIS [53]. In IF-LSGDM, positive ideal point is \(\chi _{ij}^+=\left( 1,0\right) \) and the negative ideal is \(\chi _{ij}^-=\left( 0,1\right) \). Because negative ideal point and positive ideal point of each object under each attribute are the same, we have the negative ideal and positive ideal of each object under all attributes as follows:

$$\begin{aligned}{} & {} \chi _i^+=\left( \mu _i^+,\nu _i^+\right) = \textrm{max}_i\left( \chi _{ij}\right) =\left( 1,0\right) \end{aligned}$$
(19)
$$\begin{aligned}{} & {} \chi _i^-=\left( \mu _i^-,\nu _i^-\right) = \textrm{min}_i\left( \chi _{ij}\right) =\left( 0,1\right) \end{aligned}$$
(20)

Step 2. Calculate grey relational coefficients \(g_{ij}^+\) and \(g_{ij}^-\) between object \(y_i\) and the IFPIS or IFNIS about attribute \(c_j\):

$$\begin{aligned}{} & {} g_{ij}^+=\frac{h^++\xi H^+}{d\left( \chi _{ij},\chi _i^+\right) +\xi H^+}, \end{aligned}$$
(21)
$$\begin{aligned}{} & {} g_{ij}^-=\frac{h^-+\xi H^-}{d\left( \chi _{ij},\chi _i^-\right) +\xi H^-}, \end{aligned}$$
(22)

where \(h^+= \textrm{min}_i \textrm{min}_j d\left( \chi _{ij},\chi _i^+\right) \), \(H^+= \textrm{max}_i \textrm{max}_j d\left( \chi _{ij},\right. \left. \chi _i^+\right) \),

\(h^-= \textrm{min}_i \textrm{min}_j d\left( \chi _{ij},\chi _i^-\right) \) and \(H^-= \textrm{max}_i \textrm{max}_j d\left( \chi _{ij},\right. \left. \chi _i^-\right) \). \(0<\xi <1\) is the identification coefficient. Normally, \(\xi =0.5\).

Step 3. Calculate grey relation degrees \(g_{i}^+\) and \(g_{i}^-\), respectively.

Grey relation degrees of the object \(y_i\) from the IFPIS or IFNIS are as follows:

$$\begin{aligned} g_{i}^+=\sum _{j=1}^nw_jg_{ij}^+. \end{aligned}$$
(23)
$$\begin{aligned} g_{i}^-=\sum _{j=1}^nw_jg_{ij}^-, \end{aligned}$$
(24)

where \(w_j\) is the weight of the attribute \(c_j\).

Step 4. Relative closeness of grey relation of each object is calculated as follows:

$$\begin{aligned} RC_{i}=\frac{g_{i}^+}{g_{i}^++g_{i}^-} \end{aligned}$$
(25)

Therefore, we can use \(RC_{i}\) to express probability that object \(y_i\) belongs to state D, in other words, conditional probability of object \(y_i\) is \(Pr\left( D\mid [y_i]\right) =RC_{i}\).

A novel method for LSGDM based on IFSs

In this section, the IF-TW-LSGDM method is proposed. Calculation steps are summarized and corresponding algorithm is given.

Decision-making process based on IF-TW-LSGDM considering the public’s decision evaluation

Step 1. Collect the public evaluation opinion data of LSGDM. According to actual decision-making problems, we use data crawlers, questionnaires and other methods to obtain the public comments on relevant issues from microblogs, forums and other platforms.

Step 2. Get the public decision-making opinions by using SA. Conduct initial processing on obtained the public comment data, including using Jieba for word segmentation, and using TF-IDF method to form a common feature word set. After that, on the basis of “Generation of the public decision evaluation information” section, use SA to build the public evaluation decision matrix \(E_{T_s}\) for alternatives that meet all criteria. Then standardize \(E_{T_s}\) according to Formula (7) to obtain the public decision matrix \(E_T=\left( t_{ij}\right) _{m\times n}\).

Step 3. Convert LSGDM to IF-LSGDM. According to the method proposed in “Convert LSGDM to IF-LSGDM” section, decision opinions of experts and the public are converted into IFNs, that is, \(E_k=\left( \widetilde{e_{ij}^k}=\left( \mu _{e_{ij}^k},\nu _{e_{ij}^k}\right) \right) _{m\times n}\), \(k=1,2,\ldots ,s\) and \(E_T=\left( \widetilde{t_{ij}}=\left( \mu _{t_{ij}},\nu _{t_{ij}}\right) \right) _{m\times n}\) Therefore, an LSGDM problem becomes an IF-LSGDM problem.

Step 4. Identify and manage the non-cooperative behavior of expert groups and reach consensus. The specific operation steps are given in “CRP based on IFSs” section. First, standardize the initial decision matrix of experts to obtain the decision matrix \(\dot{E_k}=\left( {\dot{e}}_{ij}^k=\left( {\dot{\mu }}_{e_{ij}^k},{\dot{\nu }}_{e_{ij}^k}\right) \right) _{m\times n}\), \(k=1,2,\ldots ,s\). Then, expert groups are clustered, and obtained subgroup weight vector \(\lambda =\left( \lambda _1,\lambda _2,\ldots ,\lambda _L\right) \), the cluster decision matrix \({\dot{E}}_l=\left( {\dot{e}}_{ij}^l\right) _{m\times n}\), \(l=1,2,\ldots ,L\), and the collect decision matrix \({\ddot{E}}=\left( \ddot{e}_{ij}\right) _{m\times n}\). Finally, identify and manage the non-cooperative behavior of expert groups. The expert subgroup with the lowest degree of consensus shall be proposed with adjustment opinions, and the uncooperative expert subgroup shall be punished with weight. After several iterations, the expected consensus is reached. The new cluster decision matrix \({\dot{E}}_l^*=\left( {\dot{e}}_{ij}^{l,*}\right) _{m\times n}\) and the new collect decision matrix \({\ddot{E}}^*=\left( \ddot{e}_{ij}^*\right) _{m\times n}\) and the new weight vector \(\lambda ^*=\left( \lambda _1^*,\lambda _2^*,\ldots ,\lambda _L^*\right) \) of expert subgroups are obtained.

Step 5. Information fusion. To integrate opinions of expert groups and the public, we use Formula (9) in “Information fusion” section to calculate the final decision matrix \(X=\left( \chi _{ij}\right) _{m\times n}\).

Step 6. Compute relative loss functions of each evaluation value represented by IFNs using the method proposed in “The relative loss functions based on IFNs” section.

Step 7. Obtain aggregated relative loss functions of each alternative. According to the method of converting IFNs into relative loss functions in “The relative loss functions based on IFNs” section, we can obtain relative loss function of alternative \(y_i\) under attribute \(c_j\), expressed as \(\textrm{loss}{\chi _{ij}}=\begin{bmatrix} \left( 0,1\right) &{}\chi _{ij}^\prime \\ \rho \chi _{ij} &{}\rho \chi _{ij}^\prime \\ \chi _{ij} &{}\left( 0,1\right) \end{bmatrix}\), where \(\rho \in \left[ 0,1\right) \). Here we use the IFWA operator given by Eq. (6) to aggregate each alternative \(y_i\) by attributes. The specific integration methods are as follows:

$$\begin{aligned} \textrm{loss}{\chi _{i}}= & {} \oplus _{j=1}^n w_j \textrm{loss}{\chi _{ij}} \nonumber \\{} & {} =\begin{bmatrix} \left( 0,1\right) &{}\oplus _{j=1}^n w_j \chi _{ij}^\prime \\ \oplus _{j=1}^n\rho w_j \chi _{ij} &{}\oplus _{j=1}^n\rho w_j \chi _{ij}^\prime \\ \oplus _{j=1}^n w_j \chi _{ij} &{}\left( 0,1\right) \end{bmatrix} \end{aligned}$$
(26)

where \(w_j\) is the weight of attribute \(c_j\), \(j=1,2,\ldots ,n\).

Step 8. Calculate the conditional probability \(Pr\left( D\mid [y_i]\right) \) of each alternative based on IFNs by using the method proposed in “Calculation method of conditional probability based on IFNs” section.

Step 9. According to decision rules (P)–(N) in “The relative loss functions based on IFNs” section, determine the domain of each alternative: \(POS\left( D\right) \), \(NEG\left( D\right) \) and \(BND\left( D\right) \).

Step 10. Rank all alternatives. We apply Eq. (27) to calculate the associated cost of each object in three domains:

$$\begin{aligned} \Re \left( y_i\right) =\left\{ \begin{aligned}&I\left( R\left( b_P\mid [y_i]\right) \right) ,&\text{ if } y_i \in POS\left( D\right) \\&I\left( R\left( b_B\mid [y_i]\right) \right) ,&\text{ if } y_i \in BND\left( D\right) \\&I\left( R\left( b_N\mid [y_i]\right) \right) ,&\text{ if } y_i \in NEG\left( D\right) \end{aligned} \right. \end{aligned}$$
(27)

where \(I\left( R\left( b_\blacklozenge \mid \left[ y_i\right] \right) \right) \) \(\left( \blacklozenge =P,B,N\right) \) can be calculated by Formulas (16)–(18).

Then, we rank all alternatives according to following rules:

(1) \(POS\left( D\right) \succ BND\left( D\right) \succ NEG\left( D\right) \);

(2) Alternatives in each domain are ranked according to the value of \(\Re \left( y_i\right) \), the smaller the value of \(\Re \left( y_i\right) \), the better the \(y_i\). For example, \(y_i,y_k \in POS\left( D\right) \), if \(\Re \left( y_i\right) <\Re \left( y_k\right) \), then \(y_i\succ y_k\).

To show our decision-making method more intuitively, the flow chart as shown in Fig. 1.

Fig. 1
figure 1

Flow chart of the IF-TW-LSGDM method

Full size image

The algorithm of the IF-TW-LSGDM method

Algorithm 1. The algorithm of the IF-TW-LSGDM method.

      Input

      (1) DMs set \(P=\left\{ p_1,p_2,\ldots ,p_s\right\} \left( s\ge 20\right) \);

      (2) Attribute set \(C=\left\{ c_1,\ldots ,c_n\right\} \), attribute

weight vector \(w=\left\{ w_1,\ldots ,w_n\right\} \);

      (3) Alternative set \(Y=\left\{ y_1,y_2,\ldots ,y_m\right\} \);

      (4) The experts’ collect decision matrix

\({\ddot{E}}^*=\left( \ddot{e}_{ij}^*\right) _{m\times n}\);

      (5) The public decision matrix

\({\dot{E}}_T=\left( {\dot{t}}_{ij}=\left( {\dot{\mu }}_{t_{ij}},{\dot{\nu }}_{t_{ij}}\right) \right) _{m\times n}\);

      (6) Two parameters \(\rho , W_{et}\).

      Output Classification and ranking results of

all alternatives.

      1   begin

      2   for \(i=1\) to m; \(j=1\) to n; do

         \(\chi _{ij}=W_{et}\ddot{e}_{ij}^*\oplus \left( 1-W_{et}\right) {\dot{t}}_{ij}\)

            \(=\left( 1-\left( 1-\ddot{\mu }_{e_{ij}^*}\right) ^{W_{et}}\left( 1-{\dot{\mu }}_{t_{ij}}\right) ^{1-W_{et}}, \right. \)

            \( \left. \left( \ddot{\nu }_{e_{ij}^*}\right) ^{W_{et}}\left( {\dot{\nu }}_{t_{ij}}\right) ^{1-W_{et}}\right) ,\)

      3   end for

      4   for \(i=1\) to m; \(j=1\) to n do

      5      \(l_{PP}^{ij}=\left( 0,1\right) \), \(l_{BP}^{ij}=\rho \left( \mu _{\chi _{ij}},\nu _{\chi _{ij}}\right) \),

      6      \(l_{NP}^{ij}=\left( \mu _{\chi _{ij}},\nu _{\chi _{ij}}\right) \),

      7      \(l_{PN}^{ij}=\left( \nu _{\chi _{ij}},\mu _{\chi _{ij}}\right) \),

      8      \(l_{BN}^{ij}=\rho \left( \nu _{\chi _{ij}},\mu _{\chi _{ij}}\right) \) and \(l_{NN}^{ij}=\left( 0,1\right) \).

      9   end for

      10   for \(i=1\) to m; \(j=1\) to n do

      \(loss{\chi _{i}}=\begin{bmatrix} \left( 0,1\right) &{}\oplus _{j=1}^n w_j \left( \nu _{\chi _{ij}},\mu _{\chi _{ij}}\right) \\ \oplus _{j=1}^n\rho w_j \left( \mu _{\chi _{ij}},\nu _{\chi _{ij}}\right) &{}\oplus _{j=1}^n\rho w_j \left( \nu _{\chi _{ij}},\mu _{\chi _{ij}}\right) \\ \oplus _{j=1}^n w_j \left( \mu _{\chi _{ij}},\nu _{\chi _{ij}}\right) &{}\left( 0,1\right) \end{bmatrix}\)

      11   end for

      12   Let \(\chi _i^+=\left( 1,0\right) , \chi _i^-=\left( 0,1\right) \)

      13   for \(i=1\) to m; \(j=1\) to n do

      14      \(g_{ij}^+=\frac{h^++\xi H^+}{d\left( \chi _{ij},\chi _i^+\right) +\xi H^+}\),

      15      \(g_{ij}^-=\frac{h^-+\xi H^-}{d\left( \chi _{ij},\chi _i^-\right) +\xi H^-}\).

      16   end for

      17   for \(i=1\) to m do

      18      \(g_{i}^+=\Sigma _{j=1}^nw_jg_{ij}^+\), \(g_{i}^-=\Sigma _{j=1}^nw_jg_{ij}^-\).

      19   end for

      20   for \(i=1\) to m do \(Pr\left( D\mid [y_i]\right) =\frac{g_{i}^+}{g_{i}^++g_{i}^-}\).

      21   end for

      22   for \(i=1\) to m do

      \(I\left( R\left( b_P\mid \left[ y_i\right] \right) \right) =1-\sqrt{\frac{\left( \mu _{\chi _{ij}}^{1-P_r}\right) ^2+ \left( \left( 1-\nu _{\chi _{ij}}\right) ^{1-P_r}\right) ^2}{2}} \)

      \(I\left( R\left( b_B\mid \left[ y_i\right] \right) \right) =1- \)

      \(\sqrt{\frac{\left( \left( 1-\mu _{\chi _{ij}}\right) ^{\rho \times P_r}\left( 1-\nu _{\chi _{ij}}\right) ^{\rho \times \left( 1-P_r\right) }\right) ^2+ \left( \mu _{\chi _{ij}}^{\rho \times \left( 1-P_r\right) }\nu _{\chi _{ij}}^{\rho \times P_r}\right) ^2}{2}}\)

      \(I\left( R\left( b_N\mid \left[ y_i\right] \right) \right) =1-\sqrt{\frac{\left( \left( 1-\mu _{\chi _{ij}}\right) ^{P_r}\right) ^2+ \left( \nu _{\chi _{ij}}^{P_r}\right) ^2}{2}}\)

      23   end for

      24   for \(i=1\) to m do

      Algorithm 1. The algorithm of the IF-TW-LSGDM method

      if \(I\left( R\left( b_P\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_B\mid \left[ y_i\right] \right) \right) \) &&

      \(I\left( R\left( b_P\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_N\mid \left[ y_i\right] \right) \right) \), then decide \(y_i\in POS\left( D\right) \);

      else if \(I\left( R\left( b_B\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_P\mid \left[ y_i\right] \right) \right) \) &&

      \(I\left( R\left( b_B\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_N\mid \left[ y_i\right] \right) \right) \), then decide \(y_i\in BND\left( D\right) \);

      else \(I\left( R\left( b_N\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_P\mid \left[ y_i\right] \right) \right) \) &&

      \(I\left( R\left( b_N\mid \left[ y_i\right] \right) \right) \le I\left( R\left( b_B\mid \left[ y_i\right] \right) \right) \), then decide \(y_i\in NEG\left( D\right) \);

      25      end if

      26   end for

      27   for \(i=1\) to m do

         if \(y_i\in POS\left( D\right) \), then \(\Re \left( y_i\right) =I\left( R\left( b_P\mid \left[ y_i\right] \right) \right) \),

         else if \(y_i\in BND\left( D\right) \), then \(\Re \left( y_i\right) =I\left( R\left( b_B\mid \left[ y_i\right] \right) \right) \),

         else \(y_i\in NEG\left( D\right) \), then\(\Re \left( y_i\right) =I\left( R\left( b_N\mid \left[ y_i\right] \right) \right) \).

      28   end for

      29   for \(i=1\) to m; \(k=1\) to m; \(i\ne k\) do

      30      Rank all objects by the following rule:

      (1) \(POS\left( D\right) \succ BND\left( D\right) \succ NEG\left( D\right) \);

      (2) The objects in each domain,

if \(\Re \left( y_i\right) <\Re \left( y_k\right) \), then\(y_i\succ y_k\).

      31   end for

      32   end begin

An illustrative example

A landslide mass is mainly composed of Quaternary eluvial deluvial colluvial breccia gravelly soil and fragmental rock soil. Its structure is loose and its pores are developed. It is a permeable heterogeneous accumulation mass. The landslide mass can be divided into upper and lower sections. The lower section is the main inundation area after the impoundment of the reservoir, with an elevation of 92–182 ms, which is mainly composed of fragmented and angular gravel containing cohesive soil; the upper section is the main accumulation area of the ancient landslide mass. According to the different causes of the slope material, this section can be divided into two elevation sections: The zone with an elevation of 181–205 m is mainly composed of slope deposit loose clayey soil containing rubble, with a longitudinal width of about 40 m. The zone with an elevation of 205–310 m is mainly colluvial loose block gravelly soil, which is the main colluvial deposit area of the landslide mass. The ground shape is complex, and human activities are mainly concentrated in this section. A large number of scarps have been formed. With the construction of the resettlement project, there have been house cracks, water inrush, ground cracks and other phenomena in this elevation section. The permeability and water content of the landslide mass are very uneven, and change obviously with rainfall. Under continuous and strong rainfall, the landslide mass is not only easy to be saturated and softened, but also makes a lot of water lag behind the contact surface or close to the dense cohesive soil layer, which is extremely unfavorable to the stability of the slope.

To ensure the stability of the landslide, it is necessary to conduct comprehensive treatment of the landslide. There are four treatment schemes, namely: earthwork cutting and load reduction, drainage works and monitoring system \(\left( y_1\right) \); Local earthwork removal, drainage works and monitoring system \(\left( y_2\right) \); Bolt shotcrete locking, drainage works and monitoring system \(\left( y_3\right) \); Anti slide pile, drainage works and monitoring system \(\left( y_4\right) \); Now, 20 experts are invited to evaluate the four programmes using six evaluation criteria. Opinions and evaluations of the surrounding people and the enthusiastic public are widely collected through questionnaires and networks. The six evaluation criteria are total investment \(\left( c_1\right) \), project income \(\left( c_2\right) \), maintenance and operation cost \(\left( c_3\right) \), governance effect \(\left( c_4\right) \), risk \(\left( c_5\right) \) and impact on the environment \(\left( c_6\right) \). The weight of each evaluation criteria is \(W=\left( 0.15,0.25,0.1,0.35,0.05,0.25\right) \). The experts’ evaluation opinions are shown in Table 5.

Table 5 Experts’ decision matrix E
Full size table

First, obtain the public evaluation decision matrix. According to “Generation of the public decision evaluation information” section, we build the public decision matrix \(E_T\) as shown in Table 6.

Table 6 Public decision matrix
Full size table

Second, convert LSGDM to IF-LSGDM according to the method proposed in “Convert LSGDM to IF-LSGDM” section, and the decision matrix based on IFS is shown in Table 7.

Table 7 Decision matrix of experts and the public
Full size table
Table 8 Collective decision matrix \({\ddot{E}}^*\) of experts
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Table 9 Final decision matrix X
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Third, the intuitionistic fuzzy C-means is used to classify experts into five clusters. The five expert subgroups after clustering are as follows: \(g_1=\{p_6, p_9, p_{16}\}\), \(g_2=\{p_1,p_5,p_{13},p_{14}, p_{17}\}\), \(g_3=\{p_4,p_7,p_{10},p_{15}, p_{18}\}\), \(g_4=\{p_2,p_3,p_{11},p_{12},p_{19}\}\) and \(g_5=\{p_8,p_{20}\}\).

Fourth, identify and manage the non-cooperative behavior of clusters and reach consensus. Through two iterations, \(GCI=0.84\) is achieved, and the collective decision matrix of expert groups is obtained as shown in Table 8.

Fifth, we use Eq. (9) in “Information fusion” section to calculate the final decision matrix \(X=\left( \chi _{ij}\right) _{m\times n}\) as shown in Table 9. Here, we let the adjustment parameter \(W_{et}=0.7\).

Sixth, suppose \(\rho =0.25\), we compute relative loss functions of each alternative base on the final decision matrix using the method proposed in “The relative loss functions based on IFNs” section. Results are shown in Table 10. In addition, the aggregated relative loss functions of each alternative are calculated using Eq. (26), as shown in Table 11.

Table 10 Relative loss functions of each alternative
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Table 11 Aggregated relative loss functions of each alternative
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Seventh, by using the method proposed in “Calculation method of conditional probability based on IFNs” section, conditional probabilities of each alternative are calculated. That is, \(Pr\left( D\mid [y_1]\right) =0.55\), \(Pr\left( D\mid [y_2]\right) =0.513\), \(Pr\left( D\mid [y_3]\right) =0.542\) and \(Pr\left( D\mid [y_4]\right) =0.536\).

Eighth, according to Eqs. (16)–(18), we can get ideal positive degrees of expected losses are \(I\left( R\left( b_\blacklozenge \mid \left[ y_1\right] \right) \right) =\{0.164, 0.180, 0.458\}\), \(I\left( R\left( b_\blacklozenge \mid \left[ y_2\right] \right) \right) =\{0.208, 0.167, 0.390\}\), \(I\left( R\left( b_\blacklozenge \mid \left[ y_3\right] \right) \right) =\{0.174, 0.176, 0.442\}\) and \(I\left( \! R\left( \! b_\blacklozenge \mid \right. \right. \left. \left. \left[ y_4\right] \!\right) \right) =\{0.178, 0.174, 0.435\}\), where \(\left( \blacklozenge =P,B,N\right) \).

Ninth, on the basis of decision rules (P)–(N) in “The relative loss functions based on IFNs” section, determine the domain of each alternative: \(POS\left( D\right) =\{y_1,y_3\}\), \(BND\left( D\right) =\{y_2,y_4\}\) and \(NEG\left( D\right) =\varnothing \).

Finally, according to Eq. (27) and ranking rules, we can get the ranking result of alternatives as following: \(y_1\succ y_3 \succ y_2\succ y_4\).

Comparisons and analyses

First, to verify the correctness and effectiveness of the IF-TW-LSGDM method, we apply this method to solve the example of Ref. [51]. Results of classification and ranking are shown in Table 12. Here, we let \(\rho =0.4\), \(W_{et}=1\).

Table 12 Comparison with Ref. [51]
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It can be obtained from Table 12, ranking result of IF-TW-LSGDM method is basically consistent with that of Reference [51], and the optimal alternative obtained is \(y_1\). This shows that the IF-TW-LSGDM method is correct and effective.

Wu et al. [51] put forward an LSGDM method based on IFS, and proposed calculation methods of experts clustering, collective decision-making matrix and weight punishment. Compared with the Reference [51], the advantages of IF-TW-LSGDM method are as follows:

(1) When evaluating alternatives, not only the opinions of experts in various fields, but also the opinions of the public have been fully considered. Sometimes experts are unable to get valuable information about decision problem from the public. Especially, in the era of rapid growth of social media, the public can share their feelings and preferences on decision problem. The mining and analysis of the public social media big data can supply important reference for decision-making.

(2) The fuzzy evaluation value in the information system is converted into IFN, and the intuitionistic fuzzy information system described by IFN is obtained. Therefore, we can take full advantage of IFSs to solve the LSGDM problem.

(3) We use TWD method to classify and rank alternatives, propose a calculation method of relative loss functions based on IFS, propose a calculation method of conditional probability, and give the classification and ranking rules based on the TWD. Not only ranking results of alternatives are obtained, but also classification results of alternatives are obtained. This can reduce mistakes in decision-making and make decision-making more reasonable.

Secondly, we take the example in “An illustrative example” section. When the parameter \(\rho \) changes, we apply the IF-TW-LSGDM method to calculate the classification and ranking results, and compare and analyze changes in the results. \(\rho \in \left[ 0,0.5\right] \) are the parameter when calculating the relative loss functions. when \(\rho \) take different values, classification and ranking results are shown in Table 13.

Table 13 Classification and ranking results with different values of \(\rho \)
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From Table 13, we can see that when the parameter \(\rho \) changes, the classification and ranking results will also change. However, when the value of \(\rho \) changes, the optimal selection result is still the same, that is \(y_2\). Thus, in terms of the best choice, the IF-TW-LSGDM method is stable. In addition, as \(\rho \) increases in the classification result, \(BND\left( D\right) \) becomes smaller and smaller, and \(POS\left( D\right) \) gradually increases.

Finally, when the number of decision-makers is less than 20 and the public opinion is not considered, the IF-TW-LSGDM method can be used to solve the MADM problem. The analysis and comparison of ranking results with other references are shown in Table 14. Here, we use the example of Ref. [54], and compare the calculation result with the results of Reference [54] and other references. The results obtained by these methods are consistent, and the IF-TW-LSGDM method is also applicable to group decision-making problems. Moreover, the IF-TW-LSGDM method can also obtain the classification results of alternatives: \(POS\left( D\right) =\{y_1,y_5\}\), \(BND\left( D\right) =\{y_3,y_4,y_6\}\) and \(NEG\left( D\right) =\{y_2\}\).

Table 14 Analysis and comparison of ranking results with other references
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Conclusions

We use TWD and IFS to solve the LSGDM problem, and By combining IFS, TWD and LSGDM, we propose a new IF-TW-LSGDM method. This novel method can classify and rank alternatives. In addition, we also give the computational procedure and corresponding algorithm description. The IF-TW-LSGDM method is applied to the decision making of landslide treatment alternatives selection. In this process, the clustering analysis, identification and management of non-cooperative behavior of experts are considered. Through experiments, we verified the effectiveness and feasibility of the IF-TW-LSGDM method. Our main contributions are as follows:

  1. (1)

    We convert the fuzzy evaluation values in LSGDM to IFNs, and get the intuitionistic fuzzy information system described by IFNs. Therefore, we make full use of the advantages of IFNs to solve the LSGDM problem.

  2. (2)

    The method of obtaining and handling the public evaluation information is given, and the aggregation method of the public evaluation information and experts evaluation information is given.

  3. (3)

    We propose a novel method based on TWD to classify and rank alternatives, and the corresponding operation steps and algorithm description are given. This has enriched the theoretical and applied research of TWD.

  4. (4)

    A effective IF-TW-LSGDM method is provided to study the selection of alternatives for landslide treatment.

We discusses data processing methods when the public evaluation information is natural language, and obtains the public’s intuitionistic fuzzy evaluation information through a method based on SA. Then, experts were clustered using the intuitionistic fuzzy C-means clustering method. The next step can be to consider using graph neural networks to directly cluster and analyze the public and experts [58].

Further research directions mainly include:

  1. (1)

    In the age of big data, more and more decision makers can participate in decision-making through the network. How to obtain the evaluation information of different decision makers more efficiently and accurately is a very important research topic [58, 59].

  2. (2)

    The IF-TW-LSGDM method can be extended to more types of fuzzy sets.

  3. (3)

    In LSGDM based on social networks, the number of experts is increasing, and the types of experts are becoming more and more complex. The research on CRP is receiving more and more attention [60]. We can consider improving the CRP model to deal with non-cooperative behavior in the social network environment.

  4. (4)

    In recent years, landslides, earthquakes and other natural disasters occur frequently. The most important content of emergency rescue is to quickly and accurately predict the demand for emergency rescue materials after an emergency. The IF-TW-LSGDM method can be considered to solve the problem of emergency material demand prediction.