1 Introduction

Major emergencies generally refer to the accidents occurring in a certain time and region and causing great losses of life and property (http://www.chinasafety.gov.cn/2007-04/20/content_232858.htm), such as the Tianjin Port “\(8 \cdot 12\)” fire and explosion incident. In the study of the risk of major emergencies, the general focus is the risk caused by the emergency disposal taken at the incident scene. When psychological factors are considered to deal with the risk of decision-making according to the prospect theory (Kahneman and Tversky 1979), regret theory (Bell 1982), etc., it means that the impact is taken into account that individual preferences have on the results of selection. This paper considers the risk caused by multiple complex preferences in large group decision-making process. Since the major participant involved in decision-making is mainly the government, this risk is similar to the decision risk faced by government in decision-making process. Bin (2012) analyses the decision-making risk of government investment projects from the perspective of the participants, identifies the risk factors, uses fuzzy clustering method to make the comparison among various risk factors’ correlations, and finally establishes an evaluation model to measure the participant risk of government investment projects. And because of the complex preferences, great conflicts are likely to exist among decision-makers’ preferences, which are one kind of potential reason of decision risk. Xuanhua et al. (2014) proposes the multi-attribute and multi-stage large group decision-making method for preference conflict optimization, which considers phase correction of decision-makers’ preferences to optimize the conflict and reduce the risk; Zhanfeng (2012) sets the anti-party system, so the extreme conflict resolution plan is proposed from the macro-level system that helps decision-makers in the mutual discussion to get consistent results. Moreover, different decision-making participants may give different decision-making values. Fasolo and Costa (2014) studies the different representations of numerical and linguistic expressions of decision-making and their effects on decision-making, whereas in the actual decision-making process different forms of decision values may also pose a risk.

Obviously, as for the decision-making of large events, merely participants’ complex preferences can lead to a very large number of risk factors. In general, these elements are scientific decision-making process, the clear responsibilities of decision-makers, the rationality of decision-making power distribution, the suitable structure of expert groups, the participants’ social responsibility, the evaluation forms, the dynamic movement of decision-makers, the conflict caused by the preference and the interpersonal relationship between the participators and so on.

With so many risks, in order to control the adverse effects, we need to have an auxiliary tool to measure and evaluate it. The previous research on risk measurement was almost in the field of finance (Junshan 2007; Chengli and Yan 2014; Ye et al. 2017; Grechuk and Zabarankin 2015), because investment portfolio gains can be directly expressed in numbers to make quantitative calculation of the risk control based on risk measure model. For emergency disposal, investment projects, site selection decisions, etc., all of them involve identifying and analyzing the risk factors as well as rating the risk of the whole decision-making process (Aven 2015). In recent years, the theory of financial risk measurement has gradually matured, and other fields have begun to study the application of risk measurement (Pender et al. 2016). In this paper, we proposed a method of measuring the risk of large group decision-making which focuses on the risk resulting from distribution of the decision-making rights, by calculating the degree of uncertainty with the information entropy theory (Xinlei 2015; Fleischhacker and Fok 2015).

This paper draws lessons from the “\(7 \cdot 23\)” Yong-Wen line large-scale railway traffic accident (http://www.chinasafety.gov.cn/newpage/Contents/Channel_21679/2011/1228/244874/content_244874.htm), the Tianjin Port “\(8\cdot 12\)” large fire and explosion incident (http://www.chinasafety.gov.cn/newpage/newfiles/201600812baogao.pdf) and other recent relevant information of emergency disposal. And on the basis of the existing emergency decision-making and risk measurement research, a new large group decision-making method is developed based on risk measurement.

2 Methodology

2.1 Problem description

According to the emergency disposal parts of the relevant large incident investigation reports in http://www.chinasafety.gov.cn/newpage/Contents/Channel_21679/2011/1228/244874/content_244874.htm and http://www.chinasafety.gov.cn/newpage/newfiles/201600812baogao.pdf, we can simplify the analysis to know that the major emergency decisions can be concluded from three aspects. First, in order to prevent more severe consequences, it is the first step to make a quick emergent decision, such as fire fighting and evacuation of the crowds. At this time, it is mainly the local government and even the relevant departments that directly make emergency decisions in a very limited time, which requires participants’ high level of the dynamic decision-making capacity. The second is the choice of rescue programs. Since the emergency scene of events may be different, the local government has prepared contingency plans as the alternatives. The third is to make decisions on medical resource scheduling, rescue compensation and other important matters. In addition, in recent years, with the increasing popularity of information technology, network public opinion (Wentao and Dachao 2011) has become a part of emergency decision-making which cannot be ignored. A part of the literature (http://www.chinasafety.gov.cn/newpage/Contents/Channel_21679/2011/1228/244874/content_244874.htm) mentioned that the misleading media reports caused the masses to mistakenly think the rescue has stopped, so the public opinion negatively affects the rescue operations. With the real-time monitoring of public opinion, the state media can announce the latest emergence plans immediately when facing some exceptional cases on the internet.

In the process of decision-making, there is generally a central leadership meeting to listen to the situation report and to make instructions. Then local government immediately establishes the accident rescue headquarters and organizes the relevant departments and experts to make emergency decision. Decision-makers of central departments contact with local headquarters through telephone or video and send representatives to the location to conduct supervision and guidance on local decisions. Therefore, a central-local decision-making model can be established, as shown in Fig. 1.

Fig. 1
figure 1

Major emergencies central-local decision mode diagram

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Assume that the central decision-maker decides the total number of local decision-makers is m. And decide the weight range of the local decision-makers in every decision-making process. We try to compute the weight distribution when the risk is the minimum and aggregate the preference information to sort the scheme.

For large group major emergency decision-making, basic symbols are provided as follows:

\(\varOmega =\{e_1 ,e_2 ,\ldots ,e_i ,\ldots ,e_m \},\)represents a collection of m local decision participants, where \(m>20\) (Liu et al. 2014);

\(A=\{a_1 ,a_2 ,\ldots ,a_l ,\ldots ,a_p \}\), represents a collection of p alternative emergency plans;

\(C=\{c_1 ,c_2 ,\ldots ,c_j ,\ldots ,c_n \},\) represents a collection of n attributes for emergency plans;

\(W=(w_1 ,w_2 ,\ldots ,w_n)^\mathrm{T},\) represents an attribute weight vector, where \(\sum \nolimits _{i=1}^n {w_i } =1\).

\({\varvec{\upomega }}=(\omega _1 ,\omega _2 ,\ldots ,\omega _m )^\mathrm{T}\), represents weight vector of decision-makers in decision group.

2.2 Information entropy risk measurement of decision participants’ rights distribution

The risks caused by the decision-makers’ right distribution in this paper mainly refer to the situation that the decision results cannot fully express the majority’s wishes when all the participants’ opinions are assembled.

Suppose that the evaluation value of the alternative \(a_{l}\) in attribute \(c_{j }\) of the member \(e_{i}\) is \(d_{lj}^{(i)} \), where \(d_{lj}^{(i)} \in [0,1]\), then the preference matrices of m members are given as follows:

$$\begin{aligned} D^{(i)}=\left[ {{\begin{array}{lllll} {d_{11}^{(i)} }&{}\quad \cdots &{}\quad {d_{1j}^{(i)} }&{}\quad \cdots &{}\quad {d_{1n}^{(i)} } \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ {d_{l1}^{(i)} }&{}\quad \cdots &{}\quad {d_{lj}^{(i)} }&{}\quad \cdots &{}\quad {d_{lj}^{(i)} } \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ {d_{p1}^{(i)} }&{}\quad \cdots &{}\quad {d_{pj}^{(i)} }&{}\quad \cdots &{}\quad {d_{pn}^{(i)} } \\ \end{array} }} \right] , \quad i=1,2,\ldots ,m_{o} \end{aligned}$$

Now the information entropy method is used to measure the risk of the entire decision-making group in the aggregation process, with the attribute weight vector W and the membership weights \({\varvec{\upomega }}\).

  1. (1)

    As all evaluations of attribute \(c_{j}\) are \(d_{lj}^{(i)} ,i=1,\ldots , m;l=1,\ldots ,p\), the interval [0, 1] is equally divided into h small intervals and the size h is given by the decision-makers. And then the sth small interval is \(\left\{ {\begin{array}{ll} ((s-1)/h,s/h],&{}\quad s>1 \\ {[}0,1/h],&{}\quad s=1 \\ \end{array}} \right. \). According to the membership weights, we can obtain the probability that the \(d_{lj}^{(i)} \) falls in the s th interval:

    $$\begin{aligned} P_{js}^l= & {} \sum \limits _{i=1}^m {\omega _i } S_{ij}^l ,\quad \hbox {where}\; S_{ij}^l =\left\{ {_{1,\quad d_{lj}^{(i)} \in ((s-1)/h,s/h]}^{0,\quad d_{lj}^{(i)} \notin ((s-1)/h,s/h]} } \right. ,\nonumber \\&j=1,2,\ldots ,n; \end{aligned}$$
    (1)
  2. (2)

    In this paper, the average uncertainty degree of evaluation information is expressed by information entropy, and the risk of right distribution when the attribute \(c_{j}\) of all schemes’ evaluation values is aggregated is calculated as follow:

    $$\begin{aligned} E_j =-\frac{1}{\ln h}\sum _{l=1}^p {\sum _{s=1}^h {\left( P_{js}^l \ln P_{js}^l \right) } }, \quad j=1,2,\ldots ,n,\nonumber \\ \end{aligned}$$
    (2)

    Then the risk vector is \(E=(E_1 ,E_2 ,\ldots ,E_n )^\mathrm{T}\);

  3. (3)

    Finally, the risk entropy of the decision-making participants resulting from the uneven distribution of rights is obtained:

    $$\begin{aligned} R=W^\mathrm{T}E. \end{aligned}$$
    (3)

2.3 Decision procedure

Step 1 The local command center reports the situation of the site to the central departments first. Then the central decision-maker decides the list of experts (\(m>20\)) of each department involved in the decision-making and gives the attributes set C and the attribute weight W.

Step 2 Local decision-makers, according to the information given by the central departments, convened above personnel to participate in decision-making. Since the evaluation value of the alternative \(a_{l}\) in attribute \(c_{j }\) of the member \(e_{i}\) is \(d_{lj}^{(i)}\), the preference matrices of m members are given as follows:

$$\begin{aligned} D^{(i)}\!=\! & {} \left[ {{\begin{array}{llll} {d_{_{11} }^{(i)} }&{}\quad {d_{_{12} }^{(i)} }&{}\quad \cdots &{}\quad {d_{_{1n} }^{(i)} } \\ {d_{_{21} }^{(i)} }&{}\quad {d_{_{22} }^{(i)} }&{}\quad \cdots &{}\quad {d_{_{2n} }^{(i)} } \\ \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots \\ {d_{_{p1} }^{(i)} }&{}\quad {d_{_{p2} }^{(i)} }&{}\quad \cdots &{}\quad {d_{_{pn} }^{(i)} } \\ \end{array} }} \right] \!=\!\left( D_1^{(i)} ,D_2^{(i)} ,\ldots ,D_n^{(i)}\right) ,\nonumber \\&i=1,2,\ldots ,m_o \end{aligned}$$

According to the calculation formula proposed by the literature (Xuanhua et al. 2009), we can obtain the preference convergence degree between the member \(e_{i_1}\) and the member \(e_{i_3}\) of the attribute \(c_{j}\):

$$\begin{aligned} r_j^{i_1 i_2 } \left( D_j^{(i_1 )} ,D_j^{(i_2 )} \right) =\frac{\left( \left| D_j^{(i_1 )} -\overline{D_j^{(i_1 )} } \right| \right) ^\mathrm{T}\left( \left| D_j^{(i_2 )} -\overline{D_j^{(i_2 )} }\right| \right) }{\left| \left| D_j^{(i_1 )} -\overline{D_j^{(i_1 )} } \right| \right| _2 \left| \left| D_j^{(i_2 )} -\overline{D_j^{(i_2 )} } \right| \right| _2 }, \end{aligned}$$

\(\hbox {where} \left\| {\left. {\bullet } \right\| } \right. _2 \) denotes the 2-norm of the vector, and

$$\begin{aligned} \overline{D_j^{(i)} } =\frac{1}{p}\sum _{l=1}^p {d_{jl}^{(i)} }. \end{aligned}$$
(4)

And the preference convergence degree between the member \(e_{i_1 } \) and the member \(e_{i_3 }\) is

$$\begin{aligned} r^{i_1 i_2 }=\sum _{j=1}^n {w_j } r_j^{i_1 i_2 } \left( D_j^{(i_1 )} ,D_j^{(i_2 )} \right) . \end{aligned}$$
(5)

Firstly, the members are randomly sorted and the first member is selected to form the aggregation \(\varOmega _1 \), which is compared with the left members in pairs to calculate the convergence degree of preferences. And determine appropriate threshold by previous study. Let these members, whose degree results are greater than the threshold, join cluster \(\varOmega _1 \). After we find all the members to join in \(\varOmega _1 \), we select the top number among the remained members according to the previous ranking. Then we continue to compare the top number member with other members which did not join the former cluster \(\varOmega _1 \)by the same calculation. Compare the result with the threshold and members having larger results take part in the cluster \(\varOmega _2 \). When convergence degrees of preferences between a decision-maker and the rest of the decision-makers are all below the threshold, the separate decision-maker becomes a cluster. The previous steps mentioned in this paragraph should be repeated until all decision-makers can join their appropriate clusters. The decision group is clustered into K clusters , which is \(\overline{\varOmega }=\{\varOmega _1 ,\varOmega _2 ,\ldots ,\varOmega _k ,\ldots , \varOmega _K \}.\)

Step 3 The previous results are reported back to the central departments, and the central decision-maker gives the weight interval of each expert in each cluster under the uniform standard. In other words, for any cluster \(\varOmega _k\) (whose number of members is denoted as \(n_k\)), the infimum set and supremum set of membership weight interval are given:

$$\begin{aligned}&\mathop {\mathrm{Inf}}\limits _k = \big \{a_{_1 }^k ,a_{_2 }^k ,\ldots a_{_g }^k ,\ldots ,a_{_{n_k } }^k \big \}, \mathop {\mathrm{Sup}}\limits _k =\{b_{_1 }^k ,b_{_2 }^k ,\ldots ,b_{_g }^k ,\ldots ,b_{n_k}^k\}, \\&\forall a_g^k \in [0,1), b_{_g }^k \in (0,1] \hbox { and } a_{g}^k <b_{g}^k ,\sum \limits _{g=1}^{n_k} {a_{_g}^k \le 1\le \sum \limits _{g=1}^{n_k} {b_{g}^k }}, \end{aligned}$$

Then we can randomly obtain the weight \(\omega _g^k\), \(g=1,2,\ldots ,q,\) of members in cluster \(\varOmega _k \), which meets the condition \((1):\omega _{g}^k \in [a_{g}^k ,b_{g}^k ],g=1,2\ldots ,n_k ;\)condition (2): \(\sum \nolimits _{g=1}^{n_k } {\omega _{g}^k =1} \).

Then, the central decision-maker provides the overall weight interval’s infimum set \(\hbox {Inf}=\{a_1 ,a_2 ,\ldots ,a_g ,\ldots ,a_K \}\) and supremum set \(\hbox {Sup}=\{b_1 ,b_2 ,\ldots b_g ,\ldots ,b_K \}\) of every cluster, where \(\forall a_g \in [0,1),b_g \in (0,1] {\textit{ and }} a_g <b_g,\sum \limits _{g=1}^K {a_g \le 1\le \sum \nolimits _{g=1}^K {b_g } } \). After aggregating cluster’s information into a decision matrix representing each cluster correspondingly, the clusters’ weights are randomly selected within the given interval, and that is \(\omega _{g}^ \in [a_{g}^ ,b_{g}],g=1,2,\ldots ,K,\hbox {where}\sum \nolimits _{g=1}^K {\omega _g =1}.\)

Step 4 According to the clustering result of decision group \(\overline{\varOmega }\), calculate the weights of experts within each cluster. For any cluster \(\varOmega _k \), we use computer to generate a set of random number \(X^{k}=(x_{_1 }^k ,x_{_2 }^k ,\ldots ,x_{_g }^k ,\ldots ,x_{n_k }^k ),\forall x_{_g }^k \in [0,1]\). According to the corresponding weight interval’s infimum and supremum, let \(y_{_g }^k =(b_{_g }^k -a_{_g }^k )x_{_g }^k +a_{_g }^k ,g=1,2,\ldots ,n_k , \quad Y^{k}=\{y_{_1 }^k ,y_{2_1 }^k ,\ldots ,y_{n_k }^k \}\), and normalized \(Y^{k}\) get the weight vector \({ {\varvec{\omega }} }^{k}=(\omega _1^k ,\omega _2^k ,\ldots \omega _{n_k }^k )^{\mathrm{T}}\). Obviously it meets the condition (2), and we have to check whether \(\varvec{\upomega }^{k}\) satisfies the condition(1). If not, we choose the new random numbers again, and continue to find the weight vector that matches the conditions; if satisfies, then take the \({\varvec{\upomega } }^{k}\) as a member weight vector. When obtain an appropriate \({\varvec{\upomega }}^{k}\), we can use the right distribution risk model of the decision-making participant proposed in this paper to calculate the risk under this weight vector:

  1. (1)

    we have already obtained K clusters, and all the members of each cluster are in the corresponding set \(\hbox {I}{_{\Omega _k } } =\{e_{1k} ,e_{2k} ,\hbox {K},e_{nk} \}\), then the Cluster \(\varOmega _k \)’s decision matrix for n attributes of m alternatives is as follows:

    $$\begin{aligned} D^{ik}=\left[ {{\begin{array}{llll} {d_{_{11} }^{ik} }&{}\quad {d_{_{12} }^{ik} }&{}\quad \cdots &{}\quad {d_{_{1n} }^{ik} } \\ {d_{_{21} }^{ik} }&{}\quad {d_{_{22} }^{ik} }&{}\quad \cdots &{}\quad {d_{_{2n} }^{ik} } \\ \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots \\ {d_{_{p1} }^{ik} }&{}\quad {d_{_{p2} }^{ik} }&{}\quad \cdots &{}\quad {d_{_{pn} }^{ik} } \\ \end{array} }} \right] ,\quad i=1,2, \ldots ,n_k \end{aligned}$$
  2. (2)

    Members in cluster \(\varOmega _k \) provide the evaluation value \(d_{lj}^{ik} ,i=1, \ldots ,n_k.\) for the alternative \(a_{l}'\)s attribute \(c_{j}\). According to formula (1), calculate the probability distributions that the \(d_{lj}^{ik} \) falls in the different intervals. Then, use the formula (2) to obtain the risk generated by the distributions mentioned above. Finally, we can get the whole risk entropy value by formula (3).

Repeat choosing random numbers for appropriate times to get teams of the weight vector, and each team’s risk entropy value is calculated. Then select the optimal weight vector \({\varvec{\omega } }_*^{k}\) corresponding to the smallest risk entropy and use weighted average operator (Yager and Alajlan 2016) to gather decision information in cluster \(\varOmega _k \). Then obtain \(D_{\varOmega _k } =\sum \nolimits _{g=1}^{n_k} {{\omega _*}_g^k D(i_g )}\). Apply the above method to each gathering processing to get each cluster’s gathering results with the minimum risk entropy value, which is equal to obtaining decision preference matrix of the number of gathering members K.

Step 5 In each given cluster’s weight interval, we select a random number to be combined, and then normalize each group of random numbers. We select multiple weight combinations that satisfy the limit interval, measure the risk of aggregating process in turn, and choose the weight vector \({{\varvec{\omega }} }^{_{*} }=(\omega _1^*,\omega _2^*,\ldots \omega _K^*)^\mathrm{T}\) when the risk is the smallest, and that is to get decision-making preference matrix that aggregates all the information \(D=\sum \nolimits _{k=1}^K {\omega _{_k }^*D_{\Omega _k } } \). Finally, according to the given attribute weight \(\mathbf{W}\), calculate the final score \(\mathbf{F}=\mathbf{W}^\mathrm{T}D\) of all programs, and sort the scores by number, which should be sent to the central decision-makers for approval and implementation.

Table 1 Personnel scheduling scheme
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3 Numerical examples

After receiving the call that a warehouse in the storage of dangerous goods broke out of fire, the Public Security Bureau 110 command center quickly sent the fire brigade to put out the fire and evacuate the crowds. Because of the rapid spread of fire and explosion, casualties and property losses continued to increase, reaching the major disaster level. The command center immediately contacted the central departments to report details of the incident.

Step 1 The central departments set up a decision-making group with four members, which led by the leader of highest level in the group to make decisions on this major incident and schedule the relevant departments to the scene to rescue personnel and make decisions (Table 1).

For 4 known different evacuation routes \(A=\{a_1 ,a_2, a_3,a_4 \}\), the central decision-making group gives five attributes which should be included in the reference based on the feedback from the site: \(c_1\): terrain convenience, \(c_2\): the suitable degree of local weather, \(c_3\): the intact degree of medical facilities, \(c_4\): the impossibility of secondary disasters, \(c_5\): adequacy of material, that is \(C=\{c_1 ,c_2 ,c_3 ,c_4 ,c_5 \}\). Each member of the central group gives the recommended value of the attribute weight and then gets averages. After that the weight of each attribute of the scheme is obtained as shown in Table 2. The central decision-making group sent 40 decision-makers \(\varOmega =\{e_1 ,e_2 ,\ldots ,e_{40} \}\) to constitute large groups of decision-making.

Table 2 Attribute weight assignment
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Step 2 Upon receiving the notice from the central government, the local command team contacted the decision-making groups to make quick evaluation on the five attributes of the four contingency plans according to the situation on site. The evaluation results are shown in Appendix 1. The degree of convergence among the decision-making preference matrices is calculated by the formula (4) and (5), and the decision-maker whose clustering degree is above the given threshold \(\gamma = 0.8\) (Xuanhua et al. 2009) is clustered. In this paper, the degree of convergence between decision-maker \(e_{1}\)’s and \(e_{2}\)’s decision matrices is taken as an example (calculated by using the MATLAB programming).

The two matrices are as follows:

$$\begin{aligned} D^{(1)}= & {} \left[ {\begin{array}{lllll} 0.4&{}0.5&{}0.7&{} 0.1&{} 0.9 \\ 0.6&{} 0.5 &{} 0.1 &{} 0.7&{} 0.2 \\ 0.5 &{}0.9 &{} 0.3 &{}0.2 &{}0.3 \\ 0.4 &{}0.5&{} 0.3 &{} 0.6 &{}0.2 \\ \end{array}} \right] \nonumber \\= & {} \left( D_1^{(1)} ,D_2^{(1)} ,D_3^{({1})} ,D_4^{({1})} ,D_5^{({1})}\right) ,\nonumber \\ D^{(2)}= & {} \left[ {\begin{array}{lllll} { 0.7} &{}{0.5} &{} {0.3} &{}{0.1} &{} {0.8} \\ { 0.6} &{}{0.4} &{} {0.2} &{} {0.7} &{} {0.6} \\ { 0.5}&{} {0.9}&{} {0.5} &{} {0.2} &{} {0.3} \\ { 0.4}&{} {0.5} &{}{0.1} &{} {0.6} &{} {0.2} \\ \end{array}}\right] \nonumber \\= & {} \left( D_1^{(2)} ,D_2^{(2)} ,D_3^{(2)} ,D_4^{(2)} ,D_5^{(2)}\right) . \end{aligned}$$

The preference coherence degree of decision-maker \(e_1\) and \(e_2\) is calculated for each attribute separately:

$$\begin{aligned}&r_j^{12} \left( D_j^{(1)} ,D_j^{(2)} \right) \nonumber \\&\quad =\frac{\left( \left| D_j^{(1)} -\overline{D_j^{(1)} } \right| \right) ^\mathrm{T}\left( \left| D_j^{(2)} -\overline{D_j^{(2)} } \right| \right) }{\left| \left| D_j^{(1)} -\overline{D_j^{(1)} } \right| \right| _2 \left| \left| D_j^{(2)} -\overline{D_j^{(2)} } \right| \right| _2 },\quad \! j=1,2,\ldots 5,\nonumber \\ \end{aligned}$$

where \(\overline{D^{(1)}_j } =\frac{1}{4}\sum \nolimits _{l=1}^4 {d_{jl}^{(1)} } \), \(\overline{D^{(2)}_j } =\frac{1}{4}\sum \nolimits _{l=1}^4 {d_{jl}^{(2)} } \). The results are shown in Table 3.

Table 3 The degree between \(e_1 \) and \(e_2 \) on each attribute
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Table 4 Decision-makers clustering results
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Table 5 Cluster weight interval
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Table 6 When \(h=5\), the best weighting result for each cluster
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Table 7 When \(h=10\), the best weighting result for each cluster
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According to the attribute weight assignment given by the central group, the total degree of the two decision-makers can be obtained:

$$\begin{aligned} r^{12}= & {} 0.245\times 0.8086+0.147\times 0.9771+0.301\times 0.3684\\&+\,0.091\times 1+0.216\times 0.9349=\hbox {0.7456.} \end{aligned}$$

When \(\gamma =0.8\), we have calculation on each two decision-makers together and cluster decision-makers whose results are greater than 0.8. The results are shown in Table 4.

Step 3 The results will be reported back to the central decision-making team. Given that the central decision-makers give the interval number \(h=5\) and \(h=10\), respectively, each weight interval is gathered for internal decision-makers of each cluster. It should be ensured that all members’ weights in a given range can be added to 1. The interval numbers are shown in Appendix 2.

The central decision group assigned the weight interval for each cluster according to the number of members, as shown in Table 5.

Step 4 Randomly select the numbers in the interval given in Appendix 2 and normalize the weight of each decision-maker to meet the conditions. Then measure the risk of cluster under the weight distribution and choose 10 effective risk entropy results to make comparisons. When the risk entropy is minimum, choose the weight distribution results for real clustering. Tables 6 and 7 show the best weight distribution of each cluster. The programming results of MATLAB are shown in Tables 8 and 9.

Table 8 When \(h=5\), the clustering result of each cluster
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Table 9 When \(h=10\), the clustering result of each cluster
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Step 5 Make a random selection from weight interval in Table 5 and ensure that the normalized weights are obtained in the weight range. Then make effective operations for over 10 times. Calculation results of risk level are shown in Tables 10 and 11. We choose a set of weights with the least risk, when \(h=5\), \({ {\varvec{\omega }} }^{{*}}=\hbox {(0.3704, 0.1017, 0.2125, 0.2924,}~\hbox {0.0055, 0.0175)}^{\mathrm{T}}\) and \(h = 10\), \({{\varvec{\omega }} }^{{*}}=\hbox {(0.3249, 0.1702,}~\hbox { 0.2227, 0.2576, 0.0145, 0.0101)}^{\mathrm{T}}\). We get the final decision evaluation matrix and calculate the final value of each scheme according to the attribute weight in Table 2. The evaluation values are shown in Tables 12 and 13.

Table 10 When \(h=5\), each cluster weight distribution and its risk
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Table 11 When \(h=10\), each cluster weight distribution and its risk
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Table 12 When \(h=5\), the final evaluation result
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Table 13 When \(h=10\), the final evaluation result
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Finally both of the two cases’ ranking results are: \(a_1>a_2>a_3 >a_4 \), which are conveyed to the central decision-making group to determine the final program selection.

4 Conclusion

This paper provides a method of multi-attribute emergency decision-making for large group, calculates the risk in the decision-making by risk measurement proposed, and reduces the risk of the process of clustering, which makes the conclusion better reflect the views of decision-makers. Through the calculation of the preference convergence and the risk of the decision-maker’s rights distribution, the risk caused by the preference difference and the distribution of rights are considered. When processing vast amounts of information, we often cannot use a precise number to represent all of our views. The fuzzy interval uses high-speed operation of the computer to find the result consistent with the demand and reflecting the views of most people. In fact, expansion research can be made on central decision-makers, for example, with large data method a more appropriate number of participants and distribution can be selected. Actually the number of people involved in decision-making can be dynamic. The method presented in this paper can deal with the situation of dynamic large groups, and the risk control of dynamic decision participants can be further studied on the basis of this paper.

There are also shortcomings. For instance, when there is too much limits of selection of the random number, it is difficult to find the weights that can fully meet the conditions from the random numbers in the selected range; with less range limits of random number, the risk level changes greatly. We can consider using numerical approximation and other methods to set a risk threshold, which can solve the appropriate weight interval until the interval’s corresponding risk value is below this threshold.