Group decision making for internet public opinion emergency based upon linguistic intuitionistic fuzzy information

With the wide use of network, the outbreak of network public opinion emergencies has changed from single to multiple. The goal of the current study is to construct the emergency group decision-making (EGDM) model for multiple network public opinion emergencies under the linguistic intuitionistic environment. First of all, we introduce a new version of Copula and Co-copula named extended Copula (EAC) and extended Co-Copula (EACC), respectively, which can be used to capture the relation of attributes (indexes) in the group decision making problems of network public opinion emergencies; Some special cases of EAC and EACC are gained to manage intuitionistic fuzzy information (IFI). Besides, the novel operational rules of linguisitic intuitionistic fuzzy numbers (LIFNs) based upon EAC and EACC are also defined under linguistic intuitionistic environment. What’s more, by integrating the Choquet integral and the proposed operational rules of LIFNs, the linguistic intuitionistic fuzzy Choquet-Copula aggregation operators (LIFCCA) are proposed together with their properties are also investigated; whilst, five specific forms of LIFCCA are obtained when EAC and EACC take different generators. Last but not least, an EGDM approach is constructed based upon proposed LIFCCA; Consequently, the validities and merits of the proposed EGDM approach are shown by comparing with existing approaches.

This work was supported in part by Sichuan Province Youth Science and Technology Innovation Team under Grant 2019JDTD0015, Application Basic Research Plan Project of Sichuan Province under Grant 2017JY0199, Scientific Research Project of Department of Education of Sichuan Province under Grant 18ZA0273 and Grant 15TD0027, Scientific Research Project of Neijiang Normal University under Grant 18TD08.

Authors and Affiliations

1. Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang, 641000, Sichuan, People’s Republic of China

   Yi Liu, Haobin Liu & Lei Xu

2. School of Mathematics and Information Sciences, Neijiang Normal University, Neijiang, 641000, Sichuan, People’s Republic of China

   Yi Liu, Haobin Liu & Lei Xu

3. School of Business, Sichuan Normal University, Chengdu, 610101, Sichuan, People’s Republic of China

   Guiwu Wei

Corresponding author

Correspondence to Yi Liu.

Conflict of interest

The authors declare that they have no conflict of interest.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

A: The Proof of Proposition 1

Proof

1. (1)

   From Theorem 1, we have

   $$\begin{aligned} \gamma a_i=\left( s_{g-\left( \varrho ^{-1}\left( \gamma \varrho \left( g-\alpha \right) \right) \right) }, s_{\varrho ^{-1}\left( \gamma \varrho \left( \beta \right) \right) }\right) , \end{aligned}$$

   therefore,

   $$\begin{aligned}&LIFCCA\left( \gamma a_{1}, \gamma a_{2}, \cdots , \gamma a_{n}\right) \nonumber \\&=\left( \begin{array}{ccc}s_{g-\left( \varrho ^{-1}\left( \gamma \left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \right) \varrho (g-\alpha _{(k)})\right) \right) }, \\ s_{\varrho ^{-1}\left( \gamma \left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \right) \varrho \left( \beta _{(k)}\right) \right) }\end{array}\right) . \end{aligned}$$

   Furthermore,

   $$\begin{aligned}&\gamma LIFCCA\left( a_{1}, a_{2}, \cdots , a_{n}\right) \\&=\gamma \left( \begin{array}{ccc}s_{g-\left( \varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \varrho (g-\alpha _{(k)})\right) \right) }, \\ s_{\varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\varrho (\beta _{(k)})\right) \right) }\end{array}\right) \\&=\left( \begin{array}{ccc}s_{g-\left( \varrho ^{-1}\left( \gamma \left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \right) \varrho (g-\alpha _{(k)})\right) \right) }, \\ s_{\varrho ^{-1}\left( \gamma \left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \right) \varrho \left( \beta _{(k)}\right) \right) }\end{array}\right) \\&=LIFCCA\left( \gamma a_{(1)}, \gamma a_{\pi (2)}, \cdots , \gamma a_{(n)}\right) . \end{aligned}$$
2. (2)

   As \(a_i \oplus _{{\mathbb {H}}}a=\left( s_{g-\left( \varrho ^{-1}\left( \varrho \left( g-\alpha _i\right) +\varrho \left( g-\alpha \right) \right) \right) }, s_{\varrho ^{-1}\left( \varrho \left( \beta _i\right) +\varrho \left( \beta \right) \right) }\right)\), then

   $$\begin{aligned} \begin{aligned}&LIFCCA\left( a_{1}\oplus _{{\mathbb {H}}}a, \cdots , a_{n}\oplus _{{\mathbb {H}}} a\right) \\&=\left( \begin{array}{ccc}s_{g-\left( \varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \left( \varrho (g-\alpha _{(k)})\right) +\varrho (g-\alpha )\right) \right) }, \\ s_{\varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \left( \varrho \left( \beta _{(k)}\right) +\varrho \left( \beta \right) \right) \right) }\end{array}\right) . \end{aligned} \end{aligned}$$

   As \(\sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) =1\), so

   $$\begin{aligned} \begin{aligned}&LIFCCA\left( a_{1}\oplus _{{\mathbb {H}}}a, \cdots , a_{n}\oplus _{{\mathbb {H}}} a\right) \\&=\left( \begin{array}{ccc}s_{g-\left( \varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \left( \varrho (g-\alpha _{(k)})\right) \right) +\varrho (g-\alpha )\right) }, \\ s_{\varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \left( \varrho \left( \beta _{(k)}\right) \right) +\varrho \left( \beta \right) \right) }\end{array}\right) . \end{aligned} \end{aligned}$$

   And

   $$\begin{aligned} \begin{aligned}&LIFCCA\left( a_{1}, \cdots , a_{n}\right) \oplus _{{\mathbb {H}}} a \\&=\left( \begin{array}{ccc}s_{g-\left( \varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \left( \varrho (g-\alpha _{(k)})\right) \right) +\varrho (g-\alpha )\right) }, \\ s_{\varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \left( \varrho \left( \beta _{(k)}\right) \right) +\varrho \left( \beta \right) \right) }\end{array}\right) . \end{aligned} \end{aligned}$$

   Therefore,

   $$\begin{aligned}&LIFCCA\left( a_{1}\oplus _{{\mathbb {H}}}a, \cdots , a_{n}\oplus _{{\mathbb {H}}} a\right) \\&\quad =LIFCCA\left( a_{1}, \cdots , a_{n}\right) \oplus _{{\mathbb {H}}} a. \end{aligned}$$
3. (3)

   It is easy to verify (3) hold from (1) and (2).

4. (4)

   If \(a_i=a=\left( s_{\alpha }, s_{\beta }\right)\) for \(i=1, \cdots , n\), then

   $$\begin{aligned}&LIFCCA\left( a_{1}, \cdots , a_{n}\right) \\&\quad =\left( \begin{array}{ccc}s_{g-\left( \varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \varrho (g-\alpha _{(k)})\right) \right) },\\ s_{\varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\varrho (\beta _{(k)})\right) \right) }\end{array}\right) .\\&\quad =\left( \begin{array}{ccc}s_{g-\left( \varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \varrho (g-\alpha )\right) \right) },\\ s_{\varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\varrho (\beta )\right) \right) }\end{array}\right) .\\&\quad =\left( s_{g-\left( \varrho ^{-1}\left( \varrho (g-\alpha )\right) \right) }, s_{\varrho ^{-1}\left( \varrho (\beta )\right) }\right) .\\&\quad =\left( s_{\alpha }, s_{\beta }\right) \\&\quad = a. \end{aligned}$$
5. (5)

   On the one hand, since \(s_{\alpha _{(k)}}\le s_{\alpha _{(k)}^{'}}, s_{\beta _{(k)}}\ge s_{\beta _{(k)}^{'}}\) for all i, we have \(\alpha _{(k)}\le \alpha _{(k)}^{'}\), and so \(g-\alpha _{(k)}\ge g-\alpha _{(k)}^{'}\). As \(\varrho\) and \(\varrho ^{-1}\) are monotonicity decreasing, we have \(\varrho \left( g-\alpha _{(k)}\right) \le \varrho \left( g-\alpha _{(k)}^{'}\right)\), furthermore,

   $$\begin{aligned}&\sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\varrho \left( g-\alpha _{(k)}\right) \right) \\&\quad \le \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\varrho \left( g-\alpha _{(k)}^{'}\right) \right) . \end{aligned}$$

   And so

   $$\begin{aligned}&\varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\varrho \left( g-\alpha _{(k)}\right) \right) \right) \\&\ge \varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\varrho \left( g-\alpha _{(k)}^{'}\right) \right) \right) . \\&g-\left( \varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\varrho \left( g-\alpha _{(k)}\right) \right) \right) \right) \\&\le g-\left( \varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\varrho \left( g-\alpha _{(k)}^{'}\right) \right) \right) \right) . \end{aligned}$$

   On the other hand, as \(\beta _{(k)}\ge \beta _{(k)}^{'}\), so \(\varrho (\beta _{(k)})\le \varrho (\beta _{(k)}^{'})\), we have,

   $$\begin{aligned}&\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \varrho (\beta _{(k)})\right) \\&\quad \le \left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \varrho (\beta _{(k)}^{'})\right) . \end{aligned}$$

   Then

   $$\begin{aligned}&\varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \varrho (\beta _{(k)})\right) \\&\quad \ge \varrho ^{-1}\left( \sum _{k=1}^{n}\left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \varrho (\beta _{(k)}^{'})\right) . \end{aligned}$$

   Therefore, \(LIFCCA\left( a_{1}, \cdots , a_{n}\right) \ge LIFCCA\left( b_1, \cdots , b_{n}\right)\).

6. (6)

   It is easy to verify the validity of (6) from (4) and (5).

\(\square\)

B: The Proof of Theorem 2.

Proof

1. (1)

   Let \({(k)}\in S_n\) s. t. \(a_{(1)}\le \cdots \le a_{(k-1)}\le a_{(k+1)}\le \cdots \le a_{(n)}\) and \(a_i=a_{(k)}\). As \(c_i\) is unnecessary, so \(\varphi \left( \gimel _{(k)}\right) =\varphi \left( \gimel _{(k)}\bigcup c_{(k+1)}\right) =\varphi \left( \gimel _{(k+1)}\right)\). Hence,

   $$\begin{aligned}&LIFCCA\left( a_{1}, \cdots , a_{n}\right) \\&\quad =\left( \oplus _{{\mathbb {H}}}\right) _{k=1}^{n}\left( \left( \varphi \left( \gimel _{(k)}\right) -\varphi \left( \gimel _{(k+1)}\right) \right) a_{(k)}\right) \\&\quad =\left( \varphi \left( \gimel _{(k)}\right) - \varphi \left( \gimel _{(k+1)}\right) \left( \gimel _{(k)}\right) \right) \\&\qquad \oplus _{{\mathbb {H}}}\left( \left( \oplus _{{\mathbb {H}}}\right) _{j=1, j\ne k}^{n}\left( \left( \varphi \left( \gimel _{(j)}\right) -\varphi \left( \gimel _{(j+1)}\right) \right) a_{(j)}\right) \right) \\&\quad =\left( \oplus _{{\mathbb {H}}}\right) _{j=1, j\ne n}^{n}\left( \left( \varphi \left( \gimel _{(j)}\right) -\varphi \left( \gimel _{(j+1)}\right) a_{(j)}\right) \right) \\&\quad =LIFCCA\left( a_1, \cdots , a_{i-1}, a_{i+1}, \cdots , a_{n}\right) . \end{aligned}$$
2. (2)

   Let \({(k)}\in S_n\) s. t. \(a_{(1)}\le \cdots \le a_{(k-1)}\le a_{(k+1)}\le \cdots \le a_{(n)}\) and \(a_j=a_{(k)}\). As \(c_j\) is independent, so \(\varphi \left( \gimel _{(k)}\right) =\varphi \left( \gimel _{(k+1)}\right) +\varphi \left( c_{(k)}\right)\). Therefore,

   $$\begin{aligned}&LIFCCA\left( a_{1}, \cdots , a_{n}\right) \\&\quad =\left( \begin{array}{ccc}s_{g-\left( \varrho ^{-1}\left( \sum _{i=1}^{n}\left( \varphi (\gimel _{(i)})-\varphi (\gimel _{(i+1)})\right) \varrho (g-\alpha _{(i)})\right) \right) }, \\ s_{\varrho ^{-1}\left( \sum _{i=1}^{n}\left( \varphi (\gimel _{(i)})-\varphi (\gimel _{(i+1)})\varrho (\beta _{(i)})\right) \right) }\end{array}\right) \\&\quad =\left( \begin{array}{ccc}s_{g-\left( \varrho ^{-1}\left( \left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\right) \varrho \left( g-\alpha _{(j)}\right) \right) \right) },\\ s_{\varrho ^{-1}\left( \left( \varphi (\gimel _{(k)})-\varphi (\gimel _{(k+1)})\varrho (\beta _{(i)})\right) +\sum _{i=1,i\ne k}^{n}\left( \varphi (\gimel _{(i)})-\varphi (\gimel _{(i+1)})\varrho (\beta _{(i)})\right) \right) }\end{array}\right) \\&\quad =\left( \begin{array}{ccc}s_{g-\left( \varrho ^{-1}\left( \varphi (\gimel _{(k)})\varrho \left( g-\alpha _{(k)}\right) \right) +\sum _{i=1,i\ne k}^{n}\left( \varphi (\gimel _{(i)})-\varphi (\gimel _{(i+1)})\varrho \left( g-\alpha _{(j)}\right) \right) \right) },\\ s_{\varrho ^{-1}\left( \left( \varphi (\gimel _{(k)})\varrho (\beta _{(k)})\right) +\sum _{i=1,i\ne k}^{n}\left( \varphi (\gimel _{(i)})-\varphi (\gimel _{(i+1)})\varrho (\beta _{(i)})\right) \right) }\end{array}\right) \\&\quad =(a_k)^{\varphi (c_{(k)})}\otimes _{{\mathbb {H}}}LIFCCA\left( a_1, a_2, \cdots , a_{k-1}, a_{k+1}, \cdots , a_{n}\right) . \end{aligned}$$

\(\square\)

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