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

Abstract

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.

Appendix

A: The Proof of Proposition 1

Proof

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

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

$\square$

B: The Proof of Theorem 2.

Proof

• (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) 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,

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

$\square$