Table 9 The third normalized decision matrix \(M_{3}^{s} = \left[ {m_{ij}^{s3} } \right]_{4 \times 5}\) by \(D_{3}\).

From: Behavioral TOPSIS technique based on probabilistic picture hesitant fuzzy probability splitting algorithm and novel interactive operations

Alternatives

\(G_{1}\)

\(G_{2}\)

\(G_{3}\)

\(G_{4}\)

\(G_{5}\)

\(\Re_{1}\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.4} \right|0.2, \cdots ,\left. {0.4} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.5} \right|0.2, \cdots ,\left. {0.5} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.5} \right|0.2, \cdots ,\left. {0.6} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.4} \right|0.2, \cdots ,\left. {0.4} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\Re_{2}\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.5} \right|0.2, \cdots ,\left. {0.6} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. 0 \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.1} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.1} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.5} \right|0.2, \cdots ,\left. {0.5} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.1} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.4} \right|0.2, \cdots ,\left. {0.4} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.5} \right|0.2, \cdots ,\left. {0.5} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.1} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\Re_{3}\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.6} \right|0.2, \cdots ,\left. {0.6} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. 0 \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.1} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.4} \right|0.2, \cdots ,\left. {0.5} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.1} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.5} \right|0.2, \cdots ,\left. {0.5} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.5} \right|0.2, \cdots ,\left. {0.5} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.5} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.4} \right|0.2, \cdots ,\left. {0.4} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\Re_{4}\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.1} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.1} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.4} \right|0.2, \cdots ,\left. {0.4} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.4} \right|0.2, \cdots ,\left. {0.4} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.2} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.3} \right|0.2, \cdots ,\left. {0.3} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.4} \right|0.2, \cdots ,\left. {0.4} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

\(\left\langle \begin{gathered} \left\{ {\left. {0.1} \right|0.2, \cdots ,\left. {0.1} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.2} \right|0.2, \cdots ,\left. {0.4} \right|0.2} \right\}, \hfill \\ \left\{ {\left. {0.5} \right|0.2, \cdots ,\left. {0.5} \right|0.2} \right\} \hfill \\ \end{gathered} \right\rangle\)

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