Introduction
Preliminaries
Circular versus disc Pythagorean fuzzy sets
Semantic interpretation
Set-theoretic and arithmetic operations on D-PFSs
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The complement operation is defined as: \(A^{c}= \{(\nu _{A}(\hbar ),\mu _{A}(\hbar ); r(\hbar )) \;\mid \; \hbar \in L \} \).
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The set containment is defined as: \( A_{1} \subseteq A_{2} \Longleftrightarrow r_{1}(\hbar ) \le r_{2}(\hbar ),\; \mu _{A_{1}}(\hbar ) \le \mu _{A_{2}}(\hbar ) \; \& \; \nu _{A_{1}}(\hbar ) \ge \nu _{A_{2}}(\hbar )\).
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The union operations are defined as$$\begin{aligned} A_{1} \cup _{\max } A_{2}&=\bigg \{ \bigg ( \max \{\mu _{A_{1}}(\hbar ),\mu _{A_{2}}(\hbar ) \}, \min \{\nu _{A_{1}}(\hbar ), \\&\qquad \nu _{A_{2}}(\hbar )\}; \max \{r_{1}(\hbar ),r_{2}(\hbar )\} \bigg ) \; \mid \; \hbar \in L \bigg \} \\ A_{1} \cup _{\min } A_{2}&=\bigg \{ \bigg ( \max \{\mu _{A_{1}}(\hbar ),\mu _{A_{2}}(\hbar ) \},\min \{\nu _{A_{1}}(\hbar ),\\ {}&\qquad \nu _{A_{2}}(\hbar )\}; \min \{r_{1}(\hbar ),r_{2}(\hbar )\} \bigg ) \;\mid \; \hbar \in L \bigg \}. \end{aligned}$$
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The intersection operations are defined as$$\begin{aligned} A_{1} \cap _{\max } A_{2}&= \bigg \{ \bigg ( \min \{\mu _{A_{1}}(\hbar ),\mu _{A_{2}}(\hbar ) \}, \max \{\nu _{A_{1}}(\hbar ),\\&\qquad \nu _{A_{2}}(\hbar )\}; \max \{r_{1}(\hbar ),r_{2}(\hbar )\} \bigg ) \mid \hbar \in L \bigg \} \\ A_{1} \cap _{\min } A_{2}&= \bigg \{ \bigg ( \min \{\mu _{A_{1}}(\hbar ),\mu _{A_{2}}(\hbar ) \}, \max \{\nu _{A_{1}}\\&\qquad (\hbar ),\nu _{A_{2}}(\hbar )\}; \min \{r_{1}(\hbar ),r_{2}(\hbar )\} \bigg ) \mid \hbar \in L \bigg \}. \end{aligned}$$
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The addition operations are defined as$$\begin{aligned}&A_{1} \oplus _{\max } A_{2} = \bigg \{ \bigg (\sqrt{ \mu _{A_{1}}^{2}(\hbar ) + \mu _{A_{2}}^{2}(\hbar ) - \mu _{A_{1}}^{2}(\hbar ) . \mu _{A_{2}}^{2}(\hbar ) }\; ,\\ {}&\quad \nu _{A_{1}}(\hbar ) . \nu _{A_{2}}(\hbar ) ; \max \{r_{1}(\hbar ),r_{2}(\hbar )\} \bigg ) \mid \hbar \in L \bigg \},\\&A_{1} \oplus _{\min } A_{2} = \bigg \{ \bigg (\sqrt{\mu _{A_{1}}^{2}(\hbar ) + \mu _{A_{2}}^{2}(\hbar ) - \mu _{A_{1}}^{2}(\hbar ) . \mu _{A_{2}}^{2}(\hbar ) }\;,\\ {}&\quad \nu _{A_{1}}(\hbar ) . \nu _{A_{2}}(\hbar ) ; \min \{r_{1}(\hbar ),r_{2}(\hbar )\} \bigg ) \mid \hbar \in L \bigg \}. \end{aligned}$$
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The multiplication operations are defined as$$\begin{aligned} A_{1} \otimes _{\max } A_{2}&= \bigg \{ \bigg ( \mu _{A_{1}}(\hbar ) . \mu _{A_{2}}(\hbar ) \;,\\&\qquad \sqrt{ \nu _{A_{1}}^{2}(\hbar ) + \nu _{A_{2}}^{2}(\hbar ) - \nu _{A_{1}}^{2}(\hbar ) \cdot \nu _{A_{2}}^{2}(\hbar )}\; ;\\&\qquad \max \{r_{1}(\hbar ),r_{2}(\hbar )\} \bigg ) \mid \hbar \in L \bigg \}\\ A_{1} \otimes _{\min } A_{2}&= \bigg \{ \bigg ( \mu _{A_{1}}(\hbar ) . \mu _{A_{2}}(\hbar ) \;,\\&\quad \sqrt{ \nu _{A_{1}}^{2}(\hbar ) + \nu _{A_{2}}^{2}(\hbar ) - \nu _{A_{1}}^{2}(\hbar ) \cdot \nu _{A_{2}}^{2}(\hbar )} \;;\\&\qquad \min \{r_{1}(\hbar ),r_{2}(\hbar )\} \bigg ) \mid \hbar \in L \bigg \}. \end{aligned}$$
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The scalar multiplication operations are defined as: for \(\lambda > 0\)$$\begin{aligned} \lambda A&= \left\{ \left( \sqrt{ 1 - (1 - \mu _{A}^{2}(\hbar ))^{\lambda } }\; ,\; \nu _{A}^{\lambda }(\hbar ) \;; r(\hbar ) \right) \;\mid \; \hbar \in L \right\} \\ A^{\lambda }&= \left\{ \left( \mu _{A}^{\lambda }(\hbar ) \;,\;\sqrt{ 1 - (1- \nu _{A}^{2}(\hbar ))^{\lambda } } \;; r(\hbar ) \right) \;\mid \; \hbar \in L \right\} . \end{aligned}$$
Operating with D-PFSs: aggregation and distances
D-PF aggregation operators
Distance measures for D-PFSs
Applications of D-PFSs in MCDM Problems
Disc Pythagorean fuzzy CODAS method
Illustrative example: selection of supermarket for fresh fruits
\(M^{1}\) | \(C_1\) | \(C_2\) | \(C_3\) | \(C_4\) | \(C_5\) |
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\(\hbar _{1}\) | (.75, .35; .04) | (.70, .19; .01) | (.75, .45; .05) | (.60, .23; .05) | (.75, .13; .05) |
\(\hbar _{2}\) | (.45, .40; .07) | (.80, .17; .06) | (.63, .55; .05) | (.70, .28; .06) | (.66, .22; .04) |
\(\hbar _{3}\) | (.60, .60; .09) | (.60, .20; .04) | (.67, .10; .06) | (.80, .17; .09) | (.76, .32; .05) |
\(\hbar _{4}\) | (.50, .56; .05) | (.70, .15; .02) | (.55, .12; .07) | (.73, .25; .07) | (.80, .40; .06) |
\(M^{2}\) | \(C_1\) | \(C_2\) | \(C_3\) | \(C_4\) | \(C_5\) |
---|---|---|---|---|---|
\(\hbar _{1}\) | (.55, .65; .04) | (.60, .25; .01) | (.55, .35; .05) | (.70, .23; .05) | (.55, .13; .05) |
\(\hbar _{2}\) | (.65, .40; .07) | (.80, .15; .06) | (.63, .45; .05) | (.40, .28; .06) | (.65, .22; .04) |
\(\hbar _{3}\) | (.70, .20; .09) | (.70, .20; .04) | (.67, .10; .06) | (.80, .17; .09) | (.85, .32; .05) |
\(\hbar _{4}\) | (.65, .56; .05) | (.70, .15; .02) | (.65, .12; .07) | (.75, .25; .07) | (.80, .40; .06) |
\(M^{3}\) | \(C_1\) | \(C_2\) | \(C_3\) | \(C_4\) | \(C_5\) |
---|---|---|---|---|---|
\(\hbar _{1}\) | (.65, .25; .04) | (.60, .19; .01) | (.85, .15; .05) | (.80, .20; .05) | (.65, .15; .05) |
\(\hbar _{2}\) | (.65, .30; .07) | (.80, .15; .06) | (.65, .25; .05) | (.70, .25; .06) | (.60, .20; .04) |
\(\hbar _{3}\) | (.50, .40; .09) | (.70, .10; .04) | (.60, .10; .06) | (.80, .15; .09) | (.55, .30; .05) |
\(\hbar _{4}\) | (.80, .15; .05) | (.70, .15; .02) | (.50, .15; .07) | (.75, .20; .07) | (.80, .20; .06) |
M | \(C_1\) | \(C_2\) | \(C_3\) | \(C_4\) | \(C_5\) |
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\(\hbar _{1}\) | (.659, .412; .04) | (.639, .212; .01) | (.727, .309; .05) | (.703, .222; .05) | (.659, .135; .05) |
\(\hbar _{2}\) | (.595, .372; .07) | (.800, .157; .06) | (.635, .417; .05) | (.614, .272; .06) | (.642, .215; .04) |
\(\hbar _{3}\) | (.626, .349; .09) | (.669, .168; .04) | (.654, .100; .06) | (.800, .165; .09) | (.772, .315; .05) |
\(\hbar _{4}\) | (.662, .403; .05) | (.700, .150; .02) | (.584, .127; .07) | (.743, .236; .07) | (.800, .336; .06) |
\(M'\) | \(C_1\) | \(C_2\) | \(C_3\) | \(C_4\) | \(C_5\) |
---|---|---|---|---|---|
\(\hbar _{1}\) | (.659, .412; .04) | (.639, .212; .01) | (.309, .727; .05) | (.703, .222; .05) | (.659, .135; .05) |
\(\hbar _{2}\) | (.595, .372; .07) | (.800, .157; .06) | (.417, .635; .05) | (.614, .272; .06) | (.642, .215; .04) |
\(\hbar _{3}\) | (.626, .349; .09) | (.669, .168; .04) | (.100, .654; .06) | (.800, .165; .09) | (.772, .315; .05) |
\(\hbar _{4}\) | (.662, .403; .05) | (.700, .150; .02) | (.127, .584; .07) | (.743, .236; .07) | (.800, .336; .06) |
\(C_1\) | \(C_2\) | \(C_3\) | \(C_4\) | \(C_5\) | |
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\(\hbar _{1}\) | (.397, .767; .04) | (.316, .733; .01) | (.122, .953; .05) | (.312, .798; .05) | (.328, .670; .05) |
\(\hbar _{2}\) | (.350, .743; .07) | (.430, .690; .06) | (.168, .934; .05) | (.262, .823; .06) | (.318, .735; .04) |
\(\hbar _{3}\) | (.372, .729; .09) | (.335, .700; .04) | (.039, .938; .06) | (.377, .763; .09) | (.407, .794; .05) |
\(\hbar _{4}\) | (.398, .761; .05) | (.355, .684; .02) | (.049, .923; .07) | (.337, .805; .07) | (.430, .804; .06) |
Authors | Method | Rankings |
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Zhang and Xu [12] | TOPSIS method |
\(\hbar _2 \succ \hbar _4\succ \hbar _3\succ \hbar _1\)
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Garg [17] | PyFEWA operator |
\(\hbar _3 \succ \hbar _4\succ \hbar _1\succ \hbar _2\)
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Ashraf et al. [18] | ST-PFWA operator |
\(\hbar _4 \succ \hbar _3\succ \hbar _1\succ \hbar _2\)
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Khan et al. [19] | PFDWA operator |
\(\hbar _4 \succ \hbar _3\succ \hbar _2\succ \hbar _1\)
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Peng and Yang [20] | MABAC method |
\(\hbar _3 \succ \hbar _4\succ \hbar _2\succ \hbar _1\)
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Akram et al. [21] | ELECTRE-II method |
\(\hbar _2 \succ \hbar _4\succ \hbar _3\succ \hbar _1\)
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Khan et al. [22] | VIKOR method |
\(\hbar _2 \succ \hbar _4\succ \hbar _3\succ \hbar _1\)
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Proposed | CODAS and \(\textrm{CPFWA}_{\min }\) |
\(\hbar _3 \succ \hbar _4\succ \hbar _2\succ \hbar _1\)
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Proposed | CODAS and \(\textrm{CPFWA}_{\max }\) |
\(\hbar _3 \succ \hbar _4\succ \hbar _2\succ \hbar _1\)
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Proposed | CODAS and \(\textrm{CPFWA}_{\max }\) |
\(\hbar _2 \succ \hbar _4\succ \hbar _3\succ \hbar _1\)
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Proposed | CODAS and \(\textrm{CPFWA}_{\min }\) |
\(\hbar _2 \succ \hbar _4\succ \hbar _3\succ \hbar _1\)
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