Abstract
A useful expansion of the intuitionistic fuzzy set (IFS) for dealing with ambiguities in information is the Pythagorean fuzzy set (PFS), which is one of the most frequently used fuzzy sets in data science. Due to these circumstances, the Aczel-Alsina operations are used in this study to formulate several Pythagorean fuzzy (PF) Aczel-Alsina aggregation operators, which include the PF Aczel-Alsina weighted average (PFAAWA) operator, PF Aczel-Alsina order weighted average (PFAAOWA) operator, and PF Aczel-Alsina hybrid average (PFAAHA) operator. The distinguishing characteristics of these potential operators are studied in detail. The primary advantage of using an advanced operator is that it provides decision-makers with a more comprehensive understanding of the situation. If we compare the results of this study to those of prior strategies, we can see that the approach proposed in this study is more thorough, more precise, and more concrete. As a result, this technique makes a significant contribution to the solution of real-world problems. Eventually, the suggested operator is put into practise in order to overcome the issues related to multi-attribute decision-making under the PF data environment. A numerical example has been used to show that the suggested method is valid, useful, and effective.




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Funding
This work was supported by the National Natural Science Foundation of China (Grant No-12071376), the Slovak Research and Development Agency (Grant No. APVV-18-0052) and the IGA project of the Faculty of Science Palacky University Olomouc (Grant No-PrF2019015).
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Appendix
Appendix
Proof of Theorem 1
For the three PFEs \(\delta\), \(\delta _1\) and \(\delta _2\), and \(\varphi , \varphi _1, \varphi _2 > 0\), as provided in Definition 6, we may obtain
- (i):
-
$$\begin{aligned}&\delta _1\bigoplus \delta _2 \\&\quad = \bigg \langle \sqrt{1-e^{-((-\log (1- \gamma ^2_{\delta _1}))^\wp +(-\log (1-\gamma ^2_{\delta _2}))^\wp )^{1/\wp }}}, \\&\quad \quad \sqrt{e^{-((-\log \upsilon ^2_{\delta _1})^\wp +(-\log \upsilon ^2_{\delta _2})^\wp )^{1/\wp }}}\bigg \rangle \\&\quad =\bigg \langle \sqrt{1-e^{-((-\log (1- \gamma ^2_{\delta _2}))^\wp +(-\log (1-\gamma ^2_{\delta _1}))^\wp )^{1/\wp }}},\\&\quad \quad \sqrt{e^{-((-\log \upsilon ^2_{\delta _2})^\wp +(-\log \upsilon ^2_{\delta _1})^\wp )^{1/\wp }}}\bigg \rangle =\delta _2 \bigoplus \delta _1 \end{aligned}.$$
- (ii):
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It is straightforward.
- (iii):
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Let \(t= \sqrt{1-e^{-((-\log (1- \gamma ^2_{\delta _1}))^\wp +(-\log (1-\gamma ^2_{\delta _2}))^\wp )^{1/\wp }}}.\)
Then \(\log (1-t^2)=-((-\log (1- \gamma ^2_{\delta _1}))^\wp +(-\log (1-\gamma ^2_{\delta _2}))^\wp )^{1/\wp }\). Using this, we get
$$\begin{aligned}&\varphi (\delta _1 \bigoplus \delta _2) \\&\quad =\varphi \bigg \langle \sqrt{1-e^{-((-\log (1- \gamma ^2_{\delta _1}))^\wp +(-\log (1-\gamma ^2_{\delta _2}))^\wp )^{1/\wp }}},\\&\quad \quad \sqrt{e^{-((-\log \upsilon ^2_{\delta _1})^\wp +(-\log \upsilon ^2_{\delta _2})^\wp )^{1/\wp }}}\bigg \rangle \\&\quad =\bigg \langle \sqrt{1-e^{-(\varphi ((-\log (1- \gamma ^2_{\delta _1}))^\wp +(-\log (1-\gamma ^2_{\delta _2}))^\wp ))^{1/\wp }}},\\&\quad \quad \sqrt{e^{-(\varphi ((-\log \upsilon ^2_{\delta _1})^\wp +(-\log \upsilon ^2_{\delta _2})^\wp ))^{1/\wp }}}\bigg \rangle \\&\quad =\bigg \langle \sqrt{1-e^{-(\varphi (-\log (1- \gamma ^2_{\delta _1}))^\wp )^{1/\wp }}}, \\&\quad \quad \sqrt{e^{-(\varphi (-\log \upsilon ^2_{\delta _1})^\wp )^{1/\wp }}}\bigg \rangle \\&\quad \bigoplus \bigg \langle \sqrt{1-e^{-(\varphi (-\log (1- \gamma ^2_{\delta _2}))^\wp )^{1/\wp }}}, \\&\quad \quad \sqrt{e^{-(\varphi (-\log \upsilon ^2_{\delta _2})^\wp )^{1/\wp }}}\bigg \rangle =\varphi \delta _1 \bigoplus \varphi \delta _2 \end{aligned}.$$ - (iv):
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$$\begin{aligned}&\varphi _1 \delta \bigoplus \varphi _2 \delta =\Big \langle \sqrt{1-e^{-(\varphi _1(-\log (1- \gamma ^2_{\delta }))^\wp )^{1/\wp }}}, \\&\quad \quad \sqrt{e^{-(\varphi _1(-\log \upsilon ^2_{\delta })^\wp )^{1/\wp }}}\Big \rangle \\&\quad \bigoplus \Big \langle \sqrt{1-e^{-(\varphi _2(-\log (1- \gamma ^2_{\delta }))^\wp )^{1/\wp }}}, \\&\quad \quad \sqrt{e^{-(\varphi _2(-\log \upsilon ^2_{\delta })^\wp )^{1/\wp }}}\Big \rangle \\&\quad =\Big \langle \sqrt{1-e^{-((\varphi _1+\varphi _2)(-\log (1- \gamma ^2_{\delta }))^\wp )^{1/\wp }}},\\&\quad \quad \sqrt{e^{-((\varphi _1+\varphi _2)(-\log \upsilon ^2_{\delta })^\wp )^{1/\wp }}}\Big \rangle \\&\quad =(\varphi _1+\varphi _2 )\delta \end{aligned}.$$
- (v):
-
$$\begin{aligned}&(\delta _1 \bigotimes \delta _2)^\varphi = \Big \langle \sqrt{e^{-((-\log \gamma ^2_{\delta _1})^\wp +(-\log \gamma ^2_{\delta _2})^\wp )^{1/\wp }}}, \\&\quad \quad \sqrt{1-e^{-((-\log (1- \upsilon ^2_{\delta _1}))^\wp +(-\log (1-\upsilon ^2_{\delta _2}))^\wp )^{1/\wp }}}\Big \rangle ^\varphi \\&\quad =\Big \langle \sqrt{e^{-(\varphi ((-\log \gamma ^2_{\delta _1})^\wp +(-\log \gamma ^2_{\delta _2})^\wp ))^{1/\wp }}}, \\&\quad \quad \sqrt{1-e^{-(\varphi ((-\log (1- \upsilon ^2_{\delta _1}))^\wp +(-\log (1-\upsilon ^2_{\delta _2}))^\wp )^{1/\wp }}}\Big \rangle \\&\quad =\Big \langle \sqrt{e^{-(\varphi (-\log \gamma ^2_{\delta _1})^\wp )^{1/\wp }}}, \\&\quad \quad \sqrt{1-e^{-(\varphi (-\log (1- \upsilon ^2_{\delta _1}))^\wp )^{1/\wp }}}\Big \rangle \\&\quad \bigoplus \Big \langle \sqrt{e^{-(\varphi (-\log \gamma ^2_{\delta _2})^\wp )^{1/\wp }}}, \\&\quad \quad \sqrt{1-e^{-(\varphi (-\log (1- \upsilon ^2_{\delta _2}))^\wp )^{1/\wp }}}\Big \rangle \\&\quad =\delta _1^\varphi \bigotimes \delta _2^\varphi \end{aligned}.$$
- (vi):
-
$$\begin{aligned}&\delta ^{\varphi _1} \bigotimes \delta ^{\varphi _2} =\Big \langle \sqrt{e^{-(\varphi _1(-\log \gamma ^2_{\delta })^\wp )^{1/\wp }}}, \\&\quad \quad \sqrt{1-e^{-(\varphi _1(-\log (1- \upsilon ^2_{\delta }))^\wp )^{1/\wp }}} \Big \rangle \\&\quad \bigotimes \Big \langle \sqrt{e^{-(\varphi _2(-\log \gamma ^2_{\delta })^\wp )^{1/\wp }}}, \\&\quad \quad \sqrt{1-e^{-(\varphi _2(-\log (1- \upsilon ^2_{\delta }))^\wp )^{1/\wp }}} \Big \rangle \\&\quad =\Big \langle \sqrt{e^{-((\varphi _1+\varphi _2)(-\log \gamma ^2_{\delta })^\wp )^{1/\wp }}},\\&\quad \quad \sqrt{1-e^{-((\varphi _1+\varphi _2)(-\log (1- \upsilon ^2_{\delta }))^\wp )^{1/\wp }}}\Big \rangle \\&\quad =\delta ^{(\varphi _1+\varphi _2 )} \end{aligned}.$$
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Senapati, T., Chen, G., Mesiar, R. et al. Multiple attribute decision making based on Pythagorean fuzzy Aczel-Alsina average aggregation operators. J Ambient Intell Human Comput 14, 10931–10945 (2023). https://doi.org/10.1007/s12652-022-04360-4
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DOI: https://doi.org/10.1007/s12652-022-04360-4