Abstract
The picture fuzzy sets (PFSs) state or model the voting information accurately without information loss. However, their existing operational laws usually generate unreasonable computing results, especially when the agreement degree (AD) or neutrality degree (ND) or opposition degree (OD) is zero. To tackle this issue, we propose the interactional operational laws (IOLs) to compute picture fuzzy numbers (PFNs), which can capture the interaction between the ADs and NDs in two PFNs, as well as the interaction between the ADs and ODs in two PFNs. Based on the proposed novel IOLs, partitioned Heronian mean (PHM) operator, and partitioned geometric Heronian mean (PGHM) operator, some picture fuzzy interactional PHM (PFIPHM), weighted PFIPHM (PFIWPHM), geometric PFIPHM (PFIPGHM), and weighted PFIPGHM (PFIWPGHM) operators are proposed in this paper. Afterwards, we investigate the properties of these operators. Using the PFIWPHM and PFIWPGHM operators, a novel multiple attribute decision-making (MADM) method with PFNs is elaborated. Finally, a study example that involves the service quality ranking of nursing facilities is provided to show the decision procedure of the proposed MADM method and we also give the comparative analysis between the proposed operators and the existing aggregation operators developed for PFNs.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96
Beliakov G, Bustince H, James S, Calvo T, Fernandez J (2012) Aggregation for Atanassov’s intuitionistic and interval valued fuzzy sets: the median operator. IEEE Trans Fuzzy Syst 20(3):487–498
Can GF, Demirok S (2019) Universal usability evaluation by using an integrated fuzzy multi criteria decision making approach. Int J Intell Comput Cybern 12(2):194–223
Cuong BC (2014) Picture fuzzy sets. J Comput Sci Cybern 30(4):409–420
Cuong BC, Hai PV (2015) Some fuzzy logic operators for picture fuzzy sets. In: Seventh international conference on knowledge and systems engineering, pp 132–137
Cuong BC, Kreinovich V (2013) Picture fuzzy sets—a new concept for computational intelligence problems. In: 2013 Third world congress on information and communication technologies 2016, pp 1-6
Cuong BC, Kreinovich V, Ngan RT (2016) A classification of representable t-norm operators for picture fuzzy sets. Departmental Tech Rep, p 1047
Garg H (2018a) Some methods for strategic decision-making problems with immediate probabilities in Pythagorean fuzzy environment. Int J Intell Syst 33(4):687–712
Garg H (2018b) New exponential operational laws and their aggregation operators for interval-valued Pythagorean fuzzy multicriteria decision-making. Int J Intell Syst 33(3):653–683
Garg H (2019a) Hesitant Pythagorean fuzzy Maclaurin symmetric mean operators and its applications to multiattribute decision-making process. Int J Intell Syst 34(4):601–626
Garg H (2019b) Novel neutrality operation–based Pythagorean fuzzy geometric aggregation operators for multiple attribute group decision analysis. Int J Intell Syst 34(10):2459–2489
Garg H, Kumar K (2020) A novel exponential distance and its based TOPSIS method for interval-valued intuitionistic fuzzy sets using connection number of SPA theory. Artificial Intell Rev 53:595–624
Jana C, Senapati T, Pal M, Yager RR (2019) Picture fuzzy Dombi aggregation operators: application to MADM process. Appl Soft Comput 74:99–109
Kakati P, Borkotokey S, Rahman S, Davvaz B (2020) Interval neutrosophic hesitant fuzzy Einstein Choquet integral operator for multicriteria decision making. Artificial Intell Rev 53:2171–2206
Khan MJ, Kumam P (2021) Distance and similarity measures of generalized intuitionistic fuzzy soft set and its applications in decision support system. Adv Intell Syst Comput 1197:355–362
Khan MJ, Kumam P, Ashraf S, Kumam W (2019a) Generalized picture fuzzy soft sets and their application in decision support systems. Symmetry 11(3):415
Khan MJ, Kumam P, Liu PD, Kumam W, Ashraf S (2019b) A novel approach to generalized intuitionistic fuzzy soft sets and its application in decision support system. Mathematics 7(8):742
Khan MJ, Kumam P, Liu PD, Kumam W, ur Rehman H (2020a) An adjustable weighted soft discernibility matrix based on generalized picture fuzzy soft set and its applications in decision making. J Intell Fuzzy Syst 38(2):2103–2118
Khan MJ, Kumam P, Deebani W, Kumam W, Shah Z (2020b) Bi-parametric distance and similarity measures of picture fuzzy sets and their applications in medical diagnosis. Egypt Inf J. https://doi.org/10.1016/j.eij.2020.08.002
Khan MJ, Kumam P, Liu PD, Kumam W (2020c) Another view on generalized interval valued intuitionistic fuzzy soft set and its applications in decision support system. J Intell Fuzzy Syst 38(4):4327–4341
Khan MJ, Phiangsungnoen S, ur Rehman H, Kumam W (2020d) Applications of generalized picture fuzzy soft set in concept selection. Thai J Math 18(1):296–314
Khan MJ, Kumam P, Deebani W, Kumam W, Shah Z (2020e) Distance and similarity measures for spherical fuzzy sets and their applications in selecting mega projects. Mathematics 8(4):519
Khan MJ, Kumam P, Alreshidi NA, Shaheen N, Kumam W, Shah Z, Thounthong P (2020f) The renewable energy source selection b remoteness index-based VIKOR method for generalized intuitionistic fuzzy soft sets. Symmetry 12(6):977
Lei Q, Xu ZS, Bustince H, Fernandez J (2016) Intuitionistic fuzzy integrals based on Archimedean t-conorms and t-norms. Inf Sci 327:57–70
Li H, Lv L, Li F, Wang L, Xia Q (2020) A novel approach to emergency risk assessment using FMEA with extended MULTIMOORA method under interval-valued Pythagorean fuzzy environment. Int J Intell Comput Cybern 13(1):41–65
Liao HC, Xu ZS (2014) Intuitionistic fuzzy hybrid weighted aggregation operators. Int J Intell Syst 29(11):971–993
Lin MW, Wei JH, Xu ZS, Chen RQ (2018a) Multiattribute group decision-making based on linguistic Pythagorean fuzzy interaction partitioned Bonferroni mean aggregation operators. Complexity, p 24
Lin MW, Xu ZS, Zhai YL, Yao ZQ (2018b) Multi-attribute group decision-making under probabilistic uncertain linguistic environment. J Oper Res Soc 69(2):157–170
Lin MW, Chen ZY, Liao HC, Xu ZS (2019) ELECTRE II method to deal with probabilistic linguistic term sets and its application to edge computing. Nonlinear Dyn 96(3):2125–2143
Lin MW, Huang C, Xu ZS (2020a) MULTIMOORA based MCDM model for site selection of car sharing station under picture fuzzy environment. Sustain Cities Soc 53:101873
Lin MW, Huang C, Xu ZS, Chen RQ (2020b) Evaluating IoT platforms using integrated probabilistic linguistic MCDM method. IEEE Int Things J 7(11):11195–11208
Lin MW, Wang HB, Xu ZS (2020c) TODIM-based multi-criteria decision-making method with hesitant fuzzy linguistic term sets. Artificial Intell Rev 53:3647–3671
Lin MW, Li XM, Chen LF (2020d) Linguistic q-rung orthopair fuzzy sets and their interactional partitioned Heronian mean aggregation operators. Int J Intell Syst 35(2):217–249
Lin MW, Xu WS, Lin ZP, Chen RQ (2020e) Determine OWA operator weights using kernel density estimation. Econ Res Ekonomska Istraživanja 33(1):1441–1464
Liu PD, You XL (2020) Linguistic neutrosophic partitioned Maclaurin symmetric mean operators based on clustering algorithm and their application to multi-criteria group decision-making. Artificial Intell Rev 53:2131–2170
Liu P, Liu Z, Zhang X (2014) Some intuitionistic uncertain linguistic Heronian mean operators and their application to group decision making. Appl Math Comput 230:570–586
Liu PD, Chen SM, Liu JL (2017) Multiple attribute group decision making based on intuitionistic fuzzy interaction partitioned Bonferroni mean operators. Inform Sci 411:98–121
Liu P, Liu J, Merigo JM (2018) Partitioned Heronian means based on linguistic intuitionistic fuzzy numbers for dealing with multi-attribute group decision making. Appl Soft Comput 62:395–422
Özkan B, Özceylan E, Kabak M, Dağdeviren M (2020) Evaluating the websites of academic departments through SEO criteria: a hesitant fuzzy linguistic MCDM approach. Artificial Intell Rev 53:875–905
Pena J, Nápoles G, Salgueiro Y (2020) Explicit methods for attribute weighting in multi-attribute decision- making: a review study. Artificial Intell Rev 53:3127–3152
Peng XD, Dai J (2020) A bibliometric analysis of neutrosophic set: two decades review from 1998 to 2017. Artificial Intell Rev 53:199–255
Peng XD, Selvachandran G (2019) Pythagorean fuzzy set: state of the art and future directions. Artificial Intell Rev 52:1873–1927
Qin JD, Liu XW, Pedrycz W (2016) Frank aggregation operators and their application to hesitant fuzzy multiple attribute decision making. Appl Soft Comput J 41:428–452
Qiyas M, Khan MA, Khan S, Abdullah S (2020) Concept of Yager operators with the picture fuzzy set environment and its application to emergency program selection. Int J Intell Comput Cybern 13(4):455–483
Rong Y, Liu Y, Pei Z (2020) Complex q-rung orthopair fuzzy 2-tuple linguistic Maclaurin symmetric mean operators and its application to emergence program selection. Int J Intell Syst 35(11):1749–1790
Tan CQ, Yi WT, Chen XH (2015) Hesitant fuzzy Hamacher aggregation operators for multicriteria decision making. Appl Soft Comput J 26:325–349
Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25(6):529–539
Wei GW (2017) Picture fuzzy aggregation operators and their application to multiple attribute decision making. J Intell Fuzzy Syst 33:713–724
Wei GW (2018) Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Fundamenta Inf 157:271–320
Wei GW, Lu M, Gao H (2018a) Picture fuzzy heronian mean aggregation operators in multiple attribute decision making. Int J Knowl Based Intell Eng Syst 22:167–175
Wei GW, Gao H, Wei Y (2018b) Some q-rung orthopair fuzzy Heronian mean operators in multiple attribute decision making. Int J Intell Syst 33(7):1426–1458
Yu DJ (2013) Intuitionistic fuzzy geometric Heronian mean aggregation operators. Appl Soft Comput 13(2):1235–1246
Yu Q, Hou FJ, Zhai YB, Du YQ (2016) Some hesitant fuzzy Einstein aggregation operators and their application to multiple attribute group decision making. Int J Intell Syst 31(7):722–746
Zadeh LA (1965) Fuzzy sets. Inform. Control 8(3):338–353
Zhan JM, Sun BZ (2020) Covering-based intuitionistic fuzzy rough sets and applications in multi-attribute decision-making. Artificial Intell Rev 53:671–701
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 61872086, 61972093 and U1805263, and Digital Fujian Institute of Big Data for Agriculture and Forestry under Grant No. KJG18019A.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A. Proof of Theorems
Appendix A. Proof of Theorems
Theorem 1
Given any two PFNs \(p_{1} = \left( {\mu_{1} ,\eta_{1} ,\nu_{1} } \right)\) and \(p_{2} = \left( {\mu_{2} ,\eta_{2} ,\nu_{2} } \right)\), then the computing result from \(p_{1} \oplus p_{2}\) is also an PFN.
Proof
Since \(0 \le \mu_{l} \le 1\), then we can derive \(1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0} \right)} \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1} \right)} \Rightarrow 0 \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} \le 1\).
Let \(g = \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)}\), then the derivative of the mathematical function \(g\) with regard to the variables \(\mu_{1}\), \(\mu_{2}\), \(\eta_{1}\), \(\eta_{2}\) can be computed and we obtain the following equations:\(g^{\prime}\left( {\mu_{1} } \right) = - \eta_{2} \le 0\), \(g^{\prime}\left( {\mu_{2} } \right) = - \eta_{1} \le 0\), \(g^{\prime}\left( {\eta_{1} } \right) = 1 - \mu_{2} - \eta_{2} \ge 0\), \(g^{\prime}\left( {\eta_{2} } \right) = 1 - \mu_{1} - \eta_{1} \ge 0\).
Hence, it can be noted that the function value of \(g\) becomes smaller as the value of variables \(\mu_{1}\) or \(\mu_{2}\) become bigger, and it becomes bigger as the value of variables \(\eta_{1}\) or \(\eta_{ 2}\) becomes bigger.
Since \(0 \le \eta_{l} \le 1\), then we can derive that\(\prod\nolimits_{l = 1}^{2} {\left( {1 - 1} \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 - 0} \right)} \le g \le \prod\nolimits_{l = 1}^{2} {\left( {1 - 0} \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0 - 1} \right)} \Rightarrow 0 \le g = \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} \le 1\).
Similarly, the OD \(\prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)}\) can also be proved to be bounded between 0 and 1.
The sum of AD, ND, and OD is
Let \(g = \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)}\), then the derivative of the function \(g\) with regard to the different variables \(\mu_{l}\), \(\eta_{l}\), \(\nu_{l}\) can be computed and we obtain the following equations:
Hence, it can be noted that the function value of \(g\) becomes bigger as the value of variables \(\mu_{l}\), \(\eta_{l}\), or \(\nu_{l}\) becomes bigger.
When \(\mu_{l} = 1\), \(\eta_{l} = 0\), \(\nu_{l} = 0\), then we can derive that
When \(\eta_{l} = 1\), \(\mu_{l} = 0\), \(\nu_{l} = 0\), then we can derive that
When \(\nu_{l} = 1\), \(\mu_{l} = 0\), \(\eta_{l} = 0\), then we can derive that
Then, it can be observed that the sum of AD, ND, and OD
The above inference process can prove Theorem 1.
Theorem 2
Given any two PFNs \(p_{1} = \left( {\mu_{1} ,\eta_{1} ,\nu_{1} } \right)\) and \(p_{2} = \left( {\mu_{2} ,\eta_{2} ,\nu_{2} } \right)\), then the computing result from \(p_{1} \otimes p_{2}\) is also an PFN.
Proof
Let \(g = \prod\limits_{l = 1}^{2} {\left( {1 - \nu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)}\), then the derivative of the function \(g\) with regard to the variables \(\mu_{1}\), \(\mu_{2}\), \(\nu_{1}\), \(\nu_{2}\) can be computed and we can get the following formulas: \(g^{\prime}\left( {\mu_{1} } \right) = 1 - \mu_{2} - \nu_{2} \ge 0\), \(g^{\prime}\left( {\mu_{2} } \right) = 1 - \mu_{1} - \nu_{1} \ge 0\), \(g^{\prime}\left( {\nu_{1} } \right) = - \mu_{2} \le 0\), \(g^{\prime}\left( {\nu_{2} } \right) = - \mu_{1} \le 0\).
Hence, the function value of \(g\) becomes bigger as the value of variables \(\mu_{1}\) or \(\mu_{2}\) becomes bigger, and it becomes smaller as the value of variables \(\nu_{1}\) or \(\nu_{ 2}\) becomes bigger.
Since \(0 \le \mu_{l} \le 1\), \(0 \le \nu_{l} \le 1\), then we can derive that
Since \(0 \le \eta_{l} \le 1\), then we can derive that \(1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0} \right)} \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)} \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1} \right)} \Rightarrow 0 \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)} \le 1\).
Similarly, the ND \(1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)}\) and OD \(1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \nu_{l} } \right)}\) can also be proved to be between 0 and 1.
The sum of AD, ND, and OD is
Let \(g = 2 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)}\), then the derivative of the function \(g\) with regard to some variables \(\mu_{l}\), \(\eta_{l}\), \(\nu_{l}\) can be computed and we can get the following formulas:\(g^{\prime}\left( {\mu_{1} } \right) = 1 - \mu_{2} - \nu_{2} \ge 0\), \(g^{\prime}\left( {\mu_{2} } \right) = 1 - \mu_{1} - \nu_{1} \ge 0\), \(g^{\prime}\left( {\nu_{1} } \right) = 1 - \mu_{2} - \nu_{2} \ge 0\), \(g^{\prime}\left( {\nu_{2} } \right) = 1 - \mu_{1} - \nu_{1} \ge 0\), \(g^{\prime}\left( {\eta_{1} } \right) = 1 - \eta_{2} \ge 0\), \(g^{\prime}\left( {\eta_{2} } \right) = 1 - \eta_{1} \ge 0\).
Hence, it can be noted that the value of \(g\) increases as the value of \(\mu_{l}\), \(\eta_{l}\), or \(\nu_{l}\) increases.
When \(\mu_{l} = 1\), \(\eta_{l} = 0\), \(\nu_{l} = 0\), then \(g \le 2 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 - 0} \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0} \right)} = 1\).
When \(\eta_{l} = 1\), \(\mu_{l} = 0\), \(\nu_{l} = 0\), then \(g \le 2 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0 - 0} \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1} \right)} = 1\).
When \(\nu_{l} = 1\), \(\mu_{l} = 0\), \(\eta_{l} = 0\), then \(g \le 2 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0 - 1} \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0} \right)} = 1\).
Hence, it can be observed that the sum of AD, ND, and OD is less than or equal to 1. The inference process can prove Theorem 2.
Theorem 3
Given three PFNs \(p = \left( {\mu ,\eta ,\nu } \right)\), \(p_{1} = \left( {\mu_{1} ,\eta_{1} ,\nu_{1} } \right)\) and \(p_{2} = \left( {\mu_{2} ,\eta_{2} ,\nu_{2} } \right)\), then we have
-
1.
\(p_{1} \oplus p_{2} = p_{2} \oplus p_{1}\);
-
2.
\(\lambda \left( {p_{1} \oplus p_{2} } \right) = \lambda p_{1} \oplus \lambda p_{2}\);
-
3.
\(\left( {\lambda_{1} + \lambda_{2} } \right)p = \lambda_{1} p \oplus \lambda_{2} p\);
-
4.
\(p_{1} \otimes p_{2} = p_{2} \otimes p_{1}\);
-
5.
\(\left( {p_{1} \otimes p_{2} } \right)^{\lambda } = p_{1}^{\lambda } \otimes p_{2}^{\lambda }\);
-
6.
\(p^{{\lambda_{1} + \lambda_{2} }} = p^{{\lambda_{1} }} \otimes p^{{\lambda_{2} }}\).
Proof
Let us define \(\delta_{1} { = }1 - \mu_{1}\), \(\delta_{2} { = }1 - \mu_{2}\), \(\varepsilon_{1} { = }1 - \mu_{1} - \eta_{1}\), \(\varepsilon_{2} { = }1 - \mu_{2} - \eta_{2}\), \(\gamma_{1} = 1 - \mu_{1} - \nu_{1}\), \(\gamma_{2} = 1 - \mu_{2} - \nu_{2}\), \(\kappa_{1} = 1 - \nu_{1}\), \(\kappa_{2} = 1 - \nu_{2}\), \(\chi_{1} = 1 - \eta_{1}\), \(\chi_{2} = 1 - \eta_{2}\).
-
(1)
\(\begin{aligned} p_{1} \oplus p_{2} = \left( {1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} ,\prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} ,\prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} } \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \left( {1 - \delta_{1} \delta_{2} ,\delta_{1} \delta_{2} - \varepsilon_{1} \varepsilon_{2} ,\delta_{1} \delta_{2} - \gamma_{1} \gamma_{2} } \right) = p_{2} \oplus p_{1} \hfill \\ \end{aligned}\);
-
(2)
\(\lambda \left( {p_{1} \oplus p_{2} } \right) = \lambda \left( {1 - \delta_{1} \delta_{2} ,\delta_{1} \delta_{2} - \varepsilon_{1} \varepsilon_{2} ,\delta_{1} \delta_{2} - \gamma_{1} \gamma_{2} } \right) = \left( {1 - \left( {\delta_{1} \delta_{2} } \right)^{\lambda } ,\left( {\delta_{1} \delta_{2} } \right)^{\lambda } - \left( {\varepsilon_{1} \varepsilon_{2} } \right)^{\lambda } ,\left( {\delta_{1} \delta_{2} } \right)^{\lambda } - \left( {\gamma_{1} \gamma_{2} } \right)^{\lambda } } \right)\).According to Definition 10, we have \(\lambda p_{1} = \left( {1 - \delta_{1}^{\lambda } ,\delta_{1}^{\lambda } - \varepsilon_{1}^{\lambda } ,\delta_{1}^{\lambda } - \gamma_{1}^{\lambda } } \right)\) and \(\lambda p_{2} = \left( {1 - \delta_{2}^{\lambda } ,\delta_{2}^{\lambda } - \varepsilon_{2}^{\lambda } ,\delta_{2}^{\lambda } - \gamma_{2}^{\lambda } } \right)\), then \(\lambda p_{1} \oplus \lambda p_{2} = \left( {1 - \left( {\delta_{1} \delta_{2} } \right)^{\lambda } ,\left( {\delta_{1} \delta_{2} } \right)^{\lambda } - \left( {\varepsilon_{1} \varepsilon_{2} } \right)^{\lambda } ,\left( {\delta_{1} \delta_{2} } \right)^{\lambda } - \left( {\gamma_{1} \gamma_{2} } \right)^{\lambda } } \right) = \lambda \left( {p_{1} \oplus p_{2} } \right)\).
-
(3)
According to Definition 10, it can be derived that \(\left( {\lambda_{1} + \lambda_{2} } \right)p = \left( {1 - \delta^{{\lambda_{1} + \lambda_{2} }} ,\delta^{{\lambda_{1} + \lambda_{2} }} - \varepsilon^{{\lambda_{1} + \lambda_{2} }} ,\delta^{{\lambda_{1} + \lambda_{2} }} - \gamma^{{\lambda_{1} + \lambda_{2} }} } \right)\), \(\lambda_{1} p = \left( {1 - \delta^{{\lambda_{1} }} ,\delta^{{\lambda_{1} }} - \varepsilon^{{\lambda_{1} }} ,\delta^{{\lambda_{1} }} - \gamma^{{\lambda_{1} }} } \right)\), and \(\lambda_{2} p = \left( {1 - \delta^{{\lambda_{2} }} ,\delta^{{\lambda_{2} }} - \varepsilon^{{\lambda_{2} }} ,\delta^{{\lambda_{2} }} - \gamma^{{\lambda_{2} }} } \right)\). Then we can get \(\begin{aligned} \lambda_{1} p \oplus \lambda_{2} p \hfill \\ = \left( {1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \delta^{{\lambda_{l} }} } \right)} ,\prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \delta^{{\lambda_{l} }} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \delta^{{\lambda_{l} }} - \delta^{{\lambda_{l} }} + \varepsilon^{{\lambda_{l} }} } \right)} ,\prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \delta^{{\lambda_{l} }} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \delta^{{\lambda_{l} }} - \delta^{{\lambda_{l} }} + \gamma^{{\lambda_{l} }} } \right)} } \right) \hfill \\ {\kern 1pt} = \left( {1 - \delta^{{\lambda_{1} + \lambda_{2} }} ,\delta^{{\lambda_{1} + \lambda_{2} }} - \varepsilon^{{\lambda_{1} + \lambda_{2} }} ,\delta^{{\lambda_{1} + \lambda_{2} }} - \gamma^{{\lambda_{1} + \lambda_{2} }} } \right) = \left( {\lambda_{1} + \lambda_{2} } \right)p \hfill \\ \end{aligned}\).
-
(4)
\(p_{1} \otimes p_{2} = \left( {\prod\nolimits_{l = 1}^{2} {\left( {1 - \nu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \nu_{l} } \right)} } \right){\kern 1pt} = p_{2} \otimes p_{2}\).
-
(5)
According to Definition 10, we can have \(\left( {p_{1} \otimes p_{2} } \right)^{\lambda } = \left( {\kappa_{1} \kappa_{2} - \gamma_{1} \gamma_{2} ,1 - \chi_{1} \chi_{2} ,1 - \kappa_{1} \kappa_{2} } \right)^{\lambda } {\kern 1pt} {\kern 1pt} = \left( {\left( {\kappa_{1} \kappa_{2} } \right)^{\lambda } - \left( {\gamma_{1} \gamma_{2} } \right)^{\lambda } ,1 - \left( {\chi_{1} \chi_{2} } \right)^{\lambda } ,1 - \left( {\kappa_{1} \kappa_{2} } \right)^{\lambda } } \right)\). Since \(p_{1}^{\lambda } = \left( {\kappa_{1}^{\lambda } - \gamma_{1}^{\lambda } ,1 - \chi_{1}^{\lambda } ,1 - \kappa_{1}^{\lambda } } \right)\) and \(p_{2}^{\lambda } = \left( {\kappa_{2}^{\lambda } - \gamma_{2}^{\lambda } ,1 - \chi_{2}^{\lambda } ,1 - \kappa_{2}^{\lambda } } \right)\), then it can be derived that \(\begin{aligned} p_{1}^{\lambda } \otimes p_{2}^{\lambda } = \left( {\prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \kappa_{l}^{\lambda } } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \kappa_{l}^{\lambda } + \gamma_{l}^{\lambda } - 1 + \kappa_{l}^{\lambda } } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \chi_{l}^{\lambda } } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \kappa_{l}^{\lambda } } \right)} } \right) \hfill \\ {\kern 1pt} = \left( {\left( {\kappa_{1} \kappa_{2} } \right)^{\lambda } - \left( {\gamma_{1} \gamma_{2} } \right)^{\lambda } ,1 - \left( {\chi_{1} \chi_{2} } \right)^{\lambda } ,1 - \left( {\kappa_{1} \kappa_{2} } \right)^{\lambda } } \right) = \left( {p_{1} \otimes p_{2} } \right)^{\lambda } \hfill \\ \end{aligned}\).
-
(6)
According to Definition 10, then we can have \(p^{{\lambda_{1} + \lambda_{2} }} = \left( {\kappa^{{\lambda_{1} + \lambda_{2} }} - \gamma^{{\lambda_{1} + \lambda_{2} }} ,1 - \chi^{{\lambda_{1} + \lambda_{2} }} ,1 - \kappa^{{\lambda_{1} + \lambda_{2} }} } \right)\). Since \(p^{{\lambda_{1} }} = \left( {\kappa^{{\lambda_{1} }} - \gamma^{{\lambda_{1} }} ,1 - \chi^{{\lambda_{1} }} ,1 - \kappa^{{\lambda_{1} }} } \right)\) and \(p^{{\lambda_{2} }} = \left( {\kappa^{{\lambda_{2} }} - \gamma^{{\lambda_{2} }} ,1 - \chi^{{\lambda_{2} }} ,1 - \kappa^{{\lambda_{2} }} } \right)\), then it can be derived that \(\begin{aligned} p^{{\lambda_{1} }} \otimes p^{{\lambda_{2} }} = \left( {\prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \kappa^{{\lambda_{l} }} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \kappa^{{\lambda_{l} }} + \gamma^{{\lambda_{l} }} - 1 + \kappa^{{\lambda_{l} }} } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \chi^{{\lambda_{l} }} } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \kappa^{{\lambda_{l} }} } \right)} } \right) \hfill \\ {\kern 1pt} {\kern 1pt} = \left( {\kappa^{{\lambda_{1} + \lambda_{2} }} - \gamma^{{\lambda_{1} + \lambda_{2} }} ,1 - \chi^{{\lambda_{1} + \lambda_{2} }} ,1 - \kappa^{{\lambda_{1} + \lambda_{2} }} } \right) = p^{{\lambda_{1} + \lambda_{2} }} \hfill \\ \end{aligned}\).
Theorem 4
Given a collection of PFNs \(P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}\) with \(p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)\), which are grouped into \(s\) clusters \(P_{1} ,P_{2} , \cdots ,P_{s}\) in which \(P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)\) and \(\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P\), then the fused result from the PFIPHM operator is still an PFN as follows:
where \(1 - \nu_{\rho } = a_{\rho }\), \(1 - \nu_{\tau } = a_{\tau }\), \(1 - \mu_{\rho } - \nu_{\rho } = b_{\rho }\), \(1 - \mu_{\tau } - \nu_{\tau } = b_{\tau }\), \(1 - \eta_{\rho } = c_{\rho }\), \(1 - \eta_{\tau } = c_{\tau }\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi\).
Proof
According to IOLs defined in Definition 10, it can be derived that
and \(p_{\tau }^{\beta } { = }\left( {\left( {1 - \nu_{\tau } } \right)^{\beta } - \left( {1 - \mu_{\tau } - \nu_{\tau } } \right)^{\beta } ,1 - \left( {1 - \eta_{\tau } } \right)^{\beta } ,1 - \left( {1 - \nu_{\tau } } \right)^{\beta } } \right)\).
Let \(1 - \nu_{\rho } = a_{\rho }\), \(1 - \nu_{\tau } = a_{\tau }\), \(1 - \mu_{\rho } - \nu_{\rho } = b_{\rho }\), \(1 - \mu_{\tau } - \nu_{\tau } = b_{\tau }\), \(1 - \eta_{\rho } = c_{\rho }\), and \(1 - \eta_{\tau } = c_{\tau }\), then
Hence, we can get \(p_{\rho }^{\alpha } \otimes p_{\tau }^{\beta } = \left( {a_{\rho }^{\alpha } a_{\tau }^{\beta } - b_{\rho }^{\alpha } b_{\tau }^{\beta } ,1 - c_{\rho }^{\alpha } c_{\tau }^{\beta } ,1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } } \right)\).
Then
Let us define that \(\left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma\), and \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi\), then we have \(\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {p_{\rho }^{\alpha } \otimes p_{\tau }^{\beta } } \right) = \left( {1 - \chi ,\chi - \varsigma ,\chi - \psi } \right)\) and \(\left( {\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {p_{\rho }^{\alpha } \otimes p_{\tau }^{\beta } } \right)} \right)^{{\frac{1}{\alpha + \beta }}} = \left( {\left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} - \psi^{{\frac{1}{\alpha + \beta }}} ,1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} ,1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)\).
Then we can continue to obtain that
Thus,
The sum of the AD, ND, and OD of the aggregation result of the PFIPHM operator is
Since \(\left( {\prod\nolimits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} { + }1} \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\nolimits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \ge 0\) and \(\left( {\prod\nolimits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \le 1\), thus, \(1 + \left( {\prod\nolimits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} { + }1} \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\nolimits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\nolimits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \ge 0\).
Since \(- \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} \le 0\), \(\left( {\prod\nolimits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} { + }1} \right)} } \right)^{{\frac{1}{s}}} \le 1\), then,
The above inference process can prove Theorem 4.
Theorem 5
(Idempotency Property). Given a collection of PFNs \(P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}\) with \(p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)\), which are grouped into \(s\) clusters \(P_{1} ,P_{2} , \cdots ,P_{s}\) in which \(P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)\) and \(\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P\), if \(p_{l} = p = \left( {\mu ,\eta ,\nu } \right)\) for each \(l\), then we have \(PFIPHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = p = \left( {\mu ,\eta ,\nu } \right)\), where \(\alpha\) and \(\beta\) are two parameters, which satisfy that \(\alpha ,\beta \ge 0\).
Proof
Since \(\mu_{\rho } { = }\mu_{\tau } { = }\mu\), \(\nu_{\rho } { = }\nu_{\tau } { = }\nu\), \(\eta_{\rho } { = }\eta_{\tau } { = }\eta\), then we have \(1 - \nu_{\rho } = 1 - \nu_{\tau } = a_{\rho } = a_{\tau } = a\), \(1 - \mu_{\rho } - \nu_{\rho } = 1 - \mu_{\tau } - \nu_{\tau } = b_{\rho } = b_{\tau } = b\), \(1 - \eta_{\rho } = 1 - \eta_{\tau } = c_{\rho } = c_{\tau } = c\).
Thus,
Bring the above equation into (2), then \(PFIPHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right){\kern 1pt} = {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {a - b,1 - c,1 - a} \right) = \left( {\mu ,\eta ,\nu } \right) = p\). Hence, the proof of Theorem 5 is finished.
Theorem 6
(Commutativity Property). Given a collection of PFNs \(P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}\) with \(p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)\), which are grouped into \(s\) clusters \(P_{1} ,P_{2} , \cdots ,P_{s}\) in which \(P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)\) and \(\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P\), if there is a permutation of \(P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)\), denoted by \(P^{\prime}_{\ell } = \left\{ {p^{\prime}_{\ell 1} ,p^{\prime}_{\ell 2} , \cdots ,p^{\prime}_{{\ell \left| {P_{\ell } } \right|}} } \right\}\), then we have \(PFIPHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = PFIPHM^{\alpha ,\beta } \left( {p^{\prime}_{1} ,p^{\prime}_{2} , \cdots ,p^{\prime}_{n} } \right)\), where \(\alpha\) and \(\beta\) are two parameters, which satisfy that \(\alpha ,\beta \ge 0\), \(\left| {P_{\ell } } \right|\) means the quantity of PFNs belonging to the cluster \(P_{\ell }\).
Proof
According to Eq. (2), we have
For each \(\ell\), we have
Thus, \(PFIPHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = PFIPHM^{\alpha ,\beta } \left( {p^{\prime}_{1} ,p^{\prime}_{2} , \cdots ,p^{\prime}_{n} } \right)\). Hence, the proof Theorem 6 is finished.
Theorem 7
Given a collection of PFNs \(P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}\) with \(p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)\), which are grouped into \(s\) clusters \(P_{1} ,P_{2} , \cdots ,P_{s}\) in which \(P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)\) and \(\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P\), then the fused result of the PFIWPHM operator is still an PFN and
where \(1 - \mu_{\rho } = a_{\rho }\), \(1 - \mu_{\tau } = a_{\tau }\), \(1 - \mu_{\rho } - \eta_{\rho } = b_{\rho }\), \(1 - \mu_{\tau } - \eta_{\tau } = b_{\tau }\), \(1 - \mu_{\rho } - \nu_{\rho } = c_{\rho }\), \(1 - \mu_{\tau } - \nu_{\tau } = c_{\tau }\), \(\left( {1 - a_{\rho }^{{\omega_{\rho } }} + c_{\rho }^{{\omega_{\rho } }} } \right) = d_{\rho }\), \(\left( {1 - a_{\tau }^{{\omega_{\tau } }} + c_{\tau }^{{\omega_{\tau } }} } \right) = d_{\tau }\), \(c_{\rho }^{{\omega_{\rho } }} = e_{\rho }\), \(c_{\tau }^{{\omega_{\tau } }} = e_{\tau }\), \(\left( {1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} } \right) = f_{\rho }\), \(\left( {1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} } \right) = f_{\tau }\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma\), and \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi\).
Proof
According to IOLs defined in Definition 10, it can be derived that
and \(\omega_{\tau } p_{\tau } = \left( {1 - \left( {1 - \mu_{\tau } } \right)^{{\omega_{\tau } }} ,\left( {1 - \mu_{\tau } } \right)^{{\omega_{\tau } }} - \left( {1 - \mu_{\tau } - \eta_{\tau } } \right)^{{\omega_{\tau } }} ,\left( {1 - \mu_{\tau } } \right)^{{\omega_{\tau } }} - \left( {1 - \mu_{\tau } - \nu_{\tau } } \right)^{{\omega_{\tau } }} } \right)\).
Let \(1 - \mu_{\rho } = a_{\rho }\), \(1 - \mu_{\tau } = a_{\tau }\), \(1 - \mu_{\rho } - \eta_{\rho } = b_{\rho }\), \(1 - \mu_{\tau } - \eta_{\tau } = b_{\tau }\), \(1 - \mu_{\rho } - \nu_{\rho } = c_{\rho }\), and \(1 - \mu_{\tau } - \nu_{\tau } = c_{\tau }\), then we have \(\omega_{\rho } p_{\rho } = \left( {1 - a_{\rho }^{{\omega_{\rho } }} ,a_{\rho }^{{\omega_{\rho } }} - b_{\rho }^{{\omega_{\rho } }} ,a_{\rho }^{{\omega_{\rho } }} - c_{\rho }^{{\omega_{\rho } }} } \right)\) and \(\omega_{\tau } p_{\tau } = \left( {1 - a_{\tau }^{{\omega_{\tau } }} ,a_{\tau }^{{\omega_{\tau } }} - b_{\tau }^{{\omega_{\tau } }} ,a_{\tau }^{{\omega_{\tau } }} - c_{\tau }^{{\omega_{\tau } }} } \right)\).
Hence, the exponential operation is
Let \(\left( {1 - a_{\rho }^{{\omega_{\rho } }} + c_{\rho }^{{\omega_{\rho } }} } \right) = d_{\rho }\), \(\left( {1 - a_{\tau }^{{\omega_{\tau } }} + c_{\tau }^{{\omega_{\tau } }} } \right) = d_{\tau }\), \(c_{\rho }^{{\omega_{\rho } }} = e_{\rho }\), \(c_{\tau }^{{\omega_{\tau } }} = e_{\tau }\), \(\left( {1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} } \right) = f_{\rho }\), \(\left( {1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} } \right) = f_{\tau }\), then it can be derived that \(\left( {\omega_{\rho } p_{\rho } } \right)^{\alpha } \otimes \left( {\omega_{\tau } p_{\tau } } \right)^{\beta } = \left( {d_{\rho }^{\alpha } d_{\tau }^{\beta } - e_{\rho }^{\alpha } e_{\tau }^{\beta } ,1 - f_{\rho }^{\alpha } f_{\tau }^{\beta } ,1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } } \right)\).
Let \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi\), then it can be derived that
Thus, we have
Finally, we obtain that
Theorem 8
Given a collection of PFNs \(P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}\) with \(p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)\), which are grouped into \(s\) clusters \(P_{1} ,P_{2} , \cdots ,P_{s}\) in which \(P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)\) and \(\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P\), then the aggregated result of the PFIPGHM operator is still an PFN, and
where \(1 - \mu_{\rho } = a_{\rho }\), \(1 - \mu_{\tau } = a_{\tau }\), \(1 - \mu_{\rho } - \eta_{\rho } = b_{\rho }\), \(1 - \mu_{\tau } - \eta_{\tau } = b_{\tau }\), \(1 - \mu_{\rho } - \nu_{\rho } = c_{\rho }\), \(1 - \mu_{\tau } - \nu_{\tau } = c_{\tau }\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} { = }\chi\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi\).
Proof
According to the IOLs presented in Definition 10, it can be derived that \(\alpha p_{\rho } = \left( {1 - \left( {1 - \mu_{\rho } } \right)^{\alpha } ,\left( {1 - \mu_{\rho } } \right)^{\alpha } - \left( {1 - \mu_{\rho } - \eta_{\rho } } \right)^{\alpha } ,\left( {1 - \mu_{\rho } } \right)^{\alpha } - \left( {1 - \mu_{\rho } - \nu_{\rho } } \right)^{\alpha } } \right)\), and \(\beta p_{\tau } = \left( {1 - \left( {1 - \mu_{\tau } } \right)^{\beta } ,\left( {1 - \mu_{\tau } } \right)^{\beta } - \left( {1 - \mu_{\tau } - \eta_{\tau } } \right)^{\beta } ,\left( {1 - \mu_{\tau } } \right)^{\beta } - \left( {1 - \mu_{\tau } - \nu_{\tau } } \right)^{\beta } } \right)\).
Let \(1 - \mu_{\rho } = a_{\rho }\), \(1 - \mu_{\tau } = a_{\tau }\), \(1 - \mu_{\rho } - \eta_{\rho } = b_{\rho }\), \(1 - \mu_{\tau } - \eta_{\tau } = b_{\tau }\), \(1 - \mu_{\rho } - \nu_{\rho } = c_{\rho }\), \(1 - \mu_{\tau } - \nu_{\tau } = c_{\tau }\), then\(\alpha p_{\rho } = \left( {1 - a_{\rho }^{\alpha } ,a_{\rho }^{\alpha } - b_{\rho }^{\alpha } ,a_{\rho }^{\alpha } - c_{\rho }^{\alpha } } \right)\), \(\beta p_{\tau } = \left( {1 - a_{\tau }^{\beta } ,a_{\tau }^{\beta } - b_{\tau }^{\beta } ,a_{\tau }^{\beta } - c_{\tau }^{\beta } } \right)\), \(\alpha p_{\rho } \oplus \beta p_{\tau } { = }\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } ,a_{\rho }^{\alpha } a_{\tau }^{\beta } - b_{\rho }^{\alpha } b_{\tau }^{\beta } ,a_{\rho }^{\alpha } a_{\tau }^{\beta } - c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)\).
Thus,
Let \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} { = }\chi\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi\), then \(\left( {\mathop \otimes \nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha p_{\rho } \oplus \beta p_{\tau } } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \left( {\chi - \varsigma ,1 - \psi ,1 - \chi } \right)\).
Thus, \(\frac{1}{\alpha + \beta }\left( {\mathop \otimes \nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha p_{\rho } \oplus \beta p_{\tau } } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} ,\left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} - \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} ,\left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} - \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)\)and
Finally,
The sum of the AD, ND, and OD of the aggregation result of the PFIPGHM operator is
and
Hence, the proof of Theorem 8 is finished.
Theorem 9
(Idempotency Property). Given a collection of PFNs \(P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}\) with \(p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)\), which are grouped into \(s\) clusters \(P_{1} ,P_{2} , \cdots ,P_{s}\) in which \(P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)\) and \(\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P\), if \(p_{l} = p = \left( {\mu ,\eta ,\nu } \right)\) for each \(l\), then we have \(PFIPGHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = p = \left( {\mu ,\eta ,\nu } \right)\), where \(\alpha\) and \(\beta\) are two parameters, which satisfy that \(\alpha ,\beta \ge 0\).
Proof
Since \(\mu_{\rho } { = }\mu_{\tau } { = }\mu\), \(\nu_{\rho } { = }\nu_{\tau } { = }\nu\), \(\eta_{\rho } { = }\eta_{\tau } { = }\eta\), then we have \(1 - \mu_{\rho } = 1 - \mu_{\tau } = a_{\rho } = a_{\tau } = a\), \(1 - \mu_{\rho } - \eta_{\rho } = 1 - \mu_{\tau } - \eta_{\tau } = b_{\rho } = b_{\tau } = b\), \(1 - \mu_{\rho } - \nu_{\rho } = 1 - \mu_{\tau } - \nu_{\tau } = c_{\rho } = c_{\tau } = c\).
Thus,
Bring the above equation into (6), then we have
Hence, the proof of Theorem 9 is finished.
Theorem 10
(Commutativity Property). Given a series of PFNs \(P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}\) with \(p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)\), which are grouped into \(s\) groups \(P_{1} ,P_{2} , \cdots ,P_{s}\) in which \(P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)\) and \(\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P\), if there is a permutation of \(P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)\), denoted by \(P^{\prime}_{\ell } = \left\{ {p^{\prime}_{\ell 1} ,p^{\prime}_{\ell 2} , \cdots ,p^{\prime}_{{\ell \left| {P_{\ell } } \right|}} } \right\}\), then we have \(PFIPGHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = PFIPGHM^{\alpha ,\beta } \left( {p^{\prime}_{1} ,p^{\prime}_{2} , \cdots ,p^{\prime}_{n} } \right)\), where \(\alpha\) and \(\beta\) are two parameters, which satisfy that \(\alpha ,\beta \ge 0\), \(\left| {P_{\ell } } \right|\) means the quantity of PFNs belonging to the group \(P_{\ell }\).
Proof
According to Eq. (6), we have
For each \(\ell\), \(\prod\nolimits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} = \prod\nolimits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi^{\prime} + \varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {\varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)}\), \(\prod\nolimits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} = \prod\nolimits_{\ell = 1}^{s} {\left( {\left( {\varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)}\), and \(\prod\nolimits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} = \prod\nolimits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi^{\prime} + \varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi^{\prime} + \varsigma^{\prime} + \psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)}\).
Thus, \(PFIPGHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = PFIPGHM^{\alpha ,\beta } \left( {p_{ 1}^{\prime } ,p_{ 2}^{\prime } , \cdots ,p_{n}^{\prime } } \right)\), which completes the proof.
Theorem 11
Given a series of PFNs \(P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}\) with \(p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)\), which are grouped into \(s\) clusters \(P_{1} ,P_{2} , \cdots ,P_{s}\) in which \(P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)\) and \(\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P\), then the aggregated result of the PFIWPGHM operator is still an PFN and
where \(1 - \nu_{\rho } = a_{\rho }\), \(1 - \nu_{\tau } = a_{\tau }\), \(1 - \mu_{\rho } - \nu_{\rho } = b_{\rho }\), \(1 - \mu_{\tau } - \nu_{\tau } = b_{\tau }\), \(1 - \eta_{\rho } = c_{\rho }\), \(1 - \eta_{\tau } = c_{\tau }\), \(1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} = d_{\rho }\), \(1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} = d_{\tau }\), \(- a_{\rho }^{{\omega_{\rho } }} { + }b_{\rho }^{{\omega_{\rho } }} { + }c_{\rho }^{{\omega_{\rho } }} = e_{\rho }\), \(- a_{\tau }^{{\omega_{\tau } }} { + }b_{\tau }^{{\omega_{\tau } }} { + }c_{\tau }^{{\omega_{\tau } }} = e_{\tau }\), \(b_{\rho }^{{\omega_{\rho } }} = f_{\rho }\), \(b_{\tau }^{{\omega_{\tau } }} = f_{\tau }\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi\).
Proof
According to the IOLs presented in Definition 10, it can be derived that \(\left( {p_{\rho } } \right)^{{\omega_{\rho } }} = \left( {\left( {1 - \nu_{\rho } } \right)^{{\omega_{\rho } }} - \left( {1 - \mu_{\rho } - \nu_{\rho } } \right)^{{\omega_{\rho } }} ,1 - \left( {1 - \eta_{\rho } } \right)^{{\omega_{\rho } }} ,1 - \left( {1 - \nu_{\rho } } \right)^{{\omega_{\rho } }} } \right)\), and \(\left( {p_{\tau } } \right)^{{\omega_{\tau } }} = \left( {\left( {1 - \nu_{\tau } } \right)^{{\omega_{\tau } }} - \left( {1 - \mu_{\tau } - \nu_{\tau } } \right)^{{\omega_{\tau } }} ,1 - \left( {1 - \eta_{\tau } } \right)^{{\omega_{\tau } }} ,1 - \left( {1 - \nu_{\tau } } \right)^{{\omega_{\tau } }} } \right)\).
Let \(1 - \nu_{\rho } = a_{\rho }\), \(1 - \nu_{\tau } = a_{\tau }\), \(1 - \mu_{\rho } - \nu_{\rho } = b_{\rho }\), \(1 - \mu_{\tau } - \nu_{\tau } = b_{\tau }\), \(1 - \eta_{\rho } = c_{\rho }\), and \(1 - \eta_{\tau } = c_{\tau }\), then
Then, we can have \(\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} = \left( {1 - \left( {1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } ,\left( {1 - a_{\rho }^{{\omega_{\rho } }} { + }b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } - \left( { - a_{\rho }^{{\omega_{\rho } }} { + }b_{\rho }^{{\omega_{\rho } }} { + }c_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } ,\left( {1 - a_{\rho }^{{\omega_{\rho } }} { + }b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } - \left( {b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } } \right)\) and \(\beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} { = }\left( {1 - \left( {1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } ,\left( {1 - a_{\tau }^{{\omega_{\tau } }} { + }b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } - \left( { - a_{\tau }^{{\omega_{\tau } }} { + }b_{\tau }^{{\omega_{\tau } }} { + }c_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } ,\left( {1 - a_{\tau }^{{\omega_{\tau } }} { + }b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } - \left( {b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } } \right)\).
Thus,
Let \(1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} = d_{\rho }\), \(1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} = d_{\tau }\), \(- a_{\rho }^{{\omega_{\rho } }} { + }b_{\rho }^{{\omega_{\rho } }} { + }c_{\rho }^{{\omega_{\rho } }} = e_{\rho }\), \(- a_{\tau }^{{\omega_{\tau } }} { + }b_{\tau }^{{\omega_{\tau } }} { + }c_{\tau }^{{\omega_{\tau } }} = e_{\tau }\), \(b_{\rho }^{{\omega_{\rho } }} = f_{\rho }\), \(b_{\tau }^{{\omega_{\tau } }} = f_{\tau }\), then \(\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} \oplus \beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} { = }\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } ,d_{\rho }^{\alpha } d_{\tau }^{\beta } - e_{\rho }^{\alpha } e_{\tau }^{\beta } ,d_{\sigma i}^{\alpha } d_{\sigma j}^{\beta } - f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)\).
Thus,
Let \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma\), \(\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi\), then we have \(\left( {\mathop \otimes \nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} \oplus \beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \left( {\chi - \varsigma ,1 - \psi ,1 - \chi } \right)\) and
Thus,
Finally,
Rights and permissions
About this article
Cite this article
Lin, M., Li, X., Chen, R. et al. Picture fuzzy interactional partitioned Heronian mean aggregation operators: an application to MADM process. Artif Intell Rev 55, 1171–1208 (2022). https://doi.org/10.1007/s10462-021-09953-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10462-021-09953-7