Abstract
The complex q-rung orthopair fuzzy (CQ-ROF) set can describe the complex uncertain information. In this manuscript, we develop the Yager operational laws based on the CQ-ROF information and Yager t-norm and t-conorm. Furthermore, in aggregating the CQ-ROF values, the power, averaging, and geometric aggregation operators have played a very essential and critical role in the environment of fuzzy set. Inspired from the discussed operators, we propose the CQ-ROF power Yager averaging (CQ-ROFPYA), CQ-ROF power Yager ordered averaging (CQ-ROFPYOA), CQ-ROF power Yager geometric (CQ-ROFPYG), and CQ-ROF power Yager ordered geometric (CQ-ROFPYOG) operators. These operators are the modified version of the Power, Yager, averaging, geometric, and the combination of these all based on fuzzy set (FS), intuitionistic FS, Pythagorean FS, q-rung orthopair FS, complex FS, complex intuitionistic FS, and complex Pythagorean FS. Moreover, we also discuss the main properties of the proposed operators. Additionally, we develop a multi-attribute decision-making (MADM) method based on the developed operators. To show the supremacy and validity of the proposed method, the comparison between the proposed method and some existing methods is done by some examples, and results show that the proposed method is better than the others in terms of generality and effectiveness.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Introduction
The MADM can be considered as a problem-evaluating procedure to get the best one from the family of alternatives. In traditional MADM, the attribute values can only be expressed by real numbers, to describe the uncertain information for the attribute values, Zadeh [1] proposed FS to describe fuzzy information. Further, Atanassov [2] improved the FS and proposed the IFS by adding one extra term based on the FS, called falsity grade with a characteristic \(0\le \overline{\overline{{\Delta }_{\widetilde{\beth }}}}\left(x\right)+\overline{\overline{{\Xi }_{\widetilde{\beth }}}}\left(x\right)\le 1\). Moreover, Yager [3, 4] modified the IFSs and proposed the PFS and Q-ROFS with two new characteristics: \(0\le {\left(\overline{\overline{{\Delta }_{\widetilde{\beth }}}}\left(x\right)\right)}^{2}+{\left(\overline{\overline{{\Xi }_{\widetilde{\beth }}}}\left(x\right)\right)}^{2}\le 1\) and \(0\le {\left(\overline{\overline{{\Delta }_{\widetilde{\beth }}}}\left(x\right)\right)}^{\Lambda }+{\left(\overline{\overline{{\Xi }_{\widetilde{\beth }}}}\left(x\right)\right)}^{\Lambda }\le 1,\Lambda \ge 1\). The Q-ROFS is a generalization of Fs, IFS, and PFS. When \(\Lambda =2,\) Q-ROFS is reduced to PFS; When \(\Lambda =1,\) Q-ROFS is reduced to IFS; and when \(\Lambda =1\) and \(\overline{\overline{{\Xi }_{\widetilde{\beth }}}}\left(x\right)=0,\) Q-ROFS is reduced to FS. Now, the Q-ROFS, PFS, IFS, and FS have received a lot of attention from various individuals because of their structure and characteristics.
Because the Q-ROFS, PFS, IFS, and FS can only express one-dimensional information, in some real decision problems, for example, to describe the amplitude and phase of a variable, they cannot describe this problem. So, it is necessary to add another dimensional information based on them. Therefore, Ramot et al. [5] extended the range of FS by utilizing the unit disc instead of the unit interval and proposed the CFS. The truth grade in CFS is expressed by a complex number where the values of real and imaginary parts are contained in the unit interval. The FS is a special case of the CFS. Furthermore, Alkouri and Salleh [6] proposed the CIFS which is massively modified from the FSs, and IFSs. In CFSs, they meet the characteristics \(0\le \overline{\overline{{\Delta }_{\widetilde{rp}}}}\left(x\right)+\overline{\overline{{\Xi }_{\widetilde{rp}}}}\left(x\right)\le 1\) and \(0\le \overline{\overline{{\Delta }_{\widetilde{ip}}}}\left(x\right)+\overline{\overline{{\Xi }_{\widetilde{ip}}}}\left(x\right)\le 1\). Further, Ullah et al. [7] and Liu et al. [8, 9] modified the CIFS and proposed the CPFS and CQ-ROFS with the characteristics: \(0\le {\overline{\overline{{\Delta }_{\widetilde{rp}}}}}^{2}\left(x\right)+{\overline{\overline{{\Xi }_{\widetilde{rp}}}}}^{2}\left(x\right)\le \mathrm{1,0}\le {\overline{\overline{{\Delta }_{\widetilde{ip}}}}}^{2}\left(x\right)+{\overline{\overline{{\Xi }_{\widetilde{ip}}}}}^{2}\left(x\right)\le 1\) and \(0\le {\overline{\overline{{\Delta }_{\widetilde{rp}}}}}^{\Lambda }\left(x\right)+{\overline{\overline{{\Xi }_{\widetilde{rp}}}}}^{\Lambda }\left(x\right)\le \mathrm{1,0}\le {\overline{\overline{{\Delta }_{\widetilde{ip}}}}}^{\Lambda }\left(x\right)+{\overline{\overline{{\Xi }_{\widetilde{ip}}}}}^{\Lambda }\left(x\right)\le 1,\Lambda \ge 1\). The FSs, IFSs, PFSs, Q-ROFSs, CFSs, CIFSs, and CPFSs are the special case of the CQ-ROFS.
Literature review
The PA operator was proposed by Yager [10] in 2001, which is the combination of two different structures such as power weighting and aggregation operators, to aggregate the finite family of attribute values into a singleton value. In addition, Yager [11] proposed Yager t-norm and t-conorm, which is a very famous and reliable operational laws. Furthermore, Xu [12] developed the simple averaging AOs for IFS. Xu and Yager [13] presented the geometric AOs for IFS. Xu [14] developed the PA operators for IFS. Rahman et al. [15] proposed the geometric AOs for PFS. Wei and Lu [16] developed the PA operators for PFS. Shahzadi and Akram [17] proposed the Yager aggregation operators for PFS. Liu and Wang [18] developed simple AOs for Q-ROFS. Riaz et al. [19] derived the geometric AOs for Q-ROFSs. Akram and Shahzadi [20] developed the Yager aggregation operators for Q-ROFS. Garg and Rani [21, 22] developed the AOs for CIFS and the geometric AOs for CIFSs. Rani and Garg [23] examined the PA operators based on the CIFS. Moreover, Akram et al. [24] developed the novel Yager AOs based on the CPFSs. Garg et al. [25] proposed the PA operators based on the CQ-ROF information. Further, some scholars developed some new and different types of operators based on prevailing concepts, for example, Senapati et al. [26] developed the Aczel-Alsina geometric AOs under the IFS. Furthermore, Garg et al. [27] examined the novel combination of Schweizer-Sklar and prioritized AOs for IFSs. Azeem et al. [28] evaluated the Einstein AOs for the complex IFSs. Mahmood and Ali [29] combined the Aczel-Alsina and power aggregation operators based on the complex IFSs, Senapati et al. [30] developed the power Aczel-Alsina AOs for IFS. Moreover, Jin et al. [31] proposed Aczel-Alsina AOs for the complex PFS and Liu et al. [32] exposed the Archimedean AOs for complex PFSs. A brief analysis of some FSs is shown in Table 1.
Two types of symbols are used in Table 1, such as \(\surd \) and \(\times \), which is represented the yes and no. From the data in Table 1, it is clear that the CQ-ROFS is a very complete and reliable structure because it contained the truth grade, falsity grade, phase term, and power conditions as well. Therefore, based on the CQ-ROFS, we develop some powerful aggregation operators.
Motivation and advantages
Now, we observed that the PA operators based on Yager t-norm and t-conorm for CQ-ROF information was not researched. However, it is very clear that the CQ-ROFSs are more general than some existing fuzzy sets, such as FSs, IFSs, PFSs, Q-ROFSs, CFSs, CIFSs, and CPFSs, and the PA operators based on Yager t-norm and t-conorm are more general than some existing averaging and geometric AOs, it is necessary to propose some new PAs for CQ-ROFSs based on Yager t-norm and t-conorm, and then to develop the MADM methods based on the proposed operators to solve real-life decision problems. The major advantages of the proposed operators are explained as
-
1.
When \(\Lambda =2\) in CQ-ROFSs, then all the proposed operators are converted into the ones based on the CPFSs.
-
2.
When \(\Lambda =1\) in CQ-ROFSs, then all the proposed operators are converted into the ones based on the CIFSs.
-
3.
When \(\overline{\overline{{\Delta }_{ip-z}}}=\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}=0\) in CQ-ROFSs, then all the proposed operators are converted into the ones based on the Q-ROFSs.
-
4.
When \(\overline{\overline{{\Delta }_{ip-z}}}=\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}=0\) and \(\Lambda =2\) in CQ-ROFSs, then all the proposed operators are converted into the ones based on the PFSs.
-
5.
When \(\overline{\overline{{\Delta }_{ip-z}}}=\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}=0\) and \(\Lambda =1\) in CQ-ROFSs, then all the proposed operators are converted into the ones based on the IFSs.
Furthermore, the power averaging\geometric aggregation operators, averaging/geometric aggregation operators, and Yager averaging\geometric aggregation operators based on the FSs, IFSs, PFSs, Q-ROFSs, CFSs, CIFSs, CPFSs, and CQ-ROFSs are the particular cases of the developed operators, which can give a big and wide space to make a valuable decision for decision-makers.
Main contributions
The main contributions of this study are shown as follows:
-
1.
We develop the main operational laws for the CQ-ROF values based on the Yager t-norm and t-conorm such as Yager operational laws.
-
2.
We propose the CQ-ROFPYA operator and CQ-ROFPYOA operator based on the CQ-ROF information and power Yager averaging operators.
-
3.
We propose the CQ-ROFPYG operator and CQ-ROFPYOG operator based on the CQ-ROF information and power Yager geometric operators.
-
4.
We also explore the major properties of the proposed operators such as idempotency, monotonicity, and boundedness.
-
5.
We develop the MADM method based on the proposed operators with the CQ-ROF information.
-
6.
Compare the proposed operators with some existing operators and show the effectiveness of the developed approaches.
The main contributes of the proposed works are stated in Fig. 1.
The main construct of this paper is shown as: In “Preliminaries”, we introduce the basic concepts and theories such as the PA operator, CQ-ROFS and their operational laws, and Yager t-norm and t-conorm. In “Power Yager aggregation operators for CQ-ROFNs” section, we propose the Yager operational laws for CQ-ROFNs. Moreover, we develop the CQ-ROFPYA, CQ-ROFPYOA, CQ-ROFPYG, and CQ-ROFPYOG operators, and discuss their fundamental properties of them. In “MADM methods based on the proposed operators” section, we propose a MADM method based on the proposed operators for CQ-ROF information. In “Comparative analysis” section, we compare the derived operators with various existing operators, and some practical examples are used to show the effectiveness of the proposed approaches. Concluding remarks are given in “Conclusion” section.
Preliminaries
For convenience, the parameters and variables used in this manuscript are briefly explained in Table 2.
Definition 1
[10] For the collection of positive integers \({\widetilde{\beth }}_{{\varvec{z}}},z=\mathrm{1,2},\dots ,p\), the PA operator is given by:
Where \(\left({\widetilde{\beth }}_{{\varvec{z}}}\right)= \sum\nolimits_{\begin{subarray}{l} z = 1 \\ z \ne y \end{subarray} }^{p} {SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)} \), which should hold the following axioms:
Definition 2
[9] The CQ-ROF set \(\widetilde{\beth }\) based on universal set \(X\) is defined by:
The \(\overline{\overline{{\Delta }_{\widetilde{\beth }}}}\left(x\right)=\left(\overline{\overline{{\Delta }_{\widetilde{rp}}}}\left(x\right),\overline{\overline{{\Delta }_{ip}}}\left(x\right)\right)\) and \(\overline{\overline{{\Xi }_{\widetilde{\beth }}}}\left(x\right)=\left(\overline{\overline{{\Xi }_{\widetilde{rp}}}}\left(x\right),\overline{\overline{{\Xi }_{\widetilde{ip}}}}\left(x\right)\right)\) are called a truth or positive and falsity or negative grades with \(0\le {\overline{\overline{{\Delta }_{\widetilde{rp}}}}}^{\Lambda }\left(x\right)+{\overline{\overline{{\Xi }_{\widetilde{rp}}}}}^{\Lambda }\left(x\right)\le 1\) and \(0\le {\overline{\overline{{\Delta }_{\widetilde{ip}}}}}^{\Lambda }\left(x\right)+{\overline{\overline{{\Xi }_{\widetilde{ip}}}}}^{\Lambda }\left(x\right)\le 1,\Lambda \ge 1\). The simplest form of refusal or neutral grade is expressed by: \({\Pi }_{rg}\left(x\right)=\left({\Pi }_{rp}\left(x\right),{\Pi }_{ip}\left(x\right)\right)=\) \(\left({\left(1-\left({\overline{\overline{{\Delta }_{\widetilde{rp}}}}}^{\Lambda }\left(x\right)+{\overline{\overline{{\Xi }_{\widetilde{rp}}}}}^{\Lambda }\left(x\right)\right)\right)}^{\frac{1}{\Lambda }},\right.\) \(\left.{\left(1-\left({\overline{\overline{{\Delta }_{\widetilde{ip}}}}}^{\Lambda }\left(x\right)+{\overline{\overline{{\Xi }_{\widetilde{ip}}}}}^{\Lambda}\left(x\right)\right)\right)}^{\frac{1}{\Lambda}}\right)\). Moreover, a CQ-ROFN is expressed by \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\).
Definition 3
[8] For any two CQ-ROFNs \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2}\), the Algebraic operational laws of them are defined by:
Definition 4
[8] For any two CQ-ROFNs \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2}\), the score and accuracy values of them are defined as
For Eq. (10) and Eq. (11), we can get the comparison method for any two CQ-ROFNs
-
(1)
when \({\nabla }_{SV}\left({\widetilde{\beth }}_{1}\right)>{\nabla }_{SV}\left({\widetilde{\beth }}_{2}\right)\Rightarrow {\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{2}\);
-
(2)
when \({\nabla }_{SV}\left({\widetilde{\beth }}_{1}\right)<{\nabla }_{SV}\left({\widetilde{\beth }}_{2}\right)\Rightarrow {\widetilde{\beth }}_{1}<{\widetilde{\beth }}_{2}\);
-
(3)
when \({\nabla }_{SV}\left({\widetilde{\beth }}_{1}\right)={\nabla }_{SV}\left({\widetilde{\beth }}_{2}\right)\Rightarrow \) when \({\nabla }_{AV}\left({\widetilde{\beth }}_{1}\right)>{\nabla }_{AV}\left({\widetilde{\beth }}_{2}\right)\Rightarrow {\widetilde{\beth }}_{1}>{\widetilde{\beth }}_{2}\); when \({\nabla }_{AV}\left({\widetilde{\beth }}_{1}\right)<{\nabla }_{AV}\left({\widetilde{\beth }}_{2}\right)\Rightarrow {\widetilde{\beth }}_{1}<{\widetilde{\beth }}_{2}\).
Further, we introduce Yager's t-norm and t-conorm (proposed by Yager [11]) which are stated as
Power Yager aggregation operators for CQ-ROFNs
In this subsection, we will develop the CQ-ROFPYA, CQ-ROFPYOA, CQ-ROFPYG, and CQ-ROFPYOG operators, and discuss their fundamental properties.
Definition 5
For any family of CQ-ROFNs \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), and \(\varphi >0,\) Yager's operational laws of them are defined as
Definition 6
For the collection of CQ-ROFNs \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), the CQ-ROFPYA operator is defined by:
where
where \(^\circ{\rm F} \left({\widetilde{\beth }}_{{\varvec{z}}}\right)= \sum\nolimits_{\begin{subarray}{l} z = 1 \\ z \ne y \end{subarray} }^{p} SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right),SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)=1-Dis\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)\), and
which should hold the following axioms:
Theorem 1
For the collection of CQ-ROFNs \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), the final value of Eq. (18) is also a CQ-ROFN, such as:
Obviously, Eq. (24) is the modified version of the power, Yager, and power Yager aggregation operators for FSs, IFSs, PFSs, Q-ROFSs, CFSs, CIFSs, CPFSs, and CQ-ROFSs.
Proof
We can prove it by mathematical induction for \(p\).
-
(1)
When \(p=2\) in Eq. (24), we have
$$\begin{aligned}{\overline{\overline{\aleph }}}_{1}{\widetilde{\beth }}_{1}=\left(\begin{array}{c}\left({\left(\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{1}\left({\overline{\overline{{\Delta }_{\widetilde{rp-1}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\right.\\\left.{\left(\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{1}\left({\overline{\overline{{\Delta }_{\widetilde{ip-1}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\right),\\ \left(\begin{array}{c}{\left(1-\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{1}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{rp-1}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\\ {\left(1-\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{1}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{ip-1}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\end{array}\right)\end{array}\right),\end{aligned}$$$$\begin{aligned}{\overline{\overline{\aleph }}}_{2}{\widetilde{\beth }}_{2}=\left(\begin{array}{c}\left({\left(\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{2}\left({\overline{\overline{{\Delta }_{\widetilde{rp-2}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\right.\\ \left.{\left(\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{2}\left({\overline{\overline{{\Delta }_{\widetilde{ip-2}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\right),\\ \left(\begin{array}{c}{\left(1-\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{2}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{rp-2}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\\ {\left(1-\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{2}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{ip-2}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\end{array}\right)\end{array}\right)\end{aligned}$$$$CQ-ROFPYA\left({\widetilde{\beth }}_{1},{\widetilde{\beth }}_{2}\right)={\oplus }_{z=1}^{2}{\overline{\overline{\aleph }}}_{z}{\widetilde{\beth }}_{{\varvec{z}}}={\overline{\overline{\aleph }}}_{1}{\widetilde{\beth }}_{1}\oplus {\overline{\overline{\aleph }}}_{2}{\widetilde{\beth }}_{2}$$$$\begin{aligned}=\left(\begin{array}{c}\left({\left(\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{1}\left({\overline{\overline{{\Delta }_{\widetilde{rp-1}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\right.\\ \left.{\left(\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{1}\left({\overline{\overline{{\Delta }_{\widetilde{ip-1}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\right),\\ \left(\begin{array}{c}{\left(1-\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{1}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{rp-1}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\\ {\left(1-\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{1}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{ip-1}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\end{array}\right)\end{array}\right)\\ \oplus \left(\begin{array}{c}\left({\left(\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{2}\left({\overline{\overline{{\Delta }_{\widetilde{rp-2}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\right.\\ \left.{\left(\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{2}\left({\overline{\overline{{\Delta }_{\widetilde{ip-2}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\right),\\ \left(\begin{array}{c}{\left(1-\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{2}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{rp-2}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\\ {\left(1-\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{2}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{ip-2}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\end{array}\right)\end{array}\right)\end{aligned}$$$$\begin{aligned}=\left(\begin{array}{c}\left({\left(\mathrm{min}\left(1,{\left(\sum_{z=1}^{2}{\overline{\overline{\aleph }}}_{z}\left({\overline{\overline{{\Delta }_{\widetilde{rp-z}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\right.\\ \left.{\left(\mathrm{min}\left(1,{\left(\sum_{z=1}^{2}{\overline{\overline{\aleph }}}_{z}\left({\overline{\overline{{\Delta }_{\widetilde{ip-z}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\right),\\ \left(\begin{array}{c}{\left(1-\mathrm{min}\left(1,{\left(\sum_{z=1}^{2}{\overline{\overline{\aleph }}}_{z}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{rp-z}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\\ {\left(1-\mathrm{min}\left(1,{\left(\sum_{z=1}^{2}{\overline{\overline{\aleph }}}_{z}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\end{array}\right)\end{array}\right),\end{aligned}$$i.e., Eq. (24) is held when \(p=2.\)
-
(2)
Suppose \(\mathrm{when }p=k\), Eq. (24) is held, then we have:
$$\begin{aligned}&CQ-ROFPYA\left({\widetilde{\beth }}_{1},{\widetilde{\beth }}_{2},\dots ,{\widetilde{\beth }}_{{\varvec{k}}}\right)\\&\quad=\left(\begin{array}{c}\left({\left(\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}\left({\overline{\overline{{\Delta }_{\widetilde{rp-z}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\right. \\ \left.{\left(\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}\left({\overline{\overline{{\Delta }_{\widetilde{ip-z}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\right),\\ \left(\begin{array}{c}{\left(1-\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{rp-z}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\\ {\left(1-\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\end{array}\right)\end{array}\right).\end{aligned}$$Then, when \(p=k+1\), we have:
$$\begin{aligned}&CQ-ROFPYA\left({\widetilde{\beth }}_{1},{\widetilde{\beth }}_{2},\dots ,{\widetilde{\beth }}_{{\varvec{k}}+1}\right)\\&\quad ={\overline{\overline{\aleph }}}_{1}{\widetilde{\beth }}_{1}\oplus {\overline{\overline{\aleph }}}_{2}{\widetilde{\beth }}_{2}\oplus ,\dots ,\oplus {\overline{\overline{\aleph }}}_{k}{\widetilde{\beth }}_{{\varvec{k}}}\\&\quad\quad\oplus {\overline{\overline{\aleph }}}_{k+1}{\widetilde{\beth }}_{{\varvec{k}}+1}\end{aligned}$$$$={\oplus }_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}{\widetilde{\beth }}_{{\varvec{z}}}\oplus {\overline{\overline{\aleph }}}_{k+1}{\widetilde{\beth }}_{{\varvec{k}}+1}$$$$\begin{aligned}&=\left(\begin{array}{c}\left({\left(\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}\left({\overline{\overline{{\Delta }_{\widetilde{rp-z}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\right. \\ \left.{\left(\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}\left({\overline{\overline{{\Delta }_{\widetilde{ip-z}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\right),\\ \left(\begin{array}{c}{\left(1-\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{rp-z}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\\ {\left(1-\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\end{array}\right)\end{array}\right)\\&\quad\oplus {\overline{\overline{\aleph }}}_{k+1}{\widetilde{\beth }}_{{\varvec{k}}+1}\end{aligned} $$$$\begin{aligned}&=\left(\begin{array}{c}\left({\left(\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}\left({\overline{\overline{{\Delta }_{\widetilde{rp-z}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\right. \\ \left.{\left(\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}\left({\overline{\overline{{\Delta }_{\widetilde{ip-z}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\right),\\ \left(\begin{array}{c}{\left(1-\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{rp-z}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\\ {\left(1-\mathrm{min}\left(1,{\left(\sum_{z=1}^{k}{\overline{\overline{\aleph }}}_{z}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\end{array}\right)\end{array}\right)\\ & \quad \oplus \left(\begin{array}{c}\left({\left(\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{k+1}\left({\overline{\overline{{\Delta }_{\widetilde{rp-k+1}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\right. \\ \left.{\left(\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{k+1}\left({\overline{\overline{{\Delta }_{\widetilde{ip-k+1}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\right),\\ \left(\begin{array}{c}{\left(1-\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{k+1}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{rp-k+1}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\\ {\left(1-\mathrm{min}\left(1,{\left({\overline{\overline{\aleph }}}_{k+1}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{ip-k+1}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\end{array}\right)\end{array}\right)\end{aligned}$$$$\begin{aligned}=\left(\begin{array}{c}\left({\left(\mathrm{min}\left(1,{\left(\sum_{z=1}^{k+1}{\overline{\overline{\aleph }}}_{z}\left({\overline{\overline{{\Delta }_{\widetilde{rp-z}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\right. \\ \left.{\left(\mathrm{min}\left(1,{\left(\sum_{z=1}^{k+1}{\overline{\overline{\aleph }}}_{z}\left({\overline{\overline{{\Delta }_{\widetilde{ip-z}}}}}^{\Lambda \Pi }\right)\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\right),\\ \left(\begin{array}{c}{\left(1-\mathrm{min}\left(1,{\left(\sum_{z=1}^{k+1}{\overline{\overline{\aleph }}}_{z}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{rp-z}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }},\\ {\left(1-\mathrm{min}\left(1,{\left(\sum_{z=1}^{k+1}{\overline{\overline{\aleph }}}_{z}{\left(1-{\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}}^{\Lambda }\right)}^{\Pi }\right)}^{\frac{1}{\Pi }}\right)\right)}^{\frac{1}{\Lambda }}\end{array}\right)\end{array}\right),\end{aligned}$$i.e., when \(p=k+1,\) Eq. (24) is also held.
-
(3)
Based on (1) and (2), we can get the conclusion when \(p\) is any positive value, Eq. (24) is held.
Furthermore, we discuss their valuable and feasible properties such as idempotency, monotonicity, and boundedness.
Property 1
(idempotency): When \({\widetilde{\beth }}_{z}=\widetilde{\beth }=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp}}}},\overline{\overline{{\Delta }_{ip}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp}}}},\overline{\overline{{\Xi }_{\widetilde{ip}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), then
Property 2
(monotonicity): When \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right)\le {\widetilde{\beth }}_{z}^{\boldsymbol{^{\prime}}\boldsymbol{^{\prime}}}=\left(\left({\overline{\overline{{\Delta }_{\widetilde{rp-z}}}}}^{\boldsymbol{^{\prime}}\boldsymbol{^{\prime}}},{\overline{\overline{{\Delta }_{ip-z}}}}^{\boldsymbol{^{\prime}}\boldsymbol{^{\prime}}}\right),\left({\overline{\overline{{\Xi }_{\widetilde{rp-z}}}}}^{\boldsymbol{^{\prime}}\boldsymbol{^{\prime}}},{\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}}^{\boldsymbol{^{\prime}}\boldsymbol{^{\prime}}}\right)\right),z=\mathrm{1,2},\dots ,p\), then
Property 3
(boundedness): When \({\widetilde{\beth }}_{z}^{-}=\left(\left(\underset{z}{\mathrm{min}}\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{min}}\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\underset{z}{\mathrm{max}}\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{max}}\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right)\) and \({\widetilde{\beth }}_{z}^{+}=\left(\left(\underset{z}{\mathrm{max}}\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{max}}\overline{\overline{{\Delta }_{ip-z}}}\right),\right.\) \(\left.\left.\underset{z}{\mathrm{max}}\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\underset{z}{\mathrm{min}}\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{min}}\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), then
Definition 7
For the collection of CQ-ROFNs \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), the CQ-ROFPYOA operator is defined by:
With \(o\left(z\right)\le o\left(z-1\right)\), where
where \(^\circ{\rm F} \left({\widetilde{\beth }}_{{\varvec{z}}}\right)= \sum\nolimits_{\begin{subarray}{l} z = 1 \\ z \ne y \end{subarray} }^{p} SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right),SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)=1-Dis\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)\), and
which should hold the following axioms:
Theorem 2
For the collection of CQ-ROFNs \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), the final value of Eq. (28) is also a CQ-ROFN, such as:
Obviously, Eq. (34) is the modified version of the power, Yager, and power Yager aggregation operators for FSs, IFSs, PFSs, Q-ROFSs, CFSs, CIFSs, CPFSs, and CQ-ROFSs. Furthermore, we discuss their valuable and feasible properties such as idempotency, monotonicity, and boundedness.
Property 4
(idempotency): When \({\widetilde{\beth }}_{z}=\widetilde{\beth }=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp}}}},\overline{\overline{{\Delta }_{ip}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp}}}},\overline{\overline{{\Xi }_{\widetilde{ip}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), then
Property 5
(monotonicity): When \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right)\le {\widetilde{\beth }}_{z}^{{^{\prime}}{^{\prime}}}=\left(\left({\overline{\overline{{\Delta }_{\widetilde{rp-z}}}}}^{{^{\prime}}{^{\prime}}},{\overline{\overline{{\Delta }_{ip-z}}}}^{{^{\prime}}{^{\prime}}}\right),\left({\overline{\overline{{\Xi }_{\widetilde{rp-z}}}}}^{{^{\prime}}{^{\prime}}},{\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}}^{{^{\prime}}{^{\prime}}}\right)\right),z=\mathrm{1,2},\dots ,p\), then
Property 6
(boundedness): When \({\widetilde{\beth }}_{z}^{-}=\left(\left(\underset{z}{\mathrm{min}}\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{min}}\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\underset{z}{\mathrm{max}}\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{max}}\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right)\) and \({\widetilde{\beth }}_{z}^{+}=\left(\left(\underset{z}{\mathrm{max}}\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{max}}\overline{\overline{{\Delta }_{ip-z}}}\right)\right.,\) \(\left.\left(\underset{z}{\mathrm{min}}\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{min}}\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), then
Definition 8
For the collection of CQ-ROFNs \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), the CQ-ROFPYG operator is defined by:
where \({\overline{\overline{\mathrm{\aleph }}}}_{z}=\frac{\left(1+\mathrm{^\circ{\rm F} }\left({\widetilde{\mathrm{\beth }}}_{{\varvec{z}}}\right)\right)}{\sum_{z=1}^{p}\left(1+\mathrm{^\circ{\rm F} }\left({\widetilde{\mathrm{\beth }}}_{{\varvec{z}}}\right)\right)},\) \(\mathrm{^\circ{\rm F} }\left({\widetilde{\mathrm{\beth }}}_{{\varvec{z}}}\right)= \sum\nolimits_{\begin{subarray}{l} z = 1 \\ z \ne y \end{subarray} }^{p} SuP\left({\widetilde{\mathrm{\beth }}}_{{\varvec{z}}},{\widetilde{\mathrm{\beth }}}_{{\varvec{y}}}\right),SuP\left({\widetilde{\mathrm{\beth }}}_{{\varvec{z}}},{\widetilde{\mathrm{\beth }}}_{{\varvec{y}}}\right)=1-Dis\left({\widetilde{\mathrm{\beth }}}_{{\varvec{z}}},{\widetilde{\mathrm{\beth }}}_{{\varvec{y}}}\right)\), and \(Dis\left({\widetilde{\mathrm{\beth }}}_{{\varvec{z}}},{\widetilde{\mathrm{\beth }}}_{{\varvec{y}}}\right)=\frac{1}{4}\left(\left|\overline{\overline{{\Delta }_{\widetilde{rp-z}}}}-\overline{\overline{{\Delta }_{\widetilde{rp-y}}}}\right|\right.\) \(\left.+\left|\overline{\overline{{\Delta }_{\widetilde{ip-z}}}}-\overline{\overline{{\Delta }_{\widetilde{ip-y}}}}\right|+\left|\overline{\overline{{\Xi }_{\widetilde{rp-z}}}}-\overline{\overline{{\Xi }_{\widetilde{rp-y}}}}\right|+\left|\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}-\overline{\overline{{\Xi }_{\widetilde{ip-y}}}}\right|\right)\), which should hold the following axioms:\((1)SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)\in \left[\mathrm{0,1}\right]\),\((2)SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)=SuP\left({\widetilde{\beth }}_{{\varvec{y}}},{\widetilde{\beth }}_{{\varvec{z}}}\right)\) and \(\left(3\right)SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)\ge SuP\left({\widetilde{\beth }}_{{\varvec{a}}},{\widetilde{\beth }}_{{\varvec{b}}}\right) \mathrm{If }\left|{\widetilde{\beth }}_{{\varvec{z}}}-{\widetilde{\beth }}_{{\varvec{y}}}\right|\le \left|{\widetilde{\beth }}_{{\varvec{a}}}-{\widetilde{\beth }}_{{\varvec{b}}}\right|\).
Theorem 2
For the collection of CQ-ROFNs \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), the final value of Eq. (38) is also a CQ-ROFN, such as:
Obviously, Eq. (39) is the modified version of the power, Yager, and power Yager aggregation operators for FSs, IFSs, PFSs, Q-ROFSs, CFSs, CIFSs, CPFSs, and CQ-ROFSs. Furthermore, we discuss their valuable and feasible properties such as idempotency, monotonicity, and boundedness.
Property 7
(idempotency): When \({\widetilde{\beth }}_{z}=\widetilde{\beth }=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp}}}},\overline{\overline{{\Delta }_{ip}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp}}}},\overline{\overline{{\Xi }_{\widetilde{ip}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), then
Property 8
(monotonicity): When \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right)\le {\widetilde{\beth }}_{z}^{{^{\prime}}{^{\prime}}}=\left(\left({\overline{\overline{{\Delta }_{\widetilde{rp-z}}}}}^{{^{\prime}}{^{\prime}}},{\overline{\overline{{\Delta }_{ip-z}}}}^{{^{\prime}}{^{\prime}}}\right),\left({\overline{\overline{{\Xi }_{\widetilde{rp-z}}}}}^{{^{\prime}}{^{\prime}}},{\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}}^{{^{\prime}}{^{\prime}}}\right)\right),z=\mathrm{1,2},\dots ,p\), then
Property 9
(boundedness): When \({\widetilde{\beth }}_{z}^{-}=\left(\left(\underset{z}{\mathrm{min}}\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{min}}\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\underset{z}{\mathrm{max}}\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{max}}\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right)\) and \({\widetilde{\beth }}_{z}^{+}=\left(\left(\underset{z}{\mathrm{max}}\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{max}}\overline{\overline{{\Delta }_{ip-z}}}\right)\right.,\) \(\left.\left(\underset{z}{\mathrm{min}}\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{min}}\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), then
Definition 9
For the collection of CQ-ROFNs \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), the CQ-ROFPYOG operator is defined by:
With \(o\left(z\right)\le o\left(z-1\right)\), where \({\overline{\overline{\aleph }}}_{z}=,\) \(^\circ{\rm F} \left({\widetilde{\beth }}_{{\varvec{z}}}\right)= \sum\nolimits_{\begin{subarray}{l} z = 1 \\ z \ne y \end{subarray} }^{p} SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right),SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)=1-Dis\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)\), and \(Dis\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)=\frac{1}{4}\left(\left|\overline{\overline{{\Delta }_{\widetilde{rp-z}}}}-\overline{\overline{{\Delta }_{\widetilde{rp-y}}}}\right|\right.\) \(\left.+\left|\overline{\overline{{\Delta }_{\widetilde{ip-z}}}}-\overline{\overline{{\Delta }_{\widetilde{ip-y}}}}\right|+\left|\overline{\overline{{\Xi }_{\widetilde{rp-z}}}}-\overline{\overline{{\Xi }_{\widetilde{rp-y}}}}\right|+\left|\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}-\overline{\overline{{\Xi }_{\widetilde{ip-y}}}}\right|\right)\), which should hold the following axioms:\(SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)\in \left[\mathrm{0,1}\right]\),\(SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)=SuP\left({\widetilde{\beth }}_{{\varvec{y}}},{\widetilde{\beth }}_{{\varvec{z}}}\right)\) and \(SuP\left({\widetilde{\beth }}_{{\varvec{z}}},{\widetilde{\beth }}_{{\varvec{y}}}\right)\ge SuP\left({\widetilde{\beth }}_{{\varvec{a}}},{\widetilde{\beth }}_{{\varvec{b}}}\right) \mathrm{If }\left|{\widetilde{\beth }}_{{\varvec{z}}}-{\widetilde{\beth }}_{{\varvec{y}}}\right|\le \left|{\widetilde{\beth }}_{{\varvec{a}}}-{\widetilde{\beth }}_{{\varvec{b}}}\right|\).
Theorem 4
For the collection of CQ-ROFNs \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), the final value of Eq. (43) is also a CQ-ROFN, such as:
Obviously, Eq. (44) is the modified version of the power, Yager, and power Yager aggregation operators for FSs, IFSs, PFSs, Q-ROFSs, CFSs, CIFSs, CPFSs, and CQ-ROFSs. Furthermore, we discuss their valuable and feasible properties such as idempotency, monotonicity, and boundedness.
Property 10
(idempotency): When \({\widetilde{\beth }}_{z}=\widetilde{\beth }=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp}}}},\overline{\overline{{\Delta }_{ip}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp}}}},\overline{\overline{{\Xi }_{\widetilde{ip}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), then
Property 11
(monotonicity): When \({\widetilde{\beth }}_{z}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right)\le {\widetilde{\beth }}_{z}^{{^{\prime}}{^{\prime}}}=\left(\left({\overline{\overline{{\Delta }_{\widetilde{rp-z}}}}}^{{^{\prime}}{^{\prime}}},{\overline{\overline{{\Delta }_{ip-z}}}}^{{^{\prime}}{^{\prime}}}\right),\left({\overline{\overline{{\Xi }_{\widetilde{rp-z}}}}}^{{^{\prime}}{^{\prime}}},{\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}}^{{^{\prime}}{^{\prime}}}\right)\right),z=1,2,\dots ,p\), then
Property 12
(boundedness): When \({\widetilde{\beth }}_{z}^{-}=\left(\left(\underset{z}{\mathrm{min}}\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{min}}\overline{\overline{{\Delta }_{ip-z}}}\right),\left(\underset{z}{\mathrm{max}}\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{max}}\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right)\) and \({\widetilde{\beth }}_{z}^{+}=\left(\left(\underset{z}{\mathrm{max}}\overline{\overline{{\Delta }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{max}}\overline{\overline{{\Delta }_{ip-z}}}\right)\right.,\) \(\left.\left(\underset{z}{\mathrm{min}}\overline{\overline{{\Xi }_{\widetilde{rp-z}}}},\underset{z}{\mathrm{min}}\overline{\overline{{\Xi }_{\widetilde{ip-z}}}}\right)\right),z=\mathrm{1,2},\dots ,p\), then
MADM methods based on the proposed operators
To get the best choice from the family of preferences, we need to develop the MADM methods based on the proposed operators. In this section, we propose the MADM methods with CQ-ROFNs based on the presented operators.
For a MADM problem, \({\widetilde{\beth }}_{1},{\widetilde{\beth }}_{2},\dots ,{\widetilde{\beth }}_{{\varvec{p}}}\) are the finite family of alternatives, and for every alternative, it is evaluated by the finite attributes such as \({C}_{1},{C}_{2},\dots ,{C}_{n},\) and the evaluation value \({\widetilde{\beth }}_{{\varvec{z}}{\varvec{j}}}\) for alternative \({\widetilde{\beth }}_{{\varvec{z}}}\) under attribute \({C}_{j}\) is expressed by a CQ-ROFN as \({\widetilde{\beth }}_{zj}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-zj}}}},\overline{\overline{{\Delta }_{ip-zj}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-zj}}}},\overline{\overline{{\Xi }_{\widetilde{ip-zj}}}}\right)\right),z=\mathrm{1,2},\dots ,p;j=\mathrm{1,2},\dots ,n\). Where \(\left(\overline{\overline{{\Delta }_{\widetilde{rp-zj}}}},\overline{\overline{{\Delta }_{ip-zj}}}\right)\) and \(\left(\overline{\overline{{\Xi }_{\widetilde{rp-zj}}}},\overline{\overline{{\Xi }_{\widetilde{ip-zj}}}}\right)\) express a truth and falsity grades with \(0\le {\overline{\overline{{\Delta }_{\widetilde{rp-zj}}}}}^{\Lambda }+{\overline{\overline{{\Xi }_{\widetilde{rp-zj}}}}}^{\Lambda }\le 1\) and \(0\le {\overline{\overline{{\Delta }_{ip-zj}}}}^{\Lambda }+{\overline{\overline{{\Xi }_{\widetilde{ip-zj}}}}}^{\Lambda }\le 1,\Lambda \ge 1\). In the following, we give the decision-making steps based on the proposed operators:
Step 1: Normalize the decision matrix
First, we need to obtain the decision matrix in which the element is expressed by a CQ-ROFN as \({\widetilde{\beth }}_{zj}=\left(\left(\overline{\overline{{\Delta }_{\widetilde{rp-zj}}}},\overline{\overline{{\Delta }_{ip-zj}}}\right),\left(\overline{\overline{{\Xi }_{\widetilde{rp-zj}}}},\overline{\overline{{\Xi }_{\widetilde{ip-zj}}}}\right)\right),z=\mathrm{1,2},\dots ,p;j=\mathrm{1,2},\dots ,n,\) then we need to convert the information in cost type to benefit type by
Step 2: Aggregate all attribute values for each alternative based on the CQ-ROFPYA operator and CQ-ROFPYG operator, and get the comprehensive value \({\widetilde{\beth }}_{z}\) by
and
Step 3: Calculate the score value for comprehensive value \({\widetilde{\beth }}_{z}\) by
Step 4: Get the ranking results according to the score values and give the best choice from the collection of finite alternatives.
In the following, we give some examples to show the decision steps.
Example 1
A university wants to select a new faculty member for Fuzzy mathematics in the department of mathematics and statistics, university of ABC, Islamabad, Pakistan. For the position of fuzzy mathematics, five candidates are applied who are expressed by \({\widetilde{\beth }}_{1},{\widetilde{\beth }}_{2},{\widetilde{\beth }}_{3},{\widetilde{\beth }}_{4}\) and \({\widetilde{\beth }}_{5}\) and are regarded as the family of alternatives. Further, they are evaluated by the following features \({C}_{1}\): fifteen-year experience, \({C}_{2}\): fifteen impact factor publications, \({C}_{3}\): age of the candidate, \({C}_{4}\): and personality and extra skills of the candidate. Based on the above MADM methods, we can solve this problem, and the decision steps are shown as follows.
Step 1: Normalize the decision matrix.
First, we get the decision matrix in which the element is expressed by the CQ-ROFNs shown in Table 3, then we normalize them. Because all the evaluation values are benefit type, they do not need to be normalized.
Step 2: Aggregate all attribute values of each alternative by the CQ-ROFPYA operator and CQ-ROFPYG operator, and the comprehensive values are shown in Table 4.
Step 3: Obtain the score values of aggregated values, which are shown in Table 5.
Step 4: Get the ranking results according to the score values which are shown in Table 6.
So, we get the same best choice is \({\widetilde{\beth }}_{1}\) according to the CQ-ROFPYA operator and the CQ-ROFPYG operator. Further, we use the proposed method to solve some decision problems with different uncertain information.
Example 2. Considering a decision problem with the CPF information and the decision matrix is shown in Table 7.
Further, we aggregate all attribute values of each alternative by the CQ-ROFPYA operator and CQ-ROFPYG operator, and the comprehensive values are shown in Table 8.
Obtain the score values of aggregated values, which are shown in Table 9.
Get the ranking results according to the score values which are shown in Table 10.
We get the same ranking result by two operators, of course, the best choice is \({\widetilde{\beth }}_{4}.\)
Example 3
Considering a decision problem with the CIF information and the decision matrix is shown in Table 11.
We aggregate all attribute values of each alternative by the CQ-ROFPYA operator and CQ-ROFPYG operator, and the comprehensive values are shown in Table 12.
Obtain the score values of aggregated values, which are shown in Table 13.
Get the ranking results according to the score values which are shown in Table 14.
We get the same ranking result by two operators, of course, the best choice is \({\widetilde{\beth }}_{5}.\)
Examples 2 and 3 can show the effectiveness of the proposed methods.
Further, using the information in Tables 3, 7, and 11, we compare the proposed methods with some famous existing methods.
Comparative analysis
To further show the advantages of the proposed methods in this manuscript, we compare them with some existing method based on some operators [12,13,14,15,16,17,18,19,20,21,22,23,24,25], including the averaging aggregation operators (AOs) for IFS from Xu [12], the geometric AOs for IFS from Xu and Yager [13], PA operators for IFS from Xu [14], the geometric AOs for PFS by Rahman et al. [15]. The PA operators for PFS by Wei and Lu [16], the Yager aggregation operators for PFS by Shahzadi and Akram [17], the simple AOs for Q-ROFS by Liu and Wang [18], the geometric AOs for Q-ROFSs from Riaz et al. [19], the Yager aggregation operators for Q-ROFS by Akram and Shahzadi [20], AOs for CIFS by Garg and Rani [21], geometric AOs for CIFSs by Garg and Rani [22], the power AOs for CIFS by Rani and Garg [23], the Yager AOs for CPFSs by Akram et al. [24], and the PA operators for CQ-ROFS by Garg et al. [25]. The comparative analysis based on the information in Table 3 is shown in Table 15.
From Table 15, we can know the best choice is \({\widetilde{\beth }}_{1}\) based on the CQ-ROFPYA operator, the CQ-ROFPYG operator, and the methods from Garg et al. [25]. They get the same best result. However, the operators in [12,13,14,15,16,17,18,19,20,21,22,23,24] can only process the uncertain information from FSs, IFSs, CIFSs, PFSs, Q-ROFS, and CPFSs, and they cannot deal with the information in CQ-ROFS, our methods can deal with all mentioned information types. In addition, compared with the methods in [25], our methods used the operational laws based on Yager t-norm and t-conorm, and [25] used the traditional operational laws. [25] is a special case of our methods. Our methods are more general than the methods in [12,13,14,15,16,17,18,19,20,21,22,23,24,25].
Further, the comparative analysis based on the information on CPFSs in Table 7 is shown in Table 16.
From Table 16, we can know that the best choice is \({\widetilde{\beth }}_{4}\) based on the CQ-ROFPYA operator, the CQ-ROFPYG operator, Yager AOs for CPFSs [24], and the PA operators for CQ-ROFS [25]. Of course, they also produce the same ranking result, this can explain the effectiveness of the proposed methods. However, the operators in [12,13,14,15,16,17,18,19,20,21,22,23] can only process the uncertain information from FSs, IFSs, CIFSs, PFSs, and Q-ROFSs, and they cannot deal with the information in CPFSs and CQ-ROFS, our methods can deal with all mentioned information types. In addition, compared with the methods in [24], our methods used Power operators, and [25] used traditional operators and CPFs. Our methods are more general than the methods in [12,13,14,15,16,17,18,19,20,21,22,23,24,25].
Moreover, the comparative analysis based on the information on CIFSs in Table 11 is shown in Table 17.
From Table 17, we can know that the best choice is \({\widetilde{\beth }}_{4}\) based on the CQ-ROFPYA operator, the CQ-ROFPYG operator, AOs for CIFS [21], geometric AOs for CIFSs [22], the power AOs for CIFS [23] Yager AOs for CPFSs [24], and the PA operators for CQ-ROFS [25]. Our method in the CQ-ROFPYA operator produced the same ranking result as the operators in [21, 23], and [24], while our method in the CQ-ROFPYG operator produced the same ranking result as the operators in [22] because they used the geometric AOs. Compared with [21], the operators in [21] only can deal with CIFS, and only used the traditional AOs; Compared with [22], the operators in [22] only can deal with CIFS, and only used the traditional geometric AOs; Compared with [23], the operators in [23] only can deal with CIFS, and only used the traditional operational laws. Our methods are more general than theirs.
Our proposed operators are based on CQ-ROFS, PA, and the Yager operational laws, and they are novel and interesting to cope with unreliable and awkward information in decision-making problems.
In addition, no special software and hardware are required due to the small amount of computation, and Excel can meet this requirement.
Conclusion
According to the PA operators and Yager t-norm and t-conorm under the consideration of CQ-ROF information, we derived the following ideas:
-
1.
We developed Yager operational laws for CQ-ROF values.
-
2.
We proposed the famous CQ-ROFPYA, CQ-ROFPYOA, CQ-ROFPYG, and CQ-ROFPYOG operators.
-
3.
We discussed some properties such as idempotency, monotonicity, and boundedness.
-
4.
We proposed a MADM methods based on the proposed operators for CQ-ROF information.
-
5.
We compared the proposed operators with some existing operators to show the effectiveness of the proposed methods.
In the future, we study the fuzzy multi-criteria decision-making problems based on the assessment of hydroponic vertical farming [33], investigation of the fuzzy Dombi EDAS technique [34], analysis of type-2 Gaussian fuzzy information [35], analysis of fuzzy secure filtering [36], interval-type-2 fuzzy non-homogeneous information [37], and fuzzy fault detection under the presence of the Markov jump systems [38], then we try to utilize them to artificial intelligence, machine learning, game theory, clustering analysis, pattern recognition, and medical diagnosis.
Availability of data and materials
Not applicable.
Abbreviations
- CQ-ROF:
-
Complex q-rung orthopair fuzzy
- CQ-ROFPYA:
-
CQ-ROF power Yager averaging
- CQ-ROFPYOA:
-
CQ-ROF power Yager ordered averaging
- CQ-ROFPYG:
-
CQ-ROF power Yager geometric
- CQ-ROFPYOG:
-
CQ-ROF power Yager ordered geometric
- MADM:
-
Multi-attribute decision-making
- FS:
-
Fuzzy sets
- IFS:
-
Intuitionistic FS
- PFS:
-
Pythagorean FS
- Q-ROFS:
-
Q-rung orthopair FS
- CFS:
-
Complex fuzzy sets
- CIFS:
-
Complex intuitionistic FS
- CPFS:
-
Complex Pythagorean FS
- CQ-ROFS:
-
Complex q-rung orthopair FS
- PA:
-
Power aggregation
- AOs:
-
Aggregation operators
References
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353
Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96
Yager RR (2013) Pythagorean fuzzy subsets. In: 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS). IEEE, pp 57–61
Yager RR (2016) Generalized orthopair fuzzy sets. IEEE Trans Fuzzy Syst 25(5):1222–1230
Ramot D, Milo R, Friedman M, Kandel A (2002) Complex fuzzy sets. IEEE Trans Fuzzy Syst 10(2):171–186
Alkouri AMDJS and Salleh AR (2012) Complex intuitionistic fuzzy sets. In: AIP conference proceedings, Vol 1482, No. 1. American Institute of Physics, pp 464–470
Ullah K, Mahmood T, Ali Z, Jan N (2020) On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition. Complex Intell Syst 6(1):15–27
Liu P, Mahmood T, Ali Z (2020) Complex q-rung orthopair fuzzy aggregation operators and their applications in multi-attribute group decision making. Information 11(1):5
Liu P, Ali Z, Mahmood T (2019) A method to multi-attribute group decision-making problem with complex q-rung orthopair linguistic information based on Heronian mean operators. Int J Comput Intell Syst 12(2):1465–1496
Yager RR (2001) The power average operator. IEEE Trans Syst Man Cybern Part A 31(6):724–731
Yager RR (1994) Aggregation operators and fuzzy systems modeling. Fuzzy Sets Syst 67(2):129–145
Xu Z (2007) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15(6):1179–1187
Xu Z, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 35(4):417–433
Xu Z (2011) Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators. Knowl-Based Syst 24(6):749–760
Rahman K, Abdullah S, Husain F and Khan MA (2016) Approaches to Pythagorean fuzzy geometric aggregation operators. Int J Comput Sci Inf Secur (IJCSIS) 14(9):174–200
Wei G, Lu M (2018) Pythagorean fuzzy power aggregation operators in multiple attribute decision making. Int J Intell Syst 33(1):169–186
Shahzadi G, Akram M, Al-Kenani AN (2020) Decision-making approach under Pythagorean fuzzy Yager weighted operators. Mathematics 8(1):70
Liu P, Wang P (2018) Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making. Int J Intell Syst 33(2):259–280
Riaz M, Razzaq A, Kalsoom H, Pamučar D, Athar Farid HM, Chu YM (2020) q-Rung orthopair fuzzy geometric aggregation operators based on generalized and group-generalized parameters with application to water loss management. Symmetry 12(8):1236
Akram M, Shahzadi G (2021) A hybrid decision-making model under q-rung orthopair fuzzy Yager aggregation operators. Granular Comput 6(4):763–777
Garg H, Rani D (2019) Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process. Arab J Sci Eng 44(3):2679–2698
Garg H, Rani D (2020) Generalized geometric aggregation operators based on t-norm operations for complex intuitionistic fuzzy sets and their application to decision-making. Cogn Comput 12(3):679–698
Rani D, Garg H (2018) Complex intuitionistic fuzzy power aggregation operators and their applications in multicriteria decision-making. Expert Syst 35(6):e12325
Akram M, Peng X, Sattar A (2021) Multi-criteria decision-making model using complex Pythagorean fuzzy Yager aggregation operators. Arab J Sci Eng 46(2):1691–1717
Garg H, Gwak J, Mahmood T, Ali Z (2020) Power aggregation operators and VIKOR methods for complex q-rung orthopair fuzzy sets and their applications. Mathematics 8(4):538
Senapati T, Chen G, Mesiar R, Yager RR (2023) Intuitionistic fuzzy geometric aggregation operators in the framework of Aczel-Alsina triangular norms and their application to multiple attribute decision making. Expert Syst Appl 212:118832
Garg H, Ali Z, Mahmood T, Ali MR, Alburaikan A (2023) Schweizer-Sklar prioritized aggregation operators for intuitionistic fuzzy information and their application in multi-attribute decision-making. Alex Eng J 67:229–240
Azeem W, Mahmood W, Mahmood T, Ali Z, Naeem M (2023) Analysis of Einstein aggregation operators based on complex intuitionistic fuzzy sets and their applications in multi-attribute decision-making. AIMS Math 8(3):6036–6063
Mahmood T, Ali Z (2023) Multi-attribute decision-making methods based on Aczel-Alsina power aggregation operators for managing complex intuitionistic fuzzy sets. Comput Appl Math 42(2):1–34
Senapati T, Simic V, Saha A, Dobrodolac M, Rong Y, Tirkolaee EB (2023) Intuitionistic fuzzy power Aczel-Alsina model for prioritization of sustainable transportation sharing practices. Eng Appl Artif Intell 119:105716
Jin H, Hussain A, Ullah K, Javed A (2023) Novel complex pythagorean fuzzy sets under Aczel-Alsina operators and their application in multi-attribute decision making. Symmetry 15(1):68
Liu P, Ali Z, Mahmood T (2023) Archimedean aggregation operators based on complex Pythagorean fuzzy sets using confidence levels and their application in decision making. Int J Fuzzy Syst 25(1):42–58
Cagri Tolga A, Basar M (2022) The assessment of a smart system in hydroponic vertical farming via fuzzy MCDM methods. J Intell Fuzzy Syst 42(1):1–12
Deveci M, Gokasar I, Castillo O, Daim T (2022) Evaluation of Metaverse integration of freight fluidity measurement alternatives using fuzzy Dombi EDAS model. Comput Ind Eng 174:108773
Tolga AC, Parlak IB, Castillo O (2020) Finite-interval-valued Type-2 Gaussian fuzzy numbers applied to fuzzy TODIM in a healthcare problem. Eng Appl Artif Intell 87:103352
Zhang Z, Song X, Sun X, Stojanovic V (2023) Hybrid-driven-based fuzzy secure filtering for nonlinear parabolic partial differential equation systems with cyber attacks. Int J Adapt Control Signal Process 37(2):380–398
Zhang X, Wang H, Stojanovic V, Cheng P, He S, Luan X, Liu F (2021) Asynchronous fault detection for interval type-2 fuzzy nonhomogeneous higher level Markov jump systems with uncertain transition probabilities. IEEE Trans Fuzzy Syst 30(7):2487–2499
Cheng P, He S, Stojanovic V, Luan X, Liu F (2021) Fuzzy fault detection for Markov jump systems with partly accessible hidden information: an event-triggered approach. IEEE Trans Cybern 52(8):7352–7361
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare they have no conflicts of interest.
Ethical approval and consent to participate
We declare that we do have no commercial or associative interests that represent a conflict of interests in connection with this manuscript. There are no professional or other personal interests that can inappropriately influence our submitted work.
Consent for publication
All authors agree to publish this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wu, X., Ali, Z., Mahmood, T. et al. Power aggregation operators based on Yager t-norm and t-conorm for complex q-rung orthopair fuzzy information and their application in decision-making problems. Complex Intell. Syst. 9, 5949–5963 (2023). https://doi.org/10.1007/s40747-023-01033-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40747-023-01033-3