Abstract
Interval-valued Pythagorean fuzzy sets (PFSs), as an extension of PFSs, have strong potential in the management of complex uncertainty in real-world applications. This study aims to develop several interval-valued Pythagorean fuzzy Frank power (IVPFFP) aggregation operators with an adjustable parameter via the integration of an isomorphic Frank dual triple. First, a special automorphism on unit interval is introduced to construct an isomorphic Frank dual triple; and this triple is further applied on the definition of interval-valued Pythagorean fuzzy Frank operational laws. Second, two IVPFFP aggregation operators with the inclusion of an adjustable parameter are defined on the basis of the proposed operational laws, and several instrumental properties are then investigated. Furthermore, some limiting cases of the proposed IVPFFP operators are analyzed with respect to the varying adjustable parameter values. Finally, an IVPFFP aggregation operator-based multiple attribute group decision-making model is developed with a practical example furnished to demonstrate its feasibility and efficiency. The power that the adjustable parameter exhibits has been leveraged to affect the final decision results, and the proposed IVPFFP operators are compared with three selected aggregation operators to demonstrate their advantages provided with a practical example.
Highlights
The term Pythagorean fuzzy set (PFS)[1,2] was coined by Yager as a powerful extension of intuitionistic fuzzy set (IFS)[3]
Sco PiA − Sco PiG (i ∈ M4) become smaller as the value of χ increases. It illustrates that the aggregation result calculated by the IVPFFPWG operator is smaller than the result obtained from the IVPFFPWA operator
The IVPFFPWA and IVPFFPWG operators reduce to the IVPFFPA and IVPFFPG operators, respectively, which can be observed in Theorem 13(1) and Theorem 17(1)
Summary
The term Pythagorean fuzzy set (PFS)[1,2] was coined by Yager as a powerful extension of intuitionistic fuzzy set (IFS)[3]. A numerical example will be provided to reveal that the Frank dual triple is not suitable for defining the generalized operations on IVPFSs. In view of the reasons mentioned before, an automorphism on [0, 1] will be introduced in this study to develop an isomorphic Frank dual triple, which includes an isomorphic Frank t-norm, an isomorphic Frank s-norm and the Pythagorean negation[1,2]. In view of the reasons mentioned before, an automorphism on [0, 1] will be introduced in this study to develop an isomorphic Frank dual triple, which includes an isomorphic Frank t-norm, an isomorphic Frank s-norm and the Pythagorean negation[1,2] This new dual triple can be used to define the Frank operations on IVPFSs. A core step of MAGDM is to aggregate multiple assessment matrixes into a synthesis assessment matrix, which is often performed by appropriately selecting aggregation operators[31,32].
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