Interval-valued (p,q,r)-spherical fuzzy sets and their applications in MCGDM and MCDM based on TOPSIS method and aggregation operators

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The (p,q,r)-Spherical fuzzy set is a useful generalization of other fuzzy structures such as fuzzy, intuitionistic fuzzy, Pythagorean fuzzy, picture fuzzy, q-rung orthopair fuzzy, spherical fuzzy, and T-spherical fuzzy sets. In this paper, we define concept of interval-valued (p,q,r)-spherical fuzzy set (IVpqr-SFS) and Set-theoretical and arithmetic operations of them. Two aggregation operators named interval-valued (p,q,r)-spherical fuzzy weighted arithmetic (IVpqr-SFWAA) and interval-valued (p,q,r)-spherical fuzzy weighted geometric (IVpqr-SFWGA) aggregation operators are introduced for IVpqr-SF numbers (IVpqr-SFN) and their properties were examined. The proposed operators are explained with examples. Furthermore, the score and accuracy functions of an IVpqr-SFN are introduced. We also define distance measures between two same types IVpqr-SFSs based on Hamming, Euclidean, and Hausdorff distance measures. Moreover, a multi-criteria group decision-making (MCGDM) method is developed based on the TOPSIS method by using the proposed distance measures and score function, and a numerical example is given to show the process of the proposed MCGDM method. In addition, a multi-criteria decision-making (MCDM) method is developed using the proposed IVpqr-SFWAA, IVpqr-SFWGA operators, and an illustrative example is given to show the process of the proposed MCDM method. Finally, a comparative analysis is presented between the proposed methods and the existing methods.