r, s, t-Spherical Fuzzy VIKOR Method and Its Application in Multiple Criteria Group Decision Making

Abstract

This study intends to significantly enhance the capacity of decision experts (DEs) to capture their judgment in a larger area. In order to accomplish this, we propound the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {r},\textsf {s},\textsf {t}$ </tex-math></inline-formula> -spherical fuzzy set ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {r},\textsf {s},\textsf {t}$ </tex-math></inline-formula> -SFS), an expansion of the t-spherical fuzzy set. In <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {r},\textsf {s},\textsf {t}$ </tex-math></inline-formula> -SFS the sum of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {r}$ </tex-math></inline-formula> th power of membership grade, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {s}$ </tex-math></inline-formula> th power of neutral grade and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {t}$ </tex-math></inline-formula> th power of non-membership grade is less than or equal to 1, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {r}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {s}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {t}$ </tex-math></inline-formula> are natural numbers. Due to the inclusion of the extra parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {r}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {s}$ </tex-math></inline-formula> , the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {r},\textsf {s},\textsf {t}$ </tex-math></inline-formula> -SFS is able to describe assessment information in a more flexible and comprehensive manner. This work begins by defining <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {r},\textsf {s},\textsf {t}$ </tex-math></inline-formula> -SFS and demonstrating that it is an extension of various existing fuzzy sets. The fundamental operations, score, and accuracy functions of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {r},\textsf {s},\textsf {t}$ </tex-math></inline-formula> -SFS are then introduced, and their mathematical features are examined. Also, we study some distance measures between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {r},\textsf {s},\textsf {t}$ </tex-math></inline-formula> -SFSs and their required properties. Next, to aggregate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textsf {r},\textsf {s},\textsf {t}$ </tex-math></inline-formula> -spherical fuzzy data, <inline-formula xmlns:mml="http://ww