Multiple attribute decision-making based on cubical fuzzy aggregation operators

The picture fuzzy set, briefly as; PFS and its extensions, spherical fuzzy set (SFS), and T-spherical fuzzy set (T-SFS) are all effective tools to express uncertain and incomplete cognitive information with membership, neutral membership, and non-membership degrees. The cubical fuzzy set (CFS) introduced in this paper, carries out uncertain and imprecise information smartly in exercising decision-making than PFS and SFS. Cubical fuzzy set (CFS)is an extension of the picture fuzzy set and spherical fuzzy set. In CFS, the membership grades satisfy the condition $${0}\le {\mu } ^{3}(x)+{\eta }^{3}\left( {x}\right) +{\nu }^{3}\left( {x}\right) \le {1}$$ instead of $${0}\le {\mu } ^{2}({x})+{\eta }^{2}\left( {x}\right) +{\nu } ^{2}\left( {x}\right) \le {1}$$ , which is the condition of a spherical fuzzy set (SFS). In the course of this article, we first devise some operations on CFS, discuss the basic properties, and propose the cubical fuzzy arithmetic and geometric aggregation operators. We introduce the concept of cubical fuzzy weighted average (CFWA) operator, cubical fuzzy ordered weighted average (CFOWA) operator, and cubical fuzzy hybrid average (CFHA) operator. In the second section, we develop cubical fuzzy weighted geometric (CFWG) operator, cubical fuzzy ordered weighted geometric (CFOWG) operator, and cubical fuzzy hybrid geometric (CFHG) operator. We define the distance measure between two CFSs and study some of its properties. In the last section, the developed operators are utilized to devise approaches for solving multiple attribute decision-making problems (MADM) in a cubical fuzzy environment. A practical example of enterprise resource planning (ERP) system selection is given to verify the developed approach and to demonstrate the practicality and effectiveness of the proposed operators.