Cubic Vague Set and its Application in Decision Making

Area of Science:

• Fuzzy Set Theory
• Information Theory
• Data Modeling

Background:

• Cubic sets are hybrid structures with unique properties.
• Existing models may not fully capture complex two-dimensional data.
• The need for advanced methods to analyze periodic real-life phenomena.

Purpose of the Study:

• To introduce and define cubic vague sets (CVSs) as a generalized hybrid structure.
• To explore the properties of internal cubic vague sets (ICVSs) and external cubic vague sets (ECVSs).
• To develop a decision-making method using a similarity measure for CVSs and analyze its application in measuring information entropy.

Main Methods:

• Definition of cubic vague sets (CVSs), internal cubic vague sets (ICVSs), and external cubic vague sets (ECVSs).
• Analysis of ICVSs and ECVSs properties under P and R-Order.
• Derivation of conditions for union and intersection operations on ICVSs and ECVSs.
• Development of a decision-making approach based on a CVS similarity measure.

Main Results:

• Properties of ICVSs and ECVSs under P and R-Order were discussed.
• It was proven that R and R-intersection of ICVSs (or ECVSs) are not necessarily ICVSs (or ECVSs).
• Conditions were derived for P-union, P-intersection, R-union, and R-intersection operations to yield ICVSs or ECVSs.
• A novel similarity measure for CVSs was proposed and demonstrated through a numerical example for entropy measurement.

Conclusions:

• Cubic vague sets (CVSs) provide a novel framework for representing and modeling two-dimensional periodic information.
• The proposed similarity measure for CVSs is valuable for quantifying information entropy.
• The study advances fuzzy set theory with a new generalized hybrid structure applicable to real-world data analysis.