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Abstract
Similarity measures, distance measures and entropy measures are some common tools considered to be applied to some interesting real-life phenomena including pattern recognition, decision making, medical diagnosis and clustering. Further, interval-valued picture fuzzy sets (IVPFSs) are effective and useful to describe the fuzzy information. Therefore, this manuscript aims to develop some similarity measures for IVPFSs due to the significance of describing the membership grades of picture fuzzy set in terms of intervals. Several types cosine similarity measures, cotangent similarity measures, set-theoretic and grey similarity measures, four types of dice similarity measures and generalized dice similarity measures are developed. All the developed similarity measures are validated, and their properties are demonstrated. Two well-known problems, including mineral field recognition problems and multi-attribute decision making problems, are solved using the newly developed similarity measures. The superiorities of developed similarity measures over the similarity measures of picture fuzzy sets, interval-valued intuitionistic fuzzy sets and intuitionistic fuzzy sets are demonstrated through a comparison and numerical examples.
Highlights
The concept of fuzzy set (FS), a tool that deals with uncertainty, was proposed by Zadeh [1] in1965
To propose different SMs, such as cosine SMs, cosine SMs based on cosine function, SMs based on cotangent function, grey SMs, set-theoretic SMs, dice SMs and generalized dice SMs for interval-valued PFSs (IvPFSs)
In this subsection some cosine SMs based on cosine function and some weighted cosine SMs based on cosine function for IvPFSs are defined
Summary
The concept of fuzzy set (FS), a tool that deals with uncertainty, was proposed by Zadeh [1] in. Xu and Chen [18] made a comparative overview of different SMs, which are proposed using Hamming distance, weighted Euclidean distance, etc This overview is extended for interval-valued intuitionistic fuzzy sets (IvIFSs). Lui et al [21] proposed interval-valued intuitionistic fuzzy ordered weighted cosine similarity measure and used these SMs to solve investment decision making problems. To propose different SMs, such as cosine SMs, cosine SMs based on cosine function, SMs based on cotangent function, grey SMs, set-theoretic SMs, dice SMs and generalized dice SMs for IvPFSs. To develop applications to strategy decision making and mineral recognition. SM, set-theoretic SM, dice SMs and generalized dice SMs. In Section 4, applications of strategy decision making and mineral recognition are established and results are studied.
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