Abstract
The intuitionistic fuzzy set introduced by Atanassov has greater ability in depicting and handling uncertainty. Intuitionistic fuzzy measure is an important research area of intuitionistic fuzzy set theory. Distance measure and similarity measure are two complementary concepts quantifying the difference and closeness of intuitionistic fuzzy sets. This paper addresses the definition of an effective distance measure with concise form and specific meaning for Atanassov’s intuitionistic fuzzy sets (AIFSs). A new distance measure for AIFSs is defined based on a distance measure of interval values and the transformation from AIFSs to interval valued fuzzy sets. The axiomatic properties of the new distance measure are mathematically investigated. Comparative analysis based in numerical examples indicates that the new distance measure is competent to quantify the difference between AIFSs. The application of the new distance measure is also discussed. A new method for multi-attribute decision making (MADM) is developed based on the technique for order preference by similarity to an ideal solution method and the new distance measure. Numerical applications indicate that the developed MADM method can obtain reasonable preference orders. This shows that the new distance measure is effective and rational from both mathematical and practical points of view.
Highlights
The theory of fuzzy set was initiated by Zadeh [1] to handle uncertainty
Even though many distance and similarity measures have been introduced for Atanassov’s intuitionistic fuzzy sets (AIFSs), an effective intuitionistic fuzzy distance measure with concise form and specific physical meaning is still desirable
We have proposed a new distance measure based on the relation between AIFSs and interval values
Summary
The theory of fuzzy set was initiated by Zadeh [1] to handle uncertainty. In a fuzzy set, the membership degree of its element is a real number in interval [0, 1]. Following the proposed axiomatic definition, they developed a new distance for AIFSs and applied it into pattern application. Chen [18] presented a continuous distance and similarity measures for AIFSs. Inspired by Szmidt and Kacprzyk’s method [19], Xu and Yager [20] improved the similarity measure of AIFSs and proposed an improved method to measure the similarity degree between AIFSs. By using the operators on intuitionistic fuzzy values (IFVs), Xia and Xu [21] proposed several similarity measures for IFVs, which can be generalized to similarity measures of AIFSs. Following analysis on the relation between entropy and intuitionistic fuzzy similarity measures, Wei et al [22] introduced a new method for constructing similarity measure based on entropy measures of interval-valued fuzzy sets, which have been proved to be equivalent to AIFSs. Based on the cosine of the angle between two vectors, Ye [23].
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