Abstract
As the complementary concept of intuitionistic fuzzy entropy, the knowledge measure of Atanassov’s intuitionistic fuzzy sets (AIFSs) has attracted more attention and is still an open topic. The amount of knowledge is important to evaluate intuitionistic fuzzy information. An entropy-based knowledge measure for AIFSs is defined in this paper to quantify the knowledge amount conveyed by AIFSs. An intuitive analysis on the properties of the knowledge amount in AIFSs is put forward to facilitate the introduction of axiomatic definition of the knowledge measure. Then we propose a new knowledge measure based on the entropy-based divergence measure with respect for the difference between the membership degree, the non-membership degree, and the hesitancy degree. The properties of the new knowledge measure are investigated in a mathematical viewpoint. Several examples are applied to illustrate the performance of the new knowledge measure. Comparison with several existing entropy and knowledge measures indicates that the proposed knowledge has a greater ability in discriminating different AIFSs and it is robust in quantifying the knowledge amount of different AIFSs. Lastly, the new knowledge measure is applied to the problem of multiple attribute decision making (MADM) in an intuitionistic fuzzy environment. Two models are presented to determine attribute weights in the cases that information on attribute weights is partially known and completely unknown. After obtaining attribute weights, we develop a new method to solve intuitionistic fuzzy MADM problems. An example is employed to show the effectiveness of the new MADM method.
Highlights
The concept of the fuzzy set [1] was developed by Zadeh to model and process uncertain information in a much better way
Theorem 1–4 illustrates that the proposed mapping KS : Atanassov’s intuitionistic fuzzy sets (AIFSs) → [0, 1] satisfies all properties in the axiomatic definition of the knowledge measure
The axiomatic definition of the knowledge measure is first proposed based on intuitive analysis on the knowledge amount
Summary
The concept of the fuzzy set [1] was developed by Zadeh to model and process uncertain information in a much better way. Following Zadeh’s work, De Luca and Termini [24] proposed a probabilistic entropy measure for fuzzy sets. First presented the entropy measure for AIFSs to quantify the intuitionism of AIFSs. Szmidt and Kacprzyk [28] developed the axioms proposed by Burillo and Bustince [27] and introduced a new entropy measure for AIFSs based on the ratio between the nearer distance and the further distance. The knowledge measure can be regarded as the dual measure of intuitionistic fuzzy entropy or uncertainty. Guo [34] proposed an axiomatic definition for the knowledge measure of AIFSs and developed a new knowledge measure following his axioms. A new method for solving MADM problems under the intuitionistic fuzzy condition is developed based on the new knowledge measure.
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