On generalized knowledge measure and generalized accuracy measure with applications to MADM and pattern recognition
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On generalized knowledge measure and generalized accuracy measure with applications to MADM and pattern recognition Surender Singh1 · Sonam Sharma1 · Abdul Haseeb Ganie1 Received: 24 May 2019 / Revised: 24 May 2020 / Accepted: 2 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this communication, we propose a generalized fuzzy knowledge measure and prove its efficiency by comparing it with some existing entropies. We also propose a generalized fuzzy accuracy measure and show some of its properties. This accuracy measure may serve as a compatibility measure between two fuzzy sets and helpful in some specific situations. We introduce a generalized fuzzy knowledge and accuracy measure-based TOPSIS for multiple-attribute decision-making problems and presents its comparison with MOORA method, VIKOR method, and a compromise-type variable weight decision-making method. The application of the proposed TOPSIS approach in multiple-attribute decision-making (MADM) is demonstrated using a numerical example. We also investigate the application and efficiency of the generalized fuzzy accuracy in pattern recognition problems. Keywords Fuzzy set · Fuzzy knowledge measure · Fuzzy accuracy measure · TOPSIS · Pattern recognition Mathematics Subject Classification 03B42 · 68T37
1 Introduction The concept of entropy measure is used to compute the amount of uncertainty in a random experiment. Formally, the idea of entropy was proposed by Shannon (1948) in the context of communication theory using probability theory. Due to various complexities in real-life Communicated by Anibal Tavares de Azevedo. * Surender Singh [email protected] Sonam Sharma [email protected] Abdul Haseeb Ganie [email protected] 1
School of Mathematics, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir 182320, India
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problems, decision-makers give their judgment in an uncertain and ambiguous environment. Thus, there is always a degree of ambiguity between the preferences of the decisionmaking and, hence, the theory obtained under such circumstances is not ideal and does not bring up the exact information to the decision-maker. To deal with such problems, Zadeh (1965) introduced the fuzzy set theory and incorporated the concept of fuzzy logic to improve upon some shortcomings of binary logic. De Luca and Termini (1971) proposed an axiomatic definition of fuzzy entropy. The entropy of the fuzzy set is considered as the amount of ambiguity/imprecision associated with that fuzzy set. After De Luca and Termini, many authors proposed the generalization of fuzzy entropy. Kosko (1986) proposed a new fuzzy entropy based on hypercube and distance between fuzzy subset A, nearest vertex Anear, and farthest vertex Afar. Hooda (2004), Bhatia and Singh (2013), Li and Liu (2008), and Bajaj and Hooda (2010b) provided parametric generalizations of fuzzy entropy. Joshi and Kumar (2017b) introduced two parametric exponential fuzzy entropy and many more. In real-
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