A new picture fuzzy information measure based on shannon entropy with applications in opinion polls using extended VIKOR
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A new picture fuzzy information measure based on shannon entropy with applications in opinion polls using extended VIKOR–TODIM approach Vikas Arya1 · Satish Kumar1 Received: 11 July 2019 / Revised: 14 May 2020 / Accepted: 13 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract Picture fuzzy set theory developed by Cuong and Kreinovich is an extension of the fuzzy set theory and intuitionistic fuzzy set theory. In this paper, we proposed a new framework for picture fuzzy entropy from a probabilistic viewpoint. The justification of the proposed axiomatic structure is established by offering a new information measure based on Shannon entropy under picture fuzzy environment and also studied its mathematical properties. Besides, we developed an algorithm for picture fuzzy set with the help of TODIM (a Portuguese acronym for Interactive Multi-Criteria Decision Making) and VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) methods to explain the multi-criteria decision-making (MCDM) problems with picture fuzzy numbers. Finally, two numerical examples are given to verify the proposed approach based on opinion surveys to anticipate the election results and the output is reasonably compared with other MCDM method existing in the literature, what’s more, the viable experiment results are gotten. Keywords Picture fuzzy set · Shannon entropy · Picture fuzzy number · Hamming distance · Extended VIKOR–TODIM Mathematics Subject Classification 94A15 · 26D15 · 94A24
1 Introduction To quantify the vague and fuzziness associated with decision making problems, fuzzy set (FS) theory proposed by Zadeh (1965) has become an effective tool. Before the evolution of FS theory, probability theory was the only way for the management of uncertain information.
Communicated by Marcos Eduardo Valle.
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Vikas Arya [email protected] Satish Kumar [email protected]
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Department of Mathematics, Maharishi Markandeshwar University, Mullana, Ambala 133207, India 0123456789().: V,-vol
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V. Arya, S. Kumar
The vagueness terms like ‘high speed’, ‘very intelligent’, etc. can be quantified using FS theory which otherwise were difficult to quantify using probability theory. Unlike classical set, FS describes the state of an element using a number lying in the interval [0, 1], called its membership degree/grade (δ). Impressed with ability and agility of FSs to handle with uncertainty and fuzziness particularly in decision-making problems, researchers from across the world started thinking to enhance the capability of FSs. A significant generalization of fuzzy set is intuitionistic fuzzy set (IFS) proposed by Atanassov (1986). An IFS is assigned by a membership grade (δ ∈ [0, 1]) and a non-membership grade (η ∈ [0, 1]) satisfying 0 ≤ (δ+η) ≤ 1. The induction of third component called ‘intuitionistic index’ (π = 1−δ−η) improved the capability of FSs to tackle uncertain information. Voting system can be a good example for an IFS. In an election, there are some people who a
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