A modified method of generating Z -number based on OWA weights and maximum entropy
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METHODOLOGIES AND APPLICATION
A modified method of generating Z-number based on OWA weights and maximum entropy Ye Tian1 · Bingyi Kang1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract How to generate Z -number is an important and open issue in the uncertain information processing of Z -number. In Kang et al. (Int J Intell Syst 33(8):1745–1755, 2018), a method of generating Z -number using OWA weight and maximum entropy is investigated. However, the meaning of the method in Kang et al. (2018) is not clear enough according to the definition of Z -number. Inspired by the methodology in Kang et al. (2018), we modify the method of determining Z -number based on OWA weights and maximum entropy, which is more clear about the meaning of Z -number. In addition, the model of generating Z -number under the environment of group decision making is well investigated based the modified model. Some numerical examples are used to illustrate the effectiveness of the proposed methodology. Keywords Z -number · OWA · Maximum entropy · Reliability · Decision making
1 Introduction Fuzzy set theory has obtained plenty of achievements in the applications of science and engineering (Rafiq et al. 2019; Ashraf et al. 2019b). Notions and frames of fuzzy sets have also richly developed, such as Pythagorean fuzzy set (Khan et al. 2019c, d, e), neutrosophic set (Ashraf et al. 2019c), hesitant fuzzy set (Fahmi et al. 2019), etc. These developed fuzzy sets and logics are widely used in many fields. Some recent works, such as the application of complex fuzzy sets in signals (Ma et al. 2019), principal component analysis based on intuitionistic fuzzy random variables (Hesamian and Akbari 2019), deal with decision problems under the framework of interval fuzzy set theory (Mohagheghi and Mousavi 2019), and some applications in the medical field (Munir et al. 2019; Romero et al. 2019; Radhakrishnan et al. 2019), failure mode and impact analysis (Li and Chen 2019; Wang et al. 2018), reliability analysis and modeling (Gao et al. 2019), decision making and assessment area (Ashraf et al. 2019a; Zeng et al. 2019; Shakeel et al. 2019; Buriboev et al. 2019; Hilletofth et al. 2019). Communicated by V. Loia.
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Bingyi Kang [email protected]; [email protected] College of Information Engineering, Northwest A&F University, Yangling 712100, Shaanxi, China
Z -number is proposed by Zadeh in 2011 to model uncertain information (Zadeh 2011), which is different from the notion of Z -numbers proposed by Mahler (1968). The inherent meaning or definition of Z -number in Zadeh (2011) is denoted as below, Z = (A, B) = Z + A, μ A · p X A is B which indicates whether X A for A is a random variable, and Z + is a indicator of a Z + -number (Zadeh 2011), where p X A is the probability of random variable X A for fuzzy set A, the membership function of fuzzy set A is denoted as μ A (x), x ∈ X A , x ∈ R, R is the real value domain, the membership function of fuzzy set B is denoted as μ B (x), domain. In addition, x ∈ X B , x ∈ R, R is t
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