Normal neutrosophic frank aggregation operators and their application in multi-attribute group decision making
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ORIGINAL ARTICLE
Normal neutrosophic frank aggregation operators and their application in multi-attribute group decision making Peide Liu1 · Peng Wang1 · Junlin Liu1 Received: 20 January 2016 / Accepted: 5 December 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2017
Abstract Normal neutrosophic set (NNS) can conveniently express random and fuzzy information, and Frank operators have the properties of generalization and flexibility. In this paper, we will extend Frank operators to Normal neutrosophic numbers (NNNs), and propose some Frank aggregation Operators for NNNs, then develop two new decision methods with NNNs. Firstly, based on Frank operators, the operational laws of NNNs are redefined, and their operational properties are proved, then the normal neutrosophic Frank averaging operator (NNFWA) and normal neutrosophic Frank weighted geometric operator (NNFWG) are developed. Further, some desirable characteristics, such as idempotency, boundedness and commutativity, are discussed in detail, and some special cases are studied. Furthermore, to deal with the multiple attribute group decision making (MAGDM) problems in which attribute values take the form of NNNs, two methods on the basis of NNFWA and NNFWG operators are developed, and they are more general and more flexible by Frank operations. Finally, an example is given to illustrate the proposed methods and demonstrate their practicality and availability. Keywords Normal neutrosophic numbers · Frank operator · Group decision making · Multi-attribute decision making.
1 Introduction MAGDM has been applied to political, military, economic and other aspects. Because most of MAGDM problems are fuzzy, the fuzzy set (FS) developed by Zadeh [1] and intuitionistic FS (IFS) proposed by Atanassov [2], have been used to multi-attribute decision making (MADM) or MAGDM problems. Now, FS and IFS have attracted a wide range of attentions [3–5]. Furthermore, Smarandache [6] developed the neutrosophic set (NS) which added an independent indeterminacy-membership on the basis of IFS. In NS, the truthmembership (or membership), indeterminacy-membership (or uncertain membership), and false-membership (or nonmembership) are completely independent. NS can represent the inconsistent, uncertain and incomplete information, and it is generalization of FS and IFS, and many researches on NS have been made [7–9]. In addition, in practical applications, the normal distribution has a wide application in many * Peide Liu [email protected] 1
School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan, Shandong 250014, China
domains. However IFS and INS do not consider the normal distribution which is most used probability distribution, so the researches on random fuzzy information have aroused widespread concern. Yang and Ko [10] firstly developed the normal fuzzy numbers (NFNs) to describe the random and fuzzy phenomena. Based on the NFNs and IFS, Wang and Li [11] gave the definition of the normal intuitionistic fuzzy numbers (NI
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