Some aggregation operators of neutrosophic Z-numbers and their multicriteria decision making method
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ORIGINAL ARTICLE
Some aggregation operators of neutrosophic Z-numbers and their multicriteria decision making method Shigui Du1,2 · Jun Ye1,2
· Rui Yong2 · Fangwei Zhang1
Received: 13 January 2020 / Accepted: 17 September 2020 © The Author(s) 2020
Abstract As the generalization of the classical fuzzy number, the concept of Z-number introduced by Zadeh indicates more ability to depict the human knowledge and judgments of both restraint and reliability as an order pair of fuzzy numbers. In indeterminacy and inconsistent environment, a neutrosophic set is described by the truth, falsity, and indeterminacy degrees, but they lack measures related to reliability. To describe the hybrid information of combining the truth, falsity and indeterminacy degrees with their corresponding reliability degrees, this paper first proposes the concept of a neutrosophic Z-number (NZN) set, which is a new framework of neutrosophic values combined with the neutrosophic measures of reliability, as the generalization of the Z-number and the neutrosophic set. Then, we define the operations of neutrosophic Z-numbers (NZNs) and a score function for ranking NZNs. Next, we present NZN weighted arithmetic averaging (NZNWAA) and NZN weighted geometric averaging (NZNWGA) operators to aggregate NZN information and investigate their properties. Regarding the NZNWAA and NZNWGA operators and the score function, a multicriteria decision making (MDM) approach is developed in the NZN environment. Finally, an illustrative example about the selection problem of business partners is given to demonstrate the applicability and effectiveness of the developed MDM approach in NZN setting. Keywords Neutrosophic Z-number set · Neutrosophic Z-number · Neutrosophic Z-number weighted arithmetic averaging operator · Neutrosophic Z-number weighted geometric averaging operator · Multicriteria decision making
Introduction It is known that fuzzy sets proposed by Zadeh [1] play an essential role in the current scientific and technical applications [2–7]. In 2011, Zadeh [8] further introduced the concept of Z-numbers to describe the restraint and reliability of the evaluation by an order pair of fuzzy numbers in uncertain situations. Compared with the classical fuzzy number,
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Jun Ye [email protected]; [email protected] Shigui Du [email protected] Rui Yong [email protected] Fangwei Zhang [email protected]
1
Institute of Rock Mechanics, Ningbo University, Ningbo 315211, People’s Republic of China
2
Department of Civil Engineering, Shaoxing University, Shaoxing 312000, People’s Republic of China
it is a more generalized notion closely related to reliability. Hence, the Z-number implies more ability to describe the human knowledge and judgments by an order pair of fuzzy numbers corresponding to the restriction and reliability. Since then, it has obtained a lot of attentions. Some researchers presented theoretical studies of Z-numbers, like Z*-numbers [9], arithmetic operations of discrete and continuous Z-numbers [10, 11], modeling of Z-number [12], approxi
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