Abstract
In real applications, most decisions are fuzzy decisions, and the decision results mainly depend on the choice of aggregation operators. In order to aggregate information more scientifically and reasonably, the Heronian mean operator was studied in this paper. Considering the advantages and limitations of the Heronian mean (HM) operator, four Heronian mean operators for bipolar neutrosophic number (BNN) are proposed: the BNN generalized weighted HM (BNNGWHM) operator, the BNN improved generalized weighted HM (BNNIGWHM) operator, the BNN generalized weighted geometry HM (BNNGWGHM) operator, and the BNN improved generalized weighted geometry HM (BNNIGWGHM) operator. Then, their propositions were examined. Furthermore, two multi-criteria decision methods based on the proposed BNNIGWHM and BNNIGWGHM operator are introduced under a BNN environment. Lastly, the effectiveness of the new methods was verified with an example.
Highlights
In the real world, there is lots of uncertain information in science, technology, daily life, and so on
Multi-criteria decision-making (MCDM) Methods Based on the BNNIGWHM and BNNIGWGHM Operator
The Similarity measures of bipolar neutrosophic sets proposed in Reference [19] with the following variables: In Table 4, we can see that the ranking results were different; Γ3 was obtained as the optimal alternative except the method in Reference [19] with λ = 0.9
Summary
There is lots of uncertain information in science, technology, daily life, and so on. In order to describe uncertain information, Zadeh [1] put forward the concept of fuzzy sets. Zhang [7] put forward an interval neutrosophic set (INS) theory. Traditional fuzzy sets could not do well in analyzing and handing uncertain information with incompatible bipolarity; this phenomenon was identified in 1994. Zhang [9] introduced incompatible bipolarity into the fuzzy set theory, Mathematics 2019, 7, 97; doi:10.3390/math7010097 www.mdpi.com/journal/mathematics. Mathematics 2019, 7, 97 and put forward the bipolar fuzzy set (BFS). Deli et al [16] put forward a bipolar neutrosophic set (BNS), which can describe bipolar information. We propose four Heronian mean operators for bipolar neutrosophic number (BNN).
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