Abstract
A single-valued and an interval neutrosophic sets are two instances of the neutrosophic set(NS), which can efficiently deal with uncertain, imprecise and inconsistent information. In the present work, the work has been done under this environment to develop some novel hybrid aggregation operators based on arithmetic and geometric aggregation operators. The preferences related to the attributes are made in the form of single-valued and interval neutrosophic numbers. Their desirable properties such as idempotency, boundedness and monotonicity are also investigated. Furthermore, a decision-making approach presents to investigate the multi-criteria decision-making problem. The effectiveness of the approach is demonstrated through a case study and a comparison analysis with some other existing methods has been done to validate the results.
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Acknowledgments
The authors are thankful to the editor and anonymous reviewers for their constructive comments and suggestions that helped us in improving the paper significantly. Also, the second author (Nancy) would like to thank the University Grant Commission, New Delhi, India for providing financial support under Maulana Azad National Fellowship scheme wide File No. F1-17.1/2017-18/MANF-2017-18-PUN-82613/(SA-III/Website) during the preparation of this manuscript.
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Appendix
Appendix
Proof of the Theorem 1:
Proof
By using the operational laws for SVNNs βj(j = 1, 2,…,n) and a real number λ ∈ [0, 1], we have
which implies that
and
Thus, by Definition 5, we have
Hence, the result.
□
Proof of the Theorem 2:
Proof
For SVNNs βj = (ζj, κj, φj)(j = 1, 2,…,n) and ω = (ω1, ω2,…,ωn)T be their normalized weight vector, then we have \(\prod \limits _{j = 1}^{n}\zeta _{j}^{\omega _{j}} \leq 1-\prod \limits _{j = 1}^{n}(1-\zeta _{j})^{\omega _{j}}\) and \(0 \leq \prod \limits _{j = 1}^{n}\zeta _{j}^{\omega _{j}} , 1-\prod \limits _{j = 1}^{n}(1-\zeta _{j})^{\omega _{j}} \leq 1\). Therefore, for a real number λ ∈ [0, 1] we have
which implies that
Similarly, for \(\prod \limits _{j = 1}^{n}\kappa _{j}^{\omega _{j}} \geq 1-\prod \limits _{j = 1}^{n}(1-\kappa _{j})^{\omega _{j}}\) and \(\prod \limits _{j = 1}^{n}\varphi _{j}^{\omega _{j}} \geq 1-\prod \limits _{j = 1}^{n}(1-\varphi _{j})^{\omega _{j}}\), we have
and
Thus, by using (20), (21) and (22) and the definition of score function, we get
Hence, H-SVNWAG(β1, β2,…,βn) ≤SVNWA(β1, β2, …,βn). Similarly, we can obtain that H-SVNWAG(β1, β2,…,βn) ≥SVNWG(β1, β2,…,βn). Thus, we get the required proof. □
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Garg, H., Nancy Some hybrid weighted aggregation operators under neutrosophic set environment and their applications to multicriteria decision-making. Appl Intell 48, 4871–4888 (2018). https://doi.org/10.1007/s10489-018-1244-9
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DOI: https://doi.org/10.1007/s10489-018-1244-9