Abstract
This study proposes a multi-attribute decision-making method for the decision-making problems with attributes and sub-attribute where the attribute weight is unknown, based on information entropy and the evaluation based on distance from average solution (EDAS) method under a refined single-valued neutrosophic set environment. First, the new distance measure, similarity measure, and neutrosophic entropy based on refined single-valued neutrosophic sets are defined. Further, the relationship between them is discussed and the attribute weights are determined based on the new neutrosophic entropy. Then, the EDAS method is used to rank and select the best alternative. Finally, two illustrative examples of typhoon disaster assessment (typhoon disaster assessment with multi-layer indicators and dynamic assessment of typhoon disaster) are presented to demonstrate the feasibility, effectiveness, and practicality of the proposed method. The advantages of the proposed method are illustrated by sensitive analysis and comparative analysis with other methods.
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Acknowledgments
This work was supported by the National Social Science Foundation of China (No. 17CGL058).
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All authors contributed to this paper. The individual responsibilities and contributions of all authors are as follows. The idea for this study was proposed by Ruipu Tan, who also wrote the paper. Wende Zhang analyzed the existing work regarding the research problem. Revision and submission of this paper was carried out by Ruipu Tan.
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Appendix
Appendix
Here, we prove the properties of Theorem 1 and Theorem 3.
(1) Proof of the three properties of Theorem 1:
Proof.
(P1) 0 ≤ TjM(xi), IjM(xi), FjM(xi), TjN(xi), IjN(xi), FjN(xi) ≤ 1, then 0 ≤ |TjM(xi) − TjN(xi)| ≤ 1, 0 ≤ |IjM(xi) − IjN(xi)| ≤ 1, 0 ≤ |FjM(xi) − FjN(xi)| ≤ 1, and 0 ≤ |TjM(xi) − TjN(xi)|λ ≤ 1, 0 ≤ |IjM(xi) − IjN(xi)|λ ≤ 1, 0 ≤ |FjM(xi) − FjN(xi)|λ ≤ 1, so \( 0\le \frac{1}{3}\left({\left|{T}_{jM}\left({x}_i\right)-{T}_{jN}\left({x}_i\right)\right|}^{\lambda }+{\left|{I}_{jM}\left({x}_i\right)-{I}_{jN}\left({x}_i\right)\right|}^{\lambda }+{\left|{F}_{jM}\left({x}_i\right)-{F}_{jN}\left({x}_i\right)\right|}^{\lambda}\right)\le 1 \),
\( 0\le {\left\{\frac{1}{3}\left({\left|{T}_{jM}\left({x}_i\right)-{T}_{jN}\left({x}_i\right)\right|}^{\lambda }+{\left|{I}_{jM}\left({x}_i\right)-{I}_{jN}\left({x}_i\right)\right|}^{\lambda }+{\left|{F}_{jM}\left({x}_i\right)-{F}_{jN}\left({x}_i\right)\right|}^{\lambda}\right)\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\le 1 \),\( 0\le \frac{1}{n}\sum \limits_{i=1}^n\frac{1}{k}\sum \limits_{j=1}^k{\left\{\frac{1}{3}\left({\left|{T}_{jM}\left({x}_i\right)-{T}_{jN}\left({x}_i\right)\right|}^{\lambda }+{\left|{I}_{jM}\left({x}_i\right)-{I}_{jN}\left({x}_i\right)\right|}^{\lambda }+{\left|{F}_{jM}\left({x}_i\right)-{F}_{jN}\left({x}_i\right)\right|}^{\lambda}\right)\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}\le 1 \).
Thus, 0 ≤ D(M, N) ≤ 1 is established.
(P2) If M = N, then TjM(xi) = TjN(xi), IjM(xi) = IjN(xi), FjM(xi) = FjN(xi), TjM(xi) − TjN(xi) = 0,
IjM(xi) − IjN(xi) = 0, FjM(xi) − FjN(xi) = 0,
so |TjM(xi) − TjN(xi)|λ + |IjM(xi) − IjN(xi)|λ + |FjM(xi) − FjN(xi)|λ = 0,
that is, D(M, N) = 0.
Conversely, if D(M, N) = 0, then |TjM(xi) − TjN(xi)|λ + |IjM(xi) − IjN(xi)|λ + |FjM(xi) − FjN(xi)|λ = 0,
TjM(xi) − TjN(xi) = 0, IjM(xi) − IjN(xi) = 0, FjM(xi) − FjN(xi) = 0, then TjM(xi) = TjN(xi), IjM(xi) = IjN(xi), FjM(xi) = FjN(xi), so M = N. Thus, we can get D(M, N) = 0 if and only if M = N.
(P3) |TjM(xi) − TjN(xi)| = |TjN(xi) − TjM(xi)|, |IjM(xi) − IjN(xi)| = |IjN(xi) − IjM(xi)|, |FjM(xi) − FjN(xi)| = |FjN(xi) − FjM(xi)|, so |TjM(xi) − TjN(xi)|λ = |TjN(xi) − TjM(xi)|λ, |IjM(xi) − IjN(xi)|λ = |IjN(xi) − IjM(xi)|λ, |FjM(xi) − FjN(xi)|λ = |FjN(xi) − FjM(xi)|λ,
Thus, D(M, N) = D(N, M) is established.
(2) Proof of the four properties of Theorem 3:
Proof.
(P1) If r is a crisp number, then r is not fuzzy; thus, its entropy is zero. In this case, for example r = 〈(1, 1, ⋯, k times), (0, 0, ⋯, k times), (0, 0, ⋯, k times)〉, or r = 〈(0, 0, ⋯, k times), (0, 0, ⋯, k times), (1, 1, ⋯, k times)〉,
So \( {E}_{RSVNN}(r)==1-2D\left(r,{r}^{\prime}\right)=1-2\frac{1}{k}\sum \limits_{j=1}^k{\left\{\frac{1}{3}\left({0.5}^{\lambda }+{0.5}^{\lambda }+{0.5}^{\lambda}\right)\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}=0. \)
Thus, we can get ERSVNN(r) = 0if r is a crisp number.
(P2) Because ERSVNN(r) = 1, that is 1 ‐ 2D(r, r') = 1, then D(r, r') = 0. Based on the Theorem 1 of Section 3.2, we can get r = r' = 〈(0.5, 0.5, ⋯, k times), (0.5, 0.5, ⋯, k times), (0.5, 0.5, ⋯, k times)〉.
If r = r' = 〈(0.5, 0.5, ⋯, k times), (0.5, 0.5, ⋯, k times), (0.5, 0.5, ⋯, k times)〉, then D(r, r') = 0, 1 ‐ 2D(r, r') = 1, so ERSVNN(r) = 1. Thus, we can get ERSVNN(r) = 1, if and only if r = r' = 〈(0.5, 0.5, ⋯, k times), (0.5, 0.5, ⋯, k times), (0.5, 0.5, ⋯, k times)〉;
(P3) If D(r, r') ≥ D(l, r'), then 2D(r, r') ≥ 2D(l, r'), 1 ‐ 2D(r, r') ≤ 1 ‐ 2D(l, r').
Thus, ERSVNN(r) = 1 ‐ 2D(r, r') ≤ ERSVNN(l) = 1 ‐ 2D(l, r'). If D(r, r') ≥ D(l, r'), ERSVNN(r) ≤ ERSVNN(l), that is, Theorem 3 (3) is established.
(P4) Because rc = 〈(F1r, F2r, ⋯, Fkr), (1 − I1r, 1 − I2r, ⋯, 1 − Ikr), (T1r, T2r, ⋯, Tkr)〉,
Thus, we can get ERSVNN(r) = ERSVNN(rc).
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Tan, Rp., Zhang, Wd. Decision-making method based on new entropy and refined single-valued neutrosophic sets and its application in typhoon disaster assessment. Appl Intell 51, 283–307 (2021). https://doi.org/10.1007/s10489-020-01706-3
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DOI: https://doi.org/10.1007/s10489-020-01706-3