Improved Symmetry Measures of Simplified Neutrosophic Sets and Their Decision-Making Method Based on a Sine Entropy Weight Model
Abstract
:1. Introduction
2. Existing SMs between Simplified Neutrosophic Asymmetry Measures and Insufficiency
3. Improved Normalized SMs of SNSs
4. Simplified Neutrosophic Sine Entropy
- (E1)
- SEk(Y) = 0 if Y = {y1, y2, …, yn} is a crisp set, i.e., yj = <1, 0, 0> or yj = <0, 0, 1> for SVNN and yj = <[1, 1], [0, 0], [0, 0]> or yj = <[0, 0], [0, 0], [1, 1]> (j = 1, 2, …, n) for INS;
- (E2)
- SEk(Y) = 1 if and only if yj = aj (j = 1, 2, …, n);
- (E3)
- If the closer an SNS Y is to the fuzziest SNS A than an SNS X, the fuzzier Y is than X, then SEk(X) ≤ SEk(Y);
- (E4)
- SEk(Y) = SEk(Yc) if Yc is the complement of Y.
- (E1)
- For a crisp set Y = {y1, y2, …, yn}, i.e., yj = <1, 0, 0> or yj = <0, 0, 1> for SVNN and yj = <[1, 1], [0, 0], [0, 0]> or yj = <[0, 0], [0, 0], [1, 1]> (j = 1, 2, …, n) for INN, by use of Equation (14) we obtain the following result:
- (E2)
- Let the sine function be f(zj) = sin(zjπ) for zj ∈ [0, 1] (j = 1, 2, …, n). By differentiating f(zj) with respect to zj and equating to zero, there exist the following results:
- (E3)
- According to Equation (16), f(zj) for zj ∈ [0, 1] (j = 1, 2, …, n) is an increasing function if zj < 0.5 and then it is a decreasing function if zj > 0.5.
- (E4)
- Since the complement of the SVNN in Y is , i.e., (αyj)c = γyj and (βyj)c = 1 − βyj (j = 1, 2,…, n) and the complement of the INN in Y is , i.e., and (j = 1, 2,…, n). Then, there is SEk(Yc) = SEk(Y) (k = 1, 2) by using Equations (14) and (15).
5. Decision-Making Method Using the Improved Weighted SMs of SNSs
- Step 1.
- Based on the concept of an ideal solution (alternative), we can determine the ideal solution where is an ideal SVNN or is an ideal INN (j = 1, 2, …, n; i = 1, 2, …, m).
- Step 2.
- Since the fuzziness/uncertainty of a criterion evaluation increases, the criterion weight should decrease. So, based on the sine entropy measure formula Equation (14) or (15) we can calculate unknown weights of each criterion by the following sine entropy weight model:
- Step 3.
- By use of Equation (12) for SVNSs or Equation (13) for INSs, the improved weighted SM between Yi (i = 1, 2, …, m) and Y* is given by:
- Step 4.
- According to the improved weighted SM values of W1(Yi, Y*) or W2(Yi, Y*), we can rank alternatives and choose the best one.
- Step 5.
- End.
6. Actual Decision Examples and Comparative Analysis
6.1. Actual Decision Example
- Step 1.
- By (j = 1, 2, 3; i = 1, 2, 3, 4), the ideal solution (ideal alternative) of SVNSs is given as:
- Step 2.
- By Equation (18), the criteria weight vector is obtained as follows:W = (w1, w2, w3) = (0.5682, 0.2952, 0.1366).
- Step 3.
- By Equation (19), the improved weighted SM values between Yi (i = 1, 2, 3, 4) and Y* can be yielded as follows:W1(Y1, Y*) = 0.9757, W1(Y2, Y*) = 0.9977, W1(Y3, Y*) = 0. 9397, and W1(Y4, Y*) = 0.9989.
- Step 4.
- The four alternatives are ranked by Y4 > Y2 > Y1 > Y3 since the SM values are W1(Y4, Y*) > W1(S2, S*) > W1(S1, S*) > W1(Y3, Y*). It is obvious that Y4 is the best scheme.
- Step 1.
- By for j = 1, 2, 3 and i = 1, 2, 3, 4, we can give the ideal solution of INSs (ideal alternative):
- Step 2.
- By Equation (18), the criteria weight vector is obtained as follows:W = (w1, w2, w3) = (0.4976, 0.3557, 0.1467).
- Step 3.
- By Equation (20), the improved weighted SM values between Yi (i = 1, 2, 3, 4) and Y* can be yielded as the following results:W2(Y1, Y*) = 0.9521, W2(Y2, Y*) = 0.9848, W2(Y3, Y*) = 0. 9145, and W2(Y4, Y*) = 0.9933.
- Step 4.
- Since the SM values are W2(Y4, Y*) > W2(Y2, Y*) > W2(Y1, Y*) > W2(Y3, Y*), the four alternatives are ranked by Y4 > Y2 > Y1 > Y3. Hence, the alternative Y4 is the best one.
6.2. Comparative Analysis
- (1)
- The improved SMs of SNSs not only indicate simpler algorithms than the existing SMs of SNSs [32], but also can overcome the insufficiency of the existing SMs of SNSs.
- (2)
- The improved MCDM method based on the sine entropy weight model can handle MCDM problems with unknown criteria weights, while the existing MCDM method [32] can only handle MCDM problems with known criteria weights. Hence, the former is superior to the latter in the MCDM problems.
- (3)
- The objective criteria weights obtained by the sine entropy weight model are more reasonable and more practicable than the known criteria weights/subjective criteria weights given by decision-makers’ preference.
- (4)
- The improved MCDM method based on the sine entropy weight model is simple and effective in simplified neutrosophic MCDM problems with unknown criteria weights.
7. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
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MCDM Method | SM Value between Yi and Y* in SVNS Setting | SM Value between Yi and Y* in INS Setting | Ranking Order in SVNS Setting | Ranking Order in INS Setting |
---|---|---|---|---|
Existing MCDM with the given weights [32] | 0.8945, 0.9964, 0.9717, 0.9730 | 0.9053, 0.9423, 0.9401, 0.9762 | Y2 > Y4 > Y3 > Y1 | Y4 > Y2 > Y3 > Y1 |
Improved MCDM method with the given weights | 0.9394, 0.9981, 0.9853, 0.9859 | 0.9472, 0.9691, 0.9675, 0.9877 | Y2 > Y4 > Y3 > Y1 | Y4 > Y2 > Y3 > Y1 |
Improved MCDM method with the sine entropy weights | 0.9757, 0.9977, 0.9397, 0.9989 | 0.9521, 0.9848, 0.9145, 0.9933 | Y4 > Y2 > Y1 > Y3 | Y4 > Y2 > Y1 > Y3 |
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Cui, W.; Ye, J. Improved Symmetry Measures of Simplified Neutrosophic Sets and Their Decision-Making Method Based on a Sine Entropy Weight Model. Symmetry 2018, 10, 225. https://doi.org/10.3390/sym10060225
Cui W, Ye J. Improved Symmetry Measures of Simplified Neutrosophic Sets and Their Decision-Making Method Based on a Sine Entropy Weight Model. Symmetry. 2018; 10(6):225. https://doi.org/10.3390/sym10060225
Chicago/Turabian StyleCui, Wenhua, and Jun Ye. 2018. "Improved Symmetry Measures of Simplified Neutrosophic Sets and Their Decision-Making Method Based on a Sine Entropy Weight Model" Symmetry 10, no. 6: 225. https://doi.org/10.3390/sym10060225