Multiple-Attribute Decision-Making Method Using Similarity Measures of Hesitant Linguistic Neutrosophic Numbers Regarding Least Common Multiple Cardinality
Abstract
:1. Introduction
2. Linguistic Neutrosophic Numbers (LNNs)
- (1)
- If S(ϑα) < S(ϑβ), then ϑα < ϑβ;
- (2)
- If S(ϑα) > S(ϑβ), then ϑα > ϑβ;
- (3)
- If S(ϑα) = S(ϑβ) and V(ϑα) < V(ϑβ), then ϑα < ϑβ;
- (4)
- If S(ϑα) = S(ϑβ) and V(ϑα) > V(ϑβ), then ϑα > ϑβ;
- (5)
- If S(ϑα) = S(ϑβ) and V(ϑα) = V(ϑβ), then ϑα = ϑβ.
3. Hesitant Linguistic Neutrosophic Numbers (HLNNs) and HLNN Set
4. LCMC-Based Distance and Similarity Measures of HLNNs
- (HP1)
- ;
- (HP2)
- if and only if;
- (HP3)
- ;
- (HP4)
- Letbe a HLNN set, thenandif.
- (HP1)
- ;
- (HP2)
- if and only if;
- (HP3)
- ;
- (HP4)
- Letbe a HLNN set, then there areandif.
5. MADM Method Using the Similarity Measure of HLNNs
6. Actual Example
7. Discussion and Analysis
7.1. Resolution Analysis
7.2. Sensitivity Analysis of Weights
8. Conclusions
- (1)
- The proposed HLNN provides a new effective way to express more decision information than existing LNNs by considering the hesitancy of DMs.
- (2)
- The proposed MADM method of HLNNs solves the MADM problems with HLNN information for the first time, as well as the gap of existing linguistic decision-making methods.
- (3)
Author Contributions
Funding
Conflicts of Interest
References
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ρ1 | Sw(g1, g*), Sw(g2, g*), Sw(g3, g*), Sw(g4, g*) 2 | Ranking Order | AV 3 | SD 4 | Best Alternative |
---|---|---|---|---|---|
1 | 0.7354, 0.7493, 0.7406, 0.7747 | g4 > g2 > g3 > g1 | 0.7500 | 0.0151 | g4 |
2 | 0.7121, 0.7224, 0.7217, 0.7525 | g4 > g2 > g3 > g1 | 0.7272 | 0.0152 | g4 |
3 | 0.6905, 0.6985, 0.7037, 0.7335 | g4 > g3 > g2 > g1 | 0.7066 | 0.0163 | g4 |
4 | 0.6710, 0.6781, 0.6867, 0.7182 | g4 > g3 > g2 > g1 | 0.6885 | 0.018 | g4 |
5 | 0.6539, 0.6608, 0.6710, 0.7061 | g4 > g3 > g2 > g1 | 0.6730 | 0.0201 | g4 |
10 | 0.5972, 0.6047, 0.6133, 0.6722 | g4 > g3 > g2 > g1 | 0.6219 | 0.0296 | g4 |
15 | 0.5690, 0.5754, 0.5817, 0.6575 | g4 > g3 > g2 > g1 | 0.5959 | 0.0358 | g4 |
20 | 0.5531, 0.5582, 0.5631, 0.6497 | g4 > g3 > g2 > g1 | 0.5810 | 0.0398 | g4 |
30 | 0.5361, 0.5397, 0.5432, 0.6417 | g4 > g3 > g2 > g1 | 0.5652 | 0.0443 | g4 |
40 | 0.5273, 0.5301, 0.5327, 0.6376 | g4 > g3 > g2 > g1 | 0.5569 | 0.0466 | g4 |
50 | 0.5220, 0.5242, 0.5264, 0.6351 | g4 > g3 > g2 > g1 | 0.5519 | 0.048 | g4 |
100 | 0.5111, 0.5123, 0.5134, 0.6301 | g4 > g3 > g2 > g1 | 0.5417 | 0.051 | g4 |
ρ | Sw(g1, g*), Sw(g2, g*), Sw(g3, g*), Sw(g4, g*) | Ranking Order | AV | SD | Best Alternative |
---|---|---|---|---|---|
1 | 0.7431, 0.7546, 0.7500, 0.7755 | g4 > g2 > g3 > g1 | 0.7558 | 0.0121 | g4 |
2 | 0.7189, 0.7278, 0.7300, 0.7529 | g4 > g3 > g2 > g1 | 0.7324 | 0.0125 | g4 |
3 | 0.6968, 0.7039, 0.7112, 0.7336 | g4 > g3 > g2 > g1 | 0.7114 | 0.0138 | g4 |
4 | 0.6769, 0.6833, 0.6938, 0.7180 | g4 > g3 > g2 > g1 | 0.693 | 0.0156 | g4 |
5 | 0.6596, 0.6659, 0.6779, 0.7057 | g4 > g3 > g2 > g1 | 0.6773 | 0.0177 | g4 |
10 | 0.6019, 0.6088, 0.6193, 0.6717 | g4 > g3 > g2 > g1 | 0.6254 | 0.0274 | g4 |
15 | 0.5727, 0.5786, 0.5865, 0.6572 | g4 > g3 > g2 > g1 | 0.5988 | 0.0341 | g4 |
20 | 0.556, 0.5608, 0.5671, 0.6495 | g4 > g3 > g2 > g1 | 0.5834 | 0.0384 | g4 |
30 | 0.5381, 0.5415, 0.5459, 0.6415 | g4 > g3 > g2 > g1 | 0.5668 | 0.0432 | g4 |
40 | 0.5289, 0.5315, 0.5349, 0.6374 | g4 > g3 > g2 > g1 | 0.5582 | 0.0458 | g4 |
50 | 0.5232, 0.5254, 0.5281, 0.6350 | g4 > g3 > g2 > g1 | 0.5529 | 0.0474 | g4 |
100 | 0.5118, 0.5128, 0.5142, 0.6300 | g4 > g3 > g2 > g1 | 0.5422 | 0.0507 | g4 |
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Cui, W.; Ye, J. Multiple-Attribute Decision-Making Method Using Similarity Measures of Hesitant Linguistic Neutrosophic Numbers Regarding Least Common Multiple Cardinality. Symmetry 2018, 10, 330. https://doi.org/10.3390/sym10080330
Cui W, Ye J. Multiple-Attribute Decision-Making Method Using Similarity Measures of Hesitant Linguistic Neutrosophic Numbers Regarding Least Common Multiple Cardinality. Symmetry. 2018; 10(8):330. https://doi.org/10.3390/sym10080330
Chicago/Turabian StyleCui, Wenhua, and Jun Ye. 2018. "Multiple-Attribute Decision-Making Method Using Similarity Measures of Hesitant Linguistic Neutrosophic Numbers Regarding Least Common Multiple Cardinality" Symmetry 10, no. 8: 330. https://doi.org/10.3390/sym10080330