An Interval-Valued Intuitionistic Fuzzy MADM Method Based on a New Similarity Measure
Abstract
:1. Introduction
2. Preliminary Knowledge
- (i)
- If and , then is no larger than , and noted by ;
- (ii)
- If and , then is equal to , and noted by .
- (i)
- The complementary set of denoted by , is ;
- (ii)
- if and only if , and ;
- (iii)
- , if and only if and .
- (i)
- ;
- (ii)
- .
- (i)
- If , then
- (ii)
- (iii)
- If , then
- (iv)
- If is a crisp set, then .
3. A New Similarity of IVIF Sets
- (i)
- If , then ,
- (ii)
- (iii)
- Let , then for any , we have
- (iv)
- When is a crisp set, that is
- (a)
- If , thenThus
- (b)
- Case (b) similar to case (a) considering , we can also prove the result .
4. A New MADM Method Based on the Proposed Similarity
4.1. Description of the MADM Problem
4.2. Weight-Determining Method
4.3. New MADM Method Based on the Proposed Similarity
5. Numerical Example
Alternatives | Attributes | ||||
---|---|---|---|---|---|
<[0.5, 0.6], [0.2, 0.3]> | <[0.3, 0.5], [0.4, 0.5]> | <[0.6, 0.7], [0.2, 0.3]> | <[0.5, 0.7], [0.1, 0.2]> | <[0.1, 0.4], [0.3, 0.5]> | |
<[0.3, 0.4], [0.4, 0.6]> | <[0.1, 0.3], [0.2, 0.4]> | <[0.3, 0.4], [0.4, 0.5]> | <[0.2, 0.4], [0.5, 0.6]> | <[0.7, 0.8], [0.1, 0.2]> | |
<[0.4, 0.5], [0.3, 0.5]> | <[0.7, 0.8], [0.1, 0.2]> | <[0.5, 0.8], [0.1, 0.2]> | <[0.4, 0.6], [0.2, 0.3]> | <[0.5, 0.6], [0.2, 0.3]> | |
<[0.3, 0.5], [0.4, 0.5]> | <[0.1, 0.2], [0.7, 0.8]> | <[0.1, 0.2], [0.5, 0.8]> | <[0.2, 0.3], [0.4, 0.6]> | <[0.2, 0.3], [0.5, 0.6]> |
Alternatives | Attributes | ||||
---|---|---|---|---|---|
<[0.4, 0.5], [0.2, 0.4]> | <[0.3, 0.4], [0.4, 0.6]> | <[0.6, 0.7], [0.1, 0.2]> | <[0.5, 0.6], [0.1, 0.3]> | <[0.1, 0.3], [0.3, 0.5]> | |
<[0.3, 0.5], [0.4, 0.5]> | <[0.1, 0.3], [0.3, 0.7]> | <[0.3, 0.4], [0.4, 0.5]> | <[0.2, 0.3], [0.6, 0.7]> | <[0.6, 0.8], [0.1, 0.2]> | |
<[0.4, 0.6], [0.3, 0.4]> | <[0.6, 0.8], [0.1, 0.2]> | <[0.7, 0.8], [0.1, 0.2]> | <[0.4, 0.6], [0.3, 0.4]> | <[0.5, 0.6], [0.2, 0.4]> | |
<[0.3, 0.4], [0.4, 0.6]> | <[0.1, 0.2], [0.6, 0.8]> | <[0.1, 0.2], [0.7, 0.8]> | <[0.3, 0.4], [0.4, 0.6]> | <[0.2, 0.4], [0.5, 0.6]> |
Alternatives | Attributes | ||||
---|---|---|---|---|---|
<[0.4, 0.7], [0.1, 0.2]> | <[0.3, 0.5], [0.3, 0.4]> | <[0.6, 0.7], [0.1, 0.2]> | <[0.5, 0.6], [0.1, 0.3]> | <[0.3,0.5], [0.4,0.5]> | |
<[0.4, 0.5], [0.2, 0.4]> | <[0.2, 0.4], [0.4, 0.5]> | <[0.4, 0.5], [0.3, 0.4]> | <[0.1, 0.2], [0.7, 0.8]> | <[0.6, 0.7], [0.2, 0.3]> | |
<[0.2, 0.4], [0.3, 0.4]> | <[0.6, 0.8], [0.1, 0.2]> | <[0.5, 0.7], [0.1, 0.3]> | <[0.5, 0.7], [0.2, 0.3]> | <[0.6, 0.8], [0.1, 0.2]> | |
<[0.3, 0.4], [0.2, 0.4]> | <[0.1, 0.2], [0.6, 0.8]> | <[0.1, 0.3], [0.5, 0.7] > | <[0.2, 0.3], [0.5, 0.7]> | <[0.1, 0.2], [0.6, 0.8]> |
Alternatives | Attributes | ||||
---|---|---|---|---|---|
<[0.6, 0.7], [0.2, 0.3]> | <[0.3, 0.4], [0.3, 0.4]> | <[0.7, 0.8], [0.1, 0.2]> | <[0.5, 0.6], [0.1, 0.3]> | <[0.1, 0.2], [0.5, 0.7]> | |
<[0.4, 0.5], [0.4, 0.5]> | <[0.1, 0.2], [0.2, 0.3]> | <[0.3, 0.4], [0.5, 0.6]> | <[0.2, 0.3], [0.4, 0.6]> | <[0.6, 0.7], [0.1, 0.2]> | |
<[0.4, 0.5], [0.3, 0.4]> | <[0.6, 0.7], [0.1, 0.3]> | <[0.5, 0.8], [0.1, 0.2]> | <[0.4, 0.5], [0.2, 0.3]> | <[0.5, 0.6], [0.3, 0.4]> | |
<[0.3, 0.4], [0.4, 0.5]> | <[0.1, 0.3], [0.6, 0.7]> | <[0.1, 0.2], [0.5, 0.8]> | <[0.2, 0.3], [0.4, 0.5]> | <[0.3, 0.4], [0.5, 0.6]> |
Alternatives | Attributes | ||||
---|---|---|---|---|---|
<[0.4385,0.6199], [0.1600, 0.2910]> | <[0.3000,0.4573], [0.3481, 0.4814]> | <[0.6116,0.7117], [0.1195, 0.2196]> | <[0.5000,0.6395], [0.1000, 0.2602]> | <[0.1323,0.3623], [0.3870, 0.5548]> | |
<[0.3502,0.4797], [0.3241, 0.4827]> | <[0.1138,0.3010], [0.2691, 0.5118]> | <[0.3379,0.4387], [0.3918, 0.4930]> | <[0.1758,0.3134], [0.5627, 0.6752]> | <[0.6395,0.7521], [0.1195, 0.2196]> | |
<[0.3516,0.4906], [0.3000, 0.4436]> | <[0.6395,0.7711], [0.1000, 0.2289]> | <[0.5213,0.7804], [0.1000, 0.2196]> | <[0.4387,0.6252], [0.2289, 0.3292]> | <[0.5392,0.6759], [0.1295, 0.3045]> | |
<[0.3000,0.4170], [0.3241, 0.4930]> | <[0.1000,0.2103], [0.6208, 0.7897]> | <[0.1000,0.2366], [0.5658, 0.7634]> | <[0.2103,0.3109], [0.4200, 0.5968]> | <[0.1849,0.3121], [0.5203, 0.6484]> |
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Ren, H.; Wang, G. An Interval-Valued Intuitionistic Fuzzy MADM Method Based on a New Similarity Measure. Information 2015, 6, 880-894. https://doi.org/10.3390/info6040880
Ren H, Wang G. An Interval-Valued Intuitionistic Fuzzy MADM Method Based on a New Similarity Measure. Information. 2015; 6(4):880-894. https://doi.org/10.3390/info6040880
Chicago/Turabian StyleRen, Haiping, and Guofu Wang. 2015. "An Interval-Valued Intuitionistic Fuzzy MADM Method Based on a New Similarity Measure" Information 6, no. 4: 880-894. https://doi.org/10.3390/info6040880
APA StyleRen, H., & Wang, G. (2015). An Interval-Valued Intuitionistic Fuzzy MADM Method Based on a New Similarity Measure. Information, 6(4), 880-894. https://doi.org/10.3390/info6040880