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Covering-based variable precision L-fuzzy rough sets based on residuated lattices and corresponding applications

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Abstract

There are many extensions of Pawlak’s rough sets. Particularly, fuzzy rough set theory based on residuated lattices is considered as one of the generalizations that extends the truth values to lattices. In this paper, we attempt to combine the idea of “variable precision” with covering-based fuzzy rough sets based on residuated lattices and present the concept of covering-based variable precision L-fuzzy rough sets. After the presentation of this concept, some basic properties of the proposed rough set model are discussed. Then, with the aid of the PROMETHEE and TOPSIS methods, we put forth a novel multi-attribute decision-making (MADM) method based on covering-based variable precision L-fuzzy rough sets (for \(L=[0,1]\)). Finally, an illustrative example is give to demonstrate the proposed decision-making method. By a sensitivity analysis along with a comparative analysis, we reveal the applicability and validity of the proposed decision-making method.

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Notes

  1. A t-norm \(\top\) is a binary function on [0, 1] if it is commutative, associative, non-decreasing and meets \(\top (1,o)=o\) for each \(o\in [0,1]\).

  2. An implicator is a binary function \(\theta\) on [0, 1] if for all \(o\in [0,1]\), \(\theta (\cdot , o)\) is non-increasing, \((o,\cdot )\) is non-decreasing and meets \(\theta (1,0)=0\), \(\theta (1,1)=\theta (0,1)=\theta (0,0)=1\).

  3. We name \(L=([0,1],\otimes ,\rightarrow ,\vee ,\wedge ,0,1)\) is a Łukasiewicz reriduated lattice if \(m\otimes n=\max \{ 0, m+n-1\}\), \(m\rightarrow n=\min \{1-m+n,1\}\) for any \(m,n\in L.\)

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Acknowledgements

The authors are extremely grateful to the editors and three reviewers for their valuable comments and helpful suggestions which helped to improve the presentation of this paper. This research is partially supported by NNSFC (11961025; 61866011).

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  1. School of Mathematics and Statistics, Hubei Minzu University, Enshi, 445000, China

    Haibo Jiang & Jianming Zhan

  2. Department of Mathematics and Physical, North China Electric Power University, Beijing, 102206, China

    Degang Chen

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Jiang, H., Zhan, J. & Chen, D. Covering-based variable precision L-fuzzy rough sets based on residuated lattices and corresponding applications. Int. J. Mach. Learn. & Cyber. 12, 2407–2429 (2021). https://doi.org/10.1007/s13042-021-01320-w

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