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Four new types of soft fuzzy rough covering models and their applications in decision-making

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Abstract

The soft fuzzy rough covering models were first proposed by Zhan and Sun (Comput Appl Math 38:149, 2019), which have garnered more interest in the last few years. Initially, they presented the notions of soft fuzzy neighborhood and the type \(\mathbb {k}\)-approximation (for \(\mathbb {k}=1)\) based on the notion of soft fuzzy covering (i.e., soft fuzzy rough covering models using soft fuzzy neighborhoods). Then, Atef and Nada (Math Comput Simul 185:452–467, 2021) further offered type \(\mathbb {k}\)-approximation (for \(\mathbb {k}=2,3,4)\) based on soft fuzzy covering (i.e., soft fuzzy rough covering models through soft fuzzy neighborhoods and soft fuzzy complement neighborhoods). However, as the notions of soft fuzzy neighborhood and soft fuzzy complement neighborhood are not able to ensure that they have the reflexive property, the type \(\mathbb {k}\)-approximation (for each \(\mathbb {k}\in \{1,2,3,4\}\)) cannot satisfy the inclusion property (which is crucial for rough set models). The present paper will give an improvement to it in a very easy-to-understand manner so that the new type \(\mathbb {k}\)-approximation has more desired properties (\(\mathbb {k}\in \{1,2,3,4\}\)). We first define the notions of reflexive soft fuzzy neighborhood and reflexive soft fuzzy complement neighborhood, along with some key findings. Then, we present four new types of approximations based on soft fuzzy covering, each of which guarantees that a fuzzy subset to be approximated both includes its lower approximation and is included in its upper approximation (we call this inclusion property) apart from having usually ideal characteristics; several examples (including miner revisions) are also presented. Finally, a thorough explanation of a novel strategy that resolves the decision-making problem is provided through the new type \(\mathbb {k}\)-approximations \((\mathbb {k}\in \{1,2,3,4\})\) based on a family of soft fuzzy coverings, and a numerical example is given to show how applicable it is.

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Notes

  1. We call the number \(1-2\inf \{|\Phi (x)-0.5| \mid x\in X\}\in [0,1]\) the fuzziness degree of an FS \(\Phi \in [0,1]^X\).

  2. Motivated from the fuzzy set on decision set found in Subsection 5.2 (Sun et al. 2017).

  3. Motivated from the Definition 3.4 found in Huang and Li (2021).

  4. (1) \({{\mathscr {P}}}\in ([0,1]^X)^I\) satisfying \(\bigvee {{\mathscr {P}}}=\bigvee \{{{\mathscr {P}}}(i) \mid i\in I\}\ge \beta _X\) is called a \(\beta \)-soft fuzzy covering (\(\beta \)-SFC or \(\beta \)-FC\({{\mathscr {P}}}\) for short), where \(\beta _X\) is a FS on X taking constant value \(\beta \) on \(X\ \ (\beta \in (0,1])\) (see, Definition 3.2 in Zhang and Zhan (2019)). (2) \(\mathcal {N}^{\beta }_{\hspace{-2.84526pt}{{\mathscr {P}}} x} =\bigwedge \{{{\mathscr {P}}}(i) \mid i\in I, {{\mathscr {P}}}(i)(x)\ge \beta \}\in [0,1]^X\) is called a \(\beta \)-soft fuzzy neighborhood of x (see, Definition 3.4 in Zhang and Zhan (2019)) and \(\mathcal {M}^{\beta }_{\hspace{-2.84526pt}{{\mathscr {P}}} x}\in [0,1]^X\), defined by \(\mathcal {M}^{\beta }_{\hspace{-2.84526pt}{{\mathscr {P}}} x}(y) =\mathcal {N}^{\beta }_{\hspace{-2.84526pt}{{\mathscr {P}}} y}(x)\ \ (\forall y\in X)\), is called a \(\beta \)-soft fuzzy complement neighborhood of x (see, Definition 4.7 in Zhang and Zhan (2019)).

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Acknowledgements

The authors are grateful to the referees for their valuable comments and suggestions.

Funding

This work was supported the National Natural Science Foundation of China (Grant no. 11771069) and the Fundamental Research Funds for the Central Universities (Grant no. GK202105007, GK202304053).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xian, 710062, China

    Ahmed Mostafa Khalil & Sheng Gang Li

  2. Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, 71524, Egypt

    Ahmed Mostafa Khalil

  3. School of Mathematics and Physics, Guangxi Minzu University, Nanning, 530006, China

    Sheng Gang Li & Heng Liu

  4. School of Science, Xi’an Polytechnic University, Xian, 710048, China

    Hu Zhao

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  1. Ahmed Mostafa Khalil
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Contributions

AMK and S-GL: Conceptualization, Methodology, Writing-original draft preparation. HL and HZ: Writing-review and editing, Funding acquisition.

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Correspondence to Sheng Gang Li.

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Khalil, A.M., Li, S.G., Liu, H. et al. Four new types of soft fuzzy rough covering models and their applications in decision-making. Comp. Appl. Math. 44, 31 (2025). https://doi.org/10.1007/s40314-024-02988-w

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