Granular Space Reduction to a β Multigranulation Fuzzy Rough Set

Algorithm 1

Find granular space reduction.

Input: a fuzzy decision information system

    𝑆 = 𝑈,𝐴𝑇∪𝐷

Output: a granular space reduction RED

 Step  1. ∀𝐴_𝑖∈𝐴𝑇, compute fuzzy similarity matrix

 Step  2. ∀𝐴_𝑖∈𝐴𝑇, compute fuzzy equivalence matrix in terms of the result of Step  1,

 then get fuzzy equivalence class

 Step  3. 𝑅𝐸𝐷←Ø

 Step  4. ∀𝐴_𝑖∈𝐴𝑇, compute sig_in(𝐴_𝑖,𝐴𝑇,𝐷)

 Step  5. 𝑅𝐸𝐷←𝐴_𝑗,sig_in(𝐴_𝑗,𝐴𝑇,𝐷)=maxsig_in𝐴_𝑖,𝐴𝑇,𝐷∶𝐴_𝑖∈𝐴𝑇

 Step  6. ∀𝐴_𝑖∈𝐴𝑇−𝑅𝐸𝐷, compute sig_out(𝐴_𝑖,𝑅𝐸𝐷,𝐷)

   If

      sig_out𝐴_𝑘,𝑅𝐸𝐷,𝐷=maxsig_out𝐴_𝑖,𝑅𝐸𝐷,𝐷∶𝐴_𝑖∈𝐴𝑇−𝑅𝐸𝐷,𝑅𝐸𝐷=𝑅𝐸𝐷∪{𝐴_𝑘}

   Until 𝛾(𝑅𝐸𝐷,𝛽,𝐷)=𝛾(𝐴𝑇,𝛽,𝐷)

   End

Step  7.  return RED