δ-Cut Decision-Theoretic Rough Set Approach: Model and Attribute Reductions

Algorithm 1

Heuristic algorithm for attribute reduction based on decision-monotonicity criterion.

Input: Decision system 𝐷𝑆=(𝑈,𝐶∪𝐷), threshold 𝛿;

Output: A decision-monotonicity reduct 𝑟𝑒𝑑.

Step  1.  𝐵←∅, 𝑀←𝐶, compute 𝐶^𝛿_(𝛼,𝛽)(𝐷_𝑖), 𝐷_𝑖∈𝜋_𝐷;

Step  2. Compute the decision-monotonicity significance for each 𝑎_𝑖∈𝐶 with 𝐷𝑀^sig_in(𝑎_𝑖,𝐶,𝛿);

Step  3.  𝐵←𝑎_𝑗 where

   𝐷𝑀^sig_in(𝑎_𝑗,𝐶,𝛿)=max{𝐷𝑀^sig_in(𝑎_𝑖,𝐶,𝛿)∶𝑎_𝑖∈𝐶};

Step  4.   While  𝑀≠∅  do

     ∀𝑎_𝑖∈𝐶−𝐵, compute 𝐷𝑀^sig_out(𝑎_𝑖,𝐵,𝛿);

     Select the maximal 𝐷𝑀^sig_out(𝑎_𝑗,𝐵,𝛿) and corresponding attribute 𝑎_𝑗;

     If  𝐷𝑀^sig_out(𝑎_𝑗,𝐵,𝛿)>0

      𝐵=𝐵∪{𝑎_𝑗};

     End

     𝑀=𝑀−{𝑎_𝑗};

   End

Step  5.  ∀𝑎_𝑖∈𝐵

   If  𝐷𝑀^sig_in(𝑎_𝑖,𝐶,𝛿)≥0

    𝐵=𝐵−{𝑎_𝑖};

   End

Step  6.  𝑟𝑒𝑑=𝐵.