A Divide-and-Conquer Approach for Solving Fuzzy Max-Archimedean 𝑡-Norm Relational Equations

Algorithm 2

Procedure for generating test matrices.

Input: 𝑚,𝑛,𝑑 and 𝑝. Note: 1/𝑚⩽𝑑⩽1/𝑝 and 𝑝⩽𝑛 are required.

Output: 𝑀=(𝑚_𝑖𝑗)_{|𝐼|×|𝐽|} where 𝑖∈𝐼={1,…,𝑚} and 𝑗∈𝐽={1,…,𝑛}

 (1)  Let  𝑃∶=1,2,…,𝑝;

 (2) Randomly divide 𝐼 into 𝑝 disjoint subsets 𝐼¹,𝐼²,…,𝐼^𝑝 such that ⋃_𝑘∈𝑃𝐼^𝑘=𝐼, and 𝐼^𝑘−𝐼^̆𝑘⩽1 for any 𝑘,𝑘̆∈𝑃;

 (3) Randomly divide 𝐽 into 𝑝 disjoint subsets 𝐽¹,𝐽²,…,𝐽^𝑝 such that ⋃_𝑘∈𝑃𝐽^𝑘=𝐽, and 𝐽^𝑘−𝐽^̆𝑘⩽1 for any 𝑘,𝑘̆∈𝑃;

 (4) Initialize 𝑚_𝑖𝑗 to 0 for all 𝑖∈𝐼 and 𝑗∈𝐽;

 (5)  For  𝑘=1 to 𝑝  do  // inject an irredundant covering to avoid Φ(𝑀)=∅.

 (6)    For  each  𝑗∈𝐽^𝑘  do

 (7)   Randomly select 𝑖 from 𝐼^𝑘, and set 𝑚_𝑖𝑗∶=1;

 (8)    End  of  for  each

 (9)  End  of  for

 (10)  Let  𝑙∶=⌈𝑚∗𝑛∗𝑑⌉−𝑛;

 (11)  While  𝑙>0  do

 (12)   Randomly select 𝑖 and 𝑗 from 𝐼 and 𝐽, respectively;

 (13)    If  𝑖∈𝐼^𝑘, 𝑗∈𝐽^𝑘 for some 𝑘∈𝑃, and 𝑚_𝑖𝑗=0  then

 (14)   let  𝑚_𝑖𝑗∶=1 and 𝑙∶=𝑙−1;

 (15)  End  of  while

 (16)  Return  𝑀;