Benchmarking in Data Envelopment Analysis: An Approach Based on Genetic Algorithms and Parallel Programming

Algorithm 4

Generate a chromosome (Method 3).

Require:   𝑋∈ℝ+^𝑚,𝑛, 𝑌∈ℝ+^𝑠,𝑛, 𝑞∈ℝ, DMU 𝑘, 𝑀𝑎𝑥𝐼𝑡𝑒𝑟∈ℤ

Ensure: Chromosome 𝑐.

 repeat

     Generate 0≤𝛽_𝑘≤1 randomly.

     Generate 𝛼_1𝑘,…,𝛼_𝑛𝑘 using Algorithm 2 and 𝑡⁻_1𝑘,…,𝑡⁻_𝑚𝑘 and 𝑡⁺_1𝑘,…,𝑡⁺_𝑠𝑘 using expression (2).

    while  ∃𝑡⁻_𝑖𝑘≤0  and ∃𝑡⁺_𝑟𝑘≤0  and  number of 𝛼_𝑗𝑘≠0>2   do

   if ∃𝑡⁻_𝑖𝑘≤0 and ∀𝑡⁺_𝑟𝑘≥0 then

    Increase 𝛽_𝑘

   end if

   if  ∀𝑡⁻_𝑖𝑘≥0 and ∃𝑡⁺_𝑟𝑘≤0   then

    Decrease 𝛽_𝑘

   end if

   if  ∃𝑡⁻_𝑖𝑘≤0 or ∃𝑡⁺_𝑟𝑘≤0  then

    Choose 𝛼_𝑗𝑘 no null randomly and make it equal to zero, decreasing the rest 𝛼_𝑗𝑘

    in 𝑝 and find the minimum 𝛼_𝑗𝑘 and modify its value in order to satisfy (3)

   end if

    end while

 until  A valid chromosome is obtained or 𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠≥𝑀𝑎𝑥𝐼𝑡𝑒𝑟  or  the number of 𝛼_𝑗𝑘

 no null is lower than 𝑛−2