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

Algorithm 2

Generate alpha.

Require:  𝑋∈ℝ+^𝑚,𝑛, 𝑌∈ℝ+^𝑠,𝑛, DMU 𝑘.

Ensure:  𝛼_1𝑘,…,𝛼_𝑛𝑘 ∈ℝ+, ∀𝛼_𝑗𝑘≥0.

   𝑉^𝑥,𝑉^𝑦∈ℝ+^𝑛 such as 𝑉^𝑥_𝑗=1𝑚^𝑚_𝑖∑₌₁𝑋_𝑖,𝑗 and 𝑉^𝑦_𝑗=1𝑠^𝑠_𝑖∑₌₁𝑌_𝑖,𝑗

  Sort 𝑉^𝑥 and 𝑉^𝑦 in decreasing.

   for  𝑗:= 1,…,𝑛  do

    if 𝑉^𝑥_𝑗 < ⌊𝑛/2⌋ and  𝑉^𝑦_𝑗 ≥ ⌊𝑛/2⌋ then

     𝛼_𝑗𝑘←0

    else if 𝑉^𝑥_𝑗 ≥ ⌊𝑛/2⌋ and 𝑉^𝑦_𝑗 < ⌊𝑛/2⌋ then

     𝛼_𝑗𝑘← Generate 0.5≤𝛼_𝑗𝑘≤1 randomly.

    else if  (𝑉^𝑥_𝑗≥𝐵𝑜𝑢𝑛𝑑𝑃𝑜𝑖𝑛𝑡 and 𝑉^𝑦_𝑗≥𝐵𝑜𝑢𝑛𝑑𝑃𝑜𝑖𝑛𝑡) or (𝑉^𝑥_𝑗<𝐵𝑜𝑢𝑛𝑑𝑃𝑜𝑖𝑛𝑡 and 

    𝑉^𝑦_𝑗<𝐵𝑜𝑢𝑛𝑑𝑃𝑜𝑖𝑛𝑡) then

     𝛼_𝑗𝑘← Generate 0 ≤𝛼_𝑗𝑘≤0.25 randomly.

    end if

   end for

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