A PSO-Based Hybrid Metaheuristic for Permutation Flowshop Scheduling Problems

Algorithm 1

The main procedure of the SA_VNS.

Begin:

   Initialization:

    Let 𝜋₀ be the input initial solution. Set 𝜋=𝜋₀, the acceptance threshold 𝑇=0.05, and

    𝑜𝑢𝑡𝑒𝑟𝑙𝑜𝑜𝑝_𝑐𝑜𝑢𝑛𝑡𝑒𝑟=0.

   while (𝑜𝑢𝑡𝑒𝑟𝑙𝑜𝑜𝑝_𝑐𝑜𝑢𝑛𝑡𝑒𝑟<𝑛/5) do

    (1) Generate a random number 𝑟 in [0, 1], and two random integer numbers 𝑤 and 𝑧.

       If 𝑟 > 0.5, generate 𝜋^′ = insert (𝜋,𝑤,𝑧); otherwise generate 𝜋^′ = interchange (𝜋,𝑤,𝑧).

    (2) Set 𝑖𝑛𝑛𝑒𝑟_𝑐𝑜𝑢𝑛𝑡𝑒𝑟=0.

    (3) while  (𝑖𝑛𝑛𝑒𝑟_𝑐𝑜𝑢𝑛𝑡𝑒𝑟<𝑛×(𝑛−1))  do

       Set 𝑘=1.

       while (𝑘≤2) do

        (1) Generate two random integer numbers 𝑤 and 𝑧.

        (2) If 𝑘=1, generate 𝜋^′′ = 𝑖𝑛𝑠𝑒𝑟𝑡(𝜋^′,𝑤,𝑧).

        (3) If 𝑘=2, generate 𝜋^′′ = 𝑖𝑛𝑡𝑒𝑟𝑐ℎ𝑎𝑛𝑔𝑒(𝜋^′,𝑤,𝑧).

        (4) If 𝑓(𝜋^′′)<𝑓(𝜋^′),thenset𝜋^′=𝜋^′′ and 𝑘=1; otherwise set 𝑘=𝑘+1.

       end while

       Set 𝑖𝑛𝑛𝑒𝑟_𝑐𝑜𝑢𝑛𝑡𝑒𝑟=𝑖𝑛𝑛𝑒𝑟_𝑐𝑜𝑢𝑛𝑡𝑒𝑟+1.

       end while

    (4) If 𝑓(𝜋^′)<𝑓(𝜋₀),set𝜋₀=𝜋^′and𝜋=𝜋^′, and then go to step 6; otherwise go to step 5.

    (5) If 𝑓(𝜋^′)≥𝑓(𝜋₀)but(𝑓(𝜋^′)−𝑓(𝜋₀))/𝑓(𝜋₀)≤𝑇,set𝜋=𝜋^′; otherwise, generate a random

      number r in [0, 1], and then set 𝜋=𝜋₀ if 𝑟>0.5 and 𝜋=𝜋^′ if 𝑟≤0.5.

    (6) Set 𝑇=𝑇×0.95 and 𝑜𝑢𝑡𝑒𝑟𝑙𝑜𝑜𝑝_𝑐𝑜𝑢𝑛𝑡𝑒𝑟=𝑜𝑢𝑡𝑒𝑟𝑙𝑜𝑜𝑝_𝑐𝑜𝑢𝑛𝑡𝑒𝑟+1.

   end while

   Report the improved solution  𝝅_𝟎.

End