A Multipopulation PSO Based Memetic Algorithm for Permutation Flow Shop Scheduling

Algorithm 2

The flowchart of MPSOMA.

Basic flowchart of MPSOMA

Step 1. 𝑡=1, Generate a initial population 𝑄;

Step 2. Evaluate the objective values of all particles in 𝑄;

Step 3. If 𝑡>𝑡_max, output the optimal sequences of job, otherwise divide 𝑄 into three

   subpopulations 𝑄₁,𝑄₂, and 𝑄₃, go to Step 4;

Step 4. Evolve each subpopulation by adopting PSO and obtain three temporary subpopulations

    𝑄^′₁, 𝑄^′₂, and 𝑄^′₃;

Step 5. Optimize each subpopulation by different local search strategies. 𝑄^′₁ and 𝑄^′₃ are updated

   by using IIS local search and 𝑄^′₂ is optimized by using VNS. Then three temporary

   subpopulations are denoted as 𝑄₁^′′, 𝑄₂^′′, and 𝑄₃^′′;

Step 6. Evaluate the objective values of all particles in 𝑄_𝑖^′′,𝑖=1,2,3 and choose the best particle

   from the three subpopulations decode them and three corresponding sequences of job

    𝜋_1best, 𝜋_2best, 𝜋_3best are obtained; the worst particle is also selected from the three subpopulations

   and the three corresponding sequences of job are denoted as 𝜋_1wor,𝜋_2wor,𝜋_3wor. Then

    𝜋_1best,𝜋_2best,𝜋_3best are used to build a probabilistic model by EDA;

Step 7. Three new sequences of job 𝜋₁,𝜋₂,𝜋₃ are sampled from the probabilistic model, if the

   maximum completion time of 𝜋_𝑖 is less than that of 𝜋_𝑖wor, then replace the worst individual of each

   sub-population by 𝜋_𝑖, here 𝑖=1,2,3;

Step 8. Evaluate the objective values of all particles in three temporary subpopulations and choose

   the best particle from three subpopulations, decode it to sequence of job 𝜋_𝑔best and SA local

   search is applied on 𝜋_𝑔best and obtain a new sequence of job 𝜋^′_𝑔best. If the maximum completion

   time of 𝜋^′_𝑔best is less than that of 𝜋_𝑔best, then use 𝜋^′_𝑔best to replace 𝜋_𝑔best;

Step 9. Encode all sequences of jobs to the positions of particles and unite the three subpopulations

   into a population, 𝑡=𝑡+1. Return to Step 2.