An Enhanced Differential Evolution Based Algorithm with Simulated Annealing for Solving Multiobjective Optimization Problems

Pseudocode 1

Pseudocode of MODESA.

Step  1. Initialization

  Initialize the values of the parameters in Table 1.

  Generate the initial population 𝑃₀ using uniform distribution.

  Generate 𝑁 opposite solutions denoted as 𝑃_obl by using OBL operation on 𝑃₀.

  Select 𝑁 best individuals as the current population by using non-dominated sorting and the

  vicinity distance based pruning method on 𝑃₀∪𝑃_obl.

  For (𝑖=0; 𝑖<𝑁; ++𝑖) {

    𝑙𝑖𝑓𝑒_𝑖=0.

  }// End For

  𝑡=0

Step  2. Main loop

  While (𝑡≤𝑇) {

    𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒=𝑇_max.

    Construct a temporary population which is denoted as 𝑃_temp and set 𝑃_temp=∅.

    For (𝑖=0; 𝑖<𝑁; ++𝑖) {

      Randomly select three distinct individuals: 𝑥₁, 𝑥₂ and 𝑥₃, which are different

      from the current individual 𝑥_𝑖 from the current population.

      Select the best individual 𝑥_best from the three distinct individuals as the base vector

      for the next mutation operation.

      Produce a trial individual 𝑤_𝑖:

        generate a random number 𝑗𝑗∈{1,…,𝑛}

        For each variable 𝑗 of 𝑤_𝑖{

            With a probability cr or 𝑗 is equal to 𝑗𝑗

           do

             𝑤_𝑗,𝑖=𝑥_𝑗,1+𝑓∗(𝑥_𝑗,2−𝑥_𝑗,3).

           otherwise

             𝑤_𝑗,𝑖=𝑥_𝑗,best.

           If (𝑤_𝑗,𝑖>𝑣_𝑗,max) 𝑤_𝑗,𝑖=𝑣_𝑗,max.

           If (𝑤_𝑗,𝑖<𝑣_𝑗,min) 𝑤_𝑗,𝑖=𝑣_𝑗,min.

        }// End For

       Check the domination status of the trial individual 𝑤_𝑖 and the current individual 𝑥_𝑖.

       If (𝑤_𝑖 dominates 𝑥_𝑖) {

        Replace the current individual 𝑥_𝑖 by the trial individual 𝑤_𝑖: 𝑥_𝑖=𝑤_𝑖.

      }

      Else if (𝑤_𝑖 and 𝑥_𝑖 are incomparable) {

        Use the proposed simulated annealing to allocate the two individuals: 𝑤_𝑖 and 𝑥_𝑖.

        SimulateAnnealing (𝑥_𝑖, 𝑤_𝑖, 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒, 𝑇_min, 𝑇_max, 𝑃_𝑡, 𝑃_temp, 𝑚𝑎𝑥_𝑙𝑖𝑓𝑒_𝑐𝑖𝑐𝑙𝑒).

         𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒=𝛾∗𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒.

      }

      Else if (𝑤_𝑖 is dominated by 𝑥_𝑖) {

        Put the trial individual 𝑤_𝑖 to the temporary population.

      }// End If

  }// End For

  Union the two population: 𝑃_𝑡∪𝑃_adv.

  If (𝑡==𝑇) {

      Use the non-dominated sorting to the union population to remove the individuals that

      are dominated by any other solutions.

    }// End If

    Select 𝑁 individuals as the next population from the union population by using the

    non-dominated sorting.

    For (𝑖=0; 𝑖<𝑁; ++𝑖) {

      If (𝑙𝑖𝑓𝑒_𝑖>0)

         𝑙𝑖𝑓𝑒_𝑖−−;

    }// End For

    Apply the vicinity distance based pruning method.

  }// End While