Constrained Multiobjective Biogeography Optimization Algorithm

Algorithm 1

CMBOA.

Step  1: Parameter setting: population size 𝑁, the size of feasible elitist archive 𝑁₁, the size of

   infeasible elitist archive 𝑁₂, maximum generation 𝑔_max. Generate an initial population 𝐴(𝑡),

   set the iterative generation 𝑡=1, and archive 𝑀(𝑡)=Φ,

Step  2: Update of the archive 𝑀(𝑡)

 Step  2.1: Divide the population 𝐴(𝑡)∪𝑀(𝑡) into the feasible set 𝑃(𝑡) and the infeasible set

     𝑄(𝑡) based on their constraint violation.

 Step  2.2: Select 𝑁₁ individuals with small fitness from 𝑃(𝑡) by individuals’ fitness sorting to

     update feasible archive 𝑃(𝑡), and select 𝑁₂ individuals with small constraint violation

     from 𝑄(𝑡) to update infeasible archive 𝑄(𝑡).

 Step  2.3: Combine 𝑃(𝑡) and 𝑄(𝑡) to gain archive set 𝑀(𝑡+1), 𝑀(𝑡+1)=𝑃(𝑡)∪𝑄(𝑡);

     If 𝑡≥𝑔_max is satisfied, output 𝑃(𝑡) and the algorithm stops; otherwise go the next step.

Step  3: Generate the offspring population

 Step  3.1: If 𝑃(𝑡)=Φ, then 𝑃(𝑡+1)=Φ, and perform the differential evolution operator on

     infeasible population 𝑄(𝑡) to obtain 𝑄(𝑡+1). Otherwise, go to Step  3.2

 Step  3.2: Implement selection operation on 𝑃(𝑡) to gain the breeding pool 𝐷(𝑡), and then

     execute migration operation on 𝐷(𝑡) to generate 𝑃(𝑡+1);

 Step  3.3: Implement the crossover and mutation on infeasible population 𝑄(𝑡) to generate 𝑄(𝑡+1).

 Step  3.4: Combine 𝑃(𝑡+1) and 𝑄(𝑡+1) to obtain the offspring population 𝐴(𝑡+1),

     𝐴(𝑡+1)=𝑃(𝑡+1)∪𝑄(𝑡+1)

Step  4: let 𝑡=𝑡+1 and return Step  2.