## 3 The Proposed Distributed Bi-Behaviors Crow Search Algorithm
### The Distributed Bi-Behaviours Crow Search algorithm (DB-CSA)
#### 3.2.1 First Variant: DB-CSA for Static MaOPs  
The key processing steps in the first variant of the proposed DB-CSA approach are shown in Figure 1 and detailed as follows:  
1. Initialization of population positions and their memories: In DB-CSA algorithm, each crow \(i\) is presented as a potential solution in the search space. The DB-CSA starts with a random initialization of position (\(X\)) and the memory (\(Mem\)) of the flock of \(N\) crows when each crow \(i\) has presented a potential solution in the search space. 2. Initialization of the archive of the non-dominated solutions: the archive (A) is initially created to store all the non-dominated solutions during the optimization process. After that, all the following steps are executed until a predefined number of iterations is reached. 3. Fitness Function Evaluation: for each crow \(i\), the fitness function \(Fit(X_{i})\) is evaluated. 4. Determine the followed crow \(i\): at each iteration, one of the main behaviors of crow \(i\) is to determine one crow \(i\) to follow by selecting a random position value between zero and the size of the flock of best crows. 5. Determine the average crow \(i\): the aggregated value of \(K\) objectives are computed as the fitness function \(Fit(X_{i}(t)\) of each crow \(i\); then, the average value of all fitness functions is selected to determine the mean solution. 6. Update the crow position using the bi-behaviours' beta-distribution profiles
7. Update the memory (\(Mem\)): the memory of each crow \(i\) is updated using Equation (4). 8. Apply the mutation operatorsUpdate the archive of non-dominated solutions: at each time \(t\) of the optimization procedure, all the non-dominated solutions are stored in the archive (A) based on the \(\varepsilon-dominance\) operator. 10. Generate OUTPUT= the best Pareto solutions from the archive (A). ```
Begin Randomly initialize the position Xof N crows with d-dimensional search space;  Initialize the memory (Mem) of each crow;  Initialize the archive (A) of non-dominated solutions;  Evaluate the position of the crows;  Initialize the flock of crows at iteration (\(t\)): \(POS(t)\leftarrow\varnothing\);  Initialize the previous flock of crows at iteration (\(t-1\)): \(POS(t-1)\leftarrow\varnothing\);  Initialize non-dominated solutions counter \(n_{p}=0\);  Initialize dominated solutions counter \(n_{q}=0\);  while iteration \(<Max_{Iterations}\) do for \(i=1\) to \(i\leq\) N do  Evaluate the fitness function of crow \(i\): \(Fit(X_{i}(t))\);  Set \(POS(t)\gets POS(t)\cup solution_{i}(t)\);  Set \(POS(t-1)\gets POS(t-1)\cup solution_{i}(t-1)\) if Dynamic Change = True then  for \(p=1\) to \(|POS(t)|\) do  for \(q=1\) to \(|POS(t-1)|\) do  Compare objective values \(POS(t)\) and \(POS(t-1)\) using dominance operator;  if \(solution_{p}\) dominates \(solution_{q}\) then \(n_{p}=n_{p}+1\);  New population \(POS(t+1)\gets POS(t+1)\cup solution_{p}\);  else if \(solution_{q}\) dominates \(solution_{p}\) then \(n_{q}=n_{q}+1\);  Re-initialize \(solution_{p}\);  end if  end for  end for  Update the archive (A) \(\leftarrow\)\(non-dominatedsolutions\) (\(POF(t)\));  end if  Choose the followed crow (\(i\)) randomly;  Determine the average crow \(i\): \(Mean(Fit(X_{i}(t)))\) if \(Fit(X_{i}(t))\geqslant Mean(Fit(X_{i}(t))\) then  Update the crow's position using Equation (8) on Beta-1 exploitation profile else  Update the crow's position using Equation (8) on Beta-2 exploration profile end if  Update the memory using Equation (4)  end For  Apply the mutation operators using Equations (9) and (10)  Update the archive (A) of non-dominated solutions end while  Return the archive (A) of the non-dominated solutions End
```  
**Algorithm 3** The DB-CSA-II Algorithm with Dynamic Optimization Process  
In the standard CSA algorithm, the crow position is updated according to the Equation (3), while the convergence and the diversity stages are treated separately, causing premature convergence.