## 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
However, this issue was treated by the new DB-CSA algorithm using a bi-behaviours beta-distribution profile to assume a dynamic and good balance between both stages. The two beta-distribution profiles are presented in Equation (8) and Figure 3, denoted by \(Beta1\_rand\) and \(Beta2\_rand\), respectively, for exploitation and exploration enhancement. The couple of beta-profiles are used to modify the original Equation (3) of the standard CSA algorithm, presenting the update process for each crow \(i\). The two profiles were proposed based on the beta-function of Alimi [77], as presented in Equations (5)-(7). The main advantage in using the beta function presented here is their capacity to produce several forms and configurations of distributions, including the normal Gaussian one. The one-dimensional Beta function is defined in Equation (5). \[\beta(x,p,q,x_{0},x_{1})= \begin{cases}\left(\dfrac{x-x_{0}}{x_{c}-x_{0}}\right) ^{p}\left(\dfrac{x_{1}-x}{x_{1}-x_{c}}\right)^{q}\text{ If }x\in[x_{0},x_{1}]\\ \text{Otherwise}\end{cases}  \tag{5}\]  
where \(p\), \(q\), \(x_{0}\) and \(x_{1}\) are real values, with \((x_{0}<x_{1})\in\mathit{IR}\), and \(x_{c}\) is detailed in Equation (6). \[x_{c}=\dfrac{(p\times x_{1})+(q\times x_{0})}{p+q} \tag{6}\]  
However, the multi-dimensional beta function provided in the mathematical definition (7), presenting m product of the one-dimensional beta function in (5). \[\beta(x)=\prod_{k=1}^{m}\beta(x_{k},p_{k},q_{k},x_{0,k},x_{1,k}) \tag{7}\]  
The dynamic switch mechanism between the bi-behaviors' Beta-1 and Beta-2 profiles are assumed using a comparison between the fitness function \(Fit(X_{i}(t))\) of each crow \(i\) and the average solution (crow). If the fitness function \(Fit(X_{i}(t))=\sum_{k=1}^{k}f_{k}\) is greater than the mean value, we assume an exploration stage for the crow optimization process using the Beta-1 behaviour in Equation (8), which is used to update the crow position. Otherwise, the second Beta-2 behaviour in Equation (8) is considered, pushing each solution to the exploitation stage. As illustrated in Figure 3, the two beta-distribution profiles are detailed as follows:  
* The first large Gaussian Beta-1 exploitation profile is used to update the crows positions which is characterized by a large standard deviation pushing the population to achieve good diversity in the search space with \(p\) and \(q\) variables of the beta-function in Equation (8), which are equal to 5. * The second narrow Gaussian Beta-2 exploration profile adapts a limited standard deviation to update the crows positions with \(p\) and \(q\) in Equation (8), which are equal to 50 allowing for a good convergence to the optimal solution over time. \[X_{i}(t+1)= \begin{cases}\left//\right.\text{\bf Beta 1 Behaviour for exploitation profile:}\\ \text{IF }Fit(X_{i}(t))\geq Mean(Fit(X_{i}(t))\text{ then:}\\ X_{i}(t)+Beta1\_rand(i)\times(Mem_{i}(t)-X_{i}(t))\\ \text{Else:}\\ \left//\right.\text{\bf Beta 2 Behaviour for exploration profile:}\\ Beta2\_rand()\end{cases} \] (8) where Beta-1 is a beta random distribution over \([0,1]\), which assimilates to a fine search step around the optimal solution, while Beta-2 is more like a random exploration mechanism performed away from the previous optimal solution, \(Mem_{i}(t)\). Both Beta-1 and Beta-2 values are determined using Equation (5) with different configurations of the two properties \(p\) and \(q\). The mutation operators in [79] are added to maintain more diversity in the flock of \(N\) rows. The nonuniform and the boundary mutation operators in Equations (9) and (10) are applied to modify the variables \(X_{i}=X_{1}^{i},X_{2}^{i},\ldots,X_{d}^{i}\) of each crow \(i\), according to the probability mutation \(P_{m}\) equal to \(1/d\), where \(d\) is the dimensional search space and \(X_{i}\in[a_{i},b_{i}]\) where \(a_{i}\) and \(b_{i}\) are the lower and upper bounds, respectively. The nonuniform mutation in Equation (9) is applied when the modulo value (mod) that divides the crow position \(i\) by three is equal to zero.