## 4 A Strategy Based on a Segmentation of the Search Domain's
### Evaluation of Subdomains Mean-Average Separation and Weight Factor Formulation
Equation (18) produces values larger than one for subdomains \(k\) belonging to the zones of the landscape with mean fitness values \(\overline{Fv}_{u,k}\) lower than the average \(\mu_{k}\), and lower than one when it is the opposite. The result would thus be lower and greater than one for the subdomains containing the peak of the prominent maximum function in Figure 8b and the deep minimum of the function in Figure 8c, respectively. Finally, Equation (20) writes the normalized version (\(I_{k}\)) of the \(I_{uk}\) values:  
\[I_{u,k}=\left(1-\frac{\overline{Fv}_{u,k}-\mu_{k}}{c_{4}\sigma_{k}}\right)^{2} \tag{18}\]  
\[\sigma_{k}=\left\{\frac{1}{S^{D}}\sum_{k=1}^{S^{D}}\left(\overline{Fv}_{u,k}- \mu_{k}\ \right)^{2}\right\}^{\frac{1}{2}} \tag{19a}\]  
\[\sigma_{D}=\left\{\frac{1}{N}\sum_{u=1}^{N}\left(Fv_{u}-\mu_{D}\ \right)^{2}\right\}^{ \frac{1}{2}} \tag{19b}\]  
Figure 7:0  
The balanced combination of the \(I_{k}\) and \(I_{k}\) descriptors must account for the overall influence of the asymmetry of the \(Fv\) distribution over the landscape. For instance, a simple arithmetic average of these descriptors would be undependable. Equation (21) writes the proposed formulation for the weight (\(W_{k}\)) factors. The resulting values allow apportioning the particles based on the potential of the subdomains to contain optima, and therefore, allow forming the initial particle distribution. In Equation (21), \(D_{f}\) introduces the overall influence of the asymmetry, while \(c_{5}\) is a constant added to fine-tune the \(I_{k}\) impact. In the present analysis, \(c_{5}\) is set at 10. The calculations associated with Equation (21) begin at position 4 in Figure 5, Part D.  
It would appear worthwhile to comment here on the adjustable constants \(c_{1}\) to \(c_{5}\) included in the model. While the proposed values are all indicated, they mainly represent arbitrary choices, since the proposed developments include no strict identification of the best options. The objective here is essentially to develop and present the basis of the model as well as illustrate its potential and versatility, as the fine-tuning of the parameters would be better examined in a dedicated study. Figure 7:1  
Figure 9 repeats the pre-probing operation illustrated in Figure 7:2 To help the illustration, the figure also displays the weighted particle distribution realized over the inverted Himmenblau function. The plot in Figure 9a shows that, even though the \(D_{f}\) measure of the Vincent function is close to zero, the weighted allotment strategy better detects the  
potential zones. Compared to Figure 7a, this plot presents an improved particle distribution, where the subdivisions containing optima are more populated. The Himmelblau function illustrated in Figure 6b describes a profile belonging to the category illustrated in Figure 8b. Thus, the weighted factor of Equation (21) leads to a completely different pre-mapping of the domain, and compared to the particle distribution presented in Figure 7b, the results of Figure 9b now mainly concentrate the particle in the subdomain situated in the optima neighbourhood. Finally, in the case of the inverted Himmelblau function, the contour area, which originally belonged to the maxima zone, now becomes the landscape region to investigate. The pre-probing strategy clearly detects this condition and concentrates the particles in the subdomains of the contour zone close to the potential minima, while the central subdivisions containing the maxima are virtually unpopulated. Figure 7:3 Pre-mapping particle distribution established based on weight factors: (**a**) 2D Vincent function, (**b**) 2D Himmelblau function and (**c**) 2D Inverted Himmelblau function.