## 4 Methodology
Subsequently, \(\vec{y}_{n}\) is determined such as

\[\vec{y}_{n}=\vec{m}\odot\hat{\vec{x}}_{z_{1}}+(1-\vec{m})\odot\hat{\vec{x}}_{ z_{2}},\quad\forall\,z_{1},z_{2}\in\{1,\ldots,M\},n\in\{M+1,\ldots,N\}, \tag{12}\]

where \(z_{1}\) and \(z_{2}\) are mutually exclusive indices from the population, i.e., \(z_{1}\
eq z_{2}\), and \(\vec{m}\in\mathbb{R}_{+}^{D}\) is the mask vector. Besides, \(M=\left\lfloor m_{p}N\right\rfloor\) is the mating pool size since \(m_{p}\in\left]0,1\right]\) is the portion of the best ranked agents from the population of size \(N\). These parameters are detailed as follows. On the one hand, \(z_{1}\) and \(z_{2}\) refer to the parent indices chosen by using a pairing scheme from the mating pool [2; 49]. The most common pairing schemes are _Random, Rank weighting, Roulette wheel_, and _Tournament_ pairing. On the other hand, the mask vector is commonly assumed in the range \([0,1]^{D}\), as we do in this work. However, some researchers incorporate values beyond such a range. This vector can be determined via diverse crossover mechanisms [49]. The most common ones are _Single-point, Two-points_, _Uniform, Blend,_ and _Linear crossover. Further information about the expressions corresponding to either pairing and crossover mechanisms can be found in [7; 31]. **Genetic Mutation (GM):**: This operator also works on a ranked version of the population \(\hat{X}(t)\) with respect to the cost function value i.e., \(\hat{X}(t)=\{\hat{\vec{x}}_{1},\ldots,\hat{\vec{x}}_{N}\}\) with \(f(\hat{\vec{x}}_{1})<\cdots<f(\hat{\vec{x}}_{N})\). Hence, the candidate position \(\vec{y}_{n}\) is calculated such as

\[\vec{y}_{n}=\vec{m}\odot\vec{x}_{n}(t)+\alpha(1-\vec{m})\odot\vec{r},\quad \forall\,n\in\{\left\lceil p_{e}N\right\rceil,\ldots,N\}, \tag{13}\]

where \(\vec{m}=H(p_{m}-\vec{q})\) is a mask vector, \(\alpha\in[0,1]\) is the spatial step size, \(\vec{r}\) and \(\vec{q}\) are vectors of i.i.d. random numbers with distribution \(\mathcal{U}(x_{l},x_{u})\) and \(\mathcal{U}(0,1)\), respectively, \(p_{e}\in[0,1]\) is the elite portion of the population (i.e., the top \(\left\lfloor p_{e}N\right\rfloor\) ranked individuals), and \(p_{m}\in[0,1]\) is the mutation probability. Furthermore, most researchers implement different probability distributions for drawing random disturbances along the problem domain [49]. **Differential Mutation (DM):** The candidate position \(\vec{y}_{n}\) is determined by using the differential mutation operator, which employs current positions of different agents in the population, as shown

\[\vec{y}_{n}=\vec{m}+\sum_{m=1}^{M}\alpha_{m}\big{(}\vec{x}_{z_{2m-1}}(t)-\vec{x}_ {z_{2m}}(t)\big{)}, \tag{10}\]

where \(\vec{m}\) is the target vector which depends of the mutation scheme expressed such that \(\text{DE}/\cdot/M\), \(M\in\{1,2,3\}\) stands the number of random differences or perturbations, and \(\alpha_{m}\in[0,3]\) is a constant factor \(\forall\,m\in\{0,\dots,M\}\). Plus, \(z_{i}\,\,\forall\,i\in\mathbb{Z}_{+}\) is an integer with uniform random distribution between \(1\) and \(N\), where \(N\) is the population size, then \(z_{i}\sim\mathcal{U}\{1,N\}\) with \(z_{i}\
otin\bigcup_{j\in\mathbb{Z}_{+}-\{i\}}z_{j}\). The most of utilised schemes are listed as follows [38]:

* _rand_: \(\vec{m}=\vec{x}_{z_{0}}(t)\),
* _best_ : \(\vec{m}=\vec{x}_{z}(t)\),
* _current_: \(\vec{m}=\vec{x}_{n}(t)\),
* _current_-to-best_: \(\vec{m}=\vec{x}_{n}(t)+\alpha_{0}(\vec{x}_{*}(t)-\vec{x}_{z_{0}}(t))\),
* _rand-to-best_: \(\vec{m}=\vec{x}_{z_{0}}(t)+\alpha_{0}(\vec{x}_{*}(t)-\vec{x}_{2M+1}(t))\), and
* _rand-to-best_-_and-current_: \(\vec{m}=\vec{x}_{z_{0}}(t)+\alpha_{0}(\vec{x}_{*}(t)-\vec{x}_{n}(t))\). **Differential Crossover (DC):** This procedure always succeeds the **DM** operation in the basic DE implementation.