# A Shadowed Type-2 Fuzzy Approach for Crossover Parameter Adaptation in Differential Evolution
## 2 Shadowed Type-2 Fuzzy Systems  
Fuzzy logic, initiated in 1965 [31; 32], is mainly based on the way we perceive the world that cannot always be defined in terms of true or false sentences, that is, it is logically applied to concepts that can take a value and any truth within a set of values. The evolution of fuzzy logic over time was type-1 fuzzy systems (T1-FS), then intervals type-2 fuzzy systems (IT2-FS), and later generalized type-2 fuzzy systems (GT2-FS); the mathematical equations of each of these systems are presented in Equations (1)-(3) respectively:  
\[\overset{\sim}{A}=\{(x,\mu_{A}(x))|x\in U,[0,1]\} \tag{1}\]  
In Equation (2) \(J_{x}\) is referred to the primary membership of \(x\) in \(\overset{\sim}{A}\). For each value of \(x\), denoted as \(x=x^{\prime}\), the two-dimensional (2D) plane formed by the axes \(u\) and \(\mu_{\overset{\sim}{A}}(x^{\prime},u)\) is known as a vertical slice of \(\overset{\sim}{A}\). A secondary membership function corresponds to a vertical slice of \(\overset{\sim}{A}\), denoted as \(\mu_{\overset{\sim}{A}}(x^{\prime},u)\), where \(x^{\prime}\in X\) and \(\forall_{u}\in J_{x^{\prime}}\subseteq[0,1]\). It can be described as follows:  
\[\tilde{\text{A}}=\{(x,u),u_{\tilde{\text{A}}}(x,u)|\forall x\in X,\forall u\in J _{x}\subseteq[0,1]\} \tag{2}\]  
A generalized type-2 fuzzy set, denoted by \(\overset{\sim}{A}\), is defined by a type-2 membership function \(\mu_{\overset{\sim}{A}}(x,u)\), where \(x\in X,u\in J_{x}\subseteq[0,1]\), and \(0\leq\mu_{\overset{\sim}{A}}(x,u)\leq 1\). It can be represented by:  
\[\overset{\sim}{A}=\left\{\left((x,u),\mu_{\overset{\sim}{A}}(x)\right)\left| \forall x\in X,\forall u\in J_{x}^{u}\subseteq[0,1]\right.\right\} \tag{3}\]  
Figure 1 depicts the type-1 and interval type-2 fuzzy sets, while Figure 2 illustrates the generalized type-2 fuzzy set, where the orange and yellow colors represent the primary membership function of a trapezoidal, the green color represents the secondary membership function of a Gaussian, and finally the blue color represents the fraction of uncertainty of the kernel and the support of the secondary membership function respectively. It is widely recognized that the utilization of generalized fuzzy systems entails a significant computational cost for their execution. In order to mitigate this, alternative approaches have been proposed that offer lower computational overhead. These alternatives include Z slices or vertical slices, as well as geometric approximations [33]. Additionally, the literature presents another option known as horizontal sections or \(\alpha\)-representation of planes [34; 35; 36]. Taking into account the alternatives discussed in the literature, this paper investigates the utilization of \(\alpha\)-planes. Equations (4) and (5) outline the structure of an \(\alpha\)-plane and the combination of an IT2-FIS with an \(\alpha\)-plane to model a GT2-FS, respectively. \[\overset{\sim}{A}_{\alpha}=\{((x,u),\alpha)|\forall x\in X,\forall u\in J_{x} \subseteq[0,1]\} \tag{4}\]  
\[\overset{\sim}{A}=\bigcup\overset{\sim}{A}_{\alpha} \tag{5}\]  
The elimination of excessive precision is one of the main ones that is sought to be carried out when using a ST2-FS, since this helps to reduce the computational cost. To achieve the elimination of excessive precision, a GT2-FS is modeled using only 2 \(\alpha\)-planes; this idea was  
Figure 1: Illustration of the type-1 and interval type-2 fuzzy sets. Figure 2: Illustration of the general type-2 fuzzy set. proposed in Pedrycz concepts, thus creating the theory of shadowed sets [37; 38; 39; 40], where the main contribution is the creation of optimal \(\alpha\)-planes. Equation (6) explains the mathematical form of the shadowed sets; this formula is built from 2 \(\alpha\)-cuts on a T1-FS, with \(\alpha\) and \(\beta\) values, producing three intervals. \[S_{\mu_{A}}(x)=\left\{ \begin{array}{cl}1,&if\ \mu_{A}(x)\geq\alpha\\ 0,&if\ \mu_{A}(x)\leq\beta\\ [0,1],&if\ \alpha\leq\mu_{A}(x)\geq\beta\end{array} \right.