## 4 Proposed Binary HHO
In this way, the shape of the TF changes over time based on the value of the controlling parameter. The purpose of employing the time-varying scheme is to obtain a better balance between exploration and exploitation through the optimization process of a BPSO. Time-varying TFs demonstrated their superiority when compared to existing static TFs based on BPSO approaches over low-dimensional and high-dimensional discrete optimization problems. Inspired by the work of [32; 34], [PERSON] et al. [35] proposed eight time-varying TFs related into two families (S-shaped and V-shaped) for developing binary versions of DA (BDA) to be used for FS. The authors demonstrated the efficiency of these time-varying TFs by comparing their performance with other static TFs as well as various wrapper-based FS approaches. In addition, three types of time-varying transfer functions were introduced in [36] for improving the performance of the binary WOA in the FS domain. WOA with time-varying TFs has shown higher effectiveness and efficiency than other popular approaches in the FS domain. In this work, considering the previous studies of the impact of TFs on the performance of binary optimization algorithms, we select the time-varying TFs, specifically V-shaped, proposed by [35], as shown in Table 1, to convert HHO to binary and apply the binary variants of HHO to the FS problem. In the time-varying form of the TFs, \(\tau\) represents a time-varying variable that begins with an initial value and progressively reduces over iterations, as shown in Equation (14)\[\tau=\tau_{max}-(\tau_{max}-\tau_{min})\times\frac{t}{T} \tag{14}\]  
where \(\tau_{min}\) and \(\tau_{max}\) represent the bounds of the \(\tau\) parameter, \(t\) denotes the current iteration, and \(T\) represents the maximum number of iterations. In this study, \(\tau_{min}\) and \(\tau_{max}\) were selected to be 0.01 and 4, respectively [35]. The original time independent V-shaped TFs are shown in Figure 2, while the time varying variants of TFs are shown in Figure 3.  
After employing the original or time-varying TFs as a first step in the binarization scheme, the real-valued solution \(\mathbb{R}^{n}\) is converted into an intermediate probability vector \([0,1]^{n}\) such that each of its element determines the probability of transforming its equivalent in \(\mathbb{R}^{n}\) into 0 or 1. In the second step, a binarization rule is applied to transform the output of TFs into a binary solution [30]. In this work, the complement binarization introduced by [PERSON] et al. [33] is applied as given in Equation (15). \[X_{j}(t+1)=\left\{ \begin{array}{ll}\backsim b_{j}&r<T(X_{j}(t))\\ b_{j}&Otherwise\end{array} \right. \tag{15}\]  
where \(\backsim\) denotes the complement, \(b_{j}\) is the current binary value for the \(j\)th element, and \(X_{j}(t+1)\) is the new binary value. It is noted that the updated binary value is set considering the current binary solution, that is, based on the probability value \(T(X_{j}(t))\), the \(j\)th element is either kept or flipped. Algorithm 1 explains the pseudo-code of the Binary HHO algorithm. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Original Family** & \multicolumn{2}{c}{**Time-Varying Family**} \\ \hline
**Name** & **Transfer Function** & **Name** & **Transfer Function** \\ \hline V1 & \(T(x)=|\text{erf}(\frac{\sqrt{\Pi}}{2}x)|\) & TV1 & \(T(x,\tau)=|\text{erf}(\frac{\sqrt{\Pi}}{2}\frac{x}{\tau})|\) \\ V2 & \(T(x)=|\tanh(x)|\) & TV2 & \(T(x,\tau)=|\tanh(\frac{x}{\tau})|\) \\ V3 & \(T(x)=|(x)/\sqrt{1+x^{2}}|\) & TV3 & \(T(x,\tau)=|(\frac{x}{\tau})/\sqrt{1+(\frac{x}{\tau})^{2}}|\) \\ V4 & \(T(x)=|\frac{2}{\Pi}\text{arc}\tan(\frac{\Pi}{2}x)|\) & TV4 & \(T(x,\tau)=|\frac{2}{\Pi}\text{arctan}(\frac{\Pi}{2}\frac{x}{\tau})|\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Original and time-varying V-shaped transfer functions. Figure 2: V-shaped transfer functions.