## 4 Modified LebTLBO (mLebTLBO) Algorithm  
LebTLBO has recently attracted the attention of scholars because of its linear character. The refined LebTLBO aims to boost its effectiveness by enhancing its both exploration and exploitation abilities. Exploration would be aided by the use of search equations stimulated by Grey Wolf optimization (GWO) [31], and exploitation would be aided by the use of Local Neighbourhood Search (LNS), which is based on understanding the best result discovered until now in the contemporary solution's small neighbourhood [32]. The explanation of the LNS concept is discussed below:  
In the teacher phase, LebTLBO improves the search space by using the current solution and local data. The local neighbourhood search model has been added to boost the search capabilities and fasten the convergence. The fundamental idea is to refine the contemporary result in the neighbourhood of the current result using the best solution observed so far. The history of the individual's neighbourhood is analyzed for updating the individual's location, and the graph of their connectivity is referred to as the neighbourhood structure. Assume \(X=(X_{1},\ X_{2},\ X_{3},\ldots,\ X_{NP})\) in the current LebTLBO population, and \(X_{i}(i\in[1,\ NP])\) is a vector. For any \(X_{i}\) vector, the radius \((2r+1<NP)\) neighbourhood is specified; that is, the \(X_{i}\) neighbourhood is made up of \(X_{i-r},\ldots,\ X_{i},\ldots,\ X_{i+r}\). For examination, we presume that the vectors are grouped into a ring topology based on their indices. The concept of a local neighbourhood model is depicted in Figure 3. As a result, the topology of the neighbourhood and the illustration of the collection of vectors are always static. The LNS model is defined in  
\[L_{i}=X_{i}+m\times\left(X_{n\_opt}-X_{i}\right)+n\times\left(X_{p}-X_{q}\right) \tag{7}\]  
where \(X_{n\_opt}\) is the best result in the \(X_{i}\) neighbourhood with \(p,q\in[i-r,i+r]\)\((p\
eq q\
eq i)\), with \(m,n\in rand()\) as the scaling factors. In the upgraded model of LebTLBO, the new ideal result is upgraded according to Equation (1) and executes the teacher phase,  
\[X_{i}^{t+1}=L_{i}^{t}+r\times\left(X_{k}^{t}-X_{m}^{t}\right) \tag{8}\]  
where \(L_{i}\) is the ideal solution upgraded by LNS, and \(X_{k}^{t}\) and \(X_{m}^{t}\) denote arbitrary solutions for the \(kth\) and \(mth\) learner, respectively, such that \(k\
eq m\) and \(r\) denotes the scaling factor, and \(r\in rand()\). To strengthen the performance of the current LebTLBO, the learning enthusiasm-based teacher phase is upgraded in the modified LebTLBO [10]. By using the mean of the first three learners' from the sets of learners, this new search equation is reinforced. These three results are per the best solution available, which is known as \(x_{teacher}\). The proposed equations in this regard are:  
\[ \begin{array}{l}x_{1}=x_{i}-A_{1}(C_{1}x_{teacher}-x_{i}),\\ x_{2}=x_{i}-A_{2}(C_{2}x_{teacher}-x_{i}),\\ \\ x_{3}=x_{i}-A_{3}(C_{3}x_{teacher}-x_{i}),\\ x_{GWO}=\frac{x_{1}+x_{2}+x_{3}}{3}\end{array}  \tag{9}\]  
where \(A_{1},\ A_{2},\ A_{3}\) and \(C_{1},\ C_{2},\ C_{3}\) refer to A and \(C\), respectively. \(A\), \(C\) are stated as  
\[ \begin{array}{l}A=2 ar_{1}-a\\ C=2\cdot r_{2}\end{array}  \tag{10}\]  
Here, \(a\) is a consequentially decreasing \([0,\ 2]\) random integer with iterations, and \(r_{1}\) and \(r_{2}\) are two uniformly distributed random numbers. These enable the search agents to expand across the search area, enhancing the LebTLBO's exploratory capabilities. The main concept for this adjustment has been influenced by GWO search equations [32]. The solution in GWO is built by regulating the parameters to get the ideal search agent location. To obtain a new solution, the same description is employed:  
\[ \begin{array}{l}x_{i,new}^{d}=\left\{\begin{array}{l}x_{GWO}^{d}+rand_{2} \Big{(}x_{teacher}^{d}-T_{F}x_{mean}^{d}\Big{)}\ if\ rand_{1}<0.5\\ x_{r_{1}}^{d}+F\Big{(}x_{r_{2}}^{d}-x_{r_{3}}^{d}\Big{)}\ otherwise\end{array}\right.\end{array}