## 3 Proposed Naked Mole-Rat Algorithm Variant
### Naked Mole-Rat Algorithm
The updated, enhanced NMRA (ENMRA) aims to improve efficiency by enhancing its fundamental exploitation and exploration capabilities. Exploration is enriched with the implementation of GWO-inspired equations [10] in basic NMRA, and local neighborhood search (LNS), along with DE equations, are used for enhanced exploitation [36]. The notion of LNS is addressed as follows. _Local Neighborhood Search (LNS)._ To boost the search space in the worker phase, NMRA employs the current solution and local information. In addition, the local neighborhood search (LNS) model is implemented to enhance the search functionality with increased convergence speed [37]. The key concept is to update the old solution to the new one in a small neighborhood using the best solution. The initial population \(X=\left(X_{1},\ X_{2},\ X_{3},\dots,\ X_{n}\right)\), \(X_{i}(i\in[1,\dots,\ n])\) is a vector (\(D\)-dimensional). The neighborhood radius \(r\) (where \(2r+1<n\)) is defined for each vector \(X_{i}\); that is, \(X_{i}\) neighborhood comprises of \(X_{i-r},\ \dots,\ X_{i},\ \dots,\ X_{i+r}\). The vectors here are structured according to their indices into a ring topology [35]. Figure 1 represents the notion of the LNS model, which is defined as:

\[L_{i}=X_{i}+m*\left(X_{n\_opt}-X_{i}\right)+n*\left(X_{p}-X_{q}\right) \tag{4}\]

where \(X_{n\_opt}\) is the optimum solution so far in the \(X_{i}\); \(m\), \(n\) \(\in\) rand () are the scaling factors and \(p\), \(q\in[i-r,\ i+r]\) (\(p\
eq q\
eq i\)). In the improved NMRA, the best solution is modified using Equation (4), and the modified solution executes the worker phase:

\[x_{i}^{t+1}=L_{i}^{t}+r*\left(x_{k}^{t}-x_{m}^{t}\right) \tag{5}\]

where \(r\)\(\in\) rand() is a scaling factor, \(L_{i}\) is the best solution modified using LNS, and \(x_{k}^{t}\)_and_\(x_{m}^{t}\) are randomly selected vectors for \(k\)th and \(m\)th NMR with \(k\
eq m\). Figure 1: Neighborhood ring topology of radius 3. _Worker Phase._ The worker phase is upgraded in the enhanced NMRA to improve the performance of conventional NMRA. Here, the total iterations are divided into two portions. The regular NMRA's general Equation (2) is utilized for the latter part of the iterations, and a new search equation is introduced for the first half. This equation is supported by considering the mean of the first three solutions that are created based on the best solution available and is represented as:

\[w_{1}=w_{i}-A_{1}(C_{1}\cdot d-w_{i}),\ w_{2}=w_{i}-A_{2}(C_{2}\cdot d-w_{i}), \ w_{3}=w_{i}-A_{3}(C_{3}\cdot d-w_{i}) \tag{6}\]

\[w_{new}=\frac{w_{1}+w_{2}+w_{3}}{3} \tag{7}\]

The main idea for this adjustment came from GWO search equations [10]. The solution in GWO is built by tweaking the parameters to the best search agent position. GWO search equations help to improve the randomness in the algorithm's worker phase search equations, and hence to overcome the poor exploration of conventional NMRA. To obtain a new solution, the same definition is employed:

\[w_{i}^{t+1}=w_{new}^{t}+\lambda(w_{j}^{t}-w_{k}^{t}) \tag{8}\]

where \(A_{1}\), \(A_{2}\), \(A_{3}\) and \(C_{1}\), \(C_{2}\), \(C_{3}\) belong to \(A\) and \(C\), respectively, and are given as:

\[A=2 ar_{1}-a,\ C=2\cdot r_{2} \tag{9}\]

where \(a\) is a random number that is linearly decreasing [0, 2] in relation to iterations and \(r_{1}\) and \(r_{2}\) are the uniformly distributed random numbers. The random number allows the agents to move around within the search space at any location, enhancing the NMRA's exploration capabilities. _Breeder Phase._ In this phase, to enhance the exploitation efficiency of NMRA, the population is separated into three sections, and the final solution is evaluated using three separate search strategies. First, the general equation similar to standard NMRA is used as follows for the first part of the population:

\[b_{i}^{t+1}=\left\{ \begin{array}{ll}b_{new}^{t}+\lambda(b_{j}^{t}-b_{k}^{t}),\ \ if\ 0\geq t\geq\ t_{max}/2\\ b_{i}^{t}+\lambda(b_{j}^{t}-b_{k}^{t}),\ \ \ \ if\frac{t_{max}}{2}+1\geq t\geq\ t_{max} \end{array} \right.