# A Distributed Energy-Throughput Efficient Cross-Layer Framework Using Hybrid Optimization Algorithm
## 2 Hybrid Marine Predator Naked Mole Rat Algorithm (MPNMRA)
The constant value \(P=0.5\) is a fixed number, and \(R\) represents a single vector of random numbers ranging from 0 to 1. This situation arises during the initial one-fourth of iterations, when the size or velocity of movement is relatively small, resulting in limited exploration capability. In this context, \(iter\) refers to the current iteration, while \(Max\_iter\) represents the maximum number of iterations. Step 2: For \(\frac{1}{4}Max\_iter<iter<\frac{1}{2}Max\_iter\)  
\[stepsize_{i}=\overrightarrow{R_{L}}\otimes\left(\overrightarrow{Elite_{i}} -\overrightarrow{R_{L}}\otimes\overrightarrow{Prey_{i}}\right)i=1,\ \ldots,\ n\]  
\[Prey_{i}=\overrightarrow{Prey_{i}}P.\overrightarrow{R}\bigotimes\overrightarrow {stepsize_{i}} \tag{3}\]  
The vector \(\overrightarrow{R_{L}}\) represents a random number vector that follows the Levy distribution, which describes [PERSON]'s movement pattern. Multiplying \(\overrightarrow{R_{L}}\) with the prey simulates the prey's movement in a Levy-like manner. Additionally, applying the stage scale to the prey's location helps simulate its movement. Since a significant portion of [PERSON]'s movement is associated with small distances, this step primarily focuses on exploitation rather than exploration. Step 3: For \(\frac{1}{2}Max\_iter<iter<\frac{3}{4}Max\_iter\)  
\[\stackrel{{\rightarrow}}{{stepsize_{i}}}=R_{B}^{\rightarrow} \otimes\left(\stackrel{{\rightarrow}}{{R_{B}}}\otimes\stackrel{{ \rightarrow}}{{Elite}}_{i}-\stackrel{{\rightarrow}}{{Prey_{i}}} \right)\ i=1,\ldots,n\]  
\[\stackrel{{\rightarrow}}{{Prey_{i}}}=Elite_{i}+P.CF\ \otimes\ \stackrel{{ \rightarrow}}{{stepsize_{i}}} \tag{4}\]  
The variable \(CF\) is an adaptive parameter that controls the step size for predatory movement, and is calculated as:  
\[CF=\left(1-\frac{iter}{Max\_iter}\right)^{\left(2 iter/Max\_iter\ \right)} \tag{5}\]  
This parameter adjusts dynamically based on the current iteration and the maximum number of iterations. By multiplying \(\stackrel{{\rightarrow}}{{R_{B}}}\) with the Elite vector, predatory movements are simulated in a Brownian manner. Meanwhile, the prey updates its location based on these predatory movements, following a Brownian motion pattern. Step 4: For \(\frac{3}{4}Max\_iter<iter<Max\_iter\)  
\[\stackrel{{\rightarrow}}{{stepsize_{i}}}=\stackrel{{ \rightarrow}}{{R_{L}}}\bigotimes\left(\stackrel{{ \rightarrow}}{{R_{L}}}\bigotimes\stackrel{{\rightarrow}}{{Elite_{i} }}-\stackrel{{\rightarrow}}{{Prey_{i}}}\right)\ \ i=1,\ \ldots,n\]  
\[\stackrel{{\rightarrow}}{{Prey_{i}}}=Elite_{i}+P.CF\bigotimes \stackrel{{\rightarrow}}{{stepsize_{i}}} \tag{6}\]  
By multiplying \(\stackrel{{\rightarrow}}{{R_{L}}}\) with the Elite vector, the movement of predators is simulated using [PERSON]'s strategy. Additionally, incorporating the step size to the Elite position helps simulate the movement of predators and assists in altering the prey's position. The generic equations have been included while maintaining the algorithm's original hierarchical organization. The new equations are used in a manner similar to that of the fundamental MPA method. In other words, according to the fundamental MPA, each equation is run for a specific number of iterations. Breeder Phase: There have been no changes made to the breeder phase for the MPNMRA; it is the same as that in the conventional NMRA breeder phase [17] and is expressed as:  
\[MN_{t+1}^{i}=(1-\lambda)MN_{t}^{i}+\lambda\left(\text{Elite}-MN_{t}^{i}\right) \tag{7}\]  
where \(MN_{t}^{i}\) represents the solution of the \(i\)th breeder rat in the \(i\)th iteration. The parameter \(\lambda\) regulates the mating frequency. On the next iteration, \(MN_{t+1}^{i}\) represents the newly developed solution. To update the fitness of these breeders, the breeding probability (\(bp\)) is used based on the initial best solution, which is represented by the Elite solution. The fundamental parameters for the MPA are \(p\), \(r\), and, \(a\), while the fundamental parameters for the NMRA are \(bp\) and \(\lambda\).