# A Distributed Energy-Throughput Efficient Cross-Layer Framework Using Hybrid Optimization Algorithm
## 2 Hybrid Marine Predator Naked Mole Rat Algorithm (MPNMRA)  
The marine predator algorithm (MPA) [16] and the naked mole-rat algorithm (NMRA) [17] are two notable advancements in the realm of nature-inspired algorithms. NMRA looks for a potential solution to the issue being studied, utilizing the mating behavior of naked mole rats with the queen, whereas MPA is governed by the various foraging methods among aquatic predators and prey. One thing unites both of these algorithms: they approach the problem under study by utilizing fundamental interactions and foraging patterns between various species. It is evident from the descriptions of MPA [16] and NMRA [17] that while both algorithms are effective, they have limitations with regard to MPA's exploitation and NMRA's exploration. The premature convergence of these algorithms leads to a stagnation of locally optimum solutions. [PERSON] et al. [18] developed a hybrid MPNMRA model that retains the basic NMRA architecture while modifying the worker phase and includes MPA's mathematical formulation. The breeder phase is left unattended. In the algorithm, the worker phase is intended for exploration, whereas the breeder phase is intended for exploitation. A hybrid MPNMRA with effective exploitation and exploration arises by combining the two algorithms. In addition to the MPA and NMRA hybridization, the parameters of both algorithms are modified using the simulated annealing \((sa)\) mutation operator [19] to make the algorithm's parameters self-adaptive. The algorithm is now self-sufficient and does not need user-based adjustment to evaluate the problem. The various phases of the hybrid MPNMR are discussed as follows:  
Initialization phase: The initialization phase of the MPNMRA begins with random initialization of search agents (preys and/or naked mole rats) within a specific range [19] and is expressed as  
\[MN_{i,j}=MN_{min,j}+U(0,1)\ \left(MN_{max,j}-MN_{min,j}\right) \tag{1}\]  
where \(i\in[1,2,3...n],j\in[1,2,3...D]\), \(MN_{i,j}\) represents the \(i\)th search agent solution for the \(j\)th dimension, \(MN_{max,j}\) and \(MN_{min,j}\) signify the upper and lower bound of the problem under study, and \(U(0,1)\) corresponds to a uniformly distributed random value in the range \([0,1]\). The fittest solution is considered as _Elite_. These initial solutions are then evaluated using the basic NMRA algorithm's original structure. The method is composed of the worker phase and the breeder phase, each of which is controlled by a set of arbitrary mathematical equations. However, the number of breeders and the proportion of workers in the population continue to be a cause for concern. Given that there are only a finite number of breeders and that they are required to mate with the queen (the optimal solution). Worker phase: From the search pool, two random solutions significantly contribute to the discovery of a nearly optimal solution during the worker phase. The worker phase in conventional NMRA is less trustworthy, and additional work has to be conducted to enhance its functional properties. The characteristic equations of MPA are thus introduced to improve the worker phase of the NMRA. Here, the worker phase of the MPNMRA has been expanded by including the generalized equations of MPA, mimicking the entire life of prey and predator for different velocity ratios as follows [16]:  
Step 1: For \(iter<\frac{1}{4}Max\_iter\)  
\[\overrightarrow{Prey_{i}}=\overrightarrow{Prey_{i}}+P.\overrightarrow{R} \otimes\overrightarrow{stepsize_{i}} \tag{2}\]  
where \(\overrightarrow{stepsize_{i}}=\overrightarrow{R_{B}}\otimes\left( \overrightarrow{Elite_{i}}-\overrightarrow{R_{B}}\otimes\overrightarrow{ Prey_{i}}\right)i=1,\ldots n\)  
The initial population \(MN\) is considered as _Prey_ in MPA. Here, the vector \(R_{B}\) represents a set of random numbers that follow a normal distribution centered around Brownian motion. The symbol \(\otimes\) denotes element-wise multiplication. The distribution of the prey mimics its movement pattern.