## 4 The Development of the MRFCO-ZTS Approach
### MRFCO-ZTS Approach
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Meaning** \\ \hline \(a_{i}\) & Attack likelihood score for user/session \(i\) (0 to 1) \\ \hline \(r_{i}\) & Risk impact if misclassification occurs (0 to 1) \\ \hline \(d_{i}\) & Computational delay or verification latency for \(i\) (in milliseconds) \\ \hline \(m_{i}\) & Misclassification penalty, 1 if misclassified, 0 otherwise \\ \hline \(\alpha,\beta,\gamma\) & Tunable weights for security, performance, accuracy \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbol definition. Using Equations (1) and (8), this can be represented as  
\[\text{J}(.)=\underset{\theta}{\text{min}}\sum_{i=1}^{M}[\alpha\cdot a_{i}\cdot r_{ i}+\beta\cdot d_{i}+\gamma\cdot m_{i}] \tag{9}\]  
In Equation (9), \(\theta\) indicates the parameters of the classification model. With the motivation of optimizing the cost function of the ZTS model for the classification of attacker/detector, a new metaheuristic algorithmic concept called Manta Ray Foraging Cost Optimization (MRFCO) is applied, which balances security, performance, and resource usage in the satellite network environment. The MRFCO is implemented based on the foraging behavior of manta rays in ocean environments. To apply MRFCO under a ZTS model for optimizing a cost function, we must align MRFO's exploration-exploitation procedure with the security objectives of ZTS, including the following: (1) finding attackers vs. detectors, (2) reducing computational delay, (3) minimizing risk while maintaining security assurance, and (4) supporting dynamic authentication decisions. In ZTS, every access request of satellite users must be authenticated continuously. This leads to complex decision-making, often involving dynamic risk scores, delay-sensitive authentication, and adaptive access control (deny, grant). Hence, an optimized cost function assists in balancing performance and achieving security. In MRFCO, manta rays contain distinctive foraging plans to find plankton and small fish using three key behaviors, i.e., chain foraging, cyclone foraging, and somersault foraging. Each signifies different phases of exploration and exploitation. Let us assume that \(z_{i}\) indicates the position of the \(i^{th}\) manta ray (solution), \(z_{best}\) defines the global best solution, \(t\) defines the current iteration, and \(T\) represents maximum iterations. Next, the different types of foraging plans considered in our developed MRFCO-ZTS are represented, as follows:  
1. _Chain foraging_: Create sequential foraging using the mathematical formulation below: \[z_{i}^{t+1}=z_{i}^{t}+\alpha_{1}\cdot(z_{best}^{t}-z_{i}^{t})+\alpha_{1}\cdot( z_{i-1}^{t}-z_{i}^{t})\] (10) In Equation (10), \(\alpha_{1},\alpha_{2}\sim U(0,1)\), which is good for global exploration. 2. _Cyclone Foraging_: Forms circular swimming to encircle, using the formula below: \[z_{i}^{t+1}=z_{i}^{t}+r\cdot(z_{best}^{t}-z_{i}^{t})\cdot\sin(\frac{\pi t}{T})\] (11) The above equation supports convergence and spiral search toward the best position. 3. _Somersault Foraging_: Local fine-tuning through somersault maneuvers, using the formula below: \[z_{i}^{t+1}=z_{i}^{t}+S\cdot(\alpha_{1}\cdot z_{best}^{t}-\alpha_{2}\cdot z_{ i}^{t})\] (12) In Equation (12), \(S\) refers to the somersault factor (e.g., 2). Using this exploitation-focused method helps refine the solutions and thus minimize the cost function of the ZTS approach for accurate classification of users. To better recognize the possible foraging, a novel classification rule has been developed and used in the developed MRFCO-ZTS approach. Based on cost minimization, the classification decision rule for satellite user '\(i\)' is mathematically described as  
\[\hat{y}=\left\{ \begin{aligned} &\text{If }\alpha\cdot A_{i}\cdot R_{i}+\beta\cdot D_{i}+\gamma\cdot m_{i}>t_{u} \text{, then }\hat{y}=1\text{ and user }i\text{ is classified as attacker}\\ &\text{Otherwise, then }\hat{y}=0\text{ and user }i\text{ is classified as detector}\end{aligned} \right.