## 3 Proposed Algorithm
### Evolutionary Programming Sine Cosine Algorithm  
The evolutionary programming sine cosine algorithm (EPSCA) is a hybrid technique that combines the features of the EP algorithm and SCA. The concept of mutation was implemented by modifying the existing features of the SCA technique. This concept has  
Figure 5: Correlation of eigenvalue sectors (pre- and post-optimised MO). four more steps in the SCA algorithm. In the first step, a random number \(v_{5}\) ranging from zero to one is updated. In the second step, an updated and best solution given by SCA is mutated based on two techniques: Levy flight or random number. In the third step, the mutated candidate's solutions and clone solutions are combined. The algorithm then updates each candidate solution. In the last step, half of the candidates are selected based on optimal solutions, and the best optimal solution is chosen. The varying positions for both phases are represented by the equation below:  
\[Y_{i,j}^{u+1}=Y_{i,u}+v_{1}^{u}\times\sin v_{2,j}\times\left|v_{3,j}R_{d,j}^{u} -Y_{i,j}^{u}\right| \tag{50}\]  
\[Y_{i,j}^{u+1}=Y_{j,u}+v_{1}^{u}\times\cos v_{2,j}\times\left|v_{3,j}R_{d,j}^{u} -Y_{i,j}^{u}\right| \tag{51}\]  
Equations (50) and (51) are combined to be applied as follows:  
\[Y_{i,j}^{u+1}= \begin{cases}Y_{i,u}+v_{1}^{u}\times\sin v_{2,j}\times\left|v_{3,j}R_{d,j}^{u}-Y_{i,j}^{u}\right|,\ \sigma_{4,j}<0.5\\ X_{j,u}+v_{1}^{u}\times\cos v_{2,j}\times\left|v_{3,j}R_{d,j}^{u}-Y_{i,j}^{u} \right|,\ \sigma_{4,j}\geq 0.5\end{cases}  \tag{52}\]  
where \(u\) denotes the current iteration; \(R_{d,j}^{u}\) is the location of the destination in the \(j\)th dimension; \(Y_{i,u}\) is the current solution at the \(u\)th iteration; \(v_{1}^{u}\), \(v_{2,j}\), and \(v_{3,j}\) are random values; \(v_{4,j}\) is a random number between 0 and 1; and \(\mid\mid\) is an absolute value. SCA has four main parameters--\(v_{1}^{u}\), \(v_{2,j}\), \(v_{3,j}\), and \(v_{4,j}\)--as shown by the aforementioned equations. The parameter \(v_{1}\) determines the subsequent location regions (or movement direction), which may be inside or outside the region between the solution and the destination. The parameter \(v_{2,j}\) specifies how far the movement should extend in either direction from the target. The value of the parameter \(v_{3}\) determines random weights for the destination to stochastically emphasise or deembasse the role of the destination in determining the distance. The parameter \(r_{4}\) then alternates equally between the sine and cosine parts of Equation (52). Equation (53), which is written as follows, changes the sine and cosine ranges to find a favourable region in a given domain and converge it to the universal optima, thereby allowing the SCA algorithm to maintain balance during exploitation and exploration:  
\[v_{1}^{u}\text{= b}-\text{u}\ \frac{b}{U} \tag{53}\]  
where \(U\) is the maximum number of iterations, \(b\) is a constant value, and \(u\) represents the current iteration. Levy flight is a generalised Brownian motion random walk method, and the duration of its steps for distance travel follows a non-Gaussian random distribution known as the Levy stable distribution. The Levy distribution can be described mathematically as follows:  
\[L_{f}(\text{z},\text{q},\text{r})= \begin{cases}\sqrt{\frac{q}{2\pi}}\exp{( \frac{-q}{2(z-r)})}\frac{1}{(z-r)^{\frac{3}{2}}},0<r<\infty\text{ and }0<z<\infty\\ 0,z\leq 0\end{cases}  \tag{54}\]  
where the scale parameter \(q\) controls the distribution scale, and the location parameter \(r\) controls the location. The offspring's standard deviation is expressed as follows:  
\[\zeta_{k}\text{= K}.\frac{G(y_{l})}{G_{max}}(p_{k}^{max}-p_{k}^{min}), \tag{55}\]  
where each parent \(y_{l}\) produces one offspring \(y_{l+n}\). Each optimised parameter \(p_{k}\) is perturbed by a Gaussian random variable \(N(0,\zeta_{1}^{2})\). \(y_{l}\) is the trial solution of fitness equation \(G(y_{l})\), and K is a scaling factor. \(G_{max}\) represents the maximum value of the fitness function.