## 4 Proposed Framework
### Optimization Using Enhanced Chaotic Sparrow Search Optimization Algorithm (ECSSOA)  
The Enhanced chaotic sparrow search algorithm (ECSSOA) is a kind of smart program that has been used in several sectors because of its distinctive qualities, including a strong capacity to do global searches, a limited number of programmable variables, and a concise architecture [28]. The density matrices are illustrated in Equation (6), determining that there are \(N\) sparrows in the D-dimensional area (1). \[X=[x_{1},x_{2},\ldots x_{N}]^{T},x_{i}=[x_{i}1,x_{i},2,\ldots x_{i}D] \tag{6}\]  
where \(x_{i}\), \(D\) denotes the \(i\)th sparrow's location in dimension \(D\). Operations generally comprise 10% to 20% of the total populace, and Equation (7) updates the position. \[x_{i,j}^{t+1}= \begin{cases}x_{i,j}^{t}exp\Big{(}\frac{-i}{\pi\cdot iter_{mm}} \Big{)}A_{2}<CD\\ \quad\quad\quad x_{i,j}^{t}+Q\cdot LA\geq CD\end{cases}  \tag{7}\]where \(t\) is the present iteration count. \(j=1,\ 2,\ \cdots,\ d\). The highest number of repetitions is represented by itermax. \(\alpha\) is a deterministic randomized number between 0 and 1. The warning and security levels for sparrows are represented by \(A2\ (A2\in[0,\ 1])\) and \(CD\ (CD\in[0.5,\ 1.0])\), correspondingly. The randomized number \(Q\) has a regular dispersion, as expected. Each member of the matrix \(L\), which has the dimensions \(d\), is 1. The product enters a wide-area searching mode when \(A2<CD\) signals that there are no biological competitors nearby and that the region is mainly secure. If \(A2>CD\) indicates that the producer is conscious of a biological competitor present, then the producer must move to a different region to feed. Equation (8) causes an update to the follower location. \[x_{i,j}^{t+1}= \begin{cases}Q\cdot exp\left(\frac{x_{worst}^{t}-x_{i,j}^{t}}{ \mu^{2}}\right)i>\frac{u}{2}\\ x_{p}^{t+1}+\left|x_{i,j}^{t}-x_{p}^{t+1}\right|\cdot A^{+}\cdot\textit{LOtherwise} \end{cases}  \tag{8}\]  
where \(x_{worst}^{t}\) denotes the location of the sparrow with the poorest adaptation. The number xp represents the location of the sparrow with the best production adaptation. Each element of the matrix represented by the letter \(A\) is given a randomized value of one or zero. \(A+=AT\ (AAT)-1\). In addition, \(i>\frac{u}{2}\) indicates that the ith sparrow is probably starving and must go to a different place to graze. Chaos is a chaotic phenomenon that occurs naturally and has been used to improve computations [29; 30]. Due to its chaotic and harmonic properties, it enhances the variety of populaces and makes it easier for the program to depart from the locally optimal. A common chaotic transfer is a cubic translation, and Equation (9) illustrates its standardized format. \[x_{n+1}=bx_{n}^{3}-cx_{n} \tag{9}\]  
where the chaotic effect factors are \(b\) and \(c\). The series produced by the cubic mapping is the chaotic series for c (2.3, 3). The highest Lyapunov exponents were calculated for 16 typical chaotic maps, including the Cube map, and the Quadratic map-based equation was modified. According to the research findings, a Cubic map has less chaotic effect than one-dimensional maps like Sine mapping and Circle mapping but is superior to those worm face maps and tented maps. Equation (10) shows the adjusted Cube map the ECSSOA uses for initialization of the inhabitant's solution. \[x_{n+1}=\rho x_{n}\Big{(}1-x_{n}^{2}\Big{)} \tag{10}\]  
where \(\rho\) is the control variable. A higher persistence value is required in the initial repetitions to extend the discoverer's worldwide searching area since the production undertakes worldwide research as rapidly feasible in the first repeats to identify the worldwide ideal option swiftly. Simultaneously, a reduced persistence value is required in the final repetitions to enhance the discoverer's localized extraction capabilities, in addition to accelerating divergence and preventing settling on the local optimum answer.