## 3 The Proposed Model
### Design of the CSRO Algorithm  
The position of the lost human is the primary motivation of the search and rescue optimization method for optimization problems, and the significance of the clue originating in this position defines the cost of the solution. Now, the best method discloses a good position with additional hints [59]. When leaving certain clues, people seek the best option across the search method. However, the search position for the individual is kept in a situation matrix (matrix \(X\)), with the equivalent size of the memory matrix, and the left clue can be saved in a memory matrix (matrix \(M\)). \(n\times d\) shows the problem parameter and \(n\) determines the individual quantity in the group. \[C=\left[ \begin{array}{c}X\\ M\end{array} \right]=\left[ \begin{array}{cccc}X_{1,1}&\ldots&X_{1,d}\\ \vdots&\ddots&\vdots\\ X_{n,l}&\ldots&X_{n,d}\\ M_{1,1}&\ldots&M_{1,d}\\ \vdots&\ddots&\vdots\\ M_{n,l}&\ldots&M_{n,d}\end{array} \right] \tag{11}\]  
From the above equation, assuming there are random clues amongst the obtained clues, the search direction can be attained by the following:  
\[sd_{i}=\left(X_{j}-C_{k}\right)\text{, Where }k\
eq i \tag{12}\]  
where \(C_{k}\) denotes a random value among 1 and 2\(N\), \(X_{i}\), and \(C_{k}\) defines the position of the \(i^{\text{th}}\) human and the \(k^{\text{th}}\) clue, respectively. \(sd_{i}\) indicates the search direction of the clues. It is noticed that \(C_{k}\) equals \(X_{j}\), \(k\
eq i\). For avoiding repetitive position searches, the parameter of \(X_{i}\) will not be modified by moving in the indicated direction [60; 61; 62; 63; 64]. The SAR approach uses a binomial crossover operator for applying to the limitations. Moreover, when the clue has greater significance compared with the current clue, a region was searched for the \(sp_{i}\) direction. Otherwise, a search for the location of the existing position in the \(sp_{i}\) is continuous. Therefore, the novel position of the \(j^{\text{th}}\) variable is expressed by the \(i\)th human, as follows:  
\[X_{i,j}=\left\{ \begin{array}{l}\left|C_{k,j}+r_{1}\times\left(X_{i,j}-C_{k, j}\right)\text{ }if\text{ }f(C_{k})>f(X_{i})\\ X_{i,j}+r_{2}\times\left(X_{i,j}-C_{k,j}\right)\text{ }if\text{ }r_{2}<sp_{i}\text{ or }j=j_{r},j-1, \ldots,d\\ X_{i}\text{ otherwise}\end{array} \right. \tag{13}\]  
where \(c_{k,j}\) denotes the position of variable j and the clue \(k\). \(j_{r}\), \(r_{1}\), and \(r_{2}\) represent three uniform random numbers within \([1,\ d]\), \([-1,1]\), and \([0,1]\), respectively. The second stage is about the individual. Here, an exploitation term has been performed regarding the human's current location [65; 66; 67; 68]. This stage employs the distinct clues connection concept from the social stage. The position, upgraded by the human, \(i\), can be attained as follows:  
\[X_{i}=X_{j}+r_{3}\times\left(C_{k}-C_{m}\right) \tag{14}\]  
where \(r_{3}\) denotes a uniformly distributed number between 0 and 1, \(C_{m}\) and \(C_{k}\) define two arbitrary numbers between 1 and 2 \(N\), respectively, and \(i\
eq k\
eq m\). They could testify that they are in the solution space afterward, solving the solution from the preceding stages. This phase is named the boundary. In such cases, the following formula is utilized when the solution is placed outside the border:  
\[X^{\prime}_{ij}=\left\{ \begin{array}{l}\frac{\left(X_{ij}+X^{\text{max}}_{j} \right)}{2}\;if\;X^{\prime}_{ij}>X^{\text{max}}_{j}\\ \frac{\left(X_{ij}+X^{\text{min}}_{j}\right)}{2}\;if\;X^{\prime}_{ij}<X^{\text{ min}}_{j}\end{array} \right. \tag{15}\]  
where \(j=1,2,\ldots,d\), \(X^{\text{min}}_{j}\), and \(X^{\text{max}}_{j}\) represent the minimal and maximal thresholds for the parameter \(j\), respectively. According to this stage, the lost human candidate is searched for on the basis of the previously elucidated technique.