## 8 Improved Chaos-Driven Hybrid Differential Evolution and Butterfly Optimization Algorithm
The use of the following two mutation operators is proposed  
* The usage of DE/best/\(1\) mutation strategy with adaptive scale factor \(F_{i}^{(\text{G})}\) from Equation (35) is proposed, which can be expressed as \[\mathbf{v}_{i}^{(\text{G}+1)}=\mathbf{x}_{best}^{(\text{G})}+F_{i}^{(\text{G}) }\Big{(}\mathbf{x}_{r_{1}}^{(\text{G})}-\mathbf{x}_{r_{2}}^{(\text{G})}\Big{)}.\] (59)
* A new DE/best-local-BOA/2, produced by hybridizing the mutation operator DE/best/2 form DE algorithm with the local search step of BOA algorithm given in Equation (46). In this regard, the term \(F\Big{(}\mathbf{x}_{r3}^{(\text{G})}-\mathbf{x}_{r4}^{(\text{G})}\Big{)}\) from the Equation (31) is replaced with the term \(\Big{(}r^{2}\times\mathbf{x}_{i}^{(\text{G})}-\mathbf{x}_{k}^{(\text{G})} \Big{)}\times\varphi_{i}\) from Equation (46). Furthermore, the adaptive scale factor \(F_{i}^{(\text{G})}\) from Equation (35) and sine chaos map-based sensory fragrance \(c^{(\text{G})}\) from Equation (48) have been introduced, providing the following expression \[\mathbf{v}_{i}^{(\text{G}+1)}=\mathbf{x}_{best}^{(\text{G})}+F_{i}^{(\text{G} )}\Big{(}\mathbf{x}_{r1}^{(\text{G})}-\mathbf{x}_{r2}^{(\text{G})}\Big{)}+ \Big{(}r^{2}\times\mathbf{x}_{j}^{(\text{G})}-\mathbf{x}_{k}^{(\text{G})} \Big{)}\times\varphi_{i}.\] (60)  
A lot of research has been proposed, which deals with the introduction of new mutation operators into the DE algorithm. In JADE, authors proposed using the DE/current-to-p-best/\(1\) mutation scheme to improve optimization performance [33]. Moreover, authors proposed using multiple mutation strategies [75]. In this way, the suggested technique presents a novel triangular mutation rule based on the convex combination vector of the triplet created by the three randomly selected vectors and the difference vector between the best and poorest individuals from the three randomly picked vectors. Combining the mutation rule with the standard mutation approach via a non-linear decreasing probability rule, authors proposed using a reflection-based mutation operation inspired by the Nelder-Mead technique and two additional fundamental mutation strategies to preserve population diversity [76]. In order to choose mutation strategies, their probabilities of success for each individual are employed using a roulette wheel selection method. Recently, the authors proposed hybridization between DE and FA algorithms with multiple mutation strategies, where the FA is introduced into the mutation scheme of DE and then randomly applied during the evolution process [16]. In this paper, we proposed using the roulette wheel selection, in order to perform the choice of the mutation operator, which is appropriate to the current optimization stage. Initially, each of the mutation operators are assigned random success rates \(s_{r}\) in the range \([0,1]\) such that \(\sum s_{r,k}=1,\forall k=1,2,3,4\). During the evolution process, the \(k\)th mutation operator, which is used to produce mutant vector \(\mathbf{v}_{i}^{(\text{G})}\) of the \(i\)th individual, is assigned value \(1\) if the produced mutant vector lead to the improvement in the optimization process, e.g., the success rate is calculated as follows  
\[s_{r,k}^{(\text{G})}= \begin{cases}1,&\text{if}f\Big{(}\mathbf{v}_{i}^{(\text{G })}\Big{)}<f\Big{(}\mathbf{x}_{i}^{(\text{G})}\Big{)}\\ 0,&\text{otherwise}\end{cases} . \tag{61}\]  
Then, the total probability of choosing the appropriate mutation scheme is calculated as follows  
\[p_{k}^{(\text{G})}=\sum_{i=1}^{N_{p}}\frac{s_{r,k}^{(\text{G})}}{N_{p,k}}, \tag{62}\]  
where \(N_{p,k}\) denotes the total number of individuals that are assigned the \(k\)th mutation operator in the current generation. Therefore, if the value of \(p_{k}\) of \(k\)th mutation operator is higher, the probability of choosing that mutation scheme increases. In this way, the pseudocode of the proposed ICDEBOA algorithm is presented in Algorithm 1, for the considered passive target localization problem.