## 8 Improved Chaos-Driven Hybrid Differential Evolution and Butterfly Optimization Algorithm  
In Section 4, it has been demonstrated that the considered passive target localization problem is a non-convex complex problem, which cannot be solved using conventional algorithms. According to a review of the literature, the DE algorithm often performs well in a global search but poorly in a local search [32]. However, applying these tactics throughout the whole evolution process might result in premature convergence. The authors have suggested utilising DE/best/1, DE/best/2, and more recently DE/current-to-p-best/1 mutation strategies to enhance local search [33]. Conversely, BOA algorithm shows excellent local search ability, however, it often prematurely converges to local optima of complex problems [37]. Therefore, the main idea of this paper is to combine good local search abilities of BOA with excellent global search of DE algorithm. In the proposed ICDEBOA algorithm, this is achieved by introducing the BOA particle movement equations into the mutation strategies of DE algorithm. Furthermore, the improvements of DE and BOA algorithms, outlined in the previous subsections, are introduced into the ICDEBOA algorithm.Therefore, in order to improve the exploration abilities of the proposed ICDEBOA algorithm, in this paper we propose the uses of the following two mutation strategies for exploration:  
* DE/rand/1 mutation strategy with adaptive scale factor \(F_{i}^{(G)}\), which is calculated according to the Equation (35). Therefore the expression for calculating the mutation vector is \[\mathbf{v}_{i}^{(G+1)}=\mathbf{x}_{\tau_{1}}^{(G)}+F_{i}^{(G)}\Bigl{(}\mathbf{ x}_{\tau_{2}}^{(G)}-\mathbf{x}_{\tau_{3}}^{(G)}\Bigr{)}.\] (57)
* Novel DE/rand-global-BOA/2 mutation strategy, which is produced by introducing the parts of BOA global search step into the DE/rand/2 mutation strategy. Therefore, to achieve this the term \(F\Bigl{(}\mathbf{x}_{\tau_{4}}^{(G)}-\mathbf{x}_{\tau_{5}}^{(G)}\Bigr{)}\) in Equation (29) is replaced with the term \(\Bigl{(}r^{2}\times\mathbf{g}^{*}-\mathbf{x}_{i}^{(G)}\Bigr{)}\times\varphi_{i}\) from Equation (45). Furthermore, the adaptive scale factor \(F_{i}^{(G)}\) given in Equation (35) is introduced instead the conventional scale factor \(F\). Additionally, the sine chaos map-based sensory fragrance \(c^{(G)}\) is employed in determining the term \(\varphi_{i}\). Therefore, the DE/rand-global-BOA/2 mutation strategy has the form \[\mathbf{v}_{i}^{(G)}=\mathbf{x}_{\tau_{1}}^{(G)}+F_{i}^{(G)}\Bigl{(}\mathbf{ x}_{\tau_{2}}^{(G)}-\mathbf{x}_{\tau_{3}}^{(G)}\Bigr{)}+\Bigl{(}r^{2}\times \mathbf{g}^{*}-\mathbf{x}_{i}^{(G)}\Bigr{)}\times\varphi_{i}.\] (58) \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Algorithm** & **10D** & **30D** & **50D** & **100D** & **Mean Ranking** & **Rank** \\ \hline BOA-c-Sine & 2.27 & 2.5 & 2.65 & 2.5 & 2.47 & 1 \\ BOA-c-Logistic & 2.76 & 2.72 & 2.62 & 2.9 & 2.75 & 2 \\ BOA-c-Picewise & 3.13 & 3.4 & 3.05 & 2.8 & 3.09 & 3 \\ BOA-c-Iterative, & 3.53 & 3.52 & 3.42 & 3.93 & 3.6 & 4 \\ BOA-c-Tent & 4.03 & 3.41 & 3.75 & 3.5 & 6.67 & 5 \\ Original BOA & 5.26 & 5.45 & 5.51 & 5.37 & 5.4 & 6 \\ Friedman \(p\)-value & \(6\times 10^{-10}\) & \(8.32\times 10^{-11}\) & \(2.96\times 10^{-12}\) & \(4.42\times 10^{-12}\) & & \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of the statistical findings of the Friedman test comparing the performance of the five chaos-improved BOA algorithms and the conventional BOA algorithm. Next, the modification is performed in order to perform local search and improve the solution accuracy.