## 4 Description of PODESCA and SBDMT  
### Structure of PODESCA  
The proposed PODESCA is constructed based on an improved hybridization between DE and SCA. In addition, QOT is used to generate the IP and also applied inside the main loop. DE was presented by [PERSON] and [PERSON] [36]. This meta-heuristic algorithm is a simple-structured and fast method which was successfully applied in several applications. The three basic procedures of DE are the \"mutation\", \"crossover\", and \"greedy selection\", which will be effectively used in the main body of PODESCA. Moreover, SCA was introduced by [PERSON] as a simple and efficient population-based algorithm [37]. It was designed by the sine and cosine functions to exploit the search space seeking better solutions. Enhanced functions of SCA will be also utilized inside the main loop of the proposed algorithm. The concept of QOT was originally evolved to accelerate the convergence and hence reduce the computational time of EAs [38; 39]. It was proved that the quasi-opposite number is closer than its opposite number to the optimal solution. This concept was used to generate the initial population for some algorithms [40; 41; 42]. However, the QOT will be used in this paper not only at the initialization stage, but also inside PODESCA's main loop. The steps of the proposed algorithm are highlighted as follows:  
Step 1: generate population of random initial solutions (PIS) and use the considered OF to evaluate it. Step 2: regenerate the PIS using QOT given by  
\[X_{\text{QO}}^{i}=\textit{rand}\Big{(}0.5*\big{(}a^{i}+b^{i}\big{)},X_{O}^{i} \Big{)}, \tag{8}\]  
where \(X_{\text{QO}}^{i}\) are the quasi-opposite points in a multi-dimensional search space (\(d\)), and \(X_{O}^{i}\) is given by  
\[X_{O}^{i}=a^{i}+b^{i}-X_{i}^{i};\ i=1,2,\ldots,d, \tag{9}\]  
where \(X_{O}^{i}\in\Big{[}a^{i},b^{i}\Big{]}\) are the opposite points in \(d\)[39; 42]. Then, evaluate the resulted PIS again using the OF. Step 3: execute the \"DE's greedy selection\" to compare both PISs obtained in steps 1 and 2 and keep the best population, which will be set as the input PIS of PODESCA's main loop. Step 4: while termination criterion is not satisfied, do  
Step 5: perform the \"DE's mutation\" on the population by  
\[S_{M}=S_{Su}+F_{DE}^{M}*\big{(}S_{Ru_{1}}-S_{Ru_{2}}+S_{Ru_{3}}-S_{Ru_{4}} \big{)}, \tag{10}\]  
where \(S_{M}\) is the mutant solution, the superior solution which is kept in each generation is denoted as \(S_{Su}\), while \(S_{Ru_{1}}  
Step 6: perform the \"DE's crossover\" by  
\[Tr_{i}^{It}=\left\{ \begin{array}{cc}S_{M_{i}}^{It+1}\;if\;r\;\leq cr\\ S_{i}^{it}\;if\;r\;\geq cr\end{array} \right., \tag{12}\]  
where \(Tr_{i}^{It}\) are trial solutions, \(cr\) denotes the crossover rate, and \(r\) is in the range \([0,1]\). Then evaluate the obtained solution using the OF. Step 7: execute the \"DE's greedy selection\" to compare both solutions obtained in steps 5 and 6 and keep the best population [36]. Step 8: perform the SCA's functions on the best population saved in step 7 by using a new enhanced SCA version which utilizes the local best solutions in addition to the global best solution in each iteration as:  
\[S_{i}^{It+1}=\left\{ \begin{array}{cc}S_{i}^{It}+Rn_{1}*\sin(Rn_{2})*\left|Rn _{3}*P_{i}^{It}-S_{i}^{It}\right|,\;Rn_{4}<0.5\\ S_{i}^{It}+Rn_{1}*\cos(Rn_{2})*\left|Rn_{3}*P_{i}^{It}-S_{i}^{It}\right|,\;Rn_ {4}\geq 0.5\end{array} \right., \tag{13}\]  
where \(S_{i}^{It}\) is the current local best solution, \(P_{i}^{It}\) is the global best solution, \(Rn_{2}\), \(Rn_{3}\), and \(Rn_{4}\) are random numbers, while \(Rn_{1}\) is modified so that the exploration is balanced and hence improved:  
\[Rn_{1}=a-It\frac{a}{It_{max}}, \tag{14}\]  
where \(a\) is a constant [37]. Then, evaluate the resulted solution using the OF. Step 9: perform the \"DE's crossover\" on the saved population in step 8 again by (12) and reevaluate it using the OF. Step 10: execute the \"DE's greedy selection\" to compare both solutions obtained in steps 8 and 9 and keep the best population.