## 3 Proposed Improved Hybrid SSA with an EDO (ISSAEDO) Algorithm
### Proposed Improved Hybrid SSA with the EDO (ISSAEDO) Algorithm
#### 3.3.1 DE Technique  
The DE technique, proposed by [PERSON] and [PERSON] in 1997 [33], is an effective population-based heuristic algorithm that employs an evolutionary process to identify the optimal solution for an optimization problem. This particular solution is notable for its simplicity in construction and comprehension, as well as its versatility in addressing various optimization problems, leading to favorable results within a fair timeframe. DE employs four fundamental processes, namely initialization, mutation, crossover, and selection, in order to identify the optimal solution for an optimization problem. The primary aim of the initialization step is to ensure comprehensive coverage of the whole search space. This is achieved by randomly setting the initial population within the specified boundaries. Subsequently, the population undergoes the processes of mutation, crossing over, and selection, which serves to augment its characteristics.  
* _Mutation step_ The objective of this procedure, alternatively referred to as a differential mutation, is to generate a mutated vector, \(v_{i}\), for every solution vector in each iteration. The modifiedvector \(v_{i}\) is generated by a random selection process, where three nominee vectors \(X_{r_{1}},X_{r_{2}},X_{r_{3}}\) are chosen from a range of options ranging from 1 to population size. The process involves computing the difference between two of the selected nominee vectors, namely \(X_{r_{2}}\) and \(X_{r_{3}}\), and afterward merging the obtained results. After multiplying the third nominee vector, Xr1, by a mutation weighting factor (W) within the interval [0, 1][34], it is subsequently added to the aforementioned difference. The numerical expression for I\(v_{i}\) is as follows: \[\overrightarrow{v_{i}}=\vec{X}_{r1}+W(\vec{X}_{r2}-\vec{X}_{r3}).\] (19) One distinguishing factor between DE and other evolutionary processes lies in this particular stage.
* _Crossover step_ To ensure the preservation of population diversity, the DE technique use this particular phase to adapt the differential mutation search strategy. The offspring vector \(u_{i}\) is generated using a crossover operation that combines values from the objective vector \(X_{i}\) and the altered vector \(v_{i}\). Based on the principles of binomial and exponential operators, the prevailing and fundamental crossover search operators can be enumerated as follows: \[u_{i,d}=\begin{cases}v_{i,d},&\text{if rand}\leq C_{R}\;\;\text{or}\;\;d=j_{ \text{rand}}\\ X_{i,d},&\text{otherwise}\end{cases}\] (20) Random numbers \(j_{\text{rand}}\) are selected from the range \([1,2,\ldots,D_{X}]\), while rand is picked from the interval \([0,1]\). This selection ensures that the modified vector will have at least one dimension. Variable \(C_{R}\), often assigned a high value (\(C_{R}=0.9\)), is the user-specified crossover rate that determines the probability of crossing over each element. A comparison is performed between variables \(C_{R}\) and rand, as elucidated in Equation (20). If the value of rand is less than or equal to \(C_{R}\), then the calculation of \(u_{i}\) is derived from the value of \(v_{i}\). If this condition is not met, \(X_{i}\) infers \(u_{i}\).
* _Selection step_ The final stage of the DE cycle is the selection phase. In this stage, the fitness function values of the offspring vector \(f(u_{i})\) to the corresponding target vector \(f(X_{i})\) are compared to determine the solution that is more appropriate for future iteration. The procedure for determining a winner is explained as \[X_{i}=\begin{cases}u_{i},&\text{if}(u_{i})<f(X_{i})\\ X_{i},&\text{otherwise}\end{cases}.\] (21) In the case where the fitness function, denoted as \(f(u_{i})\), generates a value lower than that of the target vector, \(f(X_{i})\), the offspring vector, \(u_{i}\), is reassigned to be equal to the target vector, \(X_{i}\). Alternatively, the previous target vector, denoted as \(X_{i}\), remains unchanged.