## 4 Proposed HE-DEPSO Algorithm
### Adaptive Hybrid Mutation Operator
#### 4.1.1 DE/Current-to-EHE/1 Mutation Strategy  
Within evolutionary algorithms, the best individuals in the population, also known as elite individuals, retain valuable evolutionary information to guide the population to promising regions [39; 54; 55; 56]. However, many of the proposals made use of the information from the elite individuals of the current iteration, completely forgetting the information from the previous elite individuals. This absence of information within subsequent iterations may limit the ability of new individuals to explore the search space. Because of this, historical evolution information is used to improve the ability to explore new individuals. Before applying the mutation operator, all individuals in the current population \(P_{t}\) are sorted in ascending order based on their fitness values. After reordering the population, two partitions of individuals are created. The first partition, denoted by \(\mathrm{E_{t}}\), consists of the top \(pb\%\in[0.1,1]\) best individuals or elite individuals. The second partition, denoted by \(\mathrm{NE_{t}}\), groups all non-elite individuals, i.e., \(NP-NP\cdot pb\%\). It is important to note that \(\mathrm{E_{t}}\cup\mathrm{NE_{t}}=P_{t}\). Unlike other mutation strategies [36; 39; 57], in which only elite individuals corresponding to the current iteration are used, this strategy makes use of the evolution history by incorporating two external archives of size \(NP\). The first archive, denoted by HE, stores at each iteration the elite individuals belonging to the \(\mathrm{E_{t}}\) partition. If the size of the archive HE is greater than \(NP\) then its size is readjusted. The second archive used, denoted by HL, stores the obsolete individuals (individuals discarded within the selection process) and is updated at each iteration. Similar to the HE archive, if the size of the HL file is larger than \(NP\) then its size is reset. With the information of the individuals belonging to \(\mathrm{E_{t}}\) and HE, a group of candidates can be formed using \(\mathrm{E_{t}}\cup\mathrm{HE}\) to mutate individuals in the population. On the other hand, the individuals of \(\mathrm{NE_{t}}\) and HL make up a group of individuals \(\mathrm{NE_{t}}\cup\mathrm{HL}\), whose information contributes to the exploration of the search space. The way in which a mutated vector is generated by the \"DE/current-to-EHE/1\" strategy is as follows:\[V_{i,t}=X_{i,t}+F_{i}\cdot(X_{E_{r},t}-X_{i,t})+F_{i}\cdot(X_{P_{r},t}-X_{HL_{r},t}) \tag{37}\]  
where \(X_{i,t}\) is the \(i\)-th individual at iteration \(t\), \(X_{E_{r},t}\) is a randomly selected individual from among the individuals in the group \(\text{E}_{\text{t}}\cup\text{HE}\), \(X_{P_{r},t}\) is an individual randomly selected from within the current population \(P_{t}\), \(X_{HL_{r},t}\) is an individual randomly selected from the individuals comprising the group \(\text{NE}_{\text{t}}\cup\text{HL}\), and \(F_{i}\) is the scaling factor corresponding to the \(i\)-th individual. Following Equation (37), it is possible to observe that the \"DE/current-to-EHE/1\" strategy can help the HE-DEPSO algorithm to maintain a good exploration capability and direct the mutated individuals to promising regions without leading to stagnation in local minima. This can be explained as follows:  
* The use of a randomly selected individual within the group \(\text{E}_{\text{t}}\cup\text{HE}\), i.e., \(X_{E_{r},t}\), helps to guide mutated individuals to more promising regions. However, due to the presence of the historical information of the elite individuals (HE), the mutated individuals are prevented from being directed towards the best regions found, thus allowing the maintenance of good diversity in the population and increasing the chances of targeting optimal regions. * The participation of two individuals, \(X_{P,t}\) and \(X_{L_{r},t}\), randomly selected from \(P_{t}\) and HL, respectively, promotes the diversity of mutated individuals.