## 4 Parallel Multi-Objective Energy Valley Optimizer
### Adaptive Alpha and Gamma Decay  
The basic EVO randomly selects two indices to modify the solution, replacing the sequences between these indices with the corresponding sequences from \(X_{BS}\) and \(X_{N_{S}}\), as shown in Equations (41) and (44).  
Figure 3: Flowchart of Algorithm 1.  
The variation mode in basic EVO preserves randomness during solution candidate modification, reducing the likelihood of falling into local optima. However, this mode does not effectively balance exploration and exploitation during optimization. In EVO, if the \(NEL_{i}\) of solution candidate \(X_{i}\) exceeds the enrichment boundary \(EB\) and the stability level \(SL_{i}\) exceeds the stability boundary \(SB\), it indicates that \(X_{i}\) performs relatively well within the current solution set. To balance exploration and exploitation, one strategy is to reduce changes for solutions with better performance while increasing changes for solutions with poorer performance. For high-performing solutions, most sequences close to the optimal solution are retained to preserve their excellent performance. In contrast, for lower-performing solutions, a larger portion is altered to increase the likelihood of approaching the optimal solution.  
The basic EVO is designed for single-objective optimization problems, making it easy to select the optimal solution \(X_{BS}\) in each iteration. This paper addresses a dual-objective optimization problem in energy production planning, requiring a balanced consideration of both objectives. To enhance solution diversity, this paper expands \(X_{BS}\) into a solution set, represented by the Pareto front generated in each iteration. When modifying Equation (41), solution \(X_{i}\) is varied based on each solution in the Pareto front. For example, the concept of the optimization strategy in the adaptive alpha decay process is illustrated in Figure 4.  
In Figure 4, the blue and green shadows represent the variation ranges for Solution candidate 1 and Solution candidate 2, respectively, while the dashed arrows show the ideal evolutionary direction of the solution candidates. Since Solution candidate 1 outperforms Solution candidate 2 within the solution set, its search range is larger than that of Solution candidate 2. As a result, the better-performing solution focuses more on local exploitation, while the slightly inferior solution shifts to exploitation after approaching the Pareto front following exploration. The definition of adaptive decay is as follows:  
\[X_{i}^{\text{Alpha}}=X_{i}\Big{(}X_{\text{BS}}\Big{(}x_{i}^{j}\Big{)}\Big{)}, \begin{cases}i=1,2,\dots,n.\\ j=AI_{1},\dots,AI_{2}.\end{cases} \tag{48}\]  
\[X_{i}^{\text{Gamma}}=X_{i}\Big{(}X_{\text{NB}}\Big{(}x_{i}^{j}\Big{)}\Big{)}, \begin{cases}i=1,2,\dots,n.\\ j=GI_{1},\dots,GI_{2}.\end{cases} \tag{49}\]  
\[AI_{2}-AI_{1}=GI_{2}-GI_{1}=\frac{card(PF)*d}{n} \tag{50}\]  
\[PD_{k}\prec X_{i},k=1,\dots,card(PF) \tag{51}\]  
where the length of the interval between the two indices of change depends on the relative performance of the solution candidate within the solution set. \(PF\) represents the set of  
Figure 4: The idea of adaptive alpha decay.  
solutions that dominate the solution candidate X\({}_{i}\); the more solutions there are in \(PF\), the greater the variation in the solution candidate X\({}_{i}\).