## 4 Multi-Objective Optimization Model under Uncertain Conditions
### Algorithm Design
Step 8: Fast non-dominated sorting  
Among all individuals in the initial population, the dominance relationship is first used to determine the dominance attributes between each individual in the population. \(Na\) is used to represent the number of dominant solutions \(a\), and \(Sa\) is used to represent the set of solutions dominated by dominant solutions \(a\). Initialize the hierarchical level to \(rank1\), determine all individuals of \(Na=0\) in the population, and set these individuals as \(rank1\). Subsequently, based on the set of all of the individuals controlled by \(Sa\), the individuals of \(Na=0\) at this time are searched for, and these individuals are stored in a new set as a hierarchical level, \(rank2\), by analogy until all individuals have obtained a non-dominated ranking level. Step 9: Crowding distance  
Crowding distance is an indicator of the density at the midpoint of the Pareto front. It is generally believed that the sparser the distribution of solutions, the better, and this is because if the distribution of solutions becomes tighter, after cross mutation, the solution that can be searched for is still near the previous tight interval. The calculation formula for crowding distance is as follows:  
\[Dc=\sum_{q=1}^{s}\frac{\left|f_{q}^{t+1}-f_{q}^{t-1}\right|}{f_{q}^{\max}-f_{ q}^{\min}} \tag{45}\]  
Among them, \(Dc\) represents the crowding distance, and \(s\) represents the \(s\)-th objective function value of individual \(t\). \(f_{q}^{t+1}\), \(f_{q}^{t-1}\) represent the values of individuals \(t+1\) and \(t-1\) on the \(q\)-th objective function, where \(f_{q}^{\max}\), \(f_{q}^{\min}\) are the maximum and minimum values of the \(q\)-th objective function. Step 10: Elite Selection Strategy  
The role of selection strategy is to select excellent individuals from the population for the next step of cross mutation operation, generating a new generation of offspring population. This algorithm adopts a binary tournament selection strategy based on crowding comparison operator and non-dominated ranking. Step 11: Cross and mutation operationsUsing simulated binary for crossover operation, assuming that the parents of two genes are \(P_{1j}\) and \(P_{2j}\), then:  
\[\overline{P}_{1j}=0.5\cdot\left[\left(1+\phi_{j}\right)\cdot P_{1j}+\left(1-\phi _{j}\right)\cdot P_{2j}\right] \tag{46}\]  
\[\overline{P}_{2j}=0.5\cdot\left[\left(1-\phi_{j}\right)\cdot P_{1j}+\left(1+ \phi_{j}\right)\cdot P_{2j}\right] \tag{47}\]  
Among them:  
\[\Delta j=\left\{ \begin{array}{c}\left(2u_{j}\right)^{\frac{1}{\phi+1}}\,u_{j}<0.5\\ \left[1-2\cdot\left(1-u_{j}\right)\right]^{\frac{1}{\phi+1}},\text{otherwise} \end{array} \right. \tag{48}\]  
Similarly, using simulated binary for mutation operations, assuming that the parent of a gene is \(P_{3j}\), then:  
\[\overline{P}_{3j}=P_{3j}+\Delta j \tag{49}\]  
Among them:  
\[\Delta j=\left\{ \begin{array}{c}\left(2u_{j}\right)^{\frac{1}{\phi+1}}\,u_{j}<0.5\\ \left[1-2\cdot\left(1-u_{j}\right)\right]^{\frac{1}{\phi+1}},\text{otherwise} \end{array} \right. \tag{50}\]  
Step 12: Termination Rules  
When the number of generated offspring reaches the maximum number of iterations \(Gen\) of the rule, the algorithm stops. This paper mainly introduces the changes brought by mixing the NSGA2 and MOPSO algorithms. In terms of initializing population, this paper uses the particle swarm optimization algorithm in the first stage to conduct preliminary optimization of the multi-objective problem, which is characterized by fast optimization speed and high search efficiency, and it obtains the population that has evolved to a certain extent. Then, the hybrid algorithm enters the second stage, and it takes the output results of the particle swarm optimization algorithm as part of the initial population of the NSGA2 algorithm so as to improve the overall quality of the initial population of NSGA2 algorithm, with the purpose of optimizing the initial population of the NSGA2 algorithm.