## 4 Multi-Objective Optimization Model under Uncertain Conditions
### Algorithm Design  
In the study of logistics network optimization, a non-dominant genetic algorithm (NSGA2) with elite strategy in multi-objective evolutionary algorithm (MOEA) is often used. Based on NSGA, a fast non-dominant sorting method is proposed to reduce the complexity of the algorithm. At the same time, an elite strategy is introduced in order to select the top-ranked individuals after the merging of parent-child populations as the new populations. However, the NSGA2 algorithm has the characteristic of a slow search speed, so this paper introduces the particle swarm optimization algorithm with fast computing speed in order to combine the two algorithms and for them to complement each other. The mixture of these two algorithms can make up for the shortcomings of both while preserving their respective advantages, and this is more conducive to the overall optimization effect of the emergency disposal logistics optimization network. The specific process of the improved hybrid algorithm MOPSO-NSGA2 is shown in Figure 2.  
The basic steps of the NSGA2 MOPSO hybrid algorithm are as follows:  
Step 1: Initialize particles  
Initialize the speed and position of all of the particles. Calculate the objective function of all of the particles and store the non-inferior solutions in an external archive to form a Pareto solution set. Step 2: Determine individual and group optima  
Due to the initialization process, the individual optimization of each particle is its own. The optimal selection method for the group is to use dense distance sorting, and for convenience during initialization selection, it can also be randomly selected from the Pareto solution set. Step 3: Update inertia weights  
Inertia weight is an important parameter of the particle swarm optimization algorithm, which is related to the algorithm's local and global search capabilities. At each iteration, the inertia weight is updated according to the following formula:  
\[X_{i}^{(k)}=\frac{1}{x_{\max}-x_{\min}}\frac{1}{D}\sum_{d=1}^{D}\left|g_{d}^{(k )}-x_{id}^{(k)}\right| \tag{41}\]  
\[w_{i}^{(k)}=w_{\text{start}}-(w_{\text{start}}-w_{end}){(X_{i}^{(k)}-1)}^{2} \tag{42}\]  
where \(D\) is the dimension of the solution space, \(w_{i}^{(k)}\) is the inertia weight of the \(i\) particle at time \(k\), \(w_{\text{start}}\) and \(w_{end}\) refer to the initial and ending values of \(w\), respectively, and \(x_{\max}\) and \(x_{\min}\) represent the maximum and minimum values of the particle position variables, respectively. Step 4: Update the velocity and position of particles  
Figure 2: Flow Chart of Hybrid Algorithm. Update the position and the velocity of particles in the population, and then search for the optimal solution under the guidance of \(Gb\) and \(Pb\). Among them, the position update formula is similar to the single objective particle swarm optimization algorithm:  
\[v_{id}^{(k+1)}=wv_{id}^{(k)}+c_{1}r_{1}(p_{id}^{(k)}-x_{id}^{(k)})+c_{2}r_{2}(g_ {d}^{(k)}-x_{id}^{(k)}) \tag{43}\]  
\[x_{id}^{(k+1)}=x_{id}^{(k)}+v_{id}^{(k+1)} \tag{44}\]  
where \(w\) is the inertia weight, \(c_{1}\) and \(c_{2}\) are the acceleration factor, \(r_{1}\) and \(r_{2}\) are a random number of \((0,1)\), \(p_{id}^{(k)}\) is the \(d\)-th dimensional component of the optimal position vector of the \(i\) particle at time \(k\), and \(g_{d}^{(k)}\) is the \(d\)-th dimensional component in the optimal position vector of the population at time \(k\). Step 5: Calculate the fitness value of particles  
Based on the objective function, obtain the fitness function of the particle swarm optimization algorithm. Step 6: Iteration termination judgment  
By analogy, determine whether the iteration requirements are met. If they are met, output the result. If not, return to step 3. Step 7: Initialize the population  
Use the offspring population generated by the MOPSO algorithm iteration as the initialization population of the NSGA2 algorithm.