## 3 Solution Methodology
### The Framework of the Proposed Multi-Objective Algorithm  
Based on the above improvement of the universal gravitation algorithm, the hybrid gravitational particle swarm optimization algorithm is obtained. To apply this algorithm to the multi-objective model, there are five missions that need to be finished: individual (solution) coding, initial particle swarm formation, adaptive value calculation and Pareto solution screening, individual update and update of the dominant solution set. * Coding mode  
The coding form of the solution directly affects the efficiency of the algorithm. For this reason, this paper designs a coding scheme based on multi-constraints, which has lower decoding complexity and can easily perform mutation and crossover operations. According to the feature of questions, a solution is coded as an integer value vector, which contains two pieces of code. One section codes the location of the classification detection points. The other section codes the location of the recycling and processing center. As shown in Table 1, section X is the distribution of classification detection centers in urban agglomeration, and section Y is the distribution of recycling and processing centers in urban agglomeration. For example, \(X=\{1,\,1,\,0,\,1,\,0,\,0,\,1\}\), the first element is 1, indicating that a classification detection center has been established in city 1, and the third element is 0, indicating that no classification detection center has been established in city 3, and so on. feasible solutions, each individual in the initial population is initialized by Algorithm 1. Input: _NIND_ (Particle swarm size), \(N\) (The number of urban points)  
Output: _Pop_ (Initial population)  
(1) \(i=1\);  
(2) While \(i<=NIND\) Do  
(3) Pop (\(i\)) = zeros (\(n^{\star 2}\));  
(4) \(M=\) unidrnd (\(N\))  
(5) While \(M<P_{\min}\) or \(M>P_{\max}\) Do  
(6) \(M=\) unidrnd (\(N\));  
(7) \(N=\) unidrnd (\(N\))  
(8) While \(N<D_{\min}\) or \(N>D_{\max}\) Do  
(9) \(N=\) unidrnd (\(N\));  
(10) Use above gotten \(M\) and \(N\). Choose randomly the city waiting for site selection. And the corresponding location code of the individual is changed to 1. (11) \(i\)++;  
(12) End While. **Algorithm 1. ** Initial Particle Swarm Generation  
\(\bullet\) Adaptive value calculation and Pareto solution screening  
To evolve for the better, the particle swarm must be evaluated by comparing the fitness value of each individual (site selection scheme). This scheme contains two targets. A site selection scheme of the model can be decoded for a solution X. Decode X and use the target formula to calculate the result value of two targets. For the speed and accuracy of optimizing, the result value is transformed to some extent, and then the fitness value \(f_{1}\) and \(f_{2}\) related to the two resulting values are obtained. After calculating fitness value, the next key mission is searching for the Pareto solution from the particle swarm. The Pareto optimal solution sets mean that no solution is equal to or superior to the Pareto optimal solution sets in every objective evaluation. The solutions are chosen by comparing every target to ensure that they are the Pareto optimal solution. Algorithm 2 describes how to obtain the Pareto solution by screening all the solutions. Input: _Pop_ (swarm)  
Output: \(P\) (record Pareto rank of the particle swarm); \(S\) (record particle crowding distance of the particle swarm)  
(1) Use the target formula to calculate the target value of particle swarm individuals. The transportation of waste products among logistics facilities is based on the principle of the nearest point. Only when the destination facility reaches its maximum capacity can they proceed to the next nearest point. The target value also needs to be transformed through fitness, and two targets value are transformed the fitness value with the smaller the better. (2) \(i=1\)  
(3) While \(i<=NIND\) Do  
(4) Initialize \(P(i)=0\);  
(5) For _Pop_(\(i\)), traversal the whole particle swarm.