## 3 Solution Methodology
### The Framework of the Proposed Multi-Objective Algorithm
If there is a particle \(j\) and a relationship between its fitness value and fitness value of individual i: \(f_{1}\)(i) \(>\)\(f_{1}\)(i) and \(f_{2}\)(i) \(>\)\(f_{2}\)(i), Do  
(6) \(P(i)\)++;  
(7) For particle \(i\), use the Euclidean distance between particles to calculate its crowding degree _S_(\(i\));  
(8) End While. **Algorithm 2. ** Pareto Rank and Crowding Distance Calculation of Particle Swarm Individuals  
\(\bullet\) Individual update  
To evolve in an optimal direction for the whole particle swarm, the law of individual variation is crucial. The improved universal gravitation algorithm brings in the particle swarm velocity update formula. Figure 1 shows an example in which a particle individual uses the velocity update formula to finish its own evolution. Additionally, Algorithm 3 de scribes how to obtain the updating solution of the individual particle by velocity updating. Input: _Pop_, \(P\), \(S\)  
Output: _PopNew_ (New particle swarm)  
(1) \(i\) = 1  
(2) While \(i\) = _NNND_ Do  
(3) Initialize the particle velocity _V_\({}_{i}\) = 0;  
(4) Calculate the particle accelerated velocity of individual \(i\) through the formula for calculating the acceleration of gravity. (5) The superior individuals in the dominant solution set were randomly selected as \(G_{\text{Best}}\). (6) Calculate the velocity _V_\({}_{i}\) of the particle \(i\). (7) Obtain the new solution of particle \(i\). (8) _i_++;  
(9) End While. \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \multicolumn{11}{|c|}{Gbest:} \\ \hline
1 & **0** & 1 & 1 & **0** & **0** & 1 & **0** & **0** & **0** & 1 & 1 & **0** & **0** \\ \hline \multicolumn{11}{|c|}{Pop(i):} \\ \hline
0 & **0** & **0** & 1 & **1** & **0** & **0** & **0** & **1** & **0** & **0** & 1 & **0** & **1** \\ \hline \multicolumn{11}{|c|}{Gbest-Pop(i):} \\ \hline
1 & **0** & **1** & **0** & **-1** & **0** & **1** & **0** & **-1** & **0** & **1** & **0** & **0** & **-1** \\ \hline \multicolumn{11}{|c|}{a:} \\ \hline
1 & **1** & **1** & **0** & **0** & **1** & **0** & **0** & **0** & **1** & **1** & **1** & **0** & **0** \\ \hline \multicolumn{11}{|c|}{} \\ \hline \multicolumn{11}{|c|}{0.5(Gbest-Pop(i))+0.5a:} \\ \hline
1 & 0.5 & 1 & **0** & **-0.5** & **0.5** & **0** & **-0.5** & **1** & **0.5** & **0** & **-0.5** \\ \hline \multicolumn{11}{|c|}{V(i):} \\ \hline
0 & **0** & **0** & **0** & **0** & **0** & **0** & **0** & **0** & **0** & **0** & **0** & **0** \\ \hline \multicolumn{11}{|c|}{} \\ \hline
1 & **0.5** & **1** & **0** & **-0.5** & **0.5** & **0** & **-0.5** & **0.5** & **1** & **0.5** & **0** & **-0.5** \\ \hline \multicolumn{11}{|c|}{} \\ \hline
1 & **1** & **1** & **0** & **0** & **1** & **1** & **0** & **0** & **1** & **1** & **1** & **0** & **0** \\ \hline \end{tabular}   
Figure 1: Example of individual evolution. According to Figure 1, each individual in the Pop performs evolutionary operations with \(G_{\text{best}}\) and the direction in which the individual is attracted, and then the new individual is created. All the new individuals performed by the evolutionary operations form a new swarm _PopNew_. Sometimes evolved individuals may appear to be non-feasible solutions. Shadow elements evolved new individuals, as shown in Figure 1. For this reason, some genes are adjusted by Algorithm 4 to satisfy the feasibility of solutions. As can be seen from Figure 2, the marked seventh and ninth genes in the shadow do not satisfy feasibility. After adjustment, a feasible new solution individual forms in the end. ```
Input: Pop(\(i\)) Output: PopNew(\(i\)) (1) \(M\) = Find(Pop(\(i\), 1: \(N\)) == 1) (2) If \(M<P_{\min}\) or \(M>P_{\max}\) Do (3) If \(M>P_{\max}\), the city point of after reaching the upper limit of the number of classification points is changed to 0; If \(M<P_{\min}\), the non-1 city point is randomly selected and changed to 1 until \(M\) satisfies the constraint.