## 4 Proposed Algorithm
### Nandominated Sorting Based on Reference Points
#### 4.3.2 Diversity Conservation  
In the \(Q_{ur}\) containing a large number of solutions, there will be many ranks, which leads to the existence of individuals with nondominated ranks that are too low and which are less important. These individuals not only reduce the running efficiency of the algorithm but also cause pheromone concentration changes in the poorer individuals to be recorded, thus increasing the search difficulty of the algorithm or even leading to a fall into a local optimum. To eliminate this difficulty, a routing selection strategy based on reference points is adopted. The solutions in \(Q_{ur}\) are selected according to their impact on the diversity of the algorithm. The uniformly distributed reference points are generated through the hyperplane. The individual is selected based on the association between the solutions and reference points. This strategy uses well-distributed reference points to maintain the diversity of populations. The steps are described below:  
(1) The individuals starting from nondominance rank 1 are added to set \(D\) sequentially until the size of \(D\) is larger than \(N_{max}\) after adding the individuals with nondominance rank L to \(D\). \(N_{max}\) is the maximum number of nondominated solutions. The individuals with nondominated ranks from 1 to (L-1) are added to set \(G\). Individuals with nondominant rank L are added to set \(G^{*}\). (2) Establish reference points on the hyperplane:  
The hyperplane has the same tilt for all coordinate axes and an intercept of one on each axis. For the problem with M objectives, reference points are established on an (M-1)-dimensional hyperplane. For the three-objective problem, to generate reference points, each coordinate axis is uniformly segmented into H segments. The plane perpendicular to the coordinate axis is established through the segmented points of the coordinate axis. The reference point is the intersection of planes from different coordinate axes with the hyperplane. The number of reference points is determined by Equation (31). Figure 4 shows the hyperplane of three objectives and reference points on the hyperplane when H is set to 5 (a detailed discussion of the reference points is given in Section 5.4). \[P=C_{\text{M}+H-1}^{H} \tag{31}\]  
(3) Normalize the individuals:  
First, calculate the ideal point \(z^{\text{min}}\), which is the point consisting of the minimum value of all individuals in set \(D\) on each objective. All individuals in set \(D\) are subtracted from the ideal point. Then, calculate the extreme points, which are the points with large values in one objective and small values in other objectives. M objectives have M extreme points. To find the extreme point on the \(ith\) objective, the value of the \(ith\) objective of all individuals is divided by a factor of 1, and the other objective values are divided by a factor of 10e\(-\)6. Then, excluding the \(ith\) objective value, the largest value of each individual is found. Comparing these values, the point corresponding to the smallest of these values is the extreme value point on the \(ith\) objective. The (M-1) dimensional hyperplane is formed using M extreme points. Then, the intercept between the hyperplane and the coordinate axes is calculated. Figure 5 shows the two-dimensional hyperplane formed using three extreme points. \(z_{max}^{i}\) represents the extreme point of the \(ith\) objective. \(a_{i}\) represents the intercept of the \(ith\) coordinate axis. Figure 4: Hyperplane and reference points. Figure 5: Construction of the hyperplane. Finally, the normalized objective value is calculated by Equation (32)  
\[f_{i}^{n}(x)=\frac{f_{i}(x)-z_{i}^{min}}{a_{i}-z_{i}^{min}},\ i=1,2,\ldots,M \tag{32}\]  
where \(f_{i}(x)\) represents the value of the individual in the \(ith\) objective. \(z_{i}^{min}\) represents the value of the ideal point in the \(ith\) objective. (4) Link individuals to reference points:  
The reference point is connected to the origin of the coordinates to form the connecting line.