# A Survey on Search Strategy of Evolutionary Multi-Objective Optimization Algorithms
## 3 Classification of Multi-Objective Evolutionary Algorithms from the View of Search Strategy
### Dominant Relation-Based MOEA Algorithms
#### 3.2.4 Non-Dominated Sorting Genetic Algorithm II
Parent and offspring populations are mixed and individuals are selected for new parent populations based on their non-dominance relationships and crowding levels, which in turn produce new offspring populations. This figure is adopted from [16]. the i-th solution's front surface is equal to the solution's crowding coefficient (as shown in the dotted box) in Figure 7. The calculation of the degree of crowding ensures that the population is diverse. The population must be sorted for the crowding coefficient computation using the values of each objective function in ascending sequence. As a result, for each objective function, the infinite distance value is called the boundary solution, which has the maximum and lowest values. The value for all other intermediate solutions is given as the absolute difference between the normalized values of the two nearby solutions. The other objectives have the same objectives. The distance values for each objective are added to determine the crowding coefficient values. Every objective function is normalized before the congestion factor is calculated. (2) Crowding comparison operator [85]: After performing the quick non-dominant ranking and crowding calculation, each individual \(i\) has two properties. The non-dominant ranking, \(i\_\)(rank) (the rank) and the crowding, \(i\_\)(d). These properties can be used to define a congestion comparison operator for comparing individual \(i\) with another individual \(j\). Either of the following is true and will result in an individual \(i\) winning. If the non-dominant stratum of individual \(i\) is higher in the ranking than the non-dominant stratum of individual \(j\), \(i\_\)(rank) \(<j\_\)(rank) If in the same non-dominant layer and the crowding distance of individual \(i\) is greater than that of individual \(j\), i.e., \(i\_\)(rank) \(=j\_\)(rank) and \(i\_\)(d) \(>j\_\)(d). The first requirement is to ensure that the selected individuals have a higher non-dominated rank. The second condition is to choose a region with low congestion when two individuals are tied to the same non-dominant layer according to their crowding distance. The winner can proceed to the next operation. The NSGA-II algorithm introduces an elite strategy to retain good individuals to eliminate the inferior ones. The elite strategy expands the screening range when generating the next generation of individuals by mixing the parent and offspring individuals to form a new population. Analyzing the example shown in Figure 8, where P denotes the parent population, let the individual accounts in it be n, and the Q denotes the offspring population. The steps are as follows. Elitism  
1. Create a new population by merging parental and child populations. After that, the new population is sorted non-dominantly, and in this case, the population is divided into six Pareto classes. 2. To generate a new parent population, non-dominated individuals having Pareto rank 1 are placed in a new parent collection, and then the individuals with Pareto rank 2 are added to the new parent population, and so on. Figure 7: The crowding distance about the i-th solution. The average distance between the two points on either side of point \(i\) is calculated; based on each objective function, the vertices of the rectangle are determined by their nearest neighbors, i.e., \(i-1\) and \(i+1\), and the length of the cube is equal to the congestion factor of the solution in the graph. This figure is adopted from [16]. 3. The crowding degree is calculated for all the individuals with rank \(k+1\), if there are fewer individual accounts in the set than \(n\) before all individuals with rank \(k\) are added to the new parent set, but individual accounts in the set increase after all individuals with rank \(k+1\) are added. All individuals with ranks more than \(k+1\) are deleted after these people are ranked according to how crowded they are.