4.2.2. Population Maintenance

In PC selection, all non-dominated individuals are selected from a hybrid population of new individuals resulting from PC and NPC evolutions. Therefore, it is likely to make the number of non-dominated individuals larger than the preset threshold *N* (population size), especially when the objective number is large. Therefore, an effective means of population maintenance should be added to ensure that the PC population maintains a representative (with better convergence and diversity) group of individuals.

Population maintenance is to ensure the quality of individuals in a population through niche techniques. This is also a popular technique in EAs to assess the crowding degree of each individual in the population by the location and number of individuals in the niche (objective space). The crowding degree of individual *p* in population *P* is defined as follows:

$$D(p) = 1 - \prod\_{q \in P, q \neq p} \mathcal{R}(p, q) \tag{9}$$

$$R(p,q) = \begin{cases} \ \ d(p,q)/r, & \text{if} \quad d(p,q) \le r\\ 1, & \text{otherwise} \end{cases} \tag{10}$$

where *d*(*p*, *q*) is the Euclidian distance between individuals *p* and *q*, and *r* is the radius of the niche. Due to the size of each objective is different, in order to prevent the influence of problem size, the objective value of individuals in population will be normalized by maximum and minimum normalized first when calculating the distance.

It means that each is in the other's niche when the Euclidean distance between individuals *p* and *q* is less than *r*. This point can be seen in Equations (9) and (10), and the range of this crowding degree *<sup>D</sup>*(*p*) is [0, 1]. Otherwise, there would be no effect on the crowding of these two individuals since *<sup>R</sup>*(*p*, *q*) = 1. When *d*(*p*, *q*) ≤ *r*, the larger the Euclidean distance between the two individuals is, the smaller the calculated crowding degree will be, which means that the two individuals have a good crowding degree. So, this is a good way to eliminate the more crowded (poor diversity) individuals in the population.

Since the population is constantly evolving, it is not appropriate to set a fixed niche radius *r*. The setting of *r* must be related to the evolutionary state of the population. The radius *r* of the niche in BCE was set as the average Euclidian distance from each individual to *k* closest individuals in the population. The aim is to include one or more individuals in the niche of as many individuals as possible. Here, *k* is recommended to be set to 3 for better performance. Based on this crowding degree, the most congested individual in the population (the population of non-dominated individuals selected by PC selection) is eliminated each time and the crowding degree is recalculated. This process is repeated until the number of remaining non-dominated individuals is *N*.
