Fig 1 Genetic algorithm flow chart
This paper incorporates adaptive crossover and mutation operators proposed by Srinivas, and uses the Sigmoid growth curve as the adaptive adjustment curve for crossover rate Pc and mutation ratePm . This approach is used to improve these rates, with the mathematical expressions as follows:
In the formula: Pc_max is the maximum crossover probability; Pc_min is the minimum crossover probability; Fmax is the maximum fitness value of each generation; Favg is the average fitness value of each generation; Ffit is the individual fitness value; Pm_max is the maximum mutation probability; Pm_min is the minimum mutation probability.
Adaptive Dynamic Niche Radius Technique: Niche technology is a concept derived from the natural phenomenon of like attracting like. In dealing with multimodal problems, niche technology helps maintain the diversity of solutions, thereby reducing the probability of falling into local optima. It primarily comprises two components: population division and individual fitness updating. Initially, niche populations are divided based on genetic similarity between individuals, with each niche undergoing its genetic evolution operations independently, and the optimal individual functioning solely within its own group. Subsequently, each individual in the niche updates their fitness based on the sharing function, with the updated fitness determining the optimal individual.
Hamming distance dij is a standard reflecting the genetic similarity between individuals. The division rule is as follows:
In the formula: xik andxjk represent the k-th variable of individuals i and j, respectively; m is the number of variables for each individual; N is the population size; σ radis the niche radius. Whendij <σ rad, the individuals are classified into the same niche.
This paper adopts the niche technology based on fitness sharing proposed by Goldberg. This mechanism adjusts individual fitness through the sharing function fs (dij ), which reflects the degree of closeness between individuals within a niche. The mathematical expression for this is as follows:
In the formula: fi is the fitness of individual i;fi ′ is the shared fitness of individual i;fs (dij ) is the fitness sharing function; N is the number of individuals within the niche.
The niche radius serves as a crucial criterion for dividing niches. In conventional algorithms, the use of a fixed niche radius is common. However, with the Hamming distance between individuals decreasing in the later stages of evolution, a consequent reduction in the number of niches occurs, adversely affecting population diversity. Therefore, this study introduces an adaptive niche radius, delineated by the following formula:
In the formula: t represents the iteration number; Δδ is an adaptive adjustment factor. When the number of niches becomes too small in the later stages, Δδ will increase appropriately, thereby reducing the niche radius, which in turn increases the number of niches.