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.