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Page 14 of 27 Wang et al. Intell Robot 2023;3(4):538-64 I http://dx.doi.org/10.20517/ir.2023.30

where is the amplification factor of iteration that determines the selection probability of selecting the
head genes in the later iteration stage. is the evolution factor of iteration that determines the rate of
the evolution of probability distribution in iteration. is the maximum iteration step.

At the early stage of the iteration, the probability density function will show the characteristics of ap-
proximate uniform distribution. As the iteration process proceeds, the probability density function will
gradually evolve from the uniform distribution to the exponential distribution. By using the probability
density function above, a random number with a specific distribution pattern can be achieved as follows:

1
−1 −
= ( ) = − ln[1 − (1 − ) · ] (28)

where ∼ (0, 1) is a uniformly distributed random number within [0, 1), and is the corresponding
random number with the specific distribution pattern. In summary, the location of the selected gene can
be represented as:

= ⌊ · ⌋ + 1 (29)
By using the method above, a gene can be uniformly selected in the early stage of the iteration, and
gradually evolved in the later stage to achieve precise optimization of the higher priority part for a better
optimization process.

(ii) Fine-adjusted mutation of gene
In order to make the mutation operation in traditional GA more purposeful and enable the overall pop-
ulation to effectively jump out of the local optimal solution during the iteration process, a weight matrix
W = { } is designed to represent the optimizing benefits of candidate genes at different locations,
where represents the optimizing benefits of m-th candidate gene for . The higher the optimiz-
ing benefit, the more saturated the corresponding gene is in the dominant chromosome with the same
location.

W is defined as a Zero matrix at the beginning of the iteration, which means that the optimization
benefits are unknown. The adjustment of each element in W in the iteration step is as follows:

′

∑

( + 1) = ( ) + Δ ( ) (30)
′
=1
where is the learning factor of W, is the number of all chromosomes involved in matrix adjust-
′

ment, and Δ ∈ [0, 1) represents the increased value of the optimization benefit of m-th candidate
}, the corre-
gene for of i-th chromosome. If the i-th chromosome is denoted as = { ,1 , ,2 , , ,
sponding adjustment value is defined as follows:

 0, ≠ ,

Δ ( ) = ( ) (31)
 tanh max{ }− , = ,
 max{ }−min{ }

where is the fitness of i-th chromosome, max { } and min { } are the maximum and minimum fitness
of all chromosomes. Finally, the probability distribution of candidate genes for is denoted as Prob ,
and it is defined as: