Issue 64

Y. Li et alii, Frattura ed Integrità Strutturale, 64 (2023) 250-265; DOI: 10.3221/IGF-ESIS.64.17

( ) δ NR B similarity relation that B decides as follows:

Euclidean distance function, then indicates a

{

}

( ) ( ) δ δ = ∈ × ∆ ≤ x,y x,y B NR B U U ( )

(1)

{

}

( ) δ δ = ∈ ∆ ≤ n x y x,y B B U ( )

(2)

⊆ B C and

⊆ X U ,

= , , , NDS U C D

δ

∈ k x U , Hence, X on B 's neighborhood upper and lower

Given

, for any

approximations are defined as:

{

}

( ) k

( )

δ n x B

= ∈ x

≠∅

(3)

B X

U

X



k

δ

{

}

( ) k

( )

δ n x B

= ∈ x

⊆

(4)

B X

U

X

k

δ

{

}

δ = , , , NDS U C D , for any ⊆ B C ,

=  1 2 , , l U D D D D , then the neighborhood positive region of decision

Given

and neighborhood dependency of D on B are defined as:

l

( ) j d

( ) j POS D B δ = = ∑ 1 B

(5)

δ

( ) δ

POS D

B

( ) D

δ γ B

=

(6)

U

∈ j D U D and =

 1,2, , j l

where,

Firefly algorithm Firefly algorithm is a meta-heuristic algorithm constructed to imitate the glow behavior of fireflies in nature. In this algorithm, fireflies are attracted to each other by both brightness and attractiveness. In general, the brighter firefly is more attractive, and their ability to attract each other decreases with distance [23]. To put it simply, the firefly algorithm is to locate the brightest firefly in the solution space by its luminous properties, then approach the brightest firefly, further update the position information, and repeat the process until it reaches the optimum value. The algorithm is described as: suppose the number of fireflies is n , and the search dimension is d . 1) The position of the individual i is defined as: ( ) =  1 2 d , , , i i i i x x x x . The objective function value ( ) i f x is its brightness. 2) The updating formula of attractiveness between individuals is defined as:

2

−

ij γr

= ij β β e 0

(7)

where, β 0 is the initial attraction, generally, the value is 1, γ is the absorption coefficient of the light intensity. Generally, the value is 1, ij r is the Cartesian distance between individuals: ( ) = = − ∑ 2 1 d ij ik jk k r x x (8)

3) The update formula of the position to which the individual is attracted and moved is:

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