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African vultures optimization-based hybrid neural network–proportional-integral-derivative controller...
4.3. Step 3: exploration

Equations (60)–(63) carry out this task during the
exploratory phase.

Equation 54, if Po1 ≥ randPo1
Po(i + 1) =
Equation 56, if Po1 < randPo1
Figure 7. Block diagram of the feedback control (60)
system with a neural network–proportional-
integral-derivative (PID) controller Po(i + 1) = C(i) − D(i) × Fo (61)

4. African vultures optimization
D(i) = |X × C(i) − Po(i)| (62)
algorithm

The AVOA is a modern metaheuristic optimiza-
Po(i + 1) = C(i) − Fo + rand2 × ((ub − lb)
tion technique inspired by the scavenging behav-
ior of African vultures. 40 These birds, charac- × rand3 + lb
(63)
terized by their featherless heads and ground-
dwelling habits, 41 serve as the basis for the al- where Fo signifies the vulture’s satiety level, a
measure of how full it is, determined using Equa-
gorithm’s conceptual framework. In AVOA, po-
tential solutions are evaluated, and the two opti- tion (59); Po(i + 1) denotes the vulture’s position
mal candidates are designated as the “first” and in the subsequent iteration; all random variables
“second” vultures. These elite solutions guide the (randPo1, rand2, and rand3) range between 0
iterative improvement of the population by influ- and 1; C(i) represents one of the top vultures
encing the search process. 40 (e.g., “first” or “second”), which generate a ran-
dom location X to guard food by multiplying
rand by 2(X = 2 × rand), leveraging their cur-
4.1. Step 1: selection of the most suitable rent position Po(i); and lb and ub represent the
vulture 42,43
variables’ upper and lower bounds.
After each iteration, the population is re-
evaluated, and the optimal solution is selected us- 4.4. Step 4: the initial stage of
ing Equation (58): exploitation
When |Fo| falls between 1 and 0.5, the AVOA

Best vulture1, if pi = L1 is ready to enter the first step of the utilization
C(i) = (58)
Best vulture2, if pi = L2 stage, as presented in Equations (64)-(68):
where L1 and L2 are parameters within the in-

terval (0,1), specified prior to the search process, Equation (58) if Po2 ≥ randPo2
Po(i + 1) =
and their total sum equals 1. Equation (60) if Po2 < randPo2
(64)
4.2. Step 2: vulture hunger rate
Po(i + 1) = D(i) × (F + rand4) − d(t) (65)
The hunger rate of the vultures is quantitatively
modeled using Equation (59). This parameter re-
flects the vultures’ behavioral response under con- d(t) = C(i) − Po(i) (66)
ditions of starvation or insufficient energy, leading
them to exhibit aggressive tendencies or reduced
mobility over long distances. 40 Po(i + 1) = C(i) − (S1 + S2) (67)

iterationi S1 = C(i) × ((rand5 + Po(i))/2π) × cos(Po(i))
Fo = (2 × rand1 + 1) × z × 1 − + t
max iterations S2 = C(i) × ((rand6 + Po(i))/2π) × sin(Po(i))
(59) (68)
where iterationi denotes the i − th cycle where rand4 is a random number between 0 and
max iterations is the total number of cycles, and 1.
F represents vulture satiety (the stopping thresh- When the value |Fo| is less than 0.5, the
old). Random variables zrange from −1 to 1 AVOA moves on to the second stage of the utiliza-
whereas rand1 ranges from 0 to 1. tion stage, 44 as displayed in Equations (69)–(72):
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