Biomimetics 2023, 8, 149 8 of 42
Biomimetics 2023, 8, x FOR PEER REVIEW 7 of 31
Figure 1. Schematic illustration (for the case 𝑚=2) of the exploration phase by “𝑣-subtraction” (A),
and the exploitation phase by “arithmetic mean of the 𝑣-subtractions” (B).
3.3. Repetition Process, Pseudocode, and Flowchart of SABO
After updating all the search agents, the first iteration of the algorithm is completed.
Then, based on the new values that have been evaluated for the positions of the search
agents and objective function, the algorithm enters its next iteration. In each iteration, the
best search agent is stored as the best candidate solution so far. This process of updating
the search agents continues until the last iteration of the algorithm, based on (3) to (5). In
the end, the best candidate solution that was stored during the iterations of the algorithm
is presented as the solution to the problem. The implementation steps of the SABO are
shown as a flowchart in Figure 2 and presented as a pseudocode in Algorithm 1.
Figure 2. Flowchart of SABO.
Algorithm 1. Pseudocode of SABO.
Start SABO.
1. Input problem information: variables, objective function, and
constraints.
2. Set SABO population size (N) and iterations (T).
Figure 2. Flowchart of SABO.
3.4. Computational Complexity of SABO
In this subsection, the computational complexity of the proposed SABO approach is
evaluated. The initialization steps of the SABO for dealing with an optimization problem
with
m
decision variables have a complexity that is equal to
O
(
Nm
)
, where
N
is the
number of search agents. Furthermore, the process of updating these search agents has
a complexity that is equal to
O
(
NmT
)
, where
T
is the total number of iterations of the
algorithm. Therefore, the computational complexity of the SABO is equal to
O
(
Nm
(
1 + T
))
.
4. Simulation Studies and Results
In this section, the effectiveness of the proposed SABO approach in solving optimiza-
tion problems is challenged. For this purpose, a set of fifty-two standard benchmark func-
tions is employed, consisting of seven unimodal functions (F1 to F7), six high-dimensional
multimodal functions (F8 to F13), ten fixed-dimensional multimodal functions (F14 to F23),
and twenty-nine functions from the CEC 2017 test suite [
59
]. To analyze the performance
quality of the SABO in optimization tasks, the results that were obtained from the proposed
approach have been compared with twelve well-known metaheuristic algorithms: GA,
PSO, GSA, TLBO, GWO, MVO, WOA, MPA, TSA, RSA, WSO, and AVOA. The values of
the control parameters of the competitor algorithms are specified in Table 1.
Table 1. Control parameters values.
Algorithm Parameter Value
GA
Type Real coded
Selection Roulette wheel (Proportionate)
Crossover
Whole arithmetic (Probability = 0.8,
α ∈
[
−0.5, 1.5
]
)
Mutation Gaussian (Probability = 0.05)
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