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Guan et al. Intell Robot 2024;4(1):61-73 I http://dx.doi.org/10.20517/ir.2024.04 Page 67

certain choices for handling iterative updates. If the energy of the next iteration is low, it is updated directly
to the next position. When the energy of the next iteration is higher, a certain probability to iterate also exists.
The Metropolis criterion compares the energy of the current state with that of the next step and calculates the
probability of iteration:

{
1, Δ < 0,
( ) (13)
exp(−Δ / ), Δ > 0.

( ) ( )
where Δ = − −1 isthedifferencebetweenthefitnessvalueofthecurrentiteration

and the fitness value of the previous iteration. is the system temperature, which decreases as the number of
iterations increases.

The base Metropolis criterion has a reduced convergence speed due to the particle staying at the last position.
[8]
Therefore, an IMC proposed by Yang et al. is introduced and applied to IMCPIO . For the method of
updating the position and velocity of each pigeon in the first iteration, the improvement steps are
showninFigure4. ThespecificimplementationoftheIMCPIOalgorithmthatintroducestheIMCisasfollows:

Step 1. Initialize the airspace information and the dangerous areas information.

Step 2. Initialize IMCPIO algorithm parameters, including space dimension , population size , map and
compass factor , the number of 1 and 2 for two operators, etc.

Step 3. Allocate a random position and velocity to each pigeon. Subsequently, the position , which repre-
sents the best global value, is determined by comparing the fitness of each pigeon.

Step 4. Execute the map and compass operator. Refresh the velocity and trajectory of each pigeon utiliz-
ing Equation (9). The updated positions of individuals within the boundary are filtered using the improved
Metropolis criterion. If the energy of the next iteration is low, it is updated directly to the next position. When
the energy of the next iteration is higher, there is also a certain probability to iterate. At the end of the position
update operation for each individual, evaluate the local optimal positions, compare the fitness of each pigeon,
and determine the updated .

Step 5. If the iteration count exceeds 1 , halt the map and compass operator and initiate the landmark

operator. If not, proceed to Step 4.

Step 6. All pigeons are sorted based on their fitness value. The half with higher fitness values will follow those
with lower fitness. Using Equation (10), compute and update the position . If the iteration count

exceeds 2 , the landmark operator is halted. If not, return to Step 6.

3. RESULTS
To evaluate the performance of the proposed IMPIO algorithm, we used a set of benchmark functions that
have different characteristics and compared it with the basic PIO method and the genetic algorithm (GA)
method. The benchmark functions are: Sphere (f1), which is a simple unimodal function that measures the
basic performance of the algorithm, such as convergence speed and accuracy; Rosenbrock (f2), which is a
nonlinear multimodal function that measures the algorithm’s ability to optimize in high-dimensional spaces
and escape from local optima; Ackley (f3), which is a multimodal function with one global optimum and