3.3.1. Global Search

The updating criterion of the position for the global search stage in the PSO-BOA algorithm can be represented by Equation (11) as follows:

$$\mathbf{x}\_{i}^{t+1} = \mathbf{x}\_{i}^{t} + \left(\omega \cdot \mathbf{v}\_{i}^{t} + c\_{1} \cdot rand\_{1} \cdot \left(p\_{\text{best}} - \mathbf{x}\_{i}^{t}\right) + c\_{2} \cdot rand\_{2} \cdot \left(g\_{\text{best}} - \mathbf{x}\_{i}^{t}\right)\right) f\_{\text{max}} \tag{11}$$

where *v<sup>t</sup> <sup>i</sup>* is the velocity of the *i*-th particle at the *t*-th iteration; *pbest* and *gbest* represent the local and global optimal positions of particles; and *f*max represents the current optimal scent intensity value. Generally, *c*<sup>1</sup> = *c*<sup>2</sup> = 2; *rand*<sup>1</sup> and *rand*<sup>2</sup> generate a random number that falls between 0 and 1. *ω* represents the inertia weight.

3.3.2. Local Search

The updating of the position formula for the local search stage used in the PSO-BOA algorithm can be represented by Equation (12) as follows:

$$\mathbf{x}\_{i}^{t+1} = \mathbf{x}\_{i}^{t} + \left(r^{2} \cdot \left(\mathbf{x}\_{\mathbf{a}}^{t} - \mathbf{x}\_{i}^{t}\right) - \omega \cdot \left(\mathbf{x}\_{\mathbf{b}}^{t} - \mathbf{x}\_{i}^{t}\right)\right) f\_{i} \tag{12}$$

where *x<sup>t</sup>* <sup>a</sup> and *x<sup>t</sup>* <sup>b</sup> are the spatial positions of the *a*-th and *b*-th butterflies in the *t*-th iteration; the parameter *ω* represents inertia weight; *r* generates a random number that falls between 0 and 1.

#### 3.3.3. Parameter Control Strategy

Chaos theory has numerous applications in intelligent optimization algorithms. Logistic mapping [41] is one of the classic chaotic mapping methods in chaos theory, and its representation is shown in Equation (13):

$$z\_{l+1} = \mu \, z\_l (1 - z\_l) \tag{13}$$

where *μ* is the chaotic parameter, and the value falls in [0, 4], *l* can be defined as the iteration count of the chaotic map.

The Lyapunov index [42] is a measure for distinguishing chaotic characteristics. A larger value of the Lyapunov exponent indicates a higher degree of chaos and stronger chaotic characteristics. The Lyapunov exponent is calculated by Equation (14):

$$\lambda = \lim\_{n \to \infty} \frac{1}{n\_h} \sum\_{i=0}^{n\_h - 1} \ln \left| f'(z\_i) \right| \tag{14}$$

where *λ* is the Lyapunov exponent; *nh* is the number of iterations of the map function; and *f* (·) is the first derivative of the chaotic map function.

Produce a logistic diagram and a Lyapunov exponent curve of the logistic map where parameter *μ* is within the interval (0, 4], as illustrated in Figure 7.

**Figure 7.** Logistic mapping. (**a**) Logistic mapping bifurcation diagram and (**b**) Lyapunov exponent curve.

As illustrated in Figure 7, the bifurcation of the logistic map occurs at *μ*= 3.55, and with an increase in the parameter value, the range of the map gradually expands to (0, 1). When *μ* = 4, the logistic map exhibits chaotic behavior, leading to a sequence within the range (0, 1). The maximum Lyapunov exponent of the logistic map is calculated to be 0.6839. Consequently, the parameter *μ* is set to 4.

According to the logistic mapping expression, the sensory modality *c* in the PSO-BOA algorithm can be represented by Equation (12) as follows:

$$c(t) = 4 \cdot c\_0(t-1)(1 - c\_0(t-1))\tag{15}$$

The coefficient of inertia weight directly affects the particle flight speed of the PSO algorithm. A dynamic tuning strategy is utilized to alter the local and global search capabilities of the algorithm, as depicted in Equation (16):

$$
\omega = \omega\_1 - (\omega\_1 - \omega\_2) \left(\frac{t}{T\_m}\right)^2 \tag{16}
$$

where *ω*<sup>1</sup> represents the initial inertia weight; *ω*<sup>2</sup> represents the inertia weight at the maximum number of iterations; *t* represents the current number of iterations; and *Tm* represents the maximum number of iterations. In this paper, the initial value of the inertia weight is set to 0.9, and the inertia weight value of the last iteration is set to 0.2. As the iteration progresses, the inertia weight decreases from 0.9 to 0.2. A larger inertia weight in the initial stage of the iteration can maintain the strong global search ability of the algorithm, while a smaller inertia weight in the later stage of the iteration is conducive to accurate local search and facilitates algorithm convergence.
