A novel hybrid PSO–GWO algorithm for optimization problems

Abstract

In this study, we propose a new hybrid algorithm fusing the exploitation ability of the particle swarm optimization (PSO) with the exploration ability of the grey wolf optimizer (GWO). Our approach combines two methods by replacing a particle of the PSO with small possibility by a particle partially improved with the GWO. We have evaluated our approach on five different benchmark functions and on three different real-world problems, namely parameter estimation for frequency-modulated sound waves, process flowsheeting problem, and leather nesting problem (LNP). The LNP is one of the hard industrial problems, where two-dimensional irregular patterns are placed on two-dimensional irregular-shaped leather material such that a minimum amount of the material is wasted. In our evaluations, we compared our approach with the conventional PSO and GWO algorithms, artificial bee colony and social spider algorithm, and as well as with three different hybrid approaches of the PSO and GWO algorithms. Our experimental results reveal that our hybrid approach successfully merges the two algorithms and performs better than all methods employed in the comparisons. The results also indicate that our approach converges to more optimal solutions with fewer iterations.

Appendix

In Sect. 3.1, we have given Eq. 12 for the calculation of cost and Fig. 2 for illustrating the procedure with sample polygons. In this appendix, we give the details of distance calculations needed to compute the cost.

A polygon consists of multiple edges. To calculate minimum distance of a vertex point to a polygon, $d_{\min }$, we need to calculate the minimum distances from that point to all different edges of the polygon. The minimum of these distances gives us $d_{\min }$. For this reason, we will below explain how the minimum distance of a vertex point to one of the edges is calculated.

Fig. 6

Three different cases for a point, $P_C$, and an edge, $P_AB$

In Fig. 6, we depicted three different cases for a point, $P_C$, and an edge, $P_{AB}$, where the intersection of the perpendicular line from $P_C$ to $P_{AB}$ falls into three different sections, namely: (1) outside the $P_{AB}$ and close to $P_A$, (2) over $P_{AB}$, and (3) outside the $P_{AB}$ and close to $P_B$. Here, we define a parameter t whose magnitude is calculated by dividing ${d_A}$ to ${d_{AB}}$, but we determine its sign as follows. When the intersection of the perpendicular line is outside the $P_{AB}$ and close to $P_A$, t is negative, for the other cases t is positive. In other words, $P_A$ is considered as a reference point. We can formulate the coordinates of the perpendicular intersection point, D, in terms of t and coordinates of points $P_A$ and $P_B$ as in Eq. 13. Our aim is to find t, and for this purpose, we first express the distance, $|DP_C|$, between D and $P_C$ as shown below:

$$\begin{aligned} D_{X}& = P_{A_{X}}+t \times (P_{B_{X}}-P_{A_{X}}), \nonumber \\ D_{Y}& = P_{A_{Y}}+t \times (P_{B_{Y}}-P_{A_{Y}}), \end{aligned}$$

(13)

$$\begin{aligned} |DP_C|& = \sqrt{ \begin{array}{l} ((P_{A_{X}}+t \times (P_{B_{X}}-P_{A_{X}}))-P_{C_{X}})^{2}+ \\ ((P_{A_{Y}}+t \times (P_{B_{Y}}-P_{A_{Y}}))-P_{C_{Y}})^{2}. \end{array}} \end{aligned}$$

(14)

Since $DP_C$ segment is perpendicular to the given edge only when $|DP_C|$ is minimum, we can find an equation to calculate t by minimizing $|DP_C|$. To minimize $|DP_C|$, we can take the first derivative of Eq. 14 in terms of t and equate it to zero. For the sake of simplicity, we can use the derivative of the terms inside the square root sign, since the minimization of those terms will also minimize $|DP_C|$. Such a derivation will result in Eq. 15. When we equate Eq. 15 to zero, we find Eq. 16 for t.

$$\begin{aligned}&\frac{\text {d}}{{\text {d}}t} |DP_C|^2 = 2 \times ((P_{A_{X}}+t \times (P_{B_{X}}-P_{A_{X}}))-P_{C_{X}}) \times (P_{B_{X}}-P_{A_{X}})\nonumber \\&\quad +2 \times ((P_{A_{Y}}+t \times (P_{B_{Y}}-P_{A_{Y}}))-P_{C_{Y}}) \times (P_{B_{Y}}-P_{A_{Y}}), \end{aligned}$$

(15)

$$\begin{aligned}&t = \frac{(P_{C_{X}}-P_{A_{X}}) \times (P_{B_{X}}-P_{A_{X}})+(P_{C_{Y}}-P_{A_{Y}}) \times (P_{B_{Y}}-P_{A_{Y}})}{(P_{B_{X}}-P_{A_{X}})^{2}+(P_{B_{Y}}-P_{A_{Y}})^{2}}. \end{aligned}$$

(16)

Given a point $P_C$ and an edge, after calculating t with Eq. 16, we can find minimum distance of $P_C$ to the edge, $d_{C_{\text {min}}}$, using Eq. 17. Note that, for the Cases 1 and 3 in Fig. 6, closest distances are to the points $P_A$ and $P_B$, respectively.

$$\begin{aligned} d_{C_{\text {min}}} = {\left\{ \begin{array}{ll} \sqrt{(P_{A_{X}}-P_{C_{X}})^{2}+(P_{A_{Y}}-P_{C_{Y}})^{2}} &{} t<0 \\ \sqrt{ \begin{array}{l} ((P_{A_{X}}+t \times (P_{B_{X}}-P_{A_{X}}))-P_{C_{X}})^{2}+ \\ ((P_{A_{Y}}+t \times (P_{B_{Y}}-P_{A_{Y}}))-P_{C_{Y}})^{2} \end{array}} &{} 0 \le t \le 1 \\ \sqrt{(P_{B_{X}}-P_{C_{X}})^{2}+(P_{B_{Y}}-P_{C_{Y}})^{2}} &{} t>1. \end{array}\right. } \end{aligned}$$

(17)