ESA: a hybrid bio-inspired metaheuristic optimization approach for engineering problems

Abstract

In this paper, a hybrid bio-inspired metaheuristic optimization approach namely emperor penguin and salp swarm algorithm (ESA) is proposed. This algorithm imitates the huddling and swarm behaviors of emperor penguin optimizer and salp swarm algorithm, respectively. The efficiency of the proposed ESA is evaluated using scalability analysis, convergence analysis, sensitivity analysis, and ANOVA test analysis on 53 benchmark test functions including classical and IEEE CEC-2017. The effectiveness of ESA is compared with well-known metaheuristics in terms of the optimal solution. The proposed ESA is also applied on six constrained and one unconstrained engineering problems to evaluate its robustness. The results reveal that ESA offers optimal solutions as compared to the other competitor algorithms.

Appendix: Unimodal, multimodal, and fixed-dimension multimodal benchmark test functions

1.1 Unimodal benchmark test functions

1.1.1 Sphere model

$$\begin{aligned}&F_1(z)= \sum _{i=1}^{30} z_i^2\\&\quad -100 \le z_i \le 100 , \quad f_{\min } = 0 , \quad \text {Dim} = 30 \end{aligned}$$

1.1.2 Schwefel’s problem 2.22

$$\begin{aligned}&F_2(z)= \sum _{i=1}^{30} |z_i| + \prod _{i=1}^{30} |z_i| \\ \\&\quad -10 \le z_i \le 10 , \quad f_{\min } = 0 , \quad \text {Dim} = 30 \\ \end{aligned}$$

1.1.3 Schwefel’s problem 1.2

$$\begin{aligned}&F_3(z)= \sum _{i=1}^{30} \Big (\sum _{j=1}^i z_j\Big )^2 \\&\quad -100 \le z_i \le 100 , \quad f_{\min } = 0 , \quad \text {Dim} = 30 \\ \end{aligned}$$

1.1.4 Schwefel’s problem 2.21

$$\begin{aligned}&F_4(z)= \text {max}_i\{|z_i|, 1 \le i \le 30\}\\&\quad -100 \le z_i \le 100, \quad f_{\min } = 0, \quad \text {Dim} = 30 \\ \end{aligned}$$

1.1.5 Generalized Rosenbrock’s function

$$\begin{aligned}&F_5(z)= \sum _{i=1}^{29}[100(z_{i+1} - z_i^2)^2 + (z_i - 1)^2]\\&\quad -30 \le z_i \le 30 , \quad f_{\min } = 0 , \quad \text {Dim} = 30 \\ \end{aligned}$$

1.1.6 Step function

$$\begin{aligned}&F_6(z)= \sum _{i=1}^{30}(\lfloor {z_i + 0.5}\rfloor )^2 \\&\quad -100 \le z_i \le 100 , \quad f_{\min } = 0 , \quad \text {Dim} = 30 \\ \end{aligned}$$

1.1.7 Quartic function

$$\begin{aligned}&F_7(z)= \sum _{i=1}^{30} iz_i^4 + \text {random}[0,1]\\&\quad -1.28 \le z_i \le 1.28 , \quad f_{\min } = 0 , \quad \text {Dim} = 30 \\ \end{aligned}$$

1.2 Multimodal benchmark test functions

1.2.1 Generalized Schwefel’s problem 2.26

$$\begin{aligned}&F_8(z)= v_{i=1}^{30} - z_i \text {sin} (\sqrt{|z_i|})\\&\quad -500 \le z_i \le 500 , \quad f_{\min } = -12569.5 , \quad \text {Dim} = 30 \\ \end{aligned}$$

1.2.2 Generalized Rastrigin’s function

$$\begin{aligned}&F_9(z)= \sum _{i=1}^{30} [z_i^2 - 10\text {cos}(2\pi z_i) + 10]\\&\quad -5.12 \le z_i \le 5.12 , \quad f_{\min } = 0 , \quad \text {Dim} = 30 \\ \end{aligned}$$

1.2.3 Ackley’s function

$$\begin{aligned}&F_{10}(z)= -20\text {exp} \bigg (-0.2\sqrt{\dfrac{1}{30}\sum _{i=1}^{30} z_i^2}\bigg ) \\&\quad -\, \text {exp}\bigg (\dfrac{1}{30}\sum _{i=1}^{30} cos(2\pi z_i)\bigg ) +20 +e\\&\quad -\,32 \le z_i \le 32 , \quad f_{\min } = 0 , \quad \text {Dim} = 30 \\ \end{aligned}$$

1.2.4 Generalized Griewank function

$$\begin{aligned}&F_{11}(z)= \dfrac{1}{4000} \sum _{i=1}^{30} z_i^2 - \prod _{i=1}^{30} \text {cos} \bigg (\dfrac{z_i}{\sqrt{i}}\bigg ) + 1\\&\quad -600 \le z_i \le 600 , \quad f_{\min } = 0 , \quad \text {Dim} = 30 \\ \end{aligned}$$

1.2.5 Generalized penalized functions

• $$\begin{aligned}&F_{12}(z)= \dfrac{\pi }{30}\{10\text {sin}(\pi x_1) + \sum _{i=1}^{29}(x_i - 1)^2\\&\quad \times \, [1 + 10\text {sin}^2(\pi x_{i+1})] + (x_n - 1)^2 \} \\&\quad + \sum _{i=1}^{30}u(z_i, 10, 100, 4) \\ \\&\quad -50 \le z_i \le 50 , \quad f_{\min } = 0 , \quad \text {Dim} = 30 \\ \end{aligned}$$
• $$\begin{aligned}&F_{13}(z)= 0.1\{\text {sin}^2(3\pi z_1) + \sum _{i=1}^{29}(z_i - 1)^2\\&\quad \times \,[1 + sin^2(3\pi z_i + 1)] + (z_n - 1)^2[1 + \text {sin}^2(2\pi z_{30})]\} \\&\quad + \sum _{i=1}^Nu(z_i, 5, 100, 4) \\&\quad -50 \le z_i \le 50 , \quad f_{\min } = 0 , \quad \text {Dim} = 30, \\ \end{aligned}$$

  where $x_i = 1 + \dfrac{z_i+1}{4}$

  $$\begin{aligned} u(z_i, a, k, m) = {\left\{ \begin{array}{ll} k(z_i - a)^m \quad \quad \quad z_i > a\\ 0 \quad \quad \quad \quad \quad \quad \quad -a<z_i<a\\ k(-z_i - a)^m \quad \quad z_i<-a \end{array}\right. } \\ \end{aligned}$$

1.3 Fixed-dimension multimodal benchmark test functions

1.3.1 Shekel’s Foxholes function

$$\begin{aligned}&F_{14}(z)= \bigg (\dfrac{1}{500} + \sum _{j=1}^{25}\dfrac{1}{j+\sum _{i=1}^2(z_i - a_{ij})^6}\bigg )^{-1} \\ \\&\quad -65.536 \le z_i \le 65.536 , \quad f_{\min } \approx 1 , \quad \text {Dim} = 2 \\ \end{aligned}$$

Table 24 Shekel’s Foxholes function $F_{14}$

1.3.2 Kowalik’s function

$$\begin{aligned}&F_{15}(z)= \sum _{i=1}^{11}\bigg [a_i - \dfrac{z_1(b_i^2+b_iz_2)}{b_i^2+b_iz_3+z_4}\bigg ]^2 \\ \\&\quad -5 \le z_i \le 5 , \quad f_{\min } \approx 0.0003075 , \quad \text {Dim} = 4 \\ \end{aligned}$$

1.3.3 Six-hump camel-back function

$$\begin{aligned}&F_{16}(z)= 4z_1^2 - 2.1z_1^4 + \dfrac{1}{3}z_1^6 + z_1z_2 - 4z_2^2 + 4z_2^4 \\ \\&\quad -5 \le z_i \le 5 , \quad f_{\min } = -1.0316285 , \quad \text {Dim} = 2 \\ \end{aligned}$$

1.3.4 Branin function

$$\begin{aligned}&F_{17}(z)= \bigg (z_2 - \dfrac{5.1}{4\pi ^2}z_1^2 + \dfrac{5}{\pi }z_1 - 6 \bigg )^2 + 10\bigg (1 - \dfrac{1}{8\pi }\bigg )\text {cos}z_1 + 10\\&\quad -5 \le z_1 \le 10 , \quad 0 \le z_2 \le 15 , \quad f_{\min } = 0.398 , \quad \text {Dim} = 2 \\ \end{aligned}$$

1.3.5 Goldstein–Price function

$$\begin{aligned}&F_{18}(z)= [1 + (z_1 + z_2 + 1)^2(19 - 14z_1 + 3z_1^2 \\&\quad -\, 14z_2 + 6z_1z_2 + 3z_2^2)] \\&\quad \times \, [30 + (2z_1 - 3z_2)^2 \\&\quad \times \, (18 - 32z_1 + 12z_1^2 + 48z_2 - 36z_1z_2 + 27z_2^2)] \\&\quad -\,2 \le z_i \le 2 , \quad f_{\min } = 3 , \quad \text {Dim} = 2 \\ \end{aligned}$$

1.3.6 Hartman’s family

• $$\begin{aligned}&F_{19}(z)= -\sum _{i=1}^4c_i \text {exp}\left( -\sum _{j=1}^3 a_{ij}(z_j - p_{ij})^2\right) \\&\quad 0 \le z_j \le 1 , \quad f_{\min } = -3.86 , \quad \text {Dim} = 3 \\ \end{aligned}$$
• $$\begin{aligned}&F_{20}(z)= -\sum _{i=1}^4c_i \text {exp}\left( -\sum _{j=1}^6 a_{ij}(z_j - p_{ij})^2\right) \\ \\&\quad 0 \le z_j \le 1 , \quad f_{\min } = -3.32 , \quad \text {Dim} = 6 \\ \end{aligned}$$

Table 25 Hartman function $F_{19}$

1.3.7 Shekel’s Foxholes function

• $$\begin{aligned}&F_{21}(z)= -\sum _{i=1}^5[(X - a_i)(X - a_i)^T + c_i]^{-1}\\&\quad 0 \le z_i \le 10 , \quad f_{\min } = -10.1532 , \quad \text {Dim} = 4 \\ \end{aligned}$$
• $$\begin{aligned}&F_{22}(z)= -\sum _{i=1}^7[(X - a_i)(X - a_i)^T + c_i]^{-1} \\&\quad 0 \le z_i \le 10 , \quad f_{\min } = -10.4028 , \quad \text {Dim} = 4 \\ \end{aligned}$$
• $$\begin{aligned}&F_{23}(z)= -\sum _{i=1}^{10}[(X - a_i)(X - a_i)^T + c_i]^{-1} \\&\quad 0 \le z_i \le 10 , \quad f_{min} = -10.536 , \quad Dim = 4 \\ \end{aligned}$$

Table 26 Shekel’s Foxholes functions $F_{21}, F_{22}, F_{23}$

Table 27 Hartman function $F_{20}$

1.4 CEC-2017 benchmark test functions

The detailed descriptions of 15 well-known CEC-2017 benchmark test functions (C1–C30) are mentioned in Table 28.

Table 28 IEEE CEC-2017 benchmark test functions