Table 10 Multi-objective benchmark functions (continue)
From: An improved artificial bee colony algorithm based on Bayesian estimation
Functions | Range | Functions expression, PF and PS |
|---|---|---|
UF6 | [0,1]\(\times [-1,1]^{n-1}\) | \(f_1=x_1+\max \{ 0,2(\frac{1}{2N}+\varepsilon )\sin (2N\pi x_1) \}\) |
\(\quad \quad \quad + \frac{2}{|J_1|}\left( 4\sum \nolimits _{j\in J_1}y_j^2-2\prod \nolimits _{j\in J_1}\cos \left( \frac{20y_j\pi }{\sqrt{j}}\right) +2\right) ,\ \varepsilon =0.1\) | ||
\(f_2=1-x_1+\max \{ 0,2\left( \frac{1}{2N}+\varepsilon \right) \sin (2N\pi x_1) \}\) | ||
\(\quad \quad \quad +\frac{2}{|J_2|}\left( 4\sum \nolimits _{j\in J_2}y_j^2-2\prod \nolimits _{j\in J_2}\cos \left( \frac{20y_j\pi }{\sqrt{j}}\right) +2\right) ,\ \varepsilon =0.1\) | ||
\(J_1=\{j|j\ {\text {is\ odd\ and}}\ 2\le j \le n\}\) | ||
\(J_2=\{j|j\ {\text {is\ even\ and}}\ 2\le j \le n\}\) | ||
\(y_j=x_j- \sin \left( 6\pi x_1+\frac{j\pi }{n}\right) ,\ j=2,\ldots ,n\) | ||
\({\text {PF}}:\ f_2=1-f_1,f_1\in \bigcup \nolimits _{i=1}^{N}[\frac{2i-1}{2N},\frac{2i}{2N}],\ N=2\) | ||
UF7 | [0,1]\(\times [-1,1]^{n-1}\) | \(f_1=\root 5 \of {x_1}+\frac{2}{|J_1|}\sum \nolimits _{j\in J_1}y_j^2\) |
\(f_2=1-\root 5 \of {x_1}+\frac{2}{|J_2|}\sum \nolimits _{j\in J_2}y_j^2\) | ||
\(J_1=\{j|j\ {\text {is\ odd\ and}}\ 2\le j \le n\}\) | ||
\(J_2=\{j|j\ {\text {is\ even\ and}}\ 2\le j \le n\}\) | ||
\(y_j=x_j-\sin \left( 6\pi x_1+\frac{j\pi }{n}\right) ,\ j=2,\ldots ,n\) | ||
\({\text {PF}}:\ f_2=1-f_1,\ 0\le f_1\le 1\) | ||
\({\text {PS}}:\ x_j=\sin \left( 6\pi x_1+\frac{j\pi }{n}\right) ,\ j=2,\ldots ,n,\ 0\le x_1\le 1\) |