Table 8 Different GWO algorithms’ benchmark functions simulation results.
From: Hierarchical multi step Gray Wolf optimization algorithm for energy systems optimization
Function | Dim | Range | \(\:{f}_{\text{min\:}}\) |
|---|---|---|---|
\(\:{f}_{14}\left(x\right)={\left(\frac{1}{500}+{\sum\:}_{j=1}^{25}\:\frac{1}{j+{\sum\:}_{i=1}^{2}\:{\left({x}_{i}-{a}_{ij}\right)}^{6}}\right)}^{-1}\) | 2 | \(\:[-\text{65,65}]\) | 1 |
\(\:{f}_{15}\left(x\right)={\sum\:}_{i=1}^{11}\:{\left[{a}_{i}-\frac{{x}_{1}\left({b}_{i}^{2}+{b}_{i}{x}_{2}\right)}{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}\right]}^{2}\) | 4 | \(\:[-\text{5,5}]\) | 0.00030 |
\(\:{f}_{16}\left(x\right)=4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}\) | 2 | \(\:[-\text{5,5}]\) | −1.0316 |
\(\:{f}_{17}\left(x\right)={\left({x}_{2}-\frac{5.1}{4{\pi\:}^{2}}{x}_{1}^{2}+\frac{5}{\pi\:}{x}_{1}-6\right)}^{2}+10\left(1-\frac{1}{8\pi\:}\right)\text{c}\text{o}\text{s}{x}_{1}+10\) | 2 | \(\:[-\text{5,5}]\) | 0.398 |
\(\:\begin{array}{rr}{f}_{18}\left(x\right)=&\:\left[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\left(1914{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}\right)\right]\\\:&\:\:\times\:\left[30+{\left(2{x}_{1}-3{x}_{2}\right)}^{2}\times\:\left(<span class=' convertEndash' >18-32</span>{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2}\right)\right]\end{array}\) | 2 | \(\:[-\text{2,2}]\) | 3 |
\(\:{f}_{19}\left(x\right)=-{\sum\:}_{i=1}^{4}\:{c}_{i}\text{e}\text{x}\text{p}\left(-{\sum\:}_{j=1}^{3}\:{a}_{ij}{\left({x}_{j}-{p}_{ij}\right)}^{2}\right)\) | 3 | \(\:\left[\text{1,3}\right]\) | −3.86 |
\(\:{f}_{20}\left(x\right)=-{\sum\:}_{i=1}^{4}\:{c}_{i}\text{e}\text{x}\text{p}\left(-{\sum\:}_{j=1}^{6}\:{a}_{ij}{\left({x}_{j}-{p}_{ij}\right)}^{2}\right)\) | 6 | \(\:\left[\text{0,1}\right]\) | −3.32 |
\(\:{f}_{21}\left(x\right)=-{\sum\:}_{i=1}^{5}\:{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}\) | 4 | \(\:\left[\text{0,10}\right]\) | −10.1532 |
\(\:{f}_{22}\left(x\right)=-{\sum\:}_{i=1}^{7}\:{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}\) | 4 | \(\:\left[\text{0,10}\right]\) | −10.4028 |
\(\:{f}_{23}\left(x\right)=-{\sum\:}_{i=1}^{10}\:{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}\) | 4 | \(\:\left[\text{0,10}\right]\) | −10.5363 |
