# An Improved Squirrel Search Algorithm for Global Function Optimization
## 3 The improved Squirrel Search Algorithm
### The Jumping Search Method
#### 3.2.3 Characters of the Jumping Search Method
ISSA has a smaller average rank, though it does not outperform the progressive search method and the jumping search method. In conclusion, the jumping search method has the best convergence speed and the progressive search method performs best on maintaining the population diversity. Compared with SSA, both have obvious advantages in convergence accuracy. ISSA combines the two methods together, improves the convergence speed and the convergence accuracy and maintains the population diversity as well. ISSA can find the global optimal of more benchmark functions and has the best comprehensive performance. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{4}{c}{**Rank**} \\ \cline{2-5}  & **SSA** & **Jumping Search** & **Progressive Search** & **ISSA** \\ \hline F1 & 1.5 & 3 & 1.5 & 4 \\ F2 & 4 & 1.5 & 3 & 1.5 \\ F3 & 4 & 2 & 2 & 2 \\ F4 & 4 & 2 & 2 & 2 \\ F5 & 4 & 2 & 2 & 2 \\ F6 & 2 & 4 & 1 & 3 \\ F7 & 4 & 3 & 1 & 2 \\ F8 & 2.5 & 2.5 & 2.5 & 2.5 \\ F9 & 3 & 4 & 1 & 2 \\ F10 & 3 & 4 & 1 & 2 \\ F11 & 4 & 1.5 & 3 & 1.5 \\ F12 & 4 & 1.5 & 3 & 1.5 \\ F13 & 4 & 1.5 & 3 & 1.5 \\ F14 & 4 & 1.5 & 3 & 1.5 \\ F15 & 3 & 1.5 & 4 & 1.5 \\ F16 & 4 & 1.5 & 3 & 1.5 \\ F17 & 4 & 1.5 & 3 & 1.5 \\ F18 & 4 & 1 & 3 & 2 \\ F19 & 4 & 1.5 & 3 & 1.5 \\ F20 & 4 & 1.5 & 3 & 1.5 \\ F21 & 4 & 1.5 & 3 & 1.5 \\ \hline
**Total rank** & 75 & 44 & 51 & 40 \\
**Average rank** & 3.5714 & 2.0952 & 2.4286 & 1.9048 \\
**Sort** & 1 & 3 & 2 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The results of the Friedman test. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(i\)} & \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{\(z=(R_{i}-R_{4})/\sqrt{\frac{k(k+1)}{\alpha}}=(R_{i}-R_{4})/\sqrt{\frac{4\times(4+1)}{ 6\times 21}}=(R_{i}\)}\) & \multirow{2}{*}{\(P_{i}\)} & \multirow{2}{*}{\(\alpha/(k-i)\)} \\  & & & & \\ \hline
1 & SSA & \((3.5714-1.9048)/0.3984=4.1832\) & 2e–05 & 0.0167 \\
2 & progressive search & \((2.4286-1.9048)/0.3984=1.3148\) & 0.1885 & 0.0250 \\
3 & jumping search & \((2.0952-1.9048)/0.3984=0.4779\) & 0.6348 & 0.0500 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The results of the Holm test. ### Performance Compared with Other Algorithms

To verify the advantages of ISSA, we compared it with five algorithms with better optimization results: the improved differential evolutionary algorithm--MDE (modified differential evolution, with self-adaptive parameters method) [37], the improved gravitational search algorithm--IGSA/PSO (an improved gravitational search algorithm for green partner selection in virtual enterprises) [38], the improved artificial bee colony algorithm--distABC (artificial bee colony algorithm, with distribution-based update rule) [39], the improved particle swarm optimization--ADN-RSN-PSO (all-dimension neighborhood based particle swarm optimization with randomly selected neighbors) [40] and the improved grey wolf optimization algorithm--PSO-GWO (an improved hybrid grey wolf optimization algorithm) [41]. For each algorithm mentioned above, the population number is 30, the total number of evolution times is 24,000 and each method runs 30 times independently to ensure the comparison is fair enough. The relevant parameters are set as follows:

ISSA: \(n=10\); \(N_{fg}=3\);

MDE: the crossover probability \(CR=0.4\), the mutation probability \(F\) is determined by the random number between 0 and 1. IGSA/PSO: the gravitational constant \(G_{0}=100\), \(\alpha=20\);

distABC: \(limit=\) (the population number \(\times\) dimension)/2;

ADN-RSN-PSO: the weight factor \(w=0.7298\), \(c1=c2=2.05\);

PSO-GWO: the weight factor \(c_{1}=c_{2}=2.05\); \(a_{ini}=2\), \(a_{fin}=0\); \(r_{1}\), \(r_{2}\), \(r_{3}\), \(r_{4}\) are all the random numbers between 0 and 1. Table 8 shows the optimization results of low dimensional functions (F1-F10) and Table 11 shows the ones of high dimensional functions (F11-F21).