p value – 3。01E—11⁄ 7。69E—08⁄ 3。01E—11⁄ 3。01E—11⁄ 3。01E—11⁄ 3。01E—11⁄ 3。01E—11⁄

Summary func bf – 7 5 5 6 6 6 5

func wg – 1 2 2 1 2 2 3

a   mean: The value denotes the mean function error based on 50 runs。

b   best: The value is the minimal function error among 50  runs。

c    std: The value denotes  the  standard error based  on 50  runs。

d p value: The value is a p value of a two-sample Wilcoxon rank sum test for the function error between the DFPSO algorithm and the other comparison algorithms in the corresponding column。 The p value being smaller means that the results obtained by the DFPSO algorithm are significant compared with the ones achieved by the other    algorithms。

e

m:  The  difference  between  two  samples  is  significant  at  a  level  a ¼ 0:05。

f   func b: The number of functions where the DFPSO algorithm obtains significantly better results。

g   func w: The number of functions where the DFPSO algorithm obtains significantly worse results。

Table 4

The function error values on the complex multimodal functions based on 50 runs。 The lowest values in each line for mean and best are highlighted in boldface。

Functions Index DFPSO PSO PSOw ALC-PSO SPSO2007 GPSO SPSO2011 CLPSO

f19 meana 1。95E+02 5。29E+01 5。26E+07 7。35E+01 4。90E+01 1。71E+02 1。04E+03 1。23E+02

bestb 2。61E+01 1。38E—01 1。17E—01 3。46E—02 2。30E—02 2。54E—01 1。85E+01 4。23E+01

stdc