capabilities of the proposed DFPSO algorithm, the results for the complex multimodal functions are also given in Table 4。 Similarly, the statistical results by using the Wilcoxon rank sum test for the basic multimodal functions and the complex multimodal functions are also listed in Table 3 and Table 4, respectively。 In addition, it is not true that the DFPSO algorithm can obtain the better optimization results for all the benchmark functions。 From Table 4, one can see that the number of functions where the ALC-PSO algorithm or the SPSO2007 algorithm can obtain the better results is more than the one of functions where the DFPSO algorithm can achieve the better results。 One reason is that the DFPSO algorithm is motivated and designed for the problem of odor source localization, which means that we develop the DFPSO algorithm according to the characteristics of the odor source localization problem。 If the characteristics of benchmark functions are different from the problem of odor source localization, the DFPSO algorithm maybe produces the worse results。

4。2。3。Statistical  results  on  twenty-five  benchmark functions

Usually, both single-problem and multi-problem statistical analysis procedures can be used to contrast the results achieved by the different algorithms [10,17]。 From Tables 2–4, one can see that the single-problem statistical analysis pro- cedure has been employed to deal with the results obtained over 50 runs for each given function and per pair of algorithms。 In the following, we will make use of the multi-problem analysis to cope with the results shown in Tables 2–4 for the DFPSO algorithm and the comparison algorithms。 The statistical results computed by the Wilcoxon signed ranks test based on the KEEL software [2,1] are given in Table 5 where the DFPSO algorithm shows a significant improvement over the PSO algo-

rithm, the PSOw algorithm, the GPSO algorithm, and the SPSO2011 algorithm with a level of significance a ¼ 0:05, and  over

the SPSO2007 algorithm and the CLPSO algorithm with a level of significance a ¼ 0:1。 It should be pointed out that the mean function errors in Tables 2–4 have been normalized to the interval [0,1] in order to not prioritize any function。

Table 3

The function error values on the basic 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

f11 meana 5。66E+03 3。44E+03 1。51E+03 1。32 4。37E+03 5。06E+03 5。02E+03 3。78E+01

bestb 4。01E+03 2。05E+03 1。07E+03 2。33E—02 2。19E+03 2。95E+03 2。68E+03 3。11E+01

stdc 8。54E+02 6。59E+02 2。70E+02 1。70 7。95E+02