ulations within a tolerance of 10—4 are shown in Fig。 9 and Fig。 10 for Griewank and Rastrigin, respectively。 In two figures, the

triangles denote the stability region for three algorithms, that is, three algorithms converge when the parameters fall into the triangles。

Remark 11。 It should be pointed out that the decision on piðkÞ are the same for three algorithms, that is, the Eq。 (3) is used to obtain piðkÞ。 But, three algorithms possess different controllers uiðkÞ。 In addition, because the problem of odor source localization is a two-dimension optimization problem, we consider the two-dimension functions in order to give a guidance for the parameter choice。

From Figs。 9,10, one can see that the proposed DFPSO algorithm can obtain the higher success rates (A wider stability re- gion with dark red) and lower average iterations (A wider stability region with dark blue) within a wider range of param- eters。 Hence, from more simulation results,  we  can  conclude  that  the  better  optimization  results  can  be  obtained  only  in the stability regions of three algorithms。 Moreover, the DFPSO algorithm provides a flexible mechanism of parameter choice to obtain the better results for a class of ill-posed optimization problems, which implies that the controller ui ðkÞ has an important impact  on  optimization results。  In  addition, one  can  also  see  that the  higher success  rates  and  the  lower   average

iterations are achieved for the DFPSO algorithm when the parameters fall into —2:5 < x < 0:2 and 0 < a < 10。

4。2。Twenty-five benchmark functions

In this subsection, we will test the performance capabilities of the DFPSO algorithm based on  twenty-five  benchmark functions listed in Table 1。 It is worth mentioning that f1 — f10 are the unimodal functions where f6 — f10 are from the refer- ence [38] (The readers can refer to [38] and visit the website http://www。ntu。edu。sg/home/EPNSugan/to obtain the bench- mark functions and the corresponding Matlab codes。)。 For the multimodal functions, f11 — f18 are the basic multimodal functions while f19 — f25 are the complex multimodal functions。 In the simulations, the number of dimensions of all functions is set as 30 and the maximum number of function evaluations is 1 × 10 。 Each test is repeated 50 times independently。 All simulations are carried out on a laptop with two Cores (TM) Duo CPU T6670 running at 2。20 GHz with 3 GB of RAM。 The operation  system  is  Windows  Vista  and  the  software  is  Matlab。

In order to illustrate the performance capabilities of the proposed DFPSO algorithm, seven state-of-the-art PSO variants, which include the global version PSO algorithm with a fixed inertia weight x ¼ 0:4 [36], the global version PSOw algorithm where x linearly changes from 0。9 to 0。4 [37], the ALC-PSO algorithm [7], the SPSO2007 algorithm [33,5], the generalized PSO algorithm [14], the CLPSO algorithm [21], and the SPSO2011 algorithm [33,46], are used as comparison algorithms。

The reasons of choosing the aforementioned PSO variants are stated in the following。 From the perspective of the ‘‘deci- sion-control mechanism’’, if the PSO algorithm and the PSOw algorithm are regarded as root nodes, the aforementioned algo- rithms will be pided into two classes。 The first class, which includes the GPSO algorithm and the DFPSO algorithm, is to design a new controller ui ðkÞ to adjust the particle’s behavior to improve the optimization performance of the PSO algorithm while the second class, which consists of the CLPSO algorithm, the SPSO2007 algorithm, the SPSO2011 algorithm, and the ALC-PSO algorithm, is to design a new pi ðkÞ to improve the optimization performance of the PSO   algorithm。