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Group search optimizer with intraspecific competition and lévy walk
Knowledge-Based Systems xxx (2014) xxx–xxx
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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Short Communication
Group search optimizer with intraspecific competition and lévy walk Y.Z. Li a, Q.H. Wu a,b,⇑, M.S. Li a a b
School of Electrical Engineering, South China University of Technology, Guangzhou 510641, China Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, UK
a r t i c l e
i n f o
Article history: Received 9 April 2014 Received in revised form 26 August 2014 Accepted 9 September 2014 Available online xxxx Keywords: Evolutionary algorithm Intraspecific competition Lévy walk Group search optimizer Standard benchmark functions
a b s t r a c t This short communication presents a nature-inspired optimization algorithm, group search optimizer with intraspecific competition and lévy walk (GSOICLW). The mechanism of intraspecific competition (IC) and the searching strategy of lévy walk (LW) are incorporated to the original group search optimizer (GSO) algorithm. It has been proved that GSOICLW shows a significant improvement to GSO after its test against standard benchmark functions. Ó 2014 Published by Elsevier B.V.
1. Introduction Many naturally inspired algorithms have emerged based on biological evolution and animal behaviors, which are referred to as evolutionary algorithms (EAs). Many EAs draw inspiration coming from the process of biological evolution, such as genetic algorithm (GA) [1], genetic programming (GP) [2], evolutionary programming (EP) [3] and evolutionary strategy (ES) [4]. Other EAs, simulating animal behaviors, gains much more attention. For instance, particle swarm optimization (PSO) mimics social behavior of a bird flock [5], ant colony optimization (ACO) simulates the ecological behavior of ants as for foraging [6], artificial bee colony (ABC) learns the behavior of honeybees in terms of collecting nectar [7], bacterial foraging algorithm (BFA) simulates bacterial foraging patterns [8], cuckoo optimization algorithm (COA) is inspired by the cuckoo birds’ behavior as for egg laying and breeding [9], biogeographybased optimization (BBO) is motivated by biogeography, studying the distribution of biological species [10], fruit fly optimization (FFO) algorithm is developed by simulating the food finding behavior of the fruit fly [11], group search optimizer (GSO) mimics animal’s behavior of hunting for resource [12] and et al. Among the EAs mentioned above, GSO is famous for its global searching ability, and it has been applied in many engineering optimization problems [13]. GSO consists of three types of group members: the producer, scroungers and rangers. In each genera⇑ Corresponding author at: School of Electrical Engineering, South China University of Technology, Guangzhou 510641, China. E-mail address: [email protected] (Q.H. Wu).
tion, the member conferred the best fitness value is chosen as the producer, and a number of members except the producer are randomly selected as scroungers, then the rest of members are rangers. The producer is located in the most promising area and adopts animal scanning to seek the optimal resource. Scroungers perform area copying to join the resource found by the producer, and do local searching around it. However, rangers employ ranging behavior by random walk (RW) in the searching space to increase the GSO’s chance to escape local optima. Therefore, GSO has demonstrated better performance in global searching, however, its local searching ability is not desirable, as shown in the modest performance on optimizing unimodal benchmark functions [12]. In order to improve GSO’s local searching ability while maintaining its merit in global searching, we introduce intraspecific competition (IC) [14] and lévy walk (LW) into GSO. IC is a particular form of biological phenomenon in which population members compete for scarce resources. Each member in GSO is always hunting for the optimal resource (definitely scarce) which is assumed to be at the producer’s position, therefore, IC exists inevitably in GSO’s searching process and it stimulates scroungers to closely search the local area around the producer. Moreover, it will put a diversifying effect on group members, i.e., intenser competition caused by increasing population density will compel more rangers to choose to seek another resource [15]. Rangers perform RW in GSO, however, some biologists have claimed that lévy walk (LW) is more efficient than RW for searching resources [16]. Then we replace RW with LW for rangers in GSO in order to gain better searching ability. Therefore, an improved algorithm, GSOICLW, is proposed, which incorporates IC and LW
http://dx.doi.org/10.1016/j.knosys.2014.09.005 0950-7051/Ó 2014 Published by Elsevier B.V.
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
2
Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx Producer
Scrounger
Ranger
Intraspecific Competition Levy Walk
GSOICLW
GSO
Fig. 2. The illustration of IC and LW in GSOICLW.
Fig. 1. Fuzzy membership functions as for IC and Non-IC.
Table 1 Results of mean (l), standard deviation (r) and h value of GSO, GSOLW, GSOIC, GSOICLW, PBO and BFAVP on f 1 f 21 considering 10 dimensions.
f1
l r h
f2
l r h
f3
l r h
f4
l r h
GSO
GSOLW
GSOIC
GSOICLW
PBO
2:11 1013
1:10 1013
1:37 1023
0
1:19 105
4:45 1013
13
13
23
0
5
1:21 1012 1
7:89 10 1
2:70 10 1
5:64 10 1
BFAVP
N/A
4:30 10 1
5:31 108
5:18 108
4:26 1013
0
3:24 103
1:35 107
8
9
13
0 N/A
6:25 10 1
4
1:85 107 1
6:61 10 1
8:69 10 1
1:47 10 1
3:13 105
1:77 105
1:95 107
1:72 108
4:66 104
2:63 105
4:75 105 1
1:81 105 1
2:16 107 1
2:39 108 N/A
2:32 104 1
3:96 105 1
4:34 105
3:83 105
1:44 106
4:99 108
7:94 103
4:65 105
3:71 105 1
2:72 105 1
1:92 106 1
7:28 108 N/A
1:41 103 1
3:32 105 1
l r
5.31
1.43
6:98 103
1:29 105
2.79
2.77
4.05
1.55
4.22
1
1
5:55 105 N/A
6.33
h
5:03 103 1
1
1
f6
l r
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
f7
l r
f5
h
h f8
l r h
f9
l r
f 10
l r
h
h f 11
l r h
f 12
l r h
f 13
l r h
f 14
l r h
5:87 103
5:14 103
2:52 104
3:44 105
5:62 103
1:22 103
3
3
3
5
3
7:47 103 1
2:61 10 1
2:20 10 1
1:81 10 1
1:54 10 N/A
3:24 10 1
3:72 1016
1:14 1016
4:40 1028
0
8:20 107
4:18 1016
16
16
28
0
7
2:80 10 1
1:64 10 1
0.99 0.32 1
0.58 0.26 1
1:50 10 1 0.48 0.19 1
N/A
4:10 10 1
8:67 1016 1
0:22 0:06 N/A
0.40 0.65 1
0.61 0.42 1
1
1
1
1
1
1
1:47 1016 0
1:38 1016 0
6:84 1017 0
1:84 1018 N/A
2:76 109 0
5:83 1017 0
4:64 109
1:73 1010
5:12 1015
3:27 1018
1:34 101
3:01 109
2:43 108 1
5:31 1010 1
1:85 1015 1
1:32 1018 N/A
6:37 102 1
7:77 109 1
2:93 1010
8:74 1011
1:73 1017
3:07 1019
1:25 101
1:58 109
7:59 1010 1
2:28 1011 1
1:12 1017 1
1:12 1019 N/A
6:35 102 1
3:26 109 1
4:80 1014
9:96 1015
1:15 1021
2:80 1023
5:95 106
5:36 1014
3:10 1014 1
1:87 1015 1
6:28 1021 1
1:33 1023 N/A
1:64 106 1
9:45 1014 1
4189:83
4189:83
4189:83
4189:83
4189:83
4189:83
7:53 106 0
5:32 107 0
6:54 108 0
9:92 109 N/A
5:36 105 0
1:47 105 0
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
3
Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx Table 1 (continued) GSO
l r
f 15
h
l r
f 16
h
l r
f 17
h
l r
f 18
h
l r
f 19
h
l r
f 20
h
l r
f 21
h
GSOLW
1:48 10
7
1:79 10 1
7
1:09 10
GSOIC 7
1:45 10 1
GSOICLW 13
8:97 10
7
1:10 10 1
PBO
16
3:08 107
1:28 10 N/A
3
1:04 10 1
3:50 107 1
8:88 10
13
BFAVP 3
5:29 10
17
6:03 103
5:51 103
5:60 104
4:94 105
5:59 103
5:54 103
3
3
4
5
3
1:95 103 1
2:35 10 1
1:95 10 1
3:49 10 1
1:87 10 N/A
3:49 10 1
5:71 1016
1:16 1017
8:73 1027
1:57 1032
1:41 108
3:81 1016
15
17
27
32
8
6:49 1016 1
2:16 10 1
2:42 10 1
3:64 10 1
1:07 10 N/A
4:88 10 1
2:22 1015
1:96 1016
1:27 1026
1:35 1032
3:31 106
9:87 1015
4:46 1015 1
3:99 1016 1
3:08 1026 1
5:56 1033 N/A
1:10 106 1
1:81 1014 1
3:29 107
9:52 108
1:14 108
8:32 1010
8:75 104
6:85 106
1:16 106 1
1:56 107 1
6:21 108 1
2:36 109 N/A
7:19 104 1
3:66 105 1
2:33 107
1:35 107
3:08 109
3:13 1010
0.83
0.11
3:36 107 1
3:08 107 1
6:10 108 1
6:57 109 N/A
9:47 102 1
4:57 101 1
4:06 102
3:79 102
8:89 103
2:52 103
1:78 101
1:25 101
3:37 102 1
3:08 102 1
6:43 103 1
2:83 103 N/A
1:38 101 1
1:69 101 1
The bold fonts of values indicate the best results.
based on GSO. We have evaluated GSOICLW in comparison with GSO on a set of 21 standard unimodal and multimodal benchmark functions, presented in Appendix. Simulation results prove GSOICLW achieves better local searching ability while maintaining GSO’s advantage as for global searching. 2. Group search optimizer with intraspecific competition and lévy walk 2.1. Intraspecific competition IC is defined as the struggle between population members for the scarce resource, which is divided into contest and scramble competition [14]. In contest competition, the successful competitor obtains all it needs while the rest of competitors are deprived of such resources. However, scramble competition happens when population members are crowded around the limited resources, which are not monopolized by successful competitors. This sort of IC, scramble competition, stimulates population members to compete for resources seriously. Moreover, the intensity of IC is densitydependent, i.e., competition will be more serious if population members are increasingly crowding around resources [14,17]. GSO mainly adopts the joining policy to hunt for the scarce resource [12], consequently, IC occurs in the form of scramble competition as for GSO when population members, mainly scroungers, are crowded around the producer. Here the index f ranged in ð0; 1Þ proposed by Zhan et al. [18] can well describe the population’s crowdedness in some EAs dominated by the leading member with the best fitness value, e.g., PSO and GSO. The detailed calculation steps of this index f are listed as follows. (1) Compute the mean distance di of each particle i with its position xi to all the other particles using the Euclidian metric
di ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N XD 1 X 2 ðxk xkj Þ k¼1 i N 1 j¼1;j–i
ð1Þ
where N is the population size and D is the number of particles’ dimensions, respectively.
(2) Denote di of the leading particle as dg , determine the maximum distance dmax and minimum distance dmin by comparing all di in (1). Then the index f can be defined as
f ¼
dg dmin 2 ½0; 1: dmax dmin
ð2Þ
The above equation shows that if f is small, the leading member is surrounded closely by other members [18]. Therefore, the index f is used to describe the GSO population’s crowdedness. Specifically, when f is small, GSO members are crowded around the producer, then IC happens. However, it maybe not reasonable to set an exact limen of the index f, e.g. 0.2, to determine whether GSO is in IC or not, because state transition from non-IC to IC would be nondeterministic and fuzzy, just as the phenomenon shown in [18]. Therefore, we adopt the method of fuzzy classification to determine the phrase of IC or non-IC as for GSO. We set the phrase of IC and non-IC as S1 and S2 , and their fuzzy membership functions are donated as lS1 and lS2 , depicted in Fig. 1. If lS1 ðf Þ > lS2 ðf Þ, we can say the members in GSO are in IC, and r3 ¼ ðr 31 ; r 32 ; . . . ; r 3D Þ , presented in Eq. (7) in [12], is a random vector with higher value ranged in ðf; 1Þ for scroungers to manifest this serious competition. Here r3 is called the scrounging coefficient. On the other hand, if lS1 ðf Þ < lS2 ðf Þ, GSO members are in the phrase of non-IC, then r3 is a random vector with lower value anged in ð0; fÞ. Therefore, r3 is formulated as the following.
( r 3i ¼
random 2 ðf; 1Þ random 2 ð0; fÞ
lS1 ðf Þ P lS2 ðf Þ i ¼ 1; 2; . . . ; D lS1 ðf Þ < lS2 ðf Þ
ð3Þ
where f is threshold for the scrounging coefficient to manifest IC. 2.2. Diversifying effect of intraspecific competition Richard has proposed that IC drives the diversifying effect within a population, i.e., the increasing population density leads to reduced prey availability, then some individuals will resort to alternative prey types [15]. Similar phenomena caused by IC have been also observed by other biologists [19,20]. In addition, intenser
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
4
Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx
Table 2 Results of mean (l), standard deviation (r) and h value of GSO, GSOLW, GSOIC, GSOICLW, PBO and BFAVP on f 1 f 21 considering 30 dimensions. GSO f1
l r h
f2
l r h
f3
l r h
f4
l r h
f5
l r h
f6
l r h
f7
l r h
f8
l r h
f9
l r h
f 10
l r h
f 11
l r h
f 12
l r h
f 13
l r h
f 14
l r h
f 15
l r h
f 16
l r h
f 17
l r h
f 18
l r h
f 19
l r h
f 20
l r h
GSOLW
GSOIC 1:34 10
8:54 10
2:86 10
1:26 105
5:23 1012 1
2:70 1015 N/A
5:53 104 1
1:63 105 1
5:56 105
2:99 105
1:22 106
1:38 107
2:27 102
1:54 103
5
5
6
7
3
1:83 10 1
2:86 103 1
4:51 10 1
1:96 10 N/A
15
BFAVP
5:65 109 1
3:02 10 1
12
PBO
3:81 10
9:91 10 1
9
GSOICLW
1:07 108 1
6:75 10
9
4
36.81 11.30 1
32.59 9.32 1
11.26 6.98 1
2:21 1:02 N/A
20.51 9.68 1
115.34 75.4 1
1:07 101
5:63 102
1:38 104
2:57 105
2:96 102
5:59 101
3:16 102 1
2:79 102 1
2:63 104 1
6:28 105 N/A
2:70 103 1
1:56 101 1
57.68 34.23 1
49.65 29.63 1
29.59 11.23 1
8:35 0:69 N/A
88.03 56.45 1
39.27 24.68 1
0
0
6:67 102
3:33 102
3:68 10 0
0
0
1
0
N/A
2:53 10 1
1:83 102 1
7:13 102
5:55 102
4:58 103
1:64 104
4:26 102
3:27 102
2
2
3
4
2
1:87 10 1
1:39 102 1
3:33 102 1:83 10 1
2:67 10 1
2
0 3
1:88 10 1
8:36 10 1
7:73 10 N/A
6:68 1011
5:62 1011
3:17 1017
2:98 1019
7:10 105
4:86 107
1:08 1010 1
2:77 1011 1
5:56 1017 1
1:88 1019 N/A
1:42 105 1
5:22 107 1
1.23 0.78 1
1.18 0.74 1
0.97 0.61 1
0:67 0:12 N/A
1.71 0.95 1
1.11 0.52 1
1
1
1
1
1
1
4:50 1011 0
4:80 1013 0
3:13 1015 0
2:95 1016 N/A
3:02 108 0
9:33 1010 0
2:20 105
1:23 105
5:65 1011
9:50 1012
2.16
2.07
2:58 105 1
1:87 105 1
1:73 1010 1
1:14 1012 N/A
0.57
3.84
1
1
2:31 105
1:95 105
3:72 109
2:45 1011
2.24
5.84
2:40 105 1
5:84 106 1
6:43 109 1
5:93 1011 N/A
0.54
17.50
1
1
2:55 108
1:51 109
6:59 1011
6:11 1012
3:55 104
7:70 106
8:25 108 1
2:37 109 1
2:19 1011 1
2:19 1012 N/A
8:06 105 1
1:83 105 1
12569:5
12569:5
12569:5
12569:5
7:30 102 0
6:48 104 0
2:99 105 0
6:43 106 N/A
12566:3 1.39 1
5:72 102 0
5
5
6
7
12569:5
1:75 10
2:43 10
1:27 10
3:38 102
1:96 103
4:44 105 1
1:31 105 1
8:54 106 1
2:22 107 N/A
8:31 102 1
1:98 103 1
3:36 102
2:09 102
1:96 102
6:08 103
4:30 102
2:31 102
3:37 102 1
2:37 102 1
1:03 102 1
8:54 103 N/A
4:33 102 1
2:51 102 1
1:84 1011
1:50 1012
6:17 1014
1:94 1015
1:45 107
7:94 108
5:22 1011 1
2:41 1012 1
3:38 1014 1
6:27 1016 N/A
2:98 108 1
9:15 108 1
1:16 109
5:79 1010
7:32 1011
3:01 1012
4:28 105
1:52 106
9
10
11
12
6
8:99 10 1
2:48 106 1
2:95 10
4:66 10 1
2:61 10 1
2:78 10 1
1:26 10 N/A
2:61 103
1:41 103
3:26 104
6:71 105
7:61 103
2:17 103
3
3
4
5
3
3:92 10 1
1:92 103 1
10.67 1.56 1
7.28 1.04 1
4:43 10 1 8.74 1.19 1
2:79 10 1
1:91 10 1
1:12 10 N/A
7.27 0.96 1
1.58 0.59 1
0:99 0:12 N/A
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
5
Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx Table 2 (continued)
f 21
l r h
GSO
GSOLW
GSOIC
GSOICLW
PBO
BFAVP
2.17 0.72 1
2.05 0.51 1
1.19 0.43 1
0:39 0:28 N/A
1.79 0.38 1
4.51 0.98 1
The bold fonts of values indicate the best results.
Table 3 Results of mean (l), standard deviation (r) and h value of GSO, GSOLW, GSOIC, GSOICLW, PBO and BFAVP on f 1 f 21 considering 50 dimensions. GSO f1
l r h
f2
l r h
GSOLW 6
GSOIC 6
GSOICLW 8
10
PBO
BFAVP 3
3:91 103
3:29 10
1:64 10
1:32 10
3:39 10
9:38 10
5:01 106 1
1:37 106 1
5:18 108 1
4:91 1010 N/A
1:49 103 1
2:90 103 1
1:54 103
1:03 103
6:14 104
5:14 105
1:35 103
3:24 103
3
4
4
5
2
6:25 104 1
1:33 10 N/A
4:73 10 1
335.9 90.01 1
62:61 33:35 N/A
856.32 162.35 1
1345.21 522.08 1
3:21 10 1
9:73 10 1
1:82 10 1
799.49 157.69 1
777.20 121.28 1
f3
l r
f4
l r
1.08
0.52
0.29
2:07 103
0.66
3.51
0.31
0.19
0.11
0.56
0.95
h
1
1
1
4:76 103 N/A
1
1
f5
l r
196.03 625.82 1
129.72 300.44 1
89.83 207.65 1
20:21 10:01 N/A
189.91 707.66 1
164.96 624.14 1
f6
l r
0.73 0.69 1
0.33 0.54 1
0.12 0.25 1
0 0 N/A
0.23 0.36 1
1.33 1.32 1
f7
l r
1:41 101
8:56 102
6:32 102
5:30 103
3:63 101
1:47 102
6:23 101 1
1:60 101 1
2:65 102 1
2:58 103 N/A
1:74 101 1
4:95 102 1
9:74 108
9:39 109
2:19 1011
1:36 1013
4:15 105
1:24 103
1:08 108 1
9:05 109 1
4:47 1011 1
2:41 1013 N/A
6:27 105 1
1:75 103 1
3.41 1.81 1
3.11 1.25 1
1.41 0.95 1
0:74 0:22 N/A
4.21 2.45 1
4.82 2.19 1
0:76
0:82
0:89
1
0:84
0:79
2:50 103 1
3:41 104 1
2:65 105 1
5:48 1010 N/A
6:23 103 1
9:12 103 1
1:91 102
1:33 102
2:43 103
1:12 105
7.77
7.96
2:58 102 1
1:81 102 1
4:85 103 1
2:54 105 N/A
1.93
2.36
1
1
8:43 103
1:11 103
1:59 104
1:34 105
7.44
9.47
3:18 103 1
1:29 103 1
2:85 104 1
5:93 105 N/A
1.78
2.63
1
1
1:70 105
8:20 106
7:34 107
9:45 108
1:81 102
2:72 102
4
5
7
8
2
h
h
h
h f8
l r h
f9
l r
f 10
l r
h
h f 11
l r h
f 12
l r h
f 13
f 14
f 15
l r h
7:17 10 1
2:24 10 1
3:19 10 1
l r
20829:4 215.5
20915:4 81.6
20938:4 23.7
4:01 102 1
20949:1
20896:3 79.3
20906:8 69.5
1
1
h
1
1
1
l r
9:49 102
5:96 102
3:29 103
8:28 104
1:88 102
2:71 102
2
2
3
4
1:19 10 N/A
1:22 10 1
3
5:01 102 1
l r h
f 17
3:06 10 1
1:14 103 N/A
h f 16
1:98 10 N/A
l r h
2:23 10 1
1:61 10 1
2:59 10 1
2:31 101
1:43 101
1:09 101
8:21 102
1:47 101
4:24 101
1
1
1
2
1:05 10 N/A
2:43 10 1
1
4:71 101 1
2:34 10 1
1:79 10 1
1:21 10 1
6:91 103
3:48 103
1:64 103
2:91 104
6:91 103
9:41 103
2:63 103 1
1:89 103 1
1:04 103 1
5:81 104 N/A
2:63 103 1
4:26 103 1 (continued on next page)
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Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx
Table 3 (continued) GSO
l r
f 18
h
l r
f 19
f 20
l r
f 21
l r
7:36 10 2:78 10 1
3
h
h
GSOIC 3
3:67 10
3
GSOICLW 4
2:67 10
4
1:27 10
5 5
PBO
BFAVP
2:19 10
3
3:15 103
4:47 10 1
3
5:82 103 1
1:67 10 1
3:54 10 1
2:40 10 N/A
3:92 102
2:04 102
1:57 102
7:68 103
7:61 102
2:17 102
2
2
2
3
2
1:92 102 1
2:73 10 1
h
GSOLW 3
8:22 10 1
4:51 10 1
9:25 10 N/A
3:92 10 1
21.35 10.68 1
15.53 7.88 1
8.34 3.85 1
2:35 1:75 N/A
19.02 9.87 1
34.08 12.13 1
36.15 6.39 1
29.36 5.39 1
15.38 4.19 1
6:26 2:36 N/A
31.69 10.63 1
34.29 8.36 1
The bold fonts of values indicate the best results.
IC caused by increasing population density compel more members to choose to seek other resources [15]. As a result, when GSO is in IC, the diversifying effect will happen: some members are going to seek other resources, acting as rangers. It is noted that the more crowded the population is, the more intense scramble competition becomes, and more rangers emerge. Consequently, rangers’ ratio cðf Þ should vary determined by the index f as follows, rather than being a constant (20%) in [12].
( cðf Þ ¼
1
aþbsin ðf Þ
0:2
lS1 ðf Þ P lS2 ðf Þ lS1 ðf Þ < lS2 ðf Þ
ð4Þ
where a and b are constant coefficients. 2.3. Lévy walk In GSO, rangers perform random walk (RW). However, biology experts have claimed that lévy walk (LW) is more efficient than RW by analysing experimental foraging data on selected insect, mammal and bird species [16]. Therefore, the strategy of ranging behavior in GSOICLW group is chosen as LW, and its walking length, r, drawn from a probability distribution function having a power-law tail as follows [16].
PðrÞ r l
ð1 < l < 3Þ:
ð5Þ
To be concluded, IC and LW are incorporated into GSOICLW, as shown in Fig. 2. Most GSO members, scroungers, are following the producer, and rangers adopt the RW to hunt for other resources. However, IC happens when group members, mainly scroungers, are crowded around the producer. This competition stimulates members to conduct intenser scrabble for the scare resource, and its diversifying effect leads to more rangers appear, adopting LW with longer step in the searching space. 3. Simulation studies To evaluate the performance of GSOICLW, it is tested comprehensively on 21 benchmark functions [12], compared with GSO, GSOLW (which replaces RW with LW as for GSO), GSOIC (which integrates IC into GSO), PBO (paired-bacteria optimiser [21]) and BFAVP (bacterial foraging algorithm with varying population [22]). These benchmark functions consist of unimodal functions (f 1 f 13 ) and multimodal functions (f 14 f 21 ). The experiments of all the above EAs have been repeated for 100 times independently. In each experiment, the population sizes of GSOICLW, GSOLW and GSO are set as 48, and the total iterations during each run are 3000 for all benchmark functions. Therefore, the total number of function evaluations as for each experiment is 150,000. To make fair comparisons
between GSOICLW, GSOLW, GSO, PBO and BFAVP, their total number of function evaluations are set as the same, i.e., 150,000. The parameters as for GSOICLW are selected as the following: m ¼ 0:1; n ¼ 0:3; f ¼ 0:8; a ¼ 2:0; b ¼ 3:56 and l ¼ 2. Moreover, in order to further assess the performance of GSOICLW in a stochastic search process, the Ranksum test is adopted [23]. The obtained h value is 1 shows GSOICLW performs significantly better than other algorithms, while h equals to 0 indicate the performances of two compared algorithms are not statistically different. In Table 1–3, the comparison results of mean (l), standard deviation (r) and h value are listed, as for the 21 benchmark functions, with consideration of 10, 30, 50 dimensions. It can be seen that GSOICLW can obtain much more accurate and robust results than that of other listed EAs. Especially as for unimodal functions (f 1 f 13 ), GSOICLW performs significantly better than that of GSO. The results shown in Table 1, GSOICLW performs better than other EAs as for 10-dimension benchmark functions, except all EAs can find the optima of f 6 and f 10 . Moreover, it can be seen from Tables 2 and 3 that the results obtained by all the EAs become worse, with the increase of dimensions in terms of benchmark functions. However, GSOICLW still outperforms GSO and other EAs. In Table 2, we can observe that GSOICLW can find more accurate solutions as for most of the benchmark functions, although it performs equally to other EAs regarding to f 10 . However, when the number of dimension of benchmark functions reaches to 50, although the performances of all the EAs are weaken, GSOICLW can still find the exact optima in terms of some benchmarks, i.e., f 6 and f 10 , and it shows better performances, compared with GSO, GSOLW, PBO and BFAVP. In conclusion, the simulation results have well verified that GSOICLW performs significantly better than that of GSO as for unimodal functions, which means GSOICLW gains much better local searching. On the other hand, GSOICLW also outperforms GSO for multimodal functions. Therefore, it can be proved that compared with GSO, GSOICLW achieves better local searching ability while maintaining its advantage of global searching.
4. Conclusion This paper has proposed group search optimizer with intraspecific competition and lévy walk, in which IC and LW are incorporated. It is more biologically realistic than GSO, i.e., IC makes the scrounging coefficient and rangers’ ratio adaptively vary to balance local and global searching, and LW stimulates rangers to perform more efficient searching than RW. Simulation results have demonstrated that GSOICLW achieves much better performance than that of GSO.
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
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Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx
10. Exponential Problem:
Acknowledgments The work is supported by the State Key Program of National Natural Science of China (Grant No. 51437006) and Guangdong Innovative Research Team Program (No. 201001N0104744201).
n X f 10 ðxÞ ¼ exp 0:5 x2i
! 1 6 xi 6 1
i¼1
minðf 10 Þ ¼ f 10 ð0; . . . ; 0Þ ¼ 1
Appendix A 11. High Conditioned Elliptic Function: A.1. Unimodal functions
f 11 ðxÞ ¼ 1. Sphere Function:
f 1 ðxÞ ¼
n i1 X ð106 Þn1 x2i
100 6 xi 6 100
i¼1
minðf 11 Þ ¼ f 11 ð0; . . . ; 0Þ ¼ 0
n X x2i
100 6 xi 6 100
i¼1
12. Sum of Different Power Function:
minðf 1 Þ ¼ f 1 ð0; . . . ; 0Þ ¼ 0
f 12 ðxÞ ¼
2. Schwefel’s Problem 2.22
f 2 ðxÞ ¼
n n X Y jxi j þ jxi j i¼1
minðf 12 Þ ¼ f 12 ð0; . . . ; 0Þ ¼ 0
10 6 xi 6 10
i¼1
13. Sum Squares Function:
3. Schwefel’s Problem 2 n i X X xj i¼1
f 13 ðxÞ ¼
!2
n X iðxi Þ2
10 6 xi 6 10
i¼1
minðf 13 Þ ¼ f 13 ð0; . . . ; 0Þ ¼ 0
100 6 xi 6 100
j¼1
minðf 3 Þ ¼ f 3 ð0; . . . ; 0Þ ¼ 0
A.2. Multimodal functions 14. Generalised Schwefel’s Problem 2.26:
4. Schwefel’s Problem 2.21
f 4 ðxÞ ¼ maxfjxi jg 1
100 6 xi 6 100
f 14 ðxÞ ¼¼
pffiffiffiffiffi xi sinð jxjÞ
500 6 xi 6 500
minðf 14 Þ ¼ f 14 ð418:98; . . . ; 418:98Þ ¼ 418:98 n
5. Generalised Rosenbrock’s Function n1 X 2 2 f 5 ðxÞ ¼ ½100ðxiþ1 x2i Þ þ ðxi 1Þ ;
15. Ackley’s Function:
30 6 xi 6 30
i¼1
minðf 5 Þ ¼ f 5 ð1; . . . ; 1Þ ¼ 0 6. Step Function n X ðjxi þ 0:5jÞ2
n X i¼1
minðf 4 Þ ¼ f 4 ð0; . . . ; 0Þ ¼ 0
f 6 ðxÞ ¼
1 6 xi 6 1
i¼1
minðf 2 Þ ¼ f 2 ð0; . . . ; 0Þ ¼ 0
f 3 ðxÞ ¼
n X ðjxi jÞ2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 X30 2 f 15 ðxÞ ¼ 20 exp 0:2 x i¼1 i n ! 30 1 X cosð2pxi Þ þ 20 þ e exp 30 i¼1 32 6 xi 6 32
100 6 xi 6 100
minðf 15 Þ ¼ f 15 ð0; . . . ; 0Þ ¼ 0
i¼1
minðf 6 Þ ¼ f 6 ð0; . . . ; 0Þ ¼ 0
16. Generalised Girewank Function:
7. Quartic Function n X 4 f 7 ðxÞ ¼ ixi þ randðÞ
f 16 ðxÞ ¼ 1:28 6 xi 6 1:28
30 30 Y 1 X xi x2i cos pffi þ 1 4000 i¼1 i i¼1
600 6 xi 6 600
minðf 16 Þ ¼ f 16 ð0; . . . ; 0Þ ¼ 0
i¼1
minðf 7 Þ ¼ f 7 ð0; . . . ; 0Þ ¼ 0
17. Generalised Penalised Function:
8. Axis Parallel Hyperellipsoid Function: n X 2 f 8 ðxÞ ¼ ixi
f 17 ðxÞ ¼
ð5:12 6 xi 6 5:12Þ
i¼2
i¼2
minðf 9 Þ ¼ f 9 ð0; . . . ; 0Þ ¼ 0
2
f10sin ðpy1 Þ þ
n1 X 2 2 2 ðyi 1Þ ½1 þ 10 sin ðpyiþ1 Þ þ ðyn1 Þ g i¼1
i¼1
9. Dixon Price Function: n X 2 ið2x2i xi1 Þ þ ðx1 1Þ2
n
n1 X uðxi ;10; 100;4Þ 50 6 xi 6 50 þ
minðf 8 Þ ¼ f 8 ð0; . . . ; 0Þ ¼ 0
f 9 ðxÞ ¼
p
10 6 xi 6 10
minðf 17 Þ ¼ f 17 ð1;...;1Þ ¼ 0 1 where yi ¼ 1 þ ðxi þ 1Þ; 4 8 m > kðxi 1Þ ; xi > a; < uðxi ;a;k;mÞ ¼ 0; a < xi < a; > : m kðxi 1Þ ; xi < a:
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Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx
18. Generalised Girewank Function:
f 18 ðxÞ ¼
n1 X 1 2 2 fsin ð3px1 Þ þ ðxi 1Þ2 ½1 þ sin ð3pxiþ1 Þ 10 i¼1 2
þ ðxn 1Þ2 ½1 þ sin ð2pxn Þg þ
n1 X uðxi ; 5; 100; 4Þ
50 6 xi 6 50
i¼1
minðf 18 Þ ¼ f 18 ð1; . . . ; 1Þ ¼ 0 8 m xi > a; > < kðxi 1Þ ; where uðxi ; a; k; mÞ ¼ 0; a < xi < a; > : kðxi 1Þm ; xi < a: 19. Alpine Function:
f 19 ðxÞ ¼
n X jxi sinðxi Þ þ 0:1xi j
10 6 xi 6 10
i¼1
minðf 19 Þ ¼ f 19 ð0; . . . ; 0Þ ¼ 0 20. Expansion of F10:
f 20 ðxÞ ¼ f 10 ðx1 ; x2 Þ þ þ f 10 ðxi1 ; xi Þ þ þ f 10 ðxn ; x1 Þ 100 6 xi 6 100 minðf 20 Þ ¼ f 20 ð0; . . . ; 0Þ ¼ 0 where f 10 ðxÞ ¼ ðx2 þ y2 Þ
0:25
2
0:1
½sin ð50ðx2 þ y2 Þ Þ þ 1
21. Expanded Scaffer’s Function:
f 21 ðxÞ ¼ f s ðx1 ; x2 Þ þ þ f s ðxi1 ; xi Þ þ þ f s ðxn ; x1 Þ 100 6 xi 6 100 minðf 21 Þ ¼ f 21 ð0; . . . ; 0Þ ¼ 0 0:5 2 sin ðx2 þ y2 Þ 0:5Þ where f s ðxÞ ¼ 0:5 þ 2 1 þ 0:001ðx2 þ y2 Þ References [1] H.G. Beyer, The simple genetic algorithm foundations and theory, IEEE Trans. Evol. Comput. 4 (2) (2000) 191–192.
[2] W. Banzhaf, J.R. Koza, C. Ryan, L. Spector, C. Jacob, Genetic programming, IEEE Intell. Syst. 15 (3) (2000) 74–84. [3] X. Yao, Y. Liu, G.M. Lin, Evolutionary programming made faster, IEEE Trans. Evol. Comput. 3 (2) (1999) 82–102. [4] O. Francois, An evolutionary strategy for global minimization and its markov chain analysis, IEEE Trans. Evol. Comput. 2 (3) (1998) 77–90. [5] M. Dorigo, J. Kennedy, The particle swarm: explosion, stability, and convergence in a multi-dimensional complex space, IEEE Trans. Evol. Comput. 6 (1) (2002) 58–73. [6] M. Clerc, M. Birattari, T. Stutzle, Ant colony optimization, IEEE Comput. Intell. Mag. 1 (4) (2006) 28–39. [7] D. Karaboga, B. Akay, A comparative study of artificial bee colony algorithm, Appl. Math. Comput. 214 (2009) 108–132. [8] K. Passino, Biomimicry of bacterial foraging for distributed optimization and control, IEEE Control Syst. Mag. 22 (3) (2002) 52–67. [9] R. Rajabioun, Cuckoo optimization algorithm, Appl. Soft Comput. 11 (8) (2011) 5508–5518. [10] D. Simon, Biogeography-based optimization, IEEE Trans. Evol. Comput. 12 (6) (2008) 702–713. [11] W. Pan, A new fruit fly optimization algorithm: taking the financial distress model as an example, Knowl.-Based Syst. 26 (2012) 69–74. [12] S. He, Q.H. Wu, J.R. Saunders, Group search optimizer: an optimization algorithm inspired by animal searching behavior, IEEE Trans. Evol. Comput. 13 (5) (2009) 973–990. [13] L. Li, F. Liu, Group Search Optimization for Applications in Structural Design, Springer, London, 2011. [14] A.J. Nicholson, An outline of the dynamics of animal populations, Aust. J. Zool. 2 (1954) 9–65. [15] S. Richard, I.B. Daniel, Diversity within a natural population intraspecific competition drives increased resource use diversity within a natural population, Proc. R. Soc. B 274 (2007) 839–844. [16] G.M. Viswanathan, Optimizing the success of random searches, Nature 401 (1999) 911–914. [17] A.A. Berryman, Principles of Population Dynamics and their Application, Garland Science, New York, 1999. [18] Z.H. Zhan, J. Zhang, Y. Li, H.S.H. Chung, Adaptive particle swarm optimization, IEEE Trans. Syst. Man Cybern. B Cybern. 39 (6) (2009) 1362–1381. [19] D.I. Bolnick, Intraspecific competition favours niche width expansion in drosophila melanogaster, Nature 410 (2001) 463–466. [20] R.C. MacLean, Adaptive radiation in microbial microcosms, J. Evol. Biol. 18 (2005) 1376–1386. [21] M. Li, W. Tang, Q. Wu, J. Saunders, Paired-bacteria optimiser-a simple and fast algorithm, Inform. Process Lett. 111 (2011) 809–813. [22] M. Li, T. Ji, W. Tang, Q. Wu, J. Saunders, Bacterial foraging algorithm with varying population, Biosystems 100 (2010) 185–197. [23] Q. Pan, H. Sang, J. Duan, L. Gao, An improved fruit fly optimization algorithm for continuous function optimization problems, Knowl.-Based Syst. 62 (2014) 69–83.
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Contents lists available at ScienceDirect
Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Short Communication
Group search optimizer with intraspecific competition and lévy walk Y.Z. Li a, Q.H. Wu a,b,⇑, M.S. Li a a b
School of Electrical Engineering, South China University of Technology, Guangzhou 510641, China Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, UK
a r t i c l e
i n f o
Article history: Received 9 April 2014 Received in revised form 26 August 2014 Accepted 9 September 2014 Available online xxxx Keywords: Evolutionary algorithm Intraspecific competition Lévy walk Group search optimizer Standard benchmark functions
a b s t r a c t This short communication presents a nature-inspired optimization algorithm, group search optimizer with intraspecific competition and lévy walk (GSOICLW). The mechanism of intraspecific competition (IC) and the searching strategy of lévy walk (LW) are incorporated to the original group search optimizer (GSO) algorithm. It has been proved that GSOICLW shows a significant improvement to GSO after its test against standard benchmark functions. Ó 2014 Published by Elsevier B.V.
1. Introduction Many naturally inspired algorithms have emerged based on biological evolution and animal behaviors, which are referred to as evolutionary algorithms (EAs). Many EAs draw inspiration coming from the process of biological evolution, such as genetic algorithm (GA) [1], genetic programming (GP) [2], evolutionary programming (EP) [3] and evolutionary strategy (ES) [4]. Other EAs, simulating animal behaviors, gains much more attention. For instance, particle swarm optimization (PSO) mimics social behavior of a bird flock [5], ant colony optimization (ACO) simulates the ecological behavior of ants as for foraging [6], artificial bee colony (ABC) learns the behavior of honeybees in terms of collecting nectar [7], bacterial foraging algorithm (BFA) simulates bacterial foraging patterns [8], cuckoo optimization algorithm (COA) is inspired by the cuckoo birds’ behavior as for egg laying and breeding [9], biogeographybased optimization (BBO) is motivated by biogeography, studying the distribution of biological species [10], fruit fly optimization (FFO) algorithm is developed by simulating the food finding behavior of the fruit fly [11], group search optimizer (GSO) mimics animal’s behavior of hunting for resource [12] and et al. Among the EAs mentioned above, GSO is famous for its global searching ability, and it has been applied in many engineering optimization problems [13]. GSO consists of three types of group members: the producer, scroungers and rangers. In each genera⇑ Corresponding author at: School of Electrical Engineering, South China University of Technology, Guangzhou 510641, China. E-mail address: [email protected] (Q.H. Wu).
tion, the member conferred the best fitness value is chosen as the producer, and a number of members except the producer are randomly selected as scroungers, then the rest of members are rangers. The producer is located in the most promising area and adopts animal scanning to seek the optimal resource. Scroungers perform area copying to join the resource found by the producer, and do local searching around it. However, rangers employ ranging behavior by random walk (RW) in the searching space to increase the GSO’s chance to escape local optima. Therefore, GSO has demonstrated better performance in global searching, however, its local searching ability is not desirable, as shown in the modest performance on optimizing unimodal benchmark functions [12]. In order to improve GSO’s local searching ability while maintaining its merit in global searching, we introduce intraspecific competition (IC) [14] and lévy walk (LW) into GSO. IC is a particular form of biological phenomenon in which population members compete for scarce resources. Each member in GSO is always hunting for the optimal resource (definitely scarce) which is assumed to be at the producer’s position, therefore, IC exists inevitably in GSO’s searching process and it stimulates scroungers to closely search the local area around the producer. Moreover, it will put a diversifying effect on group members, i.e., intenser competition caused by increasing population density will compel more rangers to choose to seek another resource [15]. Rangers perform RW in GSO, however, some biologists have claimed that lévy walk (LW) is more efficient than RW for searching resources [16]. Then we replace RW with LW for rangers in GSO in order to gain better searching ability. Therefore, an improved algorithm, GSOICLW, is proposed, which incorporates IC and LW
http://dx.doi.org/10.1016/j.knosys.2014.09.005 0950-7051/Ó 2014 Published by Elsevier B.V.
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
2
Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx Producer
Scrounger
Ranger
Intraspecific Competition Levy Walk
GSOICLW
GSO
Fig. 2. The illustration of IC and LW in GSOICLW.
Fig. 1. Fuzzy membership functions as for IC and Non-IC.
Table 1 Results of mean (l), standard deviation (r) and h value of GSO, GSOLW, GSOIC, GSOICLW, PBO and BFAVP on f 1 f 21 considering 10 dimensions.
f1
l r h
f2
l r h
f3
l r h
f4
l r h
GSO
GSOLW
GSOIC
GSOICLW
PBO
2:11 1013
1:10 1013
1:37 1023
0
1:19 105
4:45 1013
13
13
23
0
5
1:21 1012 1
7:89 10 1
2:70 10 1
5:64 10 1
BFAVP
N/A
4:30 10 1
5:31 108
5:18 108
4:26 1013
0
3:24 103
1:35 107
8
9
13
0 N/A
6:25 10 1
4
1:85 107 1
6:61 10 1
8:69 10 1
1:47 10 1
3:13 105
1:77 105
1:95 107
1:72 108
4:66 104
2:63 105
4:75 105 1
1:81 105 1
2:16 107 1
2:39 108 N/A
2:32 104 1
3:96 105 1
4:34 105
3:83 105
1:44 106
4:99 108
7:94 103
4:65 105
3:71 105 1
2:72 105 1
1:92 106 1
7:28 108 N/A
1:41 103 1
3:32 105 1
l r
5.31
1.43
6:98 103
1:29 105
2.79
2.77
4.05
1.55
4.22
1
1
5:55 105 N/A
6.33
h
5:03 103 1
1
1
f6
l r
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
f7
l r
f5
h
h f8
l r h
f9
l r
f 10
l r
h
h f 11
l r h
f 12
l r h
f 13
l r h
f 14
l r h
5:87 103
5:14 103
2:52 104
3:44 105
5:62 103
1:22 103
3
3
3
5
3
7:47 103 1
2:61 10 1
2:20 10 1
1:81 10 1
1:54 10 N/A
3:24 10 1
3:72 1016
1:14 1016
4:40 1028
0
8:20 107
4:18 1016
16
16
28
0
7
2:80 10 1
1:64 10 1
0.99 0.32 1
0.58 0.26 1
1:50 10 1 0.48 0.19 1
N/A
4:10 10 1
8:67 1016 1
0:22 0:06 N/A
0.40 0.65 1
0.61 0.42 1
1
1
1
1
1
1
1:47 1016 0
1:38 1016 0
6:84 1017 0
1:84 1018 N/A
2:76 109 0
5:83 1017 0
4:64 109
1:73 1010
5:12 1015
3:27 1018
1:34 101
3:01 109
2:43 108 1
5:31 1010 1
1:85 1015 1
1:32 1018 N/A
6:37 102 1
7:77 109 1
2:93 1010
8:74 1011
1:73 1017
3:07 1019
1:25 101
1:58 109
7:59 1010 1
2:28 1011 1
1:12 1017 1
1:12 1019 N/A
6:35 102 1
3:26 109 1
4:80 1014
9:96 1015
1:15 1021
2:80 1023
5:95 106
5:36 1014
3:10 1014 1
1:87 1015 1
6:28 1021 1
1:33 1023 N/A
1:64 106 1
9:45 1014 1
4189:83
4189:83
4189:83
4189:83
4189:83
4189:83
7:53 106 0
5:32 107 0
6:54 108 0
9:92 109 N/A
5:36 105 0
1:47 105 0
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
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Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx Table 1 (continued) GSO
l r
f 15
h
l r
f 16
h
l r
f 17
h
l r
f 18
h
l r
f 19
h
l r
f 20
h
l r
f 21
h
GSOLW
1:48 10
7
1:79 10 1
7
1:09 10
GSOIC 7
1:45 10 1
GSOICLW 13
8:97 10
7
1:10 10 1
PBO
16
3:08 107
1:28 10 N/A
3
1:04 10 1
3:50 107 1
8:88 10
13
BFAVP 3
5:29 10
17
6:03 103
5:51 103
5:60 104
4:94 105
5:59 103
5:54 103
3
3
4
5
3
1:95 103 1
2:35 10 1
1:95 10 1
3:49 10 1
1:87 10 N/A
3:49 10 1
5:71 1016
1:16 1017
8:73 1027
1:57 1032
1:41 108
3:81 1016
15
17
27
32
8
6:49 1016 1
2:16 10 1
2:42 10 1
3:64 10 1
1:07 10 N/A
4:88 10 1
2:22 1015
1:96 1016
1:27 1026
1:35 1032
3:31 106
9:87 1015
4:46 1015 1
3:99 1016 1
3:08 1026 1
5:56 1033 N/A
1:10 106 1
1:81 1014 1
3:29 107
9:52 108
1:14 108
8:32 1010
8:75 104
6:85 106
1:16 106 1
1:56 107 1
6:21 108 1
2:36 109 N/A
7:19 104 1
3:66 105 1
2:33 107
1:35 107
3:08 109
3:13 1010
0.83
0.11
3:36 107 1
3:08 107 1
6:10 108 1
6:57 109 N/A
9:47 102 1
4:57 101 1
4:06 102
3:79 102
8:89 103
2:52 103
1:78 101
1:25 101
3:37 102 1
3:08 102 1
6:43 103 1
2:83 103 N/A
1:38 101 1
1:69 101 1
The bold fonts of values indicate the best results.
based on GSO. We have evaluated GSOICLW in comparison with GSO on a set of 21 standard unimodal and multimodal benchmark functions, presented in Appendix. Simulation results prove GSOICLW achieves better local searching ability while maintaining GSO’s advantage as for global searching. 2. Group search optimizer with intraspecific competition and lévy walk 2.1. Intraspecific competition IC is defined as the struggle between population members for the scarce resource, which is divided into contest and scramble competition [14]. In contest competition, the successful competitor obtains all it needs while the rest of competitors are deprived of such resources. However, scramble competition happens when population members are crowded around the limited resources, which are not monopolized by successful competitors. This sort of IC, scramble competition, stimulates population members to compete for resources seriously. Moreover, the intensity of IC is densitydependent, i.e., competition will be more serious if population members are increasingly crowding around resources [14,17]. GSO mainly adopts the joining policy to hunt for the scarce resource [12], consequently, IC occurs in the form of scramble competition as for GSO when population members, mainly scroungers, are crowded around the producer. Here the index f ranged in ð0; 1Þ proposed by Zhan et al. [18] can well describe the population’s crowdedness in some EAs dominated by the leading member with the best fitness value, e.g., PSO and GSO. The detailed calculation steps of this index f are listed as follows. (1) Compute the mean distance di of each particle i with its position xi to all the other particles using the Euclidian metric
di ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N XD 1 X 2 ðxk xkj Þ k¼1 i N 1 j¼1;j–i
ð1Þ
where N is the population size and D is the number of particles’ dimensions, respectively.
(2) Denote di of the leading particle as dg , determine the maximum distance dmax and minimum distance dmin by comparing all di in (1). Then the index f can be defined as
f ¼
dg dmin 2 ½0; 1: dmax dmin
ð2Þ
The above equation shows that if f is small, the leading member is surrounded closely by other members [18]. Therefore, the index f is used to describe the GSO population’s crowdedness. Specifically, when f is small, GSO members are crowded around the producer, then IC happens. However, it maybe not reasonable to set an exact limen of the index f, e.g. 0.2, to determine whether GSO is in IC or not, because state transition from non-IC to IC would be nondeterministic and fuzzy, just as the phenomenon shown in [18]. Therefore, we adopt the method of fuzzy classification to determine the phrase of IC or non-IC as for GSO. We set the phrase of IC and non-IC as S1 and S2 , and their fuzzy membership functions are donated as lS1 and lS2 , depicted in Fig. 1. If lS1 ðf Þ > lS2 ðf Þ, we can say the members in GSO are in IC, and r3 ¼ ðr 31 ; r 32 ; . . . ; r 3D Þ , presented in Eq. (7) in [12], is a random vector with higher value ranged in ðf; 1Þ for scroungers to manifest this serious competition. Here r3 is called the scrounging coefficient. On the other hand, if lS1 ðf Þ < lS2 ðf Þ, GSO members are in the phrase of non-IC, then r3 is a random vector with lower value anged in ð0; fÞ. Therefore, r3 is formulated as the following.
( r 3i ¼
random 2 ðf; 1Þ random 2 ð0; fÞ
lS1 ðf Þ P lS2 ðf Þ i ¼ 1; 2; . . . ; D lS1 ðf Þ < lS2 ðf Þ
ð3Þ
where f is threshold for the scrounging coefficient to manifest IC. 2.2. Diversifying effect of intraspecific competition Richard has proposed that IC drives the diversifying effect within a population, i.e., the increasing population density leads to reduced prey availability, then some individuals will resort to alternative prey types [15]. Similar phenomena caused by IC have been also observed by other biologists [19,20]. In addition, intenser
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
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Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx
Table 2 Results of mean (l), standard deviation (r) and h value of GSO, GSOLW, GSOIC, GSOICLW, PBO and BFAVP on f 1 f 21 considering 30 dimensions. GSO f1
l r h
f2
l r h
f3
l r h
f4
l r h
f5
l r h
f6
l r h
f7
l r h
f8
l r h
f9
l r h
f 10
l r h
f 11
l r h
f 12
l r h
f 13
l r h
f 14
l r h
f 15
l r h
f 16
l r h
f 17
l r h
f 18
l r h
f 19
l r h
f 20
l r h
GSOLW
GSOIC 1:34 10
8:54 10
2:86 10
1:26 105
5:23 1012 1
2:70 1015 N/A
5:53 104 1
1:63 105 1
5:56 105
2:99 105
1:22 106
1:38 107
2:27 102
1:54 103
5
5
6
7
3
1:83 10 1
2:86 103 1
4:51 10 1
1:96 10 N/A
15
BFAVP
5:65 109 1
3:02 10 1
12
PBO
3:81 10
9:91 10 1
9
GSOICLW
1:07 108 1
6:75 10
9
4
36.81 11.30 1
32.59 9.32 1
11.26 6.98 1
2:21 1:02 N/A
20.51 9.68 1
115.34 75.4 1
1:07 101
5:63 102
1:38 104
2:57 105
2:96 102
5:59 101
3:16 102 1
2:79 102 1
2:63 104 1
6:28 105 N/A
2:70 103 1
1:56 101 1
57.68 34.23 1
49.65 29.63 1
29.59 11.23 1
8:35 0:69 N/A
88.03 56.45 1
39.27 24.68 1
0
0
6:67 102
3:33 102
3:68 10 0
0
0
1
0
N/A
2:53 10 1
1:83 102 1
7:13 102
5:55 102
4:58 103
1:64 104
4:26 102
3:27 102
2
2
3
4
2
1:87 10 1
1:39 102 1
3:33 102 1:83 10 1
2:67 10 1
2
0 3
1:88 10 1
8:36 10 1
7:73 10 N/A
6:68 1011
5:62 1011
3:17 1017
2:98 1019
7:10 105
4:86 107
1:08 1010 1
2:77 1011 1
5:56 1017 1
1:88 1019 N/A
1:42 105 1
5:22 107 1
1.23 0.78 1
1.18 0.74 1
0.97 0.61 1
0:67 0:12 N/A
1.71 0.95 1
1.11 0.52 1
1
1
1
1
1
1
4:50 1011 0
4:80 1013 0
3:13 1015 0
2:95 1016 N/A
3:02 108 0
9:33 1010 0
2:20 105
1:23 105
5:65 1011
9:50 1012
2.16
2.07
2:58 105 1
1:87 105 1
1:73 1010 1
1:14 1012 N/A
0.57
3.84
1
1
2:31 105
1:95 105
3:72 109
2:45 1011
2.24
5.84
2:40 105 1
5:84 106 1
6:43 109 1
5:93 1011 N/A
0.54
17.50
1
1
2:55 108
1:51 109
6:59 1011
6:11 1012
3:55 104
7:70 106
8:25 108 1
2:37 109 1
2:19 1011 1
2:19 1012 N/A
8:06 105 1
1:83 105 1
12569:5
12569:5
12569:5
12569:5
7:30 102 0
6:48 104 0
2:99 105 0
6:43 106 N/A
12566:3 1.39 1
5:72 102 0
5
5
6
7
12569:5
1:75 10
2:43 10
1:27 10
3:38 102
1:96 103
4:44 105 1
1:31 105 1
8:54 106 1
2:22 107 N/A
8:31 102 1
1:98 103 1
3:36 102
2:09 102
1:96 102
6:08 103
4:30 102
2:31 102
3:37 102 1
2:37 102 1
1:03 102 1
8:54 103 N/A
4:33 102 1
2:51 102 1
1:84 1011
1:50 1012
6:17 1014
1:94 1015
1:45 107
7:94 108
5:22 1011 1
2:41 1012 1
3:38 1014 1
6:27 1016 N/A
2:98 108 1
9:15 108 1
1:16 109
5:79 1010
7:32 1011
3:01 1012
4:28 105
1:52 106
9
10
11
12
6
8:99 10 1
2:48 106 1
2:95 10
4:66 10 1
2:61 10 1
2:78 10 1
1:26 10 N/A
2:61 103
1:41 103
3:26 104
6:71 105
7:61 103
2:17 103
3
3
4
5
3
3:92 10 1
1:92 103 1
10.67 1.56 1
7.28 1.04 1
4:43 10 1 8.74 1.19 1
2:79 10 1
1:91 10 1
1:12 10 N/A
7.27 0.96 1
1.58 0.59 1
0:99 0:12 N/A
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
5
Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx Table 2 (continued)
f 21
l r h
GSO
GSOLW
GSOIC
GSOICLW
PBO
BFAVP
2.17 0.72 1
2.05 0.51 1
1.19 0.43 1
0:39 0:28 N/A
1.79 0.38 1
4.51 0.98 1
The bold fonts of values indicate the best results.
Table 3 Results of mean (l), standard deviation (r) and h value of GSO, GSOLW, GSOIC, GSOICLW, PBO and BFAVP on f 1 f 21 considering 50 dimensions. GSO f1
l r h
f2
l r h
GSOLW 6
GSOIC 6
GSOICLW 8
10
PBO
BFAVP 3
3:91 103
3:29 10
1:64 10
1:32 10
3:39 10
9:38 10
5:01 106 1
1:37 106 1
5:18 108 1
4:91 1010 N/A
1:49 103 1
2:90 103 1
1:54 103
1:03 103
6:14 104
5:14 105
1:35 103
3:24 103
3
4
4
5
2
6:25 104 1
1:33 10 N/A
4:73 10 1
335.9 90.01 1
62:61 33:35 N/A
856.32 162.35 1
1345.21 522.08 1
3:21 10 1
9:73 10 1
1:82 10 1
799.49 157.69 1
777.20 121.28 1
f3
l r
f4
l r
1.08
0.52
0.29
2:07 103
0.66
3.51
0.31
0.19
0.11
0.56
0.95
h
1
1
1
4:76 103 N/A
1
1
f5
l r
196.03 625.82 1
129.72 300.44 1
89.83 207.65 1
20:21 10:01 N/A
189.91 707.66 1
164.96 624.14 1
f6
l r
0.73 0.69 1
0.33 0.54 1
0.12 0.25 1
0 0 N/A
0.23 0.36 1
1.33 1.32 1
f7
l r
1:41 101
8:56 102
6:32 102
5:30 103
3:63 101
1:47 102
6:23 101 1
1:60 101 1
2:65 102 1
2:58 103 N/A
1:74 101 1
4:95 102 1
9:74 108
9:39 109
2:19 1011
1:36 1013
4:15 105
1:24 103
1:08 108 1
9:05 109 1
4:47 1011 1
2:41 1013 N/A
6:27 105 1
1:75 103 1
3.41 1.81 1
3.11 1.25 1
1.41 0.95 1
0:74 0:22 N/A
4.21 2.45 1
4.82 2.19 1
0:76
0:82
0:89
1
0:84
0:79
2:50 103 1
3:41 104 1
2:65 105 1
5:48 1010 N/A
6:23 103 1
9:12 103 1
1:91 102
1:33 102
2:43 103
1:12 105
7.77
7.96
2:58 102 1
1:81 102 1
4:85 103 1
2:54 105 N/A
1.93
2.36
1
1
8:43 103
1:11 103
1:59 104
1:34 105
7.44
9.47
3:18 103 1
1:29 103 1
2:85 104 1
5:93 105 N/A
1.78
2.63
1
1
1:70 105
8:20 106
7:34 107
9:45 108
1:81 102
2:72 102
4
5
7
8
2
h
h
h
h f8
l r h
f9
l r
f 10
l r
h
h f 11
l r h
f 12
l r h
f 13
f 14
f 15
l r h
7:17 10 1
2:24 10 1
3:19 10 1
l r
20829:4 215.5
20915:4 81.6
20938:4 23.7
4:01 102 1
20949:1
20896:3 79.3
20906:8 69.5
1
1
h
1
1
1
l r
9:49 102
5:96 102
3:29 103
8:28 104
1:88 102
2:71 102
2
2
3
4
1:19 10 N/A
1:22 10 1
3
5:01 102 1
l r h
f 17
3:06 10 1
1:14 103 N/A
h f 16
1:98 10 N/A
l r h
2:23 10 1
1:61 10 1
2:59 10 1
2:31 101
1:43 101
1:09 101
8:21 102
1:47 101
4:24 101
1
1
1
2
1:05 10 N/A
2:43 10 1
1
4:71 101 1
2:34 10 1
1:79 10 1
1:21 10 1
6:91 103
3:48 103
1:64 103
2:91 104
6:91 103
9:41 103
2:63 103 1
1:89 103 1
1:04 103 1
5:81 104 N/A
2:63 103 1
4:26 103 1 (continued on next page)
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
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Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx
Table 3 (continued) GSO
l r
f 18
h
l r
f 19
f 20
l r
f 21
l r
7:36 10 2:78 10 1
3
h
h
GSOIC 3
3:67 10
3
GSOICLW 4
2:67 10
4
1:27 10
5 5
PBO
BFAVP
2:19 10
3
3:15 103
4:47 10 1
3
5:82 103 1
1:67 10 1
3:54 10 1
2:40 10 N/A
3:92 102
2:04 102
1:57 102
7:68 103
7:61 102
2:17 102
2
2
2
3
2
1:92 102 1
2:73 10 1
h
GSOLW 3
8:22 10 1
4:51 10 1
9:25 10 N/A
3:92 10 1
21.35 10.68 1
15.53 7.88 1
8.34 3.85 1
2:35 1:75 N/A
19.02 9.87 1
34.08 12.13 1
36.15 6.39 1
29.36 5.39 1
15.38 4.19 1
6:26 2:36 N/A
31.69 10.63 1
34.29 8.36 1
The bold fonts of values indicate the best results.
IC caused by increasing population density compel more members to choose to seek other resources [15]. As a result, when GSO is in IC, the diversifying effect will happen: some members are going to seek other resources, acting as rangers. It is noted that the more crowded the population is, the more intense scramble competition becomes, and more rangers emerge. Consequently, rangers’ ratio cðf Þ should vary determined by the index f as follows, rather than being a constant (20%) in [12].
( cðf Þ ¼
1
aþbsin ðf Þ
0:2
lS1 ðf Þ P lS2 ðf Þ lS1 ðf Þ < lS2 ðf Þ
ð4Þ
where a and b are constant coefficients. 2.3. Lévy walk In GSO, rangers perform random walk (RW). However, biology experts have claimed that lévy walk (LW) is more efficient than RW by analysing experimental foraging data on selected insect, mammal and bird species [16]. Therefore, the strategy of ranging behavior in GSOICLW group is chosen as LW, and its walking length, r, drawn from a probability distribution function having a power-law tail as follows [16].
PðrÞ r l
ð1 < l < 3Þ:
ð5Þ
To be concluded, IC and LW are incorporated into GSOICLW, as shown in Fig. 2. Most GSO members, scroungers, are following the producer, and rangers adopt the RW to hunt for other resources. However, IC happens when group members, mainly scroungers, are crowded around the producer. This competition stimulates members to conduct intenser scrabble for the scare resource, and its diversifying effect leads to more rangers appear, adopting LW with longer step in the searching space. 3. Simulation studies To evaluate the performance of GSOICLW, it is tested comprehensively on 21 benchmark functions [12], compared with GSO, GSOLW (which replaces RW with LW as for GSO), GSOIC (which integrates IC into GSO), PBO (paired-bacteria optimiser [21]) and BFAVP (bacterial foraging algorithm with varying population [22]). These benchmark functions consist of unimodal functions (f 1 f 13 ) and multimodal functions (f 14 f 21 ). The experiments of all the above EAs have been repeated for 100 times independently. In each experiment, the population sizes of GSOICLW, GSOLW and GSO are set as 48, and the total iterations during each run are 3000 for all benchmark functions. Therefore, the total number of function evaluations as for each experiment is 150,000. To make fair comparisons
between GSOICLW, GSOLW, GSO, PBO and BFAVP, their total number of function evaluations are set as the same, i.e., 150,000. The parameters as for GSOICLW are selected as the following: m ¼ 0:1; n ¼ 0:3; f ¼ 0:8; a ¼ 2:0; b ¼ 3:56 and l ¼ 2. Moreover, in order to further assess the performance of GSOICLW in a stochastic search process, the Ranksum test is adopted [23]. The obtained h value is 1 shows GSOICLW performs significantly better than other algorithms, while h equals to 0 indicate the performances of two compared algorithms are not statistically different. In Table 1–3, the comparison results of mean (l), standard deviation (r) and h value are listed, as for the 21 benchmark functions, with consideration of 10, 30, 50 dimensions. It can be seen that GSOICLW can obtain much more accurate and robust results than that of other listed EAs. Especially as for unimodal functions (f 1 f 13 ), GSOICLW performs significantly better than that of GSO. The results shown in Table 1, GSOICLW performs better than other EAs as for 10-dimension benchmark functions, except all EAs can find the optima of f 6 and f 10 . Moreover, it can be seen from Tables 2 and 3 that the results obtained by all the EAs become worse, with the increase of dimensions in terms of benchmark functions. However, GSOICLW still outperforms GSO and other EAs. In Table 2, we can observe that GSOICLW can find more accurate solutions as for most of the benchmark functions, although it performs equally to other EAs regarding to f 10 . However, when the number of dimension of benchmark functions reaches to 50, although the performances of all the EAs are weaken, GSOICLW can still find the exact optima in terms of some benchmarks, i.e., f 6 and f 10 , and it shows better performances, compared with GSO, GSOLW, PBO and BFAVP. In conclusion, the simulation results have well verified that GSOICLW performs significantly better than that of GSO as for unimodal functions, which means GSOICLW gains much better local searching. On the other hand, GSOICLW also outperforms GSO for multimodal functions. Therefore, it can be proved that compared with GSO, GSOICLW achieves better local searching ability while maintaining its advantage of global searching.
4. Conclusion This paper has proposed group search optimizer with intraspecific competition and lévy walk, in which IC and LW are incorporated. It is more biologically realistic than GSO, i.e., IC makes the scrounging coefficient and rangers’ ratio adaptively vary to balance local and global searching, and LW stimulates rangers to perform more efficient searching than RW. Simulation results have demonstrated that GSOICLW achieves much better performance than that of GSO.
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
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Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx
10. Exponential Problem:
Acknowledgments The work is supported by the State Key Program of National Natural Science of China (Grant No. 51437006) and Guangdong Innovative Research Team Program (No. 201001N0104744201).
n X f 10 ðxÞ ¼ exp 0:5 x2i
! 1 6 xi 6 1
i¼1
minðf 10 Þ ¼ f 10 ð0; . . . ; 0Þ ¼ 1
Appendix A 11. High Conditioned Elliptic Function: A.1. Unimodal functions
f 11 ðxÞ ¼ 1. Sphere Function:
f 1 ðxÞ ¼
n i1 X ð106 Þn1 x2i
100 6 xi 6 100
i¼1
minðf 11 Þ ¼ f 11 ð0; . . . ; 0Þ ¼ 0
n X x2i
100 6 xi 6 100
i¼1
12. Sum of Different Power Function:
minðf 1 Þ ¼ f 1 ð0; . . . ; 0Þ ¼ 0
f 12 ðxÞ ¼
2. Schwefel’s Problem 2.22
f 2 ðxÞ ¼
n n X Y jxi j þ jxi j i¼1
minðf 12 Þ ¼ f 12 ð0; . . . ; 0Þ ¼ 0
10 6 xi 6 10
i¼1
13. Sum Squares Function:
3. Schwefel’s Problem 2 n i X X xj i¼1
f 13 ðxÞ ¼
!2
n X iðxi Þ2
10 6 xi 6 10
i¼1
minðf 13 Þ ¼ f 13 ð0; . . . ; 0Þ ¼ 0
100 6 xi 6 100
j¼1
minðf 3 Þ ¼ f 3 ð0; . . . ; 0Þ ¼ 0
A.2. Multimodal functions 14. Generalised Schwefel’s Problem 2.26:
4. Schwefel’s Problem 2.21
f 4 ðxÞ ¼ maxfjxi jg 1
100 6 xi 6 100
f 14 ðxÞ ¼¼
pffiffiffiffiffi xi sinð jxjÞ
500 6 xi 6 500
minðf 14 Þ ¼ f 14 ð418:98; . . . ; 418:98Þ ¼ 418:98 n
5. Generalised Rosenbrock’s Function n1 X 2 2 f 5 ðxÞ ¼ ½100ðxiþ1 x2i Þ þ ðxi 1Þ ;
15. Ackley’s Function:
30 6 xi 6 30
i¼1
minðf 5 Þ ¼ f 5 ð1; . . . ; 1Þ ¼ 0 6. Step Function n X ðjxi þ 0:5jÞ2
n X i¼1
minðf 4 Þ ¼ f 4 ð0; . . . ; 0Þ ¼ 0
f 6 ðxÞ ¼
1 6 xi 6 1
i¼1
minðf 2 Þ ¼ f 2 ð0; . . . ; 0Þ ¼ 0
f 3 ðxÞ ¼
n X ðjxi jÞ2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 X30 2 f 15 ðxÞ ¼ 20 exp 0:2 x i¼1 i n ! 30 1 X cosð2pxi Þ þ 20 þ e exp 30 i¼1 32 6 xi 6 32
100 6 xi 6 100
minðf 15 Þ ¼ f 15 ð0; . . . ; 0Þ ¼ 0
i¼1
minðf 6 Þ ¼ f 6 ð0; . . . ; 0Þ ¼ 0
16. Generalised Girewank Function:
7. Quartic Function n X 4 f 7 ðxÞ ¼ ixi þ randðÞ
f 16 ðxÞ ¼ 1:28 6 xi 6 1:28
30 30 Y 1 X xi x2i cos pffi þ 1 4000 i¼1 i i¼1
600 6 xi 6 600
minðf 16 Þ ¼ f 16 ð0; . . . ; 0Þ ¼ 0
i¼1
minðf 7 Þ ¼ f 7 ð0; . . . ; 0Þ ¼ 0
17. Generalised Penalised Function:
8. Axis Parallel Hyperellipsoid Function: n X 2 f 8 ðxÞ ¼ ixi
f 17 ðxÞ ¼
ð5:12 6 xi 6 5:12Þ
i¼2
i¼2
minðf 9 Þ ¼ f 9 ð0; . . . ; 0Þ ¼ 0
2
f10sin ðpy1 Þ þ
n1 X 2 2 2 ðyi 1Þ ½1 þ 10 sin ðpyiþ1 Þ þ ðyn1 Þ g i¼1
i¼1
9. Dixon Price Function: n X 2 ið2x2i xi1 Þ þ ðx1 1Þ2
n
n1 X uðxi ;10; 100;4Þ 50 6 xi 6 50 þ
minðf 8 Þ ¼ f 8 ð0; . . . ; 0Þ ¼ 0
f 9 ðxÞ ¼
p
10 6 xi 6 10
minðf 17 Þ ¼ f 17 ð1;...;1Þ ¼ 0 1 where yi ¼ 1 þ ðxi þ 1Þ; 4 8 m > kðxi 1Þ ; xi > a; < uðxi ;a;k;mÞ ¼ 0; a < xi < a; > : m kðxi 1Þ ; xi < a:
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005
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Y.Z. Li et al. / Knowledge-Based Systems xxx (2014) xxx–xxx
18. Generalised Girewank Function:
f 18 ðxÞ ¼
n1 X 1 2 2 fsin ð3px1 Þ þ ðxi 1Þ2 ½1 þ sin ð3pxiþ1 Þ 10 i¼1 2
þ ðxn 1Þ2 ½1 þ sin ð2pxn Þg þ
n1 X uðxi ; 5; 100; 4Þ
50 6 xi 6 50
i¼1
minðf 18 Þ ¼ f 18 ð1; . . . ; 1Þ ¼ 0 8 m xi > a; > < kðxi 1Þ ; where uðxi ; a; k; mÞ ¼ 0; a < xi < a; > : kðxi 1Þm ; xi < a: 19. Alpine Function:
f 19 ðxÞ ¼
n X jxi sinðxi Þ þ 0:1xi j
10 6 xi 6 10
i¼1
minðf 19 Þ ¼ f 19 ð0; . . . ; 0Þ ¼ 0 20. Expansion of F10:
f 20 ðxÞ ¼ f 10 ðx1 ; x2 Þ þ þ f 10 ðxi1 ; xi Þ þ þ f 10 ðxn ; x1 Þ 100 6 xi 6 100 minðf 20 Þ ¼ f 20 ð0; . . . ; 0Þ ¼ 0 where f 10 ðxÞ ¼ ðx2 þ y2 Þ
0:25
2
0:1
½sin ð50ðx2 þ y2 Þ Þ þ 1
21. Expanded Scaffer’s Function:
f 21 ðxÞ ¼ f s ðx1 ; x2 Þ þ þ f s ðxi1 ; xi Þ þ þ f s ðxn ; x1 Þ 100 6 xi 6 100 minðf 21 Þ ¼ f 21 ð0; . . . ; 0Þ ¼ 0 0:5 2 sin ðx2 þ y2 Þ 0:5Þ where f s ðxÞ ¼ 0:5 þ 2 1 þ 0:001ðx2 þ y2 Þ References [1] H.G. Beyer, The simple genetic algorithm foundations and theory, IEEE Trans. Evol. Comput. 4 (2) (2000) 191–192.
[2] W. Banzhaf, J.R. Koza, C. Ryan, L. Spector, C. Jacob, Genetic programming, IEEE Intell. Syst. 15 (3) (2000) 74–84. [3] X. Yao, Y. Liu, G.M. Lin, Evolutionary programming made faster, IEEE Trans. Evol. Comput. 3 (2) (1999) 82–102. [4] O. Francois, An evolutionary strategy for global minimization and its markov chain analysis, IEEE Trans. Evol. Comput. 2 (3) (1998) 77–90. [5] M. Dorigo, J. Kennedy, The particle swarm: explosion, stability, and convergence in a multi-dimensional complex space, IEEE Trans. Evol. Comput. 6 (1) (2002) 58–73. [6] M. Clerc, M. Birattari, T. Stutzle, Ant colony optimization, IEEE Comput. Intell. Mag. 1 (4) (2006) 28–39. [7] D. Karaboga, B. Akay, A comparative study of artificial bee colony algorithm, Appl. Math. Comput. 214 (2009) 108–132. [8] K. Passino, Biomimicry of bacterial foraging for distributed optimization and control, IEEE Control Syst. Mag. 22 (3) (2002) 52–67. [9] R. Rajabioun, Cuckoo optimization algorithm, Appl. Soft Comput. 11 (8) (2011) 5508–5518. [10] D. Simon, Biogeography-based optimization, IEEE Trans. Evol. Comput. 12 (6) (2008) 702–713. [11] W. Pan, A new fruit fly optimization algorithm: taking the financial distress model as an example, Knowl.-Based Syst. 26 (2012) 69–74. [12] S. He, Q.H. Wu, J.R. Saunders, Group search optimizer: an optimization algorithm inspired by animal searching behavior, IEEE Trans. Evol. Comput. 13 (5) (2009) 973–990. [13] L. Li, F. Liu, Group Search Optimization for Applications in Structural Design, Springer, London, 2011. [14] A.J. Nicholson, An outline of the dynamics of animal populations, Aust. J. Zool. 2 (1954) 9–65. [15] S. Richard, I.B. Daniel, Diversity within a natural population intraspecific competition drives increased resource use diversity within a natural population, Proc. R. Soc. B 274 (2007) 839–844. [16] G.M. Viswanathan, Optimizing the success of random searches, Nature 401 (1999) 911–914. [17] A.A. Berryman, Principles of Population Dynamics and their Application, Garland Science, New York, 1999. [18] Z.H. Zhan, J. Zhang, Y. Li, H.S.H. Chung, Adaptive particle swarm optimization, IEEE Trans. Syst. Man Cybern. B Cybern. 39 (6) (2009) 1362–1381. [19] D.I. Bolnick, Intraspecific competition favours niche width expansion in drosophila melanogaster, Nature 410 (2001) 463–466. [20] R.C. MacLean, Adaptive radiation in microbial microcosms, J. Evol. Biol. 18 (2005) 1376–1386. [21] M. Li, W. Tang, Q. Wu, J. Saunders, Paired-bacteria optimiser-a simple and fast algorithm, Inform. Process Lett. 111 (2011) 809–813. [22] M. Li, T. Ji, W. Tang, Q. Wu, J. Saunders, Bacterial foraging algorithm with varying population, Biosystems 100 (2010) 185–197. [23] Q. Pan, H. Sang, J. Duan, L. Gao, An improved fruit fly optimization algorithm for continuous function optimization problems, Knowl.-Based Syst. 62 (2014) 69–83.
Please cite this article in press as: Y.Z. Li et al., Group search optimizer with intraspecific competition and lévy walk, Knowl. Based Syst. (2014), http:// dx.doi.org/10.1016/j.knosys.2014.09.005