Academc Research Internatonal ISS-L: 3-9553, ISS: 3-9944 Vol., o. 3, May 0 EVALUATIO OF THE PERFORMACES OF ARTIFICIAL BEE COLOY AD IVASIVE WEED OPTIMIZATIO ALGORITHMS O THE MODIFIED BECHMARK FUCTIOS Dlay Gök Grne Amercan Unversty TURKEY. dlaygok@gau.edu.tr Al Haydar Grne Amercan Unversty TURKEY. ahaydar@gau.edu.tr Kaml Dmller Grne Amercan Unversty TURKEY. kdmller@gau.edu.tr ABSTRACT In ths paper, the performances of the Artfcal Bee Colony (ABC) Algorthm and Invasve Weed Optmzaton (IWO) Algorthm are compared on the bass of modfed versons of fve well known benchmark functons. The modfcatons are performed on these functons n order to get rd of the symmetrcal propertes of the selected functons. Further a soluton space s shfted so that the optmal functon values are not equal to zero. The expermental results have shown that the ABC algorthm outperforms the IWO algorthm on the modfed versons of the benchmark functons. Keywords: swarm ntellgence, optmzaton algorthm, Artfcal Bee Colony Algorthm, Invasve Weed Optmzaton Algorthm ITRODUCTIO In the past decades many dfferent metaheurstc optmzaton algorthms have been developed. All of the proposed algorthms and ther modfed versons present good results for specfc types of problems. On the other hand, classcal benchmark problems are well known by the optmzaton communty and are used to examne the performance of a proposed algorthm or a proposed modfcaton on the exstng algorthms (Yao et al., 999). But, t s also recognzed that some heurstc operators of the algorthms may explot some specal propertes of these benchmark functons (Ahrar et al., 00). In ths work, fve of the classcal benchmark problems are modfed n order to get rd of some specal propertes that can be exploted by some heurstc operators. These modfcatons are amed to satsfy the followng requrements (Lang et al. 005) : () Global optmum pont s not at the orgn; () Optmum parameter value s dfferent for each varable; and (3) Optmum parameter value s not lyng n the center of the search range. Artfcal Bee Colony Algorthm s one of the recently ntroduced swarm-based algorthms whch s proposed by Karaboga (Karaboga &Akay, 009). The algorthm smulates the behavour of a honey bee swarm. In the lterature, the algorthm s appled to the optmzaton of many unmodal and multmodal numercal functons (Yao et al., 999). The algorthm s also compared wth the other well known algorthms n the lterature (Karaboga &Akay, 009), (Karaboga & Basturk, 007). Invasve Weed Optmzaton s an optmzaton algorthm whch s nspred from colonzng weeds. It s known that weeds are very robust to envronmental changes and also they can easly adapt themselves to envronmental changes. Ths algorthm s desgned n order to mmc the robustness, adaptaton and randomness of colonzng weeds. The expermental results obtaned n the lterature through the use of ths algorthm have shown that the algorthm s a powerful algorthm ( Mehraban & Lucas, 006). In ths study, these two algorthms are compared on the bass of fve modfed benchmark functons. The results that are tabulated n the expermental results secton, clearly demonstrate that the ABC algorthm performs better than IWO algorthm for many of these functons. Copyrght 0 SAVAP Internatonal 4
Academc Research Internatonal ISS-L: 3-9553, ISS: 3-9944 Vol., o. 3, May 0 ARTIFICIAL BEE COLOY ALGORITHM ABC s developed based on the observaton of the behavour of honeybees on ther fndng of nectar and sharng ths nformaton to the other honeybees n the hve. In ths algorthm, three groups of bees have been defned for fndng food source. These are called as employed bees, onlooker bees and scout bees (Karaboga &Akay, 009). The algorthm conssts of three steps whch are sendng the employed bees onto ther food sources and evaluatng ther nectar amounts; after sharng the nectar nformaton of food sources, the selecton of food source regons by the onlooker bees and evaluatng the nectar amount of the food sources; determnng the scout bees and then sendng them randomly onto possble new food sources (Karaboga&Akay, 009). In ths algorthm, the poston of a food source represents a possble soluton to the optmzaton problem and the nectar amount of a food source corresponds to the ftness of the assocated soluton (Karaboga&Akay, 009). The number of the employed bees or the onlooker bees s equal to the number of solutons n the populaton (Karaboga&Akay, 009). The code of the ABC algorthm can be gven as follows: : Intalze the populaton of solutons x, =,..., S : Evaluate the populaton 3: cycle = 4: repeat 5: Produce new solutons v for the employed bees usng the evaluaton vj = φj ( xj xkj ) and evaluate them 6: Apply the greedy selecton process for the employed bees 7: Calculate the probablty values P for the solutons x usng the evaluaton 8: Produce the new solutons v for the onlookers from the solutons evaluate them 9: Apply the greedy selecton process for the onlookers P = S n ft ft xselected dependng on n P and 0: Determne the abandoned soluton for the scout, f exsts, and replace t wth a new randomly j j j j produced soluton x usng the evaluaton x = x rand[ 0, ]( x x ) : Memorze the best soluton acheved so far : cycle = cycle 3: untl cycle = MC mn IVASIVE WEED OPTIMIZATIO ALGORITHM Ths algorthm s manly smulatng the behavour of colonzng weeds. The algorthm has manly four steps whch can be explaned brefly as follows: Intalzaton A generaton of a populaton of ntal solutons randomly n the regon of nterest. Reproducton Based on the ftness of the plant (ntal soluton), the new seeds (new solutons) wll be reproduced. The number of seeds generated for each plant s proportonal to the ftness of t. max mn Copyrght 0 SAVAP Internatonal 43
Academc Research Internatonal ISS-L: 3-9553, ISS: 3-9944 Vol., o. 3, May 0 Spatal Dspersal The new generated seeds for each plant are randomly dstrbuted near to ther parent plant. The closeness of the seeds to ther parent plant s changng as the numbers of teratons are changng. Compettve Excluson It s the mechansm of elmnatng the plants wth lower ftness n the generaton after the number of plants (solutons) reaches the maxmum number of plants n the colony. It s the rankng of the generated seeds together wth ther parent s accordng to ther ftness values and selectng the ones wth hgher ftness values. Fgure. shows the flowchart of the IWO. EXPERIMETAL RESULTS Fgure.Flow chart depctng the IWO algorthm In ths paper, our am s to compare the performances of the ABC algorthm and the IWO algorthm on some well-known benchmark functons. Fve of them are selected n such a way that some of them are unmodal and the others are multmodal. The experments that we performed have shown us that each of the two algorthms outperforms the other one on dfferent classcal benchmark functons. In order to test the effect of some specal propertes of the classcal benchmark functons on these results we modfed the selected functons. The modfcatons are done n such a way that the soluton spaces are shfted and also symmetrcal propertes of them are changed whch are the man concerns n lterature for the modfcaton of the benchmark functons (Lang et al. 005), (Ahrar et al., 00).These fve benchmark functons are modfed n ths study and are presented n Table. Copyrght 0 SAVAP Internatonal 44
Academc Research Internatonal ISS-L: 3-9553, ISS: 3-9944 Vol., o. 3, May 0 Table.Modfed Benchmark Functons Functon Range f mn Modfed Sphere Functon = k=3.5 ( x ( ) k ) ( ) [-00,00] 0.307 Modfed Rosenbrock Functon = ( ) ( ) ( ) 400 x x x [-30,30] 435 Modfed Quartc Functon 4 ( ( x ( ) k ) ) ( ) = k=0.0 random [ 0,) [-.8,.8].99749 Modfed Schwefel Functon = x [-500,500] -57.484 40 40 ( ) ( ) sn x ( ) Modfed Grewank Functon x x k = = k=0.6 ( ) ( ) ( ) ( ) k 4000 * cos [-600,600] -3535 In all the followng expermental results the dmenson of each functon s fxed to 30. Because of random ntalzaton for both of the algorthms, the programs run for 30 tmes and the best, the worst and the average of these 30 runs are presented. Table and Table 3 shows the results obtaned for the modfed sphere functon usng the IWO algorthm and the ABC algorthm respectvely wth respect to functon evaluatons. These values are obtaned by changng the value of the control parameters of each algorthm and selectng the most sutable control parameters n ther defned range. Table.The average, worst and best results of the ABC algorthm usng Modfed Sphere functon Functon evaluaton (x00) 500 000 000 3000 5000 Average 0.996 0.3078 0.3075 0.3075 0.307 Worst.7086 0.3080 0.3075 0.3075 0.307 Best 0.640 0.3076 0.3075 0.3075 0.307 Copyrght 0 SAVAP Internatonal 45
Academc Research Internatonal ISS-L: 3-9553, ISS: 3-9944 Vol., o. 3, May 0 Table 3.The average, worst and best results of IWO algorthm usng Modfed Sphere functon Functon evaluaton(x00) 500 000 000 3000 5000 Average 07.74858 39.498.08456.00790.000887 Worst 36.97338 4.36560.09756.0099.0005 Best 6.45039 33.90898.057873.00587.00068 In Fgure, we plotted the average functon value obtaned for both of the methods wth respect to the number of functon evaluatons. Ths step s performed for all 5 modfed benchmark functons and the results obtaned after 5000(x00) functon evaluatons are gven n Table 4 for both of the algorthms. Fgure.Modfed Sphere functon for ABC and IWO algorthms DISCUSSIO AD COCLUSIO It s observed that the convergence speed of the ABC algorthm s better than the convergence speed of the IWO algorthm for the modfed sphere functon as can be seen from Table and Table 3. Ths observaton s also supported by results obtaned for the other modfed functons. Table 4 shows the best functon values obtaned by a fxed number of functon evaluatons. These results also show that the ABC algorthm has better performance than the IWO algorthm. The values that are obtaned by the ABC algorthm are the exact global solutons for the modfed sphere and modfed Grewank functons whle the IWO algorthm could not reach to optmal values. For other three functons the ABC algorthm reaches values that are closer to the global optmums than the values by the IWO algorthm. Hence, from our expermental results, we deduced that the ABC algorthm performs better than the IWO algorthm for these modfed functons. We are aware of the fact that, these results may change for some other modfed functons or even for the same functons wth dfferent modfcatons. Copyrght 0 SAVAP Internatonal 46
Academc Research Internatonal ISS-L: 3-9553, ISS: 3-9944 Vol., o. 3, May 0 Table 4.Performance of ABC and IWO algorthms on modfed functons Modfed Sphere Func. Modfed RosenbrockFunc. Modfed Quartc Func. Modfed SchwefelFunc. Modfed GrewankFunc. Mnmum functon values 0.307 435.99749-57.484-3535 ABC average 0.307 435.0758708.99804-496.6-3535.00 worst 0.307 436.5740.998309-30.00-3535.00 best 0.307 435.0000.99780-567.490-3535.00 IWO average.000887 503.553835.000953-65.9-34.447 worst.0005 800.00568.00943-983.54-3339.894 best.00068 435.003865.9994688-370.0-3466.309 REFERECES Ahrar, A, Saadatmand, M. R., Sharat-Panah, M., Ata, A. A. (00). On the lmtatons of classcal benchmark functons for evaluatng robustness of evolutonary algorthms, Appled Mathematcs and Computaton, o.5, 3 39 Karaboga, D. &Akay, B. (009).A comparatve study of Artfcal Bee Colony algorthm. Appled Mathematcs and Computaton o.4, pp.08 3. Karabaga, D. &Basturk, B. (007). A powerful and effcent algorthm for numercal functon optmzaton: artfcal bee colony (ABC) algorthm. Journal of Global Optmzaton, o. 39, pp.459 47. Lang, J. J., Suganthan, P.. &Deb, K. (005). ovel Composton Test Functons for umercal Global Optmzaton. Swarm Intellgence Symposum, 005. SIS 005. Mehraban, A.R. &Lucas, C. (006).A novel numercal optmzaton algorthm nspred from weed colonzaton. Ecologcal Informatcs (006)355 366.Iran: Unversty of Tehran. Yao X., Lu, Y. & Ln, G. (999). Evolutonary Programmng Made Faster.IEEE Transactons On Evolutonary Computaton, VOL. 3, O.. Copyrght 0 SAVAP Internatonal 47