Diversion of Constant Crossover Rate DE\BBO to Variable Crossover Rate DE\BBO\L

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1 , pp Dverson of Constant Crossover Rate DE\BBO to Varable Crossover Rate DE\BBO\L Ekta 1, Mandeep Kaur 2 1 Department of Computer Scence, GNDU, RC, Jalandhar 2 Department of Computer Scence, GNDU, RC, Jalandhar 1 er_ekta@yahoo.com, 2 mandeep8@gmal.com Abstract DE\BBO s the combnaton of dfferental evoluton and bogeography based optmzaton algorthm to get a better optmzaton algorthm n terms of convergence speed. In DE\BBO, constant crossover rate has been used whch sometmes affect the performance of the hybrd algorthm leadng to ncrease n convergence speed. To cope up wth ths problem, varable crossover rate has been ntroduced n the hybrd algorthm helpng n removng the problem of constant crossover rate. Modfed algorthm has been named as DE\BBO\L n whch local search mutaton and varable crossover rate are used. Testng of BBO, DE\BBO\rand\1 and DE\BBO\L has been performed on dfferent test functons. The results reveal that DE\BBO\L wth varable crossover rate s better than DE\BBO wth constant crossover rate. Keywords: Optmzaton, Mutaton, Dfferental evoluton, Bogeography Based Optmzaton 1. Introducton Dfferental evaluaton proposed by Storn and Prce s parallel drect search way of optmzaton. The prncple of DE les n generatng the tral vector by usng some mutaton method [1-3]. The basc mutaton technque s addton of weghted dfference of two populaton vectors wth the thrd populaton vector. The resultng vector s checked aganst the objectve functon to take decson of keepng or rejectng ths vector. There are varous mutaton methods. Equatons of some mutaton methods [4] are as gven below: DE\rand\1: D,G = X r1,g + F( X r2,g X r3,g ) DE\Best\1: D,G = X best,g + F( X r1,g X r2,g ) DE\Best\2: D,G = X best,g + F( X r1,g X r2,g )+ F( X r3,g X r4,g ) Where D, G s known as donor vector whch s used to modfy the target vector X,G. The other vectors are randomly chosen from the populaton. Bogeography Based optmzaton [5] has been nfluenced from the dstrbuton of speces n geographcal area. The geographcal area has many habtat, some habtat have good features for lvng bengs to be lved whereas some have not. Habtat Sutablty Index (HSI) s used to characterze varous habtats and Sutablty Index Varables (SIV) decdes the HSI. Same way has been used n case of optmzng the real world problems. Solutons of the problem are consdered as populaton and ther goodness s decded on the bass of the value of the HSI. Immgraton and emgraton rates are used to modfy the varous solutons. The man reason behnd the development of the hybrd Dfferental evoluton wth Bogeography Based Optmzaton s the convergence speed of the orgnal BBO. Mutaton scheme of DE combned wth the BBO algorthm solves the problem of convergence speed. DE\BBO, proposed n 2010, uses the mutaton scheme wth constant crossover rate whch has to be adjusted accordng to the problem and n some cases, ISSN: IJHIT Copyrght c 2016 SERSC

2 constant crossover rate leads to ncrease n the convergence speed. Modfed DE\BBO\L wth varable crossover rate solves ths problem. Rest of the paper has been organzed as Secton 2 gves the vew of lterature survey. Secton 3 descrbes the modfcaton done n the exstng DE\BBO. Secton 4 shows the results and paper has been concluded n Secton 5 gvng future scope. 2. Related Work In 2008, Bogeography based optmzaton method [5] was ntroduced by Dan Smon. Ths method has been nspred from the natural way of dstrbuton of speces n geographcal area. The dea of spece s mgraton from one habtat to another on the bass of sutablty of that partcular habtat has been utlzed to mprove the qualty of solutons. Orgnal BBO has many defcences. To mprove orgnal BBO, many other versons of BBO have been ntroduced. Oppostonal BBO [6] was ntroduced n 2010, whch s focused on mprovng the convergence speed by usng only mgraton not mutaton. Real coded BBO [7] was ntroduced n 2010, whch uses modfed mutaton operators to mprove the performance of orgnal BBO.DE/BBO [8] was ntroduced n 2010, whch provdes better robustness and ncreased convergence speed. Blended BBO [9] was ntroduced n 2012, whch focuses on new way of changng the features of soluton. Multobjectve BBO [10] was ntroduced n 2012, whch s based on predator-prey approach. Hybrd BBO [11] was ntroduced n 2014, whch ntroduces the combnaton of varous evolutonary algorthms wth BBO. Lnearzed BBO [12] was ntroduced n 2014 for hghly non-separable problems. In 1995, Dfferental Evaluaton optmzaton method [1] was ntroduced by Stron R and Prce K. In ths, tral vector s generated from three vectors randomly selected from the populaton followed by crossover and selecton process. The lmtaton of DE s premature convergence whch has been mproved by changng ts mutaton operator and ntroducng modfed crossover processes. Varous modfed mutaton schemes are DE\best\1, DE\target-to-best\t, DE\best\2, DE\rand\2 etc. Recent work on modfcaton n mutaton scheme has been found n 2012 and 2014 whch focus on local search mutaton [13]. Ths leads to mprovement of the performance of DE n varous means lke effcency, qualty of soluton and robustness. DE\BBO [8] s based on mutaton of dfferental evoluton ntroduced n BBO algorthm. In ths paper, mutaton technques used are based on the use of constant crossover factor for all the generatons. Recent mutaton technque has been surveyed named as Local search mutaton [13] whch uses the varyng value of CR for each generaton. In ths, followng of good solutons and avodance of bad solutons has been used. In 2014, RDEL (Restart Dfferental Evoluton algorthm wth Local Search Mutaton for global numercal optmzaton) [13] was proposed whch solves the problem of choce of CR value for varyng applcaton. In ths technque, there s no need of selectng the value of CR. CR s calculated for every generaton and ths do not lead to nfluence the convergence speed of the algorthm. Local search mutaton has been ntroduced to mprove the convergence speed. Ths mutaton s based on the poston of best member and worst member of the populaton whch s nspred from PSO. The dea of learnng les behnd the poston of best member and worst member of the populaton. Ths mutaton technque can be llustrated from the followng equaton: New Vector = Member chosen at random from populaton + F1 (Best Member - Member chosen at random from populaton)+ F2( Member chosen at random from populaton Worst member) After mutaton, restart mechansm s appled accordng to the pre-defned crtera. 3. Modfed Hybrdzaton of DE/BBO 208 Copyrght c 2016 SERSC

3 As observed n lterature survey, DE\BBO uses the mutaton technques wth constant CR value. Hybrdzaton of DE and BBO s based on the concept of usng the exploraton ablty of DE and explotaton ablty of BBO. When goodness of both s combned, DE\BBO outperforms the BBO n terms of convergence speed. Algorthm 1 descrbes the process of hybrd mgraton of DE/BBO [8] as gven below: a. for G = 1 to GL // GL s generaton lmt. b. for n= 1 to PS // PS s populaton sze. c. Randomly select three members so that a 1 a 2 a 3 n d. q rand = rand(1,m) // rand s used to choose random nteger from gven numbers. e. for q=1 to m // m s populaton dmenson. f. f rand< λ n // λ s mmgraton rate. g. f rand q >CR or q==q rand // CR s crossover rate. h. J n (q)= X a1 (q)+f(x a2 (q)- X a3 (q)) // F s scalng factor.. else j. Randomly select X h wth probablty μ h k. J n (q) = X h (q) l. end m. else n. J n (q) = X n (q) o. end p. end q. end r. end In the above algorthm, CR and F are the constants used n dfferental evaluaton optmzaton where CR s crossover rate varyng from 0 to 1 and F s scalng factor varyng from 0 to 2. Ths algorthm descrbes mgraton process whch s the combnaton of mutaton of dfferental evaluaton and mgraton of bogeography based optmzaton. Algorthm 1 has lmtaton of ncrease n convergence rate when CR value tends to ts upper lmt and CR value s same for all generatons. To mprove ths lmtaton, Modfed DE\BBO\L has been proposed wth use of Local search mutaton. Local search mutaton s based on the concept that good solutons should be adopted and bad solutons should be rejected to nhert the features [13]. The CR value s dfferent for each generaton. It does not lead to ncrease n convergence speed, even f CR tends to approach ts upper lmts [13]. Algorthm 2 descrbes the concept of Modfed DE\BBO\L as gven below: a. for G= 1 to GL // GL s generaton lmt. b. Calculate CR = 0.8+( ).(1-G/GL)^4 // CR s crossover rate. c. for n= 1 to PS // PS s populaton sze. d. Calculate F1 = rand(0,1), F2 = rand(0,1) and F3 = rand(0,1) e. Randomly select three members so that a 1 a 2 a 3 n f. q rand = rand(1,m) g. for q=1 to m // m s populaton dmenson. h. f rand q <CR or q==q rand. f rand = (1-G/GL) j. J n (q)= X a1 (q)+f1(x best - X a1 (q))+f2(x a1 (q)- X worst ) k. else l. J n (q)= X a1 (q)+f3(x a2 (q)- X a3 (q)) m. end n. else o. J n (q) = X n (q) p. end q. end Copyrght c 2016 SERSC 209

4 r. end s. end 4. Expermental Results Evaluaton of BBO, DE\BBO\rand\1 and DE\BBO\L has been done by usng MATLAB software R2012b. The features of the used system are Wndow 7, RAM 3 GB and Intel 3 processor. Table 1 shows how convergence speed ncreases wth ncrease n value of constant crossover rate. Table 1 also shows the convergence speed of BBO and DE\BBO\L. In DE\BBO\L, crossover rate s not fxed to one value. It goes on changng over generaton to generaton wthout affectng the convergence speed. The results have been tested on 10 dfferent test functons by takng the best result among many smulatons for each test functon. Fgure 1 to Fgure 25 shows the convergence characterstcs of BBO, DE\BBO\rand\1 and DE\BBO\L for dfferent test functons. Table 1. Convergence Speed n Seconds of BBO, BBO\DE\rand\1 and BBO\DE\L Algorthm BBO BBO\DE\rand\1 BBO\DE\rand\1 BBO\DE\rand\2 BBO\DE\L And CR = 0.01 And CR = 0.5 And CR = 0.8 Test Functon Rosenbrock Beale Booth Brann Easom Boh Boh Boh Hump Matyas RosenBrock Booth 210 Copyrght c 2016 SERSC

5 Fgure 1. Convergence Characterstcs of BBO for Rosenbrock and Booth Test Functons Brann Easom Fgure 2. Convergence Characterstcs of BBO for Brann and Easom Test Functons Boh1 Boh2 Fgure 3. Convergence Characterstcs of BBO for Boh1 and Boh2 Test Functons Copyrght c 2016 SERSC 211

6 Boh3 Beale Fgure 4. Convergence Characterstcs of BBO for Boh3 and Beale Test Functons Hump Matyas Fgure 5. Convergence Characterstcs of BBO for Hump and Matyas Test Functons Hump Matyas Fgure 6. Convergence Characterstcs of BBO\DE\rand\1 for Hump and Matyas Test Functons and CR = Copyrght c 2016 SERSC

7 Hump Matyas Fgure 7. Convergence Characterstcs of BBO\DE\rand\1 for Hump and Matyas Test Functons and CR =0.5 Hump Matyas Fgure 8. Convergence Characterstcs of BBO\DE\rand\1 for Hump and Matyas Test Functons and CR =0.8 Boh3 Beale Fgure 9. Convergence Characterstcs of BBO\DE\rand\1 for Boh3 and Beale Test Functons and CR =0.01 Copyrght c 2016 SERSC 213

8 Boh3 Beale Fgure 10. Convergence Characterstcs of BBO\DE\rand\1 for Boh3 and Beale Test Functons and CR =0.5 Boh3 Beale Fgure 11. Convergence Characterstcs of BBO\DE\rand\1 for Boh3 and Beale Test Functons and CR =0.8 Boh1 Boh2 Fgure 12. Convergence Characterstcs of BBO\DE\rand\1 for Boh1 and Boh2 Test Functons and CR = Copyrght c 2016 SERSC

9 Boh1 Boh2 Fgure 13. Convergence Characterstcs of BBO\DE\rand\1 for Boh1 and Boh2 Test Functons and CR =0.5 Boh1 Boh2 Fgure 14. Convergence Characterstcs of BBO\DE\rand\1 for Boh1 and Boh2 Test Functons and CR =0.8 Brann Easom Fgure 15. Convergence Characterstcs of BBO\DE\rand\1 for Brann and Easom Test Functons and CR =0.01 Copyrght c 2016 SERSC 215

10 Brann Easom Fgure 16. Convergence Characterstcs of BBO\DE\rand\1 for Brann and Easom Test Functons and CR =0.5 Brann Easom Fgure 17. Convergence Characterstcs of BBO\DE\rand\1 for Brann and Easom Test Functons and CR =0.8 RosenBrock Booth Fgure 18. Convergence Characterstcs of BBO\DE\rand\1 for Rosenbrock and Booth Test Functons and CR = Copyrght c 2016 SERSC

11 RosenBrock Booth Fgure 19. Convergence Characterstcs of BBO\DE\rand\1 for Rosenbrock and Booth Test Functons and CR =0.01 RosenBrock Booth Fgure 20. Convergence Characterstcs of BBO\DE\rand\1 for Rosenbrock and Booth Test Functons and CR =0.8 RosenBrock Booth Fgure 21. Convergence Characterstcs of BBO\DE\L for Rosenbrock and Booth Test Functons Copyrght c 2016 SERSC 217

12 Brann Easom Fgure 22. Convergence Characterstcs of BBO\DE\L for Brann and Easom Test Functons Boh1 Boh2 Fgure 23. Convergence Characterstcs of BBO\DE\L for Boh1 and Boh2 Test Functons Boh3 Beale Fgure 24. Convergence Characterstcs of BBO\DE\L for Boh3 and Beale Test Functons 218 Copyrght c 2016 SERSC

13 Hump Matyas Fgure 25. Convergence Characterstcs of BBO\DE\L for Hump and Matyas Test Functons 5. Concluson and Future Scope Ths paper has been devoted to the modfcaton of DE\BBO to mprove ts lmtaton. DE\BBO wth smple mutaton technque was developed to domnate the performance of BBO. From the results, t can be seen that DE\BBO\rand\1 takes more tme to converge for hgher values of crossover rate. Example, When CR = 0.8, convergence tme for DE\BBO\rand\1 usng Rosenbrock test functon s whch s much hgher than convergence tme of BBO. Thus, ths mples that crossover rate nfluences performance of DE\BBO\rand\1. Lmtaton of DE\BBO\rand\1 has been mproved by ntroducng Local search mutaton whch uses the varable crossover rate for all generatons and mproved scheme has been named as DE\BBO\L. In DE\BBO\L, crossover value also tends to the upper lmt but t does not affect the convergence tme. The performance has been tested by usng dfferent ten test functons whch shows that DE\BBO\L outperforms BBO and DE\BBO\rand\1 n terms of convergence speed and qualty of soluton. Overall, t can be concluded that purposed DE\BBO\L algorthm domnates the performance of both BBO and DE\BBO\L and gves qualty results. A lot of future work can be done. Applcablty of the purposed algorthm can be checked for varous applcatons. Further changes can be done n DE\BBO\L by usng the restart mechansm. Comparatve study can be done wth other exstng algorthms to check the performance of DE\BBO\L. Impact of populaton sze, populaton dmenson, and generatons can be studed for DE\BBO\L. Performance of modfed DE\BBO\L can be checked on other test functons and the results can be compared. References [1] R. Storn and K. V. Prce, Dfferental evoluton- A smple and effcent adaptve scheme for global optmzaton over contnuous spaces, Techncal Report TR , ICSI, (1995). [2] R. Storn and K. V. Prce, Dfferental Evoluton- a smple and effcent heurstc for global optmzaton over contnuos spaces, Journal of Global Optmzaton, vol. 11, no. 4, (1997), pp [3] R. Storn, K. V. Prce and J. Lampnen, Dfferental Evoluton- A Practcal Approach to Global Optmzaton, Sprnger, Berln, (2005). [4] D. Swagatam, A. Ajth and K. Uday, Dfferental Evoluton Usng a Neghborhood-based Mutaton Operator, (2007). [5] D. Smon, Bogeography-based optmzaton, IEEE Transactons on Evolutonary Computaton, vol. 12, no. 6, (2008), pp [6] M. Ergezer, D. Smon and D. Du, Oppostonal Bogeography-Based Optmzaton, Proceedngs of IEEE Internatonal Conference on Systems, Man, and Cybernetcs San Antono, TX, USA, (2009). [7] W. Gong, Z. Ca, C. X. Lng and H. L, A real coded Bogeography based optmzaton wth mutaton, Appled Mathematcs and Computaton, vol. 216, (2010), pp Copyrght c 2016 SERSC 219

14 [8] A. Bhattacharya and P. K. Chattopadhyay, Hybrd Dfferental Evoluton Wth Bogeography-Based Optmzaton for Soluton of Economc Load Dspatch, IEEE TRANSACTIONS ON POWER SYSTEMS, vol. 25, no. 4, (2010). [9] S. Kanoongo and P. Jan, Blended Bogeography Based Optmzaton for dfferent economc load dspatch problem, IEEE Canadan Conference on Electrcal and Computer Engneerng (CCECE), (2012). [10] A. de Marsl, C, e Slva, L, dos S. Coelho and L. Lebensztajn, Multobjectve Bogeography- Based Optmzaton Based on predator-prey Approach, IEEE TRANSACTIONS ON MAGNETICS, vol. 48, no. 2, (2012). [11] D. Smon, M. G. H. Omran and M. Clerc, Lnearzed Bogeography-Based Optmzaton wth Rentalzaton and Local search, Informaton Scences, vol. 267, (2014), pp [12] H. Ma, D. Smon, M. Fe, X. Shu and Z. Chen, Hybrd Bogeography-Based evolutonary algorthms, Engneerng Applcatons of Artfcal Intellgence, vol. 30, (2014), pp [13] A. W. Mohamed, RDEL: Restart Dfferental Evoluton algorthm wth Local Search Mutaton for global numerc optmzaton, Egyptan Informatcs Journal, (2014). 220 Copyrght c 2016 SERSC