Binary Black Holes Algorithm

Transcription

1 Biay Black Holes al Joual of Copute Applicatios ( ) Mostafa Neati Copute Sciece Depatet, Tabai Uivesity of Babol, Ia Hossei Moei Agicultual Scieces ad Natual Resouces Uivesity of Goga Navid Bazka Copute Sciece Depatet, Tabai Uivesity of Babol, Ia ABSTRACT I this pape we adapt black holes optiizatio algoiths fo biay seach spaces by applyig a sigoid tasfoatio to the gavity ad electical foces. Black holes algoith is a Swa ispied of Black Holes fo Optiizatio Pobles. We supposes each solutio of poble as a biay black hole ad afte calculatig the gavity ad electical foces use a appig fuctio fo set it. The poposed ethod is veified usig seveal bechak pobles used i the aea of optiizatio. The expeietal esults o diffeet bechaks show that the pefoace of the poposed algoith is bette tha BPSO (Biay Paticle Swas Optiizatio), BAFS (Biay Atificial Fish Swa ) ad GA (Geetic ). Geeal Tes Atificial itelligece, Swa algoiths, Black holes algoith. Keywods Biay Black Hole, biay seach spaces, Optiizatio Poble, Sigoid Fuctio, gavity ad electical foces. 1. INTRODUCTION Thee ae diffeet ethods fo solvig a optiizatio poble. Soe of these ethods ae ispied fo atual pocesses. These ethods usually stat with a iitial set of vaiables ad the evolve to obtai the global iiu o axiu of the objective fuctio [1]. Swa itelliget algoiths have attacted sigificat attetio i ecet yeas. behavio odels the atual behavio of diffeet pheoea [2]. Itesificatio ad divesificatio ae two ai featues of the eta-heuistic algoiths. The itesificatio phase seaches aoud the cuet best solutios ad selects the best cadidates o solutios. The divesificatio phase esues that the algoith exploes the seach space oe efficietly [3]. Thee ae so ay pobles which have discete atues. Moeove i ay applicatios, solvig pobles which have cocete space ae doe i discete space. Hece, sice both cocete ad discete pobles ae solvable i a discete space, eed to have discete seach algoiths ae felt.the ajoity of Optiizatio Pobles ae defied i the discete space. Hece, poposig a efficiet algoith to solve the pobles has becoe a attactive subject i ecet yeas [4]. BPSO is a global optiizatio algoith fo discete pobles poposed by Keedy ad Ebehat [5] i I the PSO algoith, each paticle seaches fo a optial solutio to the objective fuctio i the seach space [6]. SA is aothe etaheuistic optiizatio which ca be eployed i the discete seach space. Kikpatick et al, poposed the algoith based o the way theodyaic systes go fo oe eegy level to aothe i 1983 [7]. The est of this pape is ogaized as follows: The ext sectio gives a eview about black holes algoith. The poposed algoiths (biay black holes) itoduced i sectio 3. I sectio 4 the coputatioal ad expeietal esults ae peseted to evaluate the pefoace of the poposed ethod. Fially, i Sectio 5 icludes coclusios ad discussios. 2. BLACK HOLES ALGORITHM A black hole is a egio of space-tie whose gavitatioal field is so stog that othig which etes it, ot eve light, ca escape. The theoy of geeal elativity pedicts that a sufficietly copact ass will defo space-tie to fo a black hole. Aoud a black hole thee is a atheatically defied suface called a evet hoizo that aks the poit of o etu. If aythig oves close to the evet hoizo o cosses the Schwazschild adius it will be absobed ito the black hole ad peaetly disappea. The existece of black holes ca be disceed by its effect ove the objects suoudig it [8]. It is called "black" because it absobs all the light that hits the hoizo, eflectig othig, just like a pefect black body i theodyaics [9, 10]. A black hole has oly thee idepedet physical popeties: ass, chage, ad agula oetu [11]. A chaged black hole epels othe like chages just like ay othe chaged object. The siplest black holes have ass but eithe electic chage o agula oetu. The black hole algoith poposed i the pape [12] by Neati et al. I this algoith at fist geeated a ado populatio a the evolve it i the geeatios to ea best solutio. I this algoith iitialized step is poductio of a ube of ado black holes as iitial solutio. Each of this black holes has ow positio, ass ad electical chage. The ae of this step is called big bag. Each of black holes is a solutio fo the poble. At secod step, fitess evaluated fo each of these black holes as foula (2), which f is Cost fuctio ad deteie the best black hole i the populatio ad call it global best. I thid step, evaluated the ew positio of the each black hole by calculatig the foces.i algoith each black hole attacted to the global best by gavity foce ad attacted to the local best positio by the Coulob's law, I the othe wods we assue FG (gavity foce) fo the global seach ad FQ (electicity foce) fo the local seach. FG ad FQ ae calculated by (3) ad (4) foulas. (3) (4) (1) (2) 36

2 al Joual of Copute Applicatios ( ) Whee Fg is gavitatioal foce, Fq is electical foce, is ass of global best black hole, ad is chage of local best black hole. G ad K ae costat ube. Whe Fg ad Fq wee calculated, the we ea ew positio of the black holes by foula (5). Whee ad ae the positio of i-th black hole at iteatio t+1 ad t, espectively.ad Fg is gavitatioal foce, Fq is electical foce. Ad also ado1, ado2 ae ado ube betwee [0,1]. The algoith also used of Hawkig adiatio as. At this step is the sae utatio step i geetic algoith.by hawkig adiatio the algoith escape fo tappig i local optius. I this step, by adoly we chaged the positio of black holes. With this wok the algoith escape fo tappig i local exteus. 3. BINARY BLACK HOLES ALGORITHM (PROPOSED METHOD) I ay optiizatio pobles the seachig should be pefoed i the biay space. Hece it is desiable fo these optiizatio algoiths pesetig Biay vesio. I the biay space, the exploe paticles ove i the zeo ad oe space. Coside a hypecube which labeled each of the coes with biaies. Each of these coes is oe solutio. I ou ethod, the asses ove betwee these coes to seach the solutio space. The oveet of the paticle i ay diesio chages Oes to Zeos ad vice vesa. As the basic black holes algoith [12] opeates i cotiuous ad eal ube space, it caot be used to optiize the pue biay pobles. To tackle this poble, we poposed biay black holes algoith. Fo biay seach space, we have adapted the black holes to seach i biay spaces, by applyig a sigoid tasfoatio to the gavity ad electical foces to squash this foces s ito a age [0,1], ad foce the ew values of the locatios of black holes to be 0 s o 1 s. I this pape, the Gavity ad Electical foces of each Black Hole ae cosideed as a Pobability fuctio i ay diesio ad the Black Holes ove based o these pobabilities. I fact, i the biay vesio of ou ethod, Fg+Fq epesets the pobability of beig the Oe o Zeo istead of displaceet of the paticles. The sigoid fuctio tasfos the iput, which ca have ay value betwee plus ad ius ifiity, ito a easoable value i the age betwee 0 ad 1. Gaph of sigoid fuctio showed i the figue 1. (5) The equatio fo updatig positios Eq. (5) is the eplaced by Eql (7). Whee ad is a ado ube betwee [0,1]. Based o the above the ai steps i the poposed biay black hole algoith ae suaized as follow Pseudo-code: Iput: objective fuctio Output: optial solutio Iitialize a biay populatio of black holes with ado locatios i the seach space (Biay Big Bag) While (teiatio citeia satisfy) do Fo each black hole, evaluate the objective fuctio Select the global best black hole that has the best fitess value Calculate the Sigoid fuctio by Eq. (6) Chage the locatio of each black hole accodig to Eq. (7) Do Hawkig adiatio (as utatio i algoith) Ed of while 4. THE EXPERIMENTAL RESULTS I this sectio the poposed biay black hole algoith (BBLA) is tested with bechak fuctios. Six bechak fuctios with a vaiety of coplexity ae used to evaluate the pefoace of poposed ethod. Bechak fuctio ad popeties is show o table 1. The pefoace of the poposed algoith is copaed agaist well-kow algoith like tha BPSO (Biay Paticle Swas Optiizatio), BAFS (Biay Atificial Fish Swa ) ad GA (Geetic ). The expeiets fo each fuctio u fo 10 ties ad aveage of esult is epoted. I figues 5 fo bette distictio of fou algoiths the Y-axis (fitess) is o logaithic scale.figues 3-14 show the Covegece pefoace of GA, BPSO, PFSA ad BBLA (popose algoith) o 6 Bechak fuctio (100D ad 1000D) - X-axis is geeatio ad Y-axis is fitess o logaithic scale. (7) Figue1. Gagh of sigoid fuctio (6) 37

3 al Joual of Copute Applicatios ( ) Table 1. Bechak Fuctio F Equatio Diesio s Mi Figue2. Flow chat of biay black holes algoith 38

4 al Joual of Copute Applicatios ( ) Figue3. F1 fuctio 100D Figue6. F2 fuctio 1000D Figue4. F1 fuctio 1000D Figue7. F3 fuctio 100D Figue5. F2 fuctio 100D Figue8. F3 fuctio 1000D 39

5 al Joual of Copute Applicatios ( ) Figue9. F4 fuctio 100D Figue12. F5 fuctio 1000D Figue10. F4 fuctio 1000D Figue13. F6 fuctio 100D Figue11. F5 fuctio 100D Figue14. F6 fuctio 1000D 40

6 al Joual of Copute Applicatios ( ) Table 2. Global optiizatio esults fo fuctio 1 ( ) size Diesio Aswe GA BPSO BFSA BBLA GA BPSO BFSA BBLA Table 3. Global optiizatio esults fo fuctio 2 ( ) size Diesio Aswe GA BPSO BFSA BBLA GA BPSO BFSA BBLA Table 4. Global optiizatio esults fo fuctio 3 ( ) size Diesio Aswe GA BPSO BFSA BBLA GA e+0 04 BPSO e+0 04 BFSA e+0 05 BBLA e+0 05 Table 5. Global optiizatio esults fo fuctio 4 ( ) size Diesio Aswe GA BPSO BFSA BBLA GA e+0 03 BPSO BFSA e+0 03 BBLA e

7 al Joual of Copute Applicatios ( ) Table 6. Global optiizatio esults fo fuctio 5 ( ) size Diesio Aswe GA BPSO BFSA BBLA GA BPSO BFSA BBLA Table 7. Global optiizatio esults fo fuctio 6 ( ) size Diesio Aswe GA BPSO BFSA BBLA GA BPSO BFSA BBLA CONCLUSION I this pape we adapt black holes optiizatio algoiths fo biay discete seach spaces by applyig a sigoid tasfoatio to the gavity ad electical foces. Each solutio of poble is a biay black hole ad afte calculatig the gavity ad electical foces use a appig fuctio fo set it i biay space. The expeietal esults o diffeet bechaks show that the pefoace of the poposed algoith is bette tha othe siila algoiths. To cotiue ou wok we decide to adapt black holes algoiths i ulti objective optiizatio pobles ad itoduced ulti objective black hole algoith (MBLA). 6. REFERENCES [1] Rai Rajabiou, Cuckoo Optiizatio, Applied Soft Coputig , Published by Elsevie (2011). [2] Hsig-Chih Tsai, Yog-Huag Li, Modificatio of the fish swa algoith with paticle swa optiizatio foulatio ad couicatio behavio, Applied Soft Coputig 11, , Published by Elsevie (2011). [3] Ai Hossei Gadoi, Ai Hossei Alavi, Kill hed: A ew bio-ispied optiizatio algoith, Cou Noliea Sci Nue Siulat 17, , Published by Elsevie (2012). [4] Zaha Beheshti, Siti Maiya Shasuddi, Siti Sophiayati Yuhaiz, Biay Acceleated Paticle Swa (BAPSA) fo discete optiizatio pobles, Joual of Global Optiizatio, Decebe, Published by Spige (2012). [5] Keedy, J., Ebehat, R.C.: A discete biay vesio of the paticle swa algoith. I: Poceedigs of IEEE al Cofeece o Coputatioal Cybeetics ad Siulatio, pp Olado, USA (1997). [6] H. Oapou, et al., " Dyaic Paticle Swa Optiizatio fo Multiodal Fuctio," al Joual of Atificial Itelligece (IJ-AI), Vol. 1, No. 1,. ISSN: , (2012). [7] Kikpatick, S., Gelatto, C.D., Vecchi, M.P.: Optiizatio by siulated aealig. Sciece 220, (1983). [8] L. Kape, E. Heuvel, P. Woudt, R. Giaccoi, Black hole eseach past ad futue, i: Black Holes i Biaies ad Galactic Nuclei: Diagostics, Deogaphy ad Foatio, Spige, Beli/Heidelbeg, pp. 3 15, [9] Schutz, Bead F. (2003), Gavity fo the goud up, Cabidge Uivesity Pess, ISBN [10] Davies, P. C. W,. "Theodyaics of Black Holes", Repots o Pogess i Physics, Rep. Pog. Phys., Vol. 41, Pited i Geat Bitai. [11] Heusle, M, "Statioay Black Holes: Uiqueess ad Beyod",Livig Reviews i Relativity Retieved [12] Mostafa Neati, et al., Black Holes : A Swa ispied of Black Holes fo Optiizatio Pobles, IAES al Joual of Atificial Itelligece (IJ-AI), Vol 2, No 3, Septebe (2013).. IJCA TM : 42