Algorithm of Marriage in Honey Bees Optimization Based on the Nelder-Mead Method

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1 Algorthm of Marrage Hoey Bees Optmzato Based o the Nelder-Mead Method Cheguag Yag, Je Che, Xuya Tu, Departmet of Automato, School of Iformato Scece ad Techology, Beg Isttute of Techology, Beg 8, P. R. Cha Key Laboratory of Complex System Itellget Cotrol ad Decso (Beg Isttute of Techology), Mstry of Educato, Beg 8, P. R. Cha Abstract Marrage Hoey Bees Optmzato () s a ew swarm-tellgece method, but t has the shortcomgs of low speed ad complex computato process. By chagg the structure of ad utlzg the Nelder-Mead method to perform the local characterstc, we propose a ew optmzato algorthm. The global covergece characterstc of the proposed algorthm s proved by usg the Marov Cha theory. Ad the some smulatos are doe o Travelg Salesma Problem (TSP) ad several publc evaluato fuctos. Comparg the proposed algorthm wth ad Geetc Algorthm, smulato results show that the proposed algorthm has better covergece performace. Keywords: Marrage hoey bees optmzato (), Nelder-Mead Method, Marov cha, Nelder- Mead - Marrage Hoey Bees Optmzato (NM- ), Travelg Salesma Problem (TSP). Itroducto Swarm tellgece s a ew research area. It studes the behavor of socal sects ad uses ther models to solve problems. Recetly, Based o the marrage process of hoey bees, the ew techque of Marrage Hoey Bees Optmzato () was proposed by Jaso Teo ad Husse A. Abbass[]-[] ad has bee updated by Jaso Teo, Husse A. Abbass [3] ad Omd Bozorg Haddad et al [4]-[5]. The obectve of ths paper s to crease the performace of. By combg Nelder-Mead method, a mproved algorthm s proposed ad ts covergece s aalyzed based o the theory of Marov Cha. The paper s orgazed as follows. As the bass of the study, Marrage Hoey Bees Optmzato () algorthm ad Marov Cha wth some basc theorems ad deftos are revewed respectvely Secto, Secto 3. I Secto 4 ad Secto 5, the proposed algorthm ad ts covergece aalyss are preseted. Fally, some smulatos are doe ad cocluso s gve.. Algorthm of marrage hoey bees optmzato The behavor of hoey-bees shows may features le cooperato ad commucato, so hoey-bees have aroused great terests modelg tellget behavor these years. Marrage Hoey Bees Optmzato () s a d of swarm-tellgece method. Ad such swarmtellgece has some successful applcatos. At coloy s a example ad the search algorthm s spred by ts behavor. Matg behavor of hoeybees s also cosdered as a typcal swarm-based optmzato approach. The behavor of Hoey-bees s related to the product of ther geetc potetalty, ecologcal ad physologcal evromets, the socal codtos of the coloy, ad varous pror ad ogog teractos amog these three []-[]. The fve ma processes of are: (a) the matg-flght of the quee bees wth droes ecouter at some probablstcally. (b) creatg ew broods t by the quee bees, (c) mprovg the broods ftess by worers, (d) updatg the worers ftess, ad (e) replacg the least fttest quee(s) wth the fttest brood(s). 3. Marov cha Marov cha has bee wdely appled to. Marov chas (MCs) have bee used extesvely to study covergece characterstc. Such as may methods performace were aalyzed by modelg the process as a Marov process. A Marov cha s a sequece of radom s whose probablty at a tme terval depeds upo the of the umber at the prevous tme. The

2 probabltes of a Marov cha are usually etered to a trasto matrx dcatg whch state or symbol follows whch other state or symbol. Defto [6] : A square matrx s A = a (a) f, {, K} : a>, (b) f, {, K} : a, A s postve ( A> ); A s oegatve ( A ); m (c) f A ad m Ν : A >, A s prmtve; () (d) f A ad {, K}: a =, A s stochastc. Defto [6] : If the state space S s fte ( S = ), ad the trasto probablty p ( t) are depedet from t,, S, u, v Ν, p ( u) = p ( v) () the Marov cha s sad to be fte ad homogeeous. p t s the probablty of trastog from state ( ) S to state S at t. Theorem [6] : For a homogeeous fte Marov cha, wth the trasto matrx P= ( p ), If m m Ν : P > (3) the ths Marov cha s ergodc ad wth fte dstrbuto. lm p () t = p,, S s the steady t dstrbuto of the homogeeous fte Marov Cha. Theorem [6] (The basc lmt theorem of Marov cha): If P s a prmtve homogeeous Marov cha s trasto matrx, the T T T (a)! ω > : ω P= ω, ω:a probablty vector. T (b) ϕ S( ϕ s the start state ad t's probablty vector s g ): T lm gp = ω (4) (c) From lm P = P, we ca get a lmt probablty matrx P, T t s a matrx ad t's all rows are same to ω. Theorem 3 [6] : Let P be a reducble stochastc matrx, where C: m m s a prmtve stochastc matrx ad R, T. The C C P = lm P = lm = (5) TRC T R = s a stable stochastc matrx. 4. Nelder-Mead method- marrage hoey bees optmzato (NM- ) Oe of the most mportat advatages of over Geetc Algorthm s does a local search each terato. So ca avod solely usg crossover operator ad mutato operator who s of worse local search ablty. But algorthm chooses some smple ad radom local searchg methods, such as radom wal ad radom flp [], whch wll reduce the probablty of obtag optmal soluto. So such low effcecy of Worer badly flueces the whole performace of. The Nelder-Mead method s a commoly used olear optmzato algorthm, proposed by Nelder & Mead.[7] It s a drect search method ad does ot use umercal or aalytc gradets. The mert of Nelder-Mead method s that t s ot sestve to startg s ad ether does t rely o dervatves or o cotuty of the obectve fucto. So we utlze the local search ablty ad replace the Worer of algorthm by the Nelder-Mead method. Some studes related to have bee carred out our research. Oe of them s to crease the covergece speed. Here we mae some troduce about t, because the ma wor ths paper wll based o such mproved algorthm. I algorthm, the probablty of a droe maes wth a quee s defed by the aealg fucto []. Not oly the calculato of probablty s complex, but also ts calculato partcpats are complcated. So the whole process has a large computato burde. O the other had, we have foud that wth low speed eed eough terato tmes to approach optmzato result. But several varables, such as eergy, speed, ca t mae sure about ths. As the process gog, the matg probablty becomes smaller, whch ether help the calculato process put up, or help coverge globally. So based o the orgal algorthm, we have doe some mprovemet o the orgal algorthm. That s, by radom talzg droes ad restrctg the codto of terato, the computato process wll become easer. The detal about ths mprovemet has bee dscussed other papers before. Here we further our research to mprove the performace of ad propose a algorthm of Nelder-Mead -Marrage Hoey Bees Optmzato (NM) by tag the Nelder-Mead method as the Worer. The detal of NM- s show below. Defe Q: the umber of quees D: the umber of droe M: the sperm theca sze Italze each worer wth a uque heurstc Italze each quee s geotype at radom Apply Nelder-Mead method to mprove the quee s geotype Whle the stoppg crtera s ot satsfed (Cycle Tmes bgger tha Max Cycle Number or result s good eough)

3 for quee = to Q for = to M geerate a droe radomly add spermatozoa to the quee s sperm theca geerate a brood by crossoverg the quee s geome wth the selected sperm, mutate the geerated brood s geotype use Nelder-Mead method to mprove the droe s geotype f the ew brood s better tha the worst quee replace the least-fttest quee wth the ew brood refresh the quee lst depedg o ftess ed f ed for ed for ed whle Fg. : Nelder-Mead- Marrage Hoey Bees Optmzato algorthm (NM-) I Fg., the algorthm s much easer tha that the algorthm. Ad the umber of the parameters s also less tha the later. The whole process of NM- has fewer complex probablty calculatos whch wll help crease the computato speed. I NM-, we defe three operators: Crossover, Mutato ad Heurstc. Crossover ad Mutate are same as that. But the Heurstc operator s a ew oe proposed NM-. Crossover: Crossover operator exchages the peces of gees betwee chromosomes. Through crossover, t troduces ew chromosomes to the populato, ad hece the possblty of havg ftter chromosomes. Mutato: Mutato operato alters dvdual alleles at radom locatos of radom chromosomes at a very probablty. It mght create a better or worse chromosome, whch wll ether thrve or dmsh through ext selecto. Heurstc: Heurstc operator mproves a set of broods. It help coduct local search o broods. For the good local covergece performace, we use Nelder-Mead method as the heurstc operator. 5. Covergece aalyss of NM- algorthm I ths secto, we use Marov Cha to aalyss the covergece of the Nelder-Mead-Marrage Hoey Bees Optmzato algorthm. There are oly three ways to chage from oe geerato to aother, s Crossover, Mutate ad Heurstc. These operators deped oly o the puts ad ot restrcted wth tme. The we ca get the followg theorem. Defto 3: The state space of NM- s X = x= t, t,, t t,, =,, N (6) { [ N] } where [ t t t ],, K, N s the bary bt cluster lsted tur. Defe f ( x ) as the ftess fucto based o X ad y s the ftess. So the ftess aggregate Y s Y = { y y= f( x), x X} (7) It s easy to see x X, y > (8) Defe g = Y, we ca get a ordered aggregate { y, y, K, y }, y > y > K > y (9) g g Crossover, Mutate ad Heurstc operators lead to probable trasto the state space. Ad we use three trasto matrx C, M ad H to descrbe ther fectos respectvely. Fally, we ca get Tr = C M H () where Tr s the trasto matrx of the Marov cha of the NM- algorthm. Theorem 4: The Marov Cha of NM- s fte ad homogeeous. The aggregate { x, x, K, xm } s fte. So the Marov cha composed of { x, x, K, xm } s fte. Ths fte space ca also be sad as a state space X. Wth ρ, ρ X, the probablty of trasformato from the state ρ to the state ρ at t oly depeds o ρ ad s depedet of tme. So the Marov cha of the NM- algorthm s homogeeous. Ed. Theorem 5: The trasto matrxes of the crossover probablty ( C ) ad Heurstc probablty ( H ) the NM- algorthm are all stochastc. The square matrx C s C = c. The,, K : c ad {, K }: c = () So C s stochastc. The square matrx H s H = h. The,,, K : h ad {, K }: h = () So H s stochastc. Ed Theorem 6: The trasto matrx of the NM- wth mutato probablty ( M ) s stochastc ad postve.

4 [ ] M = m s a square matrx. The,, K : m ad {, K }: m = (3) So M s stochastc. Ad the mutato has a fluece o every posto of a state vector. We ca easly ow x, x X. Each posto of x ca mutate to the of x. So the probablty of x mutate to x s postve. So M s postve. Ed Theorem 7: The Marov Cha of the NM- ( Tr ) s ergodc ad wth fte dstrbuto.. lmtr ( t) = tr >,, X t Accordg to Theorem 5, Theorem 6 ad(), Tr s postve. Ad accordg to Theorem, ths proposto s proved. Ed Defto 4: The ftess of oe geerato s the largest oe of the dvduals ths geerato. f x, x K x = max f x (4) () { ( )} K =,,..., K Defe X = { x, x,, x f( { x, x,, x} ) = y, x, x,, x X} K K K, K K K y are defed at(9), that s, the ftess of all the dvduals X s equal to y. Defto 5: For a arbtrary tal geerato X(), y s of the largest ftess, lm Pr( f( X() t ) = y ) = (5) t The the algorthm s global covergece. Theorem 8: The NM- coverges to the global optmum. We ca defe TX = X N (6) For Defto 4 ad Theorem 4, TX s a Marov Cha. I the same tme, we defe P( X ) = P X X (7) X s defed (). We ca see that P( X ) > ad = P( X ) = Defe PX (, X) s the probablty state X go to X, we ca get N N PX (, X ) = Px (, x ), x X, x X (8) = Because NM- saves the best dvdual at every geerato, so PX (, X) =, <. Ad the trasto matrx of TX s Marov Cha ca be wrte as follows: PX (, X) L PX (, X) P = M L M PX (, X) L PX (, X) L (9) PX (, X) PX (, X) O M = M M O PX (, X) L L PX (, X) For Theorem 3, PX (, X) L (, ) PX X C=, T= M O M, R= M () PX (, X) L PX (, X) PX (, X) C C P = lm P = lm = () TRC T R = For Theorem 7 ad Theorem, P s a stable radom matrx, So R =.That s lm ( P( X, X ) ) R lm R () = = M = M lm ( P( X, )) X So every state TX wll go to X, f the terato umber s bg eough, ths proposto s proved. Ed 6. Smulato To test the covergece performace of NM-, we choose orgal algorthm ad Geetc Algorthm for comparso. We dd the smulato o two parts, oe usg some popular complex Evaluato Fuctos ad the other usg Travelg Salesma Problem (TSP). 6.. Comparso o evaluato fuctos The tal s geerated radomly, ad each fgure shows the average results of tmes smulato wth oe Evaluato Fucto. Evaluato Fucto : Sphere Model f( x) = x, x (3) =

5 35 3 NM- 3 5 NM Fg. : Results of NM-, ad wth evaluato fucto. Evaluato Fucto : Schwefel s Problem 3 3 f() s = x + x, x 5 (4) = = NM Fg. 5: Results of NM-, ad wth evaluato fucto 4. Evaluato Fucto 5: Geeralzed Rosebroc s Fucto f x x x x x (7) ( ) = ( + ) + ( ), 3 = NM Fg. 3: Results of NM-, ad wth evaluato fucto. Evaluato Fucto 3: Schwefel s Problem f( x) = ( x), x (5) = Fg. 6: Results of NM-, ad wth evaluato fucto 5. Evaluato Fucto 6: Step Fucto 3 f( x) = ( x +.5 ), x (8) = 7 6 NM- 6 5 NM Fg. 4: Results of NM-, ad wth evaluato fucto 3. Evaluato Fucto 4: Schwefel s Problem 3 f( x) = max x, x (6) = Fg. 7: Results of NM-, ad wth evaluato fucto 6. Evaluato Fucto 7: Geeralzed Rastrg s Fucto

6 f x = x x + x ( ) cos( π ), 5. = NM- (9) NM dstace Fg. 8: Results of NM-, ad wth evaluato fucto 7. Evaluato Fucto 8: Acley s Fucto x f( x) = x cos( ) +, x 6 4 (3) = = Fg. : TSP wth 6 odes solved by NM-, ad..3 x 5.. NM- 3 5 NM- dstace Fg. : TSP wth 48 odes solved by NM-, ad Fg. 9: Results of NM-, ad wth evaluato fucto Travelg salesma problem A Classcal Travelg Salesma Problem (TSP) has bee a terestg problem. Gve a umber of odes ad ther dstaces of each other, a optmal travel route s to be calculated so that startg from a ode ad vst every other ode oly oce wth the total dstace covered mmzed. Here TSP based o the data form TSPLIB s solved by NM-, ad algorthm respectvely Some remars From the above, we ca see that the NM- show better performace tha ad, ot oly to solve TSP but also to optmze the evaluato fucto, ad ca eep coverge faster wth dfferet ode s umber. The smulato results show that: NM- s coverget ad eeps good performace for all these test fuctos, though these test fucto are more complex tha the ormal oes ad may have may local optmzato pots. NM- performs better tha ad. NM- coverges more qucly, especally at tal part. Partcularly, eve f the tal codto s worse tha ad, NM- ca show fer result. As for, because of the process of choosg droes wth some probablty, s performace s ot always well. It ofte eeps stayg at a for some tme ad the drops dramatcally at some.

7 Sometmes s better tha, but sometmes ot. 7. Coclusos Covergece performace s very mportat for optmzato methods. I ths paper, we proposed a algorthm of Nelder-Mead Method Hoey Bees Optmzato (NM-) to overcome the slowess of the orgal. has a set of parameters to coordate ad much of calculato tme s cost. Whle, NM- avods such complex process ad also ca reach the expect result. It geerates a droe radomly each tme ad mate wth a fte quatty of quees. So NM- ca ot oly avod the local optmum, but also crease the speed. Also NM- s easy to mplemet ad has few parameters to adust. Ad the global covergece s preserved for optmzato. Smulatg wth complex evaluato fuctos ad TSP, NM- shows better performace tha ad. The algorthm stll deserves deep study. Ad the research about NM- wll be carred out ad wll be tested ad mproved wth practcal cases the future. [5] H.S. Chag, Covergg marrage hoey-bees optmzato ad applcato to stochastc dyamc programmg. Joural of Global Optmzato, 35: 43-44, 6. [6] G. Rudolph, Covergece aalyss of caocal geetc algorthms. IEEE Trasacto Neural Networs. 5(): 96-, 994. [7] J.C. Lagaras, J.A. Reeds, M.H. Wrght ad P.E. Wrght, Covergece propertes of the Nelder- Mead smplex method low dmesos, SIAM Joural of Optmzato, 9(): -47, 998. Acowledgemet Ths wor s partally supported by Beg Prorty Laboratory Fud of Cha (Grat No.SYS75). Refereces [] H.A. Abbass, Marrage Hoey Bees Optmzato (): a haplometross polygyous swarmg approach. Cogress o Evolutoary Computato, CEC, Seoul, Korea, pp. 7-4,. [] H.A. Abbass, A sgle quee sgle worer hoey-bees approach to 3-SAT. Proceedgs of the Geetc ad Evolutoary Computato Coferece, GECCO, Sa Fracsco, USA, pp.87-84,. [3] J. Teo ad H.A. Abbass, A aealg approach to the matg-flght traectores the marrage hoey bees optmzato algorthm, Techcal Report CS4/, School of Computer Scece, Uversty of New South Wales at ADFA,. [4] O. Bozorg Haddad, A. Afshar ad A.Mguel Maro, Hoey-bees matg optmzato (HBMO) algorthm: a ew heurstc approach for water resources optmzato. Water Resources Maagemet, :66-68, 6.