Immune-inspired Evolutionary Algorithm for Constrained Optimization

Transcription

1 Immune-inpied Evolutionay Algoithm fo Contained Optimization Weiwei Zhang 1, 2 and Gay G. Yen 2 1 Depatment of Compute Science, Chongqing Univeity, Chongqing 40044, PR CHINA 2 School of Electical and Compute Engineeing, Oklahoma State Univeity, Stillwate, OK-74078, USA Abtact Thi pape popoe an atificial immune ytem baed algoithm fo olving contained optimization poblem, inpied by the pinciple of the vetebate immune ytem. The analogy between the mechanim of vetebate immune ytem and contained optimization fomulation i fit given. The population i divided into two goup- feaible individual and infeaible individual. The infeaible individual ae viewed a the inactivated immune cell appoaching the feaible egion by deceaing the containt violation wheea the feaible individual ae teated a activated immune cell eaching fo the optima. The inteaction between them though the extacted diectional infomation i facilitated mimicking the functionality of T cell. Thi mechanim not only encouage infeaible individual appoaching feaibility egion, but facilitate exploing the bounday between the feaible and infeaible egion in which optima ae often located. Thi appoach i validated and pefomance i quantified by the benchmak function ued in elated eeache though tatitical mean with thoe of the tate-of-the-at fom vaiou banche of evolutionay computation paadigm. The pefomance obtained i faily competitive and in ome cae even bette. Index Tem Atificial immune ytem, contained optimization, containt handling A I. INTRODUCTION tificial immune ytem (AIS) i a newly emeging computational paadigm inpied by the fundamental of immune ytem. 1.1 Biological Immune Sytem In biology, the human o othe vetebate immune ytem i an adaptive, ditibuted, obut, elf-egulate, and complicated ytem which contituted by cell, molecule, and ogan that potect the body againt known and unknown dieae. The immune ytem poee multilevel defene mechanim. The potection laye uch a phyical baie and innate immune ytem contitute the non-pecific epone, wheea the adaptive immune ytem i eponible fo the pecific epone. Pathogen o any molecule that timulate the immune epone ae imply called antigen in thi pape. Antigen in the envionment ae blocked fom getting into the body by phyical baie, uch a kin and hai, at fit. Undeneath the kin, thee ae phyiological baie uch a aliva and tomach acid which can kill mot micooganim ingeted in food and wate. Antigen that ecape fom the phyical baie ae tackled by the othe non-pecific epone mechanim called innate immune ytem. Innate immune cell uch a macophage can ecognize foeign antigen and fagment them into peptide. Thee peptide, expe on the uface of macophage, make it eve a antigen peent cell. Meanwhile, thee antigen peent cell ecete activating molecule uch a chemical ignal to timulate the tat of the adaptive immune epone. The adaptive immune epone i an acquied ability that eact pecifically to the attacking antigen. The adaptive immune ytem i compoed of two type of lymphocyte: B lymphocyte (B-cell) and T lymphocyte (T-cell). The main function of B-cell include the poduction and ecetion of antibodie a a epone to bind with antigen. Each B-cell i pogammed to poduce a pecific antibody. The poduction of binding antibodie i uually a way of ignaling othe cell, uch a cavenging cell, to emove the bound ubtance. The main function of T-cell include the egulation of othe cell action and diectly attack the hot infected cell. T-cell wok, pimaily, by eceting cytokine, lymphokine, o the othe chemical ubtance to maintain the egula function of the immune ytem. Moeove, the adaptive epone i alo able to memoize the attacking antigen o that it will be effectively fought againt the ame o imila antigen in the futue [1-3]. 1.2 AIS fo Contained Optimization Nomally, immune cell patol the body ciculating along the blood flow to detect if thee ae antigen peent. Once antigen i detected, the immune epone i timulated. The activated immune cell eliminate the antigen diectly o ecete ignal to ecuit othe immune cell to help. Immune cell ae uppoed to be able to ecognize the antigen patten, extact the antigen featue, lean the antigen type, elect the uitable eacting mechanim to contol the ham, and emembe the expeience. To make it imple, due to the B-cell having the capability of antigen epeentation a ame a the innate immune cell, let u omit the innate immune ytem and only conide the immune cell in adaptive immune ytem. Naïve B cell patol the body eaching

2 fo the antigen. Without activation of the chemical ignal, naïve B cell do not pefom pecific immune epone. Howeve, with the iitation of chemical ignal eceted by T-cell, naïve B cell tanfom into effecto B cell, modify thei antibody, and fight with the antigen. In piit, chemical ignal wok a contained condition which contol the function of B-cell in the immune epone. Similaly in many eal wold poblem, the deciion vaiable ae ubject to containt which limit the each fo optimal olution only in the feaible egion. A contained optimization poblem can be defined a follow, Minimize f (x) (1) ubject to: g i ( x) 0, i 1,, q (2) h i ( x) 0, i q 1,, m (3) n x x 1,, x in whee a vecto of deciion vaiable which each uppe bound, x j, j 1,, n i bounded by a lowe and a L j j j x U defining the each pace n S, q i the numbe of inequality containt while m q i the numbe of equality containt. A olution atifie all the inequality and equality containt i called a feaible olution, wheea a olution violate at leat one containt i called infeaible olution. The inequality containt that atify g i ( x) 0 at the global optimum olution ae called active containt. In thi aticle, a novel contained atificial immune algoithm i popoed baed on immune epone pinciple. A new pepective fom the ignal tanfe and the inteaction among the immune cell i popoed to handle the contained condition and optimize the poblem. The emainde of thi pape i oganized a follow. Section II eview elated wok of handling contained optimization poblem uing AIS. In Section III, the popoed algoithm i peented in detail. Section IV dicue the eult obtained fo ome elected benchmak function and compae the eult obtained by the tate-of-the-at. Finally, Section V peent ome concluding emak and elevant obevation. II. LITERATURE REVIEW Equipped with fundamental chaacteitic of biological immune ytem, AIS ha eceived appeciable attention to olving complex computational poblem in the pat few yea. Howeve, only vey few eeach wok exit in the field of containt handling uing AIS. A uvey pape which ummaized algoithm eve popoed in handing contained optimization baed on atificial immune ytem can be found in [4]. 2.1 Hybid AIS with GA and Othe Mechanim A banch of hybid AIS with GA wa popoed [5-8]. The hybid algoithm poe two laye: GA oute laye and AIS inne laye. The oute GA i eponible fo the uncontained objective function optimization, while the inne AIS embedded into the oute GA help population appoaching feaible egion. Baed on it containt-handling technique, a clonal election evolutionay tategy wa popoed [9]. Immune algoithm i alo integated with hill climbing local each in [10], and integated with paticle wam optimization (PSO) in [11] fo contained optimization poblem. In all above deign, AIS play a complementay ole in eithe handling the containt o impoving the each ability. Although ome of thee hybid algoithm how vey competitive eult, the hybid chema often adveely inceae the complexity of the deign. 2.2 AIS baed Containt Handling Algoithm AIS algoithm that ae excluively baed on poce and pinciple diectly extacted fom the biological immune ytem wee popoed [12-18] AIS baed on clonal election algoithm Cuz-Coté et al. [12] popoed an appoach baed on CLONALG [19] to containt handling uing AIS. Gauian-ditibuted mutation, Cauchy-ditibuted mutation, and contolled unifom mutation ae compaed. In [13], the feaible individual and infeaible individual ae given diffeent mutation opeation. The mutation mechanim i efficient to balance the exploation and exploitation duing the evolution of each. The algoithm i imple to implement and the expeiment eult i found bette than that of [12]. A thee algoithm applying the uncontained optimization technique to the containt optimization, it i had to olve the poblem with dicontinuou each pace o multiple diconnected feaible egion AIS baed on idiotypic netwok appoach A new algoithm which combine clonal election theoy with idiotypic netwok theoy wa developed by Wu [14-15]. In the pape, idiotypic netwok election opeato i ued to contol the numbe of good olution; omatic hypemutation and ecepto editing opeato ae contucted to exploe the each pace; and bone maow opeato i to maintain the diveity of the olution. Adaptive penalty function i intoduced to tanfom the contained optimization poblem into an uncontained one. The imulation eult how that the algoithm i effective and efficient AIS baed on T-cell model In [16-17], T-cell model to handle the contained optimization wa popoed. Baed on the matuation o development level, T-cell ae divided into thee goup: Vigin cell (VC), Effecto cell (EC), and Memoy cell (MC). Giving each goup a pecific ize of population, mutation opeato, and election pinciple, the algoithm adaptively adjut the mechanim unde diffeent

3 condition. Fom the expeiment compaion, T-cell model i poven to be vey competitive with epect to the tate-of-the-at deign. The algoithm i novel and effective, but it i complicated in the tuctue. Many deign paamete equiing additional tuning may limit it ue in application. 2.4 AIS with Special Chaacteitic An algoithm deployed the communication between innate immune epone and adaptive epone in immune ytem wa popoed in [18]. Vaccine opeato popoed in [20] i ued to exploe the each aea. 10% of the antibodie fom antibody population ae elected to fom Majo Hitocompatibility Complex (MHC) a the diection extacted fom the bette individual. The combination of the diected mutation with vaccine extacted fom the ealy tage of the algoithm give a peedy way of dicoveing the feaible egion. The pefomance of the algoithm i vey competitive, and the vaccine opeato guaantee the high feaible ucce ate. Howeve, high computational complexity i needed to guaantee the extacted vaccine ae able to help antibodie tavee the whole eaching pace without exploing the ame aea epeatedly. The optima-eaching ability of AIS ha been poven to be effective in contained optimization poblem, but the containt-handling mechanim emain deficient. The inteelation among the ytem component i aely mentioned. III. PROPOSED ALGORITHM The immune epone of B-cell to an antigen i a poce of puuing the bet binding antibody to the antigen which i conideed an optimization poce. Naïve B-cell ciculate in the body eaching fo the infectiou pot a global each, while effecto B-cell modify thei hape to match the antigen a local each. T-cell balance the global and local each though eceting chemical ignal. Additionally, the chemical ignal eceted by T-cell pefom exactly a the containt to contol the function of B-cell. Without the timulating ignal, B-cell only epond to the cetain antigen in the manne of innate immune epone in a nonpecific way. When B-cell eceive the timulating ignal, the activated B-cell will adaptively adjut themelve to eact to the antigen in a pecific manne. Accoding to the diffeent atifaction to the containt, B-cell ae epaated into two ditinct type- naïve (inactivated) B-cell and effecto (activated) B-cell. The way B-cell unde the contol of T-cell in puuing the bet binding antibody to the antigen can be conideed a a contained optimization poce. Inpied by the effective and apid epone of B-cell to antigen, an AIS algoithm i popoed heein. Containt epaate the candidate olution into two goup- infeaible and feaible which nicely match the two tate of B-cell. Infeaible individual eemble the naïve B-cell to exploe the whole each pace while feaible individual mimic the effecto B-cell to locate the exact antigen (optima). Chemical ignal play an intemediate ole of tanfeing the infomation between infeaible and feaible individual and balance the exploation and exploitation. The popoed algoithm a hown in Fig. 1 depict the flowchat a how the algoithm wok. 1) Randomly initialize P individual in the deciion pace x i, i 1,, P, whee x i the ith individual in i the n-dimenional deciion pace. 2) Evaluate the containt violation v( x i ) fo each individual in the population a q m max(0, g j ( xi )) j 1 j q 1 v( xi ) max(0, hj ( xi ) ) (4) whee i the toleance value that help tanfom the equality containt into an inequality one. 3) Categoize the population into two ditinct goup: if v ( x i ) 0, add x into infeaible goup, i IF. Othewie, add it into feaible goup, F. 4) If feaible goup F i not empty, evaluate the fitne value f ( x ) fo each feaible individual x F, 1,, F, indicate the cadinality. Then ot the individual in the acending ode accoding to thei fitne value (auming minimization poblem). a) Cloning: Fo each feaible individual x in the anking ode of, aexually poduce nc ( x ) clone P nc( x ) ound, x F (5) whee i a multiplying facto (i.e., uually 0.1), ound( ) denote the ounding opeation to the cloet intege. Since all the individual ae oted in the acending ode peviouly, the individual with highe objective value will eceive moe clone. b) Hypemutation: Hypemutate each clone at a ate inveely popotional to thei affinity uing f ( x ) f *( x ), max f ( x ) x F x ) exp( f *( x )), ( ' maxgen gen x x x N ( ) 0, 1, (6) maxgen whee N (0,1 ) i a Gauian andom numbe with zeo mean and unit vaiance, f ( x ) i the nomalized fitne of * x, i the decay contant that contol the tep ize of the mutation (default value i et to 5), maxgen i the

4 maximum numbe of geneation, and gen efe to the cuent geneation. c) Selection: Evaluate the affinity of the clone. If the bet individual of the et of mutated clone ha the highe affinity (lowe objective value) than the oiginal one, ubtitute it. Othewie, keep the oiginal individual. d) Add the newly geneated infeaible clone into the infeaible goup, IF. uing a pedefined upeion thehold (poblem dependent, 0.5 fo efeence). 7) Recepto editing: The wot 20% individual of the whole population ae ubtituted at andom (binay tounament election i adopted hee, few candidate ae geneated andomly, the one in the mot pae aea i choen). 8) If topping condition i not met go to Step 2. Othewie, output the bet one (in tem of affinity) in the feaible goup. 5) Fo each individual x IF, 1,, IF with thei containt violation v x ), pefom the following opeation. ( a) Diection extaction: if thee ae feaible individual exit in the population, chooe a feaible individual x * by oulette wheel election baed on thei fitne value, and extact the diection infomation d a follow, x x* d. (7) x x* Othewie, 10% of the bet infeaible individual ae choen to geneate the diection baed on thei containt violation. b) Location update: Infeaible individual update thei location eithe by the way of the extacted diection d, x x d U(0,1), (8) ' o jut chooe a andom velocity to move x x N(0, ), (9) ' whee [0,1 ] i an ue defined paamete contolling the effect of the diection infomation, ( 0. 1 a uggeted) U (0,1 ) i a unifom andom numbe geneated between 0 and 1, Gauian andom numbe i ued to intoduce the andom velocity. contol the magnitude of the velocity (the default value i 0.2). To futhe impove upon the exploation, the location update epeat a few time and the bet poition i choen to eplace the cuent one. Duing implementation, eithe (8) o (9) can be choen to update the individual location equally o bia one of them baed on the pefomance. The fome one i applied in the pape. 6) Suppeion: Combine the infeaible and feaible goup togethe, and delete the imila individual Fig. 1 Flowchat of the popoed algoithm Since feaible and infeaible individual take diffeent eponibility in handling contained optimization poblem, we divide them into diffeent goup and facilitate the inteaction between them in the popoed algoithm. The function of feaible individual i to each fo the optima in the feaible egion, which i egaded a an uncontained optimization each. Conideing the faily good pefomance of the epeentative algoithm in clonal election theoy- CLONALG [19] fo uncontained multimodal optimization, we boow the opeato fo optimization in feaible egion hee. The individual with highe affinity will bea moe clone and lowe mutation ate, which give moe chance to the egion with highe objective value and encouage the local each. Moeove, the iteation i conideed a a paamete to contol the convegence of the algoithm. With the inceae of geneation count, the tep ize of hypemutation will become malle and malle. Thi implie a lage mutation tep at the beginning and become malle and malle towad the end of the each poce. On the othe hand, containt may etict the feaible each pace into mall and diconnected egion. Ou pupoe i to each fo thee feaible egion and then locate the optima. So the tak of infeaible individual i on one hand, global each, to exploe the whole each pace and avoid to get tuck in the local optima. On the othe hand, appoach the feaible egion

5 a oon a poible. The location update opeato i intoduced to elocate the infeaible individual. In addition to guide infeaible individual moving along the diection in eduction of the containt violation, two choice ae offeed to each of them. Extacted diection ae ued to acceleate the infeaible individual acce the feaible egion, while andom velocity i ued to maintain the diveity. When thee i no feaible individual found, a popotion of the bet infeaible individual ae choen to guide the et of the individual which acceleate in finding the feaible egion. Othewie, diection infomation i extacted fom feaible individual to guide the infeaible individual. Once ome individual each the feaible egion, moving in the diection to thee feaible individual will acceleate the infeaible individual to appoach the neaet feaible egion. But, thi may alo attact all individual move to the few ealy dicoveed feaible egion and tuck in thee egion foeve. Then, the algoithm will loe the ability of finding the othe diconnected feaible egion and eventually lead to the pematue convegence. Theefoe, ome of the infeaible individual update thei location andomly. Suppeion and Recepto editing ae ued to delete the cowed and imila individual and make pace fo the newly geneated individual to maintain the diveity. Tounament election ued in ecepto editing i capable of etaining the bet individual and at the ame time balancing the ize of feaible and infeaible goup. IV. EXPERIMENTAL RESULTS AND DISCUSSIONS To examine the pefomance quantitatively, we applied the popoed algoithm to the contained benchmak function available fom [21]. Thee benchmak function conit of linea, nonlinea, quadatic, and cubic function. Some of thee function have high dimenional deciion pace and vey low feaibility atio. Manycontaint involved ae equality containt. Fo mot of thee teting function, it i not eay even locating a feaible egion. Each benchmak poblem i un fo 30 independent tial. Bet (b), wot (w), mean (Mn), tandad deviation(d), and ucce ate (FR) to conveging to a feaible olution ae given to compae with the othe algoithm, which ae abbeviated a b, w, Mn, Md, d, and FR in the table, epectively. All the tatitical eult wee taken only with epect to the un in which a feaible olution wa eached at the end. All the equality containt ae tanfomed into inequality containt by uing a toleance facto, h ( x) 0 ( a uggeted). To deal with equality containt, dynamic mechanim oiginally popoed in [22] i adopted hee. 35,000 objective function evaluation ae applied a a topping citeion fo each tet poblem. The tatitical eult ae ummaized in Table I. Fom Table I, we can eaily find that the popoed algoithm how vey pomiing pefomance in handling thee contained poblem. The algoithm achieve the equied accuacy level fo G01, G03, G06, G08, G11, G12, and G13. Moeove, it attain the bet eult fo G13, even bette than any known bet eult. It i wothy to note that all 30 un fo 13 tet poblem poduce 100% ucce ate in conveging to a feaible olution. In addition, the tochatic anking algoithm (SR) [23], GA-AIS c [24], AIS [12], AIS cont [13], and T-cell algoithm [17] a the efeence ae choen to compae with the popoed algoithm. To make the compaion le complicated, only the bet and mean eult ae conideed hee to judge the pefomance of the algoithm. GA-AIS c [24], AIS [12], AIS cont [13], and T-cell algoithm [17] ae the typical algoithm popoed fo contained optimization baed on AIS. The popoed algoithm ha a bette o equal pefomance when compae with GA-AIS c [24] in almot all the tet poblem except G06 and G10. Even fo G06 and G10, only mean value of GA-AIS c ae lightly bette than that of the popoed algoithm. Bet value of the popoed algoithm outpefom GA-AIS c in all the tet poblem. To compae with the AIS [12] and AIS cont [13], the eult ae imila. AIS [12] i equal to o outpefomed by the popoed algoithm except fo G04 and G05. AIS cont i totally outpefomed by the popoed algoithm. T-cell algoithm obtain vey competitive eult. T-cell algoithm i bette than the popoed algoithm in G04, G06, and G09, and ha a bette bet value in G07 and G10. Howeve, the popoed algoithm act bette o equal to T-cell algoithm in the et of 8 tet poblem (i.e., G01, G02, G03, G05, G08, G11, G12, and G13). Accoding to the compaion, the popoed algoithm i bette than o equal to thee tate-of-the-at AIS deign in mot of the 13 tet poblem. Stochatic anking algoithm (SR) [23], a well-egaded evolutionay deign fo contained optimization, i alo choen fo compaion with the popoed algoithm. Fom the Table I, SR outpefom the popoed algoithm in five tet poblem (i.e., G02, G04, G05, G07, and G09). Nonethele, the popoed algoithm outpefom SR alo in five tet poblem (i.e., G03, G06, G10, G11, and G13). Although thei pefomance favo diffeent tet poblem, the diffeence i quite mall. On the whole, the popoed algoithm i a competitive a the othe AIS-baed algoithm and the tate of the at. It need to be mentioned that, fo ome poblem, fitne evaluation of the infeaible egion i not applicable, then the algoithm a SR which baed on the evaluation of the infeaible olution will become uele. Though the eult compaion, the popoed algoithm i found faily competitive with epect to ome choen tate-of-the-at deign in the contained optimization field.

6 V. CONCLUSIONS Inpied by the fundamental of biological immune ytem, a novel atificial immune ytem i popoed to olve contained optimization in thi pape. We exploit the immune epone fom the pepective of infomation tanfe and encouage the inteaction between feaible and infeaible individual. The diection infomation extacted i ued to guide the infeaible individual effectively and efficiently towad the neaby feaible egion. Thee infeaible individual nea the bounday of feaible and infeaible egion ae eued to each the bounday thooughly. Local each opeato of feaible individual i boowed fom the taditional clonal election algoithm which had been validated to be effective in handling the uncontained optimization. The global each of infeaible individual imply employ the Gauian andom vaiable to intoduce the ditubance. The expeimental eult how the pefomance of the popoed algoithm in olving 13 fequently ued benchmak function. Though compaing with the elected epeentative algoithm fom vaiou banche of evolutionay computation paadigm, the popoed algoithm i confimed to be competitive. ACKNOWLEDGEMENTS Thi eeach i funded by the Poject No. CDJXS , uppoted by the Fundamental Reeach Fund fo the Cental Univeitie. REFERENCES [1] L. N. de Cato and J. Timmi, Atificial Immune Sytem: A New Computational Intelligence Appoach, Spinge- Velag: London, UK, [2] D. Dagupta, Atificial Immune Sytem and Thei Application, Spinge-Velag: Heidelbeg, Gemany, [3] AISWeb The online home of atificial immune ytem [4] N. Cuz-Cote, Handling containt in global optimization uing atificial immune ytem: a uvey, Studie in Computational Intelligence, vol. 198, pp , [5] P. Hajela and J.S. Yoo, Immune netwok modeling in deign optimization, In D. Cone, M. Doigo, and F. Glove, edito, New Idea in Optimization, McGaw-Hill: New Yok, NY, pp , [6] C.A. Coello Coello and N. Cuz-Cote, Hybidizing a genetic algoithm with an atificial immune ytem fo global optimization, Engineeing Optimization, vol. 36, pp , [7] H.S. Benadino, H.J. Baboa, and A.C.C. Lemonge, A hybid genetic algoithm fo contained optimization poblem in mechanical engineeing, Poceeding of IEEE Conge on Evolutionay Computation, Singapoe, pp , [8] H.S. Benadino, H.J. Baboa, A.C.C. Lemonge, and L.g. Foneca, A new hybid AIS-GA fo contained optimization poblem in mechanical engineeing, Poceeding of IEEE Conge on Evolutionay Computation, Hong Kong, China, pp , [9] W.P. Ma, L.C. Jiao, M.G. Gong, and R.H. Shang, Immune clonal election evolutionay tategy fo contained optimization, Poceeding of Pacific Rim Intenational Confeence on Atificial Intelligence, Guilin, China, pp , [10] A.R. Yıldız, A novel hybid immune algoithm fo global optimization in deign and manufactuing, Robotic and Compute-Integated Manufactuing, vol. 25, pp , [11] A.J. Ouyang, A hybid immune PSO fo contained optimization poblem, Poceeding of IEEE Intenational Confeence on Bio-Inpied Computing: Theoie and Application, Changha, China, pp , [12] N. Cuz-Cote, D. Tejo-Peez, and C.A. Coello Coello, Handling containt in global optimization uing an atificial immune ytem, In C. Jacob, M.L. Pilat, P. J. Bentley, and J.I. Timmi, Edito, ICARIS LNCS, vol. 3627, pp , [13] V.S. Aagon, S.C. Equivel, and C.A. Coello Coello, Atificial immune ytem fo olving contained optimization poblem, Revita Ibeoameicana de Inteligencia Atificial, vol. 35, pp , [14] J.Y. Wu, Atificial immune ytem fo olving contained global optimization poblem, Poceeding of IEEE Sympoium on Atificial Life, Honolulu, HI, pp , [15] J.Y. Wu, Solving contained global optimization via atificial immune ytem, Intenational Jounal on Atificial Intelligence Tool, vol. 20, pp. 1-27, [16] V.S. Aagon, S.C. Equivel, and C.A. Coello Coello, A novel model of atificial immune ytem fo olving contained optimization poblem with dynamic toleance facto, Poceeding of Mexico Intenational Confeence on Atificial Intelligence, Aguacaliente, Mexico, pp.19-29, [17] V.S. Aagon, S.C. Equivel, and C.A. Coello Coello, A modified veion of a T-cell Algoithm fo contained optimization poblem, Intenational Jounal fo Numeical Method in Engineeing, vol. 84, pp , [18] K.M. Woldemaiam and G.G. Yen, Contained optimization uing atificial immune ytem, Poceeding of IEEE Conge on Evolutionay Computation, Bacelona, Spain, pp. 1-8, [19] L.N. de Cato and F. J. Von Zuben, Leaning and optimization uing the clonal election pinciple, IEEE Tanaction on Evolutionay Computation, vol. 6, pp , [20] K.M. Woldemaiam and G.G Yen, Vaccine-enhanced atificial immune ytem fo multimodal function optimization, IEEE Tanaction on Sytem, Man, and Cybenetic, Pat B, vol. 40, pp , 2010.

7 [21] J.J. Liang, T.P. Runaon, E. Mezua-Monte, M. Clec, P.N. Suganthan, C. A. Coello Coello, and K. Deb, Poblem definition and evaluation citeia fo the CEC 2006 pecial eion on contained eal-paamete optimization, Poceeding of IEEE Conge on Evolutionay Computation, Vancouve, Canada, [22] S.B. Hamida and M. Schoenaue, ASCHEA: new eult uing adaptive egegational containt handling, Poceeding of IEEE Conge on Evolutionay Computation, Honololu, HI, pp , [23] T.P. Runaon and X. Yao, Stochatic anking fo contained evolutionay optimization, IEEE Tanaction on Evolutionay Computation, vol. 4, pp , [24] H.S. Benadino, H.J.C. Baboa, A.C.C. Lemonge, and L.G. Foneca, On GA-AIS hybid fo contained optimization poblem in engineeing, containt-handling in evolutionay optimization, Studie in Computational Intelligence, vol. 198, pp , TABLE I. COMPARISON OF EXPERIMENT RESULTS ON BENCHMARK FUNCTIONS IN [38] Function SR [23] GA- AIS c [24] AIS [12] AIS cont [13] T-cell [17] Popoed AIS G b Mn w d 0.0E E E E E-08 G b Mn w d 2.0E E E E E E-03 G b Mn w d 1.9E E E E-05 G b Mn w d 2.0E E E E E-03 G b Mn w d 3.5E E E E E E+02 FR 100% 12% 90% 75% 80% 100% G b Mn w d 1.6E E E E E E-02 FR 66% 100% 100% 100% 100% 100%

8 G b Mn w d 6.6E E E E E E-01 G b Mn w d 2.6E E E E E E-14 G b Mn w d 3.4E E E E E E-01 G b Mn w d 3.4E E E E E E+02 G b Mn w d 8.0E E E E E E-08 G12-1 b Mn w d 0.0E E E E E E-10 G b Mn w d 3.1E E E E E E-05 FR 100% 24% 100% 100% 100% 100%