NON-DOMINATED RANKED GENETIC ALGORITHM FOR SOLVING CONSTRAINED MULTI-OBJECTIVE OPTIMIZATION PROBLEMS

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1 NON-DOMINATED RANKED GENETIC ALGORITHM FOR SOLVING CONSTRAINED MULTI-OBJECTIVE OPTIMIZATION PROBLEMS OMAR AL JADAAN, LAKSHMI RAJAMANI, 3 C. R. RAO Dept. CSE, EC. Osmana Unversty, Hyderabad , Inda. Prof. Dept. CSE, EC. Osmana Unversty, Hyderabad , Inda. 3 Prof. Dept. CSE, EC. Osmana Unversty, Hyderabad , Inda. E-mal: o_jadaan@yahoo.com, lakshmraja@yahoo.com, crrcs@uohyd.ernet.n ABSTRACT Evolutonary algorthms are becomng ncreasngly valuable n solvng large-scale, realstc engneerng multobjectve optmzaton problems, whch typcally requre consderaton of conflctng and competng desgn ssues. A crtcsm of Evolutonary Algorthms mght be the lack of effcent and robust generc methods to handle constrants. The most wdespread approach for constraned search problems s to use penalty methods, because of ther smplcty and ease of mplementaton. Penalty functon s generc and applcable to any type of constrant (lnear or nonlnear). Nonetheless, the most dffcult aspect of the penalty functon approach s to fnd approprate penalty parameters. In ths paper, a method combnng the new Non-domnated Ranked Genetc Algorthm (NRGA), wth a parameterless penalty approach are exploted to devse the search to fnd Pareto optmal set of solutons, allevate the above dffcultes. The parameterless penalty approach that does not requre any penalty parameter where penalty parameters assgnment among feasble and nfeasble solutons are made wth a vew to provde a search drecton towards the feasble regon. The new Parameterless Penalty and the Non-domnated Ranked Genetc Algorthm (PP-NRGA) contnuously fnd better Pareto optmal set of solutons. Ths new algorthm have been evaluated by solvng fve test problems, reported n the mult-objectve evolutonary algorthm (MOEA) lterature. Performance comparsons based on quanttatve metrcs for accuracy, coverage, and spread are presented. Keywords: Mult-Objectve Optmzaton, Pareto Optmal Solutons, Constraned Optmzaton, Penalty Functons, Rankng.. INTRODUCTION Trade-off nformaton n the form of a Pareto optmal set of solutons s mportant n consderng competng desgn objectves when makng decsons assocated wth most engneerng problems. The Presence of multple objectves n engneerng problems, n prncple, gves rse to a Pareto set of optmal solutons, nstead of a sngle optmal soluton. In the absence of any further nformaton, one of these Pareto optmal solutons cannot be sad to be better than the other. Ths demands a user to fnd as many Pareto optmal solutons as possble. Classcal optmzaton methods (ncludng the mult-crteron decson-makng methods) suggest convertng the mult-objectve optmzaton problem to a sngle-objectve optmzaton problem by emphaszng one partcular Pareto optmal soluton at a tme. When such a method s to be used for fndng multple solutons, t has to be appled many tmes, hopefully fndng a dfferent soluton at each smulaton run. A number of multobjectve evolutonary algorthms (MOEAs) have been suggested, manly because of ther ablty to fnd multple Pareto optmal solutons n one sngle smulaton run. As evolutonary algorthms offer a relatvely more flexble way to analyze and solve realstc engneerng desgn problems, ther use n mult-crteron decson makng s becomng ncreasngly mportant. A number of multobjectve evolutonary algorthms (MOEAs) have been reported snce the early eghtes. Detaled summares of the state-of-the-art n MOEA were dscussed [][]. 640

2 In the followng, the general representaton of a standard mult-objectve problem and defntons of Pareto optmalty are presented, followed by a bref overvew of exstng MOEAs. The detals of the proposed parameterless penalty domnated rankng GA (PP-NRGA) approach PP-NRGA are then descrbed, followed by ts applcaton to a fve test problems. An extensve performance comparson of PP-NRGA and NSGA-II n solvng the test problems s presented. Fnally, concludng remarks are made wth a bref dscusson of PP-NRGA s strengths and weaknesses. II. A STANDARD MULTI-OBJECTIVE OPTIMIZATION PROBLEM A general mult-objectve optmzaton problem consstng of k competng objectves and m constrants defned as functons of decson varable set x can be represented as follows: { } Mnmze f ( x ) f ( x ), f ( x ),..., f ( x ) = k () Subject to: g ( x ) 0, = {,,..., m} () x X (3) Where x { x : j,,... n j } = = represents the decson vector, x j s the th j decson varable, X represent the decson space, g ( x ), {,..., m} are constrants, whch nclude all equalty constrants after transformng them to nequalty constrants usng (4) ( ) ε 0 h x (4) Where ε s a small tolerance. Snce the algorthm that wll be dscussed does not use gradent nformaton, t does not mater f equalty constrant (4) s non-dfferentable. F(x) s the mult-objectve vector, and f ( x l ) s the l th objectve functon. III. PARETO OPTIMAL Pareto optmal, whch s also referred as nondomnance, of a set of solutons s formally defned as follows Cohon [3]: A feasble soluton to a mult-objectve problem s Pareto optmal f there exst no other feasble soluton that wll yeld an mprovement n one objectve wthout causng degradaton n at least one other objectve. Van Veldhuzen and Lamont [] and Ztzler [9] provde a more rgorous defntons of ths and related mult-objectve termnology. Based on the defntons by Van Veldhuzen and Lamont and notatons used n Equatons (-3), the followng are defned: Pareto Domnance: A mult-objectve vector u = ( u, u,..., u k ) s sad to domnate (,,..., ) υ = υ υ υ (denoted byu f υ ) f and only k f u s partally more thanυ,.e. = {,,..., k}, u υ = {,,... k}, u < υ Pareto Optmalty: A soluton x X s sad to be Pareto optmal wth respect to X f and only f there exsts no x X for whch υ = F ( x ) domnates u = F x. ( ) Pareto Optmal Set: For a gven mult-objectve problem F(x), the Pareto optmal set P* s a set consstng of Pareto optmal solutons. P* s a subset of all the possble solutons n X. Mathematcally, P* s defned as follows: p* = x X x* X : F x * f F x (5) { ( ) ( )} Pareto Front: The Pareto front, PF* s the set that contans the evaluated objectve vectors of P*. Mathematcally PF* s defned as: PF * = u = F x x P * (6) { ( ) } IV. MULTI-OBJECTIVE EVOLUTIONARY ALGORITHMS (MOEAs) Over the past two decades, a number of dfferent EAs were suggested to solve mult-objectve optmzaton problems. Of them, VEGA [5] MOGA-III [4], SPEA [3], NSGA-II [8], Srnvas and Deb s NSGA [7], and Horn NPGA [7], Fonseca and Flemng s [5], for detaled nformaton about other MOEA algorthms readers are encouraged to refer to [],[7]. The [5],[7] and [7] algorthms demonstrated the necessary addtonal operators for convertng a smple EA to a MOEA. Two common features on all three operators were the followng: Assgnng ftness to populaton members based on non-domnated sortng. Preservng dversty among solutons of the same non-domnated front. Although they have been shown to fnd multple non-domnated solutons on many test problems, and a number of engneerng desgn problems, researchers realzed the need of ntroducng more useful operators (whch have been found useful n sngle-objectve EA s) to solve mult-objectve 64

3 optmzaton problems better. Partcularly, the nterest has been to ntroduce eltsm to enhance the convergence propertes of a MOEA. Reference [9] showed that eltsm helps n achevng better convergence n MOEAs. Among the exstng eltst MOEAs, Ztzler and Thele s SPEA [30], [3], Knowles and Corne s PAES [5], MOMGAIII [6], PAES-II [4], NSGA-II [8] are well studed. Readers are encouraged to refer to the orgnal studes. Snce evolutonary algorthms (EAs) work wth a populaton of solutons, a smple EA can be extended to mantan a dverse set of solutons. Wth an emphass for movng toward the true Pareto optmal regon, an EA can be used to fnd multple Pareto optmal solutons n one sngle smulaton run. V. PARAMETERLESS PENALTY- NON-DOMINATED RANKING GENETIC ALGORITHM (PP-NRGA) Over the years, the man crtcsms of the multobjectve evolutonary algorthms (EAs) mght be the lack of effcent and robust generc methods to handle constrants. The most wdespread approach for constraned search problems s to use penalty methods, because of ther smplcty and ease of mplementaton. The penalty functon approach s generc and applcable to any type of constrant (lnear or nonlnear). Nonetheless, the most dffcult aspect of the penalty functon approach s to fnd an approprate penalty parameters needed to gude the search towards the constraned optmum. In ths paper, a mult-objectve GA approach (Nondomnated Rankng Genetc Algorthm NRGA) and an adaptve penalty functon usng ranks as penalty parameters are exploted to devse the new approach to fnd feasble Pareto front solutons. The new approach s called Parameterless Penalty Nondomnated Rankng Genetc Algorthm PPNRGA. The adaptve penalty parameters assgnments among feasble and nfeasble solutons are made wth a vew to provde a search drecton towards the feasble regon. Two tres Rank-based Roulette Wheel selecton operator (RRWS) s used along wth the new adaptve penalty allow NRGA to contnuously fnd better feasble Pareto optmal solutons, gradually leadng the search near the true Pareto optmum solutons. NRGA wth ths constrant handlng approach have been tested on fve benchmarks problems commonly used n the lterature. In all cases, the proposed approach has been able to repeatedly fnd feasble Pareto optmal solutons closer to the true Pareto optmal solutons than NSGA-II other MOEA. VI. PROPOSED CONSTRAINT HANDLING METHOD A. Penalty Method The ntroducton of the penalty term enables us to transform constraned optmzaton problem equatons (-3) nto an unconstraned one, such as the one gven by (7): Mnmze { Ψ ( y ) = / υ ( y ), ( y ),..., / υ / υ ( y k )} (7) Where / υ = f ( x ) + rg φ g ( x j ), ( ) j =,..., m; =,..., k and φ 0 s a real-valued functon whch mposes a penalty controlled by a sequence of penalty coeffcents. Ths transformaton (.e. (7)) has been used wdely n evolutonary constraned optmzaton [9], [6]. The penalty functon method may work qute well for some problems; however, decdng an optmal (or near optmal) value for r g turns out to be a dffcult optmzaton problem tself! If r g s too small, an nfeasble soluton may not be penalzed enough. Hence, an nfeasble soluton may be evolved by an evolutonary algorthm. If r g s too large, a feasble soluton s very lkely to be found, but could be of very poor qualty. A large r g dscourages the exploraton of nfeasble regons, even n the early stages of evoluton. Ths s partcularly neffcent for problems where feasble regons n the whole search space are dsjonted. In ths case, t may be dffcult for an evolutonary algorthm to move from one feasble regon to another unless they are very close to each other. Reasonable exploraton of nfeasble regons may act as brdges connectng two or more dfferent feasble regons. The crtcal ssue here s how much exploraton of nfeasble regons (.e., how large r g s) should be consdered as reasonable. The answer to ths queston s problem dependent. Even for the same problem, dfferent stages of evolutonary search may requre dfferent r g values. Accordng to (7), dfferent r g values defne dfferent ftness functons. A ft ndvdual under one ftness functon may not be ft under a dfferent ftness functon. Fndng a near optmal r g adaptvely s equvalent to rankng ndvduals adaptvely n a populaton. Hence, the ssue becomes how to rank ndvduals accordng to ther objectve and penalty values. A novel method for rankng ndvduals wthout specfyng a r g value s proposed. Expermental studes test the 64

4 effectveness and effcency of ths method whch can be regarded as an adaptve penalty approach. It has been wdely recognzed that nether underpenalzaton nor over-penalzaton s a good constrant handlng technque, and there should be a balance between preservng feasble ndvduals and rejectng nfeasble ones [6]. In other words, rankng should be domnated by a combnaton of objectve and penalty functons, and so the penalty coeffcent r g should be wthn the bounds rg l < rg < rg u. It s worth notng that the two bounds are not fxed. They are problem dependent, and may change from generaton to generaton as they are also determned by the current populaton. VII. IMPLEMENTATION OF THE EVOLUTIONARY ALGORITHM FOR CONSTRAINED OPTIMIZATION Ths secton focuses on the detaled mplementaton of the evolutonary algorthm PP-NRGA for constraned optmzaton; also, the outcomes of testng the new approach on fve benchmark problems are hghlghted. The results are also compared wth NSGA-II other evolutonary algorthm. A. Rankng To overcome the dffculty of determnng the optmal r g a dfferent approach s suggested n ths secton to balance between the objectve and penalty functons. The followng ftness functon s ntroduced: m / υ ( x ) = f ( x ) + rank + rankg * φ( x f ) (8) j = Where =,,..., k, rank s the rank of the f objectve functon values, whch takes values n the range of [ populaton sze]. rank g Is the rank of the sum of the constrants volaton for each soluton, whch takes values from [(populaton sze+) ( * populaton sze)]. What (8) above mounts to s that mnmum ftness value and less constrants volaton nevtably leads to best ftness value. By usng rank-based roulette wheel selecton [], [4], self-adaptng s acheved wthout any extra computatonal cost. More mportantly, the motvaton of rankng comes from the need for balancng objectve and penalty functons drectly and mplctly n optmzaton. Equaton (8) provdes a convenent way of balancng n a ranked set. B. Sortng Algorthm In ths study the fast non-domnated sortng approach from [8] s used because of the comparson wth NSGA-II, and because t requres O MN computatons. Whereas any fast only ( ) sortng algorthm can be used. C. Dversty Mechansm Along wth convergence to the Pareto optmal set, t s desred that an EA mantans a good spread of solutons n the obtaned set of solutons. In the proposed PP-NRGA, crowdng dstance s used, whch does not requre any user defned parameter for mantanng dversty among populaton members. Readers are encouraged to refer to [8] for more nformaton about the crowdng dstance. D. Ranked Based Roulette Wheel Selecton The authors of [], and [3] use modfed roulette wheel selecton algorthm where each ndvdual s assgned a ftness value equal to ts rank n the populaton; the hghest rank has the hghest probablty to be selected (n case of maxmzaton). The probablty s calculated as llustrated n the followng equaton: * Rank P = (9) N *( N + ) Where N s the number of ndvduals n the front when dealng wth ndvduals, and number of fronts when dealng wth fronts. In ths study the ndvduals n a front are ranked based on ther crowdng dstance, and the fronts ranked based on the non-domnance rank. E. Man Loop Intally, a random parent populaton P s created, objectves and constrants values are evaluated, ranks of each objectve are calculated, and the ranks of the sum of the constrants volaton are computed. The constraned problem s converted to unconstraned one usng the equaton (8) for each objectve functon, then the populaton s sorted based on the non-domnaton. Each soluton s assgned a ftness (or rank) equal to ts nondomnaton level ( s the best level, s the nextbest level, and so on). Thus, mnmzaton of ftness s assumed. The crowdng dstance s computed for each soluton n each front. At frst, the usual Ranked based Roulette Wheel Selecton (RRWS), recombnaton, and mutaton operators are used to create offsprng populaton Q of sze N. Snce eltsm s ntroduced by comparng current populaton wth prevously found best nondomnated solutons, the procedure s dfferent after 643

5 ( ) the ntal generaton. We frst descrbe the l th generaton of the proposed algorthm as shown n algorthm. The step-by-step procedure shows that PP-NRGA algorthm s smple and straght forward. Frst, a combned populaton P Q s formed of sze N. Then, objectves and constrants values are evaluated, ranks of each objectve are calculated, and the ranks of the sum of the constrants volaton are computed. After that the constraned problem s converted to unconstraned one usng the equaton (8) for each objectve functon, the combned populaton s sorted accordng to non-domnaton, and the crowdng dstance s computed. Snce all prevous and current populaton members are ncluded n the combned populaton eltsm s ensured. Now, reducton (eltsm) procedure wll select N solutons out of N, t wll select the solutons wth mnmum non-domnaton rank frst, the remanng solutons whch belongs to same nondomnaton rank wll be selected based on the crowdng dstance values. The new populaton of sze N s now used for selecton (RRWS), crossover, and mutaton to create a new populaton Q of sze N. The two ters ranked based roulette wheel selecton [], and [3] s used, The frst ter wll select the front based on the non-domnaton rank, here the solutons belongng to the best non-domnated set F (best front) have the largest chance to be selected n the combned populaton. Thus, solutons from the set F are chosen wth less chance than solutons from the set F and so on. Then second ter wll select a soluton nsde the front based on the crowdng dstance value, the soluton wth larger crowdng dstance value wll have larger chance to be selected. The overall complexty of the algorthm s O MN, whch s governed by the nondomnated sortng part of the algorthm. If performed carefully, the complete populaton of sze N need not be sorted accordng to nondomnaton. As soon as the sortng procedure has found enough number of fronts to have members n Pt+, there s no reason to contnue wth the sortng procedure. The dversty among non-domnated solutons s ntroduced by usng the crowdng dstance procedure, whch s used by the ranked based roulette wheel selecton durng the populaton selecton phase. Snce solutons compete wth ther crowdng-dstance (a measure of densty of solutons n the neghborhood), no extra nchng parameter (such as σ share needed n the NSGA [7]) s requred. Although the crowdng dstance s calculated n the objectve functon space, t can also be mplemented n the parameter space, f so desred [9]. However, n all smulatons performed n ths study, the objectve functon space nchng s used. F. Survval Selecton (eltsm): After evaluatng the offsprng s ftness (nondomnated rank, crowdng dstance), parents and offsprng fght for survval as Pareto domnance s appled to the combned populaton of parents and offsprng. Then the least domnated N soluton vectors survve to make the populaton of the next generaton. VIII. TESTING AND EVALUATION OF PP-NRGA In ths secton, frst the test problems used to compare the performance of PP-NRGA wth NSGA-II are descrbed. For both NRGA and NSGA-II, the same parameter values have been chosen and have not made any effort n fndng the best parameter settng. We leave ths task for a future study. 644 A. Test Problems Test problems are chosen from the lterature. In 00, Deb and hs students [] have developed test problems for constraned mult-objectve optmzaton and suggested eght test problems. Fve of those eght problems are chosen here and

6 call them CTP, CTP, CTP3, CTP4, and CTP5. All problems have two objectve functons and constrants. These test problems are descrbed n the appendx. The smulated bnary crossover (SBX) operator and polynomal mutaton [0] are used. The crossover probablty of P c = 0.9 and a mutaton probablty of P m = 0. are used. Dstrbuton ndexes [0] for crossover and mutaton operators as η c = 5 and η m = 0, respectvely are specfed. The table I shows the remanng parameters values used n ths study. B. Performance Measures Unlke n sngle-objectve optmzaton, there are two goals n a mult-objectve optmzaton: ) Convergence to the Pareto optmal set and ) Mantenance of dversty n solutons of the Pareto optmal set. These two tasks cannot be measured adequately wth one performance metrc. Many performance metrcs have been suggested [], [7], [3], [8]. Here, two runnng performance metrcs to understand the behavor of the algorthm from [] are used n evaluatng each of the above two goals n a soluton set obtaned by a mult-objectve optmzaton algorthm. The frst metrc measures the extent of convergence to a known set of Pareto optmal solutons. Snce mult-objectve algorthms would be tested on problems havng a known set of Pareto optmal solutons, the calculaton of ths metrc s possble. Frst, we fnd a set of H = 600 unformly spaced solutons from the true Pareto optmal front n the objectve space. For each soluton obtaned wth an algorthm, the mnmum normalzed Eucldean dstance of t from H chosen solutons on the Pareto optmal front s computed. The average of these dstances s used as the frst t metrc CP (the convergence metrc). In order to keep the convergence metrc wthn [0, ] once the metrc values are calculated for all generatons, t normalze C( P ) by t maxmum value 0 (usuallyc( P )). When all obtaned solutons le exactly on chosen solutons, ths metrc takes a value of zero. Even when all solutons converge to the Pareto optmal front, the above convergence metrc does not have a value of zero. The metrc wll yeld zero only when each obtaned soluton les exactly on each of the chosen solutons. Although ths metrc alone can provde some nformaton about the spread n obtaned solutons, dfferent metrc to measure the spread n solutons obtaned by an algorthm s defned (dversty metrc). The second metrc measures the extent of spread acheved among the obtaned solutons. Here, we are nterested n gettng a set of solutons that spans the entre Pareto optmal regon. For each objectve, we calculate a dversty value as follows: the obtaned non-domnated ponts at each generaton are projected on sutable hyper-plane. The plane s dvded nto a number of small grds (or M dmensonal boxes). Dependng on whether each grd contans an obtaned nondomnated pont or not, a dversty metrc s defned. If all grds are represented wth at least one pont, the best possble (wth respect to the chosen number of grds) dversty measure s acheved. If some grds are not represented by a non-domnated pont, the dversty s poor. The parameters requred from the user are the drecton cosne of the reference plane, the number of grds ( G ) n each (M ) dmenson, and the target (or reference) set of ponts. Because of lake of space reader recommended to refer to the orgnal study [] For more detals about the runnng metrcs used. In the experments the number of grds s equal to the populaton sze and 0 f = plane to project the ponts. 645

7 Algorthm PP-NRGA : Intalze Populaton P : Generate random populaton sze N 3: Evaluate objectves values and constrants 4: Calculate the rank of objectves values, f, ( =,,..., ) for each soluton, R f [ N ] R k 5: Calculate the rank of the sum of the constrants volaton R g for each soluton ( ) ( ) Rg N + N 6: Convert the constraned problem to unconstraned one usng the equaton (8) for each objectve functon, for each soluton n P 7: Assgn Rank (level) Based on Pareto domnance sort 8: Calculate the crowdng dstance between members on each front 9: Generate offsprng Populaton Q from P 0: {Ranked based Roulette Wheel Selecton : Recombnaton and Mutaton : Evaluate objectves values and constrants} 3: for g = to G do 4: for each member of the combned populaton ( P Q ) 5: Calculate the rank of objectves values, R,(,,..., k) populaton P Q, R [ N] f = for each soluton n the combned f 6: Calculate the rank of the sum of the constrants volaton R g for each soluton n the combned populaton P Q, R ( N + ) ( 4N ) g 7: Convert the constraned problem to unconstraned one usng the equaton (8) for each objectve functon, for each soluton n P 8: Assgn Rank (level) based on Pareto sort 9: Calculate the crowdng dstance between members on each front 0: end for : (eltst) Select the members of the combned populaton based on least domnated N soluton to make the populaton P of the next generaton. Tes are resolved by takng the less crowdng dstance. : Calculate the rank of objectves values, R, (,,..., k ) f. 3: Calculate the rank of the sum of the constrants volaton R g for each soluton Rg [( N + ) ( N )]. f = for each soluton, R [ N] 4: Convert the constraned problem to unconstraned one usng the equaton (8) for each objectve functon, for each soluton n P. 5: Assgn Rank (level) Based on Pareto domnance sort. 6: Calculate the crowdng dstance between members on each front. 7: Q = Create next generaton from P {. 8: { Ranked based Roulette Wheel Selecton. 9: Recombnaton and Mutaton. 30: Evaluate objectve values and constrants }. 3: end for Fg.. CTP Fnal Generaton Fg.. CTP Fnal Generaton Fg. 3. CTP3 Fnal Generaton Fg. 4. CTP4 Fnal Generaton 646

8 Fg. 5. CTP5 Fnal Generaton Fg. 9. CTP4 Convergence Fg. 6. CTP Convergence Fg. 0. CTP5 Convergence Fg. 7. CTP Convergence Fg.. CTP Convergence Fg. 8. CTP3 Convergence Fg.. CTP Convergence 647

9 Fg. 3. CTP3 Convergence Fg. 4. CTP4 Convergence Fg. 5. CTP5 Convergence C. Dscusson of the Results Fgures -5 show the fnal generaton but these wll not gve a clear pcture about the behavor of the algorthms, for that the fgures 6-0 llustrate the convergence behavor of PP-NRGA and NSGA-II algorthms. Fgure 6 shows that the convergence metrc of PP-NRGA on problem CTP, quckly moves to zero faster than NSGA-II, thereby mplyng that startng from a random set of solutons PP-NRGA quckly approach the feasble Pareto optmal front faster than NSGA-II. A value of zero the convergence metrc mples that all nondomnated solutons match the chosen Pareto optmal solutons. After about 0 generatons, the PP-NRGA populaton comes very close to the Pareto optmal front, whereas NSGA-II took too much to get closer to the Pareto optmal front. The same s happened n the fgures 7, and 9, where n CTP3 and CTP5 NSGA-II was ahead. Fgures -5 show the dversty metrc graphs obtaned usng the two algorthms. Fgure explans that the dversty metrc ncreases exponentally tll 50 generatons n PP-NRGA and tll 60 n NSGA-II after that the dversty reman more or less the same. Although the obtaned solutons are very close to the chosen Pareto optmal front, the dversty metrc oscllates near a stable value. The same behavor of the dversty metrc n the remanng fgures, except n fgure 4 and fgure 5 where PP-NRGA performs better than NSGA-II. From above PP-NRGA out perform NSGA-II n most of the test problems. On most of the problems, PP-NRGA s able to fnd a better spread and faster convergence of solutons than NSGA-II algorthm. Fg. shows all non-domnated solutons obtaned after 00 generatons wth PP-NRGA and NSGA-II on CTP problem. The feasble Pareto optmal regon s also shown n the fgure. Ths fgure demonstrates the abltes of PP-NRGA n convergng to the feasble true front and n fndng dverse solutons n the front. In the convergence aspect PP-NRGA performed better than NSGAII n ths problem. But NSGA-II s shows better spread over the feasble Pareto front n the fnal GA populaton. Next, the non-domnated solutons on the problem CTP s shown n Fg.. Ths problem has a dscontnued pece wse Pareto optmal front. the performance of PP-NRGA s better than NSGA- II. Although PP-NRGA get closer to the true front than NSGA-II, also PP-NRGA have found a better spread and more solutons n the entre Pareto optmal regon than PP-NRGA. The problem CTP3 has vertexes Pareto optmal front. Fgure 3 shows that the overall performance of PPNRGA s same NSGA-II, but the convergence and ablty to fnd a dverse set of solutons are defntely better wth PPNRGA. Fnally, n fgure 5 and fgure 4 show that PP-NRGA fnds a better converged set of nondomnated solutons n CTP4, CTP5 compared to NSGA-II algorthm. IX. CONCLUSIONS AND FUTURE WORK Ths paper has proposed a new constrant multobjectve evolutonary algorthm called Parameterless Penalty Non-domnated Rankng Genetc Algorthm (PP-NRGA). The new eltst MOEA (PP-NRGA) have been tested on fve dfferent dffcult test problems borrowed from the lterature. The balance between the objectve and 648

10 penalty functon s acheved through rank used n ftness functon (8). The ntroducton of rank and usng that rank n the ftness functon enables the algorthm to bas toward the global Pareto front. The proposed method could get solutons closer to the global Pareto front on the fve test problems. The PP-NRGA does not ntroduce any specalzed varaton operators, and does not requre a pror knowledge about a problem snce t uses parameterless penalty coeffcent r g n a penalty functon. The procedure presented n ths paper for handlng constrants can be ntegrated to any evolutonary algorthm framework. The proposed PP-NRGA was able to mantan a better spread of solutons and converge better n the obtaned feasble non-domnated front compared to NSGA- II. However, n all problems, PP-NRGA was able to converge closer to the true Pareto optmal front. PP- NRGA mantans dversty among solutons by controllng dynamc and parameterless crowdng approach. However, the dversty preservng mechansm used n NSGA-II (whch used n PPNRGA) s found to be the best wth PP-NRGA. Wth the propertes of parameterless penalty functon, two ters ranked based roulette wheel selecton procedure, a fast non-domnated sortng procedure, and an eltst strategy, PP-NRGA can be used for solvng constrant real world problems, and should fnd ncreasng attenton and applcatons n the near future. REFERENCES [] Gary B. Lamont C. A. Coello and Davd A. Van Veldhuzen. Evolutonary Algorthms for Solvng Mult-Objectve Problems. Sprnger, second edton edton, 007. [] C. A. C. Coello. A comprehensve survey of evolutonary-based multobjectve optmzaton technques. Knowledge and Informaton System, (3):69 308, 999. [3] J.L. Cohon. Multobjectve programmng and plannng. Mathematcs n Scence and Engneerng, 40, 978. [4] Davd W. Corne, Nck R. Jerram, Joshua D. Knowles, and Martn J. Pesa-: Regon-based selecton n evolutonary multobjectve optmzaton. In Proceedngs of the Genetc and Evolutonary Computaton Conference (GECCO00, pages Morgan Kaufmann Publshers, 00. [5] Davd W. Corne, Joshua D. Knowles, and Martn J. Oates. The pareto envelopebased selecton algorthm for multobjectve optmzaton. In Proceedngs of the Parallel Problem Solvng from Nature VI Conference, pages , Pars, France, 000. Sprnger. [6] R. O. Day. Explct Buldng Block Multobjectve Evolutonary Computaton: Methods and Applcaton. PhD thess, Ar Force Insttute of Technology, AFIT/ENG, BLDG 64, 950 HOBSON WAY, WPAFB (Dayton) OH , USA, June 005. [7] K. Deb. Optmzaton for Engneerng Desgn: Algorthms and Examples. Prentce-Hall, New Delh, 995. [8] K. Deb. Mult-Objectve Optmzaton usng Evolutonary Algorthms. John Wley & Sons LTD, New York, NY, 00. [9] Kalyanmoy Deb. Mult-objectve genetc algorthms: Problem dffcultes and constructon of test problems. Evolutonary Computaton, 7:05 30, 999. [0] Kalyanmoy Deb and R. B. Agrawal. Smulated bnary crossover for contnuous search space. Complex syst., 9:5 48, Aprl 995. [] Kalyanmoy Deb and Sachn Jan. Runnng performance metrcs for evolutonary mult-objectve optmzaton. Techncal Report 00004, Kanpur Genetc Algorthm Labratory (KanGAL), 00. [] Kalyanmoy Deb, Amrt Pratap, and T. Meyarvan. Constraned test problems for mult-objectve evolutonary optmzaton. In Frst Internatonal Conference on Evolutonary Mult-Crteron Optmzaton EMO-00, pages Sprnger- Verlag, 00. [3] Marco Laumanns Eckart Zt. Ler and Lothar Thele. Spea: Improvng the strength pareto evolutonary algorthm for multobjectve optmzaton. In K. Gannakoglou, D. Tsahals, J. Peraux, P. Papalou, and T. Fogarty, edtors, Evolutonary Methods for Desgn, Optmzaton and Control wth Applcatons to Industral Problems (EUROGEN 00), pages 95 00, Athens, Greece, 00. [4] C. M. Fonseca and P. J. Flemng. Genetc algorthms for multobjectve optmzaton: Formulaton, dscusson and generalzaton. In S. Forrest, edtor, the Ffth Internatonal Conference on Genetc 649

11 Algorthms, pages 46 43, San Mateo, CA, 993. Morgan Kauffman. [5] C. M. Fonseca and P. J. Flemng. Genetc algorthms for multobjectve optmzaton: Formulaton, dscusson and generalzaton. In S. Forrest, edtor, the Ffth Internatonal Conference on Genetc Algorthms, pages Morgan Kaufmann Publshers, 993. [6] M. Gen and R. Cheng. Genetc Algorthms and Engneerng Desgn. Wley, New York, 997. [7] N. Nafplots J. Horn and D. E. Goldberg. A nched pareto genetc algorthm for multobjectve optmzaton. In Z. Mchalewcz, edtor, the Frst IEEE Conference on Evolutonary Computaton, pages 8 87, Pscataway, NJ, 994. IEEE Press. [8] Sameer Agarwal Kalyanmoy Deb, Amrt Pratap and T. Meyarvan. A fast and eltst multobjectve genetc algorthm: NSGA- II. IEEE Trans on Evolutnary Computaton, 6():8 97, Aprl 00. [9] S. Kazarls and V. Petrds. Varyng ftness functons n genetc algorthms: Studyng the rate of ncrease n the dynamc penalty terms. In Parallel Problem Solvng from Nature, pages 0, 998. [0] J. D. Knowles and D. W. Corne. Approxmatng the nondomnated front usng the pareto archved evoluton strategy. Evolutonary Computaton, 8():49 7, 000. [] C. R. Rao Omar Al Jadaan, Lakshm Rajaman. Improved selecton operator for ga. Journal of Theoretcal and Appled Informaton Technology, 4(4):6977, 008. [] Lakshm Rajaman Omar Al Jadaan, C.R. Rao. Ranked based roulette wheel selecton method. In Internatonal Symposum on Recent Advances n Mathematcs and ts Applcatons: (ISRAMA 005), Calcutta Mathematcal Socety at AE-374, Sector-, Salt Lake Cty Kolkata (Calcutta) , Inda, 005. [3] Lakshm Rajaman Omar Al Jadaan, C.R. Rao. Parametrc study to enhance genetc algorthm performance,usng ranked based roulette whell selecton method. In Internatonal Conference on Multdscplnary Informaton Scences& Technologes (InScT006), volume, pages 74 78, Merda,Span, October 006. [4] Lakshm Rajaman Omar Al Jadaan, C. R. Rao. Parametrc study to enhance genetc algorthm performance,usng ranked based roulette whell selecton method. In InScT006, volume, pages 74 78, Merda, Span, 006. [5] J. D. Schaffer. Multple Objectve Optmzaton wth Vector Evaluated Genetc Algorthms. PhD thess, Vanderblt Unversty, Nashvlle, Tennessee, 984. [6] W. Sedleck and J. Sklansky. Constraned genetc optmzaton va dynamc rewardpenalty balancng and ts use n pattern recognton. In Int. Conf. Genetc Algorthms, pages 4 49, 989. [7] N. Srnvas and K. Deb. Multobjectve functon optmzaton usng nondomnated sortng genetc algorthms. Evol. Comput., (3): 48, Fall 995. [8] E. Ztzler. Evolutonary algorthms for multobjectve optmzaton: Methods and applcatons. PhD thess, ETH 3398, Swss Federal Insttute of Technology (ETH), Zurch, Swtzerland, 999. [9] Eckart Ztzler, Kalyanmoy Deb, and Lothar Thele. Comparson of multobjectve evolutonary algorthms: Emprcal results. Evolutonary Computaton, 8:73 95, 000. [30] Eckart Ztzler and Lothar Thele. Multobjectve optmzaton usng evolutonary algorthms - a comparatve case study. In M. Schoenauer A. E. Eben, T. B ack and H.-P. Schwefel, edtors, Parallel Problem Solvng From Nature, pages Sprnger-Verlag, 998. [3] Eckart Ztzler, Lothar Thele, Marco Laumanns, Carlos M. Fonseca, and Vvane Grunert Da Fonseca. Performance assessment of multobjectve optmzers: an analyss and revew. IEEE Transactons on Evolutonary Computaton, 7:7 3,

12 APPENDIX All benchmark functons are descrbed n [8]. X. CTP Mn f ( x) = x f ( x ) g ( x ) Mn f ( x) = g( x) e Where g ( x) =.0+ x Subject to : ( ) g ( x) = f ( x) a e bf x 0 ( ) g ( x) = f ( x) a e bf x 0 Where a = 0.858, b = 0.54, a = 0.78, b = XI. CTP Mn f ( x) = x f ( x ) Mn ( ) ( ) f x = g x g( x) Where g ( x) =.0+ x Subject to: g( x) = cos( θ) f ( x) e sn( θ) f ( x) a sn b sn( )( f ( x) e) cos( ) f ( x) d c { π θ + θ } x = 0,, and θ = 0. π, a = 0., b = 0, c =, d = 6, e =. Where [ ] XII. CTP3 Mn f ( x ) = x f ( x ) Mn f ( x ) = g ( x ) g ( x ) Where g ( x ) =.0 + x Subject to: g x = cos θ f x e sn θ f x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a sn bπ sn ( θ) f ( x ) e + cos θ f x c d XIII. CTP4 Mn f ( x ) = x f ( x ) Mn f ( x ) = g ( x ) g ( x ) Where g ( x ) =.0 + x Subject to: g x = cos θ f x e sn θ f x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a sn bπ sn ( θ) f ( x ) e + cos θ f x Where [ ] x = 0,, θ = 0. π, a = 0.75, b = 0, c =, d = 0.5, e = XIV. CTP5 Mn f ( x ) = x f ( x ) Mn f ( x ) = g ( x ) g ( x ) Where g ( x ) =.0 + x Subject to: g x = cos θ f x e sn θ f x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a sn bπ sn ( θ) f ( x ) e + cos θ f x Where [ 0, ], x = θ = 0. π, a = 40, b = 0.5, c =, d =, e = s. c d c d Where [ ] x = 0,, θ = 0. π, a = 0., b = 0, c =, d = 0.5, e =. 65