ROUGH SESTS-INSPIRED EVOLUTIONARY ALGORITHM FOR ENGINEERING MULTIOBJECTIVE OPTIMIZATION PROBLEMS

Transcription

1 Volume 1, No. 1, December 013 Journal of Global Research n Mathematcal Archves RESEARCH PAPER Avalable onlne at ROUGH SESTS-INSPIRED EVOLUTIONARY ALGORITHM FOR ENGINEERING MULTIOBJECTIVE OPTIMIZATION PROBLEMS Ahmed A. EL- Sawy 1,, Mohamed A. Hussen 1,*, EL-Sayed M. Zak 1, A. A. Mousa 1, 3 1 Department of Basc Engneerng Scences, Faculty of Engneerng, Menoufa Unversty, Shbn El-Kom, Egypt. Department of Mathematcs, Faculty of Scence, Qassm Unversty, Saud Araba. 3 Department of Mathematcs, Faculty of scences, Taf Unversty, Saud Araba. 1,* Emal:moh_abdelsameea_h@yahoo.com Abstract: In ths paper we present a new optmzaton algorthm, the proposed algorthm operates n two phases: n the frst one, multobjectve verson of genetc algorthm s used as search engne n order to generate approxmate true Pareto front. Ths algorthm based on concept of coevoluton and repar algorthm for handlng nonlnear constrants. Also t mantans a fnte-szed archve of non-domnated solutons whch gets teratvely updated n the presence of new solutons based on the concept of ε -domnance. Then, n the second stage, rough set theory s adopted as local search engne n order to mprove the spread of the solutons found so far. The results, provded by the proposed algorthm for engneerng optmzaton problems, are promsng when compared wth extng well-known algorthms. Also, our results suggest that our algorthm s better applcable. 1. INTRODUCTION Most of real world optmzaton problems requre optmzng multple conflctng objectves wth each other. These problems wth more than one objectve are called Multobjectve Optmzaton Problems (MOPs). There s no sngle optmal soluton, but a set of alternatve solutons, these solutons are optmal n the wder sense that no other solutons n the search space are domnate them when all objectves are consdered. They are known as Pareto optmal solutons [1,]. MOPs naturally arse n many area of knowledge such as economcs [3-5], machne learnng [6-8] and electrcal power system [9-1]. Deb [13] classfed optmzaton methods as classcal (tradtonal) methods and evolutonary methods (evolutonary algorthms). Tradtonal multobjectve methods attempt to fnd the set of nondomnated solutons usng mathematcal programng. In tradtonal methods such as weghted method and ε-constrant method whch are most commonly used [1], the MOPs transformed n to a sngle objectve problem whch can be solved usng nonlnear optmzaton technques. On the other hand, evolutonary methods (evolutonary algorthms EAs) for MOPs optmze all objectves smultaneously and generatng a set of alternatve solutons. Many studes on evolutonary algorthms have shown that these methods can be effcently used to elmnate most of dffcultes of classcal methods. Snce they are populaton-based approaches, multple pareto optmal solutons can be found n one sngle run. These populaton-based approaches are more successful when solvng MOPs. There are many popular algorthms for solvng unconstraned MOPs. A representatve collecton of these algorthms ncludes the vector evaluated genetc algorthm by Schaffer [14], the nched Pareto genetc algorthm (NPGA) [15] and the nondomnated sortng genetc algorthm by Srnvas and Deb[16], the nondomnated sortng genetc algorthm II (NSGA-II) by Deb et al.[17], the strength Pareto evolutonary algorthm by Ztzler and Thele [18], the strength Pareto evolutonary algorthm II (SPEA-II) by Ztzler et al. [19], the Pareto archved evoluton strategy by Knowles and Corne [0] and the memetc PAES by Knowles and Corne [1]. These MOEAs dffer from each other n both explotaton and exploraton; they share the common purpose, searchng for a near-optmal, well-extended and unformly dversfed Pareto optmal front for a gven MOP [-5]. In case of constraned multobjectve optmzaton problems, there are a few evolutonary algorthms developed. Despte the developments for solve constraned optmzaton problems; there seem to be not enough studes concernng procedure for handlng constrants. For example, Fonseca [6] suggested treatng constrants as hgh-prorty objectves, and Harade [7] proposed a few effcent constrant-handlng gudelnes and desgned a Pareto descent repar method. For MOPs, a properly desgned ftness assgnment method s always requred to gude the populaton to evolve to the Pareto front, as the objectve vector cannot be used as the ftness functon value drectly. Most of the exstng MOEAs assgn the ftness functon values based on the Pareto domnance relatonshp. In ths paper we present a new optmzaton algorthm, the proposed algorthm operates n two phases: n the frst one, multobjectve verson of genetc algorthm s used as search engne n order to generate approxmate true Pareto front. Then n the second phase, rough set theory s adopted as local search engne n order to mprove the spread of the solutons found so far. The proposed algorthm s mplemented to engneerng applcatons, and the results are compared wth extng well-known algorthms. The remander of the paper s organzed as follows. In secton, we descrbe some basc concepts and defntons of MOPs. Secton 3 presents constrant multobjectve optmzaton va genetc algorthm. Applcaton on real engneerng problems are gven and dscussed n secton 4. Secton 5 ndcates our concluson. JGRMA 013, All Rghts Reserved 1

2 Ahmed A. EL- Sawy et al, Journal of Global Research n Mathematcal Archves, 1(1), December 013, BASIC CONCEPTS AND DEFINITIONS.1 Problem Formulaton General multobjectve optmzaton problem s expressed by MOP : Mn F( x) ( f ( x), f ( x),..., f ( )) T 1 m x s. t. x S x ( x, x,..., x ) T 1 n Where ( f1( x), f( x),..., fm( x)) are the m objectves functons, ( x,,..., n ) 1 x x are the n decson varables, and S the soluton (parameter) space. * Defnton 1. ( Pareto optmal soluton ): x s sad to be a Pareto optmal soluton of MOP f there exsts no other feasble x * * (.e., x S) such that, f j ( x) f j ( x ) for all j 1,,..., m and f j( x) f j( x ) for at least one objectve functon f j. m Defnton.[8]. (ε-domnance) Let f : x R and a, b a b, f and only f for {1,..., m } n R s X. Then a s sad to ε-domnate b for some ε > 0, denoted as (1 ) f ( a) f ( b ) Defnton 3. (ε -approxmate Pareto set) Let X be a set of decson alternatves and 0. Then a set x s called an ε- approxmate Pareto set of X, f any vector a x s ε-domnated by at least one vector b x,.e a x : b x such that b a Accordng to defnton, the ε value stands for a relatve allowed for the objectve values as declared n Fg. 1. Ths s especally mportant n hgher dmensonal objectve spaces, where the concept of ε -domnance can reduce the requred number of solutons consderably. Also, makes the algorthms practcal by allowng a decson maker to control the resoluton of the Pareto set approxmaton by choosng an approprate ε-value Fgure 1. Graphs vsualzng the concepts of domnance (left) and ε-domnance (rght).. Use of Rough Sets n Multobjectve Optmzaton For our proposed approach we wll try to nvestgate the Pareto front usng a Rough sets grd. To do ths, we wll use an ntal approxmate of the Pareto front (provded by any evolutonary algorthm) and wll mplement a grd n order to get more nformaton about the front that wll let to mprove ths ntal approxmaton [9]. To create ths grd, as an nput we wll have N feasble ponts dvded n two sets: the nondomnated ponts (NS) and the domnated ones (DS). Usng these two sets we want to create a grd to descrbe the set NS n order to ntensfy the search on t. Ths s, we want to descrbe the Pareto front n the decson varable space because then we could easly use ths nformaton to generate more effcent ponts and then mprove ths ntal Pareto approxmaton. Fgure shows how nformaton n objectve functon space can be translated nto nformaton n decson varable space through the use of a grd. We must note the mportance of the DS sets as n a rough sets method, where the nformaton comes from the descrpton of the boundary of the two sets NS, DS. Then the more effcent ponts provded the better. However, t s also requred to provde domnated ponts, snce we need to estmate the boundary between beng domnated and beng nondomnated. Once the nformaton s computed we can smply generate more ponts n the effcent sde. Fgure. Decson varable space (left) and objectve functon space (rght)[9].3 Structure of an Iteratve Multobjectve Search Algorthm JGRMA 014, All Rghts Reserved

3 Ahmed A. EL- Sawy et al, Journal of Global Research n Mathematcal Archves, 1(1), December 013, 1-30 The purpose of ths secton s to nformally descrbe the problem we are dealng wth. To ths end, let us frst gve a template for a large class of teratve search procedures whch are characterzed by the generaton of a sequence of search ponts and a fnte memory. Algorthm 1. Iteratve search algorthm 1. t 0 (0). A 0 (t) 3. whle termnate (A, t) false do 4. t t 1 5. f generate( ) {generate new search pont} ( t) ( t 1) ( t) 6. A update( A, f ) {update archve} 7. end whle 8. Output : A An abstract descrpton of a generc teratve search algorthm s gven n Algorthm (1) [8]. The nteger t denotes the teraton count, the n-dmensonal vector f s the sample generated at teraton t and the set A wll be called the archve at teraton t and should contan a representatve subset of the samples n the objectve space F=[ f1( x), f( x),..., fm( x )] generated so far. To smplfy the notaton, we represent samples by n-dmensonal real vectors f where each coordnate represents one of the objectve values as shown n Fg.3. The purpose of the functon the old archve set ( t 1) A. The functon Fgure 3. Block dagram of Archve/selecton strategy f generate () s to generate a new solutons n each teraton t, possbly usng the contents of ( t) ( t 1) ( t) A update( A, f ) gets the new solutons () generate and the old archve set ( t 1) A and determnes the updated one, namely A. In general, the purpose of ths sample storage s to gather useful nformaton about the underlyng search problem durng the run. Its use s usually two-fold: On the one hand t s used to store the best solutons found so far, on the other hand the search operator explots ths nformaton to steer the search to promsng regons. Ths procedure could easly be vewed as an evolutonary algorthm when the generate operator s assocated wth varaton (recombnaton and mutaton). However, we would lke to pont out that all followng nvestgatons are equally vald for any knd of teratve process whch can be descrbed as Algorthm(1) and used for approxmatng the Pareto set of multobjectve optmzaton problems. 3. CONSTRAINED MULTIOBJECTIVE OPTIMIZATION VIA GENETIC ALGORITHM In any nterestng multobjectve optmzaton problem, there exst a number of such solutons whch are of nterest to desgners and practtoners. Snce no one soluton s better than any other soluton n the Pareto-optmal set, t s also a goal n a multobjectve optmzaton to fnd as many such Pareto-optmal solutons as possble. Unlke most classcal search and optmzaton problems, GAs works wth a populaton of solutons and thus are lkely (and unque) canddates for fndng multple Pareto-optmal solutons smultaneously. There are two tasks that are acheved n a multobjectve GA. (1) Convergence to the Pareto-optmal set, and. () Mantenance of dversty among solutons of the Pareto-optmal set. Here we present a new optmzaton system, whch s based on concept of co-evoluton and repar algorthms. Also t s based on the -domnance concept. The use of -domnance also makes the algorthms practcal by allowng a decson maker to control the resoluton of the Pareto set approxmaton by choosng an approprate value. 3.1 Intalzaton Stage The algorthm uses two separate populaton, the frst populaton P conssts of the ndvduals whch ntalzed randomly satsfyng the search space (The lower and upper bounds), whle the second populaton R conssts of reference ponts whch satsfyng all constrants (feasble ponts), However, n order to ensure convergence to the true Pareto-optmal solutons, we JGRMA 014, All Rghts Reserved 3

4 Ahmed A. EL- Sawy et al, Journal of Global Research n Mathematcal Archves, 1(1), December 013, 1-30 concentrated on how eltsm could be ntroduced. So, we propose an archvng/selecton strategy that guarantees at the same tme progress towards the Pareto-optmal set and a coverng of the whole range of the non-domnated solutons. The algorthm mantans an externally fnte-szed archve A of non-domnated solutons whch gets teratvely updated n the presence of new solutons based on the concept of -domnance. 3. Repar Algorthm The dea of ths technque s to separate any feasble ndvduals n a populaton from those that are nfeasble by reparng nfeasble ndvduals. Ths approach co-evolves the populaton of nfeasble ndvduals untl they become feasble. Repar process works as follows. Assume, there s a search pont S (where S s the feasble space). In such a case the algorthm selects one of the reference ponts (Better reference pont has better chances to be selected), say r Sand creates random ponts Z from the segment defned between,r, but the segment may be extended equally [8, 3] on both sdes determned by a user specfed parameter [0,1]. Thus, a new feasble ndvdual s expressed as: z z 1. (1 ). r (1 ).. r, (1 ), [0,1] 3.3 Evolutonary Algorthm: Phase 1 ( t 0) In the frst phase, the proposed algorthm uses two separate populaton, the frst populaton P (where t s the teraton counter) conssts of the ndvduals whch ntalzed randomly satsfyng the search space (The lower and upper bounds), whle (0) the second populaton R conssts of reference ponts whch satsfyng all constrants (feasble ponts). Also, t stores ntally the (0) Pareto-optmal solutons externally n a fnte szed archve of non-domnated solutons A. We use cluster algorthm to create t 1 ( t) ( t) t 1 the next populaton P, f P A then new populaton P conssts of all ndvdual from A and the populaton P are consdered for the clusterng procedure to complete A and drectly coped to the new populaton ( t 1) P, f P A then P solutons are pcked up at random from ( t) ( t) ( t 1) P. Snce our goal s to fnd new nondomnated solutons, one smple way to combne multple objectve functons nto a scalar ftness functon [30,31] s the followng weghted sum approach m f ( x) w f ( x)... w f ( x)... w f ( x) w f ( x ) 1 1 m m j j j 1 Where x s a strng (.e., ndvdual), f( x ) s a combned ftness functon, f ( x ) s the th objectve functon When a par of strngs are selected for a crossover operaton, we assgn a random number to each weght as follows: random (.) w m, 1,,.., m random (.) Calculate the ftness value of each strng usng the random weghts accordng to he followng selecton probablty ( x) of a strng x n the populaton j 1 j w. Select a par of strngs from the current populaton P f ( x) fmn ( P ) ( t) ( t) ( x), where f () mn ( P ) mn{ f ( x) x P } t { f ( x) fmn ( P )} ( t ) x P Ths step s repeated for selectng P / Pars of strngs from the current populatons. For each selected par apply crossover operaton to generate two new strngs, for each strngs generated by crossover operaton, apply a mutaton operator wth a prespecfed mutaton probablty. The system also ncludes the survval of some of good ndvduals wthout crossover or t mutaton. The algorthm mantans a fnte-szed archve A of non-domnated solutons whch gets teratvely updated n the presence of a new solutons based on the concept of -domnance, such that new solutons are only accepted n the archve f they are not -domnated by any other element n the current archve.the use of -domnance also makes the algorthms practcal by allowng a decson maker to control the resoluton of the Pareto set approxmaton by choosng an approprate value. 3.4 Local Search Mechansm Inspred on Rough Sets Theory: Phase Upon termnaton of phase 1, we start phase, wth ntal approxmate of the Pareto front (provded by the proposed algorthm n phase1) whch noted as NS. Also all domnated solutons are marked as DS. It s worth remarkng that NS can smply be a lst of solutons. From the set NS we choose NNS ponts prevously unselected. If we do not have enough unselected ponts, we choose the rest randomly from the set DS. Next, we choose from the set DS, NDS ponts prevously unselected ( and n the same way f we do not have enough unselected ponts, we complete them n a random fashon) these ponts wll be used to approxmate the boundary between the Pareto front and the rest of the feasble set n decson varable space. We store theses ponts n the set Items and perform rough sets teratons: JGRMA 014, All Rghts Reserved 4

5 Ahmed A. EL- Sawy et al, Journal of Global Research n Mathematcal Archves, 1(1), December 013, Range Intalzaton: for each decson varable, we compute and sort (from smallest and hghest) the dfferent values t takes n the set Items. Then, for each decson varable, we have a set of rang values and combnng all these sets we have a non-unform grd n decson varable space. - Compute Atoms: we compute NNS rectangular atoms centered n the NNS effcent ponts selected. To buld a e rectangular atom assocated to a nondomnated pont x Items we compute the followng upper and lower bounds for each decson varable : Lower Bound : Mddle pont between e x and the prevous value n the set e Upper Bound : Mddle pont between x and the followng value n the set rang If there are no pervous or subsequent values n rang, we consder the absolute lower or upper bound of varable. Ths settng lets the method to explore close to the feasble set boundares. 3- Generate Offsprng: nsde each atom we randomly generate offsprng new ponts. Each of these ponts s sent to the set NS as follows. The dea s that "new solutons are only accepted n the archve f they are not ε-domnated by any other element of the current archve". If a soluton s accepted, all domnated solutons are removed. Algorthm() shows the operator for ε-approxmate Pareto set Algorthm : Operator for ε-approxmate Pareto set 1. INPUT : A, x. f x A such that x x then 3. A 4. else A 5. D { x A : x x } 6. A A { x} \ D 7. end f 8. Output : A The pseudo code of the proposed algorthm s declared n Algorthm (3) rang Algorthm 3: The proposed algorthm t=0 (0) (0). Create P, R (0) (0) 3. A nondo mn ated ( P ) (t) 3. whle termnate (A, t) false do 4. t t 1 ( t) 5. P ( t 1) ( t 1) generate( A, P ) {generate new search pont} ( t) 6. A ( t 1) ( t) update( A, P ) {update archve (algorthm 3)} 7. end whle 8. Output : A 8. A NS, Domnted Pontes DS 9. Compute "Atom" 10.Comput Upper Bound, Lower Bound 11. Generate new offsprng. 1. Update NS set Phase Phase1 13. Output NS 4. APPLICATION OF PROPSED APPROACH ON REAL ENGINEERING PROBLEMS In order to valdate the proposed approach and quanttatvely compare ts performance wth other MOEAs, we present n ths secton comparson study whch appled to the problem was chosen from the engneerng applcaton, A welded beam desgn by Deb [3] and Two-Bar Truss [33,34]. Table 1 lsts the parameter settng used n our proposed algorthm for all runs. Table 1. The parameter adopted n the mplementaton of the proposed algorthm Populaton sze 00 No. of Generaton 00 P 0.9 c P 0.0 m Selecton operator Crossover operator Mutaton operator Roulette Wheel Sngle pont Polynomal mutaton 10e-6 JGRMA 014, All Rghts Reserved 5

6 Ahmed A. EL- Sawy et al, Journal of Global Research n Mathematcal Archves, 1(1), December 013, Welded Beam: A welded beam desgn s used by Deb [3], where a beam needs to be welded on another beam and must carry a certan load F as shown n Fg.4. Fgure 4. The welded beam desgn problem It s desred to fnd four desgn parameters (thckness b, wdth t, length of weld l, and weld thckness h) for whch the cost functon of the beam and the deflecton functon at the open end are mnmum. The overhang porton of the beam has a length of 14 nch and F=6000 Ib force s appled at the end of the beam. A lttle thought wll reveal that a desgn for mnmum deflecton at the end (or maxmum rgdty of the above beam) wll make all four desgn dmensons to take large dmensons. Thus, the desgn solutons for mnmum cost and maxmum rgdty (or mnmum-end-deflecton) are conflctng to each other. In the followng, the mathematcal formulaton of the two-objectve optmzaton problem s presented: Mn f ( x) h l tb(14 l) Mn f ( x).195 / t b st.. g ( x) r( x) 0, g ( x) ( x) 0 1 g ( x) b h 0, g ( x) P ( ) c x h, b [0.15,5] l, t [0.1,10] where Pc ( ) ( ) l / 0.5( l ( h l) ) 6000 / hl 6000( l) 0.5( l ( h l) ) / t b hl( l /1 0.5( h t) ) ( t) tb 3 In the Welded Beam desgn problem, the non-lnear constrants can cause dffcultes n fndng the Pareto front. As shown n Fg. 5 and Fg. 6 our proposed approach outperformed NSGA and NSGA-II n both dstrbuton and spread f f 1 Fgure 5. Pareto optmal front of welded beam usng proposed approach JGRMA 014, All Rghts Reserved 6

7 Ahmed A. EL- Sawy et al, Journal of Global Research n Mathematcal Archves, 1(1), December 013, 1-30 Fgure 6. Pareto optmal front of welded beam usng NSGA, NSGA- II. 4. Two-Bar Truss: Fgure 7 llustrates the two-bar truss that s to be optmzed [33]. Ths problem was adapted from Krsch [34]. It s comprsed of two statonary pnned jonts, A and B, where each one s connected to one of the two bars n the truss. The two bars are pnned where the jon one another at jont C, and a 100 kn force acts drectly downward at that pont. The crosssectonal areas of the two bars are represented as x1 and x, the cross-sectonal areas of trusses AC and BC respectvely. Fnally, y represents the perpendcular dstance from the lne AB that contans the two-pnned base jonts to the connecton of the bars where the force acts (jont C). Fgure 7. Two-Bar Truss problem The problem has been modfed nto a two-objectve problem n order to show the non-nferor Pareto set clearly n two dmensons. The stresses n AC and BC should not exceed 100,000 kpa and the total volume of materal should not exceed m. The reason the objectve constrants have been mposed s that the Pareto set s asymptotc and extends from - to. As x1 and x go to zero, f volume goes to zero and f BC go to nfnty. As x1 and x go to nfnty, f volume goes to nfnty and f AC and f BC go to zero. Hence, n order to generate Pareto optmal solutons n a reasonable range, objectve constrants are mposed. The problem formulaton s shown below. Mnmze f x (16 y ) x (1 y ) Mnmze f subject to f f AC f volume BC 1 y 3 x, x volume AC (1 y ) yx (16 y ) yx 0.5 Fgure 8 declares the Pareto optmal soluton of the Two-Bar Truss obtaned from proposed algorthm. Obvously from the results, the proposed algorthm s able to mantan an almost unform set of non-domnated soluton ponts along the true Paretooptmal front. Fgure 9 shows Pareto optmal front of Two-Bar Truss (MOGA Soluton) [33, 34] and Fg.10 show Pareto optmal front usng NSGA-II [3]. It s clear that the proposed algorthm outperformed the algorthm n [3, 33, 34] JGRMA 014, All Rghts Reserved 7

8 f (x) Ahmed A. EL- Sawy et al, Journal of Global Research n Mathematcal Archves, 1(1), December 013, x f 1 (x) Fgure 8. Pareto optmal front of two- truss usng the proposed approach Fgure 9. Pareto optmal front of Two-Bar Truss (MOGA Soluton) [33, 34] Fgure 10. Pareto optmal front of of Two-Bar Truss usng NSGA- II [3] 5. CONCLUSION Fndng a good dstrbuton of solutons near the Pareto optmal front n small computatonal tme s a dream of multobjectve EAs researchers and practtoner. In ths paper a new optmzaton algorthm were present, the proposed algorthm operates n two Phases: n the frst one, multobjectve verson of genetc algorthm s used as search engne n order to generate approxmate true Pareto front. Ths algorthm based on concept of co-evoluton and repar algorthm. Also t mantans a fnte-szed archve of nondomnated solutons whch gets teratvely updated n the presence of new solutons based on the concept of ε-domnance. Then n the second phase, rough set theory s adopted as local search engne n order to mprove the spread of the solutons found so far. Our proposed approach keeps track of all the feasble solutons found durng the optmzaton. The results, provded by the proposed algorthm for engneerng applcatons, are promsng when compared wth extng well-known algorthms. Also, the non-domnated solutons n the obtaned Pareto- optmal set are well dstrbuted and have satsfactory dversty characterstcs. REFERENCES [1] K. Mettnen "Non-lnear multobjectve optmzaton" Dordrecht: Kluwer Academc Publsher; (00). [] A. A. Mousa, A new Stoppng and Evaluaton Crtera for Lnear Multobjectve Heurstc Algorthms, Journal of Global Research n Mathematcal Archves, Volume 1, No. 8, August 013 [3] QSen Ca, Defu Zhang, Bo Wu, Stehpen C.H. Leung, A Novel Stock Fore-castng Model based on Fuzzy Tme Seres and GenetcAlgorthm, Proceda Computer Scence, Volume 18, 013, Pages [4] He N, Yongqao Wang, Stock ndex trackng by Pareto effcent genetc algo-rthm, Appled Soft Computng, Volume 13, Issue 1, December 013, Pages JGRMA 014, All Rghts Reserved 8

9 Ahmed A. EL- Sawy et al, Journal of Global Research n Mathematcal Archves, 1(1), December 013, 1-30 [5] Janko Straßburg, Chrstan Gonzàlez-Martel, Vassl Alexandrov, Parallel genetc algorthms for stock market tradng rules Proceda Computer Scence, Volume 9, 01, Pages [6] Xao-We Xue, Mn Yao, Zhaohu Wu, Janhua Yang, Genetc Ensemble of Extreme Learnng Machne, Neurocomputng, In Press, Accepted Manu-scrpt, Avalable onlne 16 November 013. [7] Jadong Yang, Hua Xu, Pefa Ja, Effectve search for genetc-based machne learnng systems va estmaton of dstrbuton algorthms and embedded fea-ture reducton technques,neurocomputng, Volume 113, 3 August 013, Pages [8] Hongmng Yang, Jun Y, Junhua Zhao, ZhaoYang Dong,Extreme learnng machne based genetc algorthm and ts applcaton n power system eco-nomc dspatch Neurocomputng, Volume 10, 15 February 013, Pages [9] M.S.Osman, M.A.Abo-Snna, and A.A. Mousa " A Soluton to the Optmal Power Flow Usng Genetc Algorthm "Journal of Appled Mathematcs & Computaton (AMC) vol 155, No., 6 August (004) pp ( Top 5 Hottest Artcles, July. to Sept. 005), from M. Sc. Dssertaton. [10] Osman M.S., M.A.Abo-Snna, and A.A. Mousa "Epslon-Domnance based Multobjectve Genetc Algorthm for Economc Emsson Load Dspatch Op-tmzaton Problem, Electrc Power Systems Research 79 (009) [11] B. N. AL-Matraf, A. A. Mousa,Optmzaton methodology based on Quantum computng appled to Fuzzy practcal unt commtment problem, Internatonal Journal of Scentfc & Engneerng Research, Volume 4, Issue 11, November [1] Abd Allah A. Galal, Abd Allah A. Mousa, Bekheet N. Al-Matraf, Hybrd Ant Optmzaton System for Multobjectve Optmal Power Flow Problem Under Fuzzness, Journal of Natural Scences and Mathematcs, Vol. 6, No., pp , July 013. [13] K. Deb, Multobjectve Optmzaton Usng Evolutonary Algorthms. Chchester, U.K.: Wley, 001. [14] J.D. Schaffer, Multple objectve optmzaton wth vector evaluated genetc algorthms, n: J.J. Grefenstette, et al. (Eds.), Genetc Algorthms and Ther Applcatons, Proceedngs of the 1st Internatonal Conference on Genetc Algorthms, Lawrence Erlbaum, Mahwah, NJ, pp (1985). [15] J. Horn, N. Nafplots, D.E. Goldberg, A nched Pareto genetc algorthm for multobjectve optmzaton, n: J.J. Grefenstette et al. (Eds.), IEEE World Congress on Computatonal Intellgence, Proceedngs of the 1st IEEE Conference on Evolutonary Computaton, IEEE Press, Pscataway, NJ, pp. 8 87(1994). [16] N. Srnvas, and K. Deb, " Multobjectve Optmzaton Usng Nondomnated Sortng In Genetc Algorthms " Evolutonary Computaton,(3): 1-48 (1999). [17] K. Deb, S. Agrawal, A. Pratab, T. Meyarvan, A fast eltst non-domnated sortng genetc algorthms for multobjectve optmzaton: NSGA II, KanGAL report 00001, Indan Insttute of Technology, Kanpur, Inda, (000). [18] E. Ztzler, L. Thele, Multobjectve optmzaton usng evolutonary algorthmsa comparatve case study. In A. E. Eben, T. Back, M. Schoenauer and H. P. Schwefel (Eds.), Ffth Internatonal Conference on Parallel Problem Solvng from Nature (PPSN-V), Berln, Germany, pp ,(1998). [19] E. Ztzler, M. Laumanns, L. Thele, SPEA: Improvng the strength Pareto evolutonary algorthm for multobjectve optmzaton, n: Evolutonary methods for desgn, optmzaton and control wth applcatons to ndustral problems, EUROGEN 001, Athens, Greece, (001). [0] J.D. Knowles, D.W. Corne, The Pareto archved evoluton strategy: a new baselne algorthm for multobjectve optmzaton, n: Proceedngs of the 1999 Congress on Evolutonary Computaton, IEEE Press, Pscataway, NJ, pp ,( 1999). [1] J.D. Knowles, D.W. Corne, M-PAES: a memetc algorthm for multobjectve optmzaton, n: Proceedngs of the 000 Congress on Evolutonary Computaton, IEEE Press, Pscataway, NJ, 000, pp , (000). [] F. Jmenez, and J. L. Verdegay, Constraned multobjectve optmzaton by evolutonary algorthm. Proceedng of the nternatonal ICSC symposton on Engneerng of ntellgent systems (EIS'98),pp66-71(1998). [3] Z. Mchalewz,"Genetc Algorthms + Data Structures = Evoluton Programs",Sprnger-Verlag,3 Rd Edton (1996). [4] Z. Mchalewcz, and Schoenauer, "Evolutonary Algorthms for Constraned Parameter Optmzaton Problems", evolutonary computaton 4(1)1-3 (1996). [5] Ahmed A. EL-Sawy, Mohamed A. Hussen, EL-Sayed M. Zak, A. A. Mousa, An Introducton to Genetc Algorthms: A survey A practcal Issues, Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 1, January-014. [6] Fonseca CM, Flemng PJ. Multobjectve optmzaton and multple constrant handlng wth evolutonary algorthms: Part I: a unfed formulaton. IEEE Transactons on Systems and Cybernetcs Part A: Systems and Humans 1998;8(1):6 37. [7] Harada K, Sakuma J, Ono Isao, Kobayash S. Constrant-handlng method for mult-objectve functon optmzaton: pareto descent repar operator. In: Proceedngs of the fourth nternatonal conference on evolutonary mult- crteron optmzaton, 007. p [8] M. Laumanns, L. Thele, K. Deb, and E. Ztzler: Archvng wth Guaranteed Convergence And Dversty n Mult-objectve Optmzaton. In GECCO 00: Proceedngs of the Genetc and Evolutonary Computaton Conference, Morgan Kaufmann Publshers, New York, NY, USA, pages , July, (00) [9] Hernández-Díaz, A. G., L. V. Santana-Quntero, C. A. Coello Coello, R. Caballero and J. Molna 008) Improvng Mult-Objectve Evolutonary Algorthms by usng Rough Sets, n A. Lgeza, S. Rech, R. Schaefer and C. Cotta (eds.), Knowledge-Drven Computng: Knowledge Engneerng and Intellgent Computatons, Studes n Computatonal Intellgence, vol. 10, pp. 81:98. [30] M.S.Osman, M.A.Abo-Snna, and A.A. Mousa " An Effectve Genetc Algorthm Approach to Multobjectve Resource Allocaton Problems ( MORAPs) " Journal Of Appled Mathematcs & Computaton 163, pp , (005). [31] T. Murata, H. Ishbuch, and H. Tanaka, "Multobjectve genetc algorthm and ts applcaton to flowshop Schedulng ", Computers and Industral Engneerng, vol. 30, no 4, pp (1996). [3] K. Deb, A. Pratap, and S. Motra, (000) " Mechancal Component Desgn for Multple Objectves Usng Eltst Non-Domnated Sortng GA", Kangal Report No JGRMA 014, All Rghts Reserved 9

10 Ahmed A. EL- Sawy et al, Journal of Global Research n Mathematcal Archves, 1(1), December 013, 1-30 [33] Ruhul Sarker, Hussen A. Abbass, and Samn Karm, An Evolutonary Algorthm for Constraned Multobjectve Optmzaton Problems, at the 5th Australasa-Japan JontWorkshop Unversty of Otago, Dunedn, New Zealand [34] Krsch, U., 1981, Optmal Structural Desgn, McGraw-Hll Co., New York. JGRMA 014, All Rghts Reserved 30