International Journal of Swarm Intelligence and Evolutionary Computation

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1 ISSN: 9498 Internatonal Journal of Swarm Intellgence and Evolutonary Computaton Research Artcle Artcle Internatonal Journal of Swarm Intellgence and Evolutonary Computaton Allah, 6, 5: DOI:.47/ Open Open Access Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems Rzk M Rzk Allah* Department of Basc Engneerng Scence, ElMenoufa nversty, Shebn ElKom, Egypt Abstract We propose a novel hybrsd algorthm named, FOAFA to solve the nonlnear programmng problems (NPPs). The man feature of the hybrd algorthm s to ntegrate the strength of frut fly optmzaton algorthm (FOA) n handlng contnuous optmzaton and the mert of frefly algorthm (FA) n achevng robust exploraton. The methodology of the proposed algorthm conssts of two phases. The frst one employs a varaton on orgnal FOA employng a new adaptve radus mechansm (ARM) for explorng the whole scope around the frut fles locatons to overcome the drawbacks of orgnal FOA whch has been contnues for the nonnegatve orthant problems. The second one ncorporates FA to update the prevous locatons of frut fles to avod the premature convergence. The hybrd algorthm speeds up the convergence and mproves the algorthm s performance. The proposed FOAFA algorthm s tested on several benchmark problems and two engneerng applcatons. The numercal comparsons have demonstrated ts effectveness and effcency. Keywords: Frefly algorthm; Frut fly optmzaton algorthm; Nonlnear programmng problems Introducton Tradtonal optmzaton methods can be classfed nto two dstnct groups; drect and gradentbased methods. In drect search methods, only objectve functon and constrant value are used to gude the search, whereas gradentbased methods use the frst and/or secondorder dervatves of the objectve functon and/or constrants to gude the search process. Snce dervatve nformaton s not used, the drect search methods are usually slow, requrng many functon evaluatons for convergence. For the same reason, they can be appled to many problems wthout a major change of the algorthm. On the other hand, gradent based methods quckly converge to an optmal soluton, but are not effcent n nondfferentable or dscontnuous problems. In addton, f there s more than one local optmum n the problem, the result may depend on the selecton of an ntal pont, and the obtaned optmal soluton may not necessarly be the global optmum. Furthermore, the gradent search may become dffcult and unstable when the objectve functon and constrants have multple or sharp peaks. Therefore, to overcome these shortcomngs, a lot of research has focused on metaheurstc methods. The metaheurstc methods have many advantages compared to the tradtonal nonlnear programmng technques, among whch the followng three are the most mportant: () They can be appled to problems that consst of dscontnuous, nondfferentable and nonconvex objectve functons and/or constrants. () They do not requre the computaton of the gradents of the cost functon and the constrants. () They can easly escape from local optma. Many metaheurstc algorthms such as, genetc algorthm (GA) [], partcle swarm optmzaton (PSO) [], dfferental evoluton [] and ant colony optmzaton [4] have shown ther effcacy n solvng computatonally ntensve problems. Among the exstng metaheurstc algorthms, a wellknown branch s the FA whch s a metaheurstc search algorthm, has been developed by Yang [5,6]. FA mmcs some characterstcs of tropc frefly swarms and ther flashng behavor. A frefly tends to be attracted towards other frefles wth hgher flash ntensty. The man advantages of FA are namely ntensfcaton and dversfcaton or explotaton and exploraton: As lght ntensty decreases wth dstance, the attracton among frefles can be local or global, dependng on the absorbng coeffcent, and thus all local modes as well as global modes wll be vsted. Therefore, t has captured much attenton and has been successfully appled to solve several optmzaton problems ncludng [79]. Frut fly optmzaton algorthm (FOA) s one of the latest metaheurstc methods whch proposed by Pan [] for solvng optmzaton problems. The man nspraton of FOA s that the frut fly tself s superor to other speces n sensng and percepton, especally n osphress and vson. They feed chefly on rotten fruts. In the process of fndng food, ther osphress organs smell all knds of scents n the ar. They then fly towards the correspondng locatons. When they get close to the food locatons, they fnd foods usng ther vsons and then fly towards that drecton. FOA has many advantages such as a smple structure, ease of mplementaton and speed to acqure solutons. Therefore, FOA has been successfully appled to solve several optmzaton problems [4]. Recently, hybrdzaton s recognzed to be essental aspect of hgh performng algorthms. Pure algorthms cannot reach to an optmal soluton n a reasonable tme. In addton, the premature convergence n pure algorthms s nevtable. Thus pure algorthms are almost always nferor to hybrdzatons. To our knowledge, ths s the frst tme that the hybrdzaton between FOA and FA has been proposed for solvng NPPs. Ths hybrdzaton ams to mprove the performance of the orgnal FOA and elmnate ts drawbacks. In ths paper, we propose a novel hybrd algorthm named, FOAFA for solvng the nonlnear programmng problems and two *Correspondng author: Rzk M Rzk Allah, Department of Basc Engneerng Scence, ElMenoufa nversty, Shebn ElKom, Egypt, Emal: Receved March, 6; Accepted Aprl 9, 6; Publshed Aprl, 6 Ctaton: Allah RMR (6) Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems. Int J Swarm Intel Evol Comput 5: 4. do:.47/ Copyrght: 6 Allah RMR. Ths s an openaccess artcle dstrbuted under the terms of the Creatve Commons Attrbuton cense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal author and source are credted. ISSN: 9498 SIEC, an open access journal Volume 5 Issue 4

2 Ctaton: Allah RMR (6) Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems. Int J Swarm Intel Evol Comput 5: 4. do:.47/ Page of engneerng applcatons. The proposed FOAFA algorthm employed the merts of both frut fly optmzaton algorthm (FOA) and frefly algorthm (FA) and ts methodology conssts of two phases. The frst one employs a varaton on orgnal FOA employng a new adaptve radus mechansm for explorng the whole scope around the frut fles locatons to overcome the drawbacks of orgnal FOA whch has been contnues for the nonnegatve orthant problems. In addton, several other mprovements for orgnal FOA are consdered, such as: Intalzaton of the canddate solutons through the search space unformly and a new modulaton coeffcent. Moreover, the premature convergence of orgnal FOA degrades ts performance by reducng ts search capablty, leadng to a hgher probablty of beng trapped to a local optmum. Therefore, the second phase ncorporates the FA to update the prevous locatons of frut fles to force FOA jump out of premature convergence, because of ts strong searchng ablty. Consequently, the hybrd algorthm speeds up the convergence and mproves the algorthm s performance. The rest of the paper s organzed as follows. In Secton we descrbe some prelmnares of the nonlnear programmng problem. In Secton, we ntroduce brefly the bascs of both FOA and FA algorthms. The hybrd algorthm named, FOAFA algorthm s proposed and explaned n detals n Secton 4. The numercal experments are gven n Secton 5 to substantate the superorty of the proposed algorthm. Fnally we summarze the paper wth some comments. Nonlnear Programmng Problem (NPP) The nonlnear programmng problem can be defned as follow [5]: NPP n mn F( x), x = ( x, x,..., x ) R. : Ω x gk ( x) ; hj ( x) = ; Ω = k =,,..., ; j = +,..., P Where F(x) s the objectve functon defned on the search space S, n S R, X s called a decson varable and g k (x), h j (x) are defned the nequalty and equalty constrants, respectvely. The sets Ω S and φ = S Ω defne the feasble and nfeasble search spaces, respectvely. sually, the search space S s defned as an ndmensonal rectangle n n R as n () (domans of varables defned by ther lower and upper bounds): n { R x x x, j =,,..., n } n S = x () j j j Any pont x Ω s called a feasble soluton, otherwse, X s an nfeasble soluton. Overvew of the FOA and the FA Algorthms Frut fly optmzaton algorthm (FOA) In ths secton we descrbe the basc steps of the orgnal Frut fly optmzaton algorthm (FOA) n detals as follows []. Step : Set the man parameters of FOA and gve ntal locaton for the frut fly swarm randomly P xaxs, yaxs Step : Endow personal frut fly wth random drecton for fndng food by usng: x = X axs + Random value () Y = y axs + Random Value =,,,m where m s the swarm sze of frut fles () Step : Snce the food locaton cannot be known, calculate the dstance (Dst ) of the frut fly from the orgn and fnd out the smell concentraton judgment value (S ). Suppose that S s the recprocal (Dst ) of as follows: Dst = x + y (4) S = Dst Step 4: Calculate the smell concentraton (Smell ) of the ndvdual frut fly locaton by substtutng the smell concentraton judgment value (S ) nto the smell concentraton judgment functon (also called Ftness functon). Smell = Functon (S ) (6) Step 5: Fnd out the ndvdual frut fly wth the maxmal smell concentraton (the maxmal value of Smell) among the frut fly swarm: [ smell ndex] = max (Smell) (7) Step 6: Keep the maxmal smell concentraton value and locaton (x, y) of the frut fly. Then, the swarm fles towards that locaton. Smell = Best smell xaxs = x( ndex) (8) yaxs = y( ndex) Enter teratve optmzaton to repeat the mplementaton of step 6. When the smell concentraton s not superor to the prevous teratve smell concentraton any more, or the teratve number reaches the maxmal teratve number, the crculaton stops. Frefly algorthm (FA) The Frefly algorthm was developed by XnShe [6] and t s based on dealzed behavor of the flashng characterstcs of frefles. For smplcty, we can summarze these flashng characterstcs as the followng three rules: a) All frefles are unsex, so that one frefly s attracted to other frefles regardless of ther sex. b) Attractveness s proportonal to ther brghtness, thus for any two flashng frefles, the less brght one wll move towards the brghter one. The attractveness s proportonal to the brghtness and they both decrease as ther dstance ncreases. If no one s brghter than a partcular frefly, t wll move randomly. c) The brghtness of a frefly s affected or determned by the landscape of the objectve functon to be optmzed [6]. For smplcty we can assume that the attractveness of a frefly s determned by ts brghtness or lght ntensty whch n turn s assocated wth the encoded objectve functon. In the smplest case for an optmzaton problem, the brghtness I of a frefly at a partcular poston X can be chosen as I( x) f (x). However the attractveness s relatve, t should vary wth the dstance r j between frefly and frefly j. As lght ntensty decreases wth the dstance from ts source and lght s also absorbed n the meda, so we should allow the attractveness to vary wth degree of absorpton. Step : Attractveness and lght ntensty In the frefly algorthm, there are two mportant ssues: the varaton of the lght ntensty and the formulaton of the attractveness. We know, the lght ntensty I(r) vares wth dstance r monotoncally and exponentally, that s: (5) ISSN: 9498 SIEC, an open access journal Volume 5 Issue 4

3 Ctaton: Allah RMR (6) Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems. Int J Swarm Intel Evol Comput 5: 4. do:.47/ Page of γr I ( r) = I e (9) Where I the orgnal lght ntensty and γ s the lght absorpton coeffcent. As frefly attractveness s proportonal to the lght ntensty seen by adjacent frefles, we can now defne the attractveness β of a frefly by Equaton () γr β ( r) = β e () Where r s the dstance between each two frefles and β s ther attractveness at r =.e., when two frefles are found at the same pont of search space. The value of γ determnes the varaton of attractveness wth ncreasng dstance from communcated frefly. Step : Dstance The dstance between any two frefles and j at x and x j respectvely, the Cartesan dstance s determned by equaton () where x,k s the kth component of the spatal coordnate of x the th frefly and n s the number of dmensons. n r x x ) () j = k = (, k j, k Step : Movement The frefly movement s attracted to another more attractve (brghter) frefly j s determned by: γr x ( t + ) = x + β e ( x x ) + α( rand.5) () j Where the second term s due to the attracton whle the thrd term s randomzaton wth α beng the randomzaton parameter and r and s a random number generator unformly dstrbuted n [, ]. The Hybrd FOAFA Algorthm In ths secton, the proposed algorthm wll be presented n detal. The man reason for developng the FOAFA s to overcome the drawbacks of the orgnal FOA whch cannot handle the negatve doman that s the canddate soluton cannot be unformly generated n the problem doman. Also orgnal FOA s easy to fall nto premature convergence because the random term of the Equaton () pcks small values wthn a radus equal to one around the locaton. The methodology of the proposed FOAFA algorthm conssts of two phases. The frst one employs the FOA, where a swarm of frut fles that moves n varous drectons usng the ARM. Therefore, these moves wll follow a unform dstrbuton. The other phase ncorporates the FA to update the prevous locatons of frut fles to force FOA jump out of premature convergence, because of ts explotaton and exploraton ablty. Consequently, the hybrd algorthm speeds up the convergence and mproves the algorthm s performance. The man steps of the proposed algorthm as follows: Step : Intalze the parameters The man parameters of the proposed algorthm are the maxmum teratonnumber (T),thewarmsze(m) and dstance zone of frut fly (. The man parameters of the proposed algorthm are the maxmum teraton number (T),thewarmsze(m) and dstance zone of frut fly (. Step : Intalze the swarm locaton Intal frut fly swarm locaton s accompaned wthn the search space, that s, the algorthm assgns a random vector x axs = ( x, x,..., xn ) for th frut fly as n Equaton (): x axs = x + rand.(x x ),,,..., m. () = Where rand s a random value from the unform dstrbuton on the nterval [, ]. Step : Soluton constructon usng osphress The osphress that ntroduced n the orgnal FOA (see Equaton ) was pcked small values wthn a radus that equals one. So, we ntroduce the ARM for enhancng the dversty of solutons and achevng an effectve exploraton to the whole scope around the frut fles locatons, where the search radus decreases dynamcally wth teraton number as shown n Fgure (.e., R=, T=). Therefore, the process of the soluton constructon usng osphress s mplemented as follows: x = x axs ± ε. ( t,, (4) π π π π t ( t, = Rcos{ sn [sn( sn( ))]}, t =,,..., T. T Where R s the search radus that determnes the extent of the search subspace whch we want to cover durng the run and ε s a random number from [, ]. The random step ( t, returns a value n the range [, R] such that the probablty of ( t, beng close to ncreases as t ncreases. The soluton constructon usng osphress s demonstrated n Algorthm. Algorthm. The soluton constructon usng osphress. Input : R = x x, T, x axs. set t =, =, ε = rand(,). x = x _ axs ± ε. ( t, Rmax= Rmn= 4. f x _ axs > x, set x _ axs = x, then x = x _ axs ε. ( t, 5. elsef x _ axs < x, set x _ axs = x then x = x _ axs + ε. ( t, 6. end f 7. t = t + 8. = + Radus 9. Output: x Ω Iteratons Fgure : The varaton of ( t, versus the number of teratons. Step 4: Evaluaton The smell concentraton ( Smell ) of the ndvdual frut fly s related to objectve functon (ftness functon). In ths step, the ftness ISSN: 9498 SIEC, an open access journal Volume 5 Issue 4

4 Ctaton: Allah RMR (6) Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems. Int J Swarm Intel Evol Comput 5: 4. do:.47/ Page 4 of functon s evaluated based on the decson varable X as n Equaton (5) nstead of usng the smell concentraton judgment value (S ). Therefore, the food source wth the lowest ftness x s also found as n Equaton (6),.e., m x = arg(mn f (x ). Smell = f(x ), =,,..m (5) x =,,..., m = = arg(mn ( Smell ) (6) Step 5: pdate the swarm locaton In ths step, the swarm fles toward the obtaned locaton usng Equaton (6). If X s better than the current frut fly swarm locaton X axs, t wll replace the swarm locaton and become a new one n the next teraton,.e., x axs = x, f f (x ) < f (x axs). Step 6: Crossover operator for updatng the osphress In ths step, the crossover operator s adopted for enhancng the osphress between frut fles as follows: x = β.x ( t) + ( β ). x ( t) (8) where β = ( µ ) γ + µ s the coeffcent of modulaton, t decdes the scope of modulaton, µ [,] s random numberand sthe extendngmodulaton operator s specfed by the user. The dea of ths step s to make the new generaton of frut fles around the frut fly, where the new soluton not wthn x (t) and x (t) but the segment may be extended equally on both sdes (.e., γ as n Fgure. Therefore, the food source wth the lowest ftness x s also m found as n Equaton (6),.e., x = arg(mn= f (x ). If x s better than the current frut fly swarm locaton X axs, t wll replace the swarm locaton and become a new one n the next teraton,.e., x axs = x,f f ( x ) < f ( x axs). Step 7: FA phase for updatng the dversty The random term of the Equaton () makes the frut fles swarm fluctuates around the so far soluton; consequently, the dversty loss occurs. nder ths condton, the soluton mght be easly trapped n a local optmum or premature convergence. FA s ncorporated to update the prevous locatons of frut fles to force FOA jump out of premature convergence, because of ts explotaton and exploraton ablty. Consequently, the hybrd algorthm speeds up the convergence and mproves the algorthm s performance. In the rest of ths paper, we wll dscuss the mplementaton FA n three steps as follows (Fgure ). Step 7.: Intalzaton: Intalze a swarm of frefles wth the obtaned ant poston, where each frefly contans n varables (.e., the poston of the th frefly n the n dmensonal search space can be represented as x = ( x, x,..., xn ) ). Furthermore every member of the swarm s characterzed by ts lght ntensty (.e., ntalze each frefly wth dstnctve lght ntensty. I (x ), =,, m Step 7.: ght ntensty Calculate the lght ntensty I (x ), =,, m for each frefly whch n turn s assocated wth the encoded objectve functon, where for x ( t) x ( t) Fgure : The demonstraton of updatng the osphress. FOA phase Input: Parameters: T, m, β, I, θ, γ,α and R = x x. Set t = Intalzaton: for =,,..., m // generate m food sources x,x,...,xm x axs = x + rand.(x x ) end ( ) x = arg(mn=,,..., m ( Smell )) ( ) x axs = x oopng whle t <= T do for = to m π π π π t ( t, = R cos{ sn [sn( sn( ))]} T x = x axs ± ε. ( t, // generate new m food sources x, x,..., xm f x _ axs > x, set x _ axs = x, then x = x _ axs ε. ( t, else f x _ axs < x, set x _ axs = x then x = x _ axs + ε. ( t, f x Ω, then t = t + = + end for Smell = f ( x ), =,,..., m // evaluaton. x = arg (mn=,,..., m ( Smell )) f f ( x ) < f (x axs) then, x axs = x f f ( x axs) < f (x ) then, x axs = x axs for =,,..., m // updatng the osphress x = β.x ( t) + ( β ).x ( t) end Smell = f ( x ), =,,..., m // evaluaton. x = arg(mn=,,..., m( Smell )) f f ( x ) < f (x axs) then, x axs = x f f ( x axs) < f (x ) then, x axs = x axs FA phase Intalzaton: for =,,..., m // generate m locatons for frefles x,x,...,xm x = x + rand.(x x ) end I( x ) = f (x ), =,,..., m. // evaluaton. whle t <= T do ( t T ) αt + = αt. θ for = to m do; for j = to m do. I > I then γr x ( t + ) = x ( t) + β e ( x ( t) x ( t)) + α ( rand.5) f j end for j ; end for f x > x, then x = x else f x < x, then x = x I( x ) = f (x ), =,,..., m. // evaluaton. x = arg(mn=,,..., m I(x )) j f f ( x ) < f (x axs) then, x axs = x f f ( x axs) < f (x ) then, x axs = x axs end oopng Output: x axs Fgure : The pseudo code of the proposed FOAFA algorthm. ISSN: 9498 SIEC, an open access journal Volume 5 Issue 4

5 Ctaton: Allah RMR (6) Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems. Int J Swarm Intel Evol Comput 5: 4. do:.47/ Page 5 of mnmzaton problem the brghter frefly represents the mnmum value for I(x). Step 7.: Attractveness and movement In the frefly algorthm, the attractveness of a frefly s determned by ts lght ntensty whch n turn s assocated wth the encoded objectve functon and the dstance r j between frefly and frefly j. In addton, lght ntensty decreases wth the dstance from ts source, and lght s also absorbed n the meda, so we should allow the attractveness to vary wth the degree of absorpton. As a frefly s attractveness s proportonal to the lght ntensty seen by adjacent frefles, we can now defne the attractveness β of a frefly by Equaton (). On the other hand, the movement of a x 5 the proposed algorthm SZGA F F F F4 F5 F6 F7 F8 F9 F F Fgure 4: The graphcal representaton for the evaluaton of metrc. frefly s attracted to another more attractve (brghter) frefly J s determned by Equaton (). γr j x = x + β e (x x ) + α ( rand. 5) Snce each member of the swarm explores the problem space takng nto account results obtaned by others, therefore the randomzaton term may moves the fre fly to lose ts locaton, so we ntroduce a modfcaton on the randomzaton term that makes the frefles approached from the optmum. A further mprovement on the convergence of the algorthm s to vary the randomzaton parameter α so that t decreases gradually as the optma are approachng. For example, we can use t Tc α = ( ),,,...,. t αθ + t t = Tc (4) Where T c s the maxmum number of generatons for FA, and θ є [, ] s the randomness reducton constant. The basc steps of the proposed FOAFA algorthm can be summarzed as the pseudo code shown n Fgure 4. Results and Dscussons In ths secton, performance of the proposed FOAFA algorthm s tested on eleven benchmark problems, whch are known from the lterature and reported n Young [6] and two engneerng applcatons [7,8]. The solutons of the proposed FOAFA are compared wth the promnent algorthms that reported n Young [68]. The test problems are lsted n Table. The algorthm s coded n MATAB 7, whch has an Intel Core I 5 (.8 GHz) processor and 4 GB RAM memory. Test problems ( π ) ( π ) F = x + x.cos x.4cos 4 x +.7 Dmenson Doman x, x [.8,.8] Optmum value ( π ) ( π ) ( π ) ( π ) F = [cos x + cos.5 x.]*[. cos x + cos.5 x ] [, ] x, x F =. + ( j + ( x aj ) ) j = = x, x [65.56, 65.56] a = x [5,], x [,5].9789 F 4 x cos( ) 4 x x = + π π + 8π x + 4 x [,], x [,].6 x F5 = 4.x + x + x x + ( 4x 4) x x, x [,] F6 = + x + x + 9 4x + x 4x + 6x x + x * ( ( ) ( )) ( + ( x ) ( )) x 8 x + x + 48x 6x x + 7x 5 5 F7 = cos(( + ) x + ) * cos(( + ) x + ) = = c 8 ( c = a x x fsmulated ) F x, x [,] a, c, c (, ) f smulated x x F9 = ( x x ) + ( x ) + 9( x4 x ) + ( x) +.(( x ) + ( x4 ) ) 9 ( + + = + ) ( ) ( ) x + ( + ) x F x x = ( x ) ( + sn ( πx ))) 9 π F = ( ) sn ( πx ) ( x ) = x x x [,], =,,..., 4 [,4], =,,..., [,], =,,..., Table : Test problems. ISSN: 9498 SIEC, an open access journal Volume 5 Issue 4

6 Ctaton: Allah RMR (6) Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems. Int J Swarm Intel Evol Comput 5: 4. do:.47/ Page 6 of On the other hand, extensve expermental tests were conducted to see the effect of dfferent values on the performance of the proposed algorthm. Based upon these observatons, the followng parameters have been set as n Table. The performance evaluaton metrcs Our hybrd algorthm was tested on set of the test problems n engneerng doman. Each of the test problems was run thrty tmes ndependently, wth dfferent seeds. In order to evaluate the closeness between the obtaned optmal soluton and the theoretcal optmal soluton, as well as measurng the number of evaluatons for the proposed algorthm and the exstng algorthms, we used two dfferent metrcs. The speed: The speed wth whch the optmum s obtaned s a Number of teraton (T) The swarm sze (m) Intal attractveness (β) The lght absorpton coeffcent (γ) The randomness reducton constant (Ѳ).96 Randomzaton parameter (α).95 Table : Parameters settngs. crteron for examnng the performance. For the proposed FOAFA to be compettve, t s mportant to be able to compare ts speed wth the other exstng algorthms. As tme s a relatve noton dependng on the computer used, we preferred to measure the speed by countng the number of evaluatons of the tested problem. On the one hand, ths crteron s ndependent from the type of computer used, and on the other hand the evaluaton of the functon s the operaton takng the longest tme n the program. The error: So the test for closeness, we used the evaluaton of the error between the result gven by the algorthm and the exact value of the optmum, for each test problem. The relatve error was used each tme t was possble: E relatve F = obtaned F F optmum optmum () When the optmum to reach was, t was no longer possble to use ths expresson so we calculated the absolute error: E relatve = F F () obtaned optmum Table compares the performance of the proposed FOAFA, where the better values n the comparson are bold n font type. On the other hand, a comparson s made wth the promnent algorthms Functon value FOAFA Compared algorthms Number of functon evaluaton(nfe) Tme(s) Name Functon value NFE F (x) Best Average Best Average Best Average F.79475E SZGA.448E5.484E5.98E F SZGA F SZGA F SZGA F SZGA F SZGA F SZGA F SZGA F 9.684E7.684E SZGA E.74E F E SZGA F.E E SZGA Table : The comparson of soluton qualty..59e.565e.54e e ISSN: 9498 SIEC, an open access journal Volume 5 Issue 4

7 Ctaton: Allah RMR (6) Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems. Int J Swarm Intel Evol Comput 5: 4. do:.47/ Page 7 of from the lterature. The test problems have been solved by FOAFA for tmes. The startng values of the varables for each problem were selected randomly for all runs from the soluton space. The results found by FOAFA such as the and average functon value, numbers offuncton evaluaton (NFE) and soluton tme n seconds have been recorded n Table, whereas for the other algorthms only the functon value and numbers of functon evaluaton are gven because the soluton tmes, the and average functon value for some algorthms not gven. Frst, we start by comparng the optmum soluton of the gven the test problems for the proposed FOAFA algorthm and the exstng algorthms. Based on the obtaned results that demonstrated n Table we notes that, the proposed FOAFA s successful fndng the optmum soluton for the test problems F and F4 better than both the theoretcal optmum soluton and the obtaned by other exstng algorthms. Also the proposed FOAFA algorthm outperforms the exstng algorthms for functons F, F 6, F, F, F, F and F but equals the SZGA for the test problems F 6, F 7 and F 8. Second, as shown from Table 4, the proposed algorthm speeds up the convergence through ncluson of the lowest evaluatons, where the proposed algorthm elapsed number of evaluatons less than the other exstng algorthms for fndng the global optmum as n Fgure 4, that s the proposed algorthm s the faster than the exstng algorthms. Fnally, the last performance metrc s the relatve error or the absolute error, where as depcted from Table 4, the absolute error by usng the proposed algorthm s better than the other exsts algorthms for all test problems except SZGA for the test problem F 4 and also the relatve error s superor to the other exsts algorthms for all test problems except SZGA for the test problem F 4 The SZGA seems to better than the proposed algorthm for the test problem F 4 ths because the error s calculated related to the theoretcal optmum. In fact, the proposed algorthm outperforms the theoretcal optmum for the test problem F 4, where f the relatve error and the absolute error are calculated related to the obtaned optmum by our algorthm we fnd t.567 E5 and.5 E6 respectvely. Therefore, the smulaton results prove superorty of the proposed algorthm to those reported n the lterature, where t s completely better than the other algorthm Desgn optmzaton problems The desgn optmzaton problems are a very mportant research area n engneerng studes because real world desgn problems requre the optmzaton of relevant engneerng applcatons. In terms of robustness and effcency of the avalable methods, these methods are stll n need of mprovements. Therefore, to meet the ever ncreasng demands n the desgn of the electrcal devces, the proposed FOAFA has been substantated ts capablty for solvng these applcatons. Thus, the valdty of the proposed algorthm has been proved superorty by usng two engneerng applcatons, the lnear synchronous motor (SM) and arcored solenod [7,8]. Problems F F Absolute error Table 4: The performance assessment. Relatve error Proposed algorthm Compared algorthms Proposed algorthm Compared algorthms.6e8 F F 4.4E4 F 5 F 6 F 7 F 8 F E6.448E5.484E5 SZGA.98E5 SZGA.99999E E5 SZGA.786E9 SZGA.64E6.64E6 SZGA SZGA SZGA.99999E5.5E SZGA.55E4 SZGA SZGA.E.E SZGA SZGA E4.E4.858E4.445E4 SZGA SZGA.8E.7E.58E.6E SZGA SZGA E SZGA.74E5 SZGA F.59E SZGA.54E7 SZGA.565E F.E SZGA.E ISSN: 9498 SIEC, an open access journal Volume 5 Issue 4

8 Ctaton: Allah RMR (6) Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems. Int J Swarm Intel Evol Comput 5: 4. do:.47/ Page 8 of Desgn of SM: The lnear synchronous motor (SM) operates on the same workng prncple as that of a permanent magnet rotary D.C. motor [7]. As n a rotary motor there are two parts n a SM, one s the set of permanent magnets and the other s the armature that has conductors carryng current. The permanent magnets produce a magnetc flux perpendcular to the drecton of moton. The flow of current s n the drecton perpendcular to both the drecton of the moton and the drecton of the magnetc flux. To valdate the proposed algorthm, t s employed to optmze the geometrcal desgn of the lnear electrc actuator problem as s shown n Fgure 5(a) and 5(b). The objectve functon s to maxmum force subject to some of constrants on heat, radus, saturaton, demagnetzaton and maxmum force constrant. There are four desgn varables: the current n each slot, x, the dmensons of the slot, and x x, and the heght of the magnet, x 4. The mathematcal formulaton of the objectve functon s descrbed as follows: max F ( x) = xx ( x + x ) ( x +. 7) subject to : g( x) =. 9 * x( x4 + x ) xx 4 g( x) = x4 + x x. 66 g ( x) =. x (. 7 + x ) x. 5 g x x x x +. ) ( x) = 6. 8 * ( ) 4( 4 g () x = xx ( x + x )( x +. 7) x 8A, x, x. 8m, x. 9m. 4 The obtaned result for desgn of SM s demonstrated n Table 5, by gvng the, worst and mean values for the objectve functon. Also Fgure 6 depcts the relaton between the maxmum force and number of teratons for FOA before applyng FA and after Applyng FA (.e., the hybrd algorthm FOAFA). The results of the proposed FOA FA s superor to those obtaned usng the Deshpande [7]. Desgn of an arcored solenod: The optmzaton problem of a coreless solenod wth rectangular cross secton x x and a mean radus x Fgure 5(b) s tackled from Barba [8]. Ths problem can be formulated as: maxmze the nductance (x, x, x ) and satsfy some of constrants for the gven length k = m and k = 6 of the current carryng wre. In order to smplfy the analyss, two varables, x and x, are consdered. Correspondngly, the optmzaton problem s smplfed as: Methods FOAFA FOA Deshpande [] Best objectve functon value E4 69 Worst objectve functon value Mean objectve functon value.74 E4.698 E4 Medan objectve functon value E4 NFE 4 45 Tme (second) k x kk max F ( ). ( ), x = π x x π xx subject to : Table 5: Optmal results of the SM problem. π xx k k kk g ( ) = x 4 4π x x kk g ( x) = x >,, x,., x,. 4π x Table 6 demonstrates the result for desgn of arcored solenod, by gvng the, worst and mean values for the objectve functon. On the other hand, Fgure 7 depcts the relaton between the maxmum nductance aganst the number of teratons for FOA before applyng FA and after Applyng FA (.e., the hybrd FOAFA). The result of the proposed FOAFA s superor to those obtaned by [8]. Fgure 8 shows the graphcal representaton of the optmum soluton for the SM and ar cored solenod. It s obvous that the algorthm outperforms the other algorthms for the both problems. We concluded that the ntegrated algorthm of FOA wth FA has mproved the qualty of the founded solutons and also t guarantees the faster converge to the optmal soluton. The FOA s presented n the Methods FOAFA FOA Barba [4] Best objectve functon value Worst objectve functon value Mean objectve functon value Medan objectve functon value NFE 4477 Tme (second) Table 6: Optmal results of the arcored solenod. (a). Crosssecton of approxmately a one pole ptch long secton of SM (b). Cross secton of the solenod and desgn varables Fgure 5: The confguratons of two desgns. ISSN: 9498 SIEC, an open access journal Volume 5 Issue 4

9 Ctaton: Allah RMR (6) Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems. Int J Swarm Intel Evol Comput 5: 4. do:.47/ Page 9 of.8 x 4.8 x Maxmum Force Maxmum Force FOA FOAFA Iteratons Fgure 6: Maxmum force by usng FOA and FOAFA.. FOA CFOAFA Iteratons Maxmum Inductance FOA FOAFA Iteratons Maxmum Inductance FOA FOAFA Iteratons Fgure 7: Maxmum nductance by usng FOA and FOAFA. 8 the optmum soluton FOAFA FOA algorthms proposed by [7,8] 4 The SM problem The arcored solenod. the SM problem and arcored problem Fguge Fguge 8: 8: The The graphcal graphcal representaton representaton of of the the optmum optmum soluton. soluton. frst stage to provde the ntal soluton to optmzaton problem (close to the optmal soluton as possble) followed by FA to enhance the qualty of the soluton. However, because of ts random behavor, FOA may suffer from slow convergence. So we ntroduced the neghborhood search va adaptve radus mechansm Δ (t, that returns a value n the range [, R]. On the other hand, to reach a quck and closer result to optmal soluton, and to mprove the effcency of the FOA, hybrdzaton between FOA and FA was mplemented. In ths subsecton, a comparatve study has been carred out to assess the performance of the proposed FOAFA algorthm concernng the hybrdzaton, closeness to optmal soluton and computatonal tme. On ISSN: 9498 SIEC, an open access journal Volume 5 Issue 4

10 Ctaton: Allah RMR (6) Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems. Int J Swarm Intel Evol Comput 5: 4. do:.47/ Page of the frst hand, pure algorthms suffer from reach to an optmal soluton n a reasonable tme. Also, the snkng nto premature convergence s nevtable n pure algorthms. Consequently, our hybrdzaton algorthm has twofold feature, solvng NPPs n a reasonable tme and avodng the snkng nto the premature convergence. On the other hand, the proposed FOAFA algorthm s hghly compettve when compared wth the other exstng methods n term of calculatng the relatve, absolute error and number of evolutons. Accordngly, t provdes the faclty to search n the negatve and the nonnegatve orthant through adaptve radus mechansm. Fnally, the feasblty of usng the proposed algorthm to handle NPPs has been emprcally approved. Conclusons A novel hybrd algorthm named, FOAFA for solvng the nonlnear programmng problems and engneerng applcatons s proposed. The proposed FOAFA algorthm employed the merts of both frut fly optmzaton algorthm (FOA) and frefly algorthm (FA) and conssts of two phases. The frst one employs a varaton on orgnal FOA employng a new adaptve radus mechansm for explorng the whole scope around the frut fles locatons to overcome the drawbacks of orgnal FOA, whch has been an contnues to be a frutful paradgm for desgnng effectve the nonnegatve orthant algorthms. Moreover, the premature convergence of orgnal FOA degrades ts performance by reducng ts search capablty, leadng to a hgher probablty of beng trapped to a local optmum. Therefore, the second phase ncorporates the FA to update the prevous locatons of frut fles to force FOA jump out of premature convergence, because of ts strong searchng ablty. The hybrd algorthm speeds up the convergence and mproves the algorthm s performance. A careful observaton wll reveal the followng benefts of the proposed optmzaton algorthm. ) It can effcently safeguard the soluton from snkng nto the premature convergence through ncorporatng the FA. ) It can overcome the lack of the exploraton of the FOA whch roamng n the nonnegatve orthant by ntroducng the adaptve radus mechansm. ) It emphaszes the dversty of solutons by ncorporatng the FA that enrches the exploratory capabltes of the proposed algorthm. 4) It compettve when compared wth the other exstng algorthm. 5) It can fnd can fnd the global mnmum for the problems very effcently. 6) It can accelerate the convergence and mproves the algorthm s performance through elapsed low computatonal tme. The future work wll be focused on three drectons: () the applcaton of FOA FA to real world problems; () the extenson of the method to solve the multobjectve problems; and () another applcatons n engneerng. References. Deep K, Dpt (8) A selforganzng mgratng genetc algorthm for constraned optmzaton. Appled Mathematcs and Computaton 98: 75.. He Q, Wang (7) A hybrd partcle swarm optmzaton wth a feasblty based rule for constraned optmzaton. Appled Mathematcs and Computaton 86: Becerra R, Coello CAC (6) Cultured dfferental evoluton for constraned optmzaton. Computer Methods n Appled Mechancs and Engneerng 95: Mousa MM, ElWahed WFA, RzkAllah RM () A Hybrd Ant Colony Optmzaton Approach Based ocal Search Scheme for Multobjectve Desgn Optmzatons. Journal of Electrc Power Systems Research 8: Yang XS (9) Frefly algorthms for multmodal optmzaton, In Stochastc Algorthms: Foundaton and Applcatons. SAGA 9, ecture Notes n Computer Scences 579: Yang XS (8) NatureInspred Metaheurstc Algorthms. unver Press. 7. RzkAllah RM, Elsayed ZM, ElSawy EA () Hybrdzng ant colony optmzaton wth frefly algorthm for unconstraned optmzaton problems. Appled Mathematcs and Computaton 4: ElSawy A, Zak EM, RzkAllah RM () A Novel Hybrd Ant Colony Optmzaton and Frefly Algorthm for Solvng Constraned Engneerng Desgn Problems. Journal of Natural Scences and Mathematcs 6:. 9. ElSawy A, Zak EM, RzkAllah RM () Novel hybrd ant colony optmzatonand frefly algorthm for multobjectve optmzaton problems. Internatonal Journal of Mathematcal 4: 56.. Pan WT () A new frut fly optmzaton algorthm: takng the fnancal dstress model as an example. KnowledgeBased Systems 6: H, Guo S,, Sun J () A hybrd annual power load forecastng model based on generalzed regresson neural network wth frut fly optmzaton algorthm. KnowlBased Syst 7: n SM () Analyss of servce satsfacton n web aucton logstcs servce usng a combnaton of frut fly optmzaton algorthm and general regresson neural network. Neur Comput Appl 7: Han J, Wang P, Yang X () Tunng of PID controller based on frut fly optmzaton algorthm, n: Internatonal Conference on Mechatroncs and Automaton (ICMA) pp: Wang, Zheng X, Wang S () A novel bnary frut fly optmzaton algorthm for solvng the multdmensonal knapsack problem. KnowlBased Syst 48: Bazaraa MS, Shetly CM (979) Nonlnear Programmng Theory and Algorthms. Wley, New York. 6. YoungDoo K, SoonBum K, SeungBo Y, JaeYong K () Convergence enhanced genetc algorthm wth successve zoomng method for solvng contnuous optmzaton problems. Computers and Structures 8: Deshpande AD () Calforna near Drves, Inc, A Study Of Methods To Identfy Constrant Domnance In Engneerng Desgn Problems. MS Thess, Mechancal and Industral Engneerng Department, nversty of Massachusetts, Amherst. 8. Barba PB, Farna M, Savn A () An mproved technque for enhancng dversty n Pareto evolutonary optmzaton of electromagnetc devces. COMPE : Ctaton: Allah RMR (6) Hybrdzaton of Frut Fly Optmzaton Algorthm and Frefly Algorthm for Solvng Nonlnear Programmng Problems. Int J Swarm Intel Evol Comput 5: 4. do:.47/ ISSN: 9498 SIEC, an open access journal Volume 5 Issue 4