Journal of Applied Research and Technology ISSN: Centro de Ciencias Aplicadas y Desarrollo Tecnológico.

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1 Journal of Appled Research and Technology ISSN: Centro de Cencas Aplcadas y Desarrollo Tecnológco Méxco Lu Chun-Lang; Chu Shh-Yuan; Hsu Chh-Hsu; Yen Sh-Jm Enhanced Dfferental Evoluton Based on Adaptve Mutaton and Wrapper Local Search Strateges for lobal Optmzaton Problems Journal of Appled Research and Technology vol. 12 núm. 6 dcembre 2014 pp Centro de Cencas Aplcadas y Desarrollo Tecnológco Dstrto Federal Méxco Avalable n: How to cte Complete ssue More nformaton about ths artcle Journal's homepage n redalyc.org Scentfc Informaton System Network of Scentfc Journals from Latn Amerca the Carbbean Span and Portugal Non-proft academc project developed under the open access ntatve

2 Enhanced Dfferental Evoluton Based on Adaptve Mutaton and Wrapper Local Search Strateges for lobal Optmzaton Problems Chun-Lang Lu* 1 Shh-Yuan Chu 2 Chh-Hsu Hsu 3 and Sh-Jm Yen 4 13 Department of Appled Informaton and Multmeda Chng Kuo Insttute of Management and Health Keelung County Tawan R.O.C. *leucl@ems.cku.edu.tw 24 Department of Computer Scence and Informaton Engneerng Natonal Dong Hwa Unversty Hualen County Tawan R.O.C. ABSTRACT Dfferental evoluton (DE) s a smple powerful optmzaton algorthm whch has been wdely used n many areas. However the choces of the best mutaton and search strateges are dffcult for the specfc ssues. To allevate these drawbacks and enhance the performance of DE n ths paper the hybrd framework based on the adaptve mutaton and Wrapper Local Search (WLS) schemes s proposed to mprove searchng ablty to effcently gude the evoluton of the populaton toward the global optmum. Furthermore the effectve partcle encodng representaton named Partcle Segment Operaton-Machne Assgnment (PSOMA) that we prevously publshed s appled to always produce feasble canddate solutons for solvng the Flexble Job-shop Schedulng Problem (FJSP). Experments were conducted on comprehensve set of complex benchmarks ncludng the unmodal multmodal and hybrd composton functon to valdate performance of the proposed method and to compare wth other state-of-the art DE varants such as jde JADE MDE_pBX etc. Meanwhle the hybrd DE model ncorporatng PSOMA s used to solve dfferent representatve nstances based on practcal data for mult-objectve FJSP verfcatons. Smulaton results ndcate that the proposed method performs better for the majorty of the sngle-objectve scalable benchmark functons n terms of the soluton accuracy and convergence rate. In addton the wde range of Pareto-optmal solutons and more antt chart decson-makngs can be provded for the mult-objectve FJSP combnatoral optmzatons. Keywords: Dfferental Evoluton Wrapper Local Search Partcle Segment Operaton-Machne Assgnment Flexble Job-shop Schedulng Problem. 1. Introducton Optmzaton algorthms have become a useful technque n all-major dscplnes and engneerng applcatons [1 2]. Many practcal engneerng desgn or decson makng problems nvolve sngleobjectve or mult-objectve optmzaton. In sngleobjectve optmzaton the goal s to fnd the best desgn soluton as to the mnmum or maxmum value of the objectve functon. On the contrary the mult-objectve optmzaton gves rse to Paretooptmal solutons [3] because of the nteracton among dfferent conflctng objectves. Dfferental Evoluton (DE) algorthm s a populaton-based and stochastc optmzer frst developed by Storn and Prce [4]. Wth the advantages of smplcty less parameter and robustness the DE algorthm has been gven ncreasng attenton and wdely used n many felds such as data mnng [5] structural optmzaton [6] bogeography [7] and so on [8 9]. DE s consdered the most recent studes for solvng constraned optmzaton problems multobjectve global optmzatons and other complex real-world applcatons. More detals on the stateof-the-art nvestgaton wthn DE can be found n two surveys [10 11] and the references theren. For the classcal DE the settng of three control parameters: populaton sze Np the crossover rate Cr and the scale factor F s very senstve to the parameter settng and the choce of the best parameters s always problem-dependent [12]. In addton for a gven specfc problem t may be better to adopt dfferent parameter settngs durng dfferent generaton stages of the evoluton than use a sngle mutaton strategy wth fxed parameter Journal of Appled Research and Technology 1131

3 settngs [11]. In 2006 Brest et al. [12] presented a self-adaptve method for DE fxes the populaton sze durng the evoluton process whle adaptng the control parameters F and CR assocated wth each ndvdual. The jde reproduces new F and CR values accordng to unform dstrbutons on [0.1 1] and [0 1] respectvely. And expermental results demonstrated that jde performs remarkably better than the classc DE/rand/1/bn. Later Zhang et al. presented DE/current-to-pbest wth optonal archve and controls F and CR n an adaptve manner named JADE [13] to track the hstorcal record of success status for mutaton factors and crossover probabltes wth adaptve parameters n generatons and the external archve to store recently nferor solutons and ther dfference from current populaton provdes promsng drectons toward the optmum. In 2012 Islam et al. proposed the MDE_pBX [14] whch adds a varaton to the classcal DE/current-to-best/1 mutaton scheme by perturbng the current target vector wth the best soluton n a group of randomly selected ndvduals. The crossover operaton s performed between the current donor vector and any other ndvdual from p top-ranked ndvduals n the present generaton. Smulaton results demonstrate that MDE_pBX enhances the ablty of the basc DE for fndng solutons n search space and helps allevatng the tendency of premature convergence or stagnaton. In the recent studes [12 15] varous mutaton and controll parameters settng strateges have been presented for DE algorthm. Although a number of works can advance the search ablty of DE there s stll much room for mprovng the performance of DE. Motvated by these results the modfed mutaton scheme based on MDE_pBX n whch the mutaton operator can be adjusted dynamcally on the soluton searchng status s presented to brng several ndvduals approprately to fnd new possble solutons. Meanwhle a wrapper local search (WLS) strategy va tryng to ncrease or decrease current movng vector by the Cauchy dstrbuton s proposed to mprove the local search ablty and to balance exploraton and explotaton n the search space. Moreover the proposed hybrd DE framework ncorporated wth PSOMA method that we prevously publshed to produce feasble solutons for the mult-objectve Flexble Job-shop Schedulng Problem (FJSP) s desgned for fndng optmal solutons of mult-objectve FJSP. Compettve expermental results are observed wth respect to 15 CEC 2005 benchmark functons for sngle-objectve optmzatons and the three benchmark nstances based on practcal data were employed as mult-objectve FJSP verfcatons. The remander of ths paper s organzed as follows. Secton 2 descrbes the typcal MDE_pBX FJSP and external repostory. The PSOMA scheme adaptve mutaton method WLS and the hybrd DE model are presented n Secton 3. Experment and comparson results are provded n Secton 4. Conclusons remarks are made n Secton Related Works 2.1 Classcal Dfferental Evoluton Algorthm DE algorthm s one of the populaton-based global optmzaton algorthms has two stages ncludng ntalzaton and evoluton. After randomly ntalzs evoluton process evolves from one generaton to the next through mutaton crossover and selecton operatons untl the termnaton crtera are reached. The core of DE algorthm s the mutaton operaton whch uses a weghted random vector n each step to replace the target vector wth the better tral vector n the next generaton. The man steps of the classcal DE are summarzed as follows Intalzaton The DE begns by creatng an ntal populaton of target vectors consstng of parameter vectors are 1 2 D T denoted by X [ x x x ] (1 2 N P ) where s the ndex for ndvduals ndcates the current generaton N p s the populaton sze D s j the dmenson of the parameters and x denotes the j-th component of the -th ndvdual at the -th generaton. The ntal ndvduals are randomly determned wthn a predefned search space consderng the lower and upper bounds of each parameter as follows. x x rand(01) ( x x ) j (12 D) (1) j j j j mn max mn j j where xmn and xmax denote the lower and upper bounds respectvely and rand() s a unformly dstrbuted random number between 0 and Vol. 12 December 2014

4 2.1.2 Mutaton For the DE/rand/1/bn classcal mutaton strategy three dfferent vectors consstng of a base vector ( X ) and two dfference vectors ( X r1 and X r2 r3 ) are randomly chosen from the populatons. The scale factor F s a constant and the effectve range s usually taken from the range between 0.5 and 1 as ponted out n [4]. Mutaton operaton s then carred out by perturbng the base vector va a dfference vector scaled by a scalar factor F. The 1 2 D T donor vector V [ v v v ] (1 2 N P ) s expressed as Eq. 2. V X F( X X ) r1 r2 r3 (2) where r 1 r 2 r 3 are random and mutually dfferent ntegers and they are also dfferent wth the vector ndex Crossover To ncrease the dversty of the solutons the 1 2 D T tral vectoru [ u u u ] (1 2 N P ) s created by means of crossover operaton and s realzed between each par of target vector X and ts correspondng donor vector V. In a tral vector elements are nherted from a donor vector and a target vector. The scheme can be smply formulated for the bnomal unform crossover that s wdely used n the lterature as shown below. j j v f ( rand(01) Cr or j jrand u j D j x otherwse (3) j j j where u v and x are tral donor and target vectors from the -th vector j-th dmenson at -th generaton. C r s a user-defned probablty n the range [0 1]. j rand s a randomly chosen nteger n the range [1 D] whch ensures that the tral vector does not duplcate the target vector Selecton The selecton operaton s acheved from the target and tral vectors by comparng ther ftness values through the objectve functon f () to select the better ndvdual. In case of mnmzaton problems the tral vector and the target vector competes n ther ftness and the wnner has the chance to survve to the next generaton. U f( f( U ) f( X ) X 1 (4) X otherwse. In the current populaton target vector s updated when the newly generated tral vector gets better ftness value than ts target vector; otherwse the target vector s retaned n the populaton. DE algorthm works through a smple cycle of the stages and the pseudo-code of the classcal DE s gven below as Algorthm 1. Algorthm 1. The classcal DE algorthm. 1: Basc settng of parameters for DE. 2: Intalzaton enerate the ntal populaton. 3: Evaluate the ftness for each ndvdual. 4: whle Termnaton condton s not satsfed do 5: Mutaton. 6: Crossover. 7: Selecton. j 8: Boundary constrants for each x X. 9: end whle 10: Output lobal optmum soluton X Best. 2.2 Typcal MDE_pBX Approach Islam et al. developed an adaptve DE algorthm based on novel mutaton and crossover strateges called MDE_pBX [14] for global optmzatons. They presented a p-best crossover operaton to mprove the accuracy n search space by a large extent. Moreover the Cauchy dstrbuton and aussan dstrbuton are also ncorporated nto the generaton for scale factor F and crossover rate C r. The new mutaton operator named DE/current-togr_best/1 scheme s as follows. Journal of Appled Research and Technology 1133

5 V X F( X X X X ) gr_ best r1 r2 (5) wcr rand(01) (13) X gr_ best s the best soluton from 15% of ndvduals selected randomly n current generaton. Then normal bnomal crossover s performed between the donor vector and the randomly selected p-best vector to generate the tral vector at the same ndex. Parameter p s lnearly reduced by generatons n the followng formula. N p 1 p cel( (1 )) (6) 2 max s the current generaton number max s the maxmum number of generaton set up and cel( y ) s the celng functon returnng the lowest nteger greater than ts argument y. Fnally the parameter adaptaton schemes n MDE_pBX s ndependently generated as follows. F Cauchy( F 0.1) (7) m Eq. 7 s a random number sampled from a Cauchy dstrbuton wth locaton parameter F m formulated as Eq. 8 and the scale parameter s 0.1. Locaton parameter F m of Cauchy dstrbuton s ntalzed to be 0.5 and updated at the end of each generaton. F w F (1 w ) mean ( S ) (8) wf m F m F POW F mean rand(01) (9) x ( S ) ( ) (10) S POW F x S F F A weght factor w F s a postve constant number between 0.8 to 1. The meanpow () represents a power mean gven by Eq. 10. The adaptaton of crossover probablty Cr s smlar wth the scale factor F and the detaled descrpton as follows. Cr aussan( Cr 0.1) (11) m Cr w Cr (1 w ) mean ( S ) (12) m Cr m Cr POW Cr mean x ( S ) ( ) (14) S POW Cr x S Cr Cr where wcr s set between 0.9 to 1 Crm s ntalzed to be 0.6 and the aussan dstrbuton as shown n Eq. 11 substtutes for the Cauchy one n Eq FJSP Problem Defnton FJSP [16] s a generalzaton of the classcal JSP n whch operatons are allowed to be processed by any machne from agven set of avalable machnes. It s qute dffcult to acheve an optmal soluton due to the hgh computatonal complexty and well-known a NP-hard problem. Xa et al. [17] proposed a method whch hybrdzed PSO and smulated annealng (SA). It nvolved a varable nerta weght w and adopted a weghted concept to transform trple objectves nto sngle objectve problems. Ho et al. [18] later proposed a method to estmate bounds of dfferent types of schematc may exst correspondng to the Pareto-optmal ftness value but Ho's method dd not deal wth the dversty under the same Pareto-optmal solutons. The three mnmzaton objectves addressed n [19] ncludng the total workload of all machnes the workload of crtcal machne and the completon tme of crtcal job are consdered smultaneously. The problem s to organze the executon of n jobs J ( n) on m machnes M k ( k m) where each job J needs O operatons on the order of restrant and each workng procedure of job can be processed by multple processes of M machnes. The n total number of operatons n all jobs s O t Ot O M j means the collecton of the useable machnes about j-th operaton of -th job M j {12... m} O j k means the j-th operaton of the -th job can use k-th machne p j kmeans the requred tme of j-th operaton of -th job on the k-th machne 1 n1 j o 1 k m. The task s to fnd a set of solutons and three crtera of FJSP are consdered and descrbed as follows: Vol. 12 December 2014

6 (a) The total workload of all machnes F1 p j k (15) where the element p j k denotes the processng tme of j-th operaton of -th job n k-th machne. (b) The workload of the crtcal machne F2 W1 W2 W m max{... } (16) where W k denotes the workload of the k-th machne M k (the summaton of p j k on M k ). (c) The completon tme of the crtcal job F3 CT1 CT2 CT n max{... } (17) where CT s the completon tme of the -th job J. 2.4 External Repostory The man dfference between Sngle-Objectve Problem (SOP) and Mult-Objectve Problems (MOP) s that MOP contans more than one objectve that needs to be accomplshed smultaneously. In [20] the authors presented an external repostory (or archve) to keep the hstorcal record of the non-domnated solutons found along the search process. In Fg.1 the basc external repostory s also adopted n our proposed hybrd DE algorthm for solvng the mult-objectve FJSP. The reservaton process of non-domnated solutons s descrbed as follows. Fgure 1. Reservaton process of external repostory. Case 1. If an external repostory s empty any non-domnated soluton found currently s nserted. Case 2. If a soluton s domnated by any ndvdual n external repostory the soluton wll be dscarded. Case 3. If a soluton s non-domnated by all ndvduals or equaled by any ndvdual n external repostory the soluton wll be stored. Case 4. If a soluton domnates at least one soluton n external repostory those domnated solutons are removed from the external repostory and ths soluton wll be stored. All new solutons are compared wth solutons n the external repostory and one of four cases s corresponded each tme. N S means one of new solutons. k n S k means there exsts k solutons n external repostory. S k - means a modfed S k after some domnated solutons are removed by N S. 3. The Proposed Methods As above mentoned to the adaptve DE varants the parameters updated teratvely accordng to ther successful experence n the generatons such as JADE [13] MDE_pBX [14]. In ths work the proposed DE model based on self-adaptve mutaton (SAM) and wrapper local search (WLS) to enhance the performance of DE. In addton the effectve PSOMA [21] that we prevously publshed s successfully merged nto DE model for solvng the mult-objectve FJSP. 3.1 Self-adaptve Mutaton (SAM) Approach The orgnal DE mutaton s DE/rand/1/bn n [4]. Islam et al. proposed the mutaton of MDE_pBX although the effect of the method performs better results than many other algorthms ths dversty of soluton searchng scheme s gettng premature convergence at local optma wthn a smaller number of generatons. Takng nto consderaton of these facts and to overcome the lmtatons of ths feature the extenson of modfed mutaton searchng strategy n [14] whch s abbrevated as SAM approach can be shown below. V X F( X X X X ) gr_ better r1 r2 (18) Journal of Appled Research and Technology 1135

7 now max max mn max w w ( w w ) (19) In Eq. 18 X s known as the target vector V s known as the donor vector the scalng factor F s a postve scalng parameter for dfference vectors. The X and X r1 are two dstnct vectors pcked r2 up randomly from the current populaton and none of them s equal to X gr _ better or the target vector. The X gr _ better s dynamcally chosen from top w% ndvduals of each populaton. The lnearly decreasng nerta weght s set as Eq. 19 where now s the current teraton max s the user-defned maxmum teraton and the pre-defned lower and upper bounds of w descrbed as wmn w wmax. utlzed these random wrapper-based selected dmensons ( 1 ndcates selected 0 denotes excluded) to dentfy a sutable balance tradeoff of DE search and local search. It can save much tme than one by one the sngle dmenson search [23] and wll fne tune the searchng drecton for fndng the global best soluton. A selected dmenson Decreased search drecton Orgnal search drecton Increased search drecton Ths new proposed self-adaptve mutaton operator s a varant of the MDE_pBX. It dynamcally uses the best of a group (whose sze s w% of the lnearly decreasng populaton sze) of randomly selected solutons from current generaton to perturb the parent vector and unlke n [14] that the X better always pcks the best vector from the entre populaton fxed-rato to perturb the target vector. Ths new smple self-adaptve mutaton approach can drve the populaton to the better drecton nstead of convergence to the best ndvdual n the teraton and help DE not to fall nto local optmum quckly n smaller generaton. 3.2 Wrapper Local Search (WLS) Strategy Accordng to related works t can be found that classcal DE can perform well performance on wdely search for explorng unsearched soluton space but weakly on searchng depth. In 2010 [22] we publshed the effcent wrapper-based hybrd model for solvng the bomedcal problem and s capable of producng hgh predcton accuracy and fewer number of features selecton smultaneously. In ths study the wrapper-based feature selecton framework s properly adopted to enhance the local search performance for DE named Wrapper Local Search (WLS) strategy s nvolved to adjust the scale of movng vector va tryng to ncrease or decrease current movng vector by the Cauchy dstrbuton. In Fgure 2 the proposed WLS method Fgure 2. The WLS for one of selected dmensons. The pseudo-code of the proposed DE algorthm s descrbed as Algorthm 2. Algorthm 2. The proposed DE model combned wth SAM and WLS strateges. 1: Basc settng of parameters for the DE model. 2: Intalzaton enerate the ntal populaton. 3: Evaluate the ftness for each ndvdual. 4: whle Termnaton condton s not satsfed do 5: for =1 to N p do 6: for j=1 to D do 7: Mutaton wth SAM approach. 8: Crossover. 9: end for 10: end for 11: for =1 to N p do 12: Evaluate the offsprng. 13: Selecton. 14: end for 15: Apply WLS on the better selected ndvduals. 16: end whle 17: Output lobal optmum soluton. 3.3 PSOMA Scheme The effectve partcle encodng representaton that we prevously publshed and detaled n [21] each 1136 Vol. 12 December 2014

8 dmenson contans three components: nteger part (machne selecton) decmal part (prorty order) and real-value number (operaton number). In Fg.3 s the structure of PSOMA representaton on each dmenson. Ths encodng representaton s flexble enough for solvng the FJSP to satsfy the precedence constrants and operatons n each job by usng the structure of the proposed DE model wth real-value number. Step 2. O 1.1 s assgned to M 3. Operaton set O 2 = { O 1.1 O 2.2 O 3.1 }. Assgnment set C 2 = { O M 3 O M 2 O M 3 }. Max(O 1.1 O 2.2 O 3.1 )=Max( )=0.81. Then the { O M 3 } s assgned from C 2. Step 3. O 2.2 s assgned to M 2. Operaton set O 3 = { O 1.2 O 2.2 O 3.1 }. Assgnment set C 3 = { O M 1 O M 2 O M 3 }. Max(O 1.2 O 2.2 O 3.1 )=Max( )=0.68. Then the { O M 2 } s assgned from C 3. Fgure 3. Three components of PSOMA structure. Jobs Machnes O 1.1 O 1.2 O 1.3 O 2.1 O 2.2 O 3.1 M M M Table 1. The example of operaton schedules for FJSP. An example of FJSP (3 jobs 3 machnes and 6 operatons) s consdered and llustrated n Table 1. Three Jobs J 1 J 2 and J 3 need to be processed by at least one of three machnes M 1 M 2 and M 3. Each job contans several operatons such as Job 1 s splt nto O 1.1 O 1.2 O 1.3 ; Job 2 s splt nto O 2.1 O 2.2 etc. The executng tmes for each machne to execute each job s gven such as O 1.1 assgned to M 1 s 3 unts; O 1.2 assgned to M 2 s 4 unts etc. O 1.1 O 1.2 O 1.3 O 2.1 O 2.2 O Fgure 4. A possble canddate encodng of the ndvdual. To explan the PSOMA scheme one possble canddate encodng representaton of the ndvdual shown as Fgure 4 to gve the detaled descrpton of the decodng procedure for FJSP as follows. Step 1. O 2.1 s assgned to M 1. Operaton set O 1 = { O 1.1 O 2.1 O 3.1 }. Assgnment set C 1 = { O M 3 O M 1 O M 3 }. Max(O 1.1 O 2.1 O 3.1 )=Max( )=0.92. Then the { O M 1 } s assgned from C 1. Step 4. O 3.1 s assgned to M 3. Operaton set O 4 = { O 1.2 O 3.1 }. Assgnment set C 4 = { O M 1 O M 3 }. Max(O 1.2 O 3.1 )=Max( )=0.37. Then the { O M 3 } s assgned from C 4. Step 5. O 1.2 s assgned to M 1. Operaton set O 5 = { O 1.2 }. Assgnment set C 5 = { O M 1 }. Max(O 1.2 )=Max(0.26)=0.26. Then the { O M 1 } s assgned from C 5. Step 6. O 1.3 s assgned to M 2. Operaton set O 6 = { O 1.3 }. Assgnment set C 6 = { O M 2 }. Max(O 1.3 )=Max(0.53)=0.53. Then the { O M 2 } s assgned from C 6. As shown n Fgure 5 the operatons of all jobs are assgned to each machne done and one of the feasble antt chart soluton s provded. The precedence constrants are {O 1.1 before O 1.2 before O 1.3 } from job 1 {O 2.1 before O 2.2 } from job 2 and {O 3.1 } from job 3. The operaton order should be preserved by the dfferent machne selecton. It s obvous that each schedulng canddate soluton generated by any encodng representaton based on PSOMA should be feasble. The pseudo-code of the proposed hybrd DE model wth PSOMA scheme for solvng FJSP s shown as Algorthm 3. Tme(s) Machne M 1 O 2.1 O 1.2 M 2 O 2.2 O 1.3 M 3 O 1.1 O 3.1 Fgure 5. antt chart of operaton-machne assgnments. Journal of Appled Research and Technology 1137

9 Algorthm 3. The proposed hybrd DE model wth PSOMA scheme for FJSP. 1: Basc settng of parameters for the DE model. 2: Intalzaton enerate the ntal populaton va. 3: Evaluate the ftness for each ndvdual. 4: whle Termnaton condton s not satsfed do 5: for =1 to N p do 6: for j=1 to D do 7: Mutaton wth SAM approach. 8: Crossover. 9: end for 10: end for 11: Decodng by PSOMA for each ndvdual. 12: for =1 to N p do 13: Evaluate the offsprng. 14: Selecton. 15: end for 16: Apply WLS on the better selected ndvduals. 17: Compare antt charts wth solutons dversty. 18: Store mult-objectve non-domnated solutons and antt charts to External Repostory. 19: end whle 20: Output lobal optmum soluton(s). 4. Expermental Results 4.1 Sngle-objectve numercal benchmarks In order to verfy the performance of our approach a selected set of standard test functons from the specal sesson on real-parameter optmzaton of the IEEE Congress on Evolutonary Computatons CEC 2005 [24] were adopted for testng to related works. The global optmum (equal to functon bas) search range ntalzaton range and functon types of each test functon are llustrated n Table 2. These functons 1 to 5 are unmodal functons 6 to 12 are basc multmodal functons 13 to 14 are expanded multmodal and the hybrd composton s functon 15. In the experments 15 numercal test functons wth 30 dmensons are conducted for comparng the proposed method wth other fve related works as jde JADE and MDE_pBX reported n [14]. The ntal populaton sze s set as 100. The ftness evaluaton (FEs) s All the DE varants algorthms are mplemented on Matlab 2013a. The mean values and standard devaton are calculated. The mean and standard devaton of error values are optmzed and recorded by 25 ndependent runs. Table 2. The numercal benchmarks on CEC 2005 [23]. The experment results of 15 test functons wth 30 dmensons numercal functons are lsted n Table 3 to Table 5. The best results among the four DE approaches are shown n bold. The convergence characterstcs of the proposed method and related works are shown n Fgure 6-1 to Fgure Table 3. Expermental results of F 1 to F 5 test functons. Table 4. Expermental results of F 6 to F 10 test functons Vol. 12 December 2014

10 Fgure 6-1. Convergence curves of F 1 functon. Fgure 6-5. Convergence curves of F 5 functon. Fgure 6-2. Convergence curves of F 2 functon. Fgure 6-6. Convergence curves of F 6 functon. Fgure 6-3. Convergence curves of F 3 functon. Fgure 6-7. Convergence curves of F 7 functon. Fgure 6-4. Convergence curves of F 4 functon. Fgure 6-8. Convergence curves of F 8 functon. Journal of Appled Research and Technology 1139

11 Fgure 6-9. Convergence curves of F 9 functon. Fgure Convergence curves of F 13 functon. Fgure Convergence curves of F 10 functon. Fgure Convergence curves of F 14 functon. Fgure Convergence curves of F 11 functon. Fgure Convergence curves of F 12 functon. Fgure Convergence curves of F 15 functon. From the smulatonal results the proposed hybrd DE model gets better results such as F 2 F 5 F 11 F 12 F 14 and F 15 functons and F 1 archve the same results whch the real optmum solutons are found. In summary accordng to the results shown n Tables 3 to 5 the proposed DE model s hghly compettve to the abovementoned state-of-the-art DEs. The results of the proposed DE are better than or comparable to those of the state-of-the-art DEs n terms of the qualty of the fnal solutons Vol. 12 December 2014

12 4.2 Mult-objectve FJSP benchmarks To llustrate the effectveness and performance of the hybrd DE for the mult-objectve verfcaton three FJSP representatve benchmarks whch are problem have been conducted and to compare wth other related works such as the PSO-SA [17] and MOEA-LS [18]. For each problem the obtaned results are reported n table contans three objectves: F 1 (total workload) F 2 (crtcal machne workload) F 3 (makespan) are mentoned n Secton 2. The solutons found from three methods are shown n Table 6 to Table 8. SA and more dversty antt chart solutons can be obtaned by the hybrd DE compared wth the MOEA-LS. From the smulaton results of antt chart dversty the proposed hybrd DE approach performs sgnfcantly better than the other two PSO-SA and MOEA-LS methods n solvng all benchmarks. Afterward three of the antt charts from the soluton ( ) for problem 8 8 are exhbted solutons dversty n Fgure 7 to Fgure 9 as gvng an llustraton. Fgure 7. Soluton ( ) wth antt Chart I. Table 6. Expermental results of the FJSP problem 8 8. Table 7. Expermental results of the FJSP problem Fgure 8. Soluton ( ) wth antt Chart II. Table 8. Expermental results of the FJSP problem For example two new non-domnated solutons ( ) and ( ) can be obtaned by the proposed hybrd DE compared wth the PSO- Fgure 9. Soluton ( ) wth antt Chart III. Journal of Appled Research and Technology 1141

13 5. Concluson In ths paper an mproved Dfferental Evoluton hybrdzed adaptve mutaton (SAM) and wrapper local search (WLS) s proposed to mprove both sngle-objectve and mult-objectve optmzaton performances of DE and gude the evoluton of the populaton toward the global optmum. Meanwhle the WLS can dsturb ndvduals to help ndvduals avod trap nto local mnmum n evoluton progress. Compare the proposed method wth the publshed algorthms the expermental results show that the proposed method exhbts better performance for solvng most of the test functons from CEC 2005 benchmarks for the sngleobjectve verfcatons. In addton the wde range of Pareto-optmal solutons and the more antt charts dversty can be obtaned for the multobjectve FJSP problems. Acknowledgments Ths work was partally supported by the Natonal Scence Councl Tawan under rant NSC E References [1] Y.C. Ln Mxed-nteger constraned optmzaton based on memetc algorthm Journal of Appled Research and Technology Vol. 11 No. 2 pp [2] M. Nazr et al. PSO-A based optmzed feature selecton usng facal and clothng nformaton for gender classfcaton Journal of Appled Research and Technology Vol. 12 No. 1 pp [3]-K. Deb Mult-Objectve Optmzaton Usng Evolutonary Algorthms John Wley [4] R. Storn and K. Prce Dfferental evoluton - A smple and effcent heurstc for global optmzaton over contnuous spaces Journal of lobal Optmzaton Vol. 11 pp [5] B. Alatas et al. MODENAR: Mult-objectve dfferental evoluton algorthm for mnng numerc assocaton rules Appled Soft Computng Vol. 8 No. 1 pp [6] K. Satosh et al. Dfferental evoluton as the global optmzaton technque and ts applcaton to structural optmzaton Appled Soft Computng Vol. 11 pp [7] Boussaïd et al. Two-stage update bogeographybased optmzaton usng dfferental evoluton algorthm Computers & Operatons Research Vol. 38 pp [8] Hmmat Sngh and Laxm Srvastava Modfed Dfferental Evoluton algorthm for mult-objectve VAR management Electrcal Power and Energy Systems Vol. 55 pp [9] Wenyn ong et al. Engneerng optmzaton by means of an mproved constraned dfferental evoluton Computer Methods n Appled Mechancs & Engneerng Vol. 268 pp [10] F. Ner and V. Trronen Recent advances n dfferental evoluton: A survey and expermental analyss Artf. Intell. Rev. 33 pp [11] S. Das and P.N. Suganthan Dfferental Evoluton: A survey of the state-of-the-art IEEE Trans. on Evolutonary Computaton Vol. 15 No. 1 pp [12] J. Brest et al. Self-adaptng control parameters n dfferental evoluton: A comparatve study on numercal benchmark problems IEEE Trans. on Evolutonary Computaton Vol. 10 No. 6 pp Vol. 12 December 2014

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