Modified Bat Algorithm for the Multi-Objective Flexible Job Shop Scheduling Problem

Transcription

1 Avalable onlne at vol., no. 7, November 07, pp DOI: 0.90/jpe.7.07.p.9990 Modfed Bat Algorthm for the Mult-Objectve Flexble Job Shop Schedulng Problem Haodong Zhu a,b, Baofeng He a, Hongchan L b,* a School of Electronc Informaton Engneerng, Sas Internatonal Unversty, Zhengzhou Unversty, Xnzheng, Henan,0, Chna b School of Computer and Communcaton Engneerng, Zhengzhou Unversty of Lght Industry, Zhengzhou, Henan, 000, Chna Abstract In ths paper, a modfed bat algorthm (MBA) s proposed for solvng the mult-objectve flexble job shop schedulng problem. Three dfferent producton performance ndcators are consdered whch are the makespan, the total workload of machnes and the crtcal machne workload. Frstly, to make the algorthm adaptve to the problem, the convertng approaches are presented to mplement the converson between the contnuous poston vector and the dscrete schedulng code. Secondly, an ntalzaton scheme combnng heurstcs and random rule s ntroduced to ensure good qualty and dversty of the ntal populaton. Furthermore, fve neghborhood structures are desgned based on ndvdual postons. Then, a local search algorthm s embedded nto the BA to enhance the local searchng ablty. Fnally, smulaton results demonstrate the feasblty and effectveness of our proposed algorthm. Keywords: flexble job shop schedulng; mult-objectve combnatoral optmzaton; bat algorthm; local search (Submtted on July, 07; Revsed on September 8, 07; Accepted on October 0, 07) 07 Totem Publsher, Inc. All rghts reserved.. Introducton orkshop schedulng consders a reasonable resource allocaton to perform a collecton of producton tasks. It s vewed as an mportant decson-makng process n most manufacturng ndustres. Job shop schedulng problem (JSSP) s one of the most dffcult combnatonal optmzaton problems n the area of producton schedulng that has been proved to be a NP-hard problem. For a classcal JSSP, each job s processed wth a fxed and known processng order through all machnes to mnmze a certan crteron. As a generalzaton verson of the classc job shop schedulng problem (JSSP), the flexble job shop schedulng problem (FJSSP) provdes a closer approxmaton to real-lfe producton. In FJSSP, the restrcton of machne avalablty s reduced; that s to say, each operaton may be processed on more than one machne. Compared wth the classc JSSP, the machne assgnment to operatons should be consdered. Therefore, the complexty of the FJSSP s ncreased n a great extent, whch has already been proved to be NP-hard. In recent decades, the FJSSP has lots of researchers attenton. Due to ts complexty, meta-heurstc algorthms have become a practcal alternatve of solvng technques. Sad-Mehrabad and Fattah [7] proposed a tabu search algorthm to solve the FJSSP wth the objectve of mnmzng the makespan. Yazdan et al. [] presented a parallel varable neghborhood search algorthm to deal wth the FJSSP. Bagher et al. [] developed an artfcal mmune algorthm based on ntegrated approach. ang et al. [0] proposed an artfcal bee colony algorthm for solvng the FJSSP. Rahmat and Zandeh [] ntroduced a bogeography-based optmzaton algorthm for the FJSSP. Yuan et al. [] desgned a novel hybrd harmony search algorthm to solve the flexble job shop schedulng problem wth the crteron of makespan mnmzaton. As researchers probed more deeply and wdely, they started to study the mult-objectve FJSSP. Kaplanoğlu [8] presented an object-orented (OO) approach for mult-objectve FJSP along wth smulated annealng optmzaton algorthm. Shvasankaran et al. [8] proposed a hybrd sortng mmune smulated annealng technque (HSISAT) for solvng * Correspondng author. Tel.: ; fax: E-mal address: lhongchan@.com.

2 000 Haodong Zhu, Baofeng He, Hongchan L the FJSSP. Xa et al. [] developed a hybrd algorthm (PSO+SA) for the mult-objectve FJSSP wth the crteron to mnmze the makespan, the total workload of machnes and the workload of the crtcal machne. ang et al [] proposed a hybrd algorthm (ECO+TS) by combnng the electon campagn optmzaton and tabu search. Gao et al. [] proposed a Pareto-based groupng dscrete harmony search algorthm to deal wth the flexble job shop schedulng problem consderng two objectves: makespan and the mean of earlness and tardness. L et al. [0] developed a dscrete artfcal bee colony algorthm to solve the mult-objectve FJSSP wth mantenance actvtes. In ths paper, we proposed a modfed bat algorthm (MBA) for solvng the mult-objectve flexble job shop schedulng problem. The basc bat algorthm s a new nature-nspred meta-heurstc algorthm, whch s desgned by Yang on the bass of the echolocaton behavor of bats []. Snce t has been proposed, bat algorthm has been used for solvng varous optmzaton problems [,,]. However, n the manufacturng feld, bat algorthm s seldom used for the producton schedulng problem. Marchelvam and Prabaharam [] proposed a bat algorthm for the hybrd flow shop schedulng problem. Marchelvam et al. [] dscussed the multstage hybrd flow shop schedulng problems usng the bat algorthm. Luo et al. [] developed a dscrete bat algorthm to solve the permutaton flow shop schedulng problem. Tosun and Marchelvam [9] ntegrated the local search nto the bat algorthm to solve the permutaton flow shop schedulng problems wth the makespan crteron. In those studes, varous schedulng problems n flow shops are consdered. As far as we know, the bat algorthm s frst ntroduced to deal wth the mult-objectve FJSSP n ths paper. The man contrbuton of ths paper s to derve a modfed bat algorthm (MBA) to solve the mult-objectve flexble job shop schedulng problem. Frst, representaton methods of schedulng soluton and ndvdual poston vector are desgned. Then, the converson mechansm between the poston vector and the schedulng scheme s presented to make the bat algorthm adaptve to dscrete optmzaton algorthm. To mprove the qualty of the fnal soluton, we propose a populaton ntalzaton scheme and a varable neghborhood search algorthm. The computatonal results demonstrate that our proposed MBA s feasble and vald for the mult-objectve FJSSP.. Problem Formaton In a FJSSP, n jobs and m machnes are consdered n the workshop. Jobs are ndependent wth each other and have the same prortes. Each job conssts of a certan sequence of operatons whose processng tmes are determned by the machne assgnment. The objectve of the FJSSP ams to mnmze three ndcators, whch are makespan ( ), crtcal machne max workload ( ) and total workload of all machnes ( ). For such a system, some assumptons should be consdered as follows: () All machnes and jobs are avalable at tme 0. () Each machne can perform only one operaton smultaneously. () The process of each operaton can not be nterrupted once t starts. () Operatons of each job must be processed after ts predecessor s completed. () Setup tme of each machne s neglgble. To facltate the establshment of the mathematcal model, some symbols and varables are defned. n : the number of jobs; : the number of machnes; : the number of operatons of job ; m J O j : the th operaton of job ; : the processng tme of O j on machne k ; p jk S j j : the start tme of O j C j : the complete tme of ; O j ; k : the workload on the machne k ; : a large postve number; x jk : 0- varable, f O j s processed on machne k, x jk =; otherwse, x jk =0; yj j k : 0- varable, f O j s processed on machne k pror to Oj, yj j k =; otherwse, yj j k =0.

3 Modfed Bat Algorthm for the Mult-Objectve Flexble Job Shop Schedulng Problem 00 mn mn(max( )) () Cmax C j mn mn( ) () T m k max k k mn mn(max( )) () m s.t. Cj Sj xjk pjk,,,, n; j,,, J () k S( j) Cj 0,,,, n; j,,, J () S ( y ) C,,,,, n; j, j,,, J ; k,,, m () j j j k j S y C,,,,, n; j, j,,, J ; k,,, m (7) j j j k j m xjk,,,, n; j,,, J (8) k x 0,,,,, n; j,,, J ; k,,, m (9) jk y 0,,,,,, n; j, j,,, J ; k,,, m (0) j j k Equatons ()-() refers to the optmzaton objectve; constrant () guarantees that no preempton s allowed; constrant () ensures the precedence relatonshps between operatons; constrant () and (7) show that every machne can only process one operaton at a tme; constrant (8) presents that each operaton cannot be assgned to another machne once t starts; equatons (9) and (0) gves 0- varables. For solvng the mult-objectve optmzaton problem, there exst many approaches, whch can be classfed nto three types [9]: () the transformaton approach, whch transforms the mult-objectve problem nto a mono-objectve one by ntroducng some weght coeffcents; () the non-pareto approach, whch treats each objectve n a separated way; () the Pareto approach, whch s based on the Pareto optmzaton concept. In ths study, the objectve functon s determned by followng the frst approach, whch s consdered as the weghted sum of the three objectves,.e., mn f = w w w max, w w w.. Basc Bat Algorthm The prncple of the basc bat algorthm orgnated from the natural bat s echolocaton behavor. In the algorthm, each ndvdual s deemed as a mcro-bat n a flock. On the premse of some approxmate or dealzed rules of the actual behavor of bats, the optmzaton objectve s obtaned by contnuously updatng the ndvdual poston and velocty vector based on the varaton of pulse frequency, pulse emsson rate and pulse loudness. The steps of the basc bat algorthm are lsted below. Step. Intalze the populaton, set parameters and the termnate condton. Step. Evaluate each ndvdual and fnd the best ndvdual poston vector. Step. Update the ndvdual poston and velocty vectors followng Equatons ()-(), where respectvely the mnmum and maxmum values of the pulse frequency poston and velocty vectors of bat n the t th teraton, and [0,]. mn max mn x * fr, t x and t v fr mn and fr max are express the ndvdual fr fr ( fr fr ) () v v ( x x ) fr () t t t * x x v () t t t Step. For each ndvdual, a random number rand [0, ] s generated. If rand s greater than the pulse emsson rate r rand r,.e.,, a local search strategy s performed to the current best ndvdual poston vector to obtan a new poston x. Step. Evaluate the new ndvdual, and generate a random number rand [0, ]. If rand s smaller than the loudness *,.e., rand A, and the objectve value f( x ) f( x ), then s accepted, the pulse emsson rate and loudness A are subsequently updated followng Equatons () and (). and are constants whch determne A x r

4 00 Haodong Zhu, Baofeng He, Hongchan L the varaton of r and A. A A t t () t r 0 r ( exp( t)) () Step. Update the current best ndvdual poston vector. If the termnate condton s met, go to Step 7, otherwse, go to Step. Step 7. Termnate the procedure.. Modfed Bat Algorthm.. Schedulng Soluton Representaton In a FJSSP, machne assgnment and operaton permutaton should be consdered. The former ams to assgn an approprate machne to each operaton, whle the latter attempts to obtan a process sequence on each machne by mantanng the precedence constrants of operatons. Therefore, n ths study, a schedulng soluton conssts of two sectons, whose length q s equal to the total number of operatons. The representaton method s shown n Fgure, where the frst half secton s the operaton permutaton, and the second half s the machne assgnment. As seen from Fgure, three jobs are nvolved, each of whch contans two operatons. In the frst secton, operatons of the same job have the same element values,.e., the second means the nd operaton of Job. In the second secton, each element value represents the code of the machne assgned to the correspondng operaton. * x. Indvdual Poston Vector.. Poston Vector Representaton O O O O O O O O O O O Fgure. Schedulng soluton representaton In the proposed MBA, each ndvdual poston vector s stll a mult-dmensonal real number vector,.e., x= x(), x(),, x( l), where x ( j) [, ], j,,, l. Accordng to the representaton of the schedulng scheme, the poston vector s also composed of two sectons. The sze of the vector equals the double of the total number of operatons n the workshop,.e., l q. Fgure shows the representaton of an ndvdual poston vector. The frst half secton descrbes the nformaton of operaton sequence, whle the second half secton shows the nformaton of machne assgnment. In the two sectons, element values are stored accordng to the gven order, as shown n Fgure. O O O O O O O O O O O O O Fgure. Indvdual poston vector.. Convertng the Poston Vector to the Schedulng Soluton Due to the dscrete characterstcs of the FJSSP, the frst ssue n our algorthm s how to establsh the mappng relatonshp between the ndvdual poston vector and the schedulng soluton. The converson mechansm s shown below. () For the frst half secton, the real number poston vector s supposed to be transformed nto an operaton permutaton. Frst, the element values are rearranged n ascendng order. Accordng to the new order, the operaton permutaton can be obtaned by reorderng the operaton codes accordng to the real numbers. Fgure shows an example of the converson process.

5 Modfed Bat Algorthm for the Mult-Objectve Flexble Job Shop Schedulng Problem 00 Operaton name Indvdual poston O O O O O O Operaton code Indvdual poston Correspondng operaton Ascendng order Operaton permutaton Fgure. The converson from an ndvdual poston vector to an operaton permutaton () For the second half secton, the real number vector wll be transformed nto a machne assgnment. The converson formula s based on the approach proposed by Yuan et al. []. In Equaton (), zh ( ) x( h q) u( h) round ( ( z( h) ) ), h q () denotes the sze of the alternatve machne set of the operaton correspondng to the n the ndvdual poston vector. u ( h) [, z( h )] s the ndex of the selected machne n the alternatve machne set. s the functon that rounds the number to the nearest nteger.. Intal Schedulng Soluton h+ q th element round() The populaton of ntal schedulng solutons s a crucal factor whch affects the convergence speed and the soluton qualty n a great extent. Accordng to the two sub-problems n FJSSP, ntal schedulng solutons are generated by consderng for machne assgnment part and operaton sequencng part... Machne Assgnment Intalzaton The machne assgnment component consders how to assgn an approprate machne to each operaton. By consderng both the problem features and the objectves, global selecton (GS), local selecton (LS) and random selecton (RS) are adopted whch are proposed by Zhang et al. [7]. The GS could better explore the search space by acqurng varous ntal assgnments n dfferent runs. The LS could fnd the machne wth the shortest processng tme n alternatve machne set of each job. To ensure the dversty of ntal populaton, the RS s used to generate ntal assgnments. In ths paper, 0% of ntal solutons are generated by GS, 0% by LS, and 0% by RS... Operaton Sequencng Intalzaton The operaton sequencng component consders how to sequence the operatons assgned to each machne and determne the start/completon tme of each operaton. Fve dspatchng rules are used as follows: Shortest Processng Tme (SPT): The operaton wth the mnmal processng tme has the hghest prorty to be processed. Longest Processng Tme (LPT): The operaton wth the maxmal processng tme has the hghest prorty to be processed. Most ork Remanng (MR): The job wth the maxmal total processng tme has the hghest prorty to be processed. Most Operaton Remanng (MOR): The job wth the most remanng operatons has the hghest prorty to be processed. Random Rule (RR): The operaton sequence s obtaned by a random permutaton of operatons of all jobs... Populaton Intalzaton As mentoned above, the schedulng soluton conssts of two parts: machne assgnment and operaton sequencng. To ntal the populaton of the algorthm, the machne assgnment components are frst generated by usng the methods n Secton... For each machne assgnment, a predefned number of operaton permutatons are randomly generated accordng to the above

6 00 Haodong Zhu, Baofeng He, Hongchan L fve rules. Then, the best combnaton of the machne assgnment and ts canddate operaton permutaton s set to be an ntal schedulng soluton and added to the ntal populaton. The teraton s repeated untl all the ntal schedulng solutons have been generated... Convertng the Schedulng Soluton to the Poston Vector By consderng the searchng mechansm of the bat algorthm, the generated ntal schedulng schemes should be converted to contnuous poston vectors. Ths converson also conssts of two sectons n Secton.. () For the frst part related to operaton sequence, the converson process s llustrated n Fgure. Frstly, 0.l real numbers are randomly generated n [, ]. Then these numbers are rearranged n an ascendng order and corresponded to operatons n the ntal permutaton. The ndvdual poston vector s determned by reorderng the real numbers accordng to the operaton codes. Random number Ascendng order Intal permutaton Operaton code Indvdual poston Fgure. The converson from an ntal schedulng soluton to an ndvdual poston vector () For the second part, the converson formula (7) proposed n [] s used, whch s n fact an nverse verson of Equaton (). But when zh ( ) =, xh ( ) should be random selected n [, ].. Local Search ( u( h) ), z( h) x( h q) zh ( ) x( h q) [, ], z( h) To further mprove the local search capacty, a local search (LS) algorthm s performed on the current best ndvdual of MBA durng the searchng process. In the LS, fve neghborhood structures are systematcally changed to make the algorthm escape from the local optma... Neghborhood Structure The neghborhood structures are desgned for the searchng process as below. () Operaton permutaton neghborhood Neghborhood N : Randomly select two elements e and e correspondng to dfferent jobs n the poston vector, and then swap e wth e. (7) Neghborhood N : Randomly select two elements then nsert e before e. e and e correspondng to dfferent jobs n the poston vector, and Neghborhood N : Randomly select two elements e and e correspondng to dfferent jobs n the poston vector, and then nverse the tems between e and e. () Machne assgnment neghborhood Neghborhood N : Randomly select an element n the second secton of the canddate poston vector, whch corresponds to an operaton wth more than one alternatve machne. A machne s selected from alternatve machnes to replace the current one. A new real number wll be regenerated by Equaton (7) to replace the current value of the selected element.

7 Modfed Bat Algorthm for the Mult-Objectve Flexble Job Shop Schedulng Problem 00 Neghborhood : Randomly select an element n the second secton of the canddate poston vector, whch corresponds to an operaton wth more than one alternatve machne. The current machne for the selected operaton wll be replaced by the one wth the shortest processng tme among alternatve machnes. A new real number wll be regenerated by Equaton (7) to replace the current value of the selected element... Step of the LS N The local search s constructed by randomly selectng one approach from the three operaton permutaton neghborhoods and one method from the machne assgnment neghborhoods. The procedure starts from a gven soluton and stops when the maxmum teraton s met. The procedure of the local search s gven as below. Step. Intalzaton. Acqure the ntal poston vector Step. whle max x, set the termnate condton max and 0 x : Randomly perform an operaton permutaton neghborhood and a machne assgnment neghborhood to f f( x) f( x ) then end f x: x end whle Step. Obtan the local optma x.. x. Updatng Method of Pulse Emsson Rate and Loudness The ntal pulse emsson rate r s set to be a postve and small value. th the teraton process conductng, r wll be ncreased to. In ths paper, the updatng method of r follows Equaton (8) wth a curve smlar wth the one of Sgmod functon [9]. 0 t r t t r (8) max 0 ( ) ( exp( ( ) )) tmax By ths formula, the algorthm may explot near the current best ndvdual poston wth a large probablty to speed up the convergence process n the early teraton. In the later searchng stage, the dversty of the algorthm can be guaranteed to avod the premature [9]. For the loudness, the updatng method s expressed by Equaton (9), where denotes the objectve value of ndvdual, f max and A f mn are the maxmum and the mnmum objectve values n the current populaton. f. Steps of MBA A f f max f mn f mn (9) The detaled steps of MBA are lsted below. Step. Intalzaton. Set related parameters of the algorthm. * Step. Generate the ntal populaton, evaluate each ndvdual, and fnd the current best ndvdual poston vector x. Step. Update the ndvdual poston and velocty accordng to Equatons ()-(). Step. For each ndvdual, f rand r, the local search s performed to the current best ndvdual to generate a new poston x. * Step. Evaluate the new ndvdual, f rand A and f( x ) f( x ), then x s accepted, the pulse emsson rate r and loudness A are subsequently updated followng Equatons (8) and (9). Step. Update the current best ndvdual poston vector, and perform the local search to t to acqure a new one Step 7. Check the termnate condton t max. If t s met, go to Step 8, otherwse, go to Step. Step 8. End the procedure. * x.

8 Machne 00 Haodong Zhu, Baofeng He, Hongchan L Experment Valdatons. Parameter Settngs The benchmark nstances taken from Kacem [,7] and Brandmarte [] are used to evaluate the performance of our algorthm. Parameters are set by comparson of computatonal results under dfferent combnatons, whch are shown as follows: the populaton sze s 00; the termnate condtons t max 00 and max 0. For the objectve consdered n ths paper, the three weghts reflect the mportance of each objectve, whch are determned based on the judgments of users. All computatonal results n ths paper are obtaned based on sx dfferent combnatons of weghts whch are summarzed n Table. For each weght set, 0 trals were conducted to obtan the expermental results.. Results Comparsons Table. Dfferent eght Sets for the Objectves w w w eght set To verfy the effectveness, the proposed MBA s compared wth other algorthms n exstng lteratures, such as OO [8], HSISAT [8], PSO+SA [], ECO+TS [], FL+EA [7], MOPSO+LS [], TS [] and MATLSO []... The Fve Kacem Instances... Problem eght set Table. The Results of Kacem ( ) f max Soluton 8 Optmal Soluton 0 soluton Soluton Tme Fgure. The Gantt chart of Soluton for Kacem (Problem ).

9 Modfed Bat Algorthm for the Mult-Objectve Flexble Job Shop Schedulng Problem 007 e frst use a small-sze nstance to test the effectveness of our algorthm n whch jobs wth operatons are to be performed on machnes. The computatonal results obtaned by our MBA are shown n Table. Fgure shows the result of Soluton n the form of a Gantt chart. Each rectangle represents an operaton, under whch the character denotes the name of the operaton,.e., - refers to the second operaton of Job.... Problem 8 8 Ths s a mddle-sze nstance n whch 8 jobs wth 7 operatons are to be processed on 8 machnes. The computatonal results obtaned by our proposed algorthm are characterzed by the followng values n Table. Fgure shows the result of Soluton n the form of a Gantt chart. eght set Optmal soluton Table. The Results of Kacem (8 8) f max Soluton 7 Soluton 7 Soluton Problem 0 7 Ths s a mddle-sze nstance n whch 0 jobs wth 9 operatons are to be processed on 7 machnes. The computatonal results obtaned by our proposed algorthm are charactersed by the followng values n Table. Fgure 7 shows the result of Soluton n the form of a Gantt chart. eght set Optmal soluton f Table. The Results of Kacem (0 7) max Soluton Soluton 0 Soluton 0

10 008 Haodong Zhu, Baofeng He, Hongchan L Machne Tme Fgure. The Gantt chart of Soluton for Kacem (Problem 8 8) Machne Problem Tme Fgure 7. The Gantt chart of Soluton for Kacem (Problem 0 7). Ths s a mddle-sze nstance n whch 0 jobs wth 0 operatons are to be processed on 0 machnes. The computatonal results obtaned by our proposed algorthm are characterzed by the followng values n Table. Fgure 8 shows the result of Soluton n the form of a Gantt chart. eght set f Table. The Results of Kacem (0 0) max

11 Modfed Bat Algorthm for the Mult-Objectve Flexble Job Shop Schedulng Problem 009 Optmal soluton Soluton 8 Soluton 7 Soluton 7 Soluton Machne Problem Tme Fgure 8. The Gantt chart of Soluton for Kacem (Problem 0 0). Ths s a large-sze nstance n whch jobs wth operatons are to be processed on 0 machnes. The computatonal results obtaned by our proposed algorthm are charactersed by the followng values n Table. Fgure 9 shows the result of Soluton n the form of a Gantt chart. Table 7 shows the comparson results of the fve Kacem nstances, where - means that the correspondng result s not reported n the related lterature, and the boldface denotes the best solutons obtaned by algorthms. It can be seen from Table 7 that MBA s comparable to other algorthms for solvng the Kacem nstances. The comparson data pont out that the proposed MBA can obtan all the non domnated solutons n Kacem nstances, and. For the Kacem, MBA can obtan rcher optmal solutons than OO [7], HSISAT [8], PSO+SA [9], and ECO+TS [0]. For the Kacem nstance, approxmated non-domnated solutons were acqured by MBA. Table 7 shows that the proposed MBA algorthm performed at the same level or better wth respect to three objectve functons for the frst four nstances, when compared to the results obtaned from the other methods. Table. The Results of Kacem ( 0) eght set f C max T max Optmal soluton Soluton 9 Soluton 9

12 00 Haodong Zhu, Baofeng He, Hongchan L Machne Fgure 9. The Gantt chart of Soluton for Kacem (Problem 0). Table 7. The Comparson Results of Dfferent Algorthms PSO+S ECO+T Instance nm Objectve OO HSISAT MOGA MBA A S Kacem Kacem 8 8 Kacem 0 7 Kacem 0 0 Kacem Tme max max max max max The Three Kacem Instances wth Release Dates The second test compares the performances of FL+EA [7], MOPSO+LS [] and our MBA on three nstances wth release dates for jobs. These nstances were presented by Kacem et al. [7]. Table ndcates the comparson of the solutons acqured by three approaches, where - means that the correspondng result s not reported n the related lterature. Table 8. The Comparson Results of Instances wth Release Dates Instance nm Objectve FL+EA MOPSO+LS MBA Kacem Kacem 0 7 Kacem T max C max 8 7 T max C max T max

13 Modfed Bat Algorthm for the Mult-Objectve Flexble Job Shop Schedulng Problem 0 It can be observed from Table 8 that the computatonal results of the proposed algorthm domnate the results of the FL+ EA for solvng the Kacem and nstances. In addton, three approxmated solutons were obtaned n Kacem. The release dates of jobs are lsted as follows: () Kacem : release dates: r, r, r, r., r, r 9, r, r 7, r, r7 7, r8, r9, r0 () Kacem : release dates: r 0. () Kacem : release dates: r, r, r, r, r 9, r 7, r 7, r 8, r 9 9, r 0 0, r, r, r, r, r... The BRdata Instances For further comparson, ten BRdata nstances MK~MK0 taken from Brandmarte [] are used to compare the makespan obtaned by algorthms. Two publshed algorthms are compared wth our algorthm,.e., TS [] and MATLSO []. Table 9 suggests that the best makespan of our approach s superor to the TS algorthm for 9 ount of 0 cases. Regardng the approach MATSLO and MBA, the optmal makespan of MBA s superor to that of MATSLO for 7 out of 0 cases. To summarze, the comparson results show that our proposed algorthm outperforms most of these publshed algorthms, and s competent for the flexble job shop schedulng problems.. Conclusons Table 9. The Comparson Results of two Algorthms Instance nm TS MATSLO MBA Best Best Best MK 0 0 MK 0 0 MK MK MK MK MK7 0 7 MK8 0 0 MK MK In ths paper, a modfed bat algorthm name MBA s appled to solve the mult-objectve FJSSP. In our proposed algorthm, the convertng mechansms are presented to make the contnuous BA sutable for solvng the dscrete FJSSP. The ntalzaton approaches ntegratng heurstc and random rules are ntroduced to ntalze the populaton of the algorthm, whch makes the MBA have good qualty and dversty. To mprove the searchng ablty, fve neghborhood structures are desgned, based on whch a local search algorthm s hybrdzed wth the BA. Accordng to the smulaton results, MBA can acqure optmal or near-optmal solutons for the benchmark nstances. Some novel and effectve neghborhood structures for the FJSSP should be developed for the local search. In addton, the applcaton of BA to other knds of combnaton optmzaton problems may also be a promsng drecton. Acknowledgements The authors would lke to thank the edtors and the anonymous revewers for ther helpful comments and suggestons, whch have mproved the presentaton. Ths work was supported n part by the Scence and Technology Plan Projects of Henan Provnce of Chna under grant No.007 and No.009, the Youth Backbone Teachers Fundng Plannng Project of Colleges and Unverstes n Henan Provnce of Chna under grant No.0GGJS-08, the Key Scence Research Project of Colleges and Unverstes n Henan Provnce of Chna under grant No.A000, the Youth Backbone Teachers Tranng Targets Funded Project of Zhengzhou Unversty of Lght Industry of Henan Provnce of Chna under grant No.XGGJS0, the Ph.D.Research Funded Project of Zhengzhou Unversty of Lght Industry of Henan Provnce of Chna under grant No.00BSJJ08 and No.0BSJJ080, and the Natonal Scence Foundaton of Chna under grant No.808.

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