Improved Music Based Harmony Search Algorithm (IMBHSA) for solving Job Shop Scheduling Problems (JSSPs)

Transcription

1 Improved Musc Based Harmony Search Algorthm (IMBHSA) for solvng Job Shop Schedulng Problems (JSSPs) M.Hymavath 1*, C.S.P.Rao 2 1* Assstant Professor, Department of Mechancal Engneerng, Bengal College of Engneerng and Technology, Durgapur West Bengal Inda, E-mal: hyma.madvada07@gmal.com, 2 Professor, Department of Mechancal Engneerng, atonal Insttute of Technology, Warangal, Andhra Pradesh Inda, E-mal: csp_rao@redffmal.com Abstract In ths paper, a new meta-heurstc soluton approach for Mult-objectve Job Shop Schedulng Problems (MOJSSP) s presented. An Improved Musc Based Harmony Search algorthm (IMBHS) s a recently developed algorthm whch s conceptualzed usng the muscal process of searchng for a perfect state of harmony. It uses a stochastc search nstead of a gradent search. Musc Based Harmony Search algorthm (MBHS) and an Improved Musc Based Harmony Search (IMBHS) algorthm were proposed to mnmze the Makespan, Mnmze the Mean flow tme and Mnmze the Mean Tardness. The results are compared wth Bench Mark Solutons (BKS) and t s found that both the methods performed better n terms of the qualty of soluton but n few problems IMBHS s performng better when compared to the MBHS method and BKS solutons. The results obtaned n ths study have shown that the proposed IMBHS algorthm can be used as a new alternatve soluton technque for fndng good solutons to the JSSPs. Keywords Metaheurstcs, Job Shop Schedulng, Improved Musc Based Harmony Search. 1. Introducton A schedule s a tme based allocaton of the tasks to tme ntervals on the resources n job shop Industry. The optmal schedule s one sequence of tasks that mnmzes the overall completon tme. Ths s called the makespan. Mnmzng Tardness, tardness beng the lateness of the job f t fals to meet ts due-date, and zero otherwse. Mnmzng Mean Flow-tme whch measures the average response of the schedule to ndvdual demands of jobs for servce. Job Shop Schedulng Problems (JSSP) belongs to class of P-hard problems. Several Job Shop Schedulng Problems n varous ndustral envronments are combnatoral n nature and conventonal mathematcal tools have faled to produce optmum schedules. Thus JSSP s ever lastng feld for good research. Bruker and Garey show that the job shop schedulng s an P-hard combnatoral problem. Because of the P-hard characterstcs of job shop schedulng, t s usually very hard to fnd ts optmal soluton, and an optmal soluton n the mathematcal sense s not always necessary n practce. Researchers turned to search ts nearoptmal solutons wth all knds of heurstc algorthms. Fortunately, the searched near optmal solutons usually meet requrements of practcal problems very well. Fgure1 Schematc Dagram of Job Shop Problem 2. Job Shop Schedulng Problem Assume job (=1,2,...n) requres processng by machne k (k=1,2,... m) exactly once n ts operaton sequence (thus, each job has m operatons). Let P k be the processng tme of job on machne k, X k be the startng tme of job on machne k, Q jk be the ndcator whch takes on a value of 1 f operaton j of job requres machne k, and zero otherwse. Y hk s the varable whch takes on a value of 1 f job precedes job h on machne k, and zero otherwse. Subjected to the followng constrants a) The precedence of operatons gven by each job s to be respected. b) Sequence Constrant: Each machne can perform at most one operaton at a tme,.e., for 532-1

2 Improved Musc Based Harmony Search Algorthm (IMBHSA) for solvng Job Shop Schedulng Problems (JSSPs) a gven job I, the (j+1) th operaton may not start before the j th operaton s completed. c) Resource Constrant : The operatons cannot be nterrupted d) The startng tmes of jobs are to be respected. In ths paper, the authors have employed Improved Musc Based Harmony Search (IMBHS) algorthm developed by Geem et al n 2001 to model and solve JSSPs. The authors have also successfully mplemented Improved Musc Based Harmony Search (IMBHS) algorthm for JSSP. The two algorthms of MBHS and IMBHS frst tme appled and tested on many Job Shop Schedulng Problems Bench Mark Instances. 3. Musc Based Harmony Search Algorthm (MBHS) The harmony search algorthm (Geem et al. 2001) s one of the most recently developed optmzaton algorthm and at a same tme, t s one the most effcent algorthm n the feld of combnatoral optmzaton (Geem 2009c). Snce the emergence of ths algorthm n 2001 by Geem et al., t attracted many researchers from varous felds especally those workng on solvng optmzaton problems (Ingram and Zhang 2009). Consequently, ths algorthm guded researchers to mprove on ts performance to be n lne wth the requrements of the applcatons beng developed. In general Harmonc Search s executable n fve steps as descrbed below: Step1: Intalze the problem and algorthm parameters Consder an optmzaton problem whch s descrbed as: Mnmze the makespan, Mnmze the mean flow tme and Mnmze the mean tardness. The HS parameters are specfed n ths step. These are Harmony Memory Sze (HMS), or number of soluton vectors n the harmony memory; Harmony Memory Consderng Rate (HMCR); Ptch Adjustng Rate (PAR); and umber of Improvsatons (I) or stoppng crteron. The Harmony Memory (HM) s a memory locaton where all the soluton vectors (sets of decson varables) are stored. The parameters HMCR and PAR are used to mprove the soluton vector and these are defned n step 3. Step2: Intalze the harmony memory In ths step, the HM matrx s flled wth as many randomly generated soluton vectors as the HMS: x x... x x HM = x x... x x x x... x x x x... x x HMS 1 HMS 1 HMS 1 HMS HMS HMS HMS HMS (1) f ( x ) (2) f ( x ).. (1).... ( HMS -1) f ( x ) ( HMS ) f ( x ) Step3: Improvse a new harmony from the HM set A new harmony vector, x = (x 1, x 2,...,x n ), s generated based on three rules, namely, random selecton, memory consderaton and ptch adjustment. These rules are descrbed as follows: Random Selecton: When HS determnes the value, x for the new harmony, x = (x 1, x 2,...,x n ), t randomly pcks any value from the total value range wth a probablty of (1- HMCR). Random selecton s also used for prevous memory ntalzaton. Memory Consderaton: When MBHS determnes the value x, t randomly pcks any j value x from the HM wth a probablty of HMCR snce j= {1,2,, HMS}. 1 2 HMS x' { x, x,..., x } wth probablty HMCR (2) x' x' X wth probablty (1- HMCR ) Ptch Adjustment: Every component of the new harmony vector, x = (x 1, x 2,...,x n ),s examned to determne whether t should be ptch-adjusted. After the value x s randomly pcked from HM n the above memory consderaton process, t can be further adjusted nto neghbourng values by addng certan amount to the value, wth probablty of PAR. Ths operaton uses the PAR parameter, whch s the rate of ptch adjustment gven as follows: Yes wth probablty PAR x' (3) o wth probablty (1- PAR) The value of (1-PAR) sets the rate of dong nothng. If the ptch adjustment decson for x s yes, x s replaced as follows: x' x' ± bw (4) where, bw s the arbtrary dstance bandwdth for a contnuous desgn varable. In ths step, ptch adjustment or random selecton s appled to each varable of the ew Harmony vector. Step 4: Updatng HM If the new harmony vector, x = (x 1, x 2,..., x n ), s better than the worst harmony n the HM, from the vewpont of the objectve functon value, the new harmony s entered n the HM and the exstng worst harmony s omtted from the HM. Step 5: Checkng stoppng crteron Computaton s termnated upon satsfyng the maxmum number of mprovsatons or maxmum number of teratons, whch s the stoppng crteron. Otherwse, steps 3 and 4 are repeated. Fnally the best harmony memory vector s selected and s consdered to be the best soluton to the problem under nvestgaton

3 3.1. Improved Harmony Search Algorthm The tradtonal MBHS algorthm uses fxed value for both PAR and bw. The PAR and bw values adjusted n the ntalzaton step (Step 1) cannot be changed durng new generatons. The man drawback of ths method s that ths employs hgher number of teratons to converge at an optmal soluton. Small PAR values wth large bw values can lead to poor performance of the algorthm and consderable ncrease n the number of teratons to fnd optmum soluton. Small bw values n the fnal generatons ncrease the fne tunng of soluton vectors. Therefore, large PAR values wth small bw values usually leads to mprovement n obtanng the best soluton n the fnal generaton n whch the algorthm converges to optmal soluton vector. The IMBHS has been developed by Madhav et al. and has been successfully appled to varous benchmarkng and standard engneerng optmzaton problems. The key dfference between IMBHS and tradtonal MBHS method s n the way of adjustng PAR and bw. To mprove the performance of the MBHS algorthm and elmnate the drawbacks assocated wth the fxed values of PAR and bw, the IMBHS algorthm uses varable PAR and bwn the mprovsaton step (Step 3). The PAR values change dynamcally wth generaton number as shown n Fgure 2 and expressed as follows : PAR ( ) max PAR PAR gn PAR mn = mn + gn (5) I Fgure 2 a) Varaton of PAR versus generaton number, b) Varaton of bw versus generaton number. where, PAR s the ptch adjustng rate for each generaton, PAR mn s the mnmum ptch adjustng rate, PAR max s the maxmum ptch adjustng rate, I s the number of soluton vector generatons and gn s the generaton number. Bandwdth(bw)changes dynamcally wth generaton number as shown n Fgure 2 and defned as follows: bw( gn) = bw exp( c. gn) (6) C = Ln max bw max bw mn (7) I where, bw(gn) s the bandwdth at each generaton, bw mn and bw max are the mnmum and the maxmum bandwdths respectvely. The complete explanaton of MBHS s presented n Fgure 3. Begn Step 1: Intalzaton Intalze Harmony Memory (HM), Harmony Memory Sze (HMS), Harmony Memory Consderaton Rate (HMCR), Ptch Adjustng Rate (PAR), Maxmum number of teratons (I) Generate the Intal Populaton wth Rp and SP Step 2: Computaton Whle (ter I) Do { Generaton++; If (rand (0,1) HMCR) then Choose a value from HM for If (rand (0,1) PAR) then Adjust the value of by: X new =X old + rand (0,1) bw; End f Else Choose a random varable; X =mn+ rand (0,1) (max-mn); End f End whle If (Ft Fun (new harmony soluton) worst (ft Fun (HM)) then Accept the new harmony and replace the worst n HM wth t. End f End whle Step 3: Output optmzaton results End Fgure 3 Pseudo code for IMBHS algorthm 3.2. Implementaton of IMBHS for JSSP: The soluton to JSSP s a schedule of operaton for jobs. In the present work IMBHSA s used to fnd the optmum schedule. A HM represents feasble schedule n ths case. In the present work a drect approach Operaton based representaton s employed. Followng s the bref descrpton of the representaton. The schedule s represented n the form of a strng as shown n table 1. For example, for a three-job-three-machne problem the representaton would be as shown n table 1. Table 1 Vector Soluton representaton of JSSP soluton Inputs for the problem For solvng JSSP usng IMBHSA, the followng nputs are requred: umber of jobs (n) umber of Machnes (m) Machne order for all the jobs (M j ) Processng Tmes of all the operatons (P j ) umber of operatons (O j ) umber of teratons to be carred out (n ter ) 532-3

4 Improved Musc Based Harmony Search Algorthm (IMBHSA) for solvng Job Shop Schedulng Problems (JSSPs) Harmony Memory Sze (dependng on the problem sze) Harmony Memory Consderaton Rate (0.85) Ptch Adjustng Rate (PAR) (PAR mn =0.4,PAR max =0.99) Band Wdth (BW) (BW mn =0.0001, BW max =1) umber of Improvsatons (I=20) Maxmum populaton allowable Step wse Implementaton of IMBHS for JSSP Ths secton gves a detaled explanaton of mplementaton of the algorthm. The actual algorthm has been slghtly modfed for enhancng the performance. Improved Musc Based Harmony Search Algorthm (IMBHSA) adopts process of HM for fndng optmum solutons effcently. In ths case HMS s a potental soluton and the algorthm helps HMS to evolve and generate better populaton thus gvng rse to ftter solutons whch represent compettve schedules. Ths secton gves a detaled explanaton of mplementaton of the algorthm Intalzaton In the orgnal algorthm ntalzaton s random. n number of s are generated randomly (RP). The modfcaton s that ntally 10 s.e., 10 dfferent schedules are generated usng dspatchng rules whch present a sgnfcant optmzaton capacty (SP). Snce the choce of the ntal populaton has a hgh mpact on the speed of evoluton and the qualty of fnal results, the soluton scenaro would be focused on generatng ntal populaton usng prorty dspatchng rules. Prorty dspatchng rules are actually the most wdely used for solvng JSSP where all the operatons avalable to be scheduled are assgned a prorty.the operaton wth the hghest prorty s chosen to be sequenced. A prorty dspatchng rule s a smple mathematcal formula that, based on some processng parameters, specfes the prorty of operatons to be executed. 10 ntal schedules.e s are generated usng 10 commonly used prorty dspatchng rules gven n the Table 2. Table 2: Lst of common Prorty Dspatchng rules used to generate the ntal populaton. EXPRESSIO DESCRIPTIO Shortest Processng Tme (SPT) Longest Processng Tme(LPT) Mnmum Slack Tme Per Operaton(MISOP) The job wth shortest tme on machnes selected. p p+1 p+2 pn The job wth longest tme on machnes selected.p p+1 p+2 pn Tme remanng untl the due date Processng tmeremanng Mnmum Due Date(MIDD) Crtcal Rato(CR) Most work remanng (MWKR) Least work remanng(lwkr) Shortest remanng Mnmum Processng Tme(SRMPT) Longest remanng Maxmum Processng Tme(LRMPT) RADOM(random selecton) The job wth earlest due date s processed frst D D+1 D+2. Dn Remanng due date/remanng processng tme Select the operaton assocated wth the job of the most work remanng to be processed Select the operaton assocated wth the job of the least work remanng to be processed Mn(processng tne remanngmnmum processng tme) Max(processng tne remanngmaxmum processng tme) Select the next job to be processed randomly Performance Evaluaton The ntal populaton s made to reproduce dependng on the ftness of the ndvdual. The present problem s a Mult- Objectve optmzaton, the objectves beng mnmzng make-span, mnmzng tardness and mnmzng flow-tme. So the ftness of the ndvdual has to be evaluated consderng all the three aspects. The ndvduals are ranked based on ther ftness values(all the three). Ths rankng s done based on concept called Fuzzy-Pareto-Domnance.[3] The rankng scheme assgns domnance degrees to any set of vectors n a scale-ndependent, nonsymmetrc and set-dependent manner. Based on such a rankng scheme, the ftness values of a populaton can be replaced by the computed rankng values representng the domnatng strength of an ndvdual aganst all other ndvduals n the populaton. The three schedulng objectves used n ths mplementaton are (1)make-span of the sequence (2) mean-flow tme of the jobs, and (3)the mean tardness of jobs. MAKE-SPA MODULE (FITESS 1) In Schedulng lterature, make-span s defned as the maxmum completon tme of all jobs, or the tme taken to complete the last job on the last machne n the schedule-assumng that the processng of the frst job began at tme 0. Make

5 span s denoted by cmax and computed as cmax = max{fj}, where Fj s the flow tme for job j(the total tme taken by job j from the nstant of ts release to the shop to the tme ts processng by the last machne s over). MEA TARDIESS MODULE(FITESS 2) The lateness of a job measures the conformty of the schedule to that job s due date. Lateness s defned as the amount of tme by whch the completon tme of a job exceeds ts due-date. Mathematcally Lj=Cj-dj Lj: Lateness of job j Cj: Completon tme of job j dj: Due date of job j. Tardness of job s the lateness f t s postve else t s zero. Mean tardness(t) s defned as the average of tardness of all jobs. T = (sum of tardness of all jobs) / number of jobs MEA FLOW TIME MODULE(FITESS 3) The mean flow tme measures the average response of the schedule to ndvdual demands of jobs for servce. Mathematcally, mean flow tme s the average of the flow tmes of all jobs. Mean flow tme = sum of flow tmes of all jobs / number of jobs FUZZIFICATIO OF PARETO DOMIACE AD RAKIG In mult objectve optmzaton, the optmzaton goal s gven by more than one objectve to be extreme. Formally, gven a doman as subset of Rn, there are assgned m functons f1(x1,..., xn),..., fm(x1,..., xn). Usually, there s not a sngle optmum but rather the so-called Pareto set of nondomnated solutons. Evolutonary Computaton (EC) has been shown to be a powerful technque for mult-objectve optmzaton (EMO - Evolutonary Mult-Objectve Optmzaton). Ths bologcally nspred methodology offers both flexblty n goal specfcaton and good performance n multmodal, nonlnear search spaces. If we want to solve a hghly complex mult-objectve optmzaton problem, we mght select one of the best ranked evolutonary approaches revewed n the lterature, lke on Sorted Genetc Algorthm-II (SGA-II) and hopefully start reachng good results quckly. However, all these algorthms need domnated ndvduals n the populaton, to perform the correspondng genetc operators. For a hgher number of objectves, ths mght become a problem, snce the probablty of havng a domnated ndvdual n the populaton wll rapdly go to zero. The need for a revson of the Pareto domnance relaton for also handlng a larger number of objectves was already ponted out n a few studes, esp. gven by Farna and Amato. There, we also fnd the suggeston to use fuzzy-membershp degrees for the degree of a pont belongng to the Pareto set (so called fuzzy optmalty). Authors desgn ther revsed domnance measure n a way that the approach to the Pareto front can be regstered more early n the search. The approach was shown to work successfully n the doman of more than two objectves. It came out that the use of fuzzy concepts s frutful n ths regard The fuzzfcaton of Pareto domnance relaton can be wrtten then as follows: It s sad that vector a domnates vector b by degree µa wth µ a (a,b) =, here denotes the dmenson of vector, n the present case s 3 as we have consdered 3 objectves. It s sad that the vector a s domnated by vector b by degree µp wth µ p (a,b) = The defntons dffer n denomnator and thus are not symmetrc. Domnatng by degree µ and beng domnated by degree µ have dfferent fuzzy values. For a Pareto domnatng b, µ a (a,b) = 1 and µ p (b,a) = 1 but µ p (a,b) < 1 and µ a (b,a) < 1 We use these domnance degrees to rank a set M of mult-varate data(vectors).the vectors n ths case are that sets of three ftness values of the schedules n the populaton.the vector here s (ftness1,ftness2,ftness3). Consder two harmones or schedules a and b µ p (a,b) = s/p where s = mn(ftness1a,ftness1b) mn(ftness2a,ftness2b) mn(ftness3a,ftness3b) p = ftness1b ftness2b ftness3b Each element of M(set of schedules) s assgned the maxmum degree of beng domnated by any other element n M, and the elements of M are sorted n ncreasng order of the value of domnance degree. After sortng the populaton s ranked sequentally. So the ndvduals havng good rank(lower) are ftter members n the populaton. Wth the help of rankng scheme performance evaluaton s carred out Improvse a new memory from the HM set By usng (5), (6) and (7) equatons we are mprovsng the ew Harmony memory from HM based on HMCR, PAR and Random Choce. Agan we are we are evaluatng the objectve functon wth ths ew Harmony memory. If the soluton s better than the stored harmony, update the HM wth ew Harmony memory otherwse stored HM s the updated HM for the next generaton Checkng the stoppng crteron The stoppng crtera s one of the followng Maxmum number of teratons The populaton s worst and best ftness becomes equal 532-5

6 Improved Musc Based Harmony Search Algorthm (IMBHSA) for solvng Job Shop Schedulng Problems (JSSPs) The cycle of steps explaned s carred out untl one of the stoppng crtera s met. 4. Performance of IMBHS on Bench Mark Problems IMBHS algorthm has been coded n Matlab and runs on Intel Core2Duo Processor wth 4 GB RAM and CPU 2.00 Ghz. The bench markng problems from OR lbrary, were solved by the proposed method. The results obtaned were tabulated n table 3. In vew of the results obtaned by mplementng IMBHS to solve JSSP, t appears that IMBHS s effcent. The algorthm has been mproved by changng ts soluton codng method and hybrdzng leadng to fast convergence. The algorthm s performance has been compared to that of wth the help of some bench-mark problems and has been found to be superor to the latter. It s concluded that the applcaton of IMBHS to for solvng JSSP s a new area to be explored for compettve solutons Results and Dscussons Table 3: Results of some Bench Mark Problems Fgure 5 Comparson of JSP2 wth GA, DE, Fgure 6 Comparson of JSP3 wth GA,DE, Fgure 7 Comparson of JSP4 wth GA,DE, Concluson The performance of IMBHS algorthm s compared wth SGA-II(Deb), GA(Garen), DE(BSP Reddy) on 4 Job Shop Problems exstng n the lterature. It s found that the pareto optmal set of solutons generated by IMBHS s superor to the exstng SGA-II, GA, DE, HPSO and HAIA. Thus, IMBHS algorthm has became unque procedures to address any knd of Mult objectve JSSPs respectvely. Fgure 4 Comparson of JSP1 wth GA, DE, References Bruker P (1995), Schedulng algorthms 2nd edn. Sprnger, Berln Hedelberg ew York Garey M, et al (1976), The complexty of flow shop and job shop schedulng, Math Oper Res 1: Erschler JF, Roubellat JP, Vernhes (1976), Fndng some essental characterstcs of the feasble solutons for a schedulng problem, Operatons Research 24:

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