, 22-24 October, 2014, San Francsco, USA Genetc Algorthm based Modfcaton of Producton Schedule for Varance Mnmsaton of Energy Consumpton C. Duerden, L.-K. Shark, G. Hall, J. Howe Abstract Typcal manufacturng schedulng algorthms do not consder the energy consumpton of each job, or ts varance, when they generate a producton schedule. Ths can become problematc for manufacturers when local nfrastructure has lmted energy dstrbuton capabltes. In ths paper, a genetc algorthm based schedule modfcaton algorthm s presented. By referencng energy consumpton models for each job, adjustments are made to the orgnal schedule so that t produces a mnmal varance n the total energy consumpton n a mult-process manufacturng producton lne, all whle operatng wthn the constrants of the manufacturng lne and ndvdual processes. Emprcal results show a sgnfcant reducton n energy consumpton varance can be acheved on schedules contanng multple concurrent jobs. Index terms Energy consumpton optmsaton, Genetc algorthms, Peak energy, Schedule optmsaton I. INTRODUCTION Schedulng manufacturng jobs and ensurng that they operate wthn the capabltes of the manufacturng producton lne s fundamental n mass producton and hgh volume manufacturng. The goal of a tradtonal schedulng algorthm [1] s to allocate lmted machnery and equpment to manufacturng jobs wthout takng nto account how ths wll affect the energy consumpton at the producton lne level, and how t wll vary as the schedule s beng executed. Ths can potentally lmt the avalablty of a manufacturng producton lne as the nfrastructure can only delver a certan amount of energy at any gven tme. Ideally, for manufacturers an optmal schedule s one whch completes all requred jobs n the least amount of tme and consumes the mnmal amount of energy at any gven nstance, thus ncreasng the productvty and avalablty of the producton lne whle reducng costs. However ths s hndered by many schedulng problems beng NP-hard and therefore cannot be Manuscrpt receved June 30 th 2014; revsed July 25 th 2014. Ths work s fnancally supported by BAE Systems (Operatons) Ltd and the Engneerng and Physcal Scences Research Councl (EPSRC) as part of a CASE studentshp. C. Duerden and G. Hall are wth the Advanced Dgtal Manufacturng Research Centre, Unversty of Central Lancashre, Burnley Campus, Burnley, Lancashre, BB12 0EQ, UK (phone: 01772 896093; emal: cjduerden@uclan.ac.uk, ghall5@uclan.ac.uk). L.-K.Shark s head of the Advanced Dgtal Manufacturng Research Centre, Unversty of Central Lancashre, Preston Campus, Preston, Lancashre, PR1 2HE, UK (phone: 01772 893253; emal: lshark@uclan.ac.uk). J. Howe s head of the Centre for Energy and Power Management, Unversty of Central Lancashre, Preston Campus, Preston, Lancashre, PR1 2HE, UK (phone: 01772 894220; emal: jmhowe@uclan.ac.uk). practcally optmsed by schedulers that are based on polynomal-tme algorthms. Despte ths, work has been undertaken to nclude addtonal objectves for tradtonal schedulers to work towards. Fang et al and Pechmann et al both present methodologes for producton schedules whch am to also mnmse peak energy consumpton [2]-[4]. As does the commercal schedulng software E-PPS by Transfact [5]. Whle the work of Fang et al and Pechmann et al shows promsng results, t s concluded by Fang et al that fndng the optmal schedule s dffcult due to the complexty of the problem and ts NP-hard nature. Electrcal energy s undoubtedly one of the most valuable resources avalable to manufacturers. In 2012 UK ndustry consumed approxmately 97.82 TWh (8411 ktoe) [6]. Recently there have been numerous works on ntellgent schedulng whch ams to reduce manufacturng energy consumpton by reducng the dlng tmes of and by puttng the machne nto energy savng modes or shut them down entrely [7]-[9]. The man purpose of the latter s to ensure the machne s ready to run when the next job arrves. In ther proposed systems, by ntellgently decdng when to shut down a machne or put t nto an energy savng mode, the total energy consumpton for the producton lne can be reduced. Whle all these show promsng results, the problem wth generatng energy optmsed schedules has receved lttle attenton and appears to be plagued by ts NP-hard nature. The use of artfcal ntellgence n the generaton of manufacturng schedules has shown some promsng results. Genetc algorthms appear to be a popular choce for solvng schedulng optmsaton whch can nclude mult-objectve [10], [11] and mult-project [12] problems. Although there have been nvestgatons nto multobjectve schedulng algorthms whch produce a vald schedule whle amng to reduce the peak, or varance n the energy consumpton [10], the optmal soluton s dffcult to fnd when usng tradtonal methods and algorthms due to a very large search space. Whle artfcal ntellgence has been shown to be capable of solvng schedule optmsaton problems n an effcent manner [10]-[12], t s noted that the schedule optmsaton systems are desgned to perform the entre schedulng process. Ths may present manufactures wth a dsadvantage f a set of jobs needs to be completed as quckly as possble wth no concern for energy consumpton. An example of whch s a product order wth a short lead tme. To the authors knowledge t s unknown how well these systems can be adjusted to produce a schedule suted to the manufacturers changng preferences. Ths paper presents a genetc algorthm based schedule optmsaton system whch modfes the tmngs of a
, 22-24 October, 2014, San Francsco, USA schedule produced by a tradtonal schedulng algorthm, n order to mnmse the varance n producton lne energy consumpton wthout exceedng the overall producton deadlne. The technque used s nspred by load-shftng, a tradtonal energy optmsaton method n whch energy ntensve jobs are scheduled to run durng tmes of low energy tarffs [13]. Followng the methodology descrbed n secton 2, experments and results are presented to demonstrate the level of potental reducton n energy consumpton varance. II. METHODOLOGY In the proposed system, a schedule for a seres of manufacturng jobs s ntally generated usng tradtonal schedulng algorthms. Ths takes n a lst of jobs to be processed and produces a schedule whch ensures that a) all jobs are processed n the correct order; b) the total makespan for each process and ther chld jobs do not exceed the process deadlne; and c) at no pont does the total resource utlsaton exceed the resource constrants of the manufacturng producton lne. A schedule produced wthn these constrants wll be vald and could be executed on the proposed manufacturng producton lne, wth job order and resource allocaton already assgned. The genetc algorthm s then used to optmse that schedule wth a goal to mnmse the varance n energy consumpton. Ths s acheved by adjustng the start tmes for each jobs and referencng job specfc emprcal energy models to predct the energy varance generated by the proposed schedule. A. Gene Representaton and Genome Generaton As the goal of the genetc algorthm s to optmse when a job starts, the genomes, representng possble schedules, utlse value encodng [14] wth the value of each gene representng the start tme for a partcular job. In order to mantan the sequental order of a schedule the followng rule s specfed: Processes are ndependent and can be executed n parallel. The jobs of a process are order dependent and must be executed sequentally. Durng the generaton of the ntal populaton of canddate solutons, to ensure that all genomes comply wth ths rule and to ensure job order s mantaned, the strng of genes representng the ndvdual jobs are grouped accordng to ther parent process and job order. A relaton seed of equal length to the gene strng s then generated based on job order. Ths conssts of a number whch ncrements wth every gene and resets back to zero when a process ends. An example of a two process gene strng can be seen n Fg. 1. In ths example, the relaton seed s used by the random number generator to ensure the random numbers comply wth the job order. Let R = {r 0,, r N-1 } denote the relaton seed wth N denotng the total number of jobs, S = {s 0,, s N-1 } the job start tme, G = {g 0,, g N-1 } the genome representaton of S, and D = {d 1,, d L } the process deadlne tme wth L denotng the total number of processes. To reduce the computaton tme and to ncrease the probablty of generatng a vald schedule, the search of optmum G s concentrated n a smaller sub-space by lmtng the range of random tme generaton for each job start. If a job s denoted by and belongs to a process denoted by u wth deadlne d u, then the canddate job start tme represented s gven by { } (1) Ths ensures that, wthn a process group, the next randomly generated gene (job start tme) wll be larger than the prevous. It also permts the jobs belongng to dfferent processes to potentally run concurrently as demonstrated n Fg. 1 by jobs A1 and B2 startng at the same tme. Fg. 1. Example of a relaton seed generated based on job order and parent process order. In ths example, T = 00:01:00 and s e = 13:00:00. In order to convert between genome representaton G and a lst of job start tmes S, an encodng / decodng functon s used. In the orgnal schedule, the earlest start tme s e s used as a reference pont for encodng and decodng job start tme s to or from gene g. Encodng: Decodng: s s g e (2) T s s g T (3) e where T s the mnmal tme shft that can be appled to the job start tmes. B. Algorthm Overvew To ensure the algorthm fnds a soluton as close to the global optmum as possble, there are two separate loops as can be seen n Fg 2. The nner loop s used to smulate multple generatons of a populaton wthn the genetc algorthm. The outer loop s used to repeat the entre genetc algorthm process a predefned number of tmes. For every teraton of the outer loop, a populaton P of N p genomes s generated. The ftness of each genome s calculated before tournament selecton, whch emulates survval of the fttest whle mantanng dversty s used to buld a populaton for
, 22-24 October, 2014, San Francsco, USA reproducng. In tournament selecton, a subset of the man populaton s randomly selected and the fttest genome n that subset s placed n an nterm populaton. Ths s repeated untl the nterm populaton s the same sze as the orgnal populaton. Tournament selecton was chosen due to ts mplementaton smplcty, whch drectly affects computatonal tme. Addtonally the selecton dversty can be easly altered by adjustng the subset populaton sze. Unform crossover and mutaton s then appled to the nterm populaton. To ensure job order s mantaned durng crossover, f a gene s smaller than the one before t, and they are of the same parent process, ts value wll be modfed accordng to (4), g g C 1 (4) 1 1 where C -1 s the makespan of the job whose start tme s represented by g -1. f the optmal value s generated n an early teraton, ther clone wll survve unchanged untl the end whle the orgnal wll reproduce to see f a ftter soluton can be generated. Once the nner loop s completed, the fttest genome generated s added to a lst of fttest genomes. The outer loop then terates and a new populaton s generated. Ths conssts of N p 1 random genomes generated usng the relaton seed, and a copy of the best genome from the lst of fttest genomes. For the very frst teraton, the fttest genome s consdered to be a drect encodng of the orgnal schedule. All ths s done for several reasons: a) the orgnal schedule may produce the optmal energy varance already or may contan optmal components whch could be extracted durng crossover; b) the optmal genome from a prevous teraton may be further optmsed; and c) t ensures that a vald genome s always returned from the nner loop. The entre system s not desgned to stop once a partcular ftness has been reached. The ftness tself s the predcted energy consumpton varance, constructed from the job start tmes proposed n the genomes gene strng and the job energy models. As the mnmal varance for a manufacturng schedule cannot be known beforehand, the system s allowed to run for 1000 outer loop teratons. At ths pont t returns the global fttest genome. The confguraton used n these experments conssts of 100 teratons of the nner loop and 1000 teratons of the outer loop. Durng testng t was found that as the number of jobs ncreased, the number of outer loop teratons also had to be rased n order to ncrease the lkelhood of a genome ftter than the orgnal beng generated. Ths s due, n part, to the condtons a genome must meet to be classfed as a vald schedule. C. Ftness functon The ftness functon serves two purposes: a) determne f genome G represents a vald schedule that can be executed on the proposed manufacturng lne, and f t s vald, b) to decode the gene strng, buld a predcted energy consumpton profle, and calculate the varance. The valdty condtons for each genome are as follows: For a set of job start tme genes g wth a makespan of C, belongng to the same parent process: 1 C1 g g (5) For each job wth a parent process deadlne d j : d j g C (6) Fg. 2. Flowchart of the genetc algorthm schedule optmser. Dfferent rates of crossover probablty were nvestgated to determne the effect on the fnal outcome. Ths s dscussed further n secton three. Mutaton probablty remans fxed at 2%. Once crossover and mutaton have been appled the ftness s recalculated. The algorthm wll then return up to eltsm wth the newly generated populaton untl the nner loop s completed. As eltsm s used, the two fttest genomes are coped and ther clones bypass the crossover and mutaton process. Ths ensures that At any tme t, usage on a machne of type M k, M k,usage wth a maxmum avalablty of M k,x : M k, Usage t) M k, X ( (7) Equaton (5) ensures for jobs wthn the same parent process, the next job does not begn untl the prevous one s fnshed. Equaton (6) ensures all jobs do not exceed ther parent process deadlne. Fnally (7) ensures that usage of each machne type at any one pont n tme never exceeds
, 22-24 October, 2014, San Francsco, USA the total amount of that machne. If a partcular genome fals any of these condtons t s classfed as nvald and s assgned the maxmum ftness, n practce ths s the maxmum value of a double precson number n C#. In ths mplementaton the fttest genome s defned as the one wth the mnmum ftness value, and therefore mnmum energy consumpton varance. If a genome meets all condtons, the energy consumpton varance for that schedule s calculated by generatng a predcted energy consumpton profle. Intally an energy consumpton profle s created for each machne m n the producton lne, E m. Ths spans from s e to D Max wth tme spacng T and s ntally populated wth the dlng value for that partcular machne m Idle. Then n chronologcal order, for every job j usng that machne j m, ther assocated job specfc energy consumpton profle, j Profle s coped to E m, begnnng at the start tme denoted by the approprate gene j g. Ths process overwrtes the values currently assgned to the assocated elements of the profle. Addtonally the recorded dlng consumpton from that partcular job j Idle s subsequently assgned to the remander of the profle. Ths s because the dlng energy consumpton could have changed f the machne s now n a dfferent poston or confguraton due to the prevous job. Ths process s repeated untl the energy profles of all jobs runnng on that machne have been merged. The process descrbed above s detaled n pseudo-code below. Input: Lst of jobs each wth a representatve gene specfyng ts start tme g. The energy consumpton profles and dlng energy consumptons for each job j Profle and j Idle. A lst of machnes and ther standard dlng energy consumptons, M {m 1,..,m X } and m,idle Output: Energy profle for machne E m. For each machne m E M (s e ) to (D Max )= M,Idle For every job j whch uses m, n chronologcal order. Copy j Profle to E m, begnnng at g. Copy j Idle to E m from (g + C ) to D Max End End The system assumes that when not n operaton, each machne s left dlng. Once the predcted energy consumpton profle s generated for each machne, the total predcted energy consumpton profle for a total of X machnes can be calculated usng (8). E Total X ( t) E m m( t) (8) 1 From (8), the varance of the predcted total energy consumpton profle s gven by: DMax t se e E E 2 T E Var Total Total (9) D S Max where D Max = max{d 1,, L } and ETotal denotes the average energy consumpton. Once calculated the sample varance s assgned as the genomes ftness value. III. EXPERIMENTS AND RESULTS The proposed genetc algorthm was tested wth multple smulated schedules of ncreasng complexty. These schedules were devsed based on real schedules and ther job specfc energy consumpton profles ncorporate waveform characterstcs such as nrush currents and transents. Each schedule fle contans detals of the processes and jobs, the energy profles for each job, and the dlng power consumptons for each machne. A separate fle contans data related to the specfcaton of the ndvdual manufacturng lnes. In the frst set of experments, the performance of the proposed genetc algorthm was evaluated by modfyng low complexty schedules and comparng aganst the calculated actual optmal result. Ths actual optmal result was determned by generatng all possble combnatons of tme steps and selectng the one whch produced the lowest energy consumpton varance. For N jobs to be completed by D Max, the number of possble gene combnatons s Among the frst set of experments, a schedule contanng fve jobs wth a maxmum deadlnes of 25 tme steps was the most complex schedule consdered, where every possble combnaton was generated wth the correspondng energy consumpton profle. Ths proved that the algorthm was successful n fndng the optmum soluton among 25 5 = 9,765,635 possble solutons. In the frst set of experments, dfferent values for crossover probablty and tournament selecton sze were also nvestgated to determne the optmal values. For P Crossover, 0.65, 0.75, and 0.85 were selected. Ths range was chosen as a probablty any hgher than 85% would cause too much dsrupton to a populaton. Ths may result n a possble optmal soluton beng lost before t can be dentfed. Ths has been concluded by other authors [15]. A value less than 65% would not allow a populaton to suffcently reproduce. For N Tournament values, N p /8, N p /6 and N p /4 were nvestgated. Each combnaton was tested ten tmes to determne the number of teratons requred to return a near optmal soluton. The number of teratons presented n table I demonstrate that the algorthm works most effcently at low rates of crossover probablty wth a hgh tournament selecton sze. All results produced n the second set of experments are generated wth the algorthm set to these parameters. Table I Average number of teratons untl optmal value generated wth dfferng crossover rate and tournament selecton sze. P Crossover 0.65 0.75 0.85 N p /8 161.875 128.5 165 N Tournament N p /6 203.25 218.375 242 N p /4 101.625 123 186.625 For schedules where N 5, the global optmal soluton, whch can be calculated usng a tradtonal teratve approach, s n most cases returned after only a few outer loop teratons. However as N ncreases the amount of outer loop teratons before a proposed optmum s returned dverges. Wth a schedule of N = 10, ths has been observed to range from one to 323 teratons.
, 22-24 October, 2014, San Francsco, USA In the second set of experments, the proposed genetc algorthm was appled to more complex schedules wth N > 5. The experments show that wth schedules wth a large amount of downtme, the optmal soluton appears to be generated qucker. Ths s lkely due to the fact that whle more downtme ncreases D Max, t also ncreases the probablty of a proposed schedule beng vald n accordance wth ftness functon condton (6). Addtonally, the optmal soluton generated by repeatedly runnng the algorthm ten tmes wth the same schedule s not concse. Ths s to be expected as genetc algorthms may not fnd the most optmal soluton n the tme allotted to them. However the optmal results produced by each only vary slghtly. Wth only a small range of returned values, t can be assumed that the proposed schedule that produced t s a near optmal soluton. Table II Comparson of energy consumpton varance n the orgnal and optmsed schedules N Energy varance n orgnal schedule Energy varance n optmsed schedule Reducton % 8 143.958 52.511 63.523 10 215.111 67.185 68.767 12 237.396 69.090 70.897 15 151.928 72.077 52.558 20 76.960 26.530 65.528 30 144.673 36.395 74.843 50 236.233 42.464 82.025 IV. CONCLUSION Ths paper presents a methodology for modfyng a manufacturng producton schedule wth a goal to mnmse the varance n the producton lne energy consumpton. The use of a genetc algorthm for ndvdual job start tme manpulaton s detaled and the algorthms nternal parameters are evaluated and optmsed based on expermental data. For a seres of scheduled processes, the algorthm s found to be successful n reducng the energy consumpton varance to ts near mnmum, whle ensurng that manufacturng resource lmtatons and process deadlnes are never exceeded. Whle the global optmum soluton s not guaranteed to be returned, a soluton near the global optmum s always produced. For each potental soluton, a predcted energy consumpton profle s generated based on job specfc energy models. The varance of the predcted energy profle s then calculated. At the end of the algorthm, the soluton whch holds the mnmal varance s concluded to be the most optmal. However the accuracy of the system s entrely dependent on the accuracy and resoluton of the energy models and the mnmal tme shft that can be appled to each job, T. For the work presented n ths paper, mock energy models based on real job consumpton profles were utlsed. For real mplementatons, t s recommended that the resoluton of the models be sgnfcantly hgher than T. However, wth sutably accurate models a sgnfcant reducton n energy consumpton varance can be acheved regardless of the amount of jobs n the schedule. Through emprcal testng an average reducton percentage of approxmately 70% has been acheved. It s also seen that the reducton percentage s ndependent of the schedule job count wth a range between 8 to 50 jobs tested. If the methodologes descrbed n ths paper were mplemented n a real manufacturng producton lne wth strugglng power dstrbuton capabltes, the reduced varance could potentally allow for another process, wth a sutable optmsed energy consumpton varance, to run wthout needng to renforce the local nfrastructure. These methodologes could also allow a manufacturng producton lne to be power entrely from lmted supply resources, such as renewable energy sources. Fg 3. Comparson of energy consumpton profles produced by the orgnal and optmsed schedules. N s = 12. Orgnal varance = 237.396, optmsed varance = 69.09 Table II demonstrates the reducton n varance that can be acheved, compared to the orgnal energy consumpton varance. Fg. 3 shows the effect on the energy consumpton profle as an example. It can be seen n Fg. 3 that by redstrbutng the jobs the energy consumpton profle can alter sgnfcantly. Ths can potentally result n a large percentage reducton n the overall varance. However ths level of reducton wll be dependent on the optmsaton of the orgnal schedule, the avalable downtme, and the ndvdual job specfc energy consumpton profles. If for example, a schedule conssts of jobs whose energy consumpton changes very lttle over tme, and there s very lttle downtme before the deadlne, t may not be possble to optmse ths schedule sgnfcantly. ACKNOWLEDGEMENT The authors would lke to thank the ndustral supervsors Daman Adams and Stuart Barker of BAE Systems (Operatons) Ltd. There nsghts and dedcaton has proven nvaluable to the success of ths work. REFERENCES [1] D. Karger, C. Sten, J. Wen, Schedulng Algorthms, Algorthms and Theory of Computaton Handbook, 2 nd ed. Florda, USA: Chapman & Hall/CRC, 2009, pp 20-1 20-34. [2] K. Fang, N. Uhan, F. Zhao, J.W. Sutherland, Flow Shop Schedulng wth Peak Power Consumpton Constrants, Annals of Operatons Research, vol. 206, ssue 1, pp 115 145, Jul. 2013 [3] A. Pechmann, I. Schöler, Optmzng Energy Costs by Intellgent Producton Schedulng, Glocalzed Solutons for Sustanablty n Manufacturng, pp 293-298, May 2011
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