Real-tme Schedulg of Flexble Maufacturg Systems usg Support Vector Maches ad Neural Networks P. Prore, R. Po, J. Parreño, J. Lozao ad M. Moterrey EPI de Gó, Campus de Vesques, 33203 Gó, Asturas, Spa Abstract - Dspatchg rules are usually appled to schedule obs Flexble Maufacturg Systems (FMSs) dyamcally. Despte ther frequet use, oe of the drawbacks that they dsplay s that the state the maufacturg system s dctates the level of performace of the rule. As o rule s better tha all the other rules for all system states, t would hghly desrable to kow whch rule s the most approprate for each gve codto, ad to ths ed ths paper proposes a schedulg approach that employs Support Vector Maches (SVMs) ad backpropagato eural etworks. Usg these latter techques, ad by aalysg the earler performace of the system, schedulg kowledge s obtaed whereby the rght dspatchg rule at each partcular momet ca be determed. A module that geerates ew cotrol attrbutes s also desged order to mprove the schedulg kowledge that s obtaed. Smulato results show that the proposed approach leads to sgfcat performace mprovemets over exstg dspatchg rules. Keywords: Schedulg, Neural Networks, SVMs, FMS, Smulato Itroducto Oe of the most commoly appled solutos to the schedulg problem FMSs volves usg dspatchg rules, whch have bee evaluated for performace by may researchers (see for example, [2], [2], [22]). Almost all the above studes pot to the fact that rule performace depeds o the crtera that are chose, ad the system s cofgurato ad codtos (utlsato level of the system, relatve loadg, due date tghtess, ad so o). It would thus be terestg to be able to chage dspatchg rules at the rght momet dyamcally. The lterature descrbes two basc approaches to modfy dspatchg rules. The frst approach s to select a rule at the approprate momet by smulatg a set of pre-establshed dspatchg rules ad optg for the oe that provdes the best performace (see for example, [7], [8], [9], [23]). The secod approach, volvg artfcal tellgece, requres a set of earler system smulatos (trag ) to determe what the best rule s for each possble system state. A mache learg algorthm [] s traed to acqure kowledge through these trag, ad ths kowledge s the used to make tellget decsos real tme (see for example, [6], [9], [20]). Aytug [] ad Prore [5] provde a revew whch mache learg s appled to solvg schedulg problems. Nevertheless, there are hardly ay studes the lterature that compare the dfferet types of mache learg algorthms used schedulg problems. Ths paper therefore presets a schedulg approach that uses ad compares SVMs ad eural etworks. To mprove the maufacturg system s performace, a ew approach s also proposed whereby ew cotrol attrbutes that are arthmetcal combatos of the orgal attrbutes ca be determed. The rest of ths paper s orgased as follows. Mache learg algorthms used ths paper are frst descrbed. A approach to schedulg obs that employs mache learg s the preseted. Ths s followed by the expermetal study, whch descrbes a ew approach to determe ew cotrol attrbutes from the orgal oes. The two mache learg algorthms used are also compared. Fally, the proposed schedulg approach s compared wth the alteratve of usg a combato of dspatchg rules costatly. A summary of the results obtaed cocludes the paper. 2 Neural Networks ad Support Vector Maches Backpropagato eural etworks, or multlayer perceptro [8], whch wll be appled ths work, fgure amogst those etworks that are most well-kow ad most wdely used as patter classfers or fucto approxmators ([5], [0]). The backpropagato trag algorthm s used ths type of eural etworks. Ths algorthm calculates the most adequate coecto weghts ad thresholds so that the dfferece betwee the etwork output ad the desred oe s mmsed. Support vector maches [4] were orgally desged for bary classfcato. Let (x, y ), (x 2, y 2 ),, (x, y ) be a group of data belogg to Class or Class 2, where x R N ad the assocated labels be y = for Class ad - for Class 2 (=,, ). The formulato of SVMs s as follows:
M subect to the costrats: 2 =,..., T w w + C y ( w φ( x ) + b) ξ T ξ 0 = ξ =,..., where w s the weght vector; C s the pealty weght; ξ are o-egatve slack varables; b s a scalar, ad x are mapped to a hgher dmesoal space by a o-lear mappg fucto φ. Mappg fucto φ eeds to satsfy the followg equato: k( x, x ) = φ( x ) φ( x ) where k x, x ) s called kerel fucto. ( Mmzg w T w mples that SVMs tres to maxmse 2 2, whch represets the marg of separato betwee both w classes. The data that satsfy the equalty Eq. () are called support vectors. Moreover, by addg a set of o-egatve Lagrage multplers α ad β to geerate the Lagraga, the upper- metoed costraed optmzato problem ca be worked out wth the dual form show below: Max α 2 = = subect to the costrats: = α y 0 α C = 0 = T α α y y k( x, x ) =,..., Havg obtaed the support vectors (SVs), the decso fucto for a usee data (x) s as follows: y = sg α yk( x, x ) + b SVs () 3 Schedulg usg Neural Networks ad Support Vector Maches Two cotrastg features eed to be fulflled for a realtme schedulg system that dyamcally modfes dspatchg rules to work properly [3]:. Rule selecto must take to accout a varety of formato about the maufacturg system real tme. 2. Rule selecto must be completed fast eough for real operatos ot to be delayed. Oe way of dog ths s to employ some class of kowledge about the relatoshp betwee the maufacturg system s state ad the rule to be appled at that momet. However, oe of the most dffcult problems s precsely how ths kowledge s to be acqured. Mache learg algorthms, such as SVMs or eural etworks, are used to do ths. However, the trag ad the learg algorthm must be adequate for ths kowledge to be useful. Moreover, geeratg the trag, the attrbutes selected are crucal to the performace of the schedulg system [3]. Fgure shows a schedulg system that employs mache learg. The example geerator creates dfferet maufacturg system states va the smulato model ad choose the best dspatchg rule for each partcular state. The mache learg algorthm employs the to geerate the kowledge requred to make future schedulg decsos. The real tme cotrol system usg the schedulg kowledge, the maufacturg system s state ad performace, choose the best dspatchg rule for ob schedulg. Further may possbly be eeded order to refe the kowledge about the maufacturg system depedg o the performace of the latter. Trag ad test example geerator Smulato model Kowledge refemet Trag ad test Mache learg algorthm FMS state ad performace Schedulg kowledge Real tme cotrol system Job schedulg FMS Fgure. Geeral overvew of a kowledge-based schedulg system.
4 Expermetal Study 4. The proposed FMS The selected FMS cossts of a loadg stato, a uloadg stato, four machg cetres ad a materal hadlg system. Two types of decso are studed the FMS proposed. The frst s the selecto by the mache of parts assged to t usg the followg dspatchg rules: SPT (Shortest Processg Tme), EDD (Earlest Due Date), MDD (Modfed Job Due Date), ad SRPT (Shortest Remag Processg Tme). These rules were selected because of ther fe performace earler studes (see for example, [4]). The secod type of decso volves the selecto of the maches by the parts, as a operato ca be processed o dfferet maches. The dspatchg rules employed ths FMS are: SPT (Shortest Processg Tme), NINQ (Shortest Queue), WINQ (Work Queue), ad LUS (Lowest Utlsed Stato). 4.2 Geeratg trag ad test The cotrol attrbutes used to descrbe the maufacturg system state must frst be defed order to geerate trag ad test. I ths partcular FMS these are: F, flow allowace factor whch measures due date tghtess [2]; NAMO: umber of alteratve maches for a operato; MU: mea utlsato of the maufacturg system; U : utlsato of mache ; WIP: mea umber of parts the system; RBM: rato of the utlsato of the bottleeck mache to the mea utlsato of the maufacturg system; RSDU: rato of the stadard devato of dvdual mache utlsatos to mea utlzato. The trag ad test eeded for the learg stage are obtaed by smulato usg the WITNESS programme. The followg suppostos were made to do ths: () Jobs arrve at the system followg a Posso dstrbuto; (2) Processg tmes for each operato are sampled from a expoetal dstrbuto wth a mea of oe; (3) The actual umber of operatos of a ob s a radom varable, equally dstrbuted amog the tegers from oe to four; (4) The probablty of assgg a operato to a mache depeds o the parameters PO (percetage of operatos assged to mache ). These percetages fluctuate betwee 0% ad 40%. It s also assumed that the frst two maches have a greater workload; (5) The umber of alteratve maches for a operato vares betwee oe ad four; (6) The ob arrval rate vares so that the overall use of the system rages betwee 55% ad 95%; (7) The value of factor F fluctuates betwee oe ad te. As mea tardess ad mea flow tme the system are the most wdely used crtera to measure system performace all maufacturg systems, they are also employed ths study. I all, 00 dfferet cotrol attrbute combatos were radomly geerated, ad 00 of these were used as test. For each combato of attrbutes, mea tardess ad mea flow tme values resultg from the use of each of the dspatchg rules solato were calculated. Sxtee smulatos were actually eeded to geerate a trag or test example, as there are four rules for each of the decsos to be take. 4.3 The applcato of eural etworks Backpropagato eural etworks are partcularly used to solve classfcato problems such as the oe beg cosdered ths work. Table I provdes a summary of the results obtaed usg dfferet-szed sets of trag for the crtera of mea tardess ad mea flow tme. Geerally, t ca be see that as the umber of trag creases, test example error ( that have ot prevously bee dealt wth) decreases cosderably. Table I also shows that test error fluctuates betwee 6% ad 2% upwards of 400 for the crtero of mea tardess. Furthermore, for the crtero of mea flow tme, test error s observed to oscllate betwee 7% ad 4% upwards of 500. Errors for ths latter crtero are lower due to there beg fve dspatchg rule combatos (SPT+SPT, SPT+NINQ, SPT+WINQ, MDD+WINQ, SRPT+WINQ) that are really used. I cotrast, twelve combatos are used for the crtero of mea tardess. Table I. usg eural etworks for the crtera of mea tardess ad mea flow tme. 200 2% 9% 300 7% 0% 400 6% 0% 500 5% 7% 600 3% 5% 700 4% 6% 800 6% 6% 900 5% 5% 000 5% 4% 00 2% 4% The deal cofgurato of the eural etwork used for the crtero of mea tardess was foud to have put odes (oe for each cotrol attrbute), 6 odes the hdde layer, ad 2 odes the output layer (oe for each dspatchg rule combato). Smlarly, the deal cofgurato of the eural etwork employed for the crtero of mea flow tme was foud to have put odes, 6 odes the hdde layer, ad 5 odes the output layer. 4.4 The applcato of support vector maches The schedulg problem s essetally a mult-class classfcato problem as several dspatchg rule
combatos are employed the FMS. Ths study uses the oe-agast-oe method to exted the bary SVMs to geerate the mult-class scheduler sce ths method s more sutable for practcal use tha other methods [6]. I the same way, ths study, the radal bass fucto (RBF) ad the polyomal fucto have bee used as kerel fuctos. After several prelmary tests, t has bee decded to make use of the RBF Kerel sce t s the oe that shows a better performace. Furthermore, by employg the grd search techque o the, the best performace for the SVMs s obtaed whe C=,000 ad σ=0. Table II provdes a summary of the results obtaed for the crtera of mea tardess ad mea flow tme. Geerally, t ca be see that as the umber of creases, test example error decreases cosderably. Table II also shows that test error fluctuates betwee % ad 0% upwards of 700 for the crtero of mea tardess. Furthermore, for the crtero of mea flow tme, test error drops to % upwards of 700. Table II. usg SVMs for the crtera of mea tardess ad mea flow tme. 200 6% 6% 300 5% 5% 400 5% 2% 500 6% 2% 600 2% 2% 700 % % 800 % % 900 % % 000 0% % 00 0% % 4.5 Geeratg ew cotrol attrbutes O occasos, t s ecessary to obta arthmetcal combatos of the orgal attrbutes to mprove the schedulg kowledge. But may cases these combatos are ot kow beforehad, ad are oly foud very smple maufacturg systems after close examatos of ther smulato results. For these reasos, a module was desged whch automatcally selects the useful combatos of the orgal attrbutes by usg smulato data whch orgally provded test ad trag. To do ths, the basc arthmetc operators cosdered are addg, subtractg, multplyg ad dvdg. The pseudo-code for the geerator of the ew cotrol attrbutes s as follows:. Determato of the combatos of the preset attrbutes. 2. Geerato of ew trag ad test the lght of earler combatos. 3. Selecto of the useful combatos, whch are the decso tree ad the set of decso rules geerated by C4.5 [7]. 4. If the ew decso tree ad/or the set of decso rules has fewer classfcato errors, go back to step oe. If ot, stop the algorthm. However, the decso to cotue may also be take at step four because, eve though error may ot be mproved by the preset terato, t may well be mproved durg later terato(s). The proposed module redered the followg useful cotrol attrbute combatos for the crtero of mea tardess: U+U2, U+U4, U2+U3, U-U2, U2-U4, U3-U4 ad U3/U4. Table III shows the results obtaed for the crtero of mea tardess whe the SVMs ad the geerator module of ew cotrol attrbutes were appled. It ca be see from the results that test error oscllates betwee 0% ad 8% from 600 trag upwards, ad that the lowest test error was acheved wth 000 ad 00. The proposed module s the appled for the crtero of mea flow tme, ad the followg combatos of attrbutes were determed to be useful : U-U2, U3-U4, U/U2 ad U2/U3. The Table shows how test error drops to 0% from 700 upwards. If these results are compared wth those Table II, a mprovemet ca be see to exst. Oly sets of 600 or more were used, as lower errors are obtaed upwards of ths trag set sze. Table III. usg SVMs ad the geerator module of ew cotrol attrbutes for the crtera of mea tardess ad mea flow tme. 600 0% % 700 9% 0% 800 9% 0% 900 9% 0% 000 8% 0% 00 8% 0% usg backpropagato eural etworks ad the ew attrbutes geerated was lkewse calculated. Results are show Table IV, where t s aga clear that classfcato error drops compared to the alteratve of usg the orgal cotrol attrbutes. For the crtero of mea tardess, the backpropagato eural etwork gves a 2% test error, compared to the 8% error of the SVMs. Furthermore, for the crtero of mea flow tme, the SVMs are see to gve zero test error, whereas the backpropagato eural etwork geerates a bgger test error (2%). Fally, meto should be made of the fact that the deal eural etwork for the crtero of mea tardess has 8, 5 ad 2 euros the put,
hdde ad output layers respectvely, whlst for the crtero of mea flow tme the deal eural etwork has 5, 0 ad 5 euros. Table IV. usg eural etworks ad the geerator module of ew cotrol attrbutes for the crtera of mea tardess ad mea flow tme. 600 5% 4% 700 4% 4% 800 2% 4% 900 4% 2% 000 2% 2% 00 2% 2% 4.6 Learg-based schedulg To select the best combato of dspatchg rules accordg to the FMS s state real tme we must mplemet the schedulg kowledge the FMS smulato model. Selectg the motorg perod s aother key questo because the frequecy used to test the cotrol attrbutes determes the performace of the maufacturg system. To do ths, multples of the average total processg tme for a ob, whch our partcular case are 2.5, 5, 0 ad 20 tme uts, are take (see for example, [8], [9], [23]). I vew of the results the prevous secto, 000 were used for both performace crtera. Fve depedet replcas of 00,000 tme uts were carred out. Table V. Mea tardess ad mea flow tme for the proposed strateges. Strategy used MT MFT Strategy used MT MFT SPT+SPT 5.496 2,404 MDD+NINQ.450,2495 SPT+NINQ.2220,0438 MDD+WINQ.566,2546 SPT+WINQ.20,045 MDD+LUS 2.5326,8537 SPT+LUS 2.5920,587 SRPT+SPT 6.452 2,6549 EDD+SPT 4.7207 2,606 SRPT+NINQ.474,40 EDD+NINQ.6802,3909 SRPT+WINQ.4089,49 EDD+WINQ.6885,3948 SRPT+LUS 3.0247,7094 EDD+LUS 3.2958 2,0493 SVMs.0000.0000 MDD+SPT 4.7620 2,6750 Neural Network,0298.0055 Table V summarses the results obtaed. Mea tardess ad mea flow tme values the Table are the average of the fve replcas. Readablty has bee mproved by showg values the Table that are relatve to the lowest mea tardess ad mea flow tme obtaed (these are assged a value of oe). The motorg perod chose was 2.5 tme uts. Table V shows that the best alteratve s to employ a kowledgebased strategy ad that the SVMs geerate lower mea tardess values tha eural etwork. The combatos MDD+NINQ ad MDD+WINQ are the best of the strateges that use a fxed combato of dspatchg rules, but ther mea tardess values are hgher tha the eural etwork alteratve by betwee 4.50% ad 5.66%. Moreover, the SVMs gve better results tha the eural etwork for the crtero of mea flow tme. Table V also shows that the combatos SPT+NINQ ad SPT+WINQ geerate the least mea flow tme from amogst the strateges that apply a fxed combato of rules. However, mea flow tme values are greater tha the SVMs alteratve by betwee 4.5% ad 4.38%. Fally, the SVMs-based system s compared wth the other strateges by usg ANOVA. The cocluso s that ths schedulg system stads out above the other strateges wth a sgfcace level of less tha 0.05. 5 Coclusos A approach for schedulg usg SVMs ad eural etworks s proposed ths study. A geerator module of ew cotrol attrbutes s also corporated, ad ths reduces test error obtaed wth the mache learg algorthms leadg to better performace of the maufacturg system. The SVMs-based schedulg system s show to provde the lowest mea tardess ad mea flow tme values. Future research mght focus o usg more decso types for the proposed FMS. However, the more decso types that are used, the more smulatos are eeded to geerate the trag ad test. A smulator could usefully be corporated to decde whch rule to apply whe the schedulg kowledge provdes two or more theoretcally correct dspatchg rules. Fally, a kowledge base refemet module could also be added, whch would automatcally modfy the kowledge base whe maor chages the maufacturg system come about. 6 Refereces [] Aytug, H., Bhattacharyya, S., Koehler, G.J. ad Sowdo, J.L. (994), A revew of mache learg schedulg, IEEE Trasactos o Egeerg Maagemet, Vol. 4, pp. 65-7. [2] Baker, K.R. (984), Sequecg rules ad due-date assgmets a ob shop, Maagemet Scece, Vol. 30, pp. 093-03. [3] Che, C.C. ad Yh, Y. (996), Idetfyg attrbutes for kowledge-based developmet dyamc schedulg evromets, Iteratoal Joural of Producto Research, Vol. 34, pp. 739-755. [4] Cortes, C. ad Vapk, V. (995), Support-vector etwork, Mache Learg, Vol. 20, pp. 273-297.
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