An order batch scheduling method based on fuzzy model for cold rolling plant

Transcription

1 Producton plannng and schedulng Data warehouse Preprnts of the 8th IFAC World Congress lano (Italy) August 28 - September 2, 2 An order batch schedulng method based on fuzzy model for cold rollng plant Jun Zhao, Yng Lu, We Wang, Lqun Cong 2, Ysha Lu.Research Center of Informaton and Control, Dalan nversty of Technology, Dalan 624, Chna (e-mal: zhaoj@dlut.edu.cn, luyng8227@63.com, wangwe@dlut.edu.cn, luysha@dlut.edu.cn ) 2.Shangha Baosght Software Co. Ltd., Shangha 223, Chna (e-mal: conglqun@baosght.com ) Abstract: Ths paper proposes an orders batch schedulng method for cold rollng plant, whch frstly groups the avalable orders nto the batches, and predcts ther producton tme by means of fuzzy mode. To enhance the order delvery satsfacton, a fuzzy job shop schedulng model, n whch each order batch s vewed as a fundamental scheduled job, s establshed to represent the producton logstcs of a cold rollng plant n Shangha Baosteel. Consderng the estent nventory n front of each machne, the calculaton rules for the manufacturng tme on each machne are defned based on the varous producton scenaros. For solvng the proposed model, a partheno-genetc algorthm s employed. And, the applcaton results of the proposed model and algorthm show that the order delvery satsfacton s greatly mproved, and the schedulng workng tme s reduced. Keywords: order batch schedulng; fuzzy model; delvery satsfacton; partheno-genetc algorthm. INTRODCTION Cold rollng lne s an mportant fnshng process n steel ndustry, whose products belong to the technology-condensed and captal-condensed commodtes n the trade market. For the proft of steel enterprse, the advanced producton schedulng technque has been pad more attentons nowadays. Some studes related to the producton schedulng for tandem cold rollng lne had been carred out, whch focused manly on a certan of sngle machnes or processes. Both oon et al (999) and Lu et al (25) reported the schedulng solutons concernng the bell-type annealng process n a cold rollng lne, n whch oon et al (999) gave a mng nteger lnear programmng usng a tme slot method, and Lu et al (25) presented an optmzaton based on the dscrete event smulaton technque for the chargng optmzaton and schedulng of the annealng process. In addton, Wang et al (22) proposed a schedulng method for the optmal operatonal parameters of tandem cold mll, and Zhao et al (28) optmzed the producton plannng va a combnaton of col-mergng operaton and assgned an optmal rollng sequence for tandem cold mll. Wth respect to the ntermedate storage of cold rollng lne, the related schedulng problem was also studed n Lesten (99). Recently, Verdejo et al (29) addressed a sequencng problem n a contnuous galvanzng lne of the Spansh Steel Company, whch was solved by a heurstcs based tabu search. Tang et al (29) studed the schedulng problem of a sngle crane n annealng process of a cold rollng lne, and mentoned a two-phase heurstcs to mnmze the last col stack completon tme. Although the mentoned studes ehbted a great promoton to the manufacturng of cold rollng lne or had been partly appled to the producton practce, we dscover that there s a potental structure for the producton schedulng of cold rollng lne (see Fg.), whch conssts of not only the sngle machne schedulng, but the orders batch schedulng as well. As for the whole schedulng framework, the schedulng for sngle machne s on the bass of the orders batch schedulng soluton,.e. the results of orders batch schedulng provde the sngle machne schedulng wth the nput or the boundary condton. However, the studes n lteratures for the orders batch schedulng are rather few now. It s worth notng that such practcal optmzaton soluton s ntensvely beng requred n current steel ndustry. Although Zhao et al (28) presented an order batch schedulng model for cold rollng lne, that model completely vewed the order batch as a basc job shop schedulng model. Nevertheless, such model was somewhat nconsstence wth the manufacturng practce that cannot be entrely dealt wth by a dscrete producton mode. Sales and customer management (ERP) Order batch schedulng Sngle machne schedulng anufacturng process control system Fg.. The potental structure of producton schedulng n cold rollng lne Copyrght by the Internatonal Federaton of Automatc Control (IFAC) 54

2 Preprnts of the 8th IFAC World Congress lano (Italy) August 28 - September 2, 2 In ths study, the order batch schedulng problem of a cold rollng plant n Shangha Baosteel Co. Ltd., Chna s studed, n whch dfferent product categores and the correspondng due tme are consdered. A schedulng model wth fuzzy batch manufacturng tme and fuzzy due tme s establshed by consderng the uncertanty of batch producton on machnes. Ths model s summarzed as a specfc fuzzy Job Shop problem, to whch a trangular fuzzy number s used to denote the manufacturng tme of each batch. To evaluate the orders schedulng qualty, we take the mamzed delvery satsfacton as the objectve functon of the schedulng model, and defne a seres of calculaton prncples for the orders completon tme. A partheno-genetc algorthm (PGA) s then employed to solve the fuzzy model. In the smulaton, the real producton data from Baosteel s fnally used to verfy the proposed model and algorthm. 2. ORDER BATCHES SCHEDLING ODEL 2. Problem descrpton Takng a cold rollng plant of Baosteel Co. Ltd. as an eample, the producton machnes n the whole cold rollng lne conssts of pcklng lne (PL), tandem cold mll (TC), degreasng lne (DL), annealng lne (AL), skn passng lne (SPL), double-reduced rollng (DCR), coatng lne (CL) and fnshng lne (FL) (see Fg.2). And, each machne has ts own front nventory as a materal buffer of process. The products and the correspondng manufacturng paths can be lsted as follows: ) Hard rolled col: PL TC 2) Cold rolled col (plate): PL TC DL AL SPL 3) Super thn col (plate): PL TC DL AL SPL DCR 4) Coated col (plate): PL TC DL AL SPL CL FL 5) Super thn coated col (plate): PL TC DL AL SPL DCR CL FL Snce the producton requrements of the varous products are etremely dfferent, the devce parameters or operaton pattern has to be changed f contnuously producng the dfferent categores of products. In order to mantan the producton contnuty and reduce the adjustment cost on the machnes and the setup tme, the order groupng s just to group the orders nto some batches based on the product category constrants. At present, the order groupng and schedulng depends manly on the manual handlng by the eperenced workers. However, wth the producton development, the Baosteel s nowadays adoptng the week delvery mode to meet customers demands, mprove the delvery ablty and heghten the competton level of enterprse. Thus, the manual schedulng usually ehbts a poor capacty to fulfll the complcated requrements, and lead to a low order delvery satsfacton. Hard rolled col Cold rolled col (plate) Super thn col (plate) (Super thn) Coated col (plate) ateral DCR PL TC DL AL SPL CL FL Fg.2 The logstc stuaton and products n cold rollng plant 2.2 Order batches schedulng model In ths study, an order groupng method that combnes product category wth the order delvery week s presented. Accordng to the week delvery mode, we desgnate the due tme as Week I, Week II, Week III and etc. Then, a grouped order batch can be denoted as B pw, where p denotes the product category (p=, 2, 3, 4, 5) and w denotes the due week (w=i, II, III ). Such order groupng wll be the foundaton of the order batch schedulng model. The manufacturng tme of orders s very hard to be accurately predcted because of some uncertan events n producton, therefore t s dffcult to reasonably reflect the practcal stuaton only by the fed schedulng parameters. For eample, the completon tme of an order may be prolonged due to an unepected machne malfuncton or materal delay. So, usng fuzzy concepton could be an approprate attempt to solve such practcal schedulng problem. We vew the grouped orders as a schedulng job and map the producton of cold rollng plant nto a job shop schedulng problem (JSSP). Consderng the uncertan factors n producton, the manufacturng tme of each job on a machne ~ s represented by a trangular fuzzy number L A ( a, a, a ) where, =,2,,n denotes the nde of jobs; j=,2,, m denotes the nde of machnes; k denotes the current sequence number of the whole process of job, whch means the k-th step of job s processed on machne j; l denotes the current job s the l-th job of machne j. The varables, a, a and a, represent the shortest L manufacturng tme, the possble tme and the longest tme of a job, respectvely. And, the membershp curve s llustrated as Fg.3. In addton, we specfy the due tme of an batch s denoted by a trapezod fuzzy number ~ O D ( d, d P ), where 542

3 Preprnts of the 8th IFAC World Congress lano (Italy) August 28 - September 2, 2 O P d and d denote the optmstc due tme and pessmstc tme of job, respectvely. (See Fg.4) a ~ ( ) L a a a Fg.3 embershp functon of manufacturng tme of an order batch satsfacton of a batch as the ntersecton area between the completon tme F and the due tme of the batch D (see Fg.5). Eq.(2) denotes that the earlest start tme of job s just the ntal tme of the schedule f t s the frst job of the grouped batches and the frst work on machne j. Eq.(3) denotes that the earlest start tme of job s the completon tme of the prevous job on machne j f t s the frst step of job, whch means the succeedng job can be started on machne j only f the prevous job has been done. Eq.(4) denotes that the completon tme of each job s the fuzzy sum of the start tme and the manufacturng tme of the job. ~ d ( ) D ~ O d P d F ~ O d P d Fg.4 embershp functon of due tme of an order batch To evaluate the order delvery qualty of the schedulng, an objectve addressed by the average delvery satsfacton s adopted n ths paper (asatosh et al, 999). The model for batch schedulng of the cold rollng plant reads as follows. n ( areaf ) ( aread ) a: J () n areaf Subject to: ~ S l and k,,2,..., n, j,2,..., m (2) ~ ~,,2,..., n, j,2,..., m (3) ~ ~ ~ Fj Sj Aj,2,..., n, j,2,..., m (4) S F j ( l) l and k where, areaf denotes the area of fuzzy completon tme of job ; aread denotes the area of fuzzy due tme of job ; S ~ s the fuzzy start tme of job on machne j; ~ F s the ( k) ~ fuzzy completon tme of (k-)-th step of job ; F s the j( l) fuzzy completon tme of (l-)-th step of machne j; F ~ j s the fuzzy completon tme of job on machne j; F ~ s the fuzzy completon tme of job ; and, A ~ s the manufacturng j tme of job on machne j. Eq.() s the objectve to mamze the average delvery satsfacton of a grouped batch. We defne the delvery Fg.5 The calculaton of delvery satsfacton of each batch In general JSSP, the earlest start tme of job on machne j s usually the larger one between the completon tme of the prevous step of job and the prevous job on machne j, represented as: S ma{ F, } ( k F. However, to the ) j ( l ) proposed schedulng model, the calculaton has to be redefned snce the produced steel cols may be contnuously transported nto the nventory of net machne when the batch of orders s stll produced on current machne. The start tme of job on machne j cannot be only calculated by Fk ( ) and Fjl ( ), and the start tme of prevous step of job should be further consdered meanwhle. 2.3 Fuzzy calculaton prncple 2.3. Rankng a trangle fuzzy number Accordng to the theorem of fuzzy arthmetc defnton (Kaufman et al, 988), the rankng calculaton rules can be vewed as a rankng crteron for a trangle fuzzy number. L Rank calculaton : u 2u u Ru (). 4 Rank calculaton 2: R2( u) u f R( u ) s unavalable to evaluate two trangle fuzzy numbers. Rank calculaton 3: L R3( u) u u f both R( u ) and R2( u ) are unavalable. Here s a calculaton eample. Four trangle fuzzy numbers are u (2, 4,6), u2 (,4,7), u3 (3,6,7) and u4 (2,3,8), respectvely. sng the above rankng rules, then 543

4 Preprnts of the 8th IFAC World Congress lano (Italy) August 28 - September 2, 2 R ( u) R ( u2) R ( u4) 4, R( u3) 5.5. So, uma u3 ; R2( u) R2( u2) 4, R2( u4) 3, So, umn u4 ; and R3( u) 4, R3( u2) 6. Then, the descendng order of the trangle fuzzy numbers becomes Rank( u3) Rank( u2) Rank( u) Rank( u4) Tme calculaton prncple for order batch Gven a class of crcumstance as an eample, there mght be a part of steel cols of job that have been already produced from the prevous machne, but ths job stll are not completely fnshed; whereas, the machne j s avalable at that tme. Therefore, we propose the followng three producton scenaros to determne n ths study. Scenaro : Rank( Fj( l) ) Rank( F ( k) ), whch means that the completon tme of (l-)-th job on machne j s later than that of (k-)-th step job. (See Fg.6 (a)) Rank( Sk ( ) ) Rank( F Scenaro 2: j( l) ) Rank( F ( k) ), whch means that the completon tme of (l-)-th job on machne j s later than the start tme of (k-)-th step of job, but earler than ts completon tme. (See Fg.6 (b)) Rank( F ( ) Scenaro 3: jl ( ) ) Rank( Sk ), whch means that the completon tme of (l-)-th job on machne j s earler than the start tme of (k-)-th step of job. (See Fg.6 (c)) F k ( ) Fjl ( ) Sk ( ) Fjl ( ) F k ( ) (a) Scenaro (b) Scenaro 2 Fjl ( ) Sk ( ) (c) Scenaro 3 Fg.6 The relatonshp of fuzzy numbers n three dfferent producton scenaros We assume that the manufacturng and setup tme of a sngle steel col can be neglected. Takng the three scenaros nto account, the calculaton of the start tme of an order batch (job ) on machne j, followng rules. S, can be determned by the Rule : If the Scenaro s n the case (all steel cols n job have been placed nto the nventory of machne j before the prevous job on machne j s fnshed), then S Fjl ( ). Rule 2: If the Scenaro 2 s n the case (a part of steel cols n job have entered nto the nventory of machne j before the prevous step of job s totally fnshed), then S S F ( k) jl ( ). The operator [ ] s to obtan the larger one of the two values. Rule 3: If the Scenaro 3 s n the case (the prevous step of job just starts after the prevous job on machne j has been S S( k) F jl. totally fnshed), then ( ) 2.4 Parameters of membershp functon We gve an eperence based manufacturng tme calculaton depended upon the sze of steel cols (see Eq.(5)). Wg t( C) thwd Qv where, s the coeffcent; Wg s the weght of the steel col; th s col thckness; wd s col wdth; Q s densty of steel; v s the strp velocty of producton; and C s the compensaton value of manufacturng tme. Note that the eperence based tme predcton only shows the normal producton condton. Some abnormal stuaton should be also consdered such as machne mantenance, malfuncton and so forth. Thereby, the mentoned tme predcton wll be appled to descrbe the shortest manufacturng tme, a L. As for the possble tme a should be determned by the average producton capacty (see Eq.(6)), where t j s the accumulatve tme of current type of product on machne j, W s the accumulatve j turnout of the product category on machne j, and W s the weght of batch. Eq.(7) gves the longest manufacturng tme a calculaton consderng the machne breakdown, where T s the longest breakdown tme of machne j that j ncludes the regulatve or malfuncton machne shutdown. a t (5), t j W (6) Wj a a T (7) j 544

5 Preprnts of the 8th IFAC World Congress lano (Italy) August 28 - September 2, 2 3. SOLVING FOR THE ORDER BATCH SCHEDLING The proposed schedulng model can be regarded as an asymmetrcal travelng salesman problem, whch s wth the characterstcs of NP-hard. As t was known, t s very hard for such problem to obtan an optmal soluton wthn an acceptable computaton tme only by the generc mathematc programmng, even some ntellgent optmzaton. In ths paper, we propose a partheno-genetc algorthm (PGA) to solve ths combnatoral optmzaton. 3. partheno-genetc algorthm Partheno-genetc algorthm s a class of specal evoluton algorthm that elmnates the nfeasblty of crossover between two chromosomes organzed by ordnal strngs, and has been appled to some research felds (Katayama et al, 2). In ths study, we encode the schedulng soluton by a strng of ntegers, and the sequence of the nteger n the strng denotes the manufacturng sequence of the correspondng order batch for the cold rollng plant. We desgn the gene swap as the chef parthenogenetc operator on each ndvdual, assocated wth the substrng move. Here s an eample. Operator of gene swap: Sequence number: Parent: Chld: Operator of substrng move: Substrng: Parent: Chld: After creatng the new ndvduals, the tournament selecton s appled to establsh the new generaton as usual. 3.2 Solvng step of the schedulng The solvng step of order batch schedulng for the cold rollng plant s as follows. Step : Group the avalable orders to form a few of batches based on the order due tme and product category. Step 2: Calculate the fuzzy manufacturng tme of each batch on the machnes, and determne the fuzzy membershp functon of manufacturng tme and due tme. Step 3: Intate the status of the machnes n the cold rollng plant, and randomly generate a number of n-bt strngs represented as a set of ntal solutons. Step 4: Calculate the fuzzy completon tme and the delvery satsfacton of each batch by the proposed computaton prncples, and obtan the average satsfacton of each soluton and the current optmal soluton. Step 5: se the evoluton operator to create the new ndvdual, and obtan the best-so-far of the fuzzy schedulng soluton. Step 6: Check whether the teraton number s arrved. If so, output the optmzed order batch schedulng soluton; otherwse, go back to step DATA EXPERIENTS AND RNNING ANALYSIS A large number of eperments wth real data from the cold rollng plant of Baosteel have been employed. To verfy the effect of the proposed fuzzy schedulng model, we present an eample to ndcate the study for order delvery satsfacton compared to the manual schedulng. Table llustrates a group of four-week producton data n detal. sng the proposed method, the result of the nstance has been confrmed to be effectve by the eperenced schedulng workers of the cold rollng plant n Baosteel. Table. Informaton of order batches n detal Batch No. Product category Due week Order amount Batch weght (t) I II III IV I II III IV I II IV I II III IV I II IV In addton, we ehbt here a comparatve result for the fuzzy JSSP by usng the ntellgent algorthms, nvolvng the dscrete dfferental evoluton (DDE) (Storn et al, 997), Tabu Search (TS) and the proposed PGA. The result by eperence based manual schedulng (S) s together addressed. We select DDE because ts soluton qualty and computatonal tme are becomng more and more attractve to researchers and practtoners. The statstc comparsons are lsted n Table 2, where four groups of real data from Baosteel are used. In the statstcs, we make tmes ndependent eperments for each algorthm, and obtan the average values of the soluton. From the table, the ntellgent search algorthms, whose average delvery satsfactons fundamentally eceed 9%, get the farly better solutons than that by the manual schedulng. Furthermore, S always takes the eperenced worker over two hours to complete 545

6 Preprnts of the 8th IFAC World Congress lano (Italy) August 28 - September 2, 2 such knd of schedulng work, whch s a rather hgh labour force to humans. As for the order delvery satsfacton, the proposed PGA gets the best schedulng soluton than the other algorthms. For the computatonal tme respect, although the TS s somewhat faster than the proposed PGA, the solvng dfference can be completely accepted n the practce of producton schedulng process. All n all, one can conclude that the proposed PGA presents a sound solvng performance. Table 2. The statstcs comparson usng four methods to solve the proposed model No. Batches Average Delvery Satsfacton Computatonal Tme (on average) amount S PGA TS DDE PGA (s) TS (s) DDE (s) 7 8.3% 93.2% 9.8% 9.2% % 94.4% 9.2% 92.3% % 92.9% 9.5% 9.% % 93.9% 9.% 9.9% CONCLSIONS In ths study, a rule-based JSSP model wth fuzzy manufacturng tme s establshed for the order batch schedulng problem n the cold rollng plant of Baosteel. An evoluton algorthm s desgned to solve the fuzzy schedulng model. The runnng results usng the real producton data from Baosteel show that the proposed model and ts algorthm ehbt a better delvery satsfacton of the orders than manual schedulng or some other ntellgent optmzatons. The average delvery satsfacton of orders reaches above 9%, whch greatly meets the customer tmng demands n the market. Furthermore, snce the schedulng soluton can be obtaned wthn a dozen of seconds, the work effcency of producton schedulng wll be largely heghtened compared to the former eperence based manual method. ACKNOWLEDGEENTS Ths work s supported by the Natonal Natural Scence Foundaton of Chna (6343). The cooperaton from the cold rollng plant of Shangha Baosteel Co. Ltd., Chna n ths work s greatly apprecated. REFERENCES Katayama K., H. Narhsa. On fundamental desgn of parthenogenetc algorthm for the bnary quadratc programmng problem, Proceedngs of the Congress on Evolutonary Computaton, Seoul, 2. Kaufman A.,. Gupta. Fuzzy mathematcal models n engneerng and management scence, North-Holland, Amsterdam, 988. Lesten R. (99) Flow-shop sequencng problems wth lmted buffer storage, Internatonal Journal of Producton Research, Vol.28, pp Lu Q., W. Wang, H. Zhan, et al (25). Optmal schedulng method for a bell-type batch annealng shop and ts applcaton. Control Engneerng Practce, Vol.3, No., pp asatosh S.,. Testuya (999). An effcent genetc algorthm for job-shop schedulng problems wth fuzzy processng tme and fuzzy duedate. Computers & Industral Engneerng, Vol.36, No.2, pp oon S., A. N. Hrymak (999). Schedulng of the batch annealng process-determnstc case. Computers & Chemcal Engneerng, Vol.23, No.9, pp Storn R., K. Prce (997). Dfferental evoluton - a smple and effcent heurstc for global optmzaton over contnuous spaces, Journal of Global Optmzaton, Vol., pp Tang L., X. Xe, and J. Lu (29). Schedulng of a sngle crane n batch annealng process. Computers & Operatons Research, Vol.36, pp Verdejo V.,. P. Alarco, and. Lno Sorl (29). Schedulng n a contnuous galvanzng lne, Computers & Operatons Research, Vol.36, pp Wang D., A. K. Teu, F. G. Boer, et al. (22) Toward a heurstc optmum desgn of rollng schedules for tandem cold rollng mlls. Engneerng Applcaton of Artfcal Intellgence, Vol.3, No, 4, pp Zhao J., Q. Lu, W. Wang (28). odels and algorthms of producton schedulng n tandem cold rollng, Acta Automatca Snca, Vol.34, No.5, pp Zhao J., W. Wang, Q. Lu (28). odel and algorthm of groupng batch schedulng for cold rollng sheet producton lne, Computer Integrated anufacturng Systems, Vol.4, No., pp