A Mxed Lnear Program for a MultPart Cyclc Host Schedulng Problem Adnen El Amraou,, MareAnge Maner, Abdellah El Moudn and Mohamed Benrejeb U.R. LARA Automatque, Ecole Natonale d Ingéneurs de Tuns, Tunse. BP 7, Belvédère, Tuns, Tunsa Laboratore Systèmes et Transports, Unversté de Technologe de BelfortMontbélard, France. UTBM Ste de Belfort 9 Belfort Cedex, France {Adnen.ElAmraou, mohamed.benrejeb}@ent.rnu.tn {mareange.maner, abdellah.elmoudn}@utbm.fr Abstract. Ths paper deals wth a snglehost cyclc schedulng problem n electroplatng systems. A comprehensve mxed lnear programmng model, that can be used to fnd the optmal sequence mnmsng the cycle tme for two jobs or maxmzng the throughput of the producton system, s proposed. To mprove the throughput n the process, a refned model s developed by allowng the host to be stopped wth charge. After descrbng the problem to solve, the basc model and the refned model, benchmarks and examples are gven to llustrate the performances of proposed models. Keywords: MultProduct Cyclc Host Schedulng Problem; Electroplatng System; Mxed Lnear Programmng Model Introducton Ths paper studes the degree cyclc schedulng of two dfferent part jobs, where durng one cycle, two dfferent jobs are ntroduced nto the producton lne and two others are lvng t. Ths problem s wdely encountered n schedulng problem, such as n manufacturng and ppelne archtecture where a large producton s requred. Ths s partcularly the case n electroplatng facltes, where a host s programmed to perform a move sequence. In ths work we consder the Cyclc Host Schedulng Problem (CHSP), n whch some set of actvtes are to be repeated an ndefnte number of tme. In ths set of actvtes, the cycle tme s mnmsed and processng tme constrants as well as transport travellng tme constrants are satsfed. Ths class of problem has proven to be an NPcomplete problem []. Snce the frst model gven by Phlps and Unger [] several researchers have been nterested to the cyclc schedulng problem [5] and a large number of mathematcal! " # # $! %! " " & # $! ' % # %! % $$ # %! " ( ) *! $ " " + (,,,,.
IJSTA, Specal Issue, CEM, December 8. models [6789] and heurstc algorthms [5] were developed. Nevertheless, few works have been nterested to the multproduct case. Ptuskn n [6], for example, consders the problem that parts are processed n the same sequence, wth varous processng tmes and a sequence of dfferent parts perodcally s enterng the system. The sequence s consdered to be known, and the date of each part has to be computed. Ths problem s decomposed n several monoproduct subproblems and the soluton corresponds to a common perod. Mateo n [7] develops a branch and bound procedure whch bulds the sequence of movements progressvely. Each level of the search tree conssts of addng one tank and thus, the stages to be done on t. A lnear program s then solved at each node to check the consstency of the constrant system. In ths study, a mxed lnear programmng model s elaborated to solve sngle host schedulng problem wth multproduct part jobs. The paper s organzed as follow. Secton provdes a descrpton of the sngle host schedulng problem. The sngle host cyclc schedulng problem wth multple products s modelled n secton. Illustratve examples are presented n secton. Conclusons are gven n secton 5. Problem statements The consdered problem s a sngle host multproduct schedulng problem. It conssts n fndng the optmal move sequence of the host that mnmze the cycle perod. Ths sequence s contanng two dfferent partjobs wth the same tank route and ndvdual wndow constrants. It must respect the followng partculartes: Two knds of products are to be treated n equal quantty. Every product type must be treated through the same baths. Each processng tme have to be ncluded between a mnmum and maxmum duratons. Each one tank must receve at most one carrer at a gven tme. Host can not move more than one carrer n the same tme. Between two successve moves n the sequence, host must have enough tme to travel empty. Stock s not authorzed between two soakng operatons. P load P 5 k+ k tank tank tank tank k Fg.. Dagram of a host producton lne wth k+ tanks and k+ stages processng sequence of P processng sequence of P unload
A Mxed Lnear Program for a MultPart CHSP A. El Amraou et al. Fg. shows an example of an electroplatng lne, wth a sngle host, k+ tanks, and k+ stages used to process two dfferent types of products P and P, every cyclc perod. At each cycle, there are two dfferent jobs enterng the lne and two others are lvng. The problem treated s charactersed as follow. The lne s composed of k+ tanks where the frst tank and the last one are respectvely the load and the unload staton. In every tank, there are two soakng operatons to be done, one for every product type. Job of product s transported by the host from the frst stage (stage ), nto the followng even stage (,,, k) and the second type of product s transported by the host from the frst stage (stage ), nto the followng odd ones (, 5,, k+). The processng tme of job j n tank k, s gven by t k for the frst product f k s even and for the second product f k s odd; some tme ponts are gven n fg.. Tank t t 7 t 7 t 5 t 6 t 5 t t t t t t t t T T Fg.. Example of tme way dagram correspondng to the consdered problem t t 6. Basc model Problem parameters Let defne the followng notaton. n = the number of processng stages for the two products. s = the tank used to acheve the process of stage, =,,,, n+. a = the mnmum processng tme n stage, =,, n. b = the maxmum processng tme n stage, =,, n. d = the tme needed for a host to move a carrer from tank to tank +, =,, n+. c,j = the tme needed for one host to move empty from tank to tank j,, j =,..., n+. M = a very bg number (represents the value +). H = the sequence of host movements: H=(h [], h [],,h [n] ), wth h s the host travellng from the stage to the followng stage +, for job P f s even and for job P f s odd. These notatons are used n Jyn Lu, Yun Jang and Zhl Zhou model [].
/ IJSTA, Specal Issue, CEM, December 8. Decson varables T = the perod of the cycle. t = the startng tme, durng one cycle, of move from stage to stage +: h, =,.., n. f t < t j y,j =,j =,, n, < j. otherwse In the model there s a total of n+ contnuous and n(n)/ bnary varables to fnd. The model Usng the above notatons, the model s formulated as the followng lnear equaltes and nequaltes: Subject to: Mnmze T () d + a M( y ) t t. =,...,n, (), t t d + b + M( y ). =,...,n, (), d + a My t + T t. =,...,n, (), t + T t d + b + My. =,...,n, (5), t t d + c M( y )., j =,...,n, < j (6) j s +,s, j t t d + c My., j =,...,n, < j (7) j j s j+,s, j t d + c. =,...,n, (8) s,s T t d + c s +,s. =,...,n, (9) y + y. =,...,n, f s = s +, (),+ +,+ y + y y y. =,...,n, f s = s +, () +,+ +,,+ +, + y, j y j,., j =,...,n, < j () y y., j =,...,n, < j () j,, j
A Mxed Lnear Program for a MultPart CHSP A. El Amraou et al. t. =,...,n, (), j { } y,., j =,...,n, < j (5) T. (6) In ths model, the objectve s to mnmze the cycle tme T (). Constrants ()(6) are postve and bnary constrants. Constrants ()() ensure that y,j and y j, are defned correctly (.e., for < j f y j, = then y,j = (and vce versa)). Constrants (7)(8) means that the frst move from the loadng tank to the frst stage s equal to zero and match to the frst move n the cycle. The other constrants can be classfed nto three classes. In fact, the frst class constrants (constrants ()(5)) are related to tanks and ensure that the processng tme at each stage respects a specfc tme wndow (maxmum and mnmum processng tme). The second class of constrants (constrants (6)(9)) s assocated to host and guarantee that there s enough tme to travel. The thrd class and the last one (constrants ()()), ensures the feasblty of the cyclc sequence, analysed below. These last two constrants are added to the model to guarantee that there s no more than one carrer n each tank. If two jobs are usng the same tanks (k and k+) durng there process sequences, one of the followng cases, (a), (b) and (c) has to be consdered. Case (a): If the two processes are n the same cycle boundary The frst carrer (contanng the frst type of product) enters tank k, when the product s treated t leaves ths tank for the next stage and when ths second stage s acheved t moves away and t wll be the turn of the second carrer (contanng the second type of product). Thus, the feasble sequence s (, +, +, +). It s the same sequence than follow but the second carrer s treated frstly (the second product wll be treated n prorty) and then the feasble sequence s (+, +,, +). Case (b): f one of the processes (+ or +) crosses the cycle boundary In the stage + the frst carrer s moved nto tank k+, later n a cycle and t wll be moved away early n the next cycle. And then the feasble sequence s defned by (+, +, +, ). Smlarly, the second carrer s moved nto tank k+ for the stage +, later n a cycle and t wll be moved away early n the next cycle. And then the feasble sequence s defned by (+,, +, +). Case (c): f the two processes + and + cross the cycle boundary It s mpossble because two carrers wll be n the same tank and then the cyclc sequence wll be nfeasble. All ths cases can be seen n fg..
, IJSTA, Specal Issue, CEM, December 8. P mmerson sequence + P mmerson sequence + + tank k tank k+ Fg.. Stages usng the same consecutve tanks for dfferent jobs. Refned model In some manufactores, the host have a lttle mpact n the producton process, t s used for packagng and storng, but n ths class of problem (HSP) the sequence how to move the host s mportant and can affect the productvty and therefore the performance of the company n term of throughput and qualty. Thus, by analysng the host move n load, a new varable (w ), defned as a slack tme between the actual tme taken and the mnmum tme requred for the move, s ntroduced. In other term, the host s allowed to be stopped wth charge []. To take ths varable nto account, constrant (7) s added to the model and constrants ()(9) are substtuted wth the followng ones (constrants 85): w. =,...,n, d + a M( y, ) t t w. =,...,n, t t w d + b + M( y, ). =,...,n, d + a My, t + T t w. =,...,n, t + T t w d + b + My,. =,...,n, (7) (8) (9) () () t t d + w + C M(y )., j =,...,n, < j j s +,s j,j t t d + w + C My., j =,...,n, < j j j j s j+,s,j t d + w + C. =,...,n, s,s T t s +,s () () () d + w + C. =,...,n, (5) () (5)
. Complexty A Mxed Lnear Program for a MultPart CHSP A. El Amraou et al. The proposed nteger lnear model s functon of the number of jobs and work statons. For two jobs, n the basc model, there are (n)(n+)+(n8) processng staton constrants and materal handlng constrants. Then, the total number of constrants s O(n ). Moreover, there are two types of decsons varables. There are a total of n+ contnuous, t and T and (n)(n)/ bnary varables y,j to fnd. By consderng that host s allowed to be stopped wth charge, the number of processng staton constrants and the materal handlng constrants n the refned model s ncreased by n and thus the total number of constrants s stll O(n ). And n ths refned model there s a total of n+ contnuous, t, w and T and (n)(n)/ bnary varables y,j to fnd. Numercal examples In the am to llustrate the effcency of the model and the refned model, two examples are defned and a comparson s also made between the models, the refned model and benchmarks examples found n lterature. The modal and the refned model are solved usng commercal software, CPLEX, on a Pentum wth GHZ frequency processor.. Example Ths example s smlar to the one gven by Mateo n []. It s used here to prove the effcency of elaborated basc model. In ths example, two products must be soaked n a lne wth 5 tanks ncludng the loadng and the unloadng statons. The tme spent soakng for every part product s gven by table. The tme needed for the host to move empty between tanks of successve treatments s tme unt (t.u.). Wth load, the tme needed for the host to move s 5 t.u. Table. Data of example Stage () Job Tank (s ) a b d 5 6 7 5 5 5 5 5 8 8 75 5 5 5 5 5 5 5 5 The optmal soluton for example s gven n table.
. IJSTA, Specal Issue, CEM, December 8. Table. Results of example T 8 t t 65 t 6 t 5 t t 5 5 t 6 t 7 85 The optmal cycle length s 8 t.u. as shown n fgure and t s equal to the cycle tme fnd by []. Usng the refned model, the same result s obtaned. Tank 5 6 85 65 Fg.. Transportaton tme of example Contnuous lnes show the loaded host travellng tme between two successve stages, whle dscontnuous arcs show the unloaded host travellng tme between dfferent tanks. Table provdes the mnmum cycle tme found n lterature, obtaned from the basc model and from the refned one for the cases of 5, 6, 7, 8, 9 and tanks. These nstances were used by Mateo n [7]. T = 8 5 tme Table. Comparson wth Mateo benchmarks Problem number 5 6 tank number (k) 5 6 7 8 9 T mn lterature 68 5 6 65 69 77 T mn basc model 68 5 6 65 69 77 T mn refned model 68 5 6 65 69 77 Where T mn lterature s the best cycle tme known n lterature, T mn basc model s the best cycle tme found by the basc model and T mn refned model s the best cycle tme found by the refned model. These results show that the best soluton found n the lterature (T mn lterature) s reached by the use of the basc and the refned models and the obtaned results have been confrmed by more than 5 benchmarks.
A Mxed Lnear Program for a MultPart CHSP A. El Amraou et al. The same results are obtaned by the refned model. Ths s due to the fact that, tme requred to the host to travel between two tanks s closed to the processng tme and tme wndows are not enough large to provde flexblty to the model and thus to reduce the cycle tme.. Example Ths example s used here to prove the effcency of the elaborated refned model. In ths example, two products must be soaked n a lne wth 5 tanks ncludng the loadng and the unloadng statons. The tme spent soakng for every part product s gven by table. The tme needed for the host to move empty between tanks of successve treatments s 5 t.u. Wth load, the tme needed for the host to move s t.u. If the slack tme s forced to be zero, by usng the basc model, the two carrers enter to the lne durng a cycle and they are treated each n 8 t.u. as gven by fg. 5. Thus, the mean perod between two carrers s 5 t.u., never the less, wth nonzero slack tme, the optmal cycle tme s compacted to 7 t.u. as shown on fg. 6, and the mean perod s then reduced by 8 t.u. Moreover, n term of throughput, durng 5 t.u., n the frst case, 6 jobs for each product are obtaned however, n the second case 88 jobs for each product are obtaned, t means that throughput s mproved by,7 %. Furthermore, the optmal sequence H = (, 7,,,, 6,, 5), obtaned from table 5 and llustrated by fgure 5, s dfferent from the optmal sequence H = (, 5,, 7,,, 6, ), obtaned from table 6 and llustrated by fg. 6. It can be seen from fg. 5, that sequence H s not feasble wthout usng the slack tme. In fact, the ffth stage has to be acheved n the best at 86 t.u and n the worst at 9 t.u. In the best, the second carrer attends the unload staton at 96 t.u. and the frst stage at t.u., thus the frst carrer exceeds the maxmal requred tme. In the worst, the host moves the frst carrer from the frst stage to the second one and t attends the ffth stage at t.u. and consequently, the second carrer exceeds the upper bound. Table. Data of example Stage () Job Tank (s ) a b d 5 6 7 8 76 6 6 96 9 6 6 66 66 The optmal soluton for example s gven n tables 5 and 6.
IJSTA, Specal Issue, CEM, December 8. Table 5. Results of example usng the basc model T 8 t t 66 t 9 t 5 t t 5 8 t 6 t 7 5 Table 6. Results of the example usng the refned model T 7 t t 66 t 9 t 5 t t 5 5 T 6 t 7 5 w w w w w w 5 w 6 w 7 Tank T = 8 6 5 9 6 tme 66 7 Fg. 5. Best host movements sequence for example usng the basc model Tank Load move wth Slack tme 5 9 5 66 T = 7 tme T = 8 Fg. 6. Best host movements sequence for example usng the refned model 6 Cycle tme reducton Usng the elaborated, basc model, the tme consumed by the host to carry out a complete sequence of movements of two jobs s 8 t.u. Nevertheless, by consderng the refned model and for t.u. (95) of slack tme, the cycle tme s reduced to 7 t.u.
Conclusons A Mxed Lnear Program for a MultPart CHSP A. El Amraou et al. In ths paper, the sngle host degree cyclc schedulng for dfferent part jobs problems s studed and a mxed nteger lnear programmng model s proposed to solve ths problem. By consderng the llustratve example and usng a set of more than 5 test problems, the obtaned computatonal results have shown the effcency of the elaborated basc model to fnd satsfyng solutons for the consdered problem. Then, a refned model s elaborated wth the assumpton that the host s allowed to be stopped wth charge. Ths refned model s appled to the llustratve example n order to show how, by consderng the slack tme, the cycle tme can be reduced and producton performance can be consderably mproved. References. Le, L., Wang, T.J.: A Proof: The Cyclc Host Schedulng Problem s NPComplete, Workng Paper, No. 896. Rutgers Unversty. New Jersey (989). Phllps, L.W., Unger, P.S.: Mathematcal Programmng Soluton of a Host Schedulng Program. AIIErTransactons, Vol. 8, No.. (976) 9 5. Zhou, Z., Lng, L.: Sngle host cyclc schedulng wth multple tanks: a materal handlng soluton. Computers & Operatons Research, Vol.. () 8 89. Maner, M.A., Bloch, C.: A classfcaton for Host Schedulng Problem. Int. J. of Flexble Manufacturng Systems. Edton Kluwer. Academc Publshers, Netherland, () 755 5. Bloch, C., Maner, M.A. Notaton and Typology for the Host Schedulng Problem. IEEE Internatonal Conference on Systems, Man and Cybernetcs, Vol., Tokyo (999) 75 8 6. Spacek P., Maner M.A., El Moudn., A.: Control of an Electroplatng Lne n the Max and Mn Algebras. Int. J. of Systems Scence, Vol., No. 7 (999) 759 778 7. Shapro, G. W., Nuttle, H.L.W.: Host Schedulng for a PCB Electroplatng Faclty. IIE Transactons, Vol., No. (988) 57 67 8. Armstrong, R., Le, L., Gu, S.: A Boundng Scheme for Dervng the Mnmal Cycle Tme of a SngleTransporter NStage Process wth TmeWndow Constrants. European Journal of Operatonal Research, Vol. 75 (99) 9. Che, A., Chu, C., Levner, E.: A polynomal algorthm for degree cyclc robot schedulng, European J. of Operatonal Research, Vol. (). Baptste, P., Legeard B., Maner M.A., Varner C. : Résoluton d un problème d Ordonnancement avec la PLC, J. Européen des Systèmes Automatsés. Intellgence Artfcelle et Automatque, Vol. (996). Baptste, P., Legeard, B., Varner, C.: Host schedulng problem : an approch based on constrants logc programmng. Proceedngs of IEEE Conference on Robotcs and Automaton, Vol. (996) 9. Le, L.: Determnng optmal cyclc host Schedules n a snglehost electroplatng lne. IIE Transactons Vol. 6 (99) 5. Mateo, M., Companys, R.: Resoluton of graphs wth bounded cycle tme for the Cyclc Host Schedulng Problem. 8 th nternatonal, workshop on project management and schedulng. Valenca () 576. Lu, J., Jang, Y., Zhou, Z.: Cyclc schedulng of a sngle host n extended electroplatng lnes: a comprehensve nteger programmng soluton. IIE Transactons Vol. () 95 9
IJSTA, Specal Issue, CEM, December 8. 5. Yh, Y. : An algorthm for host schedulng problems. Int. J. of Producton Research (99) 56 6. Ptuskn, A.S.: Nowat Perodc Schedulng of Nondentcal Parts n Flexble Manufacturng Lnes wth Fuzzy Processng Tmes. Int. Workshop on Intellgent Schedulng of Robots and Flexble Manufacturng Systems. Center for Technologcal Educaton Holon (995) 7. Mateo M., Companys, R.: Host Schedulng n a chemcal lne to produce batches wth dentcal szes of dfferent products. Sxème Conférence Francophone de Modélsaton et SIMulaton, MOSIM 6. Rabat (6) 67768 8. ILOG CPLEX., User s Manuel (6)