A Hybrid Scheduling Algorithm for Multiclass Production Systems with Setup Times

Transcription

1 Hybrd Schedulng lgorthm for Multclass Producton Systems wth Setup Tmes Y. NRHRI omputer Scence and utomaton Indan Insttute of Scence angalore 0 0 INI R. SRIGOPL omputer Scence and utomaton Indan Insttute of Scence angalore 0 0 INI bstract In ths arttcle, we propose a hybrd algorthm for schedulng nonpreemptve, sngle operaton jobs n a multclass producton envronment. The objectve s to mnmze the sum of the total weghted tardness and setup costs of the schedule. We beleve the problem s NP hard, and we use an efcent suboptmal algorthm based on Lagrangaan relaxaton and smulated annealng. Our algorthm works well for a varety of schedulng problems. Introducton The problem of schedulng arses n stuatons where scarce rt:sources have to be optmally allocated to actvtes over tme. Most schedulng problems belong to the class of NP hard combnatoral optmzaton problems. ny schedulng methodology should am to PI. generate effcently near optmal solutons wth measurable performance.. perform what f analyss to examne the mpact of dynamc changes.. develop effcent methods for schedule reconfguraton to accomodat,e these changes. In the area of dscrete actvty schedulng, t s generally accepted that a gap exsts between schedulng theory and practce. Practcal methods react to dynamc changes wthout the ablty to produce good solutons and theoretcal methods produce good schedules wthout the ablty to react to dynamc changes. Recently, Luh et a [ll] and Hotomt et a [S, have developed a Lagrangan relaxaton based suboptmal algorthm for schedulng of nonpreemptve sngle/mult operaton jobs on parallel dentcal rrachnes and for job shop schedulng. Ther method performs very well n a wde varety of schedulng stuatons and s also amenable for carryng out extensve whatf analyss. However, ther methodolgy does not take nto account setup tmes and setup costs that are very mportant n multclass manufacturng system schedulng. Our present work attempts to extend the schedulng methodology to multclass producton systems comprsng parallel dentcal machnes and takng nto account setup tmes and setup costs. In a multclass producton settng, the jobs are dvded nto a number of mutually exclusve part types. Setup operatons are an mportant feature of such producton envronments. sgnfcant setup tme s ncurred when a machne changes from processng one type of parts to a dfferent type of parts. The setup tme generally ncludes tmes for fxturng tool changng and preparng the workplace. Thus, a setup cost s ncurred, snce the setup operatons do not contrbute to productvty. To mnmze the setup tmes and costs, a batch of products belongng to the same part type s manufactured after a sngle setup. Large batch szes on the other hand result n hgh nventory levels. The economc lot szng problem (LSP) [] addresses ths problem of mnmzng the sum of nventory and setup costs. The problem s known to be NP hard [lo]. The work reported n ths paper s an extenson of the work by Luh et a [ll]. The method combnes the technques of Lagrangan relaxaton and smulated annealng. The objectve s to mnmze the sum of the total weghted tardness and setup costs (assumed to be a monotoncally ncreasng functon of the setup tmes). The next secton s a survey of relevant lterature. It deals wth the schedulng of jobs n a sngle class producton envronment as descrbed by Luh et a [ll]. The secton summarses the nteger programmng formulaton of the schedulng problem and the soluton 00 $.00 0 I

2 methodology. It also brefly revews optmzaton usng smulated annealng. Secton proposes a hybrd methodology to a multclass producton settng wth setup tmes ncluded. Three examples are dscussed to test the proposed methodology and detaled numercal results are provded. Secton presents conclusons and drectons for future work. Revew of Lagrangan Relaxaton and Smulated nnealng. Lagrangan Relaxaton Lagrangan relaxaton [,, provdes an effcent way of schedulng ndependent jobs wth due dates on dentcal parallel machnes. The specal nteger programmng formulaton facltates the applcaton of the Lagrangan relaxaton technque. ecomposton of the dual problem serves to smplfy soluton at the lower level. The hgh evc.l problem s solved va a subgradent method. ynamc changes can easly be accommodated n ths approach. In ths secton, we provde a revew of the Lagrangan relaxaton technque as appled to schedulng of nonpreemptve, sngle operaton jobs on parallel dentcal machnes. The materal s mostly taken fron [ll]... Problem Formulaton n nteger programmng formulaton as descrbed n [ll] s a common way to represent a schedulng problem. The followng s a statc, dscrete tme, nteger programmng formulaton of the schedulng problem. We shall use the followng notaton. total number of jobs. tme horzon under consderaton. weght f job. processng tme of job. due date of job. number of machnes avalable at tme k (assumed to be monotoncally nondecreasng n k). begnnng tme of job. completon tme of job. nteger varable, equals f job s actve at. tme k, and 0 otherwse. T tardness of job = max (0, ). J objectve functon to be mnmzed. mong the above varables, the number of jobs N, tme horzon K, weghts of jobs {w>gl, tme requrements due dates {}gl and machne avalablty {Mk)f=(= are assumed to be gven. lso the job processng s nonpreemptve so that a contguous block of tme length t s needed to process job. The decson varables are {b}l. Once the b,s are selected, {c>=,, {~>g~ and { k } ~ ~ k k = can l easly be derved. The objectve functon of nterest s I statc and determnstc parallel machne schedulng problem can now be formulated as follows. subject to capacty constrants xt M k (k =,,...,I() () and processng tme constrants, b + = t (=,,...,N) () The sngle machne sequencng problem can be solved as a weghted bpartte matchng problem that s NP hard [lo]. onsequently, the parallel machne weghted tardness problem s also NP hard. The addtvty of the objectve functon facltates the decomposton approach... Soluton Methodology Relaxng the capacty constrants () usng Lagrange multplers rk (k =,,..., I() to form the relaxed problem, subject to (), the dual problem s wth :max L s () 0

3 subject to?r 0; Ths leads to the followng decomposed subproblems for each job (gven T). subject to R, : mn L, () h <K t+ l b, + = t, () K s assumed to be large enough to complete all the jobs. For convex programmng problems, the maxmum of the lag (dual cost) equals the mnmum of the orgnal objectve functon and a saddle pont exsts. However, there are several dffcultes n utlzng ths technque for solvng dscrete varable problems. Frst, the saddle pont may or may not exst and t may be dffcult to determne when the algorthm has termnated. Second, even f the dual optmum were obtaned, the correspondng schedule at that pont may not be feasble. Heurstc adjustment s generally requred to ensure that the once relaxed constrants are obeyed. Therefore, the varous steps to obtanng a near optmum soluton are solvng the subproblems,. solvng the dual problem,. constructng a feasble soluton, and fndng a (sub) optmal soluton. ach of these steps s dscussed n [l. The optmzed Lagrangan multplers?tk are nterpreted as a shadow prce for usng the resource (machne) at k. Therefore, they reflect the senstvty of the objectve functon wth respect to resource levels. Ths can be used to provde answers to what f questons and to reconfgure an exstng schedule when changes occur n resource avalablty. Thus, Lagrangan relaxaton has the ablty to react effectvely tcr dynamc changes and at the same tme produce good suboptmal schedules.. Smulated nnealng Smulated annealng [l, s a powerful algorthm for solvng (approxmately) combnatoral optmzaton (O) problems. Salent features of the algorthm are ts smplcty, generalty and applcablty to fnd hgh qualty solutons. The algorthm s based on an ntrgung combnaton of deas from two dfferent felds statstcal physcs and O. On the one hand, t can be vewed as an algorthm smulatng the physcal annealng process of solds to ther mnmum (ground) energy states. On the other hand, t can be consdered as a generalzaton of local search algorthms whch play an mportant role n O problems. Smulated annealng s a randomzaton technque that can be mathematcally descrbed usng Markov chans. The smulated annealng algorthm starts off wth a gven ntal soluton and contnuously tres to transform a current soluton to ts neghbours by applyng a generatng mechansm and an acceptance crteron. The acceptance crteron allows for deteroratons n a lmted way. Ths s controlled by a control parameter that plays a smlar role as temperature n the physcal annealng process. llowng deteroratons makes the smulated annealng algorthm more general than pure local search algorthms n whch only strct mprovements are allowed. The resultng effect s that the annealng algorthm can escape from a local mnmum n order to arrve at a near optmum. Thus, smulated annealng procedures present a new and promsng approach to schedulng problems. nomnal schedule or a set of schedules s vared n a small and generally random way. probablty whch s determned by the relatve change n schedule cost and control parameter s assgned to the result. Ths probablty s then used to determne whch schedule/schedules becomes nomnal for the next teraton. s n most other optmzaton problems there s some degree of enumeraton and there s no way to measure the qualty of the resultng schedule. The technque has been appled recently by Laarhoven et a [] to job shop schedulng. Hybrd Schedulng lgort hm I a multclass producton system, swtchover tmes or setup tmes can have a sgnfcant effect on the way parts are scheduled. The jobs of a gven part type need not be processed together. It s desred to fnd t schedule that mnmzes the sum of the weghted tardness and swtchover costs. Frst we determne a range of sutable values of swtchovers usng consderatons such as: too many swtchovers wll make the setup costs domnate over the tardness costs.

4 0 too few swtchoven wll make the tardness costs of delayed jobs substantal. Several complcatons arse wth the ntroducton of swtchover tmes. The Lagrangan relaxaton technque of Luh e a [ll] cannot be drectly appled because 0 For every job j, we now need to evaluate L:j, and bzj where s the part type of the job that was processed mmedately before j, (j =,..., N); ( =,...,P) where P s the total number of part types. 0 esgnng an effectve greedy heurstc to arrve at near optmum feasble schedule at the termnaton of the subgradent algort#hm s not easy. The smple smulated annealng algorthm s not lkely to yeld good results for parallel machne schedulng because an effcent perturbaton operator s not concevable. To crcumvent, ths, a hybrd approach that makes use of smulated annealng to arrve at a near optmal sequence of setup Operatons and Lagrangan relaxaton to arrve at the schedule of jobs of a part type on the machnes s developed Thv assumpton here s that once the machnes are set up for a part type, all jobs belongng to the part type arc' processed. The followng smplfyng assumptons are made regardng swtchover tmes and costs:. the swtchover tmes are the same for all classes. the setup costs depend only on the setup tmes and further, are a monotoncally ncreasng functon of the setup tmes. efnng the state of a machne at tme t to be the type of part t s processng at t, the extra data neces sary are the ntal states of the machnes and the tme nstants at whch each machne frst becomes avalable Frst, we descrbe a method to arrve at the schedule of parts of a partcular type on the machnes. Let & denote the total number of machnes. n upper bound on the plannng horzon K for schedulng jobs belongng to class = I,t (f all jobs are scheduled on a sngle machne). Let Y J ~ denote the tme machne ( =,,..., Q) frst becomes avalablr (after necessary setup operatons) Let the pernuta ton (SI, Sz,..., So) denote the sequence of machnes such that vsl vsl... "so. etermne q = sucl~ that mmj[ws, vsl +I<]; Machnes SQ+l,..,Sa can not process any jobs belongng to the part type nder consderaton. For k =:,..., K, form hfr; based on us,,... vs,, where n =,,...,&. It s here that the second assumpton regardng setup costs becomes mportant. If two or more machnes become avalable at the same tme, any machne can be chosen for processng the jobs belongng to the part type thus preventng unnecessary enumeraton at ths stage. Use Lagrangan relaxaton to arrve at the schedule of jobs and cost for each n. ach of these tasks s parallelzable. The schedule for whch the sum of the setup cost and tardness cost s mnmum s chosen and the avalablty of the machnes and states of the machnes are accordngly updated. To determne the order of the part types, hgher level smulated annealng optmzaton s carred out. The smulated annealng process wll gve us the order n whch to process the part types, takng nto account the setup tmes and setup costs. Havng obtaned the order of part types, the schedule on each machne and cost s computed usng the method dscussed n the prevous paragraph. It can easly be shown that n the global optmum schedule, jobs belongng to the same part type and havng the same processng tmes and due dates have to be processed n the decreasng order of ther weghts. These can be reordered to yeld a lower cost at the termnaton of the algorthm.. Numercal Results The examples dscussed here are adapted from the ones appearng n [ll,,.. xample There are jobs belongng to part types. They are to be scheduled on machnes that are avalable from tme nstant. Intal state of MI s gven to be and that of M to be. Job ata for xample Setup Tmes and Setup osts Setup Tme MI M z 0

5 The cost of the above schedule s unts, out of whch setup costs account for 00 unts and tardness costs equal unts... xample There are jobs belongng to part types. They are to be scheduled on machnes that are avalable from tme. Intal state of MI s gven to be, s, Ma s and M s. The cost of the (suboptmal) schedule s 0 unts. Tardness cost s 0 unts and the rest are setup costs... W J,a t, I L G for x n W t cl 0 Setup Tmes and Setup osts Joblass I [ I I I [ F I G SetupTme I 0 I 0 I 0 I 0 I 0 I 0 I 0 M.. xample ghty nne jobs belongng to part types are to be scheduled on 0 machnes. The frst fve machnes are avalable from the begnnng of the plannng horzon and the next fve are avalable from tme nstant 0. Intal states the machnes,,..., 0 are,,..., J respectvely. The cost of the schedule s unts out of whch unts are tardness costs and the rest are setup co.t.. ~ G W t b al cl F H H J J L N 0, f , ost l a s s I J K L M N O Tme ost lo00 np W _I I t 0, cl G H I J K M N Setup Tmes and Setup osts l a s s I I I I I I F I G I H Tme I 0 I 0 I 0 I 0 I 0 I 0 I 0 I 0

6 M M M M M Me M Me Ms Mn 0 onclusons In ths work, a new hybrd schedulng algorthm that uses smulated annealng and Lagrangan relaxaton has been proposed and tested for multclass producton systems consstng of dentcal parallel machnes. The technque s found to work very well for many examples. However, the two key ssues. performance evaluaton and. schedule reconfguraton n the event of dynamc changes have not been answered. Future work should concen trate on. answerng questons regardng performance evaluaton and schedule reconfguraton n the event of dynamc changes for multclass producton systems.. extendng the hybrd technque to job shop schedulng. References Schedule after Iteratons [] M. L. Fscher, dual algorthm for the one machne schedulng problem, Mahematcal Pm +I gmmmng,,, pp. 0 []. Fleahmann, The dscrete lot szng and 0 schedulng problem, uropean Journal of Oper atons Research,, 0, pp. :I 0 0 []. J. Hotomt, P.. Luh,. Max, and K. R. Pattpat, Schedulng jobs wth smple precedence constrants on parallel machnes, I ontrol Systems Magazne, 0, February 0, pp 0. []. J. Hotomt, P.. Luh, and K. R. Pattpat, practcal approach to job shop schedulng problems, I Transactons on Robotcs and utomaton,,, pp. [l]. H. L. arts and P. J. M. Van Laarhoven, Statstcal coolng general approach to solve combnatoral optmzaton problems, Phlps Journal of Research,,, pp. [] M. L. Fscher, Optmal soluton of schedulng problems usng Lagrange multplers, Opemtons Research,,, pp. [] M. L. Fscher, Lagrangan relaxaton methods for solvng nteger programmng problems, Management Scence,,, pp. [] S. Krkpatrck,.. Gelatt Jr., and M. P. Vecch, Optmzaton by smulated annealng, Scence, 0,, pp 0. [] J. M. Peter Van Laarhoven,. H. L. arts, and J. K. Lenstra, Job shop schedulng by smulated annealng, Operatons Research, 0,, pp. [lo]. L. Lawler, J. K. Lenstra,. H. G. Rnnoy Kan, and.. Shmoys, Sequencng and schedulng algorthms and complexty, Logstcs of Producton and Inventory, Volume, edted by S.. Graves,. H. G. Rnnoy Kan, and P. Zpkn,. [ll] P.. Luh,. J. Hotomt, K. R. Pattpat, and rc Max, Schedule generaton and reconfguraton for parallel machnes, I unsactons on Robotcs and utomaton,, ecember 0, pp.