Dynamic scheduling in multiproduct batch plants

Transcription

1 Computers and Chemcal Engneerng 27 (2003) 1247/ Dynamc schedulng n multproduct batch plants Carlos A. Méndez, Jame Cerdá * INTEC (UNL-CONICET), Güemes 3450, 3000 Santa Fe, Argentna Receved 6 February 2003; accepted 7 February 2003 Abstract Ths work ntroduces a novel MILP formulaton for reactve schedulng of multproduct batch plants to optmally generate updated schedules due to the occurrence of unforeseen events. It can also be used to mprove a non-optmal producton schedule before t s executed. The approach s based on a contnuous-tme problem representaton that takes nto account the schedule n progress, the updated nformaton on the batches stll to be processed, the present plant state and the tme data. To lmt the changes on the current schedule, reschedulng operatons nvolvng local reorderng and unt reallocaton of old batches as well as the nserton of new batches are just permtted. In contrast to prevous contrbutons, multple reschedulng operatons can be performed at the same tme. The MILP problem formulaton s teratvely solved untl no further mprovement on the current schedule s obtaned. Three large-sze example problems were successfully solved wth low computatonal cost. # 2003 Elsever Scence Ltd. All rghts reserved. Keywords: Reactve schedulng; Multproduct batch plants; MILP Optmzaton model; Unforeseen events 1. Introducton Most of the work reported so far on schedulng technques for multproduct batch plants (MBP) s amed at generatng a pror producton schedules assumng that plant parameters and producton requrements wll reman wthout changes throughout the tme horzon. However, an ndustral envronment s dynamc n nature and, therefore, the ntal schedule must usually be updated n mdweek because of dfferent knds of unexpected events. For nstance, changes n batch processng/setup tmes, unt breakdown/startup, late order arrvals, orders cancellatons, reprocessng of batches, delayed raw materal shpments, modfcatons n order due dates and/or customer prortes and so on. As a result, the proposed schedule may become neffcent or even nfeasble. So, the ablty to rapdly react to such unforeseen events and perodcally re-optmze the schedule on a daly or hourly bass s a key ssue n batch plant operaton. Proper adjustments to the current schedule may nclude smultaneous local reorderng of * Correspondng author. Tel.: / /77; fax: / E-mal address: jcerda@ntec.unl.edu.ar (J. Cerdá). batches at some equpment unts, reassgnment of certan batches to alternatve equpment tems due to unexpected unt falures and/or smply batch startng tme shftng. Frequently, however, batches to be processed n the current shft may not be swtched to another unt because requred raw materals have already been sent n advance to the assgned unt (Muser and Evans, 1990). As a result, a porton of the schedule should be frozen whle the remander can be subject to re-optmzaton. In the last decade, some new reactve schedulng methodologes have been reported. Hasebe, Hashmoto, and Ishkawa (1991) proposed a reorderng algorthm for the schedulng of multproduct batch plants consstng of parallel producton lnes wth a shared unt. The algorthm nvolved two reorderng operatons, the nserton of a job and the exchange of two jobs. Because exchange operatons requred longer computatonal tmes and produced worse results, the authors fnally appled a reorderng algorthm performng just nserton operatons, one at a tme. Though smultaneous reorderng of jobs generally lead to much better producton sequences, such a reschedulng opton was not allowed mostly because the search space became extremely large and the algorthm would be computatonally very /03/$ - see front matter # 2003 Elsever Scence Ltd. All rghts reserved. do: /s (03)

2 1248 C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/1259 Nomenclature Sets I orders to be scheduled (I/I old ) I old orders belongng to the current schedule (I old /I, I old I new new orders to be nserted nto the current schedule (I new /I) IA subset of old orders that can be reassgned to other processng unts durng reschedulng (IA /I old ) IS orders that can be rearranged n the current processng sequence,.e. wthout unt reallocaton (IS /I old /IA) IS old orders whose relatve locatons wth regards to an exstng order may change durng the reschedulng process: IS /{? /IS/ "/?, j old /j?,((r old /n 5/r? 5/r old /n )or(r? /n? 5/r old 5/ r? /n? ))} J processng unts J Avalable unts to process order (J /J) # J avalable unts to reallocate order I /IA (J # /J ) Parameters d due date of order /I old j unt allocated to old order /I old n the current schedule M a very large number pt j processng tme of order /I n unt j old r poston of the exstng order /I old on the processng sequence of unt j old before reschedulng ro release tme of order /I ru j ready tme of unt j /J sl slack tme of order, sl /d /Mn{pt j, j /J } su j setup tme of order /I n unt j t?j sequence-dependent setup tme between orders /I and? /I n unt j a weghtng coeffcent for earlness of order /I b weghtng coeffcent for tardness of order /I n small nteger representng the maxmum number of closer predecessors and successors that can change locaton wth order /IS on the processng sequence Varables C completon tme for order E earlness for order T tardness for order X bnary varable denotng that order /I s processed before (X? /1) or after (X? /0) order? /I, when both were allocated to the same unt bnary varable denotng the allocaton of order /(I to unt j Y j expensve. Instead, the authors consdered the possblty of aggregatng consecutve jobs of the same type before assgnng them as a block to a new locaton n the same processng sequence. Afterwards, the aggregated job was dvded agan and the nserton of jobs was contnued untl the performance ndex could no longer be mproved. Roslöf, Harjunkosk, Björkqvst, Karlsson, and Westerlund (2001) developed an MILP reorderng algorthm to mprove a non-optmal schedule or update the schedule n progress because of unforeseen events. The approach was appled to mprove a manually generated producton schedule for a paper-convertng mll producng papers of dfferent qualtes. The example nvolved 61 jobs to be processed on a sngle processng unt wth sequence-dependent setup tmes. Test runs were performed by releasng ether one or two jobs at a tme. As expected, the strategy of smultaneously releasng two jobs led to a stronger decrease of the objectve functon than the sngle-job opton but the problem sze and the computatonal effort both showed much bgger ncreases. Ths paper ntroduces a novel MILP mathematcal formulaton for the MBP reactve schedulng problem that consders the nformaton about: () the current schedule; () the present plant state; and () the devatons n plant parameters, order avalabltes and tme data from those used for generatng the schedule n progress. The approach s based on a problem representaton descrbng the producton schedule by specfyng the whole set of predecessors for every batch at the assgned equpment unt (Méndez, Hennng, & Cerdá, 2001). It can stll be appled f setup tmes are sequencedependent. In contrast to prevous contrbutons, the proposed approach allows to perform multple resche-

3 C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/ dulng operatons at the same tme. Among the operatons beng consdered, t can be mentoned the nserton of new order arrvals, the reassgnment of exstng batches to alternatve unts due to equpment falures and the reorderng and tme-shftng of old batches at the current processng sequences. To prevent reschedulng actons from dsruptng a smooth plant operaton, lmted changes n batch sequencng and unt assgnment are just permtted. The goal s to meet all the producton requrements at ether maxmum customer satsfacton or mnmum total cost by makng lmted batch relocatons to lower the overall mpact on the schedule n progress. The new reactve schedulng strategy has been appled to large-scale ndustral examples under dfferent types of unexpected events wth great success. 2. Problem defnton Gven: a) a sngle-stage multproduct batch plant wth several unts j /J workng n parallel. b) a set of sngle-batch orders /I old (excludng cancelled orders) stll to be processed durng the current schedulng horzon. c) the producton schedule n progress, by provdng the assgned unt j old and the full set of precedng batches for any exstng batch /I old on the processng sequence, before performng the reschedulng process. d) the present state of the plant ncludng nformaton on equpment breakdown and antcpated startup of unts comng from mantenance. e) up-to-date processng tmes, setup tmes, release tmes and due dates for old batches. f) new sngle-batch order arrvals /I new and ther processng/setup tmes, release tmes and due dates. g) the tme at whch each avalable equpment unt j /J wll be ready to process the next batch /I/ (I /I new ) on the processng sequence. h) new equpment unt mantenance perods defned by ther ntal/fnal tmes. ) the set of (released) old batches /IS /I old that can be locally reordered n the current processng sequence,.e. wthout unt reallocaton. j) the subset of batches? /IS wth whch a batch / IS can change ts relatve locaton,.e. swtchng from beng a predecessor to becomng a successor or vce versa. k) the set of (released) old batches /IA /I old that can be reassgned to alternatve processng equpment tems as well as the allowed unt optons (j / J # /J ) for each released batch. l) the remanng tme horzon at the reschedulng tme. The problem goal s to optmally reschedulng the old batches and nsertng new ones n such a way that all producton requrements be completed n a tmely fashon and every unt-allocaton and sequencng constrant be satsfed at the mnmum of a weghted summaton of batch earlness/tardness over the batch set /I. In ths way, the average nventory level s also mnmzed. The sets of old batches {IS, IA} beng released for reorderng and unt reallocaton, respectvely, the subset of old batches IS wth whch a batch /IS can exchange locaton and the set of unts J # to whch a batch /IA can be reassgned can be all arbtrarly chosen by the user. The set IS s easly defned by specfyng the parameter n, a small nteger gvng the number of closer predecessors or successors wth whch a batch /IS can swtch locaton n the processng sequence of the assgned unt j old. In fact, one can freeze the locaton of an old batch /IS by smply makng IS equal to the empty set and, consequently, n /0. If n / 1, then the set IS wll just nclude the drect predecessor and the drect successor of batch n the current processng sequence. Let us assume that r old denotes the current poston of the old batch n the processng sequence. When n /1, then the set IS wll comprse the old batches located at postons r old 9/1 n the current batch sequence of unt j old. When n /2, then IS wll nclude the old batches located at postons r old 9/1 and r old 9/2. In the general case, the set IS wll contan the batches wth postons r old 9/1, r old 9/2,... r old 9/n n the batch sequence of unt j old. Therefore, the set IS comprses all the batches /I old wth n /0. The number of reorderng optons for batch rses wth n but at the same tme the problem sze and the computatonal requrements wll both show a stronger ncrease. The desred value for n s the one producng the best tradeoff between those opposte trends. In turn, the set IA / I old # just comprses every old batch wth a set J ncludng unt optons other than j old. 3. Model assumptons 1) The multproduct batch plant s operated on an order-drven bass. 2) Model parameters are all determnstc. 3) Setup tmes are sequence-dependent. 4) Sngle-batch orders are just consdered. Otherwse, a batch-szng procedure should be performed to transform new product demands nto batches of gven szes before applyng the reschedulng algorthm.

4 1250 C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/1259 5) Once the processng of an order starts, t must not be nterrupted. 6) Batch splt s not allowed. 7) No resource constrants except equpment are consdered. 4. The mathematcal model 4.1. Allocaton constrants For new batches /I new to be scheduled X Y j 1 Ö I new j J (1:1) For old batches /IA that can be reassgned to other unts j /J # durng reschedulng X Y j 1 Ö IA (1:2) j J # For an exstng batch /IA, the assgned unt n the updated schedule (j new ) may be dfferent from the current one (j old ). However, the unt j old may also belong to J #, unless j old s no longer avalable durng the reschedulng horzon. The small sets IA /I old and J # /J are both selected by the user Sequencng constrants For every par of exstng batches,? /IS /I old that are currently assgned to the same unt j old and may undergo just batch reorderng operatons C t 0 j su 0 j 5C 0 pt 0 j M(1X 0) Ö IS; 0 IS ; B 0 ; j j old j old 0 (3:1) C 0 t 0 j su j 5C pt j MX 0 Ö IS; 0 IS ; B 0 ; (3:2) j j old j old 0 The notaton B/? n Eqs. (3.1) and (3.2) means that ord() s less than ord(?) n the set IS. The set IS for batch /IS s defned by specfyng the value of n,.e. the number of closer predecessors or successors on the processng sequence that can change locaton wth batch. If batch? /IS s currently a successor of batch at the processng sequence of unt j old, t may become a predecessor of after the reschedulng process. Nonetheless, an asymmetrc set IS can also be handled by defnng a par of parameters (n, n ) for each old batch /I old. For nstance, f batch? s currently a successor of batch but n /0, then the batch? does not belong to IS and should be stll processed after batch n the new schedule as requred by Eq. (3.3). For frozen old batches Q/IS, the sequencng constrants are gven by, C t 0 j su 0 j 5C pt 0 0 j Ö; 0 IS; 0 QIS ; j j old j old ; 0 (rold B r old ) (3:3) 0 where r old and r old? are the postons of the exstng orders,? on the processng sequence of the unt j old before performng reschedulng operatons Tmng constrants For an old/new batch beng frst processed at the assgned unt j after performng the reschedulng process, one of the followng constrants should be compled dependng on the batch set to whch t belongs. C ]Max[ru j ; ro ]pt j su j Ö IS I old ; (2:1) j j old C ] X (Max[ru j ; ro ]pt j su j )Y j j J # (2:2) Ö IA I old C ] X (Max[ru j ; ro ]pt j su j )Y j Ö I new (2:3) j J where ru j s the release tme for unt j,.e. the tme at whch unt j can process the next batch after reschedulng. Generally, ru j s the completon tme of the batch beng processed n unt j at the reschedulng tme For a par of batches nvolvng an old batch /IS currently assgned to unt j old and another old/new batch that can be allocated to the same unt C t 0 j su 0 j 5C pt 0 0 j M(1X 0)M(1Y 0 j ) Ö IS; 0 IA; j j old J # (4:1) 0 C 0 t 0 j su j 5C pt j MX M(1Y 0 0 j ) (4:2) Ö IS; 0 IA; j j old J # 0 Snce J #? s a small set of unts to whch the old batch? / IA can be reallocated, then ths case apples only f j old / J #?. For a new batch? /I new, t s assumed that the set of unts J? where? can be processed also ncludes the unt j old. C t 0 j su 0 j 5C pt 0 0 j M(1X 0)M(1Y 0 j ) Ö IS; 0 I new ; j j old J 0 (4:3)

5 C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/ C 0 t 0 j su j 5C pt j MX 0 M(1Y 0 j ) Ö IS; 0 I new ; j j old J 0 (4:4) For a par of old batches,? /IA that can be reassgned to the same equpment tem durng the reschedulng process C t 0 j su 0 j 5C 0 pt 0 j M(1X 0)M(2Y j Y 0 j ) (5:1) Ö; 0 IA; B 0 ; j (J # SJ # ) 0 C 0 t 0 j su j 5C pt j MX M(2Y 0 j Y 0 j ) Ö; 0 IA; B 0 ; j (J # SJ # ) (5:2) For an exstng batch /IA and a new batch? / I new that can be assgned to the same unt durng the reschedulng process 4.6. Problem objectve functon Mn X a E b T (10) I where a, b are weghtng coeffcents. When every order can be completed wthout tardness, an equvalent objectve functon s gven by, Max X I Fg. 1. A small example. C (11) C t 0 j su 0 j 5C 0 pt 0 j M(1X 0)M(2Y j Y 0 j ) (6:1) Ö IA; 0 I new ; j (J # S J 0) C 0 t 0 j su j 5C pt j MX M(2Y 0 j Y 0 j (6:2) ) Ö IA; 0 I new ; j (J # S J 0) For a par of new batches that can be assgned to the same unt durng the reschedulng process C t 0 j su 0 j 5C 0 pt 0 j M(1X 0)M(2Y j Y 0 j ) (7:1) Ö; 0 I new ; B 0 ; j (J S J 0) C 0 t 0 j su j 5C pt j MX M(2Y 0 j Y 0 j ) Ö; 0 I new ; B 0 ; j (J S J 0) (7:2) 4.4. Order tardness T ]C d Ö I (8) 5. An llustratve example In order to llustrate the proposed MILP reactve schedulng approach, a small example wll be studed (see Fg. 1). Let us consder a sngle-stage multproduct batch plant wth two parallel unts. The current schedule specfes the processng of batches {1, 4, 2} n unt A and batch {3} n unt B, but reschedulng operatons are requred to nsert a new sngle-batch order {5}. Reorderng of old batches wth just ther drect predecessor or successor are only allowed (n /1) for any batch /IS whle the new batch {5} can be assgned to any of the avalable unts. Therefore: IS/I old /{1, 2, 3, 4} and I new /{5}. Moreover, two assgnment varables (Y 5A, Y 5B ) and sx sequencng varables (X 14, X 24, X 15, X 25, X 35, X 45 ) are to be defned (see Fg. 1). The problem constrant set for ths small example s gven below Allocaton constrants Y 5A Y 5B Order earlness E ]d C Ö I (9) 5.2. Sequencng constrants between any par of old batches C 1 5C 4 pt 4A M(1X 14 )

6 1252 C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/1259 C 4 5C 1 pt 1A MX 14 C 1 5C 2 pt 2A C 2 5C 4 pt 4A M(1X 24 ) C 4 5C 2 pt 2A MX Sequencng constrants among old and new batches C 1 5C 5 pt 5A M(1X 15 )M(1Y 5A ) C 5 5C 1 pt 1A MX 15 M(1Y 5A ) C 2 5C 5 pt 5A M(1X 25 ) M(1Y 5A ) C 5 5C 2 pt 2A MX 25 M(1Y 5A ) C 3 5C 5 pt 5B M(1X 35 )M(1Y 5B ) C 5 5C 3 pt 3B MX 35 M(1Y 5B ) C 4 5C 5 pt 5A M(1X 45 )M(1Y 5A ) C 5 5C 4 pt 4A MX 45 M(1Y 5A ) 6. Usng the MILP reschedulng approach to mprovng a non-optmal soluton By embeddng preorderng rules n the rgorous mathematcal formulaton of the batch schedulng problem, compact models can easly be derved. The use of MILP compact models represents a sutable alternatve to quckly generate good producton schedules for large-scale batch facltes. Preorderng rules are ndeed sequencng rules that establsh the relatve orderng of batches at every equpment unt beforehand. Dependng on the rule beng appled, the batches are sequenced by ncreasng processng tmes (SPT-rule), by ncreasng due dates (EDD-rule), by ncreasng slacktmes (MST-rule) and so on. In ths way, the compact schedulng problem formulaton wll just nclude the allocaton varables snce the 0/1 sequencng varables are no longer requred. In other words, the major dfference between the rgorous mathematcal formulaton and a compact schedulng model s that the sequencng constrants n the latter case are only expressed n terms of assgnment varables. If a good producton schedule s to be generated from scratch by solvng an MILP compact formulaton, then there s no exstng batch (I old / ) and every batch belongs to the set I new. Then, the sequencng constrants n Eqs. (7.1) and (7.2) are just needed. Let us assume that the batches wll be arranged by ncreasng slack tmes (sl )atevery equpment unt. In such a case, the sequencng constrants n Eqs. (7.1) and (7.2) n the resultng compact formulaton wll reduce to a sngle one wth the followng form: C t 0 j su 0 j 5C pt 0 0 j M(2Y j Y 0 j ) Ö; (7?) 0 I new ; j (J S J 0); (sl Bsl 0) However, compact schedulng models usually generate good, but non-optmal, solutons snce the preorderng rule may exclude the optmal schedule from the compact feasble regon. Nevertheless, the ntal schedule provded by the compact schedule model can be mproved through reorderng and reassgnment operatons by applyng the proposed MILP reschedulng approach. Durng the reschedulng stage, all batches belong to the set I old and, therefore, I new /. In ths way, effcent producton schedules for real-world multproduct batch facltes can be found wth very low computatonal cost. 7. The reschedulng algorthm a) Defne the sets I old and I new by ncorporatng the exstng batches stll to be processed n I old and the new sngle-batch orders n I new. b) Defne the set of avalable unts J # /J for every old/ new batch and the tme perods durng whch each one can be used. c) Defne the producton schedule to be updated by specfyng the current assgned unt and the poston r old n the processng sequence for every old batch /I old. d) Defne the small sets IS and IA by specfyng the parameter n and the unt opton set J # for every old batch /I old. e) Generate the MILP reschedulng problem formulaton based on the sets {IS, IS, IA, J #, J, I old, I new }. f) Solve the MILP reschedulng formulaton to fnd the best-updated schedule through local reorderng and unt reassgnment operatons. If the soluton found s an mprovement wth regards to the one dentfed n the prevous teraton, go to step (g). Otherwse, ether stop the procedure or enlarge the sets {IS, IA, J # } by ncreasng n or the number of optons n the set J #. In the latter case, go to step (g). g) Update the sets {IS, IA, J # } and return to step (e). h) Each executon of steps (e)/(f) stands for a major teraton of the proposed reschedulng algorthm. 8. Results and dscusson 8.1. Example 1 The proposed MBP reactve schedulng approach wll be llustrated by tacklng three example problems. Example 1, frst ntroduced by Pnto and Grossmann (1995) and later studed by Ierapetrtou, Hené, and

7 Table 1 Order data Order Due date (day) Slack tme (day) Processng tme (day) Order Due date (day) Slack tme (day) Processng tme (day) U 1 U 2 U 3 U 4 U 1 U 2 U 3 U 4 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O Setup tme C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/

8 1254 C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/1259 Fg. 2. Producton schedule for example 1a reported by Pnto and Grossmann (1995). Fg. 3. Improved schedule for example 1a after the frst reschedulng step. Fg. 4. Improved schedule for example 1a after the second reschedulng step. Floudas (1999), nvolves the schedulng of a sngle-stage multproduct batch plant wth four parallel unts. Twenty-nne sngle-batch orders O 1 /O 29 are to be processed wthn a 30-day schedulng horzon. Order due dates and unt-dependent processng and setup tmes are all gven n Table 1. The proposed MILP reschedulng approach has frst been appled to mprove the best soluton reported by Pnto and Grossmann (1995) and shown n Fg. 2 (example 1a). Snce these authors appled preorderng rules to solve the problem wth a more compact mathematcal formulaton, one can expect that the soluton reported by them can be mproved through the proposed MILP reschedulng algorthm. Ierapetrtou, Hené, and Floudas (1999) have also solved Example 1 by usng an event-based approach wth preorderng rules. To mprove the schedule reported by Pnto and Grossmann (1995), we wll assume that reorderng operatons wth n /1 are just allowed durng reschedulng. Therefore, all the batches belong to the set IS and each batch can only exchange locaton wth ether ts drect predecessor or ts drect successor on the processng sequence. A total of 25 reorderng operatons (rather than one or two operatons at a tme) can potentally be performed smultaneously. By solvng the proposed MILP approach, four re-sequencng operatons descrbed n Fg. 2 are really accomplshed to mprove the objectve functon from to , thus yeldng the producton schedule shown n Fg. 3. Another major teraton of the approach produces an addtonal ncrease of the objectve functon and, consequently, an mproved schedule (see Fg. 4). A further applcaton

9 C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/ Fg. 5. Intal schedule for example 1b by embeddng the MST-rule n the proposed approach. Fg. 6. Improved schedule for example 1b after the frst reschedulng step. Fg. 7. Improved schedule for example 1b after the second reschedulng step. of the reschedulng algorthm does not yeld any extra mprovement at all. The MILP formulaton was appled agan to example 1 to ths tme establsh a very good soluton by optmally allocatng unts to the whole set of batches whle assumng that they are sequenced at every unt by ncreasng slack tmes (sl ). To do that, the mnmumslack-tme rule (MST) was embedded n the MILP model. Moreover, all the batches are assumed to belong to I new and, consequently, I old s an empty set. Smlarly to Pnto and Grossmann (1995), order completon tmes were maxmzed to reduce order earlness as much as possble. In ths way, the producton schedule shown n Fg. 5 was found (example 1b). Snce the applcaton of the mnmum-slack-tme (MST) preorderng rule for batch sequencng does not always lead to the best schedule, the proposed approach was once more appled to mprove t. Ths tme we assume that all the batches are ncluded n the set IS and local reorderng operatons wth n /1 for any batch /IS are just allowed durng reschedulng. To further reduce the total order earlness, three re-sequencng operatons outlned n Fg. 5 were performed to generate a better schedule shown n Fg. 6. Another major teraton of the approach to further mprove the updated schedule produces another ncrease n the objectve functon by performng a sngle re-sequencng operaton as ndcated n Fg. 6. The mproved schedule descrbed n Fg. 7 and Table 2 cannot be upgraded through another reschedulng step. Overall, the objectve functon ncreases from (ntal schedule) to (fnal schedule) by makng four re-sequencng operatons. Therefore, any of the schedules generated by the proposed approach s better than the one reported by Pnto and Grossmann (1995)

10 1256 Table 2 Summary of results for examples 1 /3 Order Due date (day) Completon tme (day) Order Due date (day) Completon tme (day) Example 1b (Fg. 7) Example 2 (Fg. 9) Example 3 (Fg. 11) Example 1b (Fg. 7) Example 2 (Fg. 9) Example 3 (Fg. 11) O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/1259

11 C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/ Table 3 Model szes and computatonal requrements for examples 1/3 Example Bnary vars, cont. vars, rows Objectve functon CPU tme Nodes No. 1a*/soluton reported by Pnto and Grossmann (1995) (Fg. 2) 441, 875, a 204 No. 1a*/soluton reported by Ierapetrtou, Hené, and Floudas (1999) 57,172, a 5 No. 1a*/frst reschedulng step (Fg. 3) 25, 29, b 289 No. 1a*/second reschedulng step (Fg. 4) 25, 29, b 223 No. 1b*/ths approach wth an embedded MST-rule (Fg. 5) 57, 29, b No. 1b*/frst reschedulng step (Fg. 6) 25, 29, b 592 No. 1b*/second reschedulng step (Fg. 7) 25, 29, b 363 No. 2*/nsert 11 new orders nto current schedule (Fg. 8) 147, 40, b No. 2*/frst reschedulng step (Fg. 9) 36, 40, b 7207 No. 3*/reschedule due to unt mantenance (Fg. 11) 108, 25, b a Seconds on HP wth GAMS/OSL. b Seconds on Pentum II PC (400 MHz) wth ILOG/CPLEX. Table 4 Impact of the parameter n on the problem sze and results for example 1b n Bnary vars, cont. vars, rows Objectve functon CPU tme a Nodes 1 25, 29, , 29, , 29, , 29, a Seconds on Pentum II PC (400 MHz) wth ILOG/CPLEX. featurng a value of Moreover, the problem sze shows a drastc reducton of almost one-order-ofmagntude at any teraton not only n the number of bnary and contnuous varables but also n the overall CPU tme (see Table 3). Let now study the effect of adoptng values of n greater than 1 on the reschedulng problem sze, the CPU requrements and the objectve functon mprovement per teraton. To do that, t was chosen the producton schedule shown n Fg. 5 as the ntal schedule. We wll try to mprove such an ntal soluton by applyng the reschedulng algorthm wth n /2, 3 and 4, respectvely. In ths way, the allowed number of smultaneous reorderng operatons wll almost grow by a factor n ;.e. two tmes, three tmes and four tmes, respectvely. Table 4 presents the results of such test runs after a sngle major teraton of the algorthm, ncludng the reschedulng problem sze, the objectve functon, the CPU tme requrement and the number of explored nodes. In all cases, the same updated schedule featurng an objectve functon equal to was found. It represents a slght mprovement wth regards to the one provded by the reschedulng algorthm wth n /1 after two major teratons;.e. just a 0.016% ncrease. On the other hand, the problem sze and the CPU requrements both show a polynomal growth wth n. From these results, t should be concluded that the selecton of low values for n should be favored snce t leads to a smlar mprovement wth much lower computatonal cost Example 2 Example 2 assumes that new late sngle-batch orders O 30 /O 40 have arrved before startng the executon of the producton schedule found for Example 1 (Fg. 7). Due dates and processng tmes for the new orders are gven n Table 1. In ths case, the proposed reschedulng approach s ntally appled to fully optmze batch-unt allocaton and batch sequencng for the new sngle-batch orders /I new whle just permttng local re-sequencng operatons wth n /1 for any old order /I old /IS. The new schedule found n 67.7s on a Pentum II PC (400 MHz) wth ILOG OPL studo 2.1 (ILOG, 1999), usng the embedded CPLEX mxed-nteger optmzer release, s depcted n Fg. 8. By performng a second major teraton, but ths tme allowng resequencng operatons wth n /1 for every old/new batch /IS, a better schedule has been generated (see Fg. 9 and Table 2). No further mprovement was acheved by executng another major teraton f batch reorderng wth n /1 s only permtted Example 3 Fg. 8. Intal schedule for example 2. Fnally, Example 3 assumes that the best soluton for the 40-order schedulng problem consdered n Example

12 1258 C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/1259 Fg. 9. Improved schedule for example 2 after the frst reschedulng step. be as large as days. Certanly, any unt dfferent from U 3 s processng a batch at the reschedulng tme t/14.6 days and consequently ts ready tme wll be later than Through a sngle major teraton of the proposed MILP reschedulng approach, the best-updated schedule shown n Fg. 11 and Table 2 was found n 30 s. In ths way, the total tardness s decreased from to 8.84 days whle the maxmum order tardness dropped from 2.71 to 2.34 (see Table 5). Fg. 10. Schedule n progress and reschedulng operatons for example 3. 2 has been partally executed as predcted. However, an unexpected 3-day mantenance perod for unt U 3 at tme t/14.6 days, after processng order no. 7, makes necessary to perform a reschedulng process over the set of 25 orders not yet processed (see Fg. 10). The problem goal s to mnmze a weghted combnaton of order earlness and tardness, wth hgher penaltes assocated to tardy orders (a /1, b /5). Durng reschedulng, the batches currently allocated to unt U 3 may be reassgned to other unts and sequenced n the best possble way (set IA) whle local re-sequencng operatons wth n /1 are just allowed for the remanng orders (set IS). If reschedulng s not performed and the orders allocated to U 3 should wat untl the equpment unt resumes operaton, then the overall tardness for such orders wll 9. Conclusons Large-scale schedulng packages have usually no real provson for cheaply correctng the current schedule for small to mddle-sze changes, except to make full reschedulng. Ths work ntends to provde a flexble MILP systematc tool to effcently update schedules by smultaneously makng lmted reallocaton and reorderng operatons, whle keepng the extent of the schedule adjustments under user control. The reschedulng tool can stll be used even f sequence-dependent setup tmes are to be consdered. The proposed MILP based algorthmc approach should be teratvely appled on the current schedule untl no further mprovement s obtaned. Successful soluton to three large-sze examples nvolvng from 29 to 40 orders shows that, n any case, the reschedulng steps requres a very low CPU tme. Such examples dealt wth the mprovement of an avalable non-optmal schedule, the nserton of new Fg. 11. Improved schedule n progress for example 3.

13 C.A. Méndez, J. Cerdá / Computers and Chemcal Engneerng 27 (2003) 1247/ Table 5 Comparatve table of performance measures for examples 1 /3 Performance measure (days) Example 1a, Pnto and Grossmann (1995) Example 1b (Fg. 7) Example 2 (Fg. 9) Example 3 (Fg. 11) Maxmum earlness Total earlness Average earlness Maxmum tardness Total tardness 8.84 Average tardness jobs and the update of a schedule already n progress because of an unexpected unt mantenance perod. Acknowledgements The authors acknowledge fnancal support from FONCYT under Grant , and from Unversdad Naconal del Ltoral under CAI/Ds 048 and 121. References Hasebe, S., Hashmoto, I., & Ishkawa, A. (1991). General reorderng algorthm for schedulng of batch process. Journal of Chemcal Engneerng of Japan 24 (4), 483/489. Ierapetrtou, M. G., Hené, T. S., & Floudas, C. A. (1999). Effectve contnuous-tme formulaton for short-term schedulng. 3 Multple ntermedate due dates. Industral and Engneerng Chemstry Research 38, 3445/3461. ILOG OPL studo 2.1 user s manual (1999). ILOG S.A., France. Méndez, C. A., Hennng, G. P., & Cerdá, J. (2001). A contnuous-tme approach to short-term schedulng of resource-constraned multstage batch facltes. Computers and Chemcal Engneerng 25, 701/711. Muser, R. F. H., & Evans, L. B. (1990). Batch process management. Chemcal Engneerng Progress 87 (6), 66/77. Pnto, J. M., & Grossmann, I. E. (1995). A contnuous tme mxed nteger lnear programmng model for short term schedulng of multstage batch plants. Industral and Engneerng Chemstry Research 34, 3037/3051. Roslöf, J., Harjunkosk, I., Björkqvst, J., Karlsson, S., & Westerlund, T. (2001). An MILP-based reorderng algorthm for complex ndustral schedulng and reschedulng. Computers and Chemcal Engneerng 25, 821/828.