OPTIMIZATION METHODS FOR BATCH SCHEDULING

OPTIMIZATION METHODS FOR BATCH SCHEDULING Jame Cerdá Isttuto de Desarrollo Tecológco para la Idustra Químca Uversdad Nacoal de Ltoral - CONICET Güemes 3450 3000 Sata Fe - Argeta 1

OUTLINE Problem defto Types of schedulg problems Types of schedulg methodologes Types of schedulg optmzato approaches Overvew of etwork-type dscrete ad cotuous tme models Comparso of etwork-type dscrete ad cotuous tme formulatos (bechmarkg examples) Overvew of batch-oreted cotuous tme formulatos Coclusos 2

MAIN REFERENCES Médez, C.A., Cerdá, J., Grossma, I.E., Harjukosk, I., Fahl, M. State-of-the-Art Revew of Optmzato Methods for Short-Term Schedulg of Batch Processes. Submtted to Computers & Chemcal Egeerg (July, 2005). Castro, P.M.; Barbosa-Póvoa, A.P.; Matos, H.A. & Novas, A.Q. (2004) I&EC Research, 43, 105 118. Cerdá, J.; Heg, G.P. & Grossma, I.E. (1997) I&EC Research, 36, 1695 1707. Floudas, C.A.; L, X. (2004) Computers ad Chemcal Egeerg, 28, 2109 2129. Ierapetrtou, M.G. & Floudas, C.A. (1998) I&EC Research, 37, 4341 4359. Jaak, S.L.; L, X. & Floudas, C.A. (2004) I&EC Research, 43, 2516 2533. Kodl, E; Pateldes, C.C. & Sarget, W.H. (1993) Computers ad Chemcal Egeerg, 2, 211 227. Maravelas, C.T. & Grossma, I.E. (2003) I&EC Research, 42, 3056 3074. Médez, C.A.; Heg, G.P. & Cerdá, J. (2001) Computers ad Chemcal Egeerg, 25, 701 711. Médez, C.A. & Cerdá, J. (2003) Comp. & Chem. Eg., 27, 1247 1259. Pateldes, C.C. (1994) Foudatos of Computer-Aded Process Operatos, Cache publcatos, New York, 253 274. Pto, J.M. & Grossma, I.E. (1995) I&EC Research, 34, 3037 3051. Pto, J.M. & Grossma, I.E. (1997) Computers ad Chemcal Egeerg, 21, 801 818. Pto, J.M. & Grossma, I.E. (1998). Aals of Operatos Research, 81, 433 466. Reklats, G.V. (1992). Overvew of schedulg ad plag of batch process operatos. NATO Advaced Study Isttute Batch process systems egeerg. Turkey: Atalya. 3

INTRODUCTION PROBLEM DEFINITION Schedulg s a decso-makg process thay plays a mportat role most maufacturg ad servce dustres The schedulg fucto ams to optmally allocate resources, avalable lmted supples, to processg tasks over tme. Each task requres certa amouts of specfed resources for a specfc tme terval called the processg tme Resources may be equpmet uts a chemcal plat, ruways at a arport or crews at a costructo ste Tasks may be operatos a chemcal plat, takeoffs ad ladgs at a arport, actvtes a costructo project 4

SCHEDULING DECISIONS AND GOALS A proper allocato of resources to tasks eables the compay to acheve ts objectves The objectves may take may forms such as: - mmzg the tme requred to complete all the tasks (the makespa) - mmzg the umber of orders completed after ther commtted due dates - maxmzg customer satsfacto by completg orders a tmely fasho - maxmzg plat throughput - maxmzg proft or mmzg producto costs Two elgble tasks caot geerally use the same requred resource smultaeously but oe at a tme Schedulg decsos to be made clude: - allocatg resources to tasks - sequecg tasks allocated to the same resource tem - task tmg 5

AN ILLUSTRATIVE EXAMPLE 29 Tasks - 4 Equpmet Uts - Oe-moth Perod Horzo 4.6 10.7 6

ROLLING TIME HORIZON The schedulg rollg horzo rages from 2 to 6 weeks, depedg o whether task processg tmes are o the order of a day or a week. Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 6-week rollg horzo The full schedule for a 6-week horzo mght be updated oce a week usg updated order put ad plat state. There wll be frequet correctos to the schedule mdweek to accout for ut breakdows or late order arrvals The schedulg fucto has to teract wth other decso-makg systems used the plat lke the materal requremet plag (the MRP system) The MRP system provdes formato o the weekly producto order arrvals (product, arrval tme, due date ad order sze), together wth the tasks requred to complete each order. 7

INTERACTION WITH THE MRP SYSTEM Producto Plag Master Schedulg Demad forecasts Fal product orders MRP-II Materal Requremet / Capacty Plag Producto Orders Release Dates Schedulg & Reschedulg Schedule Materal Requremets Capacty Requremets Dspatchg After the schedule has bee developed, all raw materals ad resources must be avalable at the specfed tmes MRP-II ams to guaratee that the requred raw materals ad termedates wll be avalable the rght amouts at the rght tmes, ad the plat capacty s eough to process all the requred productos orders 8

TYPES OF SCHEDULING PROBLEMS STATIC vs. DYNAMIC PROBLEMS I statc problems, all the producto orders ad ther arrval tmes are kow beforehad I dyamc problems, ew producto orders ca arrve at uexpected tmes whle the schedule s beg executed FLOW SHOP vs. JOB SHOP PROBLEMS Assume that the jobs requre to perform multple operatos o dfferet maches. - Flow shop: Every job cossts of the same set of tasks to be performed the same order. The uts are accordgly arraged producto les to mmze the movemet of materals ad mapower (multproduct plat) - Compoud Flow shop: Each ut the seres may be replaced by a set of parallel equpmet tems whch may be detcal or very dfferet. Each job goes to oe ut the frst stage, the t s trasferred to oe the secod stage ad so o. - Job shop: Producto orders have dfferet routes (requre dfferet sequeces of tasks) ad some orders may eve vst a gve ut several tmes (multpurpose plats) 9

TYPES OF BATCH PRODUCTION FACILITIES 1 2 3 Flow-shop faclty 1 2 3 4 5 6 7 8 9 10 11 18 19 20 21 2 2 23 24 25 Compoud Flow-shop faclty 1 2 3 Job-shop faclty 10

TYPES OF SCHEDULING PROBLEMS -2 MAKE-TO-STOCK vs. MAKE-TO-ORDER PRODUCTION FACILITIES MAKE-TO-STOCK FACILITIES: - A make-to-stock maufacturg faclty opt to keep stock tems for whch there s a steady demad ad o rsk of obsolescece. - Items that are produced for vetory do ot have tght due dates - The lot sze s determed by a trade-off betwee setup costs ad vetory holdg costs - Make-to-stock maufacturg plats are referred to as ope shops MAKE-TO-ORDER FACILITIES: - Make-to-order jobs have specfed due dates ad ther szes are determed by the customer - Each order s uque ad has a uque routg throughout the plat - Make-to-order maufacturg facltes are referred to as closed shops - May maufacturg plats operate partly as a make-to-stock faclty processg warehouse orders ad partly as a make-to-order faclty processg customer orders 11

TYPES OF SCHEDULING APPROACHES HEURISTIC METHODS - Basc Dspatchg Rules - Composte Dspatchg Rules ALGORITHMS OF THE IMPROVEMENT TYPE - Smulated Aealg - Geetc Algorthms - Tabu Search OPTIMIZATION APPROACHES - Dscrete Tme Models - Cotuous Tme Models Network-oreted Formulatos Batch-oreted Formulatos 12

HEURISTIC SCHEDULING METHODS BASIC DISPATCHING RULES - A basc dspatchg rule s a rule that prortzes all the jobs that are watg for processg o a mache - The prortzato scheme may take to accout jobs attrbutes ad maches attrbutes as well as the curret tme - Wheever a mache has bee freed, a dspatchg rule spects the watg jobs ad selects to process ext the job wth the hghest prorty - Dspatchg rules ca be classfed to STATIC ad DYNAMIC RULES. * A STATIC RULE s ot tme-depedet but just a fucto of the job data, the mache data or both (EDD-earlest due date frst, SPT-shortest processg tme frst) * DYNAMIC RULES are tme-depedet sce they also take to accout, addto to the job ad mache data, the curret tme (Example: MS-mmum slack tme-frst) - Dspatchg rules ca also be categorzed to two classes: LOCAL ad GLOBAL RULES * A LOCAL RULE uses oly formato related to ether the queue or the mache / workceter to whch the rule s appled * A GLOBAL RULE may use formato related to other maches, such as ether the processg tmes of the jobs or the curret queue legth o the ext mache 13

IMPROVEMENT ALGORITHMS COMPOSITE DISPATCHING RULES - Composte dspatchg rules combe a umber of basc dspatchg rules - Each basc rule the composte dspatchg rule has ts ow scalg parameter that s chose to properly scale the cotrbuto of the basc rule to the fal decso ALGORITHMS OF THE IMPROVEMENT TYPE - Start wth a complete schedule, whch may be selected arbtrarly - Try to obta a better schedule by mapulatg the curret schedule - Use local search procedures whch do ot guaratee a optmal soluto - Attempt to fd a better schedule tha the curret oe the eghborhood of the curret oe. - Two schedules are sad to be eghbors f oe ca be obtaed from the other through a well-defed modfcato scheme - The procedure ether accepts or rejects a caddate soluto as the ext schedule to move to, based o a gve acceptace-rejecto crtero - The four elemets of a mprovemet algorthm are: the schedule represetato, the eghborhood desg, the search process wth the eghborhood ad the acceptace-rejecto crtero. 14

OPTIMIZATION APPROACHES DISCRETE TIME MODELS OF THE NETWORK TYPE - State-Task-Network (STN)-based dscrete formulato - Resource-Task-Network (RTN)-based dscrete formulato CONTINUOUS TIME MODELS OF THE NETWORK TYPE - Global Tme Pots * STN-based cotuous tme formulatos * RTN-based cotuous tme formulatos - Ut-Specfc Tme Evets * STN-based ut-specfc cotuous tme formulatos BATCH-ORIENTED CONTINUOUS TIME MODELS - Tme Slot-based formulatos - Precedece-based formulatos * Ut-specfc mmedate precedece-based models * Global drect precedece-based models * Global geeral precedece-based models 15

KEY ASPECTS IN BATCH SCHEDULING (1) PROCESS TOPOLOGY: - Sgle Stage (sgle ut or parallel uts) S1 - Multple Stage (multproduct or multpurpose) - Network (batch splttg ad mxg, recyclg) 1h Heat 1 2 S2 1h Reacto1 S3 A B C 3h Reacto2 40% 70% 60% S5 10% 2h Separato 3 2h Reacto 3 90% S4 S7 (2) EQUIPMENT ASSIGNMENT - Fxed (dedcated) -Varable 30% S6 (3) EQUIPMENT CONNECTIVITY - Partal - Full (4) INVENTORY STORAGE POLICIES - Ulmted termedate storage (UIS) - No-termedate storage (NIS) - Fte termedate storage (FIS): Dedcated or shared storage uts - Zero wat (ZW) (5) MATERIAL TRANSFER - Istataeous (eglected) -Tme cosumg (o-resource, ppes, vessels) 16

KEY ASPECTS IN BATCH SCHEDULING (6) BATCH SIZE: - Fxed - Varable (mxg ad splttg operatos) (7) BATCH PROCESSING TIME -Fxed - Varable (ut / batch sze depedet) 0 Due date 1 Due date 2 (8) DEMAND PATTERNS - Due dates (sgle or multple product demads) - Schedulg horzo (fxed, mmum/maxmum requremets) Due date 3... Producto Horzo Due date NO (9) CHANGEOVERS - Noe chageover - Ut depedet - Sequece depedet (product or product/ut depedet) (10) RESOURCE CONSTRAINTS - Noe (oly equpmet) - Dscrete (mapower) - Cotuous (utltes) 17

KEY ASPECTS IN BATCH SCHEDULING (11) TIME CONSTRAINTS -Noe - No-workg perods - Mateace -Shfts (12) COSTS - Equpmet - Utltes (fxed or tme depedet) - Ivetory - Chageovers (13) Degree of certaty - Determstc - Stochastc 18

ROAD-MAP FOR OPTIMIZATION APPROACHES (A) TIME DOMAIN REPRESENTATION - Dscrete tme - Cotuous tme TASK TASK TIME (B) EVENT REPRESENTATION DISCRETE TIME - Global tme tervals CONTINUOUS TIME - Tme slots - Ut-specfc drect precedece - Global drect precedece - Global geeral precedece - Global tme pots - Ut- specfc tme evet EVENTS 1 2 3 4 5 6 TIME 7 8 9 TASK 10 11 18 19 20 21 2 2 TIME 23 24 25 (C) MATERIAL BALANCES - Lots (Order or batch oreted) - Network flow equatos (STN or RTN problem represetato) (D) OBJECTIVE FUNCTION - Makespa - Earless/ Tardess -Proft - Ivetory -Cost 19

DISCRETE TIME MODEL FEATURES (A) TIME DOMAIN REPRESENTATION - DISCRETE TIME TASK TIME (B) EVENT REPRESENTATION DISCRETE TIME - Global tme terval (C) MATERIAL BALANCES - Network flow equatos (STN or RTN problem represetato) (D) OBJECTIVE FUNCTION - Proft - Cost 20

STATE-TASK NETWORK (STN) REPRESENTATION The STN process represetato s a drected graph cosstg of three elemets: State odes: stadg for the feeds, termedates ad fal products ad represeted by crcles. Task odes: represetg the process operatos whch trasform materal from oe or more put states to oe or more output states, ad deoted by rectagles Drected arcs: lkg states ad tasks to dcate the flow of materals 21

STN-BASED DISCRETE TIME FORMULATION (Kodl et al., 1993; Shah et al., 1993) DISCRETE TIME REPRESENTATION The tme horzo s dvded to a umber of tervals of equal durato (uform tme grd). The uform tme grd s vald for all shared resources lke equpmet, utltes or mapower,.e. global tme tervals. Evets of ay type should occur at the terval boudares. It ca be regarded as evets: - the start or the ed of processg tasks - chages the avalablty of ay resource - chages the resource requremet alog the executo of a task T1 T2 T3 0 1 2 3 4 5 6 7 8 t (hr) 22

STN-BASED DISCRETE TIME FORMULATION OTHER MAJOR ASSUMPTIONS The batch processg tme for ay task s costat,.e. t does ot chage wth the batch sze. Dedcated storage taks for each fal/termedate product are avalable. Every batch of state/product s s trasferred to the assged tak (or the ext ut) mmedately after fshg the processg task A processg ut caot be used as a temporary storage devce. A batch sze chagg wth both the processg task ad the assged ut ca be selected by the model. INTERESTING PROBLEM FEATURES Product demads or bouds o product demads ( to or cubc meters) are gve Batches are to be geerated ad scheduled by solvg the model. Alteratve equpmet uts for a partcular processg task ca be avalable. The resource requremet may chage alog the task executo. Lmted resource capactes are avalable 23

STN-BASED DISCRETE TIME FORMULATION MAJOR MODEL VARIABLES a) BINARY VARIABLES: W j t task ut tme terval W j t = 1 oly f the processg of a batch udergog task ut j J s started at tme pot t. b) CONTINUOUS VARIABLES: B j t = sze of the batch (, j, t) S st R rt = avalable vetory of state s S at tme pot t = requremet of resource r (dfferet from equpmet) at tme pot t 24

STN-BASED DISCRETE TIME FORMULATION Equpmet Ut Reserved Tme Pots for Task j 1 Task (Batch a) No task ca be assged to tme pots t2 & t3 of ut j1 W, j1, t1 = 1 Just defed at the start of task j 2 Task (Batch b) Task (Batch c) W, j2, t1 = 1 Several tasks ca be assged to the tme pot t 1 but executed dfferet uts j1 & j2 0 1 3 2 4 5 6 25 Tme

STN-BASED DISCRETE TIME FORMULATION Equpmet Ut j 1 The batch sze of task s just defed at the start of the task Task (Batch a) The batch sze of task s equal to zero at the other actvated tme pots B, j1,t1 = B a B, j1, t2 = B, j1, t3 = 0 j 2 Task (Batch b) B, j2, t1 = B b 3 5 6 0 1 2 4 Tme 26

STN-BASED DISCRETE TIME FORMULATION MAJOR CONSTRAINTS ALLOCATION & SEQUENCING At most a sgle task ca be performed a partcular processg ut at ay tme pot t. BATCH SIZE The sze of a batch udergog task ut j J must be chose wth bouds. MATERIAL BALANCES The vetory of state s at tme t s equal to that stored at tme (t-1), plus the amout of s produced or receved as raw materal from exteral sources, mus the amout of s cosumed the process or delvered to the market durg tme terval t. RESOURCE BALANCES - The total demad of resource r at tme terval t s equal to the sum of the rth-resource requremets from tasks beg executed at tme t. - The overall resource requremet must ever exceed the maxmum rth-resource capacty CHANGEOVER TIMES If ut j starts processg ay task of famly f at tme t, o task of famly f ca be tated at least (cl f f + pt j ) uts of tme before tme terval t. 27

STN-BASED DISCRETE TIME FORMULATION (Kodl et al., 1993; Shah et al., 1993) jt ' t' = t pt + 1 t Ij j W 1 j,t ALLOCATION AND SEQUENCING m max V j Wjt Bjt Vj Wjt, j J, t BATCH SIZE S = S + ρ B ρ B + D p c st s( t 1) s j( t pt s t s ) s jt st st, p s c ' I j J ' I s j J m max MATERIAL BALANCE C s Sst Cs s, t R rt j J pt 1 j = µ t' = 0 ( W v B ) r, t rt' max 0 R rt R rt r, t j( t t') + rt' j( t t') RESOURCE BALANCE t CHANGEOVER TIMES W jt + W jt ' 1 j, f, f ', t f f ' I I t' = t cl pt + 1 j j f ' f j 28

STN-BASED DISCRETE TIME FORMULATION MAJOR ADVANTAGES Effcet hadlg of : - lmted resource avalabltes, oly motored at fxed, predefed tme pots - varable resource requremet alog the task executo - other tme-depedet aspects wthout compromsg model learty. Batch mxg ad splttg are allowed No bg-m costrat s requred Good computatoal performace (lower tegralty gap) Smple problem models accoutg for a wde varety of schedulg features MAJOR DISADVANTAGES Approxmate processg tmes ca lead to sub-optmal or feasble solutos. The batch sze B s a problem varable despte costat processg tmes. Hadlg of small sequece-depedet chageovers s rather awkward (very fe tme dscretzato). Sgfcat crease of the model sze for loger tme horzos 29

NETWORK-TYPE SCHEDULING EXAMPLE STATE-TASK NETWORK REPRESENTATION (STN) EQUIPMENT -HEATER - 2 REACTORS -STILL 30

STN-BASED DISCRETE TIME FORMULATION Task Ut Processg Tme (h) Heatg Heater 1 Reacto 1 2 Reacto 2 Reactor 1 Reactor 2 2 Reacto 3 1 Separato Stll 2 Number of Tme Pots: 10 (for H = 10 h) Number of Bary Varables: 80 Heatg: 1 x 1 x 10 = 10 Reactos 1, 2 & 3: 3 x 2 x 10 = 60 Separato: 1 x 1 x 10 = 10 Number of Cotuous Varables: B (80), S (60) Number of Costrats: Allocato : 40 ; Batch sze: 160 Materal Balaces: 60 Task States Produced States Cosumed Heatg Hot A Raw Materal A Reacto 1 It. AB (60%)+ P1 (40%) Hot A (40%) + It. BC (60%) Reacto 2 It. BC Raw Materals B&C (50/50) Reacto 3 Impure E Raw Materal C (20%) + It. AB (80%) Separato P2 (90%) + It. AB(10%) Impure E 31

SCHEDULING EXAMPLE STATE-TASK NETWORK REPRESENTATION (STN) EQUIPMENT -HEATER - 2 REACTORS -STILL DECISIONS Allocato Heater Reactor 1 Heatg Reacto 1 Reacto 3 Reacto 2 Sequecg Reactor 2 Separato Tmg Stll proft = 2805 32

RTN-BASED DISCRETE TIME FORMULATION (Pateldes, 1994) MAJOR FEATURES Smlarly to the STN represetato, t uses a predefed ad fxed uform tme grd that s vald for all shared resources (global tme tervals) Processg tmes are assumed to be depedet of the batch sze It s based o the Resource-Task-Network (RTN) cocept All resources (equpmet, materals, utltes) are treated the same way Its major advatage wth regards to the STN approach arses problems volvg detcal equpmet It requres to defe just a sgle bary varable rather tha multple oes for a set of equpmet uts of smlar type Each task ca be allocated to just a sgle processg ut Task duplcato s the requred to hadle alteratve uts ad ut-depedet processg tmes Chageovers have to be cosdered as addtoal tasks. 33

RTN-BASED DISCRETE TIME FORMULATION MAJOR MODEL VARIABLES a) BINARY VARIABLES: W t (oe less subscrpt) b) CONTINUOUS VARIABLES: B t, R t Sce every task ca be assged to just a sgle ut, the subscrpt j ca be elmated. MAJOR MODEL PARAMETERS µ r t = fxed amout of resource r produced/cosumed by a stace of task at tme t relatve to the startg tme terval t ν r t = coeffcet the term provdg the amout of resource r produced / cosumed by task at tme t that s proportoal to the batch sze. Whe r R stads for a processg ut, the meag of parameters µ rt ad ν rt s somewhat dfferet. 34

RTN-BASED DISCRETE TIME FORMULATION (Pateldes, 1994) R rt pt = R ( ) r( t 1) + µ rt' W ( t t') + vrt' B ( t t') ) + rt r, t I t' = 0 r max 0 R rt R rt r, t RESOURCE BALANCE m max J V r Wt Bt Vr Wt, r R, t BATCH SIZE If resource r correspods to a processg ut ad task requres pt uts of tme, the: µ r t = - 1 for t =0 (R rt decreases by oe f W t = 1) µ r t = + 1 for t =pt (R rt creases by oe f W t = 1) µ r t = 0 for ay other t (R rt remas uchaged eve f W t = 1) ad: ν r t = 0 for ay t 35

RTN-BASED DISCRETE TIME FORMULATION MAJOR ADVANTAGES Resource costrats are oly motored at predefed ad fxed tme pots All resources are treated the same way Savg bary varables for problems volvg detcal equpmet uts Effcet hadlg of lmted resource avalabltes Good computatoal performace (lower tegralty gap) Very smple models ad easy represetato of a wde varety of schedulg features MAJOR DISADVANTAGES Model sze ad complexty deped o the umber of tme tervals Costat processg tmes depedet of the batch sze Sub-optmal or feasble solutos ca be geerated due to the use of approxmate processg tmes Chageovers have to be cosdered as addtoal tasks 36

NETWORK-TYPE GLOBAL TIME CONTINUOUS MODELS (A) TIME DOMAIN REPRESENTATION - CONTINUOUS TIME TASK EVENTS TIME (B) EVENT REPRESENTATION CONTINUOUS TIME - Global tme pots (C) MATERIAL BALANCES - Network flow equatos (STN or RTN problem represetato) (D) OBJECTIVE FUNCTION - Makespa -Proft -Cost 37

STN-BASED CONTINUOUS TIME FORMULATION [Schllg & Pateldes (1996); Zhag & Sarget (1996); Mockus & Reklats (1999); Maravelas & Grossma (2003)] MAJOR FEATURES (Maravelas & Grossma, 2003) A commo tme grd that s varable ad vald for all shared resources (global tme pots) A predefed maxmum umber of tme pots (N) (a model parameter) The tme pots wll occur at a pror ukow tmes (model decsos) Every evet cludg the start ad the ed of a task must occur at a tme pot The start of several tasks ca be assged to the same tme pot but at dfferet uts ad, therefore, all must beg at the same tme T. The ed tme of a task assged to tme pot does ot ecesarly occur exactly at T They ca fsh before except those tasks followg a zero wat polcy (ZW) For storage polces other tha ZW, the equpmet ca be used as a temporary storage devce from the ed of the task to tme T Each task ca be allocated to just a sgle ut. Task duplcato s requred to hadle alteratve equpmet uts 38

STN-BASED CONTINUOUS TIME FORMULATION Cotuous Tme Represetato I Cotuous Tme Represetato II T1 T2 T3 T1 T2 T3 0 1 2 3 4 5 6 7 8 t (hr) 0 1 2 3 4 5 6 7 8 t (hr) Schllg & Pateldes, 1996 Maravelas & Grossma, 2003 MAJOR PROBLEM VARIABLES a) BINARY VARIABLES: Ws Wf = deotes allocato of the start of task to tme pot = deotes allocato of the ed of task to tme pot b) CONTINUOUS VARIABLES: T = tme for evets allocated to tme pot = ed tme of task assged to tme pot Tf Ts = start tme of task assged to tme pot Bs = batch sze of task at the start tme pot Bp = batch sze of task at the termedate tme pot Bf S s R r = batch sze of task at the completo tme pot = vetory of state s at tme pot = avalablty of resource r at tme pot 39

STN-BASED UNIT-SPECIFIC TIME-EVENT FORMULATION MAJOR CONSTRAINTS ALLOCATION CONSTRAINTS: - At most a sgle task ca be performed ut j at the evet tme - A task wll be actve at evet tme oly f t stars before or at evet tme, ad t fshes before tme evet - All tasks that start must fsh - A occurrece of task ca be started at evet pot oly f all prevous staces of task begg earler have fshed before - A occurrece of task ca fsh at evet pot oly f t starts before ad eds ot before tme pot BATCH SIZE CONSTRAINTS MATERIAL BALANCES TIMING AND SEQUENCING CONSTRAINTS STORAGE CONSTRAINTS RESOURCE CONSTRAINTS 40

STN-BASED CONTINUOUS TIME FORMULATION (GLOBAL TIME POINTS) (Maravelas ad Grossma, 2003) ALLOCATION CONSTRAINTS I j Ij Ij Ws 1 j, Wf 1 j, ' Ws ( Ws ' Wf' ) 1 j, = Wf m max V Ws Bs V Ws, m max V Wf Bf V Wf, m V Ws' Wf' Bp ' < ' Bs BATCH SIZE CONSTRAINTS V max ' < Ws ' ' Wf ', 1 + Bp( 1) = Bp + Bf, > 1 S S R T c p = Ss( 1) ρsbs + ρs Bf s, > 1 s I max s Cs s, r +1 c s I p s c c p p = Rr ( 1) µ rws + ν rbs + µ rwf + ν r Bf r, T Tf T + αws + βbs + H ( 1 Ws), Tf T + αws + βbs H ( 1 Ws ), Tf ( 1 ) T + H (1 Wf ), > ZW Tf ( 1) T H(1 Wf ) I, > 1 Ts ' Tf( 1) + cl' j, Ij, ' Ij, s S j V js 1 j J T, T Ssj CjVjs j J, s Sj, T Ss = Ssj s S, j J T s MATERIAL AND RESOURCE BALANCES TIMING AND SEQUENCING CONSTRAINTS SHARED STORAGE TASKS 41 1

STN-BASED CONTINUOUS TIME FORMULATION EXAMPLE Problem Sze (8 effectve tasks, 8 tme pots) t requres less tme pots a) Bary Varables: Ws (64) + Wf (64) = 128 b) Cotuous Varables: T(64) + Tf(64) + Bs (64) + Bf (64) + Bp (64) + S (48) = 368 c) Costrats: Allocat (104) + BSze (568) + Tme (264) + IvS (56) = 992 42

STN-BASED CONTINUOUS TIME FORMULATION MAJOR ADVANTAGES - Sgfcat reducto model sze by predefg a mmum umber of tme pots much lower tha that requred by dscrete formulatos - Hadlg of processg tmes whch vary wth the batch sze - Cosderato of a rage of schedulg aspects - Motorg of resource avalabltes just at the tme pots MAJOR DISADVANTAGES - Need of computg the mmum umber of tme pots - Model sze ad complexty both depedg o the umber of predefed tme pots - Suboptmal or feasble schedules ca be geerated f the umber of pots s smaller tha requred 43

T1 T2 T3 RTN-BASED CONTINUOUS TIME FORMULATION (Castro et al., (2001, 2004)) MAIN ASSUMPTIONS A commo tme grd for all shared resources The maxmum umber of tme pots s predefed The tme at whch each tme pot occurs s a model decso (cotuous tme doma) Tasks allocated to a certa tme pot must start at the same tme Oly zero wat tasks must fsh at a tme pot, others may fsh before Cotuous Tme Represetato II Cotuous Tme Represetato I 0 1 2 3 4 5 6 7 8 t (hr) ADVANTAGES 0 1 2 3 4 5 6 7 8 t (hr) Sgfcat reducto model sze whe the mmum umber of tme pots s predefed Varable processg tmes Resource costrats are oly motored at each tme pot A wde varety of schedulg aspects ca be cosdered a very smple model DISADVANTAGES Defto of the mmum umber of tme pots Model sze ad complexty deped o the umber of tme pots predefed Sub-optmal or feasble soluto ca be geerated f the umber of tme pots s smaller tha requred 44 T1 T2 T3

45 RTN-BASED CONTINUOUS FORMULATION (GLOBAL TIME POINTS) (Castro et al., 2004) ( ) ) ' ',(,, ' ' ' R r B W T T J I r < + β α ( ) ) ' ',(,, 1 ' ' ' ' R r B W W H T T J I I ZW r ZW r < + + β α ') ',(,, ' max ' ' m W V B W V < BATCH SIZE ( ) ( ) ( ) 1, 1) ( 1) ( ' ' ' ' ' ' 1) ( > + + + + = + > < r W W B W B W R R S r I c r p r I c r c r p r p r r r µ µ ν µ ν µ r R R R r r r, max m RESOURCE BALANCE TIMING CONSTRAINTS ),(, 1) ( max 1) ( m N I W V R W V s R r rt S + + STORAGE CONSTRAINTS 1),(, 1) ( max 1) ( m I W V R W V s R r rt S

STN-BASED UNIT-SPECIFIC TIME-EVENT FORMULATION (Ierapetrtou ad Floudas, 1998; V ad Ierapetrtou, 2000; L et al., 2002; Jaak et al., 2004). MAIN ASSUMPTIONS It s a STN-based formulato but the global tme represetato has bee relaxed Dfferet tasks assged to the same evet pot but performed dfferet uts ca be started/fshed at dfferet tmes The umber of evet pots s predefed (a model parameter) The tme pots wll occur at a pror ukow tmes (model decsos) The start ad the ed of a task must occur at a evet pot Each task ca be allocated to just a sgle ut. Task duplcato s requred to hadle alteratve equpmet uts It cosders processg tasks ad storage tasks st Evet-Based Represetato J1 J2 J3 1 2 2 2 3 3 0 1 2 3 4 5 6 7 8 t (hr) 46

STN-BASED UNIT-SPECIFIC TIME-EVENT FORMULATION (Jaak et al., 2004) MAJOR PROBLEM VARIABLES A. BINARY VARIABLES: Ws (start), W (actve), Wf (ed) 8 tasks x 10 evet pots 8 x 10 x 3 = 240 B. CONTINUOUS VARIABLES: Bs (start), B (actve), Bf (ed) Ts, Tf, Ts r, Tf r, S s, R r, R A r 8 tasks x 10 evet pots x 1 resource (8 x 10 x 6) + 30 = 510 6 states x 10 evet pots 6 x 10 = 60 570 47

STN-BASED UNIT-SPECIFIC TIME-EVENT FORMULATION MAJOR CONSTRAINTS ALLOCATION CONSTRAINTS: - At most a sgle task ca be performed ut j at the evet tme - A task wll be actve at evet tme oly f t stars before or at evet tme, ad t fshes before tme evet - All tasks that start must fsh - A occurrece of task ca be started at evet pot oly f all prevous staces of task begg earler have fshed before - A occurrece of task ca fsh at evet pot oly f t starts before ad eds ot before tme pot BATCH SIZE CONSTRAINTS MATERIAL BALANCES TIMING AND SEQUENCING CONSTRAINTS STORAGE CONSTRAINTS RESOURCE CONSTRAINTS 48

STN-BASED CONTINUOUS FORMULATION (UNIT-SPECIFIC TIME EVENT) (Jaak et al., 2004) ALLOCATION CONSTRAINTS Ij ' W 1 j, Ws ' Wf' = W, Ws Ws ' < = Wf 1 ' < ' < ' Ws ' ' < + ' < ' Wf Wf Ws Wf, MATERIAL BALANCE S s ', BATCH SIZE CONSTRAINTS V W B V W, m max max B ( 1) V 1 W ( 1) + Wf( 1), > max B ( 1) V 1 W ( 1) + Wf( 1), > B, max B + V Ws, max B V ( 1 Ws ) B, max B + V Wf, B B Bs Bs Bs, Bf Bf ( ) 1 ( ) 1 ( 1 Wf ) max Bf B V, = p c Ss( 1) + s Bf( 1) + B st sbs ρ B st s, ( 1) ρ I p s st I ST s I c s st st s I STORAGE CAPACITY max st st B C s s, I s, st 49

STN-BASED UNIT-SPECIFIC TIME EVENT FORMULATION TIMING AND SEQUENCING CONSTRAINTS (PROCESSING TASKS) Tf Ts, Tf Ts Tf Ts + H W, ' Tf Ts Ts Ts Ts ' Tf Ts ( W + Wf ), 1 ( 1) + H 1 ( 1) ( 1) > Ts α Ws α Ws + β B Tf( 1), > 1 Tf Tf Tf + β B + H + H ( Ws ) + H ( 1 Wf ) + H Wf' ' ' ',, ',( ') ' + H Wf '' ' 1 ' ' ( 1 Ws ) + H ( 1 Wf ) I ZW '',, ',( ') ( Wf Ws ), ', ', j J, 1 ' ( 1) + cl' + H 1 '( 1) ' > c p ( 1 Wf'( 1) ) s, I s, ' I s, j J, j' J', j j', 1 ( 2 Wf Ws ) ' ( 1) + H > '( 1) + H s S ZW '( 1), I c s, ' I p s, j J, j' J ', j j', > 1 50

STN-BASED UNIT-SPECIFIC TIME-EVENT FORMULATION st Tf st Ts st, p st ST Ts Tf( 1) H 1 Wf( 1) s, I s, I s, > p st ST Ts Tf( 1) + H 1 Wf( 1) s, I s, I s, > st c st ST Ts Tf st s, I, I, > 1 ( 1) s s c st ST Ts Tf st + H ( 1 Ws ) s, I, I, > 1 ( 1) s s st Ts st = Tf st, > 1 ( 1) ( ) 1 st ( ) 1 TIMING AND SEQUENCING CONSTRAINTS (STORAGE TASKS) c c Rr = µ rw + ν r B r, Ir, I r I r Tf Tf Tf Ts Ts Ts R R r r + R + R A r A r = = R I max r r R r r, = 1 A ( 1) + R ( 1) r, > 1 Tsr r 1) Tsr H 1 W ( 1) + Wf( 1) 1) Tsr H 1 W ( 1) r, I Ts H ( 1 W ) r, I r Ts + H ( 1 W ) r, I r = Tf r, 1 r, r ( ) r, Ir, 1 ( ), 1 ( > ( r > r, r, r r( 1) > RESOURCE BALANCE TIMING AND SEQUENCING OF RESOURCE USAGE 51

STN-BASED DISCRETE TIME FORMULATION MAJOR ADVANTAGES - More flexble tme decsos - Less umber of evet pots MAJOR DISADVANTAGES - Defto of evet pots - More complcated models - Model sze ad complexty deped o the umber of tme pots predefed - Sub-optmal or feasble soluto ca be geerated f the umber of tme pots s smaller tha requred - Addtoal tasks for storage ad utltes 52

COMPARISON OF DISCRETE AND STN CONTINUOUS TIME FORMULATIONS CASE STUDY: Westeberger & Kallrath (1995) Bechmark problem for producto schedulg chemcal dustry 0.31 Task 1 Task 2 Task 3 1 U1 2 U2 4 U3 5 0.2 0.7 3 Tasks 4-7 U4 Tasks 8,9 U5 9 8 7 6 10 11 Tasks 10-12 U6/U7 zw zw 0.5 zw 12 0.5 13 zw 14 Tasks 13-17 U8/U9 15 16 17 18 19 53

COMPARISON OF DISCRETE AND STN CONTINUOUS TIME FORMULATIONS PROBLEM FEATURES The process cludes flexble proportos of output states 3 & 4 (task 2), materal recycles from task 3, ad fve fal states (S15, S16, S17, S18, S19) There s eough stock of raw materal (S1) ad ulmted storage for the requred raw materal (S1) ad the fal products (S15-S19). Dfferet termedate storage modes are cosdered: - Zero-Wat trasfer polcy for states (S6, S10, S11, S13) - Fte dedcated termedate storage (FIS) polcy for the other termedate states Problem data volves oly teger processg tmes Two alteratve problem objectves are cosdered: - Mmzg makespa (Case I) - Maxmzg proft (Case II) Optos A,B: product demads of 20 tos just for three fal states have to be satsfed Opto C: mmum product demads of (10, 10, 10, 5,10) tos for states (15, 16, 17, 18 ad 19) 54

SUMMARY OF PROBLEM FEATURES 17 processg tasks, 19 states, 5 fal products 9 producto uts 0.31 Tasks 37 materal flows Task 1 Task 2 Task 3 4-7 1 2 4 U1 U2 U3 5 U4 Batch mxg / splttg 0.2 0.7 Cyclcal materal flows Tasks 8,9 3 U5 Flexble output proportos No-storable termedate products No tal stock of fal products Ulmted storage for raw materal ad fal products Sequece-depedet chageover tmes 9 8 7 6 10 11 Tasks 10-12 U6/U7 zw 0.5 zw zw 12 0.5 13 zw 14 Tasks 13-17 U8/U9 15 16 17 18 19 55

CASE I: MAKESPAN MINIMIZATION 20 20 0 20 0 0 20 0 20 20 Istace A B Formulato Dscrete Cotuous Dscrete Cotuous tme pots 30 8 9 30 7 8 bary varables 720 384 432 720 336 384 cotuous varables 3542 2258 2540 3542 1976 2258 costrats 6713 4962 5585 6713 4343 4964 LP relaxato 9.9 24.2 24.1 9.9 25.2 24.3 objectve 28 28 28 28 32 30 teratos 728 78082 27148 2276 58979 2815823 odes 10 1180 470 25 1690 63855 CPU tme (s) 1.34 108.39 51.41 4.41 66.45 3600.21 relatve gap 0.0 0.0 0.0 0.0 0.0 0.067 56

CASE I.B: MAKESPAN MINIMIZATION Dscrete model Tme tervals: 30 Makespa: 28 Cotuous model Tme pots: 7 Makespa: 32 57

CASE II.C : PROFIT MAXIMIZATION H = 24 h 10 Istace D Dscrete Cotuous Formulato LB UB tme pots 240 24 24 14 bary varables 5760 576 576 672 cotuous varables 28322 2834 2834 3950 costrats 47851 4794 4799 8476 LP relaxato 1769.9 1383.0 2070.9 1647.1 objectve 1425.8 1184.2 1721.8 1407.4 teratos 449765 3133 99692 256271 odes 5580 203 4384 1920 CPU tme (s) 7202 6.41 58.32 258.54 relatve gap 0.122 0.047 0.050 0.042 10 10 10 5 58

CASE II.C : PROFIT MAXIMIZATION H = 24 h Dscrete model Tme tervals: 240 Proft: 1425.8 Cotuous model Tme pots: 14 Proft: 1407.4 59

COMPARISON OF DISCRETE AND STN-CONTINUOUS FORMULATIONS COMMENTS For Case I, staces comprsg a larger umber of demads were ot possble to solve a reasoable tme Case I - Mmzg makespa: - Both formulatos reach the same objectve value of 28 h - 30 tme pots for the dscrete model vs. 8 pots for the cotuous formulato - 1.34 s (dscrete model) vs. 108 s (cotuous model) - The umber of tme pots s creased by oe each terato utl o mprovemet s acheved ad the reported CPU tme correspods to the last terato Case II - Maxmzg proft: - A fxed horzo legth of 24 hours was defed (loger perods caot be solved a reasoable tme) - 240 tme pots for the dscrete model vs. 14 pots for the cotuous formulato - The soluto foud through the dscrete tme model was slghtly better - Wth 14 pots the cotuous approach s faster 60

COMPARING DISCRETE VS. STN-BASED CONTINUOUS MODELS SOME PRELIMINARY CONCLUSIONS (1) Dscrete tme formulatos are usually larger, but ts smpler model structure teds to reduce the CPU tme f a reasoable umber of tme pots s proposed. (2) Dscrete tme models may geerate better ad faster solutos tha the cotuous oes wheever the tme dscretzato s a good approxmato to the real data. (3) The complex structure of cotuous tme models makes them useful oly for problems that ca be solved wth a reduced umber of tme pots (less tha 15 tme pots). (4) The model objectve fucto selected may have a otable fluece o the computatoal cost. (5) Serous lmtatos for solvg large-scale problem staces requrg a large umber of fxed/varable tme pots were observed. 61

SLOT-BASED UNIT-SPECIFIC CONTINUOUS TIME FORMULATION (Pto ad Grossma (1995, 1996); Che et. al.,2002; Lm ad Karm, 2003) MAIN ASSUMPTIONS Oe of the frst cotrbutos o batch-oreted schedulg methodologes The oto of tme slots stads for a set of predefed tme tervals of ukow durato A dfferet set of tme slots s predefed for each processg ut Batches to be scheduled are defed a pror (problem data) Every batch s to be allocated to at most a sgle tme slot No mxg ad splttg operatos are allowed It ca be appled to a multstage sequetal process wth several parallel uts at each stage Batches ca start ad fsh at ay tme durg the schedulg horzo task ut U1 U2 U3 slot Tme 62

SLOT-BASED UNIT-SPECIFIC CONTINUOUS FORMULATION A MULTISTAGE SEQUENTIAL BATCH PROCESS 1 4 5 8 job 2 6 9 3 7 reacto dryg packg Nether the batch szes or the equpmet capactes are model parameters A batch sze feasblty test s ot requred Oly batch processg tmes ad setup tmes for each product at each stage are problem data Batch processg tmes ca vary wth the selected equpmet ut 63

COMPARISON OF DISCRETE AND STN CONTINUOUS TIME FORMULATIONS PROBLEM FEATURES The process cludes flexble proportos of output states 3 & 4 (task 2), materal recycles from task 3, ad fve fal states (S15, S16, S17, S18, S19) There s eough stock of raw materal (S1) ad ulmted storage for the requred raw materal (S1) ad the fal products (S15-S19). Dfferet termedate storage modes are cosdered: - Zero-Wat trasfer polcy for states (S6, S10, S11, S13) - Fte dedcated termedate storage (FIS) polcy for the other termedate states Problem data volves oly teger processg tmes Two alteratve problem objectves are cosdered: - Mmzg makespa (Case I) - Maxmzg proft (Case II) Optos A,B: product demads of 20 tos just for three fal states have to be satsfed Opto C: mmum product demads of (10, 10, 10, 5,10) tos for states (15, 16, 17, 18 ad 19) 64

SLOT-BASED UNIT-SPECIFIC CONTINUOUS FORMULATION MAJOR CONSTRAINTS BATCH ALLOCATION: - The stage l of batch must be allocated to just a sgle tme slot SLOT ALLOCATION: - A tme slot (j,k) ca at most be assged to a sgle task (stage l of batch ) MATCHING CONSTRAINTS: - If task (,l) has bee assged to slot (j,k), the the start/ed tme of task (,l) ad the start/ed tme of slot (j,k) must be the same SLOT SEQUENCING: - The slot (k+1) at every ut j caot be started before edg the slot (j,k). No overlap of tme slots s permtted STAGE SEQUENCING: - The processg stage l+1 o batch caot be started before completg stage l SLOT TIMING: - The durato of slot (j,k) s gve by the sum of the processg tme & the setup tme for the assged task (,l), f W jkl = 1. 65

TIME-SLOT CONTINUOUS TIME FORMULATION (Pto ad Grossma (1995) j k K Tf jk Tf Tf Tf l L l j W W jkl jkl = 1, l L BATCH ALLOCATION 1 j, k K j SLOT ALLOCATION l L ( p + su ) = Ts + W j, k K SLOT TIMING jk l j k K j jkl jkl jl jl ( pjl + sujl ) l L = Ts + W, BATCH TIMING jk Ts j ( k + 1) j, k K j SLOT SEQUENCING l Ts ( l+ 1) j, k K j STAGE SEQUENCING j ( 1 Wjkl ) Tsl Ts jk, j, k K j l L ( 1 Wjkl ) Tsl Ts jk, j, k K j, l L M, M SLOT-BATCH MATCHING 66

SLOT-BASED CONTINUOUS TIME FORMULATION ADVANTAGES Sgfcat reducto model sze whe a mmum umber of tme slots s predefed Good computatoal performace Smple model ad easy represetato for sequecg ad allocato schedulg problems DISADVANTAGES Resource ad vetory costrats are dffcult to model Model sze ad complexty deped o the umber of tme slots predefed Sub-optmal or feasble soluto ca be geerated f the umber of tme slots s smaller tha requred 67

UNIT-SPECIFIC DIRECT PRECEDENCE CONTINUOUS MODEL (Cerdá et al., 1997). MAIN ASSUMPTIONS Batches to be scheduled are defed a pror No mxg ad splttg operatos are allowed (multstage sequetal processes) Batches ca start ad fsh at ay tme durg the schedulg horzo UNITS X = 1 = 1 2,3,J X 3,5,J J 2 3 5 X 1,4,J =1 X 4,6,J =1 J 1 4 6 The posto of a batch the processg sequece s defed terms of ts mmedate predecesor & ts mmedate successor ad the assged ut Defto of tme slots s ot requred Sequece-depedet setup tmes are explctely cosdered A sgle-stage sequetal batch process was studed Tme 68

SLOT-BASED CONTINUOUS TIME FORMULATION ADVANTAGES Sgfcat reducto model sze whe a mmum umber of tme slots s predefed Good computatoal performace Smple model ad easy represetato for sequecg ad allocato schedulg problems DISADVANTAGES Resource ad vetory costrats are dffcult to model Model sze ad complexty deped o the umber of tme slots predefed Sub-optmal or feasble soluto ca be geerated f the umber of tme slots s smaller tha requred 69

UNIT-SPECIFIC DIRECT-PRECEDENCE CONTINUOUS MODEL MAJOR VARIABLES A. BINARY VARIABLES: Xf j, X j batch ut batch ut batch Xf j = deotes that batch s frst processed ut j 6 batches, 1 stage 2 uts per stage (6 x 2) + (6 x 5 x 2) = 72 varables slot-based approach 36 X j = deotes that batch s processed mmedately after batch ut j B. CONTINUOUS VARIABLES: Ts, Tf Ts, Tf = start/ed tme of batch (3 slots per 6 x 2 = 12 varables Slot-based approach 24 ut) C. MODEL PARAMETERS: tp j cl j = processg tme of batch ut j = setup tme betwee batches & 70

UNIT-SPECIFIC DIRECT-PRECEDENCE CONTINUOUS MODEL MAJOR CONSTRAINTS BATCH ALLOCATION AND SEQUENCING - Oly oe batch ca be frst processed a partcular ut - A batch s frst processed or t has a sgle drect predecessor - A batch has at most a sgle drect successor the processg sequece - A batch shares the same ut wth ts drect predecessor ad ts drect successor BATCH TIMING AND SEQUENCING - A batch caot be started before edg the processg of ts drect predecessor - The completo tme of a batch ca be computed from ts startg tme by addg both the sequece-depedet setup tme ad the ut-depedet processg tme 71

UNIT-SPECIFIC DIRECT-PRECEDENCE CONTINUOUS MODEL (Cerdá et al., 1997) j J I j XF XFj j J j ' I j XFj Tf Ts = 1 j FIRST BATCH IN THE PROCESSING SEQUENCE + j X' j 1 ' I X ' j = 1 + X j + ' ' I j j' J ' I j j j' X' j' 1, = Ts + tpj XFj + j J ' I Tf ' + j J cl ' j X ' j j j J X ' j M 1 FIRST OR WITH ONE PREDECESSOR j ' J ' AT MOST ONE SUCCESSOR SUCCESSOR AND PREDECESSOR IN THE SAME UNIT X ' j PROCESSING TIME, ' SEQUENCING 72

UNIT-SPECIFIC DIRECT-PRECEDENCE CONTINUOUS MODEL MAJOR ADVANTAGES Sequecg s explctly cosdered model varables Sequece-depedet chageover tmes ad costs are easy to mplemet MAJOR DISADVANTAGES Larger umber of sequecg varables compared wth the slot-based approach Resource ad materal balaces are dffcult to model 73

GLOBAL DIRECT PRECEDENCE CONTINUOUS FORMULATION (Médez et al., 2000; Gupta ad Karm, 2003) MAIN ASSUMPTIONS Batches to be scheduled are defed a pror No mxg ad splttg operatos are allowed Batches ca start ad fsh at ay tme durg the schedulg horzo UNITS X 2,3 =1 X 3,5 =1 J J 2 3 5 X 1,4 =1 X 4,6 =1 1 4 6 Allocato varables W 2,J = 1;W 3,J = 1 ;W 5,J =1 W 1,J = 1; W 4,J = 1 ;W 6,J = 1 Tme The formulato s stll based o the mmedate or drect precedece oto The batch locato the processg sequece s gve terms of the mmedate predecessor 74

GLOBAL DIRECT PRECEDENCE CONTINUOUS FORMULATION Allocato ad sequecg decsos are separately hadled through two dfferet sets of bary varables The geeral precedece oto s ot assocated to a specfc ut,.e. t s a global oe No tme slots are predefed Sequece-depedet setup tmes are explctely cosdered MAJOR PROBLEM VARIABLES (SINGLE-STAGE BATCH PROCESS) A. BINARY VARIABLES: W j, X Xf j batch ut batch batch batch ut 6 batches, 1 stage 2 uts per stage (6 x 5) + (6 x 2 x 2) = 54 varables slot-based approach 36 X = deotes that batch s processed before batch the same ut Xf j = deotes that batch s frst processed ut j W j = deotes that batch s processed ut j but ot frst place 75

GLOBAL DIRECT PRECEDENCE CONTINUOUS FORMULATION B. CONTINUOUS VARIABLES: Ts, Tf Ts, Tf = start/ed tme of batch 6 x 2 = 12 varables Slot-based approach 24 (3 slots per ut) C. MODEL PARAMETERS: tp j = processg tme of batch ut j cl j = setup tme betwee batches & MAJOR CONSTRAINTS ALLOCATION CONSTRAINTS - At most oe batch ca be the frst processed ut j ALLOCATION-SEQUENCING MATCHING - Wheever a par of batches are related through the mmedate precedece relatoshp, both batches must be allocated to the same ut SEQUENCING CONSTRAINTS - Every batch should be ether the frst processed or drectly preceded by aother batch - Every batch has at most oly oe successor TIMING CONSTRAINTS - The edg tme of batch ca be computed from ts startg tme ad the sum of ts processg tme ad the setup tme the allocated ut - A batch ca be started after ts drect predecessor has bee completed 76

GLOBAL DIRECT PRECEDENCE CONTINUOUS FORMULATION (Médez et al., 2000) I j J j XF XF j J ' Tf Ts ' XF XF j j 1 + j J j j XFj X + W j + W ' j W j j W + X' = 1 ' 1 AT MOST ONE FIRST BATCH IN THE PROCESSING SEQUENCE = 1 ALLOCATION CONSTRAINT ' j X ' + 1, ', j J ' ( J J ) 1 X ', ', j ' = Ts + tp Tf + j J j ( XF + W ) j j ( cl' j + su' j ) W ' j M ( 1 X ' ) j J SEQUENCING-ALLOCATION MATCHING FIRST OR WITH ONE PREDECESSOR AT MOST ONE SUCCESSOR TIMING AND SEQUENCING 77

GLOBAL DIRECT PRECEDENCE CONTINUOUS FORMULATION MAJOR ADVANTAGES Sequecg s explctly cosdered model varables Chageover tmes ad costs are easy to mplemet Lower umber of sequecg varables compared wth the mmedate precedece approach MAJOR DISADVANTAGES Resource ad materal balaces are dffcult to model Large umber of sequecg varables compared wth the slot-based approach 78

GLOBAL GENERAL PRECEDENCE CONTINUOUS FORMULATION (Médez et al., 2001; Médez ad Cerdá, 2003; Médez ad Cerdá, 2004)) MAIN ASSUMPTIONS No mxg ad splttg operatos are allowed Batches ca start ad fsh at ay tme durg the schedulg horzo Batches to be scheduled are defed a pror UNITS X X 2,3 =1 2,5 =1 X 3,5 = 1 J J 2 3 5 X 1,4 =1 X 4,6 =1 1 4 6 X 1,6 =1 Tme Allocato varables Y 2,J =1;Y 3,J = 1 ; Y 5,J =1 Y 1,J =1;Y 4,J = 1 ; Y 6,J = 1 The geeralzed precedece oto exteds the mmedate precedece cocept The batch locato s gve terms of ot oly the mmedate predecessor but also of all the batches processed before the same ut The geeral precedece oto s ot related to a specfc ut but a global oe No tme slots are predefed 79

GLOBAL GENERAL PRECEDENCE CONTINUOUS FORMULATION INTERESTING FEATURES OF THE APPROACH Allocato ad sequecg decsos are dvded to two dfferet sets of bary varables Oly oe sequecg varable s requred to defe the relatve locato of ay par of batch tasks that ca be allocated to the same resource Dfferet types of shared reewable resources such as processg uts, storage taks, utltes ad mapower ca be treated the same way Such reewable resources ca be effcetly hadled through the same set of sequecg varables wthout compromsg the soluto optmalty Sequece-depedet setup tmes are explctely cosdered 80

GLOBAL GENERAL PRECEDENCE CONTINUOUS FORMULATION Ut Ut 3 batch a Oe-Stage batch c W a3 = W b3 = W c3 = 1 X ab = X ac = X bc = 1 batch b Tme Batches a ad b are the (drect/o-drect) predecessors of batch c 81

GLOBAL GENERAL PRECEDENCE CONTINUOUS FORMULATION MAJOR VARIABLES (SINGLE-STAGE BATCH PROCESS) A. BINARY VARIABLES: W j, X < batch ut batch batch 6 batches, 1 stage 2 uts per stage (6 x 5)/2 + (6 x 2) = 27 varables slot-based approach 36 X = deotes that batch s processed before batch the same ut B. CONTINUOUS VARIABLES: Ts, Tf Ts, Tf = start/ed tme of batch 6 x 2 = 12 varables Slot-based approach 24 (3 slots per ut) C. MODEL PARAMETERS: tp j = processg tme of batch ut j cl j = setup tme betwee batches & 82

GLOBAL GENERAL PRECEDENCE CONTINUOUS FORMULATION MAJOR PROBLEM VARIABLES (MULTISTAGE BATCH PROCESS) A. BINARY VARIABLES: W l j, X l l stage 6 batches, 3 stages 2 uts per stage 3 x (6 x 5)/2 + (6 x 2 x 3) = 81 varables slot-based approach 108 batch ut batch stage stage batch < X l l = 1, f task (,l ) s processed before task (,l) the same ut X l l = 0, f task (,l ) s processed after task (,l) the same ut B. CONTINUOUS VARIABLES: Ts l, Tf l Ts l, Tf l = start/ed tme of batch task (,l) 6 x 3 x 2 = 36 varables Slot-based approach 72 (3 slots per ut) C. MODEL PARAMETERS: tp l j = processg tme of batch task (,l) ut j cl l l j = setup tme betwee batch tasks (,l ) & (,l) 83

GLOBAL GENERAL PRECEDENCE CONTINUOUS FORMULATION MAJOR CONSTRAINTS ALLOCATION CONSTRAINTS Every batch task should be allocated to oly oe processg ut SEQUENCING CONSTRAINTS - Defed for every par of tasks that ca be allocated to the same ut - Assume that tasks (,l) & (,l ) were allocated to the same ut. If task (,l) s processed before, the task (,l ) starts after completg task (,l). - Otherwse, task (,l ) s eded before startg task (,l) TECHNOLOGICAL CONSTRAINT - The stage (l+1) of batch ca be started oly f stage l has bee completed TIMING CONSTRAINTS - The completo tme of task (,l) ca be computed from ts startg tme by addg to t both the task processg tme ad the sequece-depedet setup tme 84

GLOBAL GENERAL PRECEDENCE CONTINUOUS FORMULATION (Médez ad Cerdá, 2003) j Tf J l l W lj = 1, l L = Tsl + tp j J l lj W lj, l L ALLOCATION CONSTRAINT PROCESSING TIME Ts ( 1 X l, ' l' ) M ( 2 Wlj W ' l' j ), ', l L, l' L ' j J l, ' l' ' l' Tfl + cll, ' l' + su' l' M, SEQUENCING CONSTRAINTS Ts l Tf ( 2 Wlj W ' l ' j ), ', l L, l' L ' j J l, ' l ' ' l ' + cl' l ', l + sul M X l, ' l' M, Ts l Tf( l 1), l L, l > 1 STAGE PRECEDENCE 85

GLOBAL GENERAL PRECEDENCE CONTINUOUS FORMULATION ADVANTAGES Geeral sequecg s explctly cosdered model varables Chageover tmes ad costs are easy to mplemet Lower umber of sequecg decsos Sequecg decsos ca be extrapolated to other resources DISADVANTAGES Materal balaces are dffcult to model 86

CASE STUDY FOR COMPARING BATCH-ORIENTED APPROACHES 1 Batches to be Processed B1 B2 B3...... B11 B12 Tme Horzo = 30 days 2 3 4 Extruders Oe operator crew per extruder Lmted Mapower: Case 1: 4 operator crews Case 2: 3 operator crews Case 3: 2 operator crews 87

CASE STUDY FOR COMPARING BATCH-ORIENTED APPROACHES-2 OPTIMAL SCHEDULES (a) wthout mapower lmtato (b) 3 operator crews 88

CASE STUDY FOR COMPARING BATCH-ORIENTED APPROACHES-3 OPTIMAL SCHEDULES (b) 3 operator crews (c ) 2 operator crews 89

CASE STUDY FOR COMPARING BATCH-ORIENTED APPROACHES-4 Case Study Evet represetato Bary vars, cot. vars, costrats Objectve fucto CPU tme 2.a Tme slots & preorderg 100, 220, 478 1.581 67.74 (113.35)* Geeral precedece 82, 12, 202 1.026 0.11 b 2.b Tme slots & preorderg Geeral precedece 289, 329, 1156 2.424 2224 (210.7)* 127, 12, 610 1.895 7.91 b 2.c Tme slots & preorderg 289, 329, 1156 8.323 76390 (927.16)* Geeral precedece 115, 12, 478 7.334 35.87 b 90

BATCH REACTIVE SCHEDULING Roslöf, Harjukosk, Björkqvst, Karlsso ad Westerlud (2001); Médez ad Cerdá (2003, 2004) A dustral evromet s dyamc ature ad the proposed schedule must usually be updated mdweek because of uexpected evets. Dfferet types of uexpected evets may happe lke: - chages batch processg/setup tmes - ut breakdow/startup - late order arrvals ad/or orders cacellatos - reprocessg of batches - delayed raw materal shpmets - modfcatos order due dates ad/or customer prortes To prevet reschedulg actos from dsruptg a smooth plat operato, lmted chages batch sequecg ad ut assgmet are just permtted. The goal s to meet all the producto requremets stll to acheve uder the ew codtos by makg lmted low-cost chages batch sequecg ad assgmet. 91

BATCH REACTIVE SCHEDULING Roslöf, Harjukosk, Björkqvst, Karlsso ad Westerlud (2001); Médez ad Cerdá (2003, 2004) A mmum deterorato of the problem objectve (mmum makespa, mmum tardess) s pursued. PROBLEM DATA - the schedule progress - the preset plat state - curret vetory levels - preset resource avalabltes - curret tme data - uexpected evets - allowed reschedulg actos - The crtero to be optmzed RESCHEDULING ACTIONS Proper adjustmets to the curret schedule may clude: - smultaeous local reorderg of old batches at some equpmet uts - reassgmet of certa old batches to alteratve equpmet tems due to uexpected ut falures - serto of ew batches - batch tme shftg 92

BATCH REACTIVE SCHEDULING CONTINUOUS APPROACH Médez ad Cerdá (2003, 2004) It s based o the global geeral precedece oto It s a batch-oreted teratve approach It allows smultaeous serto ad reallocato of ew/old batches as well as the resequecg of old batches at each terato It cosders sequece-depedet chageovers ad lmted reewable resources A lmted umber of reschedulg actos ca be appled order to reduce the problem sze as much as the scheduler wats At each terato of the reschedulg algorthm, two steps are sequetally executed: - the assgmet step durg whch ew batches are serted ad a lmted umber of old batches ca be smultaeously reallocated - the sequecg step where eghborg batches the same queue ca exchage locatos, ad the procedure s repeated utl o mprovemet the objectve fucto s observed 93

REACTIVE SCHEDULING Problem Varables New batch to be serted 94

REACTIVE SCHEDULING EXAMPLE A sgle-stage multproduct batch plat wth four uts workg parallel Forty batches are to be processed wth a 30-day schedulg horzo. A uexpected 3-day mateace perod for ut U 3 at tme t = 14.6 d makes ecessary to perform a reschedulg process At the reschedulg tme, 25 batches are stll to be processed 95

REACTIVE SCHEDULING EXAMPLE UNEXPECTED 3-DAY MAINTAINANCE OF UNIT U3 BATCH RESEQUENCING SELECTED BATCH REALLOCATION SCHEDULE IN PROGRESS 96

REACTIVE SCHEDULING EXAMPLE NEW IMPROVED SCHEDULE Total Tardess Oly batch tme shftg ut U3: 13.55 d New mproved schedule: 8.84 d 97

CONCLUSIONS Curret optmzato models are able to solve moderate-sze batch processes Small examples ca be solved to optmalty Dscrete-tme models are computatoally more effectve tha cotuous-tme models of the etwork type Dffcult selecto of the umber of tme or evet pots etwork-oreted cotuous tme formulatos Network-oreted cotuous-tme models become quckly computatoally tractable for schedulg of medum complexty process etworks. Problems wth more tha 150 tme tervals are dffcult to solve usg dscrete tme models Problems wth more tha 15 tme or evet pots appear tractable usg etwork-oreted cotuous tme models. Depedg o the objectve fucto, dfferet computatoal performaces are observed Batch-oreted cotuous approaches are computatoally more effcet but usually requre to frst solve the batchg problem (a decomposto approach) Combg other approaches wth mathematcal programmg (hybrd methods) for solvg large scale problems looks very promsg 98