Scheduling of a pipeless multi-product batch plant using mixed-integer programming combined with heuristics

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1 Scheduling f a pipeless ulti-prduct batch plant using ixed-integer prgraing cbined with heuristics Sebastian Panek 1, Sebastian Engell 1, Cathrin Lessner 2 1 University f Drtund Prcess Cntrl Lab (BCI-AST), University f Drtund, Drtund 44221, Gerany 2 Axx Sftware AG Axx Sftware AG, Munich, Gerany Abstract The wrk presented here deals with shrt-ter scheduling in the cheical industry. It prpses an alternative del t STN and RTN in the fr f a MILP frulatin which describes the plant at a lwer level f detail. The ai is t reduce the cplexity f the del and thus t increase the slutin efficiency. The ain siplificatin cpared t STN/RTN is the that prduct stcks and ass balance cnstraints are ignred because discrete fixed-size batches are assued. The prpsed MILP frulatin is used fr the scheduling f a real-life exaple prvided by a custer f Axx, a lacquer prductin in a pipeless plant. Three types f recipes that invlve cycles, rigid tiing cnstraints between individual peratins, and parallel allcatins f statinary and bile units are cnsidered. Several dzens f prductin rders f different lacquer types in varius clrs with hundreds f peratins have t be scheduled. A special slutin prcedure cbines the slutin f the MILP prble in tw separate steps with three different heuristics which further reduce the del cplexity. Several prble instances f the lacquer prductin are slved t denstrate the the successful applicatin. Keywrds: supply chain anageent, scheduling, planning, ultiprduct batch prcessing, pipeless plant 1. Intrductin Research effrts in the past 10 years in the area f batch plant scheduling (batch sizing, resurce allcatin, sequencing) have led t pwerful general delling fraewrks as STN (Kndili et al., 1993; Shah et al., 1993) and RTN (Schilling et al., 1996). They enable the user t express a large variety f cnstraints that ight arise in the prcessing industry, such as capacity cnstraints n units and prducts, tiing cnstraints between peratins, sequence-dependent changever prcedures, varius strage plicies, cplex prductin recipes etc. Within bth fraewrks, the representatin f tie turned ut t be crucial fr the perfrance f the slvers f the atheatical dels. This led t cntinuus tie STN/RTN representatins with unifr and nn-unifr

2 grids (e.g. Ierapetritu et. al., 1998). Many recent publicatins fcused n this prble by prpsing iprved delling and slutin appraches. Fr slver perfrance, it is ften beneficial t reduce the cplexity f the del in ters f the nuber f equatins and variables. This is ften dne by refining and adapting the cnstraints t explit special prperties f the plant and the prductin schee. A discussin f varius tie representatins can be fund in (Fludas et. al., 2004). We refer t tw recent cntinuus-tie delling appraches here. The first ne is sltbased and suitable fr netwrk-based prductin schees invlving ass balances and prduct flws (Pint et. al. 1994). The secnd ne is based n rder-riented sequential prcesses withut tie slts (Manne, 1960; Cerda et. al. 1997). The wrk presented here is strngly related t the latter apprach. It ais at vercing the difficulties entined abve by ffering an rder-based delling schee. It prpses an alternative t STN and RTN in the fr f a different MILP frulatin which describes the plant at a lwer level f detail. The ai is t reduce the representatin f tie t a iniu and thus t increase the slutin efficiency. An explicit tie representatin is avided by establishing a inial set f event pints withut predefined rder. Sequencing decisins are encded using dedicated binary variables which crrespnd t typical binary disjunctins arising in all sequencing prbles. This leads t an efficient del frulatin in which the resurce assignent and the sequencing are decided in parallel. The ain siplificatin cpared t STN/RTN is the neglect f prduct stcks and ass balance cnstraints. Discrete fixedsize batches are assued instead. This wrk als suggests a tw step slutin prcedure. It cbines the slutin f the MILP prble in tw separate steps with three different heuristics which further reduce the del cplexity. The rest f this paper is structured as fllws: after stating the prble the delling apprach is described in detail. The slutin prcedure is discussed in the fllwing sectin and the results f nuerical experients are given afterwards. 2. Prble Stateent The prpsed MILP frulatin is used fr the scheduling f a real-life exaple prvided by a custer f Axx, a lacquer prductin in a pipeless plant. The plant facilities cnsist f 5 bile ixing vessels and 9 statinary prcessing units. The statinary units are: pre-dispersin line, ain dispersin line, special pre-dispersin unit, dse distributr, tw equal dse spinners, tw equal filling statins and a labratry fr quality checks. Tw ixing vessels have capacities f 19,000 litres, the ther three vessels can hld 20,000 litres. The plant tplgy with traffic n liited rutes, pssible cllisins and different distances is nt cnsidered here. Three basic recipes fr lacquers, each with 6-8 peratins are given. The recipes invlve cycles, rigid tiing cnstraints between individual peratins, and parallel allcatins f statinary and bile units. Three types f cnstraints invlve either 1) starting r 2) ending r 3) ending and starting dates f individual peratins within the recipes. Each f thse cnstraints establishes a link between tw peratins and frces the t either start r t end within a tie windw defined by the cnstraint. The cycles in the prductin cprise tw steps: dsing and quality checks. The terinatin f a cycle depends n the result f the last quality check. During the prcessing steps, n aterial

3 flws fr and t the vessels are cnsidered such that the aunt f prduct in the batch reains cnstant. Raw aterials and the strage fr end prducts are unliited and available at any tie. The arket deand is represented by 29 prductin rders fr different prducts and with irregular release and due dates spread ver several nths. The bjective is t iniize the verall prductin cst which cnsists f penalties fr issing the due dates. The recipe fr a standard etallic lacquer is depicted in Fig. 1. The ain prcessing steps are: ccupatin f a bile ixing vessel, dsing, quality check, crrectin, a secnd quality check and filling. The ther recipes Dsing Quality check Crrectin Quality check Filling Mixing vessel Figure 1. The recipe graph f etallic lacquer have a siilar structure but invlve tw additinal pre-prcessing steps. The ixing vessel hlds the prduct during all steps and is thus allcated in parallel. 3. Mdelling Apprach 3.1 Siplificatins and Assuptins Se siplificatins with respect t the riginal prble are assued. As the deanded prduct aunts and vessel capacities are quite siilar and d nt vary uch, we assue equal ixing vessels with sufficient capacities t hld ne batch f each prduct. Anther siplificatin is related t the cycles. The results f the quality checks in the labratry are a-priri unknwn. Since the fllwing steps depend n the, the verall nuber f prcessing steps is als unknwn. We avid stchastic delling and assue that the first quality check always fails and the secnd ne always succeeds. Thus, the lp in the recipe is always taken twice and we get 29 batches f different lacquer types in varius clurs with 202 peratins in ttal that have t be scheduled. Since the lab is an unliited resurce quality checks ay take place at any tie arbitrarily ften, but require a fixed aunt f tie. This allws us t reve all quality checks and the lab fr the del. Apprpriate tiing cnstraints are established t ake sure that the necessary inial aunt f tie passes between the dsing, crrectin and filling peratins. This reduces the ttal nuber f peratins t 144 and the nuber f statinary units t The MILP Frulatin The key idea f the frulatin discussed here was riginally prpsed by (Manne, 1960) and develped further in (Cerda et. al. 1997). The ain cntributin is the cnsideratin f additinal cnstraints, siplificatins and re cplex recipes. It uses cntinuus event pints fr the starting and the ending dates f the peratins instead f tie grids and slts. Fr each pair f peratins and that can be prcessed n the sae unit, a binary variable is intrduced t ensure utual exclusin in resurce utilizatin. It is set t 1 if precedes and set t 0 in the ppsite case. Since resurce assignent is als subject f ptiizatin, we define that this

4 binary variable is 0 if either r are assigned t an alternative unit (and n cllisin can ccur). The resurce assignent is delled by additinal binary variables. Starting and ending dates f peratins are expressed by real variables. If an peratin can be prcessed n alternative units, further cpies f thse variables are intrduced fr each pssible unit Sets The fllwing sets are used: jbs J, peratins O, peratins f jb j O, j J, units j (achines) M, alternative units fr peratin M, peratins t be executed n unit : O, peratins f jb j that can be prcessed n unit O, j J, M. j Paraeters and Cnstants Release and due dates f jbs: rel, due, recipes f jbs: rec. Miniu and axiu j j j + peratin duratins n achine : dur, dur, M, O. Three types f tiing cnstraints between starting and ending dates f peratins are defined: ss (starting-starting cnstraints), es (ending-starting) and ee (ending-ending). Every cnstraint defines a tie windw [ t, t + ] which liits the difference f bth ' ' crrespnding event pints (e.g. the end f the dispersin and the beginning f the dsing). The tie windws fr the three types f cnstraints are dented by: ss, ss +, es, es +, ee, ee + t t t t t t, R; ;, ' O. The rder f subsequent peratins f a jb is ' ' ' ' ' ' established using es cnstraints. A safe tie hrizn fr Big-M cnstraints is H R Variables Machine allcatin variables: a {0,1}, M, O. Variables representing starting and ending dates f peratins: s, e R, M, O. Variables t encde precedence relatins f peratins: p {0,1}, M,, ' O. ' Equatins Every peratin ust be prcessed n ne achine and started after the release date f the jb: O : a = 1, j J, O : s rel, j j M M Starting and ending dates f peratin n achine are set t 0 if anther achine is assigned t : M, O : s H a, e H a. Miniu and axiu + peratin duratins: M, O : s + dur a e, s + dur a e. Tiing cnstraints between peratins: j J,, ' O j : s + t s, ' es, e + t s ', ' ss ss s + t + s ' ' ' ' M M M M M M ' ' ' es e t + ee ee + ' s e + t e, e + t + e ' ' ' ' ' M M M M M M. ' If and are prcessed n and precedes then ust nt precede : M,, ' O : p + p a, p + p a, p + p a + a 1 ' ' ' ' ' ' ' ' Operatins n the sae achine ust nt verlap: M,, ' O : '

5 e s H(1 p ) + H(2 a a ), e s H p H(2 a a ) ' ' ' ' ' ' e s H p + H(2 a a ), e s H(1 p ) H(2 a a ) ' ' ' ' ' ' The bjective functin The bjective functin penalizes the accuulated tardiness f the final peratins f all jbs. These are the ixing vessel peratins MV which finish later than the filling j peratins because f the cleaning f the vessels: in Ω= ax{ due e, 0}. j MV j 4. Slutin Prcedure The cercial package GAMS/Cplex was used. Paraeter studies f varius Cplex paraeters yielded that dpriind = 1 clearly increased the slutin perfrance. This setting was used fr the prcedure described belw, the ther paraeters kept their default values. The heuristics used here are siilar t thse in (Cerda et. al., 1997). H1 - Nn-vertaking f nn-verlapping jbs: if due < rel then M, O, ' O : e s j j' j j' ' H2 - Nn-vertaking f jbs with equal recipes: if rec = rec due < due then M, O, ' O : e s j j' j j' j j' ' H3 - Earliest due date: if due < due then M, O, ' O : e s. j j' j j' ' Nte that H1-H3 preserve enugh degrees f freed t keep the prble nn-trivial. H1 and H2 d nt exclude pssibly ptial schedules. The 2-step slutin prcedure is: 1. Apply heuristics H3 t the prble by fixing the crrespnding p variables. Slve the prble and save the integer slutin which represents a valid schedule. 2. Relax the variables fixed in step 1. Apply H1 and H2 by fixing the crrespnding p variables. Slve the prble using the slutin fr step 1 as initial slutin. The slutin btained in step 1 is always a valid slutin in step 2 and therefre can be used as an initial slutin. This is because p variables fixed t 1 in H1 and H2 wuld als be fixed in H3. Being the st restrictive heuristics H3 fixes re p variables t 1 than H1 and H2 tgether. Bth steps were ipleented within the sae GAMS del. j J 5. Experiental Results The del and the slutin prcedure were ipleented in the GAMS 21.3 language and slved with Cplex 9.0 n a 2.4 GHz Athln achine with 1 GB f ery. Bth steps f the slutin prcedure were liited t 20 in. cputatin tie and a ptiality gap f 5%. In rder t investigate the scalability, several prble instances ranging fr 10 t 29 jbs were slved. The jb table was srted accrding t due dates and fr each instance, the first jbs fr the table were taken. The results are shwn in Table 1. The first clun gives the prble size and the next 5 cluns shw the nuber f variables and equatins f the prble as already reduced by Cplex. The final tw cluns shw the slutin ties f bth steps in secnds. All prble instances culd be slved t ptiality (accuulated tardiness and lwer bund equal t 0) with reasnable cputatinal effrt (less than 700 secnds). While the nuber f

6 real variables des nt change between steps 1 and 2, the nuber f discrete variables increases cnsiderably when the heuristics are switched fr H3 t H1+H2. Table 1.Results f bth steps1 and 2 in the scalability experients. #Jbs #Equs.1+2 #Vars. 1 #Vars. 2 #disc. Vars 1 #disc. Vars 2 Tie 1 Tie The first step f the slutin prcedure was always sufficient t cpute ptial slutins. N better slutins culd be derived within the secnd step. The tie spend in step 2 was needed t deterine that the ptial slutin had been fund in step 1. Hence, the EDD heuristics (H3) led t ptial slutin in all test cases. 6. Suary, Cnclusins and Rearks This wrk denstrates a successful applicatin f a MILP frulatin fr shrt and id-ter scheduling prbles t a real-life exaple fr the cheical industry. An exaple prble with re than 80,000 equatins and 9,000 variables culd be slved t ptiality in less than 700 secnds. Althugh less pwerful (in ters f delling strength) than the widely used STN/RTN apprach, ur del is sufficiently expressive t del sequencing cnstraints, resurce assignent, varius tiing cnstraints and cplex recipes with parallel peratins. Sequence-dependant changever prcedures and re cplex bjective functins, invlving strage csts and akespan, can als be accdated in the prpsed frulatin. Unlike STN/RTN, the batch sizes ust be planned in a separate pre-prcessing step. The slutin efficiency was iprved by intrducing a tw-step slutin prcedure invlving heuristics. The authrs gratefully acknwledge financial supprt fr the AMETIST prject (IST ). References A. S. Manne, 1960, On the jb-shp scheduling prble, Op. Res. 8, Issue 2 (1960), E. Kndili, C. Pantelides and R. Sargent, 1993, A general algrith fr shrt-ter scheduling f batch peratins i. ilp frulatin, Cp. and Che. Eng. 17 (1993), N. Shah, C. C. Pantelides and R. Sargent, 1993, A general algrith fr shrt-ter scheduling f batch peratins ii. cputatinal issues, Cp. and Che. Eng. 17 (1993), n. 2, J. M. Pint and I. E. Grssann, 1994, Optial cyclic scheduling f ultistage cntinuus ultiprduct plants, Cp. and Che. Eng. 18 (1994), G. Schilling and C. Pantelides, 1996, A siple cntinuus-tie prcess scheduling frulatin and a nvel slutin algrith, Cp. and Che. Eng. 20 (1996), J. Cerda, G. Henning and I. Grssann, 1997, A Mixed-Integer Prgraing Mdel fr Shrt- Ter Scheduling f Single-Stage Multiprduct Batch Plants with Parallel Lines, Ind. and Eng. Che. Res. 36 (1997), M. Ierapetritu and C. Fludas, 1998, Effective cntinuus-tie frulatin fr shrt-ter scheduling. 1. ultipurpse batch prcesses, Ind. and Eng. Che. Res. 37 (1998), C. Fludas and X. Lin, 2004, Cntinuus-tie versus discrete-tie appraches fr scheduling f cheical prcesses: a review, Cp. and Che. Eng. 28 (2004),