An Application of MILP-based Block Planning in the Chemical Industry

Transcription

1 The Eghth Internatonal Symposum on Operatons Research and Its Applcatons (ISORA 09) Zhangjaje, Chna, September 20 22, 2009 Copyrght 2009 ORSC & APORC, pp An Applcaton of MILP-based Block Plannng n the Chemcal Industry Hans-Otto Günther Dept. of Producton Management, Techncal Unversty of Berln, Strasse des 17. Jun 135, Berln, Germany, e-mal: hans-otto.guenther@tu-berln.de Abstract Producton processes n the chemcal ndustry often consst of an ntal stage n whch a basc chemcal product s produced and a subsequent stage whch produces the fnal product varants accordng to demand forecasts or customer orders. Typcally, a fllng and packagng process s ntegrated nto ths fnal producton stage. For schedulng the producton and packagng actvtes a mxed-nteger lnear programmng (MILP) model s proposed whch s based on the defnton of setup famles and the producton of product varants n a pre-defned sequence. The model determnes the sze and the tme phasng of the ndvdual producton lots. In contrast to conventonal lot-szng and schedulng models, the proposed mzaton model s based on a contnuous representaton of tme. Keywords Producton plannng and schedulng Lot szng Mxed-nteger lnear programmng model chemcal ndustry 1 Introducton Consderng the producton of pharmaceutcals as an example, two major producton stages can be dstngushed. In the ntal stage a basc chemcal product s produced through a number of tme-consumng chemcal processng steps whle n the subsequent stage the basc chemcal product s processed further and the fnal product varants are produced accordng to demand forecasts or customer orders. In the chemcal and pharmaceutcal ndustry, dfferent varants of a product are acheved by adjustng the process parameters and the mx of nput materals as well as usng a varety of packagng formats accordng to the end users requrements and country-specfc regulatons. In ths paper we focus on the fnal producton and packagng stage of the entre chemcal producton process whch often conssts of a number of fllng and packagng lnes. Ths combned producton and packagng process typcally consttutes the bottleneck n the fnal stage of the producton system. Such producton systems can be found n many branches of the process ndustres, e.g. n the chemcal or consumer goods ndustres. Specfcally n the chemcal and pharmaceutcal ndustry, parallel lnes are establshed for each package type, e.g. lqud and sold drugs. A lne usually produces a number of product types and packages them nto ndvdual unts for shpment to customer warehouses. Each product type corresponds to a specfc recpe whch determnes the ngredents and the processng condtons whle product varants are defned by the specfc packagng form. Typcally, producton lnes are set up for a

2 104 The 8th Internatonal Symposum on Operatons Research and Its Applcatons specfc product type requrng a major setup such that the changeover between the ndvdual product varants can be accomplshed wth only a mnor setup or cleanng operaton. The ntenton of ths paper s to develop an mzaton model for short-term schedulng and lot szng n sngle-stage chemcal producton. The model formulaton s based on the block plannng concept proposed by Günther et al. (2006). The remander of ths artcle s organzed as follows. In the next secton the major characterstcs of the block plannng concept are explaned. In the subsequent Secton 3, a block plannng model based on mxed-nteger lnear programmng s developed. Fnally, conclusons are drawn and the practcal applcablty of the proposed modellng approach s dscussed. 2 Block plannng In short-term producton plannng, decsons have to be made on lot szes and ther tmng and sequencng on the varous manufacturng and packagng lnes. In partcular, n process-related ndustres there s often a natural sequence n whch the varous products are to be produced n order to mnmze total changeover tme and to mantan product qualty standards. For example, setups are sequenced from products wth hgh to low purty requrements or from the brghter color of a product to the darker. Hence, famles of products can be dentfed whch are produced n a gven sequence under the same basc equpment setup. In ths case, major setups are ncurred for changng over between product famles whle only mnor setups are needed for swtchng to another product wthn the same famly. Snce conventonal lot-szng and schedulng models do not suffcently reflect the condtons gven n process ndustres we propose an alternate approach, called block plannng, for schedulng producton orders on a contnuous tme scale wth demand elements beng assgned to dstnct delvery dates. Moreover, ssues lke defnton of setup famles wth consderaton of major and mnor setup tmes and multple non-dentcal producton lnes wth dedcated product-lne assgnments are addressed n a realstc way. The model formulaton ams at supportng operatve decsons n a sngle-stage producton system. So far, n the scentfc lterature the applcaton of block plannng concepts and the development of correspondng mzaton models are wdely neglected though block plannng concepts are easy to mplement and reflect manageral practce prevalent n make-and-pack producton. Block plannng approaches have been developed for specfc applcatons, e.g. har dye producton (cf. Günther et al. 2006) or n fresh food ndustres (cf. Lütke Entrup et al. 2005). Recently, an ntegrated model for producton and dstrbuton plannng n the fast movng consumer goods ndustry has been presented by Blgen and Günther (2010). In the chemcal ndustry applcaton nvestgated here, specfc combned fllng and packagng lnes are used for the varous packagng types, e.g. lqud and sold forms of drugs. Ths makes t possble to subdvde the entre plannng problem nto lne-specfc sub-problems. Hence, n the followng we consder only one sngle lne. For the fllng and packagng lne at hand, the block pattern gven n Fg. 1 llustrates the sequence of product types whch are produced each after a major setup.

3 An Applcaton of MILP-based Block Plannng n the Chemcal Industry 105 Once the lne s set up ths way, a famly of dfferent product varants s processed n the pre-defned sequence each wth a producton sub-lot of varable sze. When changng over from one product varant to another, a mnor setup tme s ncurred. Further blocks are appended n a cyclcal fashon untl the end of the plannng horzon s reached. Fgure 1: Block pattern for a producton lne. The major characterstcs of the basc block plannng concept upon whch the development of the MILP model n the next secton s based can be summarzed as follows. Gven the assgnment of products to setup famles, ed setup sequences of products are defned based on human expertse and technologcal requrements. Each block corresponds to a setup famly and s scheduled n a cyclcal fashon. The composton of the ndvdual blocks s not necessarly the same. Bnary decson varables ndcate whether a product s set up or not and contnuous decson varables reflect the lot sze of each product n the block. Dependng on the development of demand over tme, the lot szes of an ndvdual product may vary from block to block. As a result, also the tme needed to complete a block s varable. The start-off and completon dates of a block are not drectly lnked to the perod boundares. Hence, a block s allowed to start n an earler perod as soon as the predecessor block has been completed. However, the executon of a block must be fnshed before the end of the assgned perod. Typcally, a major setup operaton s performed before startng or after completng a block (e.g. for cleanng the manufacturng equpment), whle only a mnor setup operaton s requred when changng between products wthn the same block (e.g. for provson of materal or for adjustng the processng condtons). Macro-perods, e.g. weeks, are used for the assgnment of blocks whle the assgnment of external demand elements s based on mcro-perods, e.g. days. For each product nventory balances are updated on a perodc bass accordng to the producton output and the gven external demand. The usual objectve functon s to mnmze total nventory holdng and setup costs. Major constrants arse from the avalable producton capactes and the satsfacton of external demand.

4 106 The 8th Internatonal Symposum on Operatons Research and Its Applcatons 3 Model formulaton In ths paper, a flexble block plannng approach based on a mxed-nteger lnear programmng (MILP) model s proposed whch ntroduces a consderable degree of flexblty for determnng the length of an entre block, varyng the producton quanttes for ndvdual products varants wthn a block, and for schedulng the start-off and completon tmes of all blocks and producton lots. The MILP model formulaton uses a contnuous tme representaton to model the producton runs for all products wthn a setup famly whle a dscrete tme representaton s used for the assgnment of blocks to macro-perods and for the daly assgnment of demand elements. Bnary decson varables refer to the setup of blocks, producton lots for packagng forms, and sub-lots for product types. Contnuous varables are used to model the lot szes and sub-lot szes, nventory levels and the tmng of the varous producton actvtes. The objectve functon mnmzes major setup costs for producton lots, mnor set up costs for sub-lots, and nventory holdng costs. The notaton used n the model formulaton s gven as follows. Indces and sets I blocks 1,...,last( I ) I ed blocks ( 1,...,last( I ) ) I onal blocks ( frst( I ),...,last( I ) ) p P products p P() products whch are produced n block p P( j) products whch belong to product famly j j J product famles t, perods wth and ndcatng the earlest start and latest feasble completon tme, respectvely, of block Parameters ls p mnmum sub-lot sze of product p n the ed block M p maxmum sub-lot sze of product p a p unt producton tme for product p s p mnor setup tme per sub-lot of product p S major setup tme for block d pt external demand of product p assgned to the end of perod t mn c mnor setup cost per sub-lot of product p p maj c major setup cost for block nv c p nventory holdng cost per unt of product p per perod Decson varables and domans xp 0 sub-lot sze of product p n block y j 01, =1, f product famly j s assgned to onal block p 01, =1, f product p s set up n block (0, otherwse) 01, = 1, f onal block s actve,.e. a product famly s assgned to t (0, otherwse)

5 An Applcaton of MILP-based Block Plannng n the Chemcal Industry 107 z t 01, =1, f block has been fnshed up to the end of perod t (0, otherwse) qpt 0 output of product p avalable at the end of perod t Fpt 0 nventory level of product p at the end of perod t 0 start tme of block 0 end tme of block 0 duraton of block The followng MILP model formulaton s based on the dstncton of ed and onal blocks and does not rely on a strct block to perod assgnment. Snce each product famly must be scheduled at least once n the course of the plannng horzon gven that a postve net-requrement for at least one tem wthn the product famly exsts we consder the setup of one block of each product famly as beng ed, but stll leave the start-off tme of the block open. Further setups of a product famly are consdered onal because the number and the sze of the respectve blocks as well as ther tmng have to be determned by the mzaton model. In order to obtan the sequence of the ed blocks, the correspondng product famles are sorted n ascendng order of ther run-out-tmes. The second part of the producton schedule contans the onal blocks. As a flexble and practce-orented approach we propose that a menu of onal blocks s defned by the human planner leavng open the assgnment of product famles to blocks and the sub-lot szes of the ndvdual products. Snce not all of the allowed onal blocks must be utlzed, actve and non-actve blocks can be dstngushed. The constrants of the block plannng model are the followng. Setup constrants for ed blocks: Product famles whose stocks deplete durng the plannng horzon requre at least one setup,.e. the value of the correspondng bnary setup varable s set to one n constrant (1) and, accordng to constrant (2), the sub-lot szes of all products belongng to the product famly must meet the mnmum lot sze necessary to cover demand over the pre-determned run-out tme of the ntal stock. p 1 I,pP()ls p 0 (1) xp ls p I,pP()ls p 0 (2) Setup constrants for onal blocks: Accordng to (3) block can only be actve, f ts predecessor block -1 s actve. In other words, non-actve blocks are assgned to the end of the schedule. Constrant (4) ensures that exactly one product famly s assgned to an onal block, f the block s actve, and no product famly s assgned, f the block s not actve. Accordng to (5) bnary setup varables for the producton sub-lots are allowed to take values of one only f the respectve product famly j s assgned to the block. Constrant (6) models the relatonshp between the sze of the sub-lot and the bnary setup varable. The sze of the sub-lot s enforced to zero f no correspondng setup operaton s performed. I wth 1 (3) 1 last( I )

6 108 The 8th Internatonal Symposum on Operatons Research and Its Applcatons yj jj pp( j) p yj P( j ) I (4) I, j J (5) xp M p p I (6) Block schedule: Constrant (7) models the duraton of a block whch results from the major setup tme for the block, the mnor setup tmes for all sub-lots and the tme requred for producng the sub-lot szes. Accordng to (8) a block s allowed to start as soon as the predecessor block has been completed. Constrants (9) and (10) mpose tme wndows for the earlest start-off and for the latest completon tme of a block. Constrant (11) defnes the end tme of block. sp p ap xp S I (7) pp 1 1 2,...,last( I ) wth 1 0 (8) I (9) I (10) I (11) Matchng producton output and demand: Snce the assgnment of product famles to blocks s not gven n advance, addtonal decson varables and constrants are ntroduced n order to trace the completon of producton actvtes over tme and to match the resultng producton output aganst external demand. Whle a contnuous tme representaton s used to schedule the producton runs for all products wthn a product famly, demand elements are assgned to the end of a perod. Socalled heavy-sde varables (cf. Grunow et al., 2003) are ntroduced whch ndcate f block has been fnshed up to a partcular perod t. The followng logcal constrants (12) and (13) enforce the heavsde varables to zero for all perods pror to the completon perod of the block and to one for the completon perod and all succeedng perods. For all perods t pror to the completon perod of block, the rghthand sde n (12) s greater than zero and less than one and n (13) less than zero. Hence the bnary heavsde varables are enforced to zero. For the remanng perods the rght-hand sde n (12) s greater than one and n (13) between zero and one. As a result, the heavsde varables are enforced to one. Ths way the completon of a producton sub-lot s ndcated by a swtch from zero to one n the perodc development of the heavsde varables. t zt 1 I,t,..., (12) t zt I,t,..., (13) The followng constrants are needed to derve the quanttes of fnal products from the varous onal blocks dependng on the assgned product famly. Accordng to (14) and (15) only at the perod, n whch the values of the heavsde varables swtch from zero to one, output from the producton process s avalable. For

7 An Applcaton of MILP-based Block Plannng n the Chemcal Industry 109 all other perods the respectve varables takes a value of zero. Constrant (16) makes sure that the daly output quanttes match the producton sub-lot sze. Constrant (17) contans the nventory balances. qpt M p zt I,pP,t (14) qpt M p zt z,t 1 I, p P,t 1,..., (15) qpt xp I, p P,t,..., (16) t Fpt Fp,t1 qpt dpt pp,t T wth Fp0 gven (17) Objectve functon: The entre mzaton model conssts of constrants (1) to (17) and the objectve functon (18) stated below. The objectve functon ams to mnmze total costs comprsed of major and mnor setup costs and nventory holdng costs. maj mn nv mn c c c F 4 Conclusons (18) p p p pt I I pp pptt In ths paper an MILP-based block plannng approach for producton schedulng n sngle-stage producton systems has been presented. The sze of the MILP model depends on the number of product famles and specfc product varants, the densty of the tme grd whch s mposed for the assgnment of demand elements, and on the number of blocks n the schedule. Despte ts general applcablty, a key factor whch has a consderable mpact on the computatonal effort s the ntroducton of daly tme perods and the use of heavsde varables whch are needed to trace the completon date of the producton lots. Especally, when a large number of small-szed demand elements have to be consdered, the computatonal complexty of the model soluton s accordngly ncreased and CPU tmes rse sgnfcantly. Nevertheless, ntal numercal tests showed that mal solutons can be obtaned wthn a few mnutes of CPU tme. In any case, the framework of mxed-nteger lnear programmng provdes consderable flexblty for the ntegraton of applcaton-specfc features whch s not gven for conventonal lot szng and schedulng models known from the academc lterature. Combned wth the human planner s expertse on the defnton of setup famles and the natural sequence n whch the varous products are to be produced n order to mnmze total changeover tme and to prevent qualty losses, block plannng represents a practcal and effcent way to support decson makers n practce. References [1] Blgen B, Günther H-O (2010) Integrated producton and dstrbuton plannng n the fast movng consumer goods ndustry: A block plannng applcaton. OR Spectrum (to appear) [2] Grunow M, Günther H-O, Yang G (2003) Plant coordnaton n pharmaceutcs supply

8 110 The 8th Internatonal Symposum on Operatons Research and Its Applcatons networks. OR Spectrum, 25: [3] Günther H-O, Grunow M, Neuhaus U (2006) Realzng block plannng concepts n make-and-pack producton usng MILP modellng and SAP APO. Internatonal Journal of Producton Research, 44: [4] Lütke Entrup M, Günther H-O, van Beek P, Grunow M, Seler T. (2005) Mxed nteger lnear programmng approaches to shelf-lfe-ntegrated plannng and schedulng n yoghurt producton. Internatonal Journal of Producton Research, 43: