An MILP model for plannng of batch plants operatng n a campagn-mode Yanna Fumero Insttuto de Desarrollo y Dseño CONICET UTN yfumero@santafe-concet.gov.ar Gabrela Corsano Insttuto de Desarrollo y Dseño CONICET UTN gcorsano@ santafe-concet.gov.ar Jorge M. Montagna Insttuto de Desarrollo y Dseño CONICET UTN mmontagna@ santafe-concet.gov.ar Abstract A mxed nteger lnear programmng for the detaled producton plannng of multproduct batch plants s proposed n ths work. New tmng decsons are ncorporated to the model takng nto account that an operaton mode based n campagns s adopted. For plants operatng n a regular fashon along a tme horzon, ths operaton mode assures a more effcent producton management. In addton, sequencedependent changeover tmes and dfferent unt szes for parallel unts n each stage are consdered. Gven the plant confguraton and unt szes, the total amount of each product to be produced and the product recpes, the proposed model determnes the number of batches that compose the producton campagn and ther szes, the assgnment and sequencng of batches n each unt, and the tmng of batches n each unt n order to mnmze the campagn cycle tme. The proposed model provdes a useful tool for solvng the optmal campagn plannng of nstalled facltes. Keywords: multproduct batch plants; producton campagn; plannng; schedulng; MILP model. 1 Introducton Multproduct batch plants are characterzed by ther flexblty to manufacture multple products usng the same equpment. These plants consst of a collecton of processng unts where batches of the varous products are produced by executng a set of operatons. These operatons can be characterzed by a processng tme and they do not nvolve both smultaneous feed and removal of products from the unt durng ths processng tme. Unts that perform the same operaton are grouped n a producton stage, and they can operate n parallel mode (n phase or out of phase). In multproduct batch plants, every product follows the same sequence through all the processng stages (Voudours and Grossmann, 1992). Assumng a gven plant,.e., ts confguraton and the unt szes are known, dfferent producton problems can be posed dependng of the contemplated scenaro. In partcular, when products demands can be accurately forecasted durng a relatvely long tme horzon due to a stable context, more effcent management and control of the producton resources can be attaned f the plant s operated n a perodc
or cyclc way,.e. n a campagn-mode. In ths case, the campagn conssts of several batches of dfferent products that are gong to be manufactured and the same pattern s repeated at a constant frequency over a tme horzon. Ths campagn-based operaton mode has several advantages, for example, more standardzed producton durng certan perods of tme, easer and proftable operatons decsons, more effcent operaton control, and adequate nventory levels wthout generatng excessve costs and mnmzng the possblty of stock-outs. Under ths context, a cyclc schedulng problem must be addressed. Ths type of schedulng s used for products manufacturng wth relatvely constant demand durng a plannng horzon, whch lead to a more regular producton mode and t s more approprate for a make-to-stock producton polcy. From the computatonal pont of vew, the cyclc schedulng allows reducng the sze of the overall schedulng problem, whch s often ntractable. On the other hand, from the modelng pont of vew, one of the man dfferences between cyclc schedulng based on mxed product campagns (MPCs) and short-term schedulng s the adopted objectve functon. Whle the most of approaches for short-term schedulng dealt wth makespan mnmzaton, tardness or earlness, the most approprate performance measure for the schedulng problem usng MPCs of cyclc repetton s the mnmzaton of the campagn cycle tme (Fumero et al., 2012). Takng n mnd that n a plannng context the campagn wll be repeated over the tme horzon, consecutve campagns have to be overlapped n order to reduce dle tmes between them as much as possble. Accordng to Maravelas (2012), the schedulng problem n the context of batch process nvolves the followng decsons: () selecton and szng of batches to be carred out; () assgnment of batches to process unts; () sequencng of batches on unts; and (v) tmng of batches. Takng nto account the combnatoral nature of the problem, most of the exstng approaches n the process systems engneerng lterature consder a specal case of the problem, where the number and sze of batches s fxed,.e. the lot-szng problem s solved frst and then obtaned batches are used as nputs n the schedulng model. The schedulng problem usng MPCs was scarcely addressed n the lterature. Besdes the paper of Fumero et al. (2012), Brewar and Grossmann (1989) developed slot-based formulatons MILP for schedulng of multproduct batch plants usng producton campagns, consderng dfferent transfer polces (unlmted ntermedate storage, UIS, and zero wat, ZW) and where the number and sze of batches are data problem. They determned the optmal campagn cycle tme, for smple plants ncludng only one unt per processng stage. In Fumero et al. (2011) two MILP models for the smultaneous desgn and schedulng of a mult-stage batch plant are proposed. The parallel unts are consdered dentcal and no changeover tmes are taken nto account. The rest of the papers that menton the use of campagns, do not refer to the determnaton of batches and ts cyclc sequencng, as t s managed n ths work. In ths work, the detaled plannng problem of multstage batch plants wth an operaton based on MPCs s addressed usng a MILP model. It s assumed that the plant manager must produce known demands usng a cyclc campagn durng a tme horzon. Nondentcal parallel unts, ZW transfer polcy and sequence-dependent changeover tmes are consdered. Gven the plant confguraton and unt szes, the total amount of each product to be produced n the campagn and the product recpes, the approach determnes the number of batches that compose the producton campagn and ther szes, the batches assgnment to unts, the sequencng of batches n each unt for each stage, and the ntal and fnal tmes of the batches processed n each unt n order to mnmze the campagn cycle tme. Wth the am of reducng the combnatoral complexty assocated to the schedulng decsons, addtonal constrants are consdered n order to elmnate equvalent symmetrc solutons. Then, the schedulng approach through MPCs consderng sequence-dependent changeover tmes for multstage batch plants wth nondentcal parallel unts s effcently solved. 2 Problem descrpton
The problem addressed n ths artcle deals wth a multproduct batch plant where J denotes the set of processng stages that compose the plant and K the set of all unts n the plant. K j represents the set of nondentcal parallel batch unts that operate out-of-phase n stage j, so K = K 1 K 2 K J. A set I of products must be manufactured n the plant followng the same sequence of stages. The total amount requred of each product n the campagn, Q ( I), whch allows mantanng adequate stocks levels takng nto account the estmated demands, s a model parameter. Q can be fulflled wth one or more batches, therefore an ndex b s ntroduced to denote the bth batch requred to meet producton of the correspondng product. In each stage, there are not restrctons about parallel unt szes and, therefore, dfferent unt szes are admtted. Then, V k s used to denote the sze of unt k. The processng tme of each batch of product n unt k, t k, and the sze factor SF j that denotes the requred capacty of unts n stage j to produce one mass unt of fnal product, are problem data. Consderng the demand of product, the non-dentcal parallel unt szes for each stage, the equpment utlzaton mnmum rate for product at each unt, denoted by α k, and the sze factors of product n each stage, the mnmum and maxmum numbers of batches requred to fulfll the demand of product can be calculated n order to ensure soluton optmalty. Thus, the mnmum and maxmum numbers of batches of product for the campagn are calculated, respectvely, as followng: LOW Q UP Q NBC = max and NBC = mn B B max mn where B = mn max and B = max mn α k are the maxmum and mnmum j J k K j SF j J k K j j SFj feasble batch szes for product. The upper bound for the number of batches of each product n the UP NBC campagn allows to propose a set of generc batches assocated to that product, IB, where IB =. Intermedate storage tanks are not allowed. Therefore, takng nto account the confguraton of the plant, there s no batch splttng or mxng,.e. each batch s treated as a dscrete entty throughout the whole process. It s assumed that a batch cannot wat n a unt after fnshng ts processng. Therefore, the ZW transfer polcy between stages s adopted,.e., after beng processed n stage j, a batch b s mmedately transferred to the next stage j+1. Besdes, batch transfer tmes between unts are assumed very small compared to process operaton tmes and, consequently, they are ncluded n the processng tmes. Sequence-dependent changeover tmes, c k, are consdered between consecutve batches processed n the same unt k, even of the same product. Ths transton tme corresponds to the preparaton or cleanng of the equpment to perform the followng batch processng. It s necessary for varous reasons: ensure products qualty, mantan the equpment, safety reasons, etc. For schedulng decsons, an asynchronous slot-based contnuous-tme representaton has been used. The slots correspond to tme ntervals of varable length where batches wll be assgned. In each slot l of a specfc unt k at most one batch b of product can be processed and, f no product s assgned to slot l, ts length wll be zero. The number of slots that must be postulated for unt k of stage j, denoted by L kj, can be approxmated consderng the estmaton on the maxmum number of batches of each product at the campagn. Then, the number of slots postulated for all unts of each stage s the same and t s gven by: UP L NBC k, j = I Although ths value s an overestmaton, a major approxmaton cannot be proposed takng nto account that the parallel unts are dfferent and, on the other hand, the number and szes of batches to be scheduled are optmzaton varables, unlke the most of schedulng approaches presented n the lterature where they are consdered as parameters. However, the lower bound on the number of batches of product at the
campagn, NBC LOW, strongly reduces the number of possble combnatons and consequently mproves the computatonal performance of the model. As prevously stated, the problem conssts of solvng smultaneously two decson levels often addressed sequentally. Through a holstc approach, the selecton and szng of batches of each product, the assgnment of batches to unts n each stage, the producton sequence of assgned batches n each unt and ntal and fnal processng tmes for batches that compose the campagn n each processng unt are jontly determned. 3 Mathematcal formulaton 3.1 Batches selecton and szng constrants The number of batches of product that must be manufactured n the campagn s a model varable. Then, a bnary varable z b s ntroduced, whch takes value 1 f batch b of product s selected to satsfy the demand requrements of that product and 0 otherwse. Let B b be the sze of batch b of product and Q the demand of product that must be fulflled, then: Q B I (1) = b IB b Takng nto account that the sze of unt k denoted by V k and the sze factor SF j are model parameters, f batch b of product s processed n unt k of stage j the followng nequaltes lmt the sze B b of batch b between the mnmum and maxmum processng capactes of unt k: α k Bb I,b IB,k {unts of stage j used to process batch b} (2) SF SF j j where α k s the mnmum flled rate requred to process product n unt k. Due to the unts selected to process the batches of each product are optmzaton varables and ther szes are dfferent, Eq. (2) must be expressed through a varable that ndcates ths selecton, as t wll see later. Besdes, wthout loss the generalty and n order to reduce the number of alternatve solutons, the selecton of batches of a same product as well as the assgned szes to them are made n ascendng and descendng numercal order, respectvely, that s: zb+ 1 zb I,b IB,b + 1 IB (3) Bb+ 1 Bb I,b IB,b + 1 IB (4) 3.2 Assgnment and Sequencng constrants Selected batches must be assgned, n each stage, to specfc slots n the unts. Then, the bnary varable Y bkl s ntroduced, whch takes value 1 f batch b s assgned to slot l n unt k and 0 otherwse. Although ths varable s enough for formulatng the schedulng problem, the bnary varable X kl, whch specfes the slots set utlzed n unt k for processng batches, wll be also used n order to reduce the search space and, therefore, to mprove the computatonal performance. Logcal relatons can be defned among bnary varables z b, X kl and Y bkl. In fact, f slot l of unt k s not utlzed, then none of the proposed batches s processed n t. Moreover, f slot l of unt k s utlzed, then only one of the proposed batches s processed n t. Then, the followng constrant s mposed: Y = X j J,k K, 1 l L (5) I b IB bkl kl j On the other hand, f batch b of product s selected (.e. z b = 1), then ths batch s processed, n each stage j, n only one slot of some of the avalable unts at the stage. Ths condton s guaranteed by:
Y bkl k K 1 l L j = z b j J, I,b IB Wthout loss of generalty and n order to reduce the search space, t s assumed that slots of each unt are consecutvely used n ascendng numercal order. Hence, the slots of zero length take place at the end of each unt. Eq. (7) establshes that for each unt k, slot l+1 s only used f slot l has been already allocated: X kl X kl+ 1, j J,k K j, 1 l L (7) Fnally, varable Y bkl allow correctly expressng the nequaltes posed n (2) as: α k Ybkl Bb I,b I, 1 l L (8) SF B b j SF j + M (1 Y ) I,b IB 1 bkl (9) 1 l L where scalar M 1 s a suffcently large number. 3.3. Tmng constrants Nonnegatve contnuous varables, TI kl and TF kl, are used to represent the ntal and fnal processng tmes, respectvely, of the proposed slots n each unt k. When slot l s not the last slot used n unt k of stage j for processng one batch, that s, f Y b kl+1 take value 1 for some b, fnal processng tme TF kl of slot l n unt k s constraned by: TFkl TI kl + (tk + c'k )Y bkl Yb'kl + 1 j J,k K j, 1 l < L (10) = I ' I b IB b' IB ' b b' A nonnegatve varable YY blb l k s defned to elmnate the blnear products, whch takes value 1 f Y bkl = 1 and Y b' kl + 1 = 1, and 0 otherwse, so (10) s represent usng Bg-M expressons: TFkl TIkl ( tk + c' k ) YY blb' l + 1 k M 2( X kl + 1 ) j J,k K j, 1 l < L (11a) 1 I ' I b IB b' IB' b b' 1 TFkl + TIkl + ( tk + c' k ) YY blb' l + 1 k M 2( X kl + 1 1) j J,k K j, l < L (11b) I ' I b IB b' IB' b b' On the other hand, when the sequence of slots used n unt k s 1, 2,... l,.e. slot l s the last slot used at unt k of stage j to process some batch, takng nto account that the campagn can be cyclcal repeated over a tme horzon, the fnal processng tme TF kl s calculated consderng the changeover tme requred for processng the batch assgned to slot 1 n unt k of stage j. Constrants analogous to (11a) and (11b) are posed for ths case. Constrants to avod the overlappng between the processng tmes of dfferent slots n a unt as well as to match the ntal tmes of empty slots wth the fnal tme of the prevous slot are added to the formulaton. In order to assure ZW transfer polcy, constrants of Bg-M type are ncluded, dependng f slot l s or s not the last slot used at unt k for processng one batch. Due to space reasons, ths set of constrants s not provded n ths manuscrpt, but nterested readers can request t to authors. Fnally, takng nto account that slots of each unt are used n ascendng numercal order, the expresson for the cycle tme of the campagn, CT, s gven by: CT TFkL TI k1, j J,k K j (12) 3.4. Objectve functon The problem goal s to mnmze the cycle tme of the producton campagn that fulflls the demands requrements, subject to prevous constrants. (6)
4 Example The consdered batch plant conssts of three stages wth nondentcal parallel unts wth known szes that operate out-of-phase, as s llustrated n Fgure 1. Avalable unts at each stage are denoted by the sets: K 1 = {1}, K 2 = {2, 3}, and K 3 = {4, 5}, respectvely. Products A, B, and C have to be processed through all stages before beng converted nto fnal products. The requred amounts n the campagn are Q A = 10500, Q B = 6000 and Q C = 9500. Data on processng tmes and sze factors of each product are shown n Table 1, whle the sequence-dependent changeover tmes are gven n Table 2. Consderng the non-dentcal parallel unt szes at each stage, the sze factors for each product n each stage and assumng that the equpment utlzaton mnmum rate s 0.50 for all products and equpment tems, the mnmum feasble batch szes for products A, B and C are: B mn = 0. 5max 5714 kg, 5000 kg,5000 kg = 2857 { } kg { 6666 kg, 4285kg,5555kg} 3333kg { 5714 kg, 4615 kg, 4545 kg} 2857 kg A, B mn B = 0. 5max =, B mn C = 0. 5max =. Stage 1 Stage 2 Stage 3 4000 L 4200 L 3000 L 3000 L 2500 L Fgure 1. Plant structure Table 1. Processng tmes and sze factors of products Processng tme: t k (h) Sze factor: SF j (L/kg) Product Stage 1 Stage 2 Stage 3 Stage 1 Stage 2 Stage 3 1 2 3 4 5 k = 1 k = 2, 3 k = 4, 5 A 14 25 20 7 6 0.70 0.60 0.50 B 16 18 18 5 3 0.60 0.70 0.45 C 12 15 12 4 3 0.70 0.65 0.55 Table 2. Sequence-dependent changeover tmes Product Sequence-dependent changeover tme: c k (h) Stage 1 Stage 2 Stage 3 k = 1 k = 2, 3 k = 4, 5 A B C A B C A B C A 0 0.5 0.3 0.25 0.3 0.4 0 0.6 0.6 B 0.8 0 0.6 2.2 0.25 0.8 0.8 0 0.8 C 1 0.5 0 3 1.5 0.25 2 1.5 0
Then, consderng the campagn demands for all products, the maxmum number of batches of each product at the campagn s four, two and four for products A, B and C, respectvely. Thus, the sets of proposed batches are {b 1, b 2, b 3, b 4 }, {b 5, b 6 }, and {b 7, b 8, b 9, b 10 } for products A, B and C, respectvely, and consequently a total of ten batches must be postulated to guarantee the global optmalty of the soluton. Also, the maxmum feasble batch szes for all products allow determnng the mnmum number of batches of every product at the campagn. In ths case, the maxmum feasble batch szes for all products are: B max A B max B = mn = mn { 5714 kg, 7000 kg, 6000 kg} = 5714 kg { 6666 kg, 6000 kg, 6666 kg} = 6000 kg { 5714 kg, 6461 kg, 5454 kg} = 5454 kg B max C = mn then, the requred mnmum number of batches for products A and C s two, whle for product B s one. The model under these assumptons comprses 52052 constrants, 9167 contnuous varables and 555 bnary varables. It was mplemented and solved usng GAMS, va CPLEX 12.5 solver, n 42.77 CPU seconds wth a 0% of optmalty gap. The optmal campagn cycle tme s equal to 70.4 hours and t nvolves two batches of product A (b 1, b 2 ), one of B (b 5 ), and two of C (b 7, b8),.e. the demands of all products are fulfll wth the mnmum number of batches. The optmal producton sequence obtaned n each batch unt for the dfferent stages, consderng sequence-dependent changeover tmes, s llustrated n the Gantt chart of Fgure 2. Takng nto account that the optmal campagn s cyclcally repeated over a tme horzon, the changeover tmes between products processed n the last and frst slot of each unt must be ncluded n the optmzaton n order to acheve the accurate overlap of successve campagns. For ths example, as t can be seen from Fgure 2, changeover tmes between pars of campagns are: c AC1 = 0.3 h for unt of stage 1; c AC2 = c AC3 = 0.4 h for unts of stage 2; and c AC4 = c AC5 = 0.6 h for unts of stage 3. Unts Campagn cycle tme = 70.4 h Stage 1 1 b7 b5 b8 b1 b2 Stage 2 2 3 b7 b5 b8 b1 b2 Stage 3 4 5 b7 b5 b8 b1 b2 0 12 27 41.1 53.1 Product A Product B Product C Changeovers tmes 70 81.5 88.7 90.5 96.7 Tme (h) Fgure 2. Gantt chart of the producton campagn Batches b 1 and b 2 satsfy the total requred demand of product A wth szes of 5500 kg and 5000 kg, respectvely; batch b 5 wth sze equal to 6000 kg s only selected to accomplsh the campagn demand of product B; whle batches b 7 and b 8 are requred to acheve the producton of C wth szes equal to 4955 kg and 4545 kg, respectvely. The capactes used n each unt of the dfferent stages for processng the selected batches are resumed n Table 3. The batches that reach the maxmum capactes are hghlghted n boxes shaded n gray. Batch b 2 of product A s processed n unts 1, 3 and 5 and ts sze s the maxmum possble to be processed n unts 3 and 5 of stages 2 and 3, respectvely. Then, batch b 1 fulflls the requred amount of that product occupyng approxmately 96%, 79% and 92% of the capacty of unts 1, 2 and 4, respectvely. Batch b 5 of product B s processed n unts 1, 2 and 4 and ts sze s the maxmum
possble to be processed n unt 2 of stage 2. On the other hand, two batches of product C are processed for meetng ts demand. Batch b 8 s processed n unts 1, 3 and 5 usng 80%, 98.5% and 100% of ther capactes, respectvely; whle batch b 7 fulflls the requred amount of that product n the campagn. Table 3. Capactes used n each unt of each stage Stage 1 Stage 2 Stage 3 Product Batch k = 1 k = 2 k = 3 k = 4 k = 5 A b 1 3850 3300 2750 b 2 3500 3000 2500 B b 5 3600 4200 2700 C b 7 3468.2 3220.4 2725 b 8 3181.8 2954.5 2500 5 Conclusons In ths work, the optmal producton plannng of multstage batch plants wth nondentcal parallel unts that operate n campagn-mode s faced. Schedulng s modeled accordng to campagn-based operaton mode n such way that the campagn cycle tme mnmzaton s an approprate optmzaton crteron. Sequence-dependent changeover tmes are consdered for each ordered par of products n each unt of the dfferent stages. Takng nto account the complexty of the smultaneous nvolved decsons, some addtonal constrants that elmnate equvalent symmetrc solutons mantanng the model generalty are consdered, n order to reduce the search space and therefore mprove the computatonal performance. Also, varous equatons are reformulated n order to keep the problem lnear and assure the global optmalty of the soluton. Through the example the capabltes of the proposed formulaton are shown. Wth the proposed formulaton, an nterestng problem has been solved. Many tmes, n made-to-stock contexts, the campagn-based operaton mode s an approprate alternatve that allows takng advantage of the avalable resources wth an ordered producton management. The proposed model smultaneously solves lot-szng and schedulng problems n reasonable computng tme. Thus, ths approach can be appled n real producton systems that operate n campagn-mode takng nto account the assumed suppostons as far as dfferent unt szes, changeovers, etc. References 1. C. T. Maravelas. General framework and modelng approach classfcaton for chemcal producton schedulng. AIChE Journal 2012, 58(6): 1812-1828, 2012. 2. D. Brewar and I. E. Grossmann. Incorporatng schedulng n the optmal desgn of multproduct batch plants. Computers and Chemcal Engneerng, 13, 141-161, 1989. 3. V.T. Voudours, I.E. Grossmann. Mxed-Integer Lnear Programmng Reformulatons for Batch Process Desgn wth Dscrete Equpment Szes. Ind. Eng. Chem. Res. 1992,31, 1315-1325. 4. Y. Fumero, G. Corsano, J. M. Montagna. Schedulng of multstage multproduct batch plants operatng n a campagn-mode. Industral Engneerng and Chemcal Research, 5: 3988 4001, 2012. 5. Y. Fumero, G. Corsano, J. M. Montagna. Detaled Desgn of Multproduct Batch Plants Consderng Prodcton Schedulng. Industral Engneerng and Chemcal Research, 50 (10), 6146-6160, 2011.