A Hybrid MILP/CP Decomposition Approach for the Continuous Time Scheduling of Multipurpose Batch Plants

Transcription

1 A Hybrd MILP/CP Decomposton Approach for the Contnuous Tme Schedulng of Multpurpose Batch Plants Chrstos T. Maravelas, Ignaco E. Grossmann Carnege Mellon Unversty, Department of Chemcal Engneerng Pttsburgh, PA Submtted March 2003 Revsed Manuscrpt Submtted December 2003 Abstract A hybrd Mxed-Integer Lnear Programmng (MILP)/Constrant Programmng (CP) decomposton algorthm s proposed for the short-term schedulng of batch plants that rely on the State Task Network representaton. The decsons about the type and number of tasks performed, as well as the assgnment of unts to tasks are made by the MILP master problem. The CP subproblem checks the feasblty of a specfc assgnment and generates nteger cuts for the master problem. A graph-theoretc preprocessng that determnes tme wndows for the tasks and equpment unts s also performed to enhance the performance of the algorthm. To exclude as many nfeasble confguratons as possble, three classes of nteger cuts are generated. Varous objectve functons such as the mnmzaton of assgnment cost, the mnmzaton of makespan for fxed demand and the maxmzaton of proft for a fxed tme horzon can be accommodated. Varable batch-szes and duratons, dfferent storage polces, and resource constrants are taken nto account. The proposed framework s very general and can be used for the soluton of almost all batch schedulng problems. Numercal results show that for some classes of problems, the proposed algorthm can be two to three orders of magntude faster than standalone MILP and CP models. 1. Introducton The problem of short term schedulng of general (or multpurpose) batch plants has receved consderable attenton n the lterature. Kondl et al. (1993) proposed the dscrete-tme State Task Network (STN) representaton, and Shah et al. (1993) proposed a reformulaton. The equvalent Resource Task Network (RTN) representaton was proposed by Panteldes (1994). Contnuous tme representatons, whch are more general but potentally dffcult to solve, were proposed by Sargent and Zhang (1995), and Mockus and Reklats (1999). Based on the STN and RTN representatons, several other contnuous tme MILP models have been proposed snce then (e.g. Ierapetrtou and Floudas, 1998; Lee et al., 2001; Castro et al., 2001; Maravelas and Grossmann, 2003a). Although the performance of these models n several test problems has proved to be encouragng, ther major lmtaton s that for large problems they can become computatonally expensve due to the bg-m constrants that are used for tme matchng. Ths has led prevous workers to consder specal cases of the general multpurpose batch plant problem. Some of the common assumptons are, (a) no resource constrants other than equpment, and (b) no batch splttng and mxng. Furthermore, the computatonal performance of STN- and RTN-based models s poor when the objectve s the mnmzaton of makespan for a gven demand. For short-term schedulng, however, the demands are usually fxed, whch mples that the mnmzaton of makespan or the mnmzaton of producton cost s often a more realstc objectve. Constrant Programmng (CP) s a new modelng and soluton paradgm that has proved to be very effectve for solvng certan classes of problems (van Hentenryck, 1989; Marrot and Stuckey, 1999; Hooker, 2000). To whom all correspondence should be addressed: E-mal: grossmann@cmu.edu, Tel: , Fax:

2 Constrant Programmng s partcularly effectve for solvng feasblty problems and seems to be better than tradtonal MILP approaches n dscrete optmzaton problems where fndng a feasble soluton s dffcult. The lack of an obvous relaxaton, however, makes CP worse for loosely constraned problems, where the focus s on fndng the optmal soluton among many feasble ones and provng optmalty. Constrant Programmng has been successfully used for some classes of schedulng problems (Baptste et al., 2001). In the general short-term schedulng of multpurpose batch plants the number of tasks (jobs) s not known a pror: t s a decson varable that s to be determned by the optmzaton. Moreover, n ts general form (varable processng tmes and batch szes, recycle streams, batch splttng/mxng, resources other than equpment, changeover tmes) the schedulng problem s loosely constraned and a standalone CP approach s not effcent. Thus, n ths work we have combned the tradtonal MILP approach wth the CP approach n order to take advantage of ther complementary strengths. We use MILP to optmze and fnd partal solutons and CP to check feasblty and produce complete solutons. Smlar approaches have been proposed by Jan and Grossmann (2001), and Harjunkosk and Grossmann (2002) for the restrcted cases of mnmzaton of assgnment cost of sngle and multstage batch plants, respectvely, yeldng consderable computatonal mprovements compared to standalone MILP and CP models. The proposed hybrd MILP/CP method conssts of two phases and s based on the MILP model by Maravelas and Grossmann (2003a). In the frst phase we determne the earlest start tmes (EST) and the latest fnsh tmes (LFT) of tasks and unts, and derve strong nteger cuts that exclude nfeasble subconfguratons. In the second phase we use an teratve scheme where we solve a master MILP model and a CP subproblem. The master problem yelds a set of tasks to be performed and ther assgnments to unts. The subproblem checks feasblty and derves a feasble schedule, f one exsts, for the assgnment obtaned by the master problem. Integer cuts are added n the master problem to exclude the current assgnment. Three objectve functons are consdered: (a) mnmzaton of makespan subject to satsfyng a gven demand, (b) maxmzaton of proft for a fxed tme horzon, and (c) mnmzaton of cost subject to satsfyng gven orders wth due dates. Numercal examples are presented that show that orders of magntude reducton n computatonal effort can be acheved wth the proposed hybrd approach. 2. Background 2.1. Mxed-Integer Formulatons for Multpurpose Batch Plants In ths paper we address the schedulng of multpurpose batch plants. An example s shown n Fgure 1, where raw materal S1 s heated to form ntermedate S2, whch s then used for the producton of ntermedate S3 (Reacton 1), ntermedate S4 and fnal product S5 (Reacton 2). Intermedates S3 and S4 are mxed to produce fnal product S7 (Reacton 3) and ntermedate S3 s purfed to gve fnal product S6 and S2 that s recycled (Separaton). Multpurpose batch plants exhbt a number of features that make ther modelng challengng. As shown n Fgure 1, unlke tradtonal schedulng problems where each batch can be vewed as an entty movng throughout the plant, we may have batch splttng (e.g. a batch of S2 can be used for Reacton 1 and Reacton 2), batch mxng (e.g. ntermedates S3 and S4 are mxed to produce one batch of S7) or recycle streams. Thus, general batch plants can be vewed as the generalzaton of all possble plant confguratons. Furthermore, the duraton and the batch sze of the tasks may not be constant. Moreover, resource constrants other than those on equpment unts may be present (e.g. manpower, coolng water, etc.), and dfferent storage polces (UIS/FIS/NIS/ZW) can be appled for the varous chemcals. In order to account for all these ssues, we need to montor the level of nventores and the level of resource consumpton throughout the tme horzon. To do so, we need to partton the tme horzon nto a suffcently large number of perods and enforce equpment unt, utlty and nventory feasblty for all tme perods, whch leads to large and dffcult to solve formulatons. 2

3 S1 1h Heat 1h Reacton1 S2 3h Reacton2 70% S4 40% 60% 10% 2h S3 Separaton 2h Reacton 3 90% S6 S7 30% S5 Fgure 1: Multpurpose batch plant Kondl et al. (1993) formulated the STN representaton for batch schedulng wth an MILP usng a dscretetme representaton (Fgure 2a), where the tme horzon s dvded nto H ntervals of equal duraton, common for all unts, and where the tasks begn and fnsh at a tme pont. Ths means that the duraton of the ntervals must be equal to the greatest common factor of the processng tmes of tasks and that the dscretetme representaton can only be used when the processng tmes are constant. The assumpton of constant processng tmes s not always realstc, whle the length of the ntervals may be so small that t ether leads to a prohbtve number of ntervals renderng the resultng model unsolvable, or else t requres approxmatons whch may compromse the feasblty and optmalty of the soluton. It should be noted that the Resource Task Network (RTN) formulaton by Panteldes (1994) provdes a more compact representaton but shares smlar lmtatons as the STN model. To crcumvent the above cted dffcultes, two dfferent contnuous-tme representatons have been proposed n whch the tme horzon s dvded nto tme ntervals of unequal and unknown duraton, common for all unts. In the contnuous tme representaton I (Fgure 2b), each task must start and fnsh exactly at a tme pont (Zhang and Sargent, 1995; Schllng and Panteldes, 1996; Mockus and Reklats, 1999), whle n the representaton II (Fgure 2c), each task must start at a tme pont but t need not fnsh at a tme pont (Castro et al., 2001). In both representatons, the number of tme ponts s determned wth an teratve procedure, durng whch the number of tme ponts s ncreased by one untl there s no mprovement n the objectve functon. Snce tme ponts are not fxed, constrants that match a tme pont wth the start (or fnsh) of a task are necessary. These constrants are bg-m constrants that result n poor LP relaxatons. On the other hand, the contnuous-tme representaton accounts for varable processng tmes, and s more realstc than the dscrete-tme representaton. It also requres sgnfcantly fewer tme ntervals and hence leads to smaller problems. A combnaton of the two types of contnuous-tme representatons was proposed by Maravelas and Grossmann (2003a). An alternatve approach s the event-pont representaton (Fgure 2d) proposed by Ierapetrtou and Floudas (1998), where tme events are not common for all unts. In ths approach, the tme horzon s dvded nto a number of events whch are dfferent for each unt, subject to certan sequencng constrants. Snce events need not be common among unts, the number of events necessary n ths approach s usually smaller than the number of perods requred n other contnuous-tme representatons, and the proposed model s faster compared to contnuous-tme STN formulatons wth common tme grd. However, as has been shown n Maravelas and Grossmann (2003a), ths representaton may not always yeld accurate representatons of batch operatons. Fnally, a number of papers address specal cases of multpurpose batch plants (Rodrgues et al., 2000; Mendez et al., 2000; Mendez and Cerda, 2000; Mendez et al., 2001; Lee et al., 2001). In most of these papers, specal assumptons are made to allow the development of specal MILP models that are easer to solve. Some of the common assumptons are, (a) the plant has some specal confguraton, (b) no batch splttng and mxng s allowed, and (c) there are no resource constrants other than those on equpment unts. The objectve functon n most of these approaches s the maxmzaton of proft for a fxed tme horzon. In short-term schedulng, however, the demand s usually fxed and more meanngful objectves are to mnmze the makespan for a fxed demand, or mnmze the producton cost for fxed demands wth due dates. As has 3

4 been shown, however, the computatonal performance of STN-based MILP models s poor when the objectve s the mnmzaton of makespan. For ths reason we consder Constrant Programmng as an alternatve soluton technque. (a) Dscrete-Tme Representaton (b) Contnuous-Tme Representaton I U1 U2.. UN H-1 H t (c) Contnuous-Tme Representaton II U1 U2.. UN H-1 H t (d) Event-Pont Representaton U1 U2.. UN U1 U2.. UN k k k k H-1 H t Fgure 2: Tme Representatons. t 2.2. Constrant Programmng and MILP/CP Integraton Schemes Constrant Programmng (van Hentenryck, 1989; Hooker, 2000) was orgnally developed to solve feasblty problems, but t has been extended to solve optmzaton problems as well. Constrant programmng (CP) s based on performng a tree enumeraton by reducng at each node the domans of the varables, whch can be contnuous, general nteger, boolean and bnary. If an empty doman s found the doman s pruned. Branchng s performed whenever a doman of an nteger, bnary or boolean varable has more than one element, or when the bounds of the doman of a contnuous varable do not le wthn a tolerance. Whenever a soluton s found, or a doman of a varable s reduced, new constrants are added. The search termnates when no further nodes must be examned. Constrant programmng algorthms are very effcent for some classes of problems, among whch schedulng s a promnent one. Due to the general nature of the batch plants that appear n chemcal ndustry, however, a standalone CP model would not be effectve for solvng ths class of problems. Ths s due to the fact that the type and number of tasks to be performed n a general batch plant are optmzaton decsons, whch mples that the number of potental actvtes n the CP formulaton may be very large and, moreover, the large number of alternatve producton paths makes the CP problem loosely constraned. The computatonal performance of a standalone CP model that we developed was ndeed very poor. Specfcally, the computatonal tme requred for a well studed example of Kondl et al. (1993), that nvolves fve tasks, four unts and nne states, by a standalone CP model s more than 300 CPU sec, whereas the same problem s solved n less than 1 CPU sec by any contnuous-tme MILP model. Models that ntegrate MILP and CP have also appeared recently. The motvaton for ths ntegraton follows from the fact that MILP and CP have complementary strengths that can be exploted smultaneously. Mathematcal programmng technques are effcent n fndng optmal solutons and provdng good bounds, whle the CP language s more expressve and the CP search technques are often more effcent n solvng feasblty problems. The ntegraton between MILP and CP can be acheved n two ways (Hooker, 2002; van Hentenryck, 2002): (a) By combnng MILP and CP constrants nto one hybrd model. In ths case a hybrd algorthm s also needed for the soluton of the model. (b) By decomposng the orgnal problem nto two subproblems: one MILP and one CP problem. Each model s solved separately and nformaton obtaned whle solvng one subproblem s used for the soluton of the other subproblem. 4

5 More nformaton on CP can be found n Marrot and Stuckey (1999) and Hooker (2000). Constraned-based schedulng algorthms can be found n Baptste et al. (2001), and a short descrpton of MILP/CP ntegraton schemes can be found n Jan and Grossmann (2001). 3. Problem Statement For the problem addressed n ths paper we assume that we are gven: () a fxed or varable tme horzon () the avalable unts and storage tanks, and ther capactes () the avalable utltes and ther upper lmts (v) the producton recpe (mass balance coeffcents, utlty requrements) (v) the processng tmes and changeover tmes (v) the amounts of avalable raw materals; the demand of fnal products and ther due dates (v) the prces of raw materals and fnal products The goal s then to determne: () the type and number of tasks performed () the assgnment of equpment unts to tasks () the sequencng and tmng of tasks takng place n each unt (v) the batch sze and duraton of tasks (v) the amount of resources allocated to each task (v) the amount of raw materals purchased and the amount of fnal products sold The objectve can be, (a) the maxmzaton of producton, ncome or proft for a fxed tme horzon, or (b) the mnmzaton of makespan for a specfed demand, or (c) the mnmzaton of cost for specfed orders wth due dates. 4. Proposed Hybrd Algorthm As mentoned above, several STN-based MILP models have been proposed for the schedulng of batch plants. Dscrete-tme models, whle computatonally more effectve than the contnuous-tme models, often requre approxmatons that may gve nfeasble or suboptmal solutons. Furthermore, the schedulng of medum complexty process networks (10-20 tasks, states) becomes ntractable when the number of ntervals s above 60. Contnuous-tme and event-based models, whle more general n terms of task duratons, become computatonally ntractable even more quckly. Specfcally, medum-complexty STN networks, problems wth more than 15 ntervals are ntractable. Another shortcomng of contnuous-tme STN models s that they perform reasonably well only for the maxmzaton of proft over a fxed tme horzon. Ther computatonal performance for other objectves s often poor (Maravelas and Grossmann, 2003a). The dffculty n solvng STN schedulng problems led us to develop a hybrd MILP/CP method that explots the complementary strengths of MILP and CP. We use MILP to optmze the hgh level decsons, and CP to determne a feasble detaled schedule. Specfcally, we propose an teratve scheme where we terate between a MILP master problem and a CP subproblem, n a smlar fashon as n Jan and Grossmann (2001). However, n ths work the type and number of tasks to be performed and the assgnment of tasks to equpment unts are determned n the master MILP problem, whle the CP subproblem s used to derve a feasble schedule for the assgnment obtaned by the master problem. At each teraton, one or more specalzed nteger cuts are added to the master problem to exclude nfeasble or prevously obtaned assgnments. For a maxmzaton problem, the relaxed master problem provdes an upper bound and the 5

6 subproblem, when feasble, provdes a lower bound. To enhance the performance of the algorthm, preprocessng s performed before the man teratve scheme. Preprocessng s used to determne Earlest Start Tmes (EST) and Latest Fnsh Tmes (LFT) of both tasks and unts, and to create strong nteger cuts that are added a pror n the cut-pool of the master problem. Moreover, the proposed method can be used to obtan more than one feasble solutons; we just need to, (a) store all feasble solutons found durng the executon, and (b) keep teratng after the convergence of the bounds. For the mplementaton of the method we used ILOG s OPL Studo 3.5. (Appendx A). A smplfed flow dagram of the proposed algorthm for the maxmzaton of proft s shown n Fgure 3. The flow dagrams for the mnmzaton of makespan and cost are smlar MILP Master Problem For the master MILP problem an aggregated STN representaton has been used wth no tme ponts, or equvalent contnuous tme ntervals. Only assgnment, batch sze and mass balance constrants are ncluded, and snce there are no tme ponts, mass balance constrants are expressed once (for the total amounts) at the end of the tme horzon for each state. Specal nteger cuts are added to exclude prevously found sets of tasks. Resource constrants other than equpment are not consdered. Preprocessng I Calculate EST, LFT, j Preprocessng II Decompose nto subnetworks Derve Integer Cuts II and III Solve MILP Master Problem max proft s.t. Assgnment constrants Total mass balances Integer Cuts Obtan UB Fx tasks/assgnments Add Integer Cuts I Solve CP Subproblem max proft s.t. ALL CONSTRAINTS w/ fxed tasks/assgnments Obtan LB NO UB LB? YES Fgure 3: Schematc dagram of the proposed hybrd MILP/CP algorthm. In order to decouple unts from tasks, we use the followng rule: f a task can be performed n both unts j and j, then two tasks (performed n unt j) and (performed n unt j ) are defned (see Ierapetrtou and Floudas, 1998). As explaned above, the number of tmes each task s carred out s to be determned by the optmzaton, and thus for each task we postulate a maxmum number of copes,.e. an upper bound on the number of batches of task that can be carred out n any feasble soluton. A strct upper bound C MAX on the 6

7 number of copes of task s gven by equaton (1) where H s an upper bound on the length of the tme horzon and D MIN s the mnmum duraton of task. In practce, though, we can use our knowledge about the process network to postulate a smaller number of copes. C MIN H D MAX = C = / (1) For each copy c of task we defne the bnary Z c, whch s equal to 1 f the c th copy of task s carred out. We also defne ts duraton D c and batch sze B c. For each state, we defne ts nventory level Ŝ s at the end of the schedulng horzon. The master MILP problem s shown n Appendx C to be a relaxaton of the MILP model of Maravelas and Grossmann (2003a) gven n Appendx B. Hence, the master MILP problem (MP) provdes an upper bound to the proft and a lower bound to the cost or makespan. It conssts of equatons (2) to (10): I ( j) c D c MS j (2) D c = α Z + β B, c C (3) c c B Sˆ MIN s Z c B B Z, c C (4) c MAX c O I = S0s + ρ Bc Bc s s ρ (5) s O( s) c I ( s) c S ˆ d s FP (6) s s S ˆ C s INT (7) s s Z < c+ 1 Z c, c C, c C (8) Integer Cuts Objectve Functon (10) Z c { 0,1}, D 0, B 0, Sˆ 0 c c s Constrant (2) s a relaxed assgnment constrant whch enforces that the sum of the duratons of the tasks assgned to a unt does not exceed the schedulng horzon MS, where I(j) s the set of tasks that can be assgned to unt j. The duraton of copy c of task s a functon of ts batch sze [constrant (3)], and the batch sze of copy c of task s bounded through constrant (4). The amount of state s at the end of the tme horzon Ŝ s s calculated by (5) to be equal to the ntal amount S0 s plus the amount produced, mnus the amount consumed, where ρ I s and ρ O s are the mass fractons for consumpton and producton, respectvely, of state s by task. Note that constrant (5) ensures that the net producton of state s n non-negatve, but snce we do not montor and restrct the nventory level of state s durng the entre schedulng horzon, a soluton of the master problem may mply that the level of state s s at some pont negatve. If state s corresponds to a fnal product (s FP), Ŝ s must be greater than the demand d s [constrant (6)]; f t corresponds to an ntermedate (s INT) t must be less than the capacty C s of the storage tank of state s [constrant (7)]. Constrant (8) s used to elmnate symmetrc assgnments by enforcng the condton that copy c+1 of task can be carred out only f copy c s carred out. At a specfc teraton k, constrants (9) nclude all the nteger cuts that have been added durng preprocessng and n prevous teratons. The objectve functon, whch as noted above provdes a bound, can be the maxmzaton of proft for a fxed tme horzon, the mnmzaton of makespan for fxed demand, or the mnmzaton of producton cost for fxed demand and due dates. The exact form of the objectve functon s gven n secton 4.5. (9) 7

8 4.2. CP Subproblem In ths paper we model CP subproblems usng the modelng language of ILOG s OPL Studo 3.5, whch has a number of global constrants and specal constructs specfcally developed for schedulng applcatons, for whch a short descrpton can be found n Appendx A. The descrpton of the CP subproblem, hence, s made n terms of these constructs and constrants. For each equpment unt j we defne a unary resource called Unt[j] and for each resource r (e.g. coolng water) we defne a dscrete resource Utlty[r] wth a maxmum capacty R r MAX. Furthermore, for each state s we defne a reservor called State[s] wth capacty C s and ntal level S0 s. For each bnary Z c that s equal to 1 n the current optmal soluton of the master problem (.e. copy c of task s carred out) we defne an actvty called Task[,c] wth duraton D c. We also defne a dummy actvty MS wth zero duraton and no resource requrements. The reason we ntroduce MS, s because Constrant Programmng s more effcent when the objectve functon s a functon of one or few varables. Moreover, f a dummy actvty s not used, the objectve functon for the mnmzaton of makespan wll be the mnmzaton of the maxmum fnsh tme (.e. a mn max problem), whch wll requre the ntroducton of addtonal constrants. If there are orders wth due dates, and D s the set of orders for fnal products, D(s) s the set of orders for state s (.e. D= s D(s)), and for each d D, AD d s the amount due and TD d s the due date, we also defne D actvtes, called Order[d], wth zero duraton that are used for the representaton of the orders. The correspondng declaratons n OPL modelng language are: UnaryResource Unt[j n Unts]; (11) DscreteResource Utlty[r n Utltes] (R MAX r ); (12) Reservor State[s n States] (C s, S0 s ); (13) Actvty Task[ n Tasks, c n Copes] (D c ); (14) Actvty MS (0); (15) Actvty Order[d n Orders] (0); (16) The CP subproblem (SP) conssts of equatons (17) to (32): B D D MIN c c B B, c C (17) c MAX c = α + β B ZW, c C (18a) c α + β B ZW, c C (18b) B B R I cs O cs cr I = ρ B, c C, s (19) s c O = ρ B, c C, s (20) s r c r c = γ + δ B, c C (21) O B ds s FP (22) cs c Task[,c] requres Unt[j] j, I(j), c C (23) Task[,c] requres R cr Utlty[r], c C, r (24) 8

9 Task[,c] consumes B I cs State[s], c C, s (25) Task[,c] produces B O cs State[s], c C, s (26) Order[d].start = TD d d (27) Order[d] consumes AD d State[s] s, d D(s) (28) Task[,c].end MS.start, c C (29) Task[,c] precedes Task[,c+1], c< C (30) Process Network Specfc Constrants (31) (Optonal) Objectve Functon (32) The batch sze of actvtes s bounded by equaton (17). If Zero Wat storage polcy apples, the product of a task must be mmedately transferred to the next task, and thus the duraton of the task s exactly equal to ts processng tme as n (18a); for any other storage polcy the materal can be temporarly stored n the equpment unt, whch means that the tme durng whch equpment unt s used by Task[,c] can be greater than the actual processng tme [constrant (18b)]. The amount of reservor State[s] consumed/produced by actvty Task[,c] s calculated by equaton (19)/(20); the amount of dscrete resource Utlty[r] requred by actvty Task[,c] s calculated n (21); the condton that the amount of fnal products should meet the demand s enforced by (22), where d s s the total demand for state s. Parameter d s s ether gven (n the case of fxed demand wth no due dates) or calculated by, d s = AD d D( s) d s Specal CP constructs and global constrants are used n equatons (23) to (31). Constrant (23) enforces that all tasks n I(j) are assgned to unary resource Unt[j]. The consumpton of R cr unts of dscrete resource Utlty[r] by actvty Task[,c] s enforced n (24), and the consumpton/producton of B I cs /B O cs unts of reservor State[s] by Task[,c] s enforced by constrant (25)/(26). Orders wth due dates are modeled through constrants (27) and (28). Each order s executed at ts due tme [constrant (27)], and the amount delvered s equal to the amount due [constrant (28)]. In (29) the end tme of all actvtes s restrcted to be smaller than the start of actvty MS, and MS has, (a) fxed fnsh tme equal to H when the objectve s the maxmzaton of proft over a fxed tme horzon, or the mnmzaton of cost for fxed demand and due dates, and (b) a varable fnsh tme when the objectve s the mnmzaton of makespan for fxed demand wth no due dates. Constrant (30) s a symmetry-breakng constrant that reduces the number of possble confguratons by mposng a sequence between copes of the same task. Constrants that descrbe some specal features of the process network, or a specal structure that can be exploted are ncluded n (31). An example of such a feature s a Zero Wat state that s produced by only one task A and consumed by only one task B. In such a case, we can nfer that every tme task A takes place t must be mmedately followed by task B, and that the batchszes of consecutve batches of tasks A and B should be equal. Therefore, we can add the followng constrants that greatly enhance the computatonal effcency of the CP solver: Task[A,c].end = Task[B,c].start c (31a) BS Ac =BS Bc c (31b) Dependng on the nature of the problem (constant vs. varable processng tmes) and the objectve functon, we may want to solve the CP subproblem as one feasblty problem, as successve feasblty problems or as an optmzaton problem (detals n secton ). If the CP s an optmzaton problem we add the objectve (32) whose exact form s gven n secton

10 4.3. Tme Doman Reducton The performance of the proposed model depends on how fast we solve models (MP) and (SP), and the number of teratons needed to generate solutons and prove optmalty. It s crucal, consequently, to exclude nfeasble or suboptmal assgnments of tasks as soon as possble. Preprocessng enhances the performance of the algorthm by (a) reducng the domans of certan varables, and (b) creatng strong cuts that are added n the cut-pool of the master problem and are used to elmnate a pror a number of potental confguratons Earlest Start Tme and Latest Fnsh Tme of Unts To llustrate the doman reducton for unts, consder the example of Fgure 4, where unts U1, U2 and U3 are used for the producton of products A, B and C, for whch we assume that the duraton of all tasks s 2 hours, the tme horzon s 12 hours, and that at most two copes of the same task can take place (.e. C =2 ). Specfcally, for unt U2, replacng duratons from (3) nto (2) we get: 2Z TA2,1 +2Z TA2,2 +2Z TB2,1 +2Z TB2,2 +2Z TC2,1 +2Z TC2,2 12 whch mples that all 26 possble assgnments are feasble snce the constrant s satsfed f all Z c are equal to 1. In the case where there are no ntermedates SA1, SB1 and SC1 at t=0, however, no task can be carred out n unt U2 before t=2,.e. before one of the states used as nput n unt U2 s produced. Hence, the Earlest Start Tme (EST) for any task n unt U2 s 2 hours. Smlarly, snce the tme horzon s 12 hours, any amount of ntermedates SA2, SB2 and SC2 produced after t=10 wll not be used for the producton of fnal products, whch means that n all realstc solutons all tasks assgned to U2 must fnsh at or before t=10;.e. the Latest Fnsh Tme (LFT) of unt U2 s 10. When the tme horzon s not fxed nstead of usng LFT we use the Shortest Tal (ST) whch s the dfference between the varable schedulng horzon and the LFT. In ths example, the Shortest Tal (ST) of unt U2 s 2 hours. FA TA1 SA1 TA2 SA2 TA3 A FB TB1 SB1 TB2 SB2 TB3 B FC TC1 SC0 TC2 SC2 TC3 C U1 U2 U3 Fgure 4: Flow dagram of Example. Usng ths nsght, we can rewrte the assgnment constrant for unt U2 as follows: 2Z TA2,1 +2Z TA2,2 +2Z TB2,1 +2Z TB2,2 +2Z TC2,1 +2Z TC2,2 LFT U2 - EST U2 = (H-ST U2 ) EST U2 = 8 whch allows only up to four tasks assgned to unt U2,.e. only 22 out of the total 26 assgnments. Note that smlar tghtenng can be also performed when processng tmes are varables; n the example of Fgure 4, we need only use the mnmum, nstead of the fxed, processng tmes of tasks TA1, TB1 and TC1 to calculate the EST of unt U2, and the mnmum processng tme of tasks TA3, TB3 and TC3 to calculate the ST of unt U3. In the general case assgnment constrant (2) s tghtened as follows, I ( j) c D c ( MS ST ) EST j (33) j j 10

11 where ST j and EST j are the Shortest Tal and Earlest Start Tme of unt j, respectvely, and MS s the tme horzon (equal to H n the case of fxed tme horzon). To calculate the EST and ST of unt j we need to calculate frst the EST and ST of the tasks performed n j, whch s descrbed n the next secton Earlest Start Tme and Latest Fnsh Tme of Tasks To llustrate the doman reducton for tasks, consder the example of Fgure 5. The duraton of all tasks s 2 hours, the tme horzon s 16 hours, each task can be carred out at most three tmes and unt U1 s used for tasks T1 and T8. If no ntermedates are avalable at t=0, the EST and LTF for task T1 are 0 and 4 hours, respectvely, and for task T8 EST s equal to 6 and LFT s equal to 10. Ths means that we can wrte the followng constrants, 2Z T1,1 +2Z T1,2 +2Z T1,3 4 0 = 4 (34) 2Z T8,1 +2Z T8, +2Z T8, = 4 (35) Furthermore, the EST and LTF for unt U1 are 0=mn{EST T1,EST T8 } and 10=max{LFT T1,LFT T8 }, respectvely, and the assgnment constrant for unt U1 s, (2Z T1,1 + 2Z T1,2 +2Z T1,3 ) + (2Z T8,1 +2Z T8,2 +2Z T8,3 ) 10 0 = 10 (36) Note that the LHS of constrant (36) s the sum of the LHSs of constrants (34) and (35), whereas the RHS of constrant (36) s larger than the sum of the sum of the RHSs of constrants (34) and (35), whch mples that constrant (36) s a relaxaton of constrants (34) and (35). In general, thus, we can get a tghter formulaton by addng the constrants that restrct the sum of the processng tmes of the copes of the same task. Thus, for the general case, we add the followng constrant for each task n I(j), Dc ( MS ST ) EST I ( j) (37) c where ST and EST are the Shortest Tal and the Earlest Start Tme of task, respectvely. We can therefore replace equaton (2) by equatons (33) and (37), and hence the master problem (MP) conssts of equatons (3) (10), (33) and (37). F1 S1 S2 INT1 S3 P1 T1 T2 T3 T4 T5 S4 ADD F2 S5 INT S6 T6 T7 T8 T Unt U1 Fgure 5: Process network of Example 2. Note that for a general batch plant the calculatons for EST and ST (or LFT) are not equvalent to a tme wndow calculatons for flow-shop batch plants. The man reasons for ths are: (a) batch splttng and mxng s allowed, and (b) nventory for ntermedate states may be avalable at t=0. In prncple, EST and ST of tasks can be determned by nspecton, but here we use a general, graph-theoretc algorthm that reles on dentfyng shortest and longest paths wthn the STN. We defne parameter AT s as the tme at whch state s becomes avalable. States that are avalable at the begnnng of the tme horzon have AT s =0. In order to calculate the EST of a task we need to know the AT s of all the states that are consumed by task. In order to calculate the AT of a state we need to know the EST and the (mnmum) processng tme of all tasks producng ths state. If SI() s the set of states consumed by task 11

12 , O(s) s the set of tasks that produce state s, DT s s the earlest delvery tme of state s (zero f state s s avalable at t=0), and D MIN s the mnmum duraton of task, the procedure, n summary, s as follows: The EST of task s calculated by: EST = max s SI() {AT s } The AT of state s s calculated by: AT s = mn{dt s, mn O(s) {EST + D MIN }} For the calculaton of the Shortest Tal ST of task we follow smlar rules but we start from the fnal products (s FP), and for each state s S we calculate the mnmum tme MT s needed for state s to be used for the producton of a fnal product. Is SO(s) s the set of states produced by task, and I(s) s the set of tasks consumng state s, the summary of the backward procedure for the calculaton of the ST of tasks (f there are no recycle streams) s the followng: The ST of task s calculated by: ST = mn s SO() {MT s } The MT of state s s calculated by: MT s = mn I(s) {ST + D MIN } If ET (UT) s the set of examned (unexamned) tasks, and ES (US) s the set of examned (unexamned) states, and D MIN s the mnmum (or fxed) processng tme of task, a straghtforward computer mplementaton of the procedure for networks wth varable or constant processng tmes and no recycle streams s the followng: Calculaton of EST: Intalzaton: ET =, UT = I, ES = {s S S0 s > 0}, US = S\ES, AT s =0 s ES Untl UT = do For all UT If ES SI() = SI() then EST() = max s SI() {AT s } ET = ET {} UT = UT\{} For all s US If ET O(s) = O(s) then AT(s) = mn{dt s, mn O(s) {EST + D MIN }} ES = ES {s} US = US\{s} Calculaton of ST: Intalzaton: ET =, UT = I, ES = FP, US = S\FP, MT s =0 s ES Untl UT = do For all UT If ES SO()=SO() then ST() = mn s SO() {MT s } ET = ET {} UT = UT\{} For all s US If ET I(s) = I(s) then MT(s) = mn I(s) {ST + D MIN } ES = ES {s} US = US\{s} An example of ts applcaton s presented n Appendx D. In case there are recycle streams and RC s the set of recycled states, the calculaton s more complcated. We ntally exclude states n RC (.e. US=S\(FP RC), calculate EST and ST for all states and then re-calculate takng nto account the recycled states. Havng calculated the EST and ST of all tasks, the EST and ST of unts are easly calculated by, EST j = mn{est I(j)} (38) ST j = mn{st I(j)} (39) 12

13 4.4. Integer Cuts The preprocessng descrbed n the prevous secton reduces the number of potental confguratons by tghtenng the exstng and addng new assgnment constrants n the MILP master problem (MP). Another way to reduce the number of potental assgnments s by generatng nteger cuts that explctly forbd some nfeasble or suboptmal confguratons that cannot be excluded by the tghtenng of domans. Whle one smple nteger cut s added at each teraton of the proposed algorthm, some nteger cuts can be nferred durng preprocessng. The smple nteger cuts and two more classes of cuts derved durng preprocessng are dscussed next Integer Cuts I The frst class of nteger cuts comprses of the smplest and weakest cuts that exclude the current assgnment, and ts general form s: k k k Z c + (1 Z c ) B + N 1 or Z c Z c B 1 (40) k (, c) B (, c) N k k (, c) B (, c) N where B k and N k are the sets of (,c) pars for whch Z c = 1 and Z c =0, respectvely, n the current teraton k. The nteger cut n (40) excludes only the current assgnment. A stronger cut s one that excludes the current assgnment k and any other assgnment that s a superset of assgnment k;.e. any assgnment l for whch B l B k. The general form of ths cut s: (, c) B Z c B k k 1 Whle stronger, the nteger cuts n (41) may cutoff feasble solutons. If economy of scale holds for batchszes and processng tmes, then they can be safely used when the tme horzon s fxed and the current assgnment B k s nfeasble, snce any assgnment B l wth B l B k wll also be nfeasble. But they cannot be used f the current assgnment s feasble, because an assgnment B l wth B l B k may yeld a hgher proft. A case where they can be added n every teraton s when the objectve s the mnmzaton of makespan and processng tmes are constant; n ths case, f an assgnment B k satsfes the demand and yelds a makespan MS k, then any assgnment l that ncludes more tasks would yeld a makespan MK l MK k Integer Cuts II The motvaton behnd these cuts s to decompose the STN nto subnetworks n order to dentfy nfeasble assgnments of tasks as early as possble. To llustrate the dervaton of ths class of nteger cuts consder the process network shown n Fgure 6. There, raw materal RM1 s converted nto ntermedate INT1 (va tasks T11, T12, T13 or T14 that are performed n unt U1), raw materal RM2 s converted nto ntermedate INT2 (va tasks T21 or T22 that are performed n unt U2), ntermedates INT1 and INT2 are fed n a 4:1 rato for task T31 n unt U1 to produce fnal product P1, and ntermedate INT2 s converted nto fnal product (through task T32 also n unt U3). Each task has a constant cost c and a constant processng tme pt shown n Fgure 6. The capacty of unts U1, U2 and U3 s 5, 10 and 10 kg respectvely. There s one order of 5 kg of P1 to be met at t=6 and one order of 5 kg of to be met at t=4. The objectve s to fnd a schedule of mnmum cost that satsfes the demand and the due dates. The Earlest Start Tme (EST) for all tasks performed n unts U1 and U2 s 0, whle for tasks T31 and T32 t s 2. Snce the latest due date s at t=6, the schedulng horzon s 6 hours the Latest Fnsh Tme (LFT) of all tasks performed n unts U1 and U2 s 4 and for T31and T32 n unt U3 s 6 (or ST U1 =ST U2 =2 and ST U3 =0). Usng these values, the proposed teratve scheme yelds the seres of assgnments shown n Fgure 7 (tasks that are carred out are hghlghted). The frst three assgnments nclude one of the tasks performed n U1, and tasks T22, T31 and T32 and they are all nfeasble. The reason for the nfeasblty s that the duraton of task T22 s 3 hours, whch means that ntermedate INT2 becomes avalable at t=3, and thus the order of k (41) 13

14 cannot be met on tme. We can easly nfer that n any feasble soluton, task T21 must be carred out because ths s the only way ntermedate INT2 becomes avalable at t=2. Hence, f we had ths nsght before startng the teratve scheme and we had fxed bnary Z T21,1 to 1, we would get the 4 th assgnment at the frst teraton. Note that smlar reasonng can also be appled when processng tmes are varables, usng the mnmum nstead of the fxed processng tme of tasks. RM1 T11 T12 T13 T14 c=2.0 pt=4 c=2.2 pt=4 c=2.4 pt=4 c=4 pt=2 RM2 INT1 T21 T22 c=3 pt=2 c=2 pt=3 80% 20% INT2 T31 c=4 pt=2 T32 c=4 pt=2 P1 U1 U2 U3 Fgure 6: Flow dagram for nteger cuts. a) RM1 T11 T12 T13 T14 INT1 RM2 T21 T22 80% 20% INT2 T31 T32 P1 Iteraton 1: Cost = 12.0 INFEASIBLE ASSIGNMENT Cut: Z T11,1 +Z T22,1 +Z T31,1 +Z T32,1 3 b) RM1 U1 U2 U3 INT1 T11 80% T31 T12 20% T13 RM2 T21 T32 INT2 T14 T22 P1 Iteraton 2: Cost = 12.2 INFEASIBLE ASSIGNMENT Cut: Z T12,1 +Z T22,1 +Z T31,1 +Z T32,1 3 U1 U2 U3 c) RM1 T11 T12 T13 T14 INT1 RM2 T21 T22 80% 20% INT2 T31 T32 P1 Iteraton 3: Cost = 12.4 INFEASIBLE ASSIGNMENT Cut: Z T13,1 +Z T22,1 +Z T31,1 +Z T32,1 3 d) RM1 U1 U2 U3 INT1 T11 80% T31 T12 20% T13 RM2 T21 T32 INT2 T14 T22 P1 Iteraton 4: Cost = 13.0 FEASIBLE ASSIGNMENT U1 U2 U3 Fgure 7: Assgnments of Master Problem Based on ths smple example we can n general decompose the process network nto smaller subnetworks and try to nfer general rules (fxed assgnments or nteger cuts) that hold true n any soluton. By subnetworks we mean a subset of tasks that, (a) can be easly dentfed (e.g. wth a procedure smlar to the one descrbed n the next paragraph), and (b) are lkely to gve useful nformaton (.e. nformaton that holds true for larger sets of tasks and deally for the entre process network). Such a subnetwork can be, for example, the set of tasks used for the producton of one ndvdual product. In the example of Fgure 6 the process network can be decomposed nto the two subnetworks shown n Fgure 8. Whle no nference can be made by the subnetwork for the producton of P1, the second subnetwork reveals that n order to meet the due date for product we need to carry out task T21 (.e. Z T21,1 =1 n every feasble soluton). Havng fxed Z T21,1, we then obtan the optmal assgnment (Fgure 7d) at the frst teraton. Snce (a) the capacty of unt U2 s 10 kg, (b) the processng tme of task T21 does not depend on the batch sze and (c) for both orders we 14

15 need 6 kg of INT2, we could have further nferred that no other task s needed to be carred out n unt U2 (.e. Z T21,c = Z T22,c = 0, c). RM1 T11 T12 T13 T14 c=2.0 pt=4 c=2.2 pt=4 c=2.4 pt=4 c=4 pt=2 RM2 INT1 T21 T22 c=3 pt=2 c=2 pt=3 80% 20% INT2 T31 c=4 pt=2 P1 U1 U2 U3 (a) Subnetwork for the producton of P1 RM2 T21 T22 c=3 pt=2 c=2 pt=3 INT2 T32 c=4 pt=2 U2 U3 (b) Subnetwork for the producton of Fgure 8: Subnetworks of process network of Fgure 6. In general, any network can be decomposed n FP subnetworks, and each subnetwork SN conssts of a subset of tasks SNI I and a subset of states SNS S. To dentfy such a subnetwork, we start from a fnal product s* FP and, backtrackng, we add to SNI and SNS all the tasks and states that are nvolved n the producton of s*. If O(s) s the set of tasks that produce state s, SI() s the set of nput states to task, and RM s the set of raw materal states, the formal procedure for the dentfcaton of the subnetwork that corresponds to fnal product s* s the followng: SUBNET_PROC: Intalzaton: SNI =, SNS = {s*}, CI = O(s*) Untl CI = For all CI SNI = SNI {}, CI = CI\{} For all s SI() SNS = SNS {s}, CI = CI O(s) For each subnetwork dentfed by SUBNET_PROC, we apply the proposed teratve algorthm and derve cuts that have the general form of constrant (40) and exclude confguratons that are nfeasble for the specfc subnetwork;.e. Integer Cuts II have the form of (40), where SNI B k. Moreover, snce Integer Cuts II have the form of equaton (40), they exclude only one assgnment and, thus, can be used for both constant and varable processng tmes. Network-specfc constrants that nclude only the Z c bnares, smlar to constrants n (31) of the CP subproblem, can be added also n the master problem and grouped as Integer Cuts II. For the case of Zero Wat state that s produced by only one task A and consumed by only one task B, for nstance, we can add the followng equaton that allows assgnments that consst of the same number of copes of tasks A and B: Ac = c Z Z (42) c Bc Note, that the restrcton for zero-wat storage polcy for one or more states makes, n general, contnuous MILP models very dffcult to solve due to the addtonal tme ponts needed for the fnsh tme of the tasks that produce ZW states. In contrast, ths restrcton makes the CP subproblem of the proposed approach easer to solve because t becomes more tghtly constraned. 15

16 Dependng on the plant topology, other sub-networks can also be studed. The equatons added n ths step (fxng of varables or nteger cuts) are Integer Cuts II. Any other cut that can be extracted from the structure of the process network (based on heurstc rules, pror knowledge of the process network or any other source) can be grouped n ths class Integer Cuts III In batch plants t s very common to have a number of dentcal parallel unts n one stage, or smlar tasks that are carred out n the same unt. The smlarty of these tasks s that they have the same processng tme but slghtly dfferent utlty requrements or cost. Tasks T11, T12 and T13 of the network n Fgure 6, for example, are smlar because they have equal processng tmes but dfferent costs. Ths smlarty usually mples that f one of these tasks leads to an nfeasble assgnment, every other smlar task wll also gve an nfeasble assgnment. The hgh frequency at whch these confguratons appear n many problems led us to develop a new class of nteger cuts. The am s to exclude smlar assgnments n as few teratons as possble. To motvate ths class of cuts, consder the prevous process network (Fgure 6), wth two orders of 5 kg each for P1 due at t=4 and t=10, and one order of 5 kg of due at t=8. The doman reducton for tasks and unts yelds the results of Table 1. Table 1: Earlest start and latest fnsh tmes of Example. T11 T12 T13 T14 T21 T22 T31 T32 U1 U2 U3 EST LFT Assumng that the tme horzon of the two subnetworks used for the producton of P1 and s 10 and 8 hours, respectvely, the study of the two subnetworks does not reveal any addtonal nformaton (.e. we cannot derve any nteger cut of type II). Thus, the teratve scheme yelds the assgnments that are shown n Fgure 9, and the optmal soluton s found n the 8 th teraton. Note that assgnments 1, 2 and 3 are practcally the same because the processng tmes of tasks T11, T12 and T13 are the same. Smlarly, assgnments 4, 5 and 6 are dentcal to each other n terms of processng tmes. To exclude these two groups of smlar assgnments, however, we need the frst sx nteger cuts shown n Fgure 9. It would be very useful, hence, to develop a class of nteger cuts that excludes a whole group of smlar assgnments. To do so, we need to dentfy the classes of smlar equpment unts and defne new bnary varables. The general procedure for dervng the thrd class of nteger cuts s the followng: (a) Identfy classes of smlar tasks;.e. tasks that perform dentcal operatons (same nput and output states and same converson factors) and wth equal processng tmes. The set of smlar tasks that belong to class g s denoted by I(g). (b) Defne new bnary varables Y gcn for each class g of smlar tasks; bnary Y gcn s 1 f, n the current assgnment, there are n of the c th copes of tasks n I(g). The value of the Y gcn bnares s determned by the followng equatons that lnk the class bnares Y gcn to the task bnares Z c, Z c = n Ygcn g, c (43) I ( g) Y gcn n= 1.. I ( g ) n= 1.. I ( g ) = 1 g, c In the example of Fgure 6, for nstance, f A s the class of tasks {T11, T12, T13} and n the current assgnment there are two copes of task T11 (Z T11,c =1, for c=1, 2), and one copy of task T12 (Z T12c =1, for c=1), we wll have Y A12 =1, Y A21 =1. (44) 16

17 (c) Develop nteger cuts n the mxed (Y gcn, Z c ) space: for any assgnment that ncludes a task I(g) use equatons (43) and (44) to calculate bnares Y gcn and develop a new cut (cut III) by replacng bnares Z c wth bnary Y gnc n the correspondng nteger cut I. Any nteger cut that ncludes Y gcn bnares s called an Integer Cut III. RM1 T11 T12 T13 T14 INT1 RM2 T21 T22 80% 20% INT2 T31 T32 P1 Iteraton 1: Cost = 12.0 INFEASIBLE ASSIGNMENT Cut: Z T11,1 +Z T22,1 +Z T31,1 +Z T32,1 3 U1 U2 U3 RM1 T11 T12 T13 T14 INT1 RM2 T21 T22 80% 20% INT2 T31 T32 P1 Iteraton 2: Cost = 12.2 INFEASIBLE ASSIGNMENT Cut: Z T12,1 +Z T22,1 +Z T31,1 +Z T32,1 3 RM1 U1 U2 U3 T11 INT1 80% T31 T12 20% T13 RM2 T21 T32 INT2 T14 T22 P1 Iteraton 3: Cost = 12.4 INFEASIBLE ASSIGNMENT Cut: Z T13,1 +Z T22,1 +Z T31,1 +Z T32,1 3 U1 U2 U3 RM1 T11 T12 T13 T14 INT1 RM2 T21 T22 80% 20% INT2 T31 T32 P1 Iteraton 4: Cost = 13.0 INFEASIBLE ASSIGNMENT Cut: Z T11,1 +Z T21,1 +Z T31,1 +Z T32,1 3 U1 U2 U3 RM1 T11 T12 T13 T14 INT1 RM2 T21 T22 80% 20% INT2 T31 T32 P1 Iteraton 5: Cost = 13.2 INFEASIBLE ASSIGNMENT Cut: Z T12,1 +Z T21,1 +Z T31,1 +Z T32,1 3 U1 U2 U3 RM1 T11 T12 T13 T14 INT1 RM2 T21 T22 80% 20% INT2 T31 T32 P1 Iteraton 6: Cost = 13.4 INFEASIBLE ASSIGNMENT Cut: Z T13,1 +Z T21,1 +Z T31,1 +Z T32,1 3 U1 U2 U3 RM1 T11 T12 T13 T14 INT1 RM2 T21 T22 80% 20% INT2 T31 T32 P1 Iteraton 7: Cost = 14.0 INFEASIBLE ASSIGNMENT Cut: Z T14,1 +Z T22,1 +Z T31,1 +Z T32,1 3 U1 U2 U3 RM1 T11 T12 T13 T14 INT1 RM2 T21 T22 80% 20% INT2 T31 T32 P1 Iteraton 8: Cost = 15.0 FEASIBLE ASSIGNMENT U1 U2 U3 Fgure 9: Assgnments derved by the master problem. 17

18 In the example of Fgure 6, tasks T11, T12 and T13 make up a class of smlar tasks. Let A be ths class. The seven frst assgnments of Fgure 9 and the correspondng nteger cuts are gven n the second and thrd column, respectvely, of Table 2. Table 2: Assgnments of Fgure 9 and nteger cuts I and III. Iter. Tasks of assgnment Integer Cut I Equaton (43) Integer Cut III 1 st T11, T22, T31, T32 Z T11,1 +Z T22,1 +Z T31,1 +Z T32,1 3 Z T11,1 =1 Y A,1,1 =1 Y A,1,1 +Z T22,1 +Z T31,1 +Z T32,1 3 2 nd T12, T22, T31, T32 Z T12,1 +Z T22,1 +Z T31,1 +Z T32,1 3 Z T12,1 =1 Y A,1,1 =1 3 rd T13, T22, T31, T32 Z T13,1 +Z T22,1 +Z T31,1 +Z T32,1 3 Z T13,1 =1 Y A,1,1 =1 4 th T11, T21, T31, T32 Z T11,1 +Z T21,1 +Z T31,1 +Z T32,1 3 Z T11,1 =1 Y A,1,1 =1 Y A,1,1 +Z T21,1 +Z T31,1 +Z T32,1 3 5 th T12, T21, T31, T32 Z T12,1 +Z T21,1 +Z T31,1 +Z T32,1 3 Z T12,1 =1 Y A,1,1 =1 6 th T13, T21, T31, T32 Z T13,1 +Z T21,1 +Z T31,1 +Z T32,1 3 Z T13,1 =1 Y A,1,1 =1 7 th T14, T21, T31, T32 Z T14,1 +Z T21,1 +Z T31,1 +Z T32,1 3 Usng equatons (43) and (44) we can calculate the value of bnary Y Anc at each teraton (shown n the fourth column) and develop the new nteger cuts III by replacng Z c for tasks n A by bnary Y Acn (ffth column). Note that the sx frst assgnments correspond to only two dfferent assgnments n the (Z c, Y gcn ) space. The frst nteger cut III excludes the frst three assgnments and the second cut excludes the next three assgnments. Thus, we could have found the optmal assgnment n only four teratons f we had added the two nteger cuts of the ffth column. Integer Cuts III can be used for both constant and varable processng tmes Objectve Functons and Propertes of Upper and Lower Bounds Maxmzaton of Proft over a Fxed Tme Horzon Master problem objectve: Subproblem objectve: s FP max ζ sŝ s (10a) max ζ B (32a) s FP I ( s) c C where ζ s s the prce of fnal product s FP, MS n constrants (33) and (36), s equal to the fxed tme horzon H, and the dummy actvty MS n the CP subproblem s fxed to start at t=h. The maxmzaton of producton s a specal case of ths objectve wth ζ s =1 for all s. Snce the objectve s to maxmze proft, assgnments that are feasble wth regard to each unt separately are obtaned by the master problem, and the upper bound provded by the master problem s often very large. Even feasble assgnments yeld upper bounds that are usually much hgher than ther actual objectve value (calculated by the subproblem). The frst CP subproblems are usually nfeasble and the optmal soluton usually corresponds to the soluton of one of the frst feasble assgnments, but a sgnfcant number of addtonal teratons may be needed to prove optmalty. A typcal graph of the upper and lower bounds s gven n Fgure 10a, where some assgnments are nfeasble, the upper-lower bound gap s large, and many addtonal teratons are needed to prove optmalty Mnmzaton of Makespan for Fxed Demand (wth no Due Dates) Master problem objectve: Subproblem objectve: s O cs mn MS (10b) mn MS. end (32b) where MS n constrants (33) and (34) s a varable, and MS.end s the fnshng tme of the dummy task MS n the CP subproblem. 18