Cyclic short-term scheduling of multiproduct batch plants using continuous-time representation

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1 Computers and Chemical Engineering (00) Cyclic short-term scheduling of multiproduct batch plants using continuous-time representation D. Wu, M. Ierapetritou Department of Chemical and Biochemical Engineering, Rutgers University, 9 Brett Road, Piscataway, NJ 0, USA Received October 00; received in revised form October 00; accepted April 00 Available online May 00 Abstract The idea of cyclic scheduling is commonly utilized to address short-term scheduling problems for multiproduct batch plants under the assumption of relatively stable operations and product demands. It requires the determination of optimal cyclic schedule, thus greatly reducing the size of the overall scheduling problems with large time horizon. In this paper a new cyclic scheduling approach is proposed based on the state-task network (STN) representation of the plant [Comput. Chem. Eng. (99) ] and a continuous-time formulation [Ind. Eng. Chem. Res. (99a) ]. Assuming that product demands and prices are not fluctuating along the time horizon under consideration, the proposed formulation determines the optimal cycle length as well as the timing and sequencing of tasks within a cycle. This formulation corresponds to a non-convex mixed integer nonlinear programming (MINLP) problem, for which local and global optimization algorithms are used and the results are illustrated for various case studies. 00 Elsevier Ltd. All rights reserved. Keywords: Cyclic scheduling; Continuous-time formulation; MINLP. Introduction Scheduling problems for batch plant received great attention during the past two decades. Short-term scheduling problem involves the determination of the optimal timing and sequencing of tasks through a time horizon of few days up to few weeks. Thus scheduling optimally allocates the competitive resources of the batch plant such that the overall economic profit is maximized. Early publications presented mathematical models based on discrete time formulation which descretizes the entire time horizon into a number of intervals of equal duration and determines the production decisions at each of these intervals (Kondili, Pantelides, & Sargent, 99). The main limitations of time discretization method are that (a) they require all the tasks to start and finish at the boundaries of time intervals, thus resulting in sub-optimal solutions and (b) they require a large number Corresponding author. Tel.: addresses: danwu@so .rutgers.edu (D. Wu), marianth@so .rutgers.edu (M. Ierapetritou). Tel.: +--0; fax: +--. of binary variables due to unnecessary time discretization that results in large mathematical models difficult to solve. Towards developing efficient mathematical models to address the short-term scheduling problem, attention has been given to continuous-time representations. Zhang and Sargent (99) presented a mixed integer nonlinear programming (MINLP) formulation based on the resource-task network (RTN) representation (Pantelides, 99), which they linearize to MILP problem. Mockus and Reklaitis (99) proposed MINLP formulation using state-task network (STN) representation (Kondili et al., 99) employing a Bayesian heuristic solution approach. Ierapetritou and Floudas (99a, 99b) developed a continuous-time formulation for the short-term scheduling problem that requires fewer number of variables and constraints. Other recent works include slot-based pseudo-continuous-time formulation (Pinto & Grossmann, 99), continuous-time MILP model with a novel branch-and-bound algorithm (Schilling & Pantelides, 999) and hybrid generalized disjunctive MILP model (Maravelias & Grossmann, 00). Although model development and decomposition solution techniques greatly advanced the range of applications 009-/$ see front matter 00 Elsevier Ltd. All rights reserved. doi:0.0/j.compchemeng

2 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) Nomenclature Indices i tasks j units n event points representing the beginning of a task s states Sets I I j I s IS J J i N S Parameters price s r s ST max s U Vij max Vij min α ij β ij ρ p si,ρc si Variables B(i, j, n) d(s, n) H ST(s, n) STIN(s) T f (i,j,n) T s (i,j,n) wv(i, n) yv(j, n) tasks tasks which can be performed in unit j tasks which produce or consume state s subset of all involved intermediate states s units units which are suitable for performing task i event points within the time horizon set of all involved states s price of state s average market requirement for state s available maximum storage capacity for state s upper bound of cycle time length denotes the maximum capacity of unit j when processing task i denotes the minimum amount of material processed by task i required to start operating unit j constant term of processing time of task i at unit j variable term of processing time of task i at unit j expressing the time required by the unit to process one unit of material proportion of state s produced, consumed from task i, respectively amount of material undertaking task i in unit j at event point n amount of state s being delivered to the market at event point n time horizon for a single cycle amount of state s stored at event point n amount of state s imputed initially time when task i, which starts at event point n finishes in unit j time when task i starts in unit j at event point n binary variables that assign the beginning of task i at event point n binary variables that assign the utilization of unit j at event point n of scheduling approaches, the computational complexity of the resulting mathematical models is still prohibitive when realistic case studies and large time horizons are considered. The expansion of time horizon results in the incorporation of a large number of decision variables corresponding to the allocation of resources. A unified continuous-time model has been developed by Orcun, Altinel, and Hortacsu (00) to address the detailed scheduling over an expanding planning horizon. The corresponding MINLP problem is reformulated as MILP using linearization techniques, but the resulted model is still addressed as extremely difficult almost impossible to solve. Similar computational barrier is met when the continuous formulation proposed by Ierapetritou and Floudas is applied to longer time horizon (Wu & Ierapetritou, 00). Campaign mode of operation can be selected in batch plant operation when demands and operating conditions are relatively stable. In this case, plants focus on producing only a subset of products over a certain time period. This results in big saving in operations due to effective management of frequent changes and is easy to implement. In the same context, cyclic scheduling (periodic scheduling) is developed to make the operation decisions easier and profitable. It establishes an operation schedule and makes it executed repeatedly. Apart of the advantage of easy management of plant operations, mathematically the problem

3 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) is limited to a smaller time horizon and can be thus solved more efficiently. Shah, Pantelides, and Sargent (99) modified the formulation of Kondili et al. (99) formulation and extended it to the periodic scheduling of batch plants using a discrete time representation. Zhu and Majozi (00) proposed a decomposition strategy for planning problem that implies cyclic schedules without addressing the issue of obtaining the optimal cycle length. Schilling and Pantelides (999) presented a periodic scheduling formulation which is based on their earlier work on continuous-time representation for short-term scheduling problem. Due to the difficulty in linearizing the nonlinear function, a branch-and-bound algorithm that branches on both discrete and continuous variables was proposed. More recently, Castro, Barbosa-Povoa, and Matos (00) modified their short-term scheduling formulation to fit periodic scheduling requirement for an industrial application. The proposed model is based on the basic RTN representation and encounters the main limitation of the prohibitive model size. Research in this paper extends the work by Ierapetritou and Floudas (99a,b) based on the STN representation, and develops the cyclic scheduling formulation with the inherited advantage of using few binary variables. The cyclic scheduling problem is stated in Section as well as the continuous-time representation. A motivating example is reviewed in Section to illustrate the complexity of the scheduling problem. The proposed mathematical formulation is presented in detail in Section and applied to different examples in Section, where results are compared with existing approaches.. Basic concepts of the proposed approach.. Scheduling problem The scheduling problem considered in this paper is defined as follows. Given are: (i) the production recipe (i.e. the processing times for each task at the suitable units, and the amounts of the materials required for the production of each product); (ii) the available units and their capacity limits; (iii) the available storage capacity for each of the materials; (iv) the time horizon under consideration; (v) the market requirements of products. The objective is to determine the optimal operational plan to meet a specified economic criterion such as maximum profit or minimum cost while satisfying all the production requirements. It should be noted however, that the product demands are considered at the end of time horizon and all of the above constraints are fixed within the time horizon under consideration for the campaign operation... Periodic scheduling approach The idea of periodic scheduling is frequently utilized for the solution of the operation problem described in Section.. Although the optimal solution of the overall problem in general implies that the schedule should not be periodic (Pantelides, 99), one has to balance against the computational complexity of solving non-cyclic schedules for a long time horizon. The presented cyclic scheduling approach resides on the following assumption. For the case that the time horizon is long compared with the duration of individual tasks, a sub-schedule exists with a much smaller time horizon, the periodic execution of which achieves production very close to the optimal one without periodicity assumption. Thus the size of the problem is reduced to a much smaller one that can be efficiently solved since only a smaller time horizon is used. Besides its computation efficiency the proposed operation plan is easier to implement in campaign operation mode since it assumes repetition of the same schedule. In this approach, the variables include the cycle time length as well as the detailed schedule of this period, which are defined as unit period and unit schedule, respectively. Unit schedule requires certain amounts of intermediates at the beginning. These intermediates will be produced and stored at the end of the same period, so as to preserve the material balance across the boundaries and assure the continuity of operation as shown in Fig.. It should be noticed that in cyclic scheduling, each processing unit may have an individual cycle as long as the cycle time is equal to the duration of the unit period, thus all the units do not necessarily share the same starting and ending time points as illustrated in Fig. a. This concept can be Units Time Fig.. Cyclic schedule.

4 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) Units Units (a) Time (b) Time Fig.. Unit schedule. found in in Shah s discrete time representations for periodic scheduling problem as wrap-around (Shah et al., 99). Schilling and Pantelides (999) incorporated the same concept into their continuous-time formulation based on the RTN representation. In this paper, the same concept is used together with the continuous-time representation using the idea of event points as will be presented in detail in Section. Fig. a illustrates the unit schedule that corresponds to the cyclic schedule of Fig.. When a larger time period has to be scheduled using the unit schedule, overlapping is allowed in order to achieve better resource utilization. In this way the equivalent unit schedule is determined as shown in Fig. b. Note that by using this idea better schedules are determined since tasks are allowed to cross the unit schedule boundaries... Continuous-time approach The continuous-time representation proposed by Ierapetritou and Floudas (99a) is used in this work that avoids the shortcoming of prepostulating unnecessary time slots or intervals. In contrast to the previously presented continuous-time formulations, this approach decouples the unit events from the task events and results in significantly fewer binary and continuous variables that can be efficiently solved using available MILP solvers.. Motivating example In this motivating example (Kondili et al., 99), two different products are produced through five processing stages, heating, reaction, reaction and reaction, and separation of product from impure E as shown in the state task network (STN) representation of the plant flowsheet in Fig.. The data for this example are presented in Table. The processing times are allowed to vary within ±% around the mean values as shown in Table. When large time horizon is considered, the size of the model becomes intractable. For example considering a time horizon of h, the formulation of Ierapetritou and Floudas involves constraints, continuous variables and binary variables using event points. It takes 9.9 CPU seconds on PentiumIII 0 MHz using GAMS/CPLEX. to get a solution of 0.9 objective function value which cannot be proved optimum since further increase of event points causes computational infeasibility. When the same formulation is used for a time horizon of h, a Product Heating 0% IntAB 0% 0% Reaction Feed A Hot A 0% IntBC Impure E 0% Separation 0% Reaction 0% Reaction 90% Feed B 0% 0% Product Feed C Fig.. State task network representation for motivating example.

5 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) Table Data for motivating example Units Capacity Suitability Mean processing time ( τ ij ) Heater 00 Heating.0 Reactor 0 Reaction.0,.0,.0 Reactor 0 Reaction.0,.0,.0 Still 00 Separation.0 States Storage capacity Initial amount Price Feed A Unlimited Unlimited 0.0 Feed B Unlimited Unlimited 0.0 Feed C Unlimited Unlimited 0.0 Hot A IntAB IntBC Impure E Product Unlimited Product Unlimited feasible schedule cannot be obtained for the whole time horizon. These results point to the importance of developing a new approach for the fast solution of scheduling problem supporting campaign operation.. Proposed formulation In the following sub-section the proposed mathematical model is presented in detail... Mathematical model Allocation constraints i I j wv(i, n) = yv(j, n), j J, n N () V min ij Capacity constraints wv(i, n) B(i, j, n) Vij max wv(i, n), i I, j J i, n N () Storage constraints ST(s, n) ST max s, s S, n N () Material balances ST(s, n) = ST(s, n ) d(s, n) + ρ p si B(i, j, n ) i I s j J i + ρsi c B(i, j, n), i I s j J i s S, n N () where ρ c si 0,ρ p si 0 represent the proportion of state s consumed or produced from task i, respectively. Duration constraints T f (i,j,n)= T s (i,j,n)+ α ij wv(i, n) + β ij B(i, j, n), i I, j J i, n N () where α ij,β ij are the constant and variable terms of the processing time of task i at unit j. Sequence dependent changeovers could also be easily incorporated. In this case, the changeover time is considered by incorporating logic expressions such as those in the following sequence constraints (Ierapetritou & Floudas, 99b). Note that the sequence constraints still maintain their linearity since the extra terms involve products of binary variables and the changeover parameters. Sequence constraints: same task in the same unit T s (i,j,n+ ) T f (i,j,n) U( wv(i, n) yv(j, n)), i I, j J i, n N, n N () T s (i,j,n+ ) T s (i, j, n), i I, j J i, n N, n N () T f (i,j,n+ ) T f (i, j, n), i I, j J i, n N, n N () Sequence constraints: different tasks in the same unit T s (i,j,n+ ) T f (i,j,n) U( wv(i,n) yv(j, n)), j J, i I j, i I j, i i, n N, n N (9) Sequence constraints: different tasks in different units T s (i,j,n+ ) T f (i,j,n) U( wv(i,n) yv(j, n)), j, j J, i I j, i I j, i i, j j, n N, n N (0) Sequence constraints: completion of previous tasks T s (i,j,n+ ) n N,n n i I j,i i (T f (i,j,n ) T s (i,j,n )), i I, j J i, n N, n N () These constraints can be classified as mass-related constraints () () and time-related constraints () (). Constraints () state that only one task can be performed in the same unit at each event point n. Constraints () enforce the requirement for minimum amount Vij min of batch size B(i, j, n), in order for a unit j to start processing task i, and enforce the maximum capacity Vij max of a unit j when task i is performed at event point n (i.e. wv(i, n) equals one). All B(i, j, n) variables take the value of zero when wv(i, n) equals zero. Maximum storage capacity for each state s is

6 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) represented as upper bound to storage of state s at each event point n in constraints (). Material balances () state that the amount of material of state s at event point n is equal to that at event point n adjusted by any amounts produced or consumed between the event points n and n and the amount delivered to the market at event point n. The time-related constraints () () are very important since they enforce the optimal ordering and timing of all the tasks that satisfy the mass-related requirements. Constraints () assume a variation of / around the mean value of the processing time τ ij. Thus α ij takes the value of / τ ij and corresponds to the minimum processing time τij min and β ij = (τij max τ min ij τij min )/(Vij max Vij min ), where τij max = / τ ij and = / τ ij and expresses the time required by the unit j to process one unit of material while performing task i. Note that different variations of the processing time duration can be considered in the similar manner. The duration constraints express the dependence of the time duration of task i in unit j at event point n from the amount of material being processed. Note that when wv(i, n) equals zero, the last two terms become zero due to the capacity constraints () and hence T f (i,j,n) = T s (i,j,n). The sequence constraints enforce the recipe requirements of starting and finishing times of different tasks within a cycle. The sequence constraints that deal with tasks between cycles will be introduced in the next part of this section. The sequence constraints () () state that task i starting at event point n + should start after the end of the same task performed at the same unit j which has already started at event point n. U corresponds to the upper bound of cycle length H. Constraints (9) establish the relationship between the starting time of a task i at point n + and the end time of task i at event point n when these tasks take place at the same unit. Similarly, constraints (0) represent the order of different tasks i, i that are performed in different units j, j but take place consecutively according to the production recipe. The sequence constraints () represent the requirement of a task i to start after the completion of all the tasks performed at previous event points in the same unit j. In order to represent the features of cyclic scheduling, the following constraints are added to enforce the continuity in plant operation between cycles. Material balances between cycles STIN(s) = ST(s, n), s IS, n = N () Constraints () represent the key feature of periodic scheduling. The intermediates stored at the last event point of the previous cycle should equal the amount of material needed to start the next cycle in order to maintain smooth operation without any accumulation or shortage in between. Raw materials and products are calculated based on the consumed or produced amounts in unit schedule. Demand constraints d(s, n) r s H, s S () n N where r s represents the average requirement of state s. Constraints () express the requirement of meeting demand for all products. Note that the requirements for the time horizon under consideration are assumed to be evenly distributed to each cycle. Cycle timing constraints Unlike the sequence constraints () (0) which describe the sequence of tasks within the same cycle, cycle timing constraints express the sequence relationship of the last task in the previous cycle and the first task in the current cycle so as to maintain continuity of operation between cycles. Cycle timing constraints: tasks in the same unit T s (i,j,n0) T f (i,j,n) H, j J, i I j, i I j, n = N () where n0 stands for the first event point in the current cycle. T f (i,j,n) H corresponds to the time of last event point in the previous cycle. Constraints () represent that task i performing at the beginning of the cycle has to start after the finishing of task i in the same unit at the previous cycle. Since only one task can take place in the same unit at each event point n, constraints () also express the correct recipe sequence for the same unit. Cycle timing constraints: tasks in the different units T s (i,j,n0) T f (i,j,n) H, j, j J, i I j, i I j, i i, n = N () Constraints () represent the requirement of the first task in a new cycle to start after the completion of the tasks in different units at previous cycle based on the recipe requirements. Similar to constraints (0), these constraints are written for the tasks that should take place consecutively in different units and ensure the correct sequence of tasks between cycles. Time horizon constraints T f (i,j,n) H, i I, j J i, n N () T s (i,j,n) H, i I, j J i, n N () Since the starting points of a cycle are not necessarily synchronized for all units, some units may start performing tasks later than others. The maximum idle time, however would not be greater than a cycle period given constraints () and (). Therefore the time horizon constraints () and () represent the requirement of each task i to start and finish before two cycle lengths H. Cycle length constraints (T f (i,j,n) T s (i, j, n)) H, j J () n N i I j The cycle length constraints () state that the duration of all tasks performed in the same unit must be less than the cycle length H, which ensures that cycle of each unit cannot be longer than a cycle length.

7 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) Objective: maximization of average profit n price sd(s, n) H s (9) The objective function for the planning problem is to maximize the production in terms of profit due to product sales. Assuming cyclic scheduling this objective is transformed to maximizing the average profit as shown in (9). The average profit is considered to express the profit over the whole time horizon which depends on the cycle time and the overall production during each cycle. Note that the objective function involves fractional terms d(s, n)/h, thus giving rise to a MINLP problem. Alternative objectives can be also incorporated to express different scheduling targets such as cost minimization... Scheduling problem decomposition Although the idea of cyclic scheduling is to overlook the start-up and finishing periods, in order to obtain a feasible solution for short-term scheduling problem, especially when long time horizons are considered where such a solution is a challenge, a detailed consideration of start-up and finishing periods is proposed in this work. Thus the overall time horizon is divided into three periods, the initial period when the necessary amounts of intermediates are produced to start the cyclic schedule, the main period when cyclic scheduling is applied and the final period to wrap up all the intermediates. The initial and final periods are bounded by a time range and solved independently. The sum of time lengths of all three periods equals that of the whole time horizon. Given the optimal cycle length resulted from solving the cyclic scheduling problem described in Section., the problem for initial period is solved first with the objective function of minimum makespan so as to ensure the existence of feasible solution in order to provide those intermediates for cyclic scheduling. Then the same problem is solved with the objective of maximizing the profit with the time horizon obtained from the solution of the makespan minimization problem. The problem for the final period can be solved in parallel once the time horizon for the cycle length and the initial period are determined. The intermediates considered for the final period are obtained from the unit schedule and the time horizon is the time left for the planning problem. Both initial and final problems are using the set of constraints for short-term scheduling problem and correspond to MILP problems since the time horizon is fixed. The overall approach is schematically shown in Fig.. Solve the periodic scheduling problem Obtain optimal cyclic schedule Determine the number of cycles Calculate the initial input of intermediates and starting time of each unit Solve the minization of makespan problem providing the intermediates for cyclic schedule Solve the maximization of production problem with the same time horizon obtained from the above step Generate the schedule for the initial period Solve the maximization of production problem for the final period with the fixed time horizon Generate the schedule for the final period Incorporate schedule of initial period, cycle operations and final period to obtain the schedule for the time horizon under consideration Fig.. Flow chart of proposed approach.

8 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) Table Solution for motivating example Cycle time range (h) Number of event points Objective function value Optimal cycle time (h) CPU time (s) Case studies.. Results of motivating example The model developed in Section corresponds to a mixed integer nonlinear programming problem. The nonlinearities appear only at the objective function as fractional terms of continuous variables. GAMS Corporation Inc. (GAMS)/discrete and continuous optimizer (DICOPT) is used in this paper that uses outer approximation/equality relaxation/augmented penalty (Grossmann, Viswanathan, Vecchietti, Raman, & Kalvelagen, 00) as a MINLP solution procedure. Since this is a nonconvex problem, OA/ER/AP theoretically can not guarantee global optimality. GAMS/branch-and-reduce optimization navigator (BARON) (Sahinidis & Tawarmalani, 00) is used when a proven global optimum is required. BARON is based on conventional branch-and-bound algorithm and integrates range reduction techniques which contract the search space at each node together with a number of compound branching schemes that accelerate convergence of standard branching strategies. BARON guarantees to provide global optimality to the type of MINLP problems involving fractional terms. This example is solved with both DICOPT and BARON, and the solution obtained is the same. The problem is solved on a Linux system with processor PentiumIII 0 MHz. In order to determine the optimal schedule and cycle length, the following strategy is considered. Instead of considering the whole cycle time range, for example h, several sub-ranges are considered, 0, up to h and the resulting problems are solved independently. The advantages of such an approach are that: (i) each of the sub-period problem utilizes adequate number of event points, resulting in efficient solution procedure; (ii) it generates a number of scheduling alternatives with different cycle lengths that can be utilized when additional requirements such as work shift constraints have to be considered; (iii) each sub-problem can be solved independently and thus parallelization can be easily achieved. Event points correspond to either the initiation of a task or the beginning of unit utilizationis. Thus the number of event points define the maximum number of tasks that are allowed to take place during the time horizon. Since the number of variables and constraints are proportional to the number of event points, the optimal number of event points (i.e. enough to describe the optimal solution while avoiding excessive computation) should be used. For example, four event points are used for the subrange of h. Since consideration of additional event points does not improve the objective value, four is the optimal number of event point for this sub-range. Similarly, six event points are optimal for the sub-range of separ_ reactor_ reactor_ heater_ Fig.. Optimal cyclic schedule for motivating example.

9 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) 9 Table Computational statistics for motivating example Relative optimality criterion 0.0 Cycle time range (h) Number of event points Binary variables Continuous variables 99 Constraints 0 Optimal cycle time (h).09 Objective function value.9 CPU time (s). 0 h. As shown in Table, the final optimal cycle length obtained is.90 h with the objective value of 9.09 units. The optimal schedule is shown in Fig.. One could argue that it is possible to find a better objective value if an even larger range is considered since increase of the cycle time allows more complex cyclic schedule, but this contradicts the basic objective of the cyclic scheduling which is to achieve a good periodic schedule using affordable computational time. It should be noticed that the increased computational time of the problem with the sub-range of 0 h is mainly due to the master MILP problem that requires almost 99% of the computational time. Additional computational information for this example is presented in Table for the problem with cycle time range of h. DICOPT, CONOPT and CPLEX. are the corresponding MINLP, NLP and MILP optimization solvers used. As pointed out in Section, the scheduling problem is computationally intractable when a time horizon of h is considered. Therefore the approach presented in Section. is applied here. The time horizon is divided in three periods, the initial period when the necessary amounts of intermediates are produced to start the periodic schedule, the main period when cyclic scheduling is applied and the final period to wrap up all intermediates. First the cyclic scheduling problem was solved to optimality, obtaining an average profit of 9.09 with an optimal cycle of.9 h. Then the initial input of intermediates and starting time of each unit were calculated. Six cycles were then determined to leave enough time for the initial and final periods to cover the necessary production. The problem of makespan minimization was then solved for the initial period to obtain the shortest time in order to provide the intermediates to cyclic operation. The problem of production maximization for the initial and final periods are then solved simultaneously based on the time horizon calculated from the makespan minimization problem for the initial period. It required 90. CPU seconds to solve the makespan minimization problem,. and. CPU seconds to solve the production maximization problems for the initial and final period problems, respectively. The overall objective function value representing the total profit over the whole time horizon is Figs. and illustrate the schedules for the period of [0, ] that involves the initial period of. h together with a cycle of operations and for the period of [0, ] that involves one cycle and the final period of.9 h. Note that the batch sizes in these figures are round off to one decimal digit for clarity in presentation of the schedules. A large time horizon of 0 days is considered next for the same example. The proposed approach generates the initial and final periods and determines the optimal cyclic operation performed during the rest of the time horizon. The overall objective function value obtained is.99e corresponding to 9 cycles of operation. Note that the schedule for the initial period is the same as that obtained for a time horizon of h shown in Fig. since the initial period is not affected by the overall time horizon considered. The production maximization problem for the final period requires 9. CPU seconds and the result in shown in Fig. for the period of [, 0] that involves one cycle and the final period of. h... Results of example This example was considered by Schilling and Pantelides (999) and is similar to the motivating example except the following differences: (i) there is no heating process in example ; (ii) hot A has both storage capacity and supply for 000; (iii) reaction in reactor produces Int AB only; (iv) all the units have an identical maximum capacity of 0 and different minimum capacities as 0, 0, 0 for reactor, reactor and still, respectively; (v) the processing times of all tasks are those of motivating example multiplied by 0; (vi) the price for product is instead of 0 in motivating example. The proposed formulation is applied to this example in order to compare the results with those presented by Schilling and Pantelides. In order to obtain the global optimal solution, GAMS/BARON is utilized as MINLP solver. In the cycle length range of 0 0 h considered by Schilling and Pantelides, the solution procedure results in approximately (with roundoff errors) the same objective value and schedule. The required computation is only. CPU seconds on PentiumIII 0 MHz using GAMS/BARON. The optimal cycle length is. h with an objective value of.9 producing. units of product and units of product per cycle. The optimal Gantt-Chart is shown in Fig. 9. Schilling and Pantelides used their own branch-and-bound algorithm and a parallel computation scheme due to the complexity of the linearized constraints. They implemented the parallel computing with a network of seven Sun ultra workstations and reported CPU seconds for this example. In a recent work Castro et al. (00) presented a different RTN periodic continuous-time formulation and applied to this example. These results are compared in Table. Note that significant less number of variables are required using the proposed formulation. Castro et al. (00) reported a computational time of. CPU seconds using DICOPT as MINLP solver on a PentiumIII GHz machine. When a longer range of 0 0 h is solved, they obtain a sub-optimal solution corresponding to an objective value of.0 and cycle length of.0,

10 separ_ reactor_ reactor_ heater_ Fig.. Schedule of initial period for h of motivating example. 0 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00)

11 separ_ 0.0 reactor_ 0.0 reactor_ heater_ Fig.. Schedule of final period for h of motivating example D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00)

12 separ_ reactor_ reactor_ heater_ Fig.. Schedule of final period for 0 days of motivating example. D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00)

13 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) separ_. reactor_ reactor_ Fig. 9. Optimal cyclic schedule for example. Table Results of example Proposed approach Formulation of Schilling and Pantelides Formulation of Castro et al. Cycle time range (h) Event points or time slots Binary variables Continuous variables Constraints 0 Optimal cycle time (h)... Objective function value.9.. Table Solution for example Cycle time range (h) Number of event points Objective function value Optimal cycle time (h) CPU time (s) compared to the optimal solution obtained using the proposed formulation that corresponds to an objective value of.9 and cycle length of.0. Table presents the optimal cycle lengths for different time ranges solved using GAMS/DICOPT. Although GAMS/DICOPT is only a local optimization solver, results with the sub-range up to 0 h are proved to correspond to the global optimal solutions obtained using the global solver GAMS/BARON. It should be noticed that by confining the cycle time to be less than 0 h a sub-optimal solution is obtained since the optimal cycle length is found to be. h... Results of example In this example, four products are produced through eight tasks using three raw materials. The problem involves six intermediates and six different units. The STN representation for this process is shown in Fig. 0, and the required data are presented in Table. The proposed approach was applied for this example with the results shown in Table. The Table Data for example Units Capacity Suitability (task) , ,.0 States Storage capacity Initial amount Price Feed Unlimited Unlimited 0.0 Intermediate Intermediate Intermediate Intermediate Intermediate Intermediate Products Unlimited 0.0 Mean processing time ( τ ij )

14 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) Product Int 0. Int 9 Product Feed Task Int 0. Task 0. Task 0. Task Feed Task 0. Feed Int 0. Int 0. Task Task Product Product Task Int 0. Fig. 0. State task network representation for example. Table Solution for example Cycle time range (h) Number of event points Objective function value Optimal cycle time (h) CPU time (s) optimal cycle obtained in. CPU seconds corresponds to 0. h with an objective value of 9. and the optimal schedule shown in Fig.. Considering a scheduling problem of time horizon of 0 days for this example, the proposed approach determines cycles of operation, schedules for an initial period of.9 h and a final period of. h as shown in Figs. and, respectively. The overall objective function value is.0e and the CPU time for solving initial and final periods is 0. and. CPU seconds, respectively. This proposed cyclic scheduling approach can be applied to planning problem with larger size, because the time unit unit unit unit unit unit Fig.. Optimal cyclic schedule for example.

15 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) unit unit unit unit unit unit. 0 0 Fig.. Schedule for initial period for 0 days of example. unit unit unit unit unit unit Fig.. Schedule for final period for 0 days of example. horizon does not affect the computational complexity of the cyclic scheduling. However, since cyclic scheduling assumes stable operating conditions this approach is not suitable for production facilities with large variability of production parameters.. Conclusions and future directions This paper presents a new framework for the solution of scheduling problem based on a continuous-time formulation and the idea of operating periodicity. It determines the optimal cyclic schedule as well as the optimal cycle length for multipurpose batch plant. Compared with existing approaches, the proposed formulation results in fewer number of variables and constraints. Although the proposed model corresponds to a nonconvex MINLP problem, it is shown that it can be efficiently solved by GAMS/DICOPT that achieves the global optimal solutions which are verified by utilizing the global optimization algorithms. Based on the proposed approach, current research aims at the development of a hierarchical framework to enable the solution of the planning and scheduling problem when the demand and price vary substantially within the time horizon under consideration.

16 D. Wu, M. Ierapetritou / Computers and Chemical Engineering (00) Acknowledgements The authors would like to acknowledge financial support by the NSF Career Award (CTS-990). References Castro, P., Barbosa-Povoa, A., & Matos, H. (00). Optimal periodic scheduling of batch plants using RTN-based discrete and continuous-time formulations: A case study approach. Industrial and Engineering Chemical Research,, 0. Grossmann, I., Viswanathan, J., Vecchietti, A., Raman, R., & Kalvelagen, E. (00). GAMS/DICOPT: A discrete continuous optimiization package. GAMS Corporation Inc. Ierapetritou, M., & Floudas, C. (99a). Effective continuous-time formulation for short-term scheduling.. Multipurpose batch processes. Industrial and Engineering Chemical Research,, 9. Ierapetritou, M., & Floudas, C. (99b). Effective continuous-time formulation for short-term scheduling.. Continuous and semicontinuous processes. Industrial and Engineering Chemical Research,, 0. Kondili, E., Pantelides, C., & Sargent, R. (99). A general algorithm for short-term scheduling of batch operations. I. Milp formulation. Computers and Chemical Engineering,,. Maravelias, T., & Grossmann, I. (00). New continuous-time state task network formulation for the scheduling of multipurpose batch plants. Industrial and Engineering Chemical Research,, 0 0. Mockus, L., & Reklaitis, G. (99). Mathematical programming formulation for scheduling of batch operaions based on nonuniform time discretization. Computers and Chemical Engineering,,. Orcun, S., Altinel, I., & Hortacsu, O. (00). General continuous time models for production planning and scheduling of batch processing plants: Mixed integer linear program formulations and computational issues. Computers and Chemical Engineering,, 9. Pantelides, C. (99). Unified frameworks for optimal process planning and scheduling. In Proceedings of the Second Conference on FOCAPO (pp. ). Pinto, J., & Grossmann, I. (99). A continuous-time mixed-integer linear-programming model for short-term scheduling of multistage batch plants. Industrial and Engineering Chemical Research,, 0 0. Sahinidis, N., & Tawarmalani, M. (00). GAMS/BARON.0: Global optimization of mixed-integer nonlinear programs. GAMS Corporation Inc. Schilling, G., & Pantelides, C. (999). Optimal periodic scheduling of multipurpose plants. Computers and Chemical Engineering,,. Shah, N., Pantelides, C., & Sargent, R. (99). Optimal periodic scheduling of multipurpose batch plants. Annual Operation Research,, 9. Wu, D., & Ierapetritou, M. (00). Decomposition approaches for the efficient solution of short-term scheduling problem. Computers and Chemical Engineering,,. Zhang, X., & Sargent, R. (99). The optimal operation of mixed production facilities General formulation and some solution approaches for the solution. In Proceedings of the Fifth International Symposium on Process Systems Engineering (pp. ). Zhu, X., & Majozi, T. (00). Novel continuous time milp formulation for multipurpose batch plants.. Integrated planning and scheduling. Industrial and Engineering Chemical Research, 0,.