A Simple New Continuous-Time Formulation for Short-Term

A new continuous-time formulation for scheduling short-term multipurpose batch processes is presented. The formulation gives rise to a mixed-integer l...
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PROCESS DESIGN AND CONTROL A Simple New Continuous-Time Formulation for Short-Term Scheduling of Multipurpose Batch Processes Nikolaos F. Giannelos and Michael C. Georgiadis* Centre for Research and Technology-Hellas, Chemical Process Engineering Research Institute, P.O. Box 361, Thermi-Thessaloniki 57001, Greece

A new continuous-time formulation for scheduling short-term multipurpose batch processes is presented. The formulation gives rise to a mixed-integer linear programming (MILP) model. The state-task network (STN) representation forms the basis of the proposed approach. A number of event points is prepostulated, which is the same for all tasks in the process. Event times are defined by the ends of task execution, and they are generally different for different tasks of the process, giving rise to a nonuniform time grid. The necessary time monotonicity for single tasks is guaranteed by means of simple duration constraints. Suitable sequencing constraints, applicable to batch tasks involving the same state, are also introduced, so that state balances are properly posed in the context of the nonuniform time grid. The expression of duration and sequencing constraints is greatly simplified by hiding all unit information within the task data. Three benchmark problems are used to illustrate the efficiency and applicability of the new formulation. Results are shown to compare favorably with existing continuous-time formulations in terms of model size and computational effort. 1. Introduction Scheduling in process systems generally refers to the allocation of resources (process units, materials, utilities) in time for the manufacturing of one or more products. Reviews of the batch scheduling problem have been presented by Reklaitis,1 Pantelides,2 Shah,3 and Pinto and Grossmann.4 In multipurpose batch scheduling, multiple resources are shared by multiple processing steps (tasks) for the production of multiple products. The batch scheduling problem is usually concerned with short-term production, but operation of batch facilities in a cyclic (campaign) mode in time is also common when little or no production target uncertainty is present in the process. Process representation is a major issue in model development for batch scheduling problems. Significant contributions in this area include the well-known statetask network (STN) introduced by Kondili5 and presented compactly in Kondili et al.6 and Shah et al.7 A subsequent general representation appeared in the form of the resource-task network (RTN) of Pantelides.2 The RTN treats units as additional resources, allowing for a unified representation of material states, utilities, and process units. Traditionally, STN- and RTN-based formulations have employed a simple discretization of the time horizon into equal-length intervals. Continuous-time formulations emerged as an effort to eliminate unnecessary time periods in hopes of reducing the problem size and computational cost. Continuous formulations can * To whom correspondence should be addressed. Tel.: +3031-498143.Fax: +30-31-498180.E-mail: [email protected].

be classified as uniform and nonuniform grid schemes. Uniform continuous-time representations employ a single time grid for all states/resources in the process. The continuous-time formulations of Mockus and Reklaitis,8 Zhang and Sargent,9,10 and Schilling and Pantelides11 (also recently used by Castro et al.12) fall under the uniform time grid category. Nonuniform time grids are employed by Karimi and McDonald,13 Zentner and Reklaitis,14 Pinto and Grossmann,15,16 and Ierapetritou and Floudas17,18 (also used in Ierapetritou et al.19). In this work, a nonuniform time grid is devised and tailored to a new simple formulation for the multipurpose short-term batch scheduling problem. The remainder of this paper presents the underlying concepts (section 2), mathematical model (section 3), and model predictions for a number of standard test problems reported previously in the literature (section 4). A comparative exposition with respect to other continuoustime formulations examining the same benchmark problems is also provided (section 5). 2. Conceptual Details of the Formulation The state-task network (STN) representation was chosen as the basis of this work. Within STN, several concepts relating to tasks, continuous time, task durations, and state-task interactions are developed as building blocks of the formulation. These are detailed below. 2.1. Task Representation. A process task that can be performed in k distinct units is treated in the formulation as k distinct tasks. For example, if the physical process task representing the simple reaction A f B can be executed in three different vessels, then

10.1021/ie010399f CCC: $22.00 © 2002 American Chemical Society Published on Web 04/05/2002

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Figure 4. State consumed by multiple tasks.

Figure 1. Continuous-time representation.

Figure 5. State produced by multiple tasks.

Figure 2. Time representation for disjunctive tasks. Figure 6. Intermediate state. Nonpermissible event combination.

Figure 3. Buffer times.

three different model tasks are considered. This task representation scheme effectively hides all unit data within tasks, as shown in section 3.2, and it constitutes a salient feature of the proposed modeling approach. 2.2. Continuous-Time Representation. A fixed number of event points is prepostulated, (t1, t2, ..., tn-1, tn), which is the same for all tasks in the process. Any task can then be executed at most n times during the time horizon of interest. The end of task execution marks the placement of an event point on the continuous-time domain for the particular task at hand. Event times can be different for different tasks in the process. Figure 1 illustrates this concept. The case of tasks that can never coincide at any point in time (because, for example, they can only be performed in the same unit) can, in principle, be treated within the general framework of event points mentioned above. For these sets of tasks, though, a simpler event time topology can be applied without loss of generality, as shown in Figure 2. 2.3. Task Durations. In continuous-time formulations, it is common practice to assume batch-sizedependent durations θi for all tasks in the process

θi ) ai + biBi where ai is the size-independent contribution to task duration and bi is the term dependent on the batch size, Bi. The present work utilizes size-dependent task durations, with the addition of a relaxation term (buffer time) θbuf i

θi ) ai + biBi + θbuf i The incentive for this relaxation can best be explained by virtue of Figure 3. In Figure 3, a production task ends at time tp, and a consuming task starts at time tc, with tc > tp. By prolonging the production task sufficiently,

and also by artificially shifting the depleting task earlier in time, a single time boundary/interface is created at time ti, where material balances and intermediate storage constraints can be applied. Of course, buffer times for tasks producing unstable states can always be set to zero, so that a zero-wait policy can be applied. 2.4. State-Task Interactions. The time representation presented in section 2.2 is clearly of the nonuniform type. Consequently, interactions between tasks involving the same material state are not trivial to model, and extreme care is required in posing material balance and intermediate storage constraints. Material balances are typical multiperiod-type equations; because the location of event points is determined dynamically by the model and the location of event points can be different for different tasks in the process, sequencing constraints are required to ensure that a material balance can be written for any state of the process without ambiguity. To this end, state-task interactions in this work are broken down into three general classes. This state-task classification scheme, combined with the simple task duration relaxation technique of section 2.3, provides the means by which properly defined material balance expressions can be posed. 2.4.1. States Consumed by Multiple Tasks. This case is shown in Figure 4. Given that state s is consumed at the beginning of tasks i and ii, the nonuniform time grid has to be constrained so that the starting times of these tasks for event points tn are the same for both tasks engaged in consuming the state. When the actual starting times of tasks i and ii are not identical, buffer times allow the equality to hold. The desired functionality is enabled via simple sequencing constraints. 2.4.2. States Produced by Multiple Tasks. This case is shown in Figure 5. Here, a similar argument applies to the ending times of tasks i and ii producing state s, given that states are produced at the end of task execution. The relaxed durations allow tasks i and ii to end at the same time. Another set of sequencing constraints is used for this purpose. 2.4.3. Intermediate States. This situation is shown in Figure 6. A nonpermissible event combination appears alongside. With respect to Figure 6, if the ending time of task i for event point tn-1 is greater than the

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starting time of task ii for event point tn, then no sensible multiperiod expression can be applied to determine the amount of s available through production by i and depletion by ii. A third set of constraints is required to this end. 3. Mathematical Model 3.1. Nomenclature. The sets, parameters, and variables used in this work are all defined in the Nomenclature section. 3.2. Constraints. With the data and variables defined in the Nomenclature section, the complete model formulation is presented below. 3.2.1. Batch Size Constraints. min max xi,tBi,t e Bi,t e xi,t Bi,t , ∀ i ∈ I, t ∈ T

(1)

Equation 1 ensures that batch sizes are kept within known lower and upper limits. For tasks that are not executed (xit ) 0), the corresponding batch sizes are forced to equal zero. 3.2.2. Task Duration Constraints.

τit g τi,t-1 + θbuf it + (ai xit + bi Bit), ∀ i ∈ I, t ∈ T (2) τit e C

, ∀ i ∈ I, t ∈ T, t ) tn

MAX

(3)

Equation 2 imposes the required time monotonicity on event points corresponding to the same task. The combination of eqs 2 and 3 ensures that the time horizon of interest is never violated by any task of the process. 3.2.3. Sequencing Constraints.

τit ) τii,t, ∀ s ∈ S; i, ii ∈ Ips ; i ) HEAD(Ips ); ii * i; t ∈ T (4) τit - θbuf it - (aixit + biBit) ) buf - (aiixii,t + biiBii,t) τii,t - θii,t

∀ s ∈ S; i, ii ∈ Ics; i ) HEAD(Ics); ii * i; t ∈ T (5) buf τi,t-1 ) τii,t - θii,t - (aiixii,t + biiBii,t)

∀ s ∈ S; i ) HEAD(Ips ); ii ) HEAD(Ics); t ∈ T; t > 1 (6) Equation 4 forces the ending times of tasks producing the same state to be equal. Equation 5 forces the starting times of tasks consuming the same state to be equal. Equation 6 is the sequencing constraint for tasks where a state appears as an intermediate. The HEAD () operator marks the first task element in the task sets appearing in eqs 4-6. Equation 6 need only be applied to one task element of the Ics and Ips sets; for all other task instances appearing in Ics and Ips , the required functionality is imposed by virtue of eqs 4 and 5. 3.2.4. Equipment Allocation Constraints.

∑ xit e 1,

∀ u ∈ U, t ∈ T

(7)

i∈Iu

τit ) τii,t ∀ u ∈ U; i, ii ∈ Iu; i ) HEAD(Iu); ii * i; t ∈ T (8)

Figure 7. STN for example 1. Table 1. Task Specifications for Example 1 task i

Bmin i

Bmax i

ai

bi

Mix React Sep

0 0 0

100 75 50

3.000 2.000 1.000

0.0300 0.0266 0.0200

Equation 7 ensures that any piece of process equipment (unit) is not assigned to multiple tasks concurrently. Coupled with eq 8, eq 7 enables the simple event point topology shown in Figure 2. 3.2.5. Storage and Material Balance Constraints.

STst e STmax , ∀ s ∈ SFIS, t ∈ T s

(9)

STend e STmax , ∀ s ∈ SFIS s s

(10)

STst ) STs,t-1 +

∑ pisBi,t-1 + ∑ cisBit,

i∈Ips

STend ) STst + s

∑ pisBit,

i∈Ics

∀ s ∈ S, t ∈ T (11)

∀ s ∈ S, t ∈ T, t ) tn (12)

i∈Ip

Equations 9 and 10 express aggregated storage constraints on some or all states of the process. Equations 9 and 10 imply the existence of dedicated intermediate storage for every state s ∈ SFIS. Equations 11 and 12 are the usual STN-based material balance multiperiod expressions. As a reminder, initial amounts for all states are available in ST0s . We reiterate that the functional form of the intermediate storage and material balance equations 9-12, given the nonuniform nature of the time grid, is sensible by virtue of the duration and sequencing constraints in eqs 2 and 4-6, which effectively transform the nonuniform time grid to a uniform one for batch tasks involving the same material state. 3.3. Objective Function. A typical economic objective function completes the formulation outlined above

max z )

∑ vsSTend s

(13)

s∈S

subject to constraints 1-12. It is a straightforward task to employ alternative optimization objectives. In particular, minimization of the time horizon (makespan) can be effected by turning parameter CMAX into a variable, placing an upper limit on it, and imposing variables desired lower limits (demands) on the STend s of interest. 4. Example Problems Three example problems are used to illustrate the performance of the model. The same example problems are considered in Castro et al.,12 Ierapetritou and Floudas,17 Schilling and Pantelides,11 and Zhang.20 For Zhang,20 results are reported for the smaller variable event time (VET) formulation. The STN for example 1 is shown in Figure 7. Process data are summarized in Tables 1 and 2. The process consists of three simple tasks, each performed in a dedicated unit.

Ind. Eng. Chem. Res., Vol. 41, No. 9, 2002 2181 Table 2. State Specifications for Example 1 state s

ST0s

STmax s

value

s1 s2 s3 s4

∞ 0 0 0

∞ 0 0 ∞

0 0 0 1

Table 3. Comparison of Different Models for Example 1 indicators

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events int vars cont vars constraints relaxation solution nodes CPU s

4 12 51 75 100 71.518 5 0.033

Castro Ierapetritou Schilling and et al.12 and Floudas17 Pantelides11 Zhang20 5 28 95 120 100 71.518 31 0.25

5 15(6) 105(48) 108(58) 100 71.518 13 0.05(0.03)

6 46 157 220 170.79 71.47 418 -

7 48 187 263 149.99 71.45 528 21.9

Figure 10. STN for example 2. Table 4. Task Specifications for Example 2 task i

Bmin i

Bmax i

ai

bi

Heat Ra1 Ra2 Rb1 Rb2 Rc1 Rc2 Sep

0 0 0 0 0 0 0 0

100 50 80 50 80 50 80 200

0.667 1.333 1.333 1.333 1.333 0.667 0.667 1.333

0.00667 0.02700 0.01700 0.02700 0.01700 0.01330 0.00833 0.00667

Table 5. State Specifications for Example 2

Figure 8. Gantt chart for example 1.

state s

ST0s

STmax s

value

feeda feedb feedc hota intab intbc impe prod1 prod2

∞ ∞ ∞ 0 0 0 0 0 0

∞ ∞ ∞ 100 200 150 200 ∞ ∞

0 0 0 0 0 0 0 10 10

Table 6. Comparison of Different Models for Example 2 indicators

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Castro et al.12

events int vars cont vars constraints relaxation solution nodes CPU s

4 32 142 241 1804.35 1480.06 23 0.15

5 80 226 297 1804.36 1480.06 60 0.32

Ierapetritou Schilling and and Floudas17 Pantelides11 Zhang20 5 40 260 374 1732.36 1503.15 51 0.28

6 130 386 587 2783.14 1480.05 1230 -

7 147 497 741 2258.71 1497.69 9575 1027.5

Figure 9. Gantt chart for example 1 with buffer times.

The results, and a comparison with previously published studies of the same problem, are shown in Table 3. The results of the proposed formulation were obtained on a Dec Alpha 500/500 workstation using GAMS/ CPLEX.21,22 CPU times for other formulations are included for completeness only. The optimal schedule is shown in Figures 8 and 9. Figure 9 indicates that only one buffer time was used in the case of the first instance of the separation task (Sep). Event points are also shown for all tasks. This formulation yields exactly the same value for the objective function as the models of Castro et al.12 and Ierapetritou and Floudas17 by utilizing one less event point. The other models produce essentially the same optimum, as minute differences can be attributed to the values of the duration parameters actually implemented by the authors. In terms of model size, the formulation compares favorably with all other models and is marginally worse than the overall-best reduced model of Ierapetritou and Floudas,17 for which results are indicated in parentheses in Table 3. It is noted that the

reduced model was obtained after a manual postprocessing step on the full model, by taking advantage of the STN structure and by eliminating certain variables through direct substitution from their defining equations. The STN for example 2 is shown in Figure 10. All reaction steps (Ra, Rb, Rc) can be performed in two units. This is depicted in the task specifications for the process, Table 4. Two valuable products are manufactured, and storage constraints are placed on all intermediates, as indicated in the state specifications for the process, Table 5. The minimum batch size for the mixing tasks is set to five units to avoid unrealistic processing of almost-zero amounts of material. Computational statistics for example 2 are presented in Table 6, and the best schedule appears in Figures 11 and 12. The optimum in the objective function as predicted by this formulation is also found by the models of Castro et al.12 and Schilling and Pantelides,11 which require one and two more event points, respectively. Zhang20 reports a higher objective value, possibly resulting from the use of different parameter values.

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events int vars cont vars constraints relaxation solution nodes CPU s

5 30 127 182 72.80 58.545 13 0.066

Castro Ierapetritou Schilling and et al.12 and Floudas17 Pantelides11 Zhang20 7 99 284 388 94.75 58.55 178 0.50

7 36 223 289 80.00 58.81 60 0.19

8 126 414 645 80.04 58.55 11945 -

8 119 426 605 88.12 57.72 1836 176.2

Figure 11. Gantt chart for example 2.

Figure 14. Gantt chart for example 3.

Figure 12. Gantt chart for example 2 with buffer times.

Figure 13. STN for example 3. Figure 15. Gantt chart for example 3 with buffer times.

Table 7. Task Specifications for Example 3 task i

Bmin i

Bmax i

ai

bi

React Mix1 Mix2 Filt Sep1 Sep2

0 5 5 0 0 0

20 20 20 20 20 20

17.330 2.670 2.670 4.000 5.330 5.330

0.870 0.130 0.130 0.200 0.270 0.270

Table 8. State Specifications for Example 3 state s

ST0s

STmax s

value

feed add rprod blend slurry prod

∞ ∞ 0 0 0 0

∞ ∞ 100 100 100 ∞

0 0 0 0 0 2

two units, and the final separation step (Sep) can be performed in two units as well. This is indicated in Table 7. The results for example 3 are presented in Table 9, and the best schedule is shown in Figures 14 and 15. The value of the objective function is identical to the ones reported by Castro et al.12 and Schilling and Pantelides.11 Ierapetritou and Floudas17 report a better value, which, as in example 2, is due to the task duration parameter values actually used by the authors. The proposed formulation compares favorably with all models in terms of model size and computational effort. The formulation requires only five event points and produces the best linear relaxation reported so far. 5. Discussion and Significance

Ierapetritou and Floudas17 report a better objective than all other models. However, it has recently been established beyond doubt23,24 that the improved objective obtained by Ierapetritou and Floudas17 is merely an artifact of slightly differentsyet favorablesparameter values actually used by these authors rather than the ones listed in their original work.17 The STN for example 3 appears in Figure 13. Task and state specifications are compiled in Tables 7 and 8. The mixing processing step (Mix) can be performed in

In this section, the proposed formulation is critically reviewed by comparison to existing continuous-time formulations of the short-term multipurpose batch scheduling problem. An interesting feature of the formulation is the elimination of unnecessary event points through the use of a nonuniform time grid and a simple relaxation of the task durations. Duration and sequencing constraints effectively transform the nonuniform time grid to a uniform one for the purposes of material balance and storage constraints, ensuring that the mass conserva-

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tion and batch sizing problem is well-posed throughout the continuous-time domain. Also, the formulation relies on a low-resolution time grid, as only the ending times of task execution are utilized. A5 as a consequence, the proposed approach relies on a trade-off between modest computational cost and acceptable modeling rigor. With respect to the resolution of the time grid, the approach of Schilling and Pantelides11 is clearly superior, as both starting and ending times of tasks are taken into account for the placement of event points on a simple uniform time scale. This fact, coupled with the RTN representation, has resulted in a comprehensive formulation, applicable to a variety of scheduling objectives as reported recently in the literature (Schilling,25 Schilling and Pantelides26), but at the added expense of increased computational times even when resorting to specialized solution techniques and parallel processing hardware. The approach taken by Zhang20 in terms of time grid resolution is comprehensive as well. The resulting models are nonlinear, however, and problem sizes are increased even further by the application of exact linearization techniques. The formulation of Ierapetritou and Floudas17 utilizes a low-resolution nonuniform time grid. The formulation relies on the concept of decoupling task events from unit events and has been applied to a variety of short-term scheduling problems.18,19 Sequencing constraintssalbeit conceptually different from the ones documented in the present worksare used in the context of the nonuniform time grid. The formulation of Castro et al.12 is a relaxation of the original formulation of Schilling and Pantelides,11 in which material states are allowed to remain in process units after their processing has been completed. The relaxation approach carries obvious similarities with the introduction of buffer times in the proposed formulation. For highly constrained problems, where units have to be allocated to other processing tasks immediately, the formulation of Castro et al.12 can always employ more event points, at which stage it becomes identical to the formulation of Schilling and Pantelides,11 and no computational gains can materialize. 6. Conclusions This work presented a simple new formulation to the short-term multipurpose batch scheduling problem. Because of the low-resolution nonuniform time grid employed, the method produces small model sizes while preserving material balance and storage limitations unambiguously. The formulation was shown to perform well on small-size example problems. Extensions of the underlying nonuniform time grid concept to continuous processes are under way, and results on larger-scale case studies found in the literature (mostly continuous and semicontinuous processes) will be the objective of future work. Nomenclature Sets/Indices I/i, ii ) tasks S/s ) states T/t ) event points U/u ) units Ics ) tasks consuming state s

Ips ) tasks producing state s Iu ) tasks that can be performed in unit u SFIS ) states of finite intermediate storage Parameters Bmax ) maximum allowable batch size for task i i ) minimum allowable batch size for task i Bmin i ) maximum allowable storage for state s STmax s ST0s ) initial amount of state s vs ) value/price of state s ci,s ) fraction of state s consumed in batch recipe i pis ) fraction of state s produced by batch recipe i ai ) constant term of task i duration bi ) batch-size-dependent term of task i duration CMAX ) time horizon Variables xit ) 1 if task i terminates at event point t, 0 otherwise Bit ) batch size of task i terminating at event point t STst ) amount of state s at the start of event point t ) amount of state s at the end of the time horizon STend s τit ) real time corresponding to event point t for task i θbuf it ) buffer time of task i for event point t

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(15) Pinto, J. M.; Grossmann, I. E. A Continuous Time Mixed Integer Linear Programming Model for Short-Term Scheduling of Multistage Batch Plants. Ind. Eng. Chem. Res. 1995, 34, 3037. (16) Pinto, J. M.; Grossmann, I. E. An Alternative MILP Model for Short-Term Scheduling of Batch Plants with Preordering Constraints. Ind. Eng. Chem. Res. 1996, 35, 338. (17) Ierapetritou, M. G.; Floudas, C. A. Effective ContinuousTime Formulation for Short-Term Scheduling. 1. Multipurpose Batch Processes. Ind. Eng. Chem. Res. 1998, 37, 4341. (18) Ierapetritou, M. G.; Floudas, C. A. Effective ContinuousTime Formulation for Short-Term Scheduling. 2. Continuous and Semicontinuous Processes. Ind. Eng. Chem. Res. 1998, 37, 4360. (19) Ierapetritou, M. G.; Hene, T. S.; Floudas, C. A. Effective Continuous-Time Formulation for Short-Term Scheduling. 3. Multiple Intermediate Due Dates. Ind. Eng. Chem. Res. 1999, 38, 3446. (20) Zhang, X. Algorithms for Optimal Process Scheduling Using Nonlinear Models. Ph.D. Thesis, Imperial College of Science, Medicine, and Technology, London, 1995. (21) Brooke, A.; Kendrick, D.; Meeraus, A.; Raman, R. GAMSs A User’s Guide. GAMS Development Corporation: Washington, DC, 1998.

(22) GAMS/CPLEX 6.6 User Notes. In GAMSsThe Solver Manuals; GAMS Development Corporation: Washington, DC, 2000. (23) Ierapetritou, M. G.; Floudas, C. A. Comments on “An Improved RTN Continuous-Time Formulation for the Short-Term Scheduling of Multipurpose Batch Plants”. Ind. Eng. Chem. Res. 2001, 40, 5040. (24) Castro, P.; Barbosa-Po´voa, A. P. F. D.; Matos, H. Reply to Comments on “An Improved RTN Continuous-Time Formulation for the Short-Term Scheduling of Multipurpose Batch Plants”. Ind. Eng. Chem. Res. 2001, 40, 5042. (25) Schilling, G. Algorithms for Short-Term and Periodic Process Scheduling and Rescheduling. Ph.D. Thesis, Imperial College of Science, Medicine, and Technology, London, 1997. (26) Schilling, G.; Pantelides, C. C. Optimal Periodic Scheduling of Multipurpose Plants. Comput. Chem. Eng. 1999, 23, 635.

Received for review May 2, 2001 Revised manuscript received January 25, 2002 Accepted February 1, 2002 IE010399F