Ind. Eng. Chem. Res. 2009, 48, 2947–2964
2947
Novel Unified Modeling Approach for Short-Term Scheduling Munawar A. Shaik† and Christodoulos A. Floudas*,‡ Department of Chemical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi, 110016, India, and Department of Chemical Engineering, Princeton UniVersity, Princeton, New Jersey 08544
Unit-specific event-based continuous time models have gained significant attention in the past decade for their advantages of requiring less number of event points, smaller problem size, and hence, a better computational performance. In the literature, different models had been proposed for short-term scheduling problems involving with and without resource constraints using unit-specific event-based formulations. For scheduling problems involving no resource constraints, generally the unit-specific event-based models do not allow tasks to continue over multiple events unlike in models that account for resource constraints explicitly. In this work, we emphasize the necessity for allowing tasks to take place over multiple event points even for simpler scheduling problems involving no resource constraints. We propose a novel short-term scheduling model using three-index binary and continuous variables that efficiently merges both the problems involving resources and no resource constraints into a unified, generic common framework. The proposed approach is based on state-task-network (STN) representation and is suitable for both batch and continuous plants, although we focus only on batch plants in this paper. Detailed computational case studies are presented to demonstrate the efficacy of the proposed model. 1. Introduction During the last two decades, numerous formulations have been proposed in the literature based on continuous-time representation, due to their established advantages over discretetime representations. Floudas and Lin1,2 and Mendez et al.3 presented extensive reviews comparing various discrete- and continuous-time-based short-term scheduling formulations. On the basis of the time representation used, the different continuous-time models in the literature were broadly classified into slot-based, global-event-based, unit-specific event-based, and precedence-based formulations. In the slot-based models,4,5 the time horizon is represented in terms of ordered blocks of unknown, variable lengths, or time slots. Global-event-based models6,7 use a set of events that are common across all units, and the event points are defined for either the beginning or end (or both) of each task in each unit. Unit-specific event-based models8–10 define events on a unit basis, allowing tasks corresponding to the same event point but in different units to take place at different times. For sequential processes, other alternative approaches based on precedence relationships have also been developed11–13 which do not require the postulation of any events or time slots. In this paper our focus is on formulations based on unit-specific event-based continuous time representation. Floudas and co-workers8–10,14–26 developed unit-specific event-based models for a variety of problems involving design, synthesis, short-term, medium-term, reactive scheduling, and scheduling under uncertainty. A detailed comparison of various continuous-time models for short-term scheduling of batch and continuous plants was performed recently by Shaik et al.23 and Shaik and Floudas.24,25 They concluded that, due to heterogeneous locations of event points used, the unit-specific eventbased models always require less event points and exhibit favorable computational performance compared to both slotbased and global-event-based models. For continuous plants, * To whom correspondence should be addressed. E-mail: floudas@ titan.princeton.edu.. Tel.: +1-609-258-4595. Fax: +1-609-258-0211. † Indian Institute of Technology. ‡ Princeton University.
Shaik and Floudas24 presented an improved model compared to Ierapetritou and Floudas9 that outperforms the other models considered in their study. In chemical process operations, short-term scheduling problems can be broadly classified into two categories, those that involve resource constraints such as utilities (cooling water, steam, and so on), manpower etc., and the simple cases that do not involve resources. For batch plants that do not have resource considerations, it was found23 that the modified model of Ierapetritou and Floudas8 as presented in Shaik et al.,23 outperforms the other models both in terms of least problem size and fast computational performance. Similarly, for batch plants with resource constraints, the enhanced model of Janak et al.10,15 was found10,15,23 to perform well. However, for an additional instance of the following example, involving a recycle stream, shown in Figure 1, it is observed that the model of Ierapetritou and Floudas8 yields a suboptimal solution as discussed below. 1.1. Motivating Example. Consider the second example discussed in Shaik et al.,23 in which two different products are produced through five processing stages: heating, reactions 1, 2, and 3, and separation, as shown in the STN representation of the plant flow sheet in Figure 1. Since each of the reaction tasks can take place in two different reactors, each reaction is represented by two separate tasks. The processing time of task i on unit j is assumed to be a linear function, Ri + βiB, of its
Figure 1. State-task network representation for motivating example.
10.1021/ie8010726 CCC: $40.75 2009 American Chemical Society Published on Web 02/17/2009
2948 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
Figure 2. Gantt chart for motivating example using the JLF model.
Figure 3. State-task network representation for example 1.
Figure 4. Gantt chart for example 2b for maximization of profit using the unified model.
Figure 5. State-task network representation for example 3.
batch size, B. The relevant data23 of the constant (Ri) and variable (βi) coefficients for processing times of different tasks (i), the suitable units (j), and their minimum (Bmin i ) and maximum (Bmax i ) batch sizes are shown in Table 1. The initial stock level for all intermediates is assumed to be zero and unlimited storage capacity (UIS) is assumed for all states. The prices of products 1 and 2 are $10/mu. For the objective of maximization of profit and a time horizon (H) of 10 h, this example is solved using the unitspecific event-based model of Ierapetritou and Floudas8 (I&F), the global-event-based models of Castro et al.7 (CBMN), and Maravelias and Grossmann6 (M&G), and using the slot-based model of Sundaramoorthy and Karimi5 (S&K). All the resulting mixed-integer linear programming (MILP) models are solved in GAMS27 distribution 21.1 using CPLEX 8.1.0 on the same computer (3 GHz Pentium 4 with 2 GB
RAM) as in Shaik et al.23 and Shaik and Floudas.25 Table 2 provides a comparison of different models in terms of the problem statistics such as the number of binary and continuous variables, number of constraints, CPU time taken to solve to the specified integrality gap, the number of nodes taken to reach the optimal solution, the objective function at the relaxed node, and so forth. It should be noted that for the S&K model, n event points are shown to represent n - 1 slots for a valid comparison with the other global-event and unit-specific event-based models. In the CBMN model, there is an additional parameter (∆t) that defines a limit on the maximum number of events over which a task can occur. For this case, the slot-based and global-event-based models require at least eight event points and are able to find the global optimal solution of $1962.7, compared to the unitspecific event-based I&F model, which gives a suboptimal
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 2949
Figure 6. Gantt chart for example 3b for maximization of profit using the unified model.
Figure 7. State-task network representation for example 4.
solution of $1943.2 with six events and with higher events as well. When this case is solved using the enhanced unitspecific event-based model of Janak et al.10,15 (JLF), it found the global optimal solution of $1962.7 using six events, as shown in Table 2. The Gantt chart for the JLF model is shown in Figure 2. The reason for this exception can be attributed to the fact that the I&F model does not allow a task to continue over several events, while the JLF model is an enhanced version of the I&F model that allows tasks to take place over multiple event points in order to accurately account for the resource considerations such as utility requirements. Although, there are no resource constraints in this example, it can be observed from the Gantt chart of Figure 2 for the JLF model that this
schedule will not be feasible for the I&F model. According to the constraint for different tasks in different units of the I&F model, for state s5, the consuming task (i ) 7) at event N5 should start after the end time of the recycle producing task (i ) 8) at event N4, which is clearly not the case in the global optimal solution of Figure 2. This constraint becomes relaxed in the JLF model because although the recycle producing task (i ) 8) starts at event N4 it does not end at event N4, but it continues over the next event and ends at event N5 in the global optimal solution. This feature is not feasible for the I&F model, and hence, it yields a suboptimal solution for this case. The models of S&K, M&G, and CBMN allow tasks to take place over multiple events, and hence, are able to find the global optimal solution. Moreover, in these models the events/slots are globally aligned, and hence, they do not require the above-mentioned sequencing constraint for different tasks in different units, which is generally required for the unit-specific event-based models. As shown later, similar problems occur with the RTN-based model of Shaik and Floudas,25 as they also do not allow tasks to occur over multiple events. This example demonstrates that, although, there are no resource constraints, in some cases it is a requirement for the unit-specific event-based models as well to allow tasks to take
Figure 8. Gantt chart and resource utilization levels for JLF model in example 4b for profit maximization.
2950 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
Figure 9. Gantt chart and resource utilization levels for unified model in example 4b for profit maximization. Table 1. Data of Coefficients of Processing Times of Tasks, Limits on Batch Sizes of Units for Motivating Example unit (j)
RI
βI
heater reactor 1 reactor 2 reactor 1 reactor 2 reactor 1 reactor 2 separator
0.667 1.334 1.334 1.334 1.334 0.667 0.667 1.3342
0.00667 0.02664 0.01665 0.02664 0.01665 0.01332 0.008325 0.00666
task (i) heating (i ) reaction 1 (i ) (i ) reaction 2 (i ) (i ) reaction 3 (i ) (i ) separation (i )
1) 2) 3) 4) 5) 6) 7) 8)
Table 3. Data of Coefficients of Processing Times of Tasks, Limits on Batch Sizes of Units for Example 1
Bimin (mu) Bimax (mu) -----------------
100 50 80 50 80 50 80 200
task (i) task1 (i ) 1) (i ) 2) task2 (i ) 3) task3 (i ) 4) (i ) 5)
place over multiple events in order to find the global optimal solution. Now, to understand such cases, let us examine the following constraint for different tasks in different units that is used in the I&F model as presented in Shaik et al.23 ts(i, j, n + 1) g ts(i', j', n) + Ri'j′w(i', j', n) + βi'j′b(i', j', n) H(1 - w(i', j', n)) ∀s, i, i', j, j' ∈ suitij, suiti'j′, i * i', j * j', Fsi < 0, Fsi' > 0, ∀n < N(A) It states that the consuming task at the current event should start after the end time of the producing task at the previous event that processes the same state, which need not be true if there is sufficient material for the consuming task to start production, which happens to be the case in the particular instance of Figure 2. The amount of state s5 produced by
unit (j)
Rij
βij
unit 1 unit 2 unit 3
1.333 1.333 1.000
0.01333 0.01333 0.00500
100 150 200
unit 4 unit 5
0.667 0.667
0.00445 0.00445
150 150
Bijmin (mu)
Bijmax (mu)
the recycle stream (i ) 8) at event N4 is not necessary for starting the consuming task (i ) 7) at event N5. So, this constraint needs to be relaxed depending on whether there is sufficient amount for the consuming task to start production or not, which is implicitly realized by allowing the tasks to take place over multiple events. Having realized the requisite for the unit-specific eventbased models to allow tasks to occur and continue over more than one event point for the simpler case of no resources as well, now, we focus on the computational performance of the JLF model in Table 2. It has weak LP relaxation and requires a large number of constraints, nonzeros, and CPU time, compared to the other competitive models of S&K and CBMN. Recently, Janak and Floudas19 proposed additional rigorous bounding constraints and a model involving partial task splitting to improve the computational performance. Because, the JLF model was originally developed to handle
Table 2. Computational Results for Motivating Example under Maximization of Profit for UIS model
events
CPU time (s)
nodes
RMILP ($)
MILP ($)
binary variables
continuous variables
constraints
nonzeros
S&K M&G CBMN (∆t ) 1) (∆t ) 2) (∆t ) 3) I&F
8 8
105.5 507.64
88679 184605
2690.6 2690.6
1962.7 1962.7
84 112
433 609
456 1402
1615 4884
8 8 8 6 7 6
1.82 81.95 207.43 2.16 43.73 322.20
6449 194968 366226 6713 101415 138714
2690.6 3136.3 3136.3 3078.4 3551.8 5284.5
1860.7a 1959a 1962.7 1943.2a 1943.2a 1962.7
56 104 144 34 42 68
170 218 258 140 165 386
189 261 321 267 318 1500
760 1238 1635 859 1046 5874
JLF a
Suboptimal solution.
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 2951 Table 4. Computational Results for Example 1 under Maximization of Profit with UIS model
events
CPU time (s)
nodes
RMILP ($)
S&K M&G CBMN (∆t ) 1) I&F S&F this work (∆n ) 0)
5 5 5 4 4 4
0.05 0.03 0.01 0.01 0.01 0.01
13 2 0 1 1 4
2000.0 2000.0 2000.0 2000.0 2000.0 2000.0
S&K M&G CBMN (∆t ) 1) (∆t ) 2) I&F S&F this work (∆n ) 0)
7 7
0.68 0.76
746 656
3384.3 3548.4
7 7 5 5 5
0.04 0.27 0.02 0.02 0.01
49 479 15 15 15
3361.1 3703.1 3000.0 3000.0 3000.0
S&K M&G CBMN (∆t ) 1) (∆t ) 2) I&F S&F this work (∆n ) 0)
9 9
26.83 29.52
27176 26514
4481.0 4563.8
9 9 6 6 6
0.23 10.32 0.03 0.02 0.03
606 21874 24 28 35
4419.9 5237.6 4000.0 4000.0 4000.0
MILP ($)
binary variables
continuous variables
constraints
nonzeros
40 40 20 10 10 10
215 195 70 48 68 68
192 520 86 69 84 89
642 1425 274 176 239 223
2628.2 2628.2
60 60
315 285
300 760
1000 2350
2600.4a 2628.2 2628.2 2628.2 2628.2
30 55 15 15 15
100 125 62 87 87
124 174 92 107 117
410 650 245 333 308
3463.6 3463.6
80 80
415 375
408 1000
1358 3415
3301.6a 3463.6 3463.6 3463.6 3463.6
40 75 20 20 20
130 165 76 106 106
162 232 115 130 145
546 886 314 427 393
Example 1a (H ) 8) 1840.2 1840.2 1840.2 1840.2 1840.2 1840.2
Example 1b (H ) 10)
Example 1c (H ) 12)
Example 1d (H ) 16) S&K M&G CBMN (∆t ) 2) (∆t ) 3) I&F S&F this work (∆n ) 0) a
12 12
5328.22 37675.13
3408476 17465450
6312.6 6332.8
5038.1 5038.1
110 110
565 510
570 1360
1895 5275
12 12 9 9 9
1086.08 3911.14 1.76 1.46 1.84
1642027 4087336 6596 5487 6261
7737.6 7737.6 6601.5 6600.9 6600.9
5000.0a 5038.1 5038.1 5038.1 5038.1
105 150 35 35 35
225 270 118 163 163
319 409 184 199 229
1240 1680 521 709 648
Suboptimal solution.
the more general case of problems with resource constraints, it can be observed that it does not reduce well, in terms of problem statistics, to the case of no resources. With this motivation, in this study we propose a generic unified modeling approach for short-term scheduling of batch plants using unit-specific event-based continuous-time representation, which (i) can handle problems with resource constraints by allowing tasks to take place over multiple events, (ii) merges both scheduling problems with and without resources into a common framework, and (iii) efficiently reduces to the simpler case of no resources. The proposed unified model for short-term scheduling of batch plants is described in the next section. In section 3, we consider examples of problems with and without resources and evaluate the computational performance of the proposed model against other continuoustime models from literature.
it may end at a later event n + ∆n (∆n ) 1,...), which would be similar to the model of Janak et al.10,15 In the model of Castro et al.7 (CBMN) also such parameter (∆t) was defined. However, unlike in the proposed model, in all the slot-based and global-event-based models,5–7 a task that starts at an event does not end at the same event, leading to a minimum value of one for a parameter similar to ∆n in other models. The mathematical model has the following allocation, capacity, material balance, duration, sequencing, resource balance, and storage related constraints. 2.1. Allocation Constraints. In every equipment unit, at the most, one task can start at each event as given by constraint 1.
∑ ∑ i∈Ij
w(i, n, n') e 1
∀j ∈ J, n ∈ N
(1)
n'∈N nen'en+∆n
2. Mathematical Formulation Nomenclature is given in the appendix of this work. A task which can be performed in different units is considered as multiple separate tasks each suitable in one unit. The three index binary variable w(i,n,n′) defines the assignment of task i that starts at event n and ends at event n′ (n′ g n). To exercise control on the maximum number of multiple events over which a task is allowed to continue, a parameter, ∆n, is defined such that n e n′ e n + ∆n, ∆n ) 0, 1,.... So, the task i that starts at event n may end either at the same event point n (∆n ) 0), which would be similar to I&F model; or
Similarly constraint 2 states that, in every equipment unit at the most one task can end at each event.
∑ i∈Ij
∑
w(i, n, n') e 1∀j ∈ J, n' ∈ N, ∆n > 0 (2)
n∈N n'-∆nenen'
Constraint 3 states that if a task starts at an event, a different task cannot end at the same event when both have the same suitable equipment unit, because only the task that starts at an event can end at the same event.
2952 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
∑ ∑
i'∈Ij
∑
w(i', n', n) e 1 -
n'∈N
w(i, n, n')
n'∈N
i*i' n-∆nen'en
nen'en+∆n
∀i ∈ Ij, j ∈ J, n ∈ N, ∆n > 0 (3)
Constraint 4 states that in a given equipment unit a new task can start only if the total number of tasks that started earlier matches the total number of tasks ending.
∑
Otherwise, if ∆n is nonzero, then the finish time of a task that started at an earlier event is calculated from eqs 10-12. ∀i ∈ I, n ∈ N, ∆n > 0
T f(i, n) g T s(i, n)
(10)
T f(i, n') g T s(i, n) + Riw(i, n, n') + βib(i, n, n') - M(1 - w(i, n, n')) ∀i ∈ I ,n, n' ∈ N, n e n' e n + ∆n, ∆n > 0 (11)
w(i, n, n') e
n'∈N nen'en+∆n
∑
∑
n'∈N
n''∈N
1-
∑ ∑ w(i', n', n'') + j∈Ji i′∈Ij
n' 0 (12)
2.5. Sequencing Constraints. 2.5.1. Same Task in the Same Unit.
∑ ∑ w(i', n'', n')
∀i ∈ I, n ∈ N, n < N (13)
T s(i, n + 1) g T f(i, n)
j∈Ji i'∈Ij
n'∈N,n' 0
∑
+
(14)
n'∈N
n-∆nen'en
2.5.2. Different Tasks in the Same Unit. T s(i, n + 1) g T f(i', n)
∀i, i' ∈ Ij, i * i', j ∈ J, n ∈ N, n < N (15)
2.5.3. Different Tasks in Different Units.
n'∈N
ST(s, n) ) ST0(s) +
n'∈N
b(i, n', n - 1) +
n'∈N
n-1-∆nen'en-1
∑F ∑
∑ ( [
M 1-
∀s ∈ S, n ) 1 (8)
2.4. Duration Constraints. If ∆n ) 0, then the finish time of a task that started at the same event is calculated from eq 9. T f(i, n) ) T s(i, n) + Riw(i, n, n) + βib(i, n, n) ∀i ∈ I, n ∈ N, ∆n ) 0 (9)
(
T s(i, n + 1) g T f(i', n) - M 1 -
∑ n'∈N n-∆nen'en
)
w(i', n', n)
∀s ∈ S, i ∈ Isc, i' ∈ Isp, i ∈ Ij, i' ∈ Ij', i * i', j, j' ∈ J, j * j', n ∈ N, n < N (16) For different tasks that produce or consume the same state, the start time of the consuming task at the next event is enforced to be later than the finish time of the producing task at the current event, provided the producing task is finishing at the current event. If the producing task continues to the next event then the constraint is relaxed as explained in the motivating example. 2.6. Tightening Constraint. The sum of the durations of all tasks suitable in each unit should be less than the scheduling time horizon.
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 2953
∑∑ ∑ i∈Ij n∈N
2.7.2. Sequencing of Utility Related Tasks. Constraint 19 is similar to the constraint for same task in the same unit written for each utility
(Riw(i, n, n') + βib(i, n, n')) e H
n'∈N nen'en+∆n
∀j ∈ J (17)
2.7. Resource Constraints. For scheduling problems involving resource considerations, the following auxiliary constraints need to be added without the need for any additional changes to the basic framework in the proposed model. 2.7.1. Resource Balance. The consumption of each utility at an event by all suitable active tasks is constrained by the maximum availability in eq 18
∑ i∈Iu
[( γiu
∑
∑ n''∈N
w(i, n', n'') -
∑
∑
n′′∈N
)]
w(i, n'', n')
n'∈N,n'67000b >67000c >67000d 6.19 5.29 5.06
17270000 9060850 80602289 14962 12006 10960
S&K M&G CBMN (∆t ) 1) (∆t ) 2) I&F S&F this work (∆n ) 0)
10 10
809.58 8866.64
231810 1402457
10 10 8 8 8
15.04 206.30 16.35 12.45 13.68
33503 315632 32106 22839 25814
a
RMILP ($) Example 1730.9 1730.9 1730.9 1812.1 1730.9 1730.9
Example 3343.4 3343.4 4409.5 3788.3 3301.0 3301.0
4318.8 4579.4 4435.0 4291.7 4291.7
MILP ($)
3658.1a 3738.4 3738.4 3738.4 3738.4
Suboptimal solution. b Relative gap, 1.59%. c Relative gap, 5.12%. d Relative gap, 2.58%.
2954 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
enforced to occur before Tuts (u,n), if the task is active. If any of these tasks are finishing at the previous event, then the end times are enforced to be equal to Tuts (u,n). T f(i, n - 1) g Tuts(u, n) -
[ (
M 1-
∑
∑
n'∈N
n''∈N
w(i, n', n'') -
n'en-1 n'en''en'+∆n
∑
∑
)]
w(i, n'', n')
n′′∈N n'∈N,n'
