- Email: [email protected]
Short-term Scheduling of a Multipurpose Batch Plant Considering Degradation Effects
¨ Anton A. Kiss, Edwin Zondervan, Richard Lakerveld, Leyla Ozkan (Eds.) Proceedings of the 29th European Symposium on Computer Aided Process Engineering c 2019 Elsevier B.V. All rights reserved. June 16th to 19th , 2019, Eindhoven, The Netherlands. http://dx.doi.org/10.1016/B978-0-12-818634-3.50203-4
Short-term Scheduling of a Multipurpose Batch Plant Considering Degradation Effects Ouyang Wua,b , Giancarlo Dalle Avec,d , Iiro Harjunkoskic , Lars Imslandb , Stefan Marco Schneidera , Ala E.F. Bouaswaiga and Matthias Rotha a Automation
Technology, BASF SE, 67056 Ludwigshafen, Germany of Engineering Cybernetics, NTNU, 7491 Trondheim, Norway c ABB Corporate Research Germany, 68526 Ladenburg, Germany d Dept. of Biochemical & Chemical Engineering, TU Dortmund, 44221 Dortmund, Germany [email protected] b Department
Abstract Fouling is a typical type of degradation in the process industries, which results in significant negative effects on the efficiency of plants. This paper considers a case study, where production tasks with different recipes contribute to the batch-to-batch fouling evolution depending on their sequence. The scheduling of the production tasks is improved by explicit consideration of the degradation effects, resulting in an optimized production sequence. A set of continuous-time MILP-based scheduling models integrated with degradation models are employed to formulate this batch scheduling problem. The proposed formulations are tested for different problem sizes of the case study illustrating the effectiveness of each of the proposed approaches. Keywords: multipurpose batch scheduling, sequence-dependent degradation, continuous-time MILP formulation, disjunctive model, precedence model
1. Introduction Multipurpose chemical batch plants are often a challenge to schedule. These processes typically produce a large number of products requiring several processing steps. When scheduling such processes several factors must be taken into account including, but not limited to, process throughput, storage constraints, equipment condition, and maintenance concerns. When creating scheduling models, one of the major decisions is the time representation chosen for the problem. The time representation can either be continuous or discrete. Continuous time problems in general result in far fewer variables than their discrete counterparts and provide an explicit representation of timing decisions. Continuous time scheduling formulations are broadly classified into precedence, single, or multiple time grid based models (M´endez et al., 2006). Continuous time models have been applied to many different scenarios. Castro and Grossmann (2012) used generalized disjunctive programming to derive generic continuous-time scheduling models. One aspect of scheduling that has been gaining attention in literature lately is that of integrating equipment condition into production scheduling. Vieira et al. (2017) studied the optimal planning of a continuous biopharmaceutical process with decaying yield. They formulated the problem as a continuous single-time grid formulation making both production scheduling and maintenance scheduling decisions. Other applications of combined maintenance and production scheduling can be found in Biondi et al. (2017) and Dalle Ave et al. (2019) who studied the joint production and maintenance problem of steel plant scheduling under different conditions. The goal of this work is to investigate the production scheduling of a chemical batch plant with unit degradation. A batch reactor case study considering fouling is presented in which the fouling
1214
O. Wu et al.
prolongs the time needed to complete a batch and is highly influenced by the sequence of products produced in the reactor. The problem is formulated using a set of continuous-time MILP-based scheduling models which will be described and compared in the remainder of this work.
2. Problem description This case study focuses on a batch polymerization process. Firstly, the product monomers are mixed in a single tank before being homogenized and dispatched to the reactors. Once the monomers have reacted they are dispatched via a shared piping system to storage. The major cause of degradation in this process is fouling in the reactors; polymer residues are accumulated on the inner surfaces of the reactors, heat exchangers, and pipes. This causes reduced heat transfer from the product to the coolant thereby decreasing cooling efficiency and prolonging batch duration. Moreover, the flow resistance due to fouling leads to an increased pressure drop over the heat exchanger. A pressure-based key performance indicator (KPI) was developed to indicate the degree of fouling for each batch run using a state estimation approach, and in this approach many interfering factors due to batch production are excluded from the fouling KPI (Wu et al., 2019). The batch-to-batch behaviors of the fouling evolution discussed in Wu et al. (2018, 2019) indicate that different batch recipes contribute to the fouling evolution differently. Hence, a linear model based the fouling KPI is developed to describe the effects of batch sequences on the batch-to-batch fouling growth, given as: fk, j = ARk−1, j j · fk−1, j + BRk−1, j
j
(1)
where, Rk, j = r indicates Recipe r at the kth batch of Unit j; fk, j is the fouling KPI for the kth batch of Unit j, which is assumed to follow recipe-based unit-specific linear dynamics {(Ar j , Br j ) | r = 1, 2, ..., |R|, j = 1, 2, ..., |J|}. The recipe determines the total processing time at each unit, and the reaction duration, which is also affected by the coolant temperature and the fouling. The degree of fouling is approximated using eq. (1), and the coolant temperature is considered as an external disturbance due to uncertainties among the cooling capacities and the demands. Moreover, the reactors and other units are not identical, which also leads to differences in the processing time. A linear model structure is applied to fit the processing time as a function of the fouling KPI and the coolant temperature given the recipe Rk, j in Unit j, the coolant temperature is assumed to be fixed for the scheduling scenario. Therefore, the processing time is calculated using only the fouling KPI as eq. (2) shows: Duk, j = ADRk, j j · fk, j + BDRk, j
j
(2)
where, Duk, j is the reaction processing time at the kth batch of Unit j; {ADr j , BDr j } are the model parameters for Recipe r at Unit j. The degradation impairs the batch production capacity and therefore needs to be explicitly considered in production scheduling. Provided the quantitative models in eqs. (1) and (2), a new scheduling problem considering degradation effects is formulated by integrating the degradation models into existing scheduling optimization approaches. In the next section, continuous-time MILP approaches using precedence concepts are presented for the aforementioned batch scheduling problem.
3. Methodology In this section, four continuous-time MILP scheduling formulations are presented which integrate the sequence-dependent degradation model using precedence-based constraints. Furthermore, no changeover time and no intermediate storage are considered for the process specifications. Note that while this work focuses on a particular case study, the methodology presented here is generic and could be adapted to other scenarios.
Short-term Scheduling of a Multipurpose Batch Plant Considering Degradation Effects
1215
3.1. Nomenclature I, J, L, R Jl , Jde , Lde T sil , Teil T pi j , Trl Tai j fi j , FI j , Rri Ar j , Br j ADr j , BDr j Xiiui j , Wi j Xiiim l , Yi j , Wi j Xiig l , Yi j Si j
Sets of batch orders (i = 1, 2, ..., |I|), units ( j = 1, 2, ..., |J|), stages (l = 1, 2, ..., |L|) and batch recipes (r = 1, 2, ..., |R|), Sets of units belonging to Stage l, units suffered from degradation and stages containing units suffered from degradation, Start time and end time of Order i in Stage l, Processing time of Order i at Unit j, transfer time in Stage l, Additional processing time of Order i at Unit j due to degradation, Degradation (fouling) KPI for Order i at Unit j, initial fouling KPI at Unit j, recipe binary indicator for Order i using Recipe r, Degradation model parameters for Recipe r at Unit j, see eq. (1), Duration model parameters for Recipe r at Unit j, see eq. (2), Sequencing binary variables for a unit-specific immediate precedence model, Sequencing binary variables for an immediate precedence model, Sequencing binary variables for a general precedence model, Absolute position of Order i in the sequence of Unit j,
3.2. Sequence-dependent degradation The continuous-time immediate precedence fits the form of the sequence-dependent degradation model as shown in eq. (1). Using a unit-specific immediate precedence model, the degradation initialization and propagation are presented in the form of disjunctive constraints as eqs. (3) and (4) show: if Order i is immediately processed after Order i at Unit j (Xiui i j = 1), then the batch degradation KPI ( fi j ) is calculated from its previous value ( fi j ) using eq. (1); if Order i is processed first in the sequence of unit j (Wi j = 1), then fi j equals an initial value of the fouling KPI. To avoid nonlinear constraints, the processing time of Order i at Unit j is further divided into a varying duration due to fouling (Tai j ) and a constant duration (T pi j ) according to eq. (2), where the calculations of Tai j and T pi j are presented in eq. (4). ⎡ ⎤ ⎡ ⎤ Xiui i j Wi j ⎣ fi j ASi j · fi j + BSi j ⎦ , ⎣ fi j FI j ⎦ (3) i∈I, j∈Jde i,i ∈I:i =i , j∈Jde fi j FI j fi j ASi j · fi j + BSi j
Tai j = ∑r∈R Rri · ADr j · fi j , T pi j = ∑r∈R Rri · BDr j , ∀i ∈ I, j ∈ Jde
(4)
where, ASi j = ∑r∈R Rri Ar j , and BSi j = ∑r∈R Rri Br j . 3.3. Unit-specific immediate precedence model [M1] A single-stage unit-specific immediate precedence model is proposed in Cerd´a et al. (1997), and an adapted version for multistage problems considering unit degradation is presented as follows: given the sequence binaries (Xiui i j = 1, if Order i is processed immediate after Order i at Unit j, Wi j = 1 if Order i is processed in the first place of the sequence at Unit j), the unit-specific immediate precedence constraints are presented in eqs. (5) and (6); the order timing constraints for a single order are presented in eq. (7); the order sequencing and timing constraints for two orders in a single unit are presented using disjunctive constraints as shown in eq. (8). The order sequencing and timing constraints for two orders in neighboring stages considering no immediate storage are presented in eq. (9) (the transfer procedure occupies both units in the two stages); the degradation and duration model constraints are presented in eqs. (3) and (4). ∑i∈I Wi j = 1, ∀ j ∈ J ∑
j ∈Jl
(5)
Wi j + ∑i ∈I:i =i, j ∈Jl Xiui i j = 1, ∑i ∈I:i =i, j ∈Jl Xiiui j 1, ∀ i ∈ I, l ∈ L
Teil =
T sil + ∑ j∈Jl [T pi j (Wi j + ∑i ∈I:i =i Xiui i j ) + Tai j ],
(6)
T si(l+1) Teil + Trl , ∀ i ∈ I, l ∈ L (7)
O. Wu et al.
1216
i,i ∈I:i =i , l∈L
∑ j∈Jl Xiui i j Tei l + Trl T sil − Tr(l−1)
(8)
i,i ∈I:i =i , l∈L:l<|I|
∑ j∈Jl Xiui i j T sil T si (l+1) + Trl
(9)
3.4. Immediate precedence model [M2] In contrast to M1, the immediate precedence model (Gupta and Karimi, 2003) uses two types of im
binary variables {Xiim
il ,Yi j } (Xi il = 1 indicates Order i is immediately processed after Order i in some units of Stage l; Yi j = 1 indicates Order i is assigned to Unit j), which fits into the sequencedependent degradation in eq. (3) by replacing Xiui i j with Xiim
il ∧Yi j ∧Yi j . The order assignment and immediate precedence constraints are presented in eqs. (5), (10) and (11), and the order timing constraints for a single order are presented in eq. (12). Lastly, the order sequencing and timing constraints for two orders in a single unit and neighboring stages are presented in eq. (13). im ∑ j∈Jl Yi j = 1, ∑ j∈Jl Wi j + ∑i ∈I:i =i Xiim
il = 1, ∑i ∈I:i =i,l∈L Xii l 1, ∀ i ∈ I, l ∈ L
(10)
Yi j Yi j + 1 − Xiiim l − Xiim
il , ∀ i, i ∈ I : i = i , j ∈ Jl , l ∈ L
(11)
Teil = T sil + ∑ j∈Jl (T pi jYi j + Tai j ), T si(l+1) Teil + Trl , ∀ i ∈ I, l ∈ L ⎡ ⎤ Xiim
il Xiim
il ⎣ Tei l + Trl T sil − ⎦ , T sil T si (l+1) + Trl i,i ∈I:i =i , l∈L i,i ∈I:i =i ,l∈L:l<|I| Tr(l−1)
(12) (13)
3.5. Hybrid precedence model [M3] When comparing to M1 and M2, general precedence models proposed by M´endez and Cerd´a (2003) generally prevail with less computation cost. These models use binary variables Xig il = 1, Yi j = 1, Yi j = 1 to represent that Order i is processed after i at Unit j in Stage l. But unlike M1 and M2, it does not provide immediate precedence relations. Given that the sequence-dependent degradation only occurs in certain stages, a hybrid precedence model is developed by replacing M2 with general precedence constraints in the stages requiring no immediate precedence information (l ∈ Ln = L ∧ ¬Lde ). The precedence constraints are presented in eq. (14), and the order timing constraints for a single order are the same as M2 as eq. (12) shows. The order sequencing and timing constraints for two orders in the same stage or neighboring stages are presented in eq. (15). g g ∑ j∈Jl Yi j = 1, Xii l + Xi il = 1, ∀i, i ∈ I : i = i , l ∈ L − Lde ⎡ ⎤ Xig il ∧Yi j ∧Yi j Xig il ⎣ Tei l + Trl T sil − Tr(l−1) ⎦ , T si l + Trl T sil i,i ∈I:i =i,l∈Ln i, i ∈ I : i = i, T sil T si (l+1) + Trl
(14) (15)
j ∈ Jl , l ∈ Ln
3.6. General precedence based model [M4] A general precedence based model for sequence-dependent changeovers is proposed in Aguirre et al. (2012, 2017), which also can be adapted to the scheduling problem considering sequencedependent degradation. The general precedence model presented in M3 is extended to all stages, and the immediate precedence binary variables Xiim
il , Wi j are introduced for the stages and units having sequence-dependent behaviors ( j ∈ Jde , l ∈ Lde ). By introducing a continuous variable Si j , defined as the absolute position of batch i in the sequence of Unit j, the immediate precedence variables are connected to the general precedence model using eq. (16). ⎡ ⎤ Xig il ∧Yi j ∧Yi j ⎣ ⎦ , Yi j Si j ∑i∈I Yi j , ∀i ∈ I, j ∈ Jde Si j Si j + 1 (16) i,i ∈I:i =i , j∈Jl ,l∈Lde Xiim
il + Si j − 1 Si j + 1
Short-term Scheduling of a Multipurpose Batch Plant Considering Degradation Effects
1217
3.7. Big-M reformulation and objective function The disjunctive programming constraints in the precedence models are reformulated into the MILP frameworks using the big-M approach. More details can be found in Castro and Grossmann (2012). This objective function considered in this work is the minimization of makespan considering fouling-influenced batch time.
4. Case study: computational results Size 12(6,6) 15(7,8) 18(7,8,3) 25(6,5,4,5,5)
M1 3495.02 (53%) 4315.79 (70%) 5164.13 (79%) 7237.93 (91%)
M2 3495.02(74sec) 4266.73(979sec) 5589.43(84%) 13339.37(94%)
M3 3495.02(37%) 4282.61(42%) 5116.97(46%) 7034.91(51%)
M4 3495.02(37%) 4266.73(42%) 5164.13(47%) 7375.89(53%)
Table 1: Computational results of different formulations: objective function and optimality gap Referring to the aforementioned case studies, the monomer make-up section and the reaction section are taken as the two production stages with one monomer vessel (U1) in Stage L1 and two non-identical reactors (U2 and U3) in Stage L2. The process topology can be viewed in Figure 1. Due to the lack of intermediate storage, the homogenization section (L1-Tr) is modeled as a transfer procedure between the stages. This transfer occupies both the discharging mixer as well as the reactor that is being charged, which is modeled using eq. (9) for model M1. Additionally, the only pipeline (L2-Tr) transfers product from either U2 or U3 into storage. To prevent multiple orders overlapping in L2-Tr, the ordering and timing constraints are used in a similar way as shown in eq. (8) by introducing an artificial precedence binary for this unit. In this case study, when minimizing makespan, it is not possible for jobs to overtake one another in L2 and therefore the precedence binary variable for L2-Tr same one for L1. In addition, an operational threshold is added on the degradation indicator.
Figure 1: The topology of of the batch process Each of the four models defined earlier were tested for varying problem sizes. Take 15(7,8) for example, 15 refers to the total amount of product demand, and the values inside the parentheses (7,8) refer to the demands for different recipes. The parameters of the scheduling problem are obtained from the case study and the historical data. The models are implemented in GAMS 25.1 and solved using CPLEX 12.8, and the time limit is set to 1200 CPU seconds. The Gannt chart and fouling curves for the 25-order case using M3 are illustrated in fig. 2; the normalized fouling KPI data are presented for two units (U2 and U3) with different symbols denoting the recipes from R1 to R5, and the threshold for the normalized fouling KPI is one. Furthermore, the computational results are presented in table 1 with objective values and relative optimality gaps. Results show that the small-size problems using M2 are solved to zero gap before 20 minutes but the computation cost increases dramatically with larger problem size; M3 has better computational performance (smaller value of objective function and smaller relative optimality gap) for the problem sizes 18(7,8,3) and 25(6,5,4,5,5).
5. Conclusions A short-term batch scheduling problem considering sequence-dependent degradation was formulated using different continuous-time MILP methods. Taking an industrial case study as example,
1218
O. Wu et al. Gantt Chart
Batch-to-batch Fouling Evolution
0.4
U2 U3 R1 R2 R3 R4 R5
Unit
U3
7
8
15
U2
2
13
4
14
3
6
22
1
12
19
20
21
23
16
17
25
5
10
9
24
11
18
L1-Tr
U1
Fouling KPI (%)
L2-Tr 0.3
0.2
0.1
0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
1
Time
2
Time
3
4
Figure 2: Result of the 25-order problem using M3 the recipe-based degradation models are integrated with the precedence models to improve the scheduling of batch production. Four integrated MILP formulations are compared by running the scheduling models with different problem sizes, and the hybrid precedence model is shown to have the best computation performance for relatively large-size problems. However, the investigated formulations do not find a provably optimal solution within 20 minutes; future work involves improving the computational performance of such models. Acknowledgments: Financial support is gratefully acknowledged from the Marie Skłodowska Curie Horizon 2020 EID-ITN project “PRONTO”, Grant agreement No 675215.
References A. M. Aguirre, S. Liu, L. G. Papageorgiou, 2017. Mixed integer linear programming based approaches for mediumterm planning and scheduling in multiproduct multistage continuous plants. Industrial & Engineering Chemistry Research 56 (19), 5636–5651. A. M. Aguirre, C. A. M´endez, P. M. Castro, C. De Prada, 2012. Milp-based approach for the scheduling of automated manufacturing system with sequence-dependent transferring times. In: Computer Aided Chemical Engineering. Vol. 30. Elsevier, pp. 477–481. M. Biondi, G. Sand, I. Harjunkoski, 2017. Optimization of multipurpose process plant operations: A multi-time-scale maintenance and production scheduling approach. Computers & Chemical Engineering 99, 325 – 339. P. M. Castro, I. E. Grossmann, 2012. Generalized disjunctive programming as a systematic modeling framework to derive scheduling formulations. Industrial & Engineering Chemistry Research 51 (16), 5781–5792. J. Cerd´a, G. P. Henning, I. E. Grossmann, 1997. A mixed-integer linear programming model for short-term scheduling of single-stage multiproduct batch plants with parallel lines. Industrial & Engineering Chemistry Research 36 (5), 1695–1707. G. Dalle Ave, J. Hernandez, I. Harjunkoski, L. Onofri, S. Engell, 2019. Demand side management scheduling formulation for a steel plant considering electrode degradation. In: IFAC International Symposium on Dynamics and Control of Process Systems (DYCOPS). S. Gupta, I. Karimi, 2003. An improved milp formulation for scheduling multiproduct, multistage batch plants. Industrial & engineering chemistry research 42 (11), 2365–2380. C. A. M´endez, J. Cerd´a, 2003. An milp continuous-time framework for short-term scheduling of multipurpose batch processes under different operation strategies. Optimization and Engineering 4 (1-2), 7–22. C. A. M´endez, J. Cerd´a, I. E. Grossmann, I. Harjunkoski, M. Fahl, 2006. State-of-the-art review of optimization methods for short-term scheduling of batch processes. Computers & Chemical Engineering 30 (6-7), 913–946. M. Vieira, T. Pinto-Varela, A. P. Barbosa-Pvoa, 2017. Production and maintenance planning optimisation in biopharmaceutical processes under performance decay using a continuous-time formulation: A multi-objective approach. Computers & Chemical Engineering 107, 111 – 139. O. Wu, A. E. F. Bouaswaig, S. M. Schneider, F. Moreno Leira, L. Imsland, M. Roth, 2018. Data-driven degradation model for batch processes: a case study on heat exchanger fouling. In: Computer Aided Chemical Engineering. Vol. 43. Elsevier, pp. 139–144. O. Wu, L. Imsland, E. Brekke, S. M. Schneider, A. E. F. Bouaswaig, M. Roth, 2019. Robust state estimation for fouling evolution in batch processes using the em algorithm. In: IFAC International Symposium on Dynamics and Control of Process Systems (DYCOPS).
Short-term Scheduling of a Multipurpose Batch Plant Considering Degradation Effects Ouyang Wua,b , Giancarlo Dalle Avec,d , Iiro Harjunkoskic , Lars Imslandb , Stefan Marco Schneidera , Ala E.F. Bouaswaiga and Matthias Rotha a Automation
Technology, BASF SE, 67056 Ludwigshafen, Germany of Engineering Cybernetics, NTNU, 7491 Trondheim, Norway c ABB Corporate Research Germany, 68526 Ladenburg, Germany d Dept. of Biochemical & Chemical Engineering, TU Dortmund, 44221 Dortmund, Germany [email protected] b Department
Abstract Fouling is a typical type of degradation in the process industries, which results in significant negative effects on the efficiency of plants. This paper considers a case study, where production tasks with different recipes contribute to the batch-to-batch fouling evolution depending on their sequence. The scheduling of the production tasks is improved by explicit consideration of the degradation effects, resulting in an optimized production sequence. A set of continuous-time MILP-based scheduling models integrated with degradation models are employed to formulate this batch scheduling problem. The proposed formulations are tested for different problem sizes of the case study illustrating the effectiveness of each of the proposed approaches. Keywords: multipurpose batch scheduling, sequence-dependent degradation, continuous-time MILP formulation, disjunctive model, precedence model
1. Introduction Multipurpose chemical batch plants are often a challenge to schedule. These processes typically produce a large number of products requiring several processing steps. When scheduling such processes several factors must be taken into account including, but not limited to, process throughput, storage constraints, equipment condition, and maintenance concerns. When creating scheduling models, one of the major decisions is the time representation chosen for the problem. The time representation can either be continuous or discrete. Continuous time problems in general result in far fewer variables than their discrete counterparts and provide an explicit representation of timing decisions. Continuous time scheduling formulations are broadly classified into precedence, single, or multiple time grid based models (M´endez et al., 2006). Continuous time models have been applied to many different scenarios. Castro and Grossmann (2012) used generalized disjunctive programming to derive generic continuous-time scheduling models. One aspect of scheduling that has been gaining attention in literature lately is that of integrating equipment condition into production scheduling. Vieira et al. (2017) studied the optimal planning of a continuous biopharmaceutical process with decaying yield. They formulated the problem as a continuous single-time grid formulation making both production scheduling and maintenance scheduling decisions. Other applications of combined maintenance and production scheduling can be found in Biondi et al. (2017) and Dalle Ave et al. (2019) who studied the joint production and maintenance problem of steel plant scheduling under different conditions. The goal of this work is to investigate the production scheduling of a chemical batch plant with unit degradation. A batch reactor case study considering fouling is presented in which the fouling
1214
O. Wu et al.
prolongs the time needed to complete a batch and is highly influenced by the sequence of products produced in the reactor. The problem is formulated using a set of continuous-time MILP-based scheduling models which will be described and compared in the remainder of this work.
2. Problem description This case study focuses on a batch polymerization process. Firstly, the product monomers are mixed in a single tank before being homogenized and dispatched to the reactors. Once the monomers have reacted they are dispatched via a shared piping system to storage. The major cause of degradation in this process is fouling in the reactors; polymer residues are accumulated on the inner surfaces of the reactors, heat exchangers, and pipes. This causes reduced heat transfer from the product to the coolant thereby decreasing cooling efficiency and prolonging batch duration. Moreover, the flow resistance due to fouling leads to an increased pressure drop over the heat exchanger. A pressure-based key performance indicator (KPI) was developed to indicate the degree of fouling for each batch run using a state estimation approach, and in this approach many interfering factors due to batch production are excluded from the fouling KPI (Wu et al., 2019). The batch-to-batch behaviors of the fouling evolution discussed in Wu et al. (2018, 2019) indicate that different batch recipes contribute to the fouling evolution differently. Hence, a linear model based the fouling KPI is developed to describe the effects of batch sequences on the batch-to-batch fouling growth, given as: fk, j = ARk−1, j j · fk−1, j + BRk−1, j
j
(1)
where, Rk, j = r indicates Recipe r at the kth batch of Unit j; fk, j is the fouling KPI for the kth batch of Unit j, which is assumed to follow recipe-based unit-specific linear dynamics {(Ar j , Br j ) | r = 1, 2, ..., |R|, j = 1, 2, ..., |J|}. The recipe determines the total processing time at each unit, and the reaction duration, which is also affected by the coolant temperature and the fouling. The degree of fouling is approximated using eq. (1), and the coolant temperature is considered as an external disturbance due to uncertainties among the cooling capacities and the demands. Moreover, the reactors and other units are not identical, which also leads to differences in the processing time. A linear model structure is applied to fit the processing time as a function of the fouling KPI and the coolant temperature given the recipe Rk, j in Unit j, the coolant temperature is assumed to be fixed for the scheduling scenario. Therefore, the processing time is calculated using only the fouling KPI as eq. (2) shows: Duk, j = ADRk, j j · fk, j + BDRk, j
j
(2)
where, Duk, j is the reaction processing time at the kth batch of Unit j; {ADr j , BDr j } are the model parameters for Recipe r at Unit j. The degradation impairs the batch production capacity and therefore needs to be explicitly considered in production scheduling. Provided the quantitative models in eqs. (1) and (2), a new scheduling problem considering degradation effects is formulated by integrating the degradation models into existing scheduling optimization approaches. In the next section, continuous-time MILP approaches using precedence concepts are presented for the aforementioned batch scheduling problem.
3. Methodology In this section, four continuous-time MILP scheduling formulations are presented which integrate the sequence-dependent degradation model using precedence-based constraints. Furthermore, no changeover time and no intermediate storage are considered for the process specifications. Note that while this work focuses on a particular case study, the methodology presented here is generic and could be adapted to other scenarios.
Short-term Scheduling of a Multipurpose Batch Plant Considering Degradation Effects
1215
3.1. Nomenclature I, J, L, R Jl , Jde , Lde T sil , Teil T pi j , Trl Tai j fi j , FI j , Rri Ar j , Br j ADr j , BDr j Xiiui j , Wi j Xiiim l , Yi j , Wi j Xiig l , Yi j Si j
Sets of batch orders (i = 1, 2, ..., |I|), units ( j = 1, 2, ..., |J|), stages (l = 1, 2, ..., |L|) and batch recipes (r = 1, 2, ..., |R|), Sets of units belonging to Stage l, units suffered from degradation and stages containing units suffered from degradation, Start time and end time of Order i in Stage l, Processing time of Order i at Unit j, transfer time in Stage l, Additional processing time of Order i at Unit j due to degradation, Degradation (fouling) KPI for Order i at Unit j, initial fouling KPI at Unit j, recipe binary indicator for Order i using Recipe r, Degradation model parameters for Recipe r at Unit j, see eq. (1), Duration model parameters for Recipe r at Unit j, see eq. (2), Sequencing binary variables for a unit-specific immediate precedence model, Sequencing binary variables for an immediate precedence model, Sequencing binary variables for a general precedence model, Absolute position of Order i in the sequence of Unit j,
3.2. Sequence-dependent degradation The continuous-time immediate precedence fits the form of the sequence-dependent degradation model as shown in eq. (1). Using a unit-specific immediate precedence model, the degradation initialization and propagation are presented in the form of disjunctive constraints as eqs. (3) and (4) show: if Order i is immediately processed after Order i at Unit j (Xiui i j = 1), then the batch degradation KPI ( fi j ) is calculated from its previous value ( fi j ) using eq. (1); if Order i is processed first in the sequence of unit j (Wi j = 1), then fi j equals an initial value of the fouling KPI. To avoid nonlinear constraints, the processing time of Order i at Unit j is further divided into a varying duration due to fouling (Tai j ) and a constant duration (T pi j ) according to eq. (2), where the calculations of Tai j and T pi j are presented in eq. (4). ⎡ ⎤ ⎡ ⎤ Xiui i j Wi j ⎣ fi j ASi j · fi j + BSi j ⎦ , ⎣ fi j FI j ⎦ (3) i∈I, j∈Jde i,i ∈I:i =i , j∈Jde fi j FI j fi j ASi j · fi j + BSi j
Tai j = ∑r∈R Rri · ADr j · fi j , T pi j = ∑r∈R Rri · BDr j , ∀i ∈ I, j ∈ Jde
(4)
where, ASi j = ∑r∈R Rri Ar j , and BSi j = ∑r∈R Rri Br j . 3.3. Unit-specific immediate precedence model [M1] A single-stage unit-specific immediate precedence model is proposed in Cerd´a et al. (1997), and an adapted version for multistage problems considering unit degradation is presented as follows: given the sequence binaries (Xiui i j = 1, if Order i is processed immediate after Order i at Unit j, Wi j = 1 if Order i is processed in the first place of the sequence at Unit j), the unit-specific immediate precedence constraints are presented in eqs. (5) and (6); the order timing constraints for a single order are presented in eq. (7); the order sequencing and timing constraints for two orders in a single unit are presented using disjunctive constraints as shown in eq. (8). The order sequencing and timing constraints for two orders in neighboring stages considering no immediate storage are presented in eq. (9) (the transfer procedure occupies both units in the two stages); the degradation and duration model constraints are presented in eqs. (3) and (4). ∑i∈I Wi j = 1, ∀ j ∈ J ∑
j ∈Jl
(5)
Wi j + ∑i ∈I:i =i, j ∈Jl Xiui i j = 1, ∑i ∈I:i =i, j ∈Jl Xiiui j 1, ∀ i ∈ I, l ∈ L
Teil =
T sil + ∑ j∈Jl [T pi j (Wi j + ∑i ∈I:i =i Xiui i j ) + Tai j ],
(6)
T si(l+1) Teil + Trl , ∀ i ∈ I, l ∈ L (7)
O. Wu et al.
1216
i,i ∈I:i =i , l∈L
∑ j∈Jl Xiui i j Tei l + Trl T sil − Tr(l−1)
(8)
i,i ∈I:i =i , l∈L:l<|I|
∑ j∈Jl Xiui i j T sil T si (l+1) + Trl
(9)
3.4. Immediate precedence model [M2] In contrast to M1, the immediate precedence model (Gupta and Karimi, 2003) uses two types of im
binary variables {Xiim
il ,Yi j } (Xi il = 1 indicates Order i is immediately processed after Order i in some units of Stage l; Yi j = 1 indicates Order i is assigned to Unit j), which fits into the sequencedependent degradation in eq. (3) by replacing Xiui i j with Xiim
il ∧Yi j ∧Yi j . The order assignment and immediate precedence constraints are presented in eqs. (5), (10) and (11), and the order timing constraints for a single order are presented in eq. (12). Lastly, the order sequencing and timing constraints for two orders in a single unit and neighboring stages are presented in eq. (13). im ∑ j∈Jl Yi j = 1, ∑ j∈Jl Wi j + ∑i ∈I:i =i Xiim
il = 1, ∑i ∈I:i =i,l∈L Xii l 1, ∀ i ∈ I, l ∈ L
(10)
Yi j Yi j + 1 − Xiiim l − Xiim
il , ∀ i, i ∈ I : i = i , j ∈ Jl , l ∈ L
(11)
Teil = T sil + ∑ j∈Jl (T pi jYi j + Tai j ), T si(l+1) Teil + Trl , ∀ i ∈ I, l ∈ L ⎡ ⎤ Xiim
il Xiim
il ⎣ Tei l + Trl T sil − ⎦ , T sil T si (l+1) + Trl i,i ∈I:i =i , l∈L i,i ∈I:i =i ,l∈L:l<|I| Tr(l−1)
(12) (13)
3.5. Hybrid precedence model [M3] When comparing to M1 and M2, general precedence models proposed by M´endez and Cerd´a (2003) generally prevail with less computation cost. These models use binary variables Xig il = 1, Yi j = 1, Yi j = 1 to represent that Order i is processed after i at Unit j in Stage l. But unlike M1 and M2, it does not provide immediate precedence relations. Given that the sequence-dependent degradation only occurs in certain stages, a hybrid precedence model is developed by replacing M2 with general precedence constraints in the stages requiring no immediate precedence information (l ∈ Ln = L ∧ ¬Lde ). The precedence constraints are presented in eq. (14), and the order timing constraints for a single order are the same as M2 as eq. (12) shows. The order sequencing and timing constraints for two orders in the same stage or neighboring stages are presented in eq. (15). g g ∑ j∈Jl Yi j = 1, Xii l + Xi il = 1, ∀i, i ∈ I : i = i , l ∈ L − Lde ⎡ ⎤ Xig il ∧Yi j ∧Yi j Xig il ⎣ Tei l + Trl T sil − Tr(l−1) ⎦ , T si l + Trl T sil i,i ∈I:i =i,l∈Ln i, i ∈ I : i = i, T sil T si (l+1) + Trl
(14) (15)
j ∈ Jl , l ∈ Ln
3.6. General precedence based model [M4] A general precedence based model for sequence-dependent changeovers is proposed in Aguirre et al. (2012, 2017), which also can be adapted to the scheduling problem considering sequencedependent degradation. The general precedence model presented in M3 is extended to all stages, and the immediate precedence binary variables Xiim
il , Wi j are introduced for the stages and units having sequence-dependent behaviors ( j ∈ Jde , l ∈ Lde ). By introducing a continuous variable Si j , defined as the absolute position of batch i in the sequence of Unit j, the immediate precedence variables are connected to the general precedence model using eq. (16). ⎡ ⎤ Xig il ∧Yi j ∧Yi j ⎣ ⎦ , Yi j Si j ∑i∈I Yi j , ∀i ∈ I, j ∈ Jde Si j Si j + 1 (16) i,i ∈I:i =i , j∈Jl ,l∈Lde Xiim
il + Si j − 1 Si j + 1
Short-term Scheduling of a Multipurpose Batch Plant Considering Degradation Effects
1217
3.7. Big-M reformulation and objective function The disjunctive programming constraints in the precedence models are reformulated into the MILP frameworks using the big-M approach. More details can be found in Castro and Grossmann (2012). This objective function considered in this work is the minimization of makespan considering fouling-influenced batch time.
4. Case study: computational results Size 12(6,6) 15(7,8) 18(7,8,3) 25(6,5,4,5,5)
M1 3495.02 (53%) 4315.79 (70%) 5164.13 (79%) 7237.93 (91%)
M2 3495.02(74sec) 4266.73(979sec) 5589.43(84%) 13339.37(94%)
M3 3495.02(37%) 4282.61(42%) 5116.97(46%) 7034.91(51%)
M4 3495.02(37%) 4266.73(42%) 5164.13(47%) 7375.89(53%)
Table 1: Computational results of different formulations: objective function and optimality gap Referring to the aforementioned case studies, the monomer make-up section and the reaction section are taken as the two production stages with one monomer vessel (U1) in Stage L1 and two non-identical reactors (U2 and U3) in Stage L2. The process topology can be viewed in Figure 1. Due to the lack of intermediate storage, the homogenization section (L1-Tr) is modeled as a transfer procedure between the stages. This transfer occupies both the discharging mixer as well as the reactor that is being charged, which is modeled using eq. (9) for model M1. Additionally, the only pipeline (L2-Tr) transfers product from either U2 or U3 into storage. To prevent multiple orders overlapping in L2-Tr, the ordering and timing constraints are used in a similar way as shown in eq. (8) by introducing an artificial precedence binary for this unit. In this case study, when minimizing makespan, it is not possible for jobs to overtake one another in L2 and therefore the precedence binary variable for L2-Tr same one for L1. In addition, an operational threshold is added on the degradation indicator.
Figure 1: The topology of of the batch process Each of the four models defined earlier were tested for varying problem sizes. Take 15(7,8) for example, 15 refers to the total amount of product demand, and the values inside the parentheses (7,8) refer to the demands for different recipes. The parameters of the scheduling problem are obtained from the case study and the historical data. The models are implemented in GAMS 25.1 and solved using CPLEX 12.8, and the time limit is set to 1200 CPU seconds. The Gannt chart and fouling curves for the 25-order case using M3 are illustrated in fig. 2; the normalized fouling KPI data are presented for two units (U2 and U3) with different symbols denoting the recipes from R1 to R5, and the threshold for the normalized fouling KPI is one. Furthermore, the computational results are presented in table 1 with objective values and relative optimality gaps. Results show that the small-size problems using M2 are solved to zero gap before 20 minutes but the computation cost increases dramatically with larger problem size; M3 has better computational performance (smaller value of objective function and smaller relative optimality gap) for the problem sizes 18(7,8,3) and 25(6,5,4,5,5).
5. Conclusions A short-term batch scheduling problem considering sequence-dependent degradation was formulated using different continuous-time MILP methods. Taking an industrial case study as example,
1218
O. Wu et al. Gantt Chart
Batch-to-batch Fouling Evolution
0.4
U2 U3 R1 R2 R3 R4 R5
Unit
U3
7
8
15
U2
2
13
4
14
3
6
22
1
12
19
20
21
23
16
17
25
5
10
9
24
11
18
L1-Tr
U1
Fouling KPI (%)
L2-Tr 0.3
0.2
0.1
0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
1
Time
2
Time
3
4
Figure 2: Result of the 25-order problem using M3 the recipe-based degradation models are integrated with the precedence models to improve the scheduling of batch production. Four integrated MILP formulations are compared by running the scheduling models with different problem sizes, and the hybrid precedence model is shown to have the best computation performance for relatively large-size problems. However, the investigated formulations do not find a provably optimal solution within 20 minutes; future work involves improving the computational performance of such models. Acknowledgments: Financial support is gratefully acknowledged from the Marie Skłodowska Curie Horizon 2020 EID-ITN project “PRONTO”, Grant agreement No 675215.
References A. M. Aguirre, S. Liu, L. G. Papageorgiou, 2017. Mixed integer linear programming based approaches for mediumterm planning and scheduling in multiproduct multistage continuous plants. Industrial & Engineering Chemistry Research 56 (19), 5636–5651. A. M. Aguirre, C. A. M´endez, P. M. Castro, C. De Prada, 2012. Milp-based approach for the scheduling of automated manufacturing system with sequence-dependent transferring times. In: Computer Aided Chemical Engineering. Vol. 30. Elsevier, pp. 477–481. M. Biondi, G. Sand, I. Harjunkoski, 2017. Optimization of multipurpose process plant operations: A multi-time-scale maintenance and production scheduling approach. Computers & Chemical Engineering 99, 325 – 339. P. M. Castro, I. E. Grossmann, 2012. Generalized disjunctive programming as a systematic modeling framework to derive scheduling formulations. Industrial & Engineering Chemistry Research 51 (16), 5781–5792. J. Cerd´a, G. P. Henning, I. E. Grossmann, 1997. A mixed-integer linear programming model for short-term scheduling of single-stage multiproduct batch plants with parallel lines. Industrial & Engineering Chemistry Research 36 (5), 1695–1707. G. Dalle Ave, J. Hernandez, I. Harjunkoski, L. Onofri, S. Engell, 2019. Demand side management scheduling formulation for a steel plant considering electrode degradation. In: IFAC International Symposium on Dynamics and Control of Process Systems (DYCOPS). S. Gupta, I. Karimi, 2003. An improved milp formulation for scheduling multiproduct, multistage batch plants. Industrial & engineering chemistry research 42 (11), 2365–2380. C. A. M´endez, J. Cerd´a, 2003. An milp continuous-time framework for short-term scheduling of multipurpose batch processes under different operation strategies. Optimization and Engineering 4 (1-2), 7–22. C. A. M´endez, J. Cerd´a, I. E. Grossmann, I. Harjunkoski, M. Fahl, 2006. State-of-the-art review of optimization methods for short-term scheduling of batch processes. Computers & Chemical Engineering 30 (6-7), 913–946. M. Vieira, T. Pinto-Varela, A. P. Barbosa-Pvoa, 2017. Production and maintenance planning optimisation in biopharmaceutical processes under performance decay using a continuous-time formulation: A multi-objective approach. Computers & Chemical Engineering 107, 111 – 139. O. Wu, A. E. F. Bouaswaig, S. M. Schneider, F. Moreno Leira, L. Imsland, M. Roth, 2018. Data-driven degradation model for batch processes: a case study on heat exchanger fouling. In: Computer Aided Chemical Engineering. Vol. 43. Elsevier, pp. 139–144. O. Wu, L. Imsland, E. Brekke, S. M. Schneider, A. E. F. Bouaswaig, M. Roth, 2019. Robust state estimation for fouling evolution in batch processes using the em algorithm. In: IFAC International Symposium on Dynamics and Control of Process Systems (DYCOPS).