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New approaches for scheduling of multitasking multipurpose batch processes in scientific service facilities
Anton Friedl, Jiří J. Klemeš, Stefan Radl, Petar S. Varbanov, Thomas Wallek (Eds.) Proceedings of the 28th European Symposium on Computer Aided Process Engineering June 10th to 13th, 2018, Graz, Austria. © 2018 Elsevier B.V. All rights reserved. https://doi.org/10.1016/B978-0-444-64235-6.50181-9
New approaches for scheduling of multitasking multipurpose batch processes in scientific service facilities Nikolaos Rakovitis, Jie Li,* Nan Zhang School of Chemical Enginneering and Analytical Science,University of Manchester, Manchester M13 9PL,UK [email protected]
Abstract Scheduling of multitasking multipurpose batch industry has not gained adequate attention in the literature. In this work, two novel mathematical models for scheduling of multitasking multipurpose batch processes in scientific service facilities are developed. The first model is developed based on unit-specific event-based approach, while the second is based on task-specific event-based approach. By solving a number of examples with both proposed models and the existing ones in the literature, it seems that both proposed models reduce the model size. The proposed unit-specific eventbased model is superior to others, significantly reducing the model size and requiring at least one order of magnitude less computational time to generate the optimum solution, especially for the case of makespan minimization. Keywords: Scheduling, Multitasking, Scientific service facilities, Mixed-integer linear programming
1. Introduction Scientific service facilities examine a number of samples from different customers for their physical and chemical properties using a number of units. Each unit can examine one property and process samples from more than one customer simultaneously due to its large capacity, which allows multitasking. Due to the high competitive market, scientific service seeks ways to minimize the use of units and raw materials. Even though, process scheduling has been considered in the last three decades (Harjunkoski et al., 2014), most of the existing models only allow single tasking and they cannot be directly applied to this class of problem. Recently, Patil et al. (2015) developed a discrete-time model for scheduling scientific service facilities. Lagzi et al. (2017a) used the global event time approach for the same problem. However, both models require large amount of computational time due to their large model sizes. The advantages of the unit-specific event-based approach have been well established in the literature (Shaik and Floudas, 2009; Li and Floudas, 2010), where the scheduling horizon is divided based on units. Although there are some models that divide the scheduling horizon based on tasks, they are also classified as unit-specific event-based models (Shaik and Floudas 2009), which are actually task specific. In this work, two novel models based on unit-specific and task-specific event-based approaches are developed for this problem. Both models allow multitasking to take
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place in a unit. New tightening constraints are developed for the task-specific eventbased model. The computational results show that the proposed unit-specific eventbased model is superior to the proposed task-specific event-based model and the existing ones (Patil et al., 2015; Lagzi et al., 2017a) with smaller model size and at least one magnitude less computational time to generate the optimum solution, especially for the case of makespan minimization.
2. Problem description A scientific service facility (Figure 1) examines Pr (Pr = 1, 2, …, Pr) properties, using J (J = 1, 2, …, J) units. Each unit is able to process only one property denoted by a set Jpr. The facility has to examine m sample groups, each one for different properties. In each unit more than one groups can be processed simultaneously. Given the total sample groups, the number of samples in each group, the processing path, the scheduling horizon H, as well as the capacity and processing time of each unit, the scheduling problem is to determine the optimal production schedule, including batch sizes, allocations, sequences, timings on processing units. It is assumed that each unit requires a fix time to examine a property, regardless of the samples that are processed. Furthermore, unlimited storage capacity (UIS) for all samples is considered.
Figure 1 STN representation of scientific service facility
3. Mathematical formulations 3.1. Unit-specific event-based model We first develop a novel unit-specific event-based model for scheduling multitasking batch processes. 3.1.1. Allocation constraints A unit must start and end only at one event point.
∑
∑
n −∆n ≤ n ′ ≤ n n ≤ n ′′ ≤ n ′ +∆n
w j , n′, n′′ ≤ 1
∀j , n, ∆n > 0
(1)
3.1.2. Capacity constraints The total batch sizes taking place in a unit should be constrained at an event point.
∑b
i∈I j
i , j , n , n′
≤ B max w j , n , n′ j
∀j , n, n ≤ n′ ≤ n + ∆n
(2)
New approaches for scheduling of multitasking multipurpose batch processes
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3.1.3. Material balance constraints
STs , n = STs , n −1 + ∑ ρi , s ∑
∑
j n −1−∆n ≤ n ′ ≤ n −1
i∈I SP
ST = ST 0 s + ∑ ρi , s ∑ s,n
∑
j n ≤ n ′ ≤ n + ∆n
i∈I SC
bi , j , n′, n −1 + ∑ ρi , s ∑
∑
j n ≤ n ′ ≤ n +∆n
i∈I SC
bi , j , n , n′ ∀s, n > 1 ∀s, n = 1
bi , j , n , n′
(3) (4)
3.1.4. Duration constraints The finish time of unit j must be after the start time of the unit at the same event point plus the processing time required to process task. ∀j , n, n ≤ n′ ≤ n + ∆n
Tf j , n′ ≥ Ts j , n + α j w j , n , n′
(5)
3.1.5. Time matching constraints A state is available to be consumed after the finish time of the production task. Ts , n ≥ Tf j , n − M 1 − ∑ w j , n′, n n −∆n ≤ n′≤ n
∀s ∈ S in , j , n, ∑ ρi , s > 0
(6)
i∈I j
The start time of a unit, that consumes a state s at event point n+1, should be after the time that the state is available to be consumed at event n. Ts , n ≤ Ts j , n +1 + M 1 − w j , n +1, n′ ∑ n +1≤ n′≤ n +1+ ∆n
∀s ∈ S in , j , n < N , ∑ ρi , s < 0
(7)
i∈I j
3.1.6. Sequencing constraints The start time of unit j at event n+1 must be after the finish time at the event n.
∀j , n < N
Ts j , n +1 ≥ Tf j , n
(8)
The time that a state is available at a time event point should be earlier that the time that the same state is available at the next event point. ∀s ∈ S in , n < N
Ts , n ≤ Ts , n +1
(9)
3.1.7. Objective functions Both maximization of productivity during a specific scheduling horizon and minimization of makespan have been considered. z = ∑ ps ∑ ∑∑ s
i∈I SP
∑
n n ≤ n ′ ≤ n + ∆n
j
ρi , s bi , j , n , n′
MS ≥ Tf j , N
∀s ∈ S in , n < N
(10)
∀j
(11)
In the latter case all properties in all samples should be examined at the last event point STs , N + ∑ ρi , s ∑ i∈I SP
j
∑
N −∆n ≤ n ′ ≤ N
bi , j , n′, N ≤ Ds
∀s
(12)
N. Rakovitis et al.
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The task-specific event-based model of Shaik and Floudas (2009) for scheduling of single-tasking multipurpose batch processes was modified to consider multitasking. In this section, the modified constraints are only presented. 3.2.1. Allocation constraints Shaik and Floudas (2009) introduced several allocation constraints where only one task is processed in a unit in event n. We replace these constraints with a new set of allocation constraints, which allows multitasking.
∑
∑
n −∆n ≤ n ′ ≤ n n ≤ n ′′ ≤ n ′ +∆n
wi , n′, n′′ ≤ 1
∀j , n, ∆n > 0
(13)
A binary variable yj,n,n′ is defined as one if unit j is active from event n to n'.
y j , n, n′ ≥ wi , n, n′
∀i ∈ I j , j , n, n ≤ n′ ≤ n + ∆n
(14)
y j , n , n′ ≤ ∑ wi , n , n′
∀j , n, n ≤ n′ ≤ n + ∆n
(15)
i∈I j
3.2.2. Capacity constraints In multitasking, the summation of the batch sizes of all tasks that are able to be processed in a unit should be constrained.
∑b
i∈I j
i , n , n′
≤ B max y j , n , n′ j
∀j , n, n ≤ n′ ≤ n + ∆n
(16)
3.2.3. Tightening constraints The tightening constraints in Shaik and Floudas (2009) are only effective for single tasking in a unit. New tightening constraints below are introduced for multitasking.
Tjf j , n′ ≥ Tjs j , n + [max (α i )] y j , n , n′
∀j , n, n ≤ n′ ≤ n + ∆n
(17)
Tjs j , n +1 ≥ Tjf j , n
∀j , n < N
(18)
i∈I j
4. Computational results Five examples were solved using proposed models as well as the existing models of Patil et al. (2015) and Lagzi et al. (2017a). In example 1, a small-scale facility presented in Patil et al. (2015) was considered, while for the next 3 examples a random facility was generated. The last example uses a representation of an actual scientific service facility as presented by Lagzi et al. (2017b). Both maximization of productivity and minimization of makespan were considered as objective. All examples were solved in a machine using GAMS 24.6.1. CPLEX 12 in an Intel® Core™ i5-2500 3.3 GHz and 8 GB RAM running windows 7. The comparative results with maximization of productivity are given on Table 1. Both proposed models require less number of event points, which decrease the model size.
New approaches for scheduling of multitasking multipurpose batch processes
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However, the unit-specific event-based model leads to significantly smaller model size and therefore less computational time. The discrete-time model of Patil et al. (2015) also seems competitive due to the fact that it has the tightest relaxation. In cases of very detailed processing times, though such as example 4, the model size makes it infeasible to generate a feasible solution after one hour. By comparing continuous-time models, it seems that the proposed models improve the relaxation of the problem. Table 1 Comparative results for examples 1-5 for maximization of productivity as objective
Ex.
Model
1
La Pb T-Sc U-Sd L P T-S U-S L P T-S U-S L P T-S U-S L P T-S U-S
2
3
4
5
a
Event points 8 96 7 7 8 480 7 7 20 480 17 17 6 5 5 3 480 3 3
CPU time (s) 3600e 0.826 87.94 0.218 162.0 2.886 4.602 0.234 3600f 217.6 3600g 103.0 2000 >3600 10.12 0.125 0.202 7.270 0.046 0.094
RMILP
MILP
4822 3147 3424 3424 1404 1279 1404 1404 3259 2330 2744 2744 4495 3382 3382 846 666 846 846
3047i 3147 3147 3147 1254 1254 1254 1254 1194h 2330 2300h 2330 3100 3100 3100 666 666 666 666
Discrete Variables 2196 5150 770 427 540 6264 154 105 6300 32445 2312 1275 1456 460 260 3744 71220 966 702
Continuous Variables 2709 4417 1219 623 585 6721 337 221 7350 38401 4030 2687 3500 1216 650 4920 64375 1768 669
Constraints 9402 5387 5102 1211 2341 10569 672 348 26491 45135 16946 4715 6424 2730 703 16330 65850 4617 2284
Lagzi et. al. 2017. b Patil et. al. 2015. c Task-specific model. d Unit-specific model. e Relative gap 36.8%. f Relative gap 63.4%. g Relative gap 16.2%. h Suboptimal solution
The results for the same examples using minimization of makespan as objective are presented on Table 2. In that case, the unit-specific event-based model is superior to others with smaller model size and tighter relaxation. Consequently, it generates the optimum solution in at least one order of magnitude less computational time. The superiority of the model is also illustrated in example 3, where the unit-specific eventbased model generates the optimum solution in less than one minute, while reported models are not even able to generate a feasible solution after one hour. The task-specific event-based model has also a better relaxation and leads to smaller model sizes than the models previously reported in the literature.
5. Conclusions In this work, two novel mathematical models, based on task-specific and unit-specific event-based approaches for scheduling of multitasking batch process in scientific service facilities were presented. Both mathematical models lead to smaller model size with less number of event points, variables and constraints than previous reported formulations. The unit-specific event-based model is superior model than other models
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with significantly smaller model, much tighter relaxation and at least one magnitude less computational time to generate optimal solution, especially for the case of makespan minimization. Table 2 Comparative results for examples 1-5 for minimization of makespan as objective
Ex.
Model
1
La Pb T-Sc U-Sd L P T-S U-S L P T-S U-S L P T-S U-S L P T-S U-S
2
3
4
5
Event points 8 250 8 8 8 600 7 7 52 52 7 6 6 13 13 13
CPU time (s) 3600e 93.49 3600f 0.187 119.7 13.74 1.482 0.219 >3600 >3600 3600g 27.58 3600h >3600 2.325 0.281 3600i >3600 3600j 3600k
RMILP
MILP
99.50 23.90 125.13 934.38 88.09 4.02 137.39 307.50 167.75 1562.00 95.15 223.50 840.56 14.25 169.00 624.38
1065.0 1065.0 1065.0 1065.0 555.0 555.0 555.0 555.0 1966.0 1696.0 1001.7 1001.7 1001.7 7431.0 5131.0 5131.0
Discrete Variables 2196 14544 880 488 540 8064 154 105 7072 3900 1664 552 312 13104 4186 3042
Continuous Variables 2709 11501 1393 717 585 6401 337 221 12325 8357 4000 1459 786 18340 7658 3039
Constraints 9411 25636 5978 1816 2343 26642 685 403 53425 18160 7364 3418 1128 59434 24112 12581
a Lagzi et. al. 2017 with objective function MS ≥ TN. b Patil et. al. 2015 with objective function MS ≥ [(t – 1) + Tr(p) – 1]⋅yj,t. c Task-specific model. d Unit-specific model. e Relative gap 25.9%. f Relative gap 25.6%. g Relative gap 89.6%. h Relative gap 19.0% i Relative gap 99.7%. j Relative gap 78.9%. k Relative gap 62.5%.
References I. Harjunkoski, C. Maravelias, P. Bongers, P. Castro, S. Engell, I. Grossmann, J. Hooker, C. Méndez, G. Sand, J. Wassick, 2014, Scope for industrial application of production scheduling models and solution methods, Computers and Chemical Engineering, 62(5), 161-193 S. Lagzi, R. Fukasawa, L. Ricardez-Sandoval, 2017a, A multitasking continuous time formulation for short-term scheduling of operations in multipurpose plants, Computers & Chemical Engineering, 97, 135-146 S. Lagzi, D. Lee, R. Fukasawa, L. Ricardez-Sandoval, 2017b, A Computational Study of Continuous and Discrete Time Formulations for a Class of Short-Term Scheduling Problems for Multipurpose Plants, Industrial & Engineering Chemistry Research, 56(31), 8940-8953 J. Li, C. Floudas, 2010, Optimal event point determination for short-term scheduling of multipurpose batch plants via unit-specific event-based continuous-time approaches, Industrial & Engineering Chemistry Research, 49(16), 7446-7469 B. Patil, R. Fukasawa, L. Ricardez-Sandoval, 2015, Scheduling of operations in a large-scale Scientific services facility via multicommodity flow and an optimization-based algorithm, Industrial & Engineering Chemistry Research, 54(5), 1628-1639 M. Shaik, C. Floudas, 2009, Novel Unified Modeling Approach for Short-Term Scheduling, Industrial & Engineering Chemistry Research, 48(6), 2947-2964
New approaches for scheduling of multitasking multipurpose batch processes in scientific service facilities Nikolaos Rakovitis, Jie Li,* Nan Zhang School of Chemical Enginneering and Analytical Science,University of Manchester, Manchester M13 9PL,UK [email protected]
Abstract Scheduling of multitasking multipurpose batch industry has not gained adequate attention in the literature. In this work, two novel mathematical models for scheduling of multitasking multipurpose batch processes in scientific service facilities are developed. The first model is developed based on unit-specific event-based approach, while the second is based on task-specific event-based approach. By solving a number of examples with both proposed models and the existing ones in the literature, it seems that both proposed models reduce the model size. The proposed unit-specific eventbased model is superior to others, significantly reducing the model size and requiring at least one order of magnitude less computational time to generate the optimum solution, especially for the case of makespan minimization. Keywords: Scheduling, Multitasking, Scientific service facilities, Mixed-integer linear programming
1. Introduction Scientific service facilities examine a number of samples from different customers for their physical and chemical properties using a number of units. Each unit can examine one property and process samples from more than one customer simultaneously due to its large capacity, which allows multitasking. Due to the high competitive market, scientific service seeks ways to minimize the use of units and raw materials. Even though, process scheduling has been considered in the last three decades (Harjunkoski et al., 2014), most of the existing models only allow single tasking and they cannot be directly applied to this class of problem. Recently, Patil et al. (2015) developed a discrete-time model for scheduling scientific service facilities. Lagzi et al. (2017a) used the global event time approach for the same problem. However, both models require large amount of computational time due to their large model sizes. The advantages of the unit-specific event-based approach have been well established in the literature (Shaik and Floudas, 2009; Li and Floudas, 2010), where the scheduling horizon is divided based on units. Although there are some models that divide the scheduling horizon based on tasks, they are also classified as unit-specific event-based models (Shaik and Floudas 2009), which are actually task specific. In this work, two novel models based on unit-specific and task-specific event-based approaches are developed for this problem. Both models allow multitasking to take
N. Rakovitis et al.
1034
place in a unit. New tightening constraints are developed for the task-specific eventbased model. The computational results show that the proposed unit-specific eventbased model is superior to the proposed task-specific event-based model and the existing ones (Patil et al., 2015; Lagzi et al., 2017a) with smaller model size and at least one magnitude less computational time to generate the optimum solution, especially for the case of makespan minimization.
2. Problem description A scientific service facility (Figure 1) examines Pr (Pr = 1, 2, …, Pr) properties, using J (J = 1, 2, …, J) units. Each unit is able to process only one property denoted by a set Jpr. The facility has to examine m sample groups, each one for different properties. In each unit more than one groups can be processed simultaneously. Given the total sample groups, the number of samples in each group, the processing path, the scheduling horizon H, as well as the capacity and processing time of each unit, the scheduling problem is to determine the optimal production schedule, including batch sizes, allocations, sequences, timings on processing units. It is assumed that each unit requires a fix time to examine a property, regardless of the samples that are processed. Furthermore, unlimited storage capacity (UIS) for all samples is considered.
Figure 1 STN representation of scientific service facility
3. Mathematical formulations 3.1. Unit-specific event-based model We first develop a novel unit-specific event-based model for scheduling multitasking batch processes. 3.1.1. Allocation constraints A unit must start and end only at one event point.
∑
∑
n −∆n ≤ n ′ ≤ n n ≤ n ′′ ≤ n ′ +∆n
w j , n′, n′′ ≤ 1
∀j , n, ∆n > 0
(1)
3.1.2. Capacity constraints The total batch sizes taking place in a unit should be constrained at an event point.
∑b
i∈I j
i , j , n , n′
≤ B max w j , n , n′ j
∀j , n, n ≤ n′ ≤ n + ∆n
(2)
New approaches for scheduling of multitasking multipurpose batch processes
1035
3.1.3. Material balance constraints
STs , n = STs , n −1 + ∑ ρi , s ∑
∑
j n −1−∆n ≤ n ′ ≤ n −1
i∈I SP
ST = ST 0 s + ∑ ρi , s ∑ s,n
∑
j n ≤ n ′ ≤ n + ∆n
i∈I SC
bi , j , n′, n −1 + ∑ ρi , s ∑
∑
j n ≤ n ′ ≤ n +∆n
i∈I SC
bi , j , n , n′ ∀s, n > 1 ∀s, n = 1
bi , j , n , n′
(3) (4)
3.1.4. Duration constraints The finish time of unit j must be after the start time of the unit at the same event point plus the processing time required to process task. ∀j , n, n ≤ n′ ≤ n + ∆n
Tf j , n′ ≥ Ts j , n + α j w j , n , n′
(5)
3.1.5. Time matching constraints A state is available to be consumed after the finish time of the production task. Ts , n ≥ Tf j , n − M 1 − ∑ w j , n′, n n −∆n ≤ n′≤ n
∀s ∈ S in , j , n, ∑ ρi , s > 0
(6)
i∈I j
The start time of a unit, that consumes a state s at event point n+1, should be after the time that the state is available to be consumed at event n. Ts , n ≤ Ts j , n +1 + M 1 − w j , n +1, n′ ∑ n +1≤ n′≤ n +1+ ∆n
∀s ∈ S in , j , n < N , ∑ ρi , s < 0
(7)
i∈I j
3.1.6. Sequencing constraints The start time of unit j at event n+1 must be after the finish time at the event n.
∀j , n < N
Ts j , n +1 ≥ Tf j , n
(8)
The time that a state is available at a time event point should be earlier that the time that the same state is available at the next event point. ∀s ∈ S in , n < N
Ts , n ≤ Ts , n +1
(9)
3.1.7. Objective functions Both maximization of productivity during a specific scheduling horizon and minimization of makespan have been considered. z = ∑ ps ∑ ∑∑ s
i∈I SP
∑
n n ≤ n ′ ≤ n + ∆n
j
ρi , s bi , j , n , n′
MS ≥ Tf j , N
∀s ∈ S in , n < N
(10)
∀j
(11)
In the latter case all properties in all samples should be examined at the last event point STs , N + ∑ ρi , s ∑ i∈I SP
j
∑
N −∆n ≤ n ′ ≤ N
bi , j , n′, N ≤ Ds
∀s
(12)
N. Rakovitis et al.
1036 3.2. Task-specific event-based model
The task-specific event-based model of Shaik and Floudas (2009) for scheduling of single-tasking multipurpose batch processes was modified to consider multitasking. In this section, the modified constraints are only presented. 3.2.1. Allocation constraints Shaik and Floudas (2009) introduced several allocation constraints where only one task is processed in a unit in event n. We replace these constraints with a new set of allocation constraints, which allows multitasking.
∑
∑
n −∆n ≤ n ′ ≤ n n ≤ n ′′ ≤ n ′ +∆n
wi , n′, n′′ ≤ 1
∀j , n, ∆n > 0
(13)
A binary variable yj,n,n′ is defined as one if unit j is active from event n to n'.
y j , n, n′ ≥ wi , n, n′
∀i ∈ I j , j , n, n ≤ n′ ≤ n + ∆n
(14)
y j , n , n′ ≤ ∑ wi , n , n′
∀j , n, n ≤ n′ ≤ n + ∆n
(15)
i∈I j
3.2.2. Capacity constraints In multitasking, the summation of the batch sizes of all tasks that are able to be processed in a unit should be constrained.
∑b
i∈I j
i , n , n′
≤ B max y j , n , n′ j
∀j , n, n ≤ n′ ≤ n + ∆n
(16)
3.2.3. Tightening constraints The tightening constraints in Shaik and Floudas (2009) are only effective for single tasking in a unit. New tightening constraints below are introduced for multitasking.
Tjf j , n′ ≥ Tjs j , n + [max (α i )] y j , n , n′
∀j , n, n ≤ n′ ≤ n + ∆n
(17)
Tjs j , n +1 ≥ Tjf j , n
∀j , n < N
(18)
i∈I j
4. Computational results Five examples were solved using proposed models as well as the existing models of Patil et al. (2015) and Lagzi et al. (2017a). In example 1, a small-scale facility presented in Patil et al. (2015) was considered, while for the next 3 examples a random facility was generated. The last example uses a representation of an actual scientific service facility as presented by Lagzi et al. (2017b). Both maximization of productivity and minimization of makespan were considered as objective. All examples were solved in a machine using GAMS 24.6.1. CPLEX 12 in an Intel® Core™ i5-2500 3.3 GHz and 8 GB RAM running windows 7. The comparative results with maximization of productivity are given on Table 1. Both proposed models require less number of event points, which decrease the model size.
New approaches for scheduling of multitasking multipurpose batch processes
1037
However, the unit-specific event-based model leads to significantly smaller model size and therefore less computational time. The discrete-time model of Patil et al. (2015) also seems competitive due to the fact that it has the tightest relaxation. In cases of very detailed processing times, though such as example 4, the model size makes it infeasible to generate a feasible solution after one hour. By comparing continuous-time models, it seems that the proposed models improve the relaxation of the problem. Table 1 Comparative results for examples 1-5 for maximization of productivity as objective
Ex.
Model
1
La Pb T-Sc U-Sd L P T-S U-S L P T-S U-S L P T-S U-S L P T-S U-S
2
3
4
5
a
Event points 8 96 7 7 8 480 7 7 20 480 17 17 6 5 5 3 480 3 3
CPU time (s) 3600e 0.826 87.94 0.218 162.0 2.886 4.602 0.234 3600f 217.6 3600g 103.0 2000 >3600 10.12 0.125 0.202 7.270 0.046 0.094
RMILP
MILP
4822 3147 3424 3424 1404 1279 1404 1404 3259 2330 2744 2744 4495 3382 3382 846 666 846 846
3047i 3147 3147 3147 1254 1254 1254 1254 1194h 2330 2300h 2330 3100 3100 3100 666 666 666 666
Discrete Variables 2196 5150 770 427 540 6264 154 105 6300 32445 2312 1275 1456 460 260 3744 71220 966 702
Continuous Variables 2709 4417 1219 623 585 6721 337 221 7350 38401 4030 2687 3500 1216 650 4920 64375 1768 669
Constraints 9402 5387 5102 1211 2341 10569 672 348 26491 45135 16946 4715 6424 2730 703 16330 65850 4617 2284
Lagzi et. al. 2017. b Patil et. al. 2015. c Task-specific model. d Unit-specific model. e Relative gap 36.8%. f Relative gap 63.4%. g Relative gap 16.2%. h Suboptimal solution
The results for the same examples using minimization of makespan as objective are presented on Table 2. In that case, the unit-specific event-based model is superior to others with smaller model size and tighter relaxation. Consequently, it generates the optimum solution in at least one order of magnitude less computational time. The superiority of the model is also illustrated in example 3, where the unit-specific eventbased model generates the optimum solution in less than one minute, while reported models are not even able to generate a feasible solution after one hour. The task-specific event-based model has also a better relaxation and leads to smaller model sizes than the models previously reported in the literature.
5. Conclusions In this work, two novel mathematical models, based on task-specific and unit-specific event-based approaches for scheduling of multitasking batch process in scientific service facilities were presented. Both mathematical models lead to smaller model size with less number of event points, variables and constraints than previous reported formulations. The unit-specific event-based model is superior model than other models
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with significantly smaller model, much tighter relaxation and at least one magnitude less computational time to generate optimal solution, especially for the case of makespan minimization. Table 2 Comparative results for examples 1-5 for minimization of makespan as objective
Ex.
Model
1
La Pb T-Sc U-Sd L P T-S U-S L P T-S U-S L P T-S U-S L P T-S U-S
2
3
4
5
Event points 8 250 8 8 8 600 7 7 52 52 7 6 6 13 13 13
CPU time (s) 3600e 93.49 3600f 0.187 119.7 13.74 1.482 0.219 >3600 >3600 3600g 27.58 3600h >3600 2.325 0.281 3600i >3600 3600j 3600k
RMILP
MILP
99.50 23.90 125.13 934.38 88.09 4.02 137.39 307.50 167.75 1562.00 95.15 223.50 840.56 14.25 169.00 624.38
1065.0 1065.0 1065.0 1065.0 555.0 555.0 555.0 555.0 1966.0 1696.0 1001.7 1001.7 1001.7 7431.0 5131.0 5131.0
Discrete Variables 2196 14544 880 488 540 8064 154 105 7072 3900 1664 552 312 13104 4186 3042
Continuous Variables 2709 11501 1393 717 585 6401 337 221 12325 8357 4000 1459 786 18340 7658 3039
Constraints 9411 25636 5978 1816 2343 26642 685 403 53425 18160 7364 3418 1128 59434 24112 12581
a Lagzi et. al. 2017 with objective function MS ≥ TN. b Patil et. al. 2015 with objective function MS ≥ [(t – 1) + Tr(p) – 1]⋅yj,t. c Task-specific model. d Unit-specific model. e Relative gap 25.9%. f Relative gap 25.6%. g Relative gap 89.6%. h Relative gap 19.0% i Relative gap 99.7%. j Relative gap 78.9%. k Relative gap 62.5%.
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