An improved approach to scheduling multipurpose batch processes with conditional sequencing

Anton A. Kiss, Edwin Zondervan, Richard Lakerveld, Leyla Özkan (Eds.) Proceedings of the 29th European Symposium on Computer Aided Process Engineering June 16th to 19th, 2019, Eindhoven, The Netherlands. © 2019 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/B978-0-12-818634-3.50232-0

An improved approach to scheduling multipurpose batch processes with conditional sequencing 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 multipurpose batch processes has gained much attention in the past decades. Numerous models using different mathematical modelling approaches have been proposed. The model size and computational performance largely depend on the number of time points, slots or event points required. Most existing models still require a great number of time points, slots or event points mainly because a consumption task always takes place after its related production tasks regardless of whether it consumes materials from the related production tasks. Although two existing models have been developed to overcome this limitation, they can either lead to real time violation or generate suboptimal solutions in some cases. In this work, we develop an improved unit-specific event-based model in which a consumption task takes place after its related production task only if it consumes materials from the production task. A consumption task starts immediately after its related production task completes only if there is no enough storage for materials produced from the producing task. We also allow production and consumption tasks related to the same state to take place at the same event points. The results show that the proposed model generates same or better solutions than existing models with less number of event points and less computational time. Keywords: Scheduling, programming.

Multipurpose

batch

process,

Mixed-integer

linear

1. Introduction Scheduling of multipurpose batch processes has received much attention during the past three decades (Harjunkoski et. al. 2014). A number of mathematical models have been developed using discrete-time and continuous-time modelling approaches. The continuous-time modelling approaches include slot-based, global event-based, sequence based, and unit-specific event-based approaches. The model size and the computational performance largely depend on the number of time points, slots or event points required. Most existing models still require a great number of time points, slots, or event points to generate the optimal solution since a consuming task must start after the related producing tasks even if it does not consume materials from the producing tasks. In addition, production and consumption tasks related to the same state are not allowed to take place at the same time points, slots, or event points. To tackle these issues, Seid and Majozi (2012) imposed consuming tasks to take place after related production tasks if materials produced from the production units are used by any related consuming task. However, their model could lead to storage violations in real time. Vooradi and Shaik

1388

N. Rakovitis et al.

(2013) enforced a consuming task to take place after a related production task only if this consuming task uses materials from the specific production task with introduction of a high number of additional binary variables, leading to an intractable model size. Furthermore, materials are not allowed to be stored in a processing unit if it is idle, leading to suboptimality in some cases. These two efforts do not allow production and consumption tasks related to the same state to take place at the same time points, slots, or event points yet. Recently, Shaik and Vooradi (2017) and Rakovitis et al. (2018) did allow related production and consumption tasks to take place at the same event points. They still impose a consuming task must always start after the related producing tasks. In this work, we develop an improved unit-specific event-based approach for this scheduling problem, which requires less number of event points than existing formulations. We use the definition of recycling tasks from Rakovitis et al. (2018) and allow related non-recycling producing and consumption tasks to take place at the same event points. Furthermore, the proposed model sequence a processing unit which process a consumption task with its related production task, only if it consumes materials from the unit that process that task, and it force the finish time of a consuming task to be equal to the start time of the production task, if materials produced by the producing task are not able to be stored. Finally, we allow processing units to store the materials that were produced during the events that the unit is inactive. The computational results demonstrate that the proposed model generates same or better solutions using less number of event points and less computational time.

2. Problem description Figure 1 illustrates a multipurpose batch process facility involving S (s = 1, 2, …, S) states, J (j = 1, 2, …, J) processing units and I (i = 1, 2, …, I) tasks. The states include feed materials Sfeed, intermediates SIN and final products SFP. Each processing unit can process Ij tasks, but process at most one task at a time. Two storage policies including unlimited or finite intermediate storage (UIS or FIS) for intermediate states are considered, while unlimited storage is available for feed materials and products. Unlimited amount of feed materials is assumed during the scheduling horizon. Finally, unlimited wait policy (UW) is considered for all batches. Given production recipes, minimum/maximum unit capacities, processing times, storage policy for each intermediate and the scheduling horizon, the scheduling problem is to determine the optimal allocation of tasks to units, the start and end times of each task, task sequences in a processing unit as well as the inventory profiles in order to maximize productivity.

Figure 1 STN representation of a multipurpose batch process facility

An improved approach to scheduling multipurpose batch processes with conditional sequencing

1389

3. Mathematical formulation The mathematical formulation is developed using unit-specific event-based modelling approach because the advantages of this modelling approach have been well established in the literature. In the model, we have similar allocation constraints, capacity constraints, material balance constraints, and duration constraints to those of Rakovitis et al. (2018), which are not presented here. Next, we introduce new features of the proposed model. 3.1. Different tasks in different units We define continuous variables ܾܿ௜ǡ௜ ᇲ ǡ௦ǡ௡ as material s produced from task i consumed by task i′ at event n. Thus, the material consumed should not exceed total available materials.

−¦ j′

§ · ¨ ρ s ,i ′ ⋅ ¦ bi ′, j ′, n, n′ ¸ ≤ STs , n −1 + ¦P ¦C bci ,i ′, s , n n ≤ n′ ≤ n +Δn ¹ , i ′∈I j′ © i∈I s i ′∈I s

¦

i ′∈ICs

∀s ∈ S IN , n

(1)

The amount of materials from i to i′ (i.e., ܾܿ௜ǡ௜ᇲ ǡ௦ǡ௡ ) should not exceed the production amount of task i or consumption amount of task i′.

¦ bc

i ,i ′, s , n

≤ ρ s ,i ⋅

¦ bc

i ,i ′, s , n

≤ − ρ s ,i ′ ⋅

i ′∈ICs

i∈I sP

¦

n −Δn ≤ n′ ≤ n

bi , j , n′, n

¦

n ≤ n′ ≤ n + Δn

bci ,i ′, s , n ≤ z j , j ′, s , n ⋅ ρ s ,i ⋅ Bimax ,j

bi ′, j ′, n,n′

∀s ∈ S IN , i ∈ I sP , i ∉ I Re , j , n

(2)

∀s ∈ S IN , i ′ ∈ ICs , j, n

(3)

∀s ∈ S IN , i ∈ I sP , i ∈ I j , i′ ∈ ICs , i ′ ∈ I j ′ j , j ′, j ≠ j ′, n

§ · Ts , j , n ≥ Tf j , n − M ⋅ ¨ 1 − ¦ wi , j , n′, n ¸ ¦ ¨ i∈I P ,i∈I n −Δn ≤ n ′≤ n ¸ s j © ¹

∀s ∈S IN , j, n

(4) (5)

§ · Ts , j , n ≤ Ts j ′, n + M ⋅ ¨ 2 − ¦ wi ′, j ′, n , n′ − z j ′, j , s , n ¸ ¦ ¨ ¸ i ′∈I Cs , i∈I j n ≤ n ′ ≤ n + Δn © ¹ ∀s ∈ S IN , j, j ′, i ∈ I j , i ∈ I sP , i ∉ I Re , n

(6)

An active consumption task i’ should be processed in a unit after the time that state s is available at the previous event n. § · Ts , j , n ≤ Tf j ′, n +1 + M ⋅ ¨ 1 − ¦ wi ′, j ′, n +1, n′ ¸ ¨ i ′∈I C ,i ′∈I n +1≤ n¦ ¸ ′ ≤ n +1+Δn s j′ © ¹ ∀s ∈ S IN , j , j ′, i ∈ I sP , i ∈ I j , i ∉ I Re , n < N

(7)

3.2. Sequencing constraints for limited intermediate storage policies For FIS states, it should be examined whether the materials produced can be stored.

¦ ¦ρ

i∈I sP ,i∈I j ,i∉I Re

j

i,s

¦

n −Δn ≤ n ′≤ n

bi , j , n′,n + STs , n ≤ STsmax + ¦ ¦ bsi ,i′, s ,n i∈I Ps

∀s ∈SFIS , n

(8)

i ′∈ICs

If they cannot be stored then they either have to be consumed immediately or stored in the unit, processing consuming task i’ at event n, must take a non-zero value.

1390

N. Rakovitis et al.

bsi ,i′, s , n ≤ Bimax , j ( xi , i ′ , s , n + ui , i ′ , s , n )

∀s ∈ S FIS , j, j ′, i ∈ I sP , i ′ ∈ I Cs , n

(9)

xi ,i ′, s , n + ui ,i′, s , n ≤ 1

∀s ∈ S FIS , j, j ′, i ∈ I sP , i ′ ∈ I Cs , n

(10)

∀s ∈ S FIS , j, i ∈ I sP , i ∈ I j , i ∉ I Re , n

(11)

∀s ∈ S FIS , i ′ ∈ I Cs , j ′, n

(12)

ρ s ,i ⋅

¦

n −Δn ≤ n′ ≤ n

− ρ s ,i ′ ⋅

¦

bi , j , n′, n ≥

n ≤ n′ ≤ n +Δn

¦ bs

i , i ′, s , n

i ′∈ICs

bi ′, j ′, n , n′ ≥ ¦ bsi ,i′, s , n i∈I Ps

If bsi ,i ′, s , n takes a non-zero value, then the start time of consuming task as well as the finish time of the producing task are enforced to be equal. § · Ts , j , n ≤ Tf j , n + M ⋅ ¨1 − xi ,i ′, s , n ¸ ¨ i∈I P ,i¦ ¸ Re s ∈I j , i∉I © ¹

∀s ∈S FIS , j, n

(13)

§ · Ts , j , n ≥ Ts j ′, n − M ⋅ ¨ 1 − ¦ xi ,i′, s , n ¸ ¨ i ′∈IC ¸ s © ¹

∀s ∈ S FIS , j, j ′, i ∈ I Ps , i ∈ I j , i ∉ I Re , n

(14)

In order to avoid real time storage violations, between producing and consumption tasks occurring at the previous event the following constraint is introduced.

§ · Tf j ′, n +1 ≥ Ts j , n − M ⋅ ¨ 1 − wi′, j ′, n′, n +1 ¸ ∀s ∈ S FIS , j , j ′, i′ ∈ I Cs , i ′ ∈ I j , n < N ¦ © n +1−Δn ≤ n′≤ n +1 ¹

(15)

3.3. Allow production units to hold materials We define ‫ݏݑ‬௝ǡ௡ as the extra amount of materials stored in a processing unit j at n.

STs ,n ≤ STsmax + ¦

¦

us j ,n

∀s ∈ S FIS , n

(16)

j i∈I Ps ,i∈I j

The amount of materials stored in a unit at n cannot exceed the amount produced at n−1.

us j , n ≤

¦ ¦

s∈S FIS i∈I j , i∈I sP

¦

n −1−Δn ≤ n′≤ n

ρi , s bi , j ,n′, n−1 + us j ,n −1 ∀j , n > 1

(17)

If a unit j holds some material at n, then it cannot process any task at this event.

· º § us j , n ≤ ª« max ( Bimax wi , j , n′, n′′ ¸ ¦ ¦ , j )» ⋅ ¨1 − ¦ ¨ ¸ ∈ i I j ¬ ¼ © i∈I j n −Δn ≤ n′≤ n n ≤ n′′≤ n′ +Δn ¹

∀j , n

(18)

We define binary variables ‫ݏݔ‬௝ǡ௡ to denote if a unit j holds materials.

º us j , n ≤ ª« max ( Bimax , j ) » ⋅ xs j , n ¬ i∈I j ¼

∀j , n

(19)

Finally, if a unit holds some material at n, the finish time of all consuming tasks must be equal to the start time of that unit.

Ts , j , n ≤ Tf j , n + M ⋅ (1 − xs j ,n )

∀s ∈ S FIS , j , i ∈ I sP , i ∈ I j , n

(20)

An improved approach to scheduling multipurpose batch processes with conditional sequencing

§ · Ts , j , n ≥ Ts j ′, n − M ⋅ ¨ 2 − xs j , n − ¦ wi′, j ′, n, n′ ¸ © ¹ n ≤ n′ ≤ n +Δn ∀s ∈ S FIS , j, j ′, i ∈ I sP , i ∈ I j , i ∉ I Re , i′ ∈ ICs , i′ ∈ I j ′ , n

1391

(21)

4. Computational results We solve five examples including four well-established examples in the literature (Shaik and Floudas, 2009; Li and Floudas, 2010) to illustrate the capability of the proposed model. Both UIS and FIS policies are considered. We also compare the performance of the proposed model with the models of Li and Floudas (2010) (denoted as LF) and Vooradi and Shaik (2013) (denoted as VS). The computational results are given in Tables 1-2. From Tables 1-2, it can be observed that the proposed model reduces the total number of event points required for all examples compared to the LF model. This is because as it is depicted in figure 2 unit J1 is able to process 100 units of intermediate state using the proposed formulation since 50 of them can be temporary stored in the processing unit before being processed by J2, while by using existing formulations only 60 units can be processed since they cannot be stored in the processing unit and the storage capacity is 10. Compared to the VS model, the proposed model reduces the number of event points required for most examples except Examples 2 and 3 because all tasks in these two examples have to be treated as recycling task. The proposed model required less computational time for all examples compared to the LF and VS models especially for Example 3 which is reduced by 55% with UIS policy and 15% with FIS policy. More significantly, the proposed model generates a better solution than LF and VS models for Example 5. This is because a production unit is allowed to store materials produced. Therefore, the amount of materials produced can exceed the storage capacity. The optimal schedule for the proposed model is illustrated in Figure 2.

Figure 2 STN representation of a multipurpose batch process facility

5. Conclusions In this work, an improved unit-specific event-based formulation for multipurpose batch processes has been presented. The proposed model allows related non-recycling production and consumption units to take place at the same event. Furthermore, a unit processing a consuming task is sequenced with a related production task, only if the unit processing the consuming task consumes materials from the unit processing the producing task. Additionally, a consumption task is allowed not to start immediately

1392

N. Rakovitis et al.

after the finish of the related production task if storage is available. Finally, a processing unit is allowed to store materials if it is idle. The results demonstrate that the proposed model is able to generate same or better solutions, using less events, which can significantly reduce the computational time, especially in computationally expensive problems. Table 1 Comparative results for examples 1-5 with UIS and FIS policy

Ex.

Model

1

LFa VSb U-Sc LF VS U-S LF VS U-S LF VS U-S LF VS U-S

2

3

4

5

Event points 5 5 3 6d 5 5 8d 7 7 5 5 2 3 3 2

CPU time (s) 0.1 0.3 0.1 3.7 0.2 0.2 1244 1179 546 0.1 0.2 0.2 0.1 0.1 0.1

UIS RMILP 3000.00 3000.00 3000.00 2730.66 2436.69 2436.69 3618.64 3369.69 3369.69 80.0000 80.0000 80.0000 500.000 500.000 500.000

MILP 2628.19 2628.19 2628.19 1962.69 1962.69 1962.69 2358.20 2358.20 2358.20 58.9870 58.9870 58.9870 500.000 500.000 500.000

Event points 6 5 3 6d 5 5 9d 7 7 5 5 2 3 3 2

CPU time (s) 0.2 0.2 0.2 4.4 0.3 0.4 3600e 950 821 0.1 0.2 0.1 0.1 0.1 0.1

FIS RMILP 3973.92 3000.00 3000.00 2730.66 2436.69 2436.69 3618.64 3369.69 3369.69 80.0000 80.0000 80.0000 500.000 500.000 500.000

MILP 2628.19 2628.19 2628.19 1962.69 1962.69 1962.69 2345.31f 2358.20 2358.20 58.9870 58.9870 58.9870 300.000 300.000 500.000

a

Li et. al. 2010. b Vooradi and Shaik 2013. c Proposed unit-specific model d ǻn = 1 eRelative gap 7.11%. f Suboptimum solution.

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 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 Reshearch, 49(16), 7446-7469 N. Rakovitis, J. Li, N. Zhang, 2018, A novel modelling approach to scheduling of multipurpose batch processes, Computer Aided Chemical Engineering, 44, 1333-1338 R. Seid , T. Majozi, 2012, A robust mathematical formulation for multipurpose batch plants, Chemical Engineering Science, 68(1), 36-53 M. Shaik, C. Floudas, 2009, Novel Unified Modeling Approach for Short-Term Scheduling, Industrial & Engineering Chemistry Research, 48(6), 2947-2964 M. Shaik M., R. Vooradi, 2017, Short-term scheduling of batch plants: Reformulation for handling material transfer at the same event, Industrial & Engineering Chemistry 56(39), 11175-11185 R. Vooradi, M. Shaik, 2013, Rigorous unit-specific event based model for short term scheduling of batch plants using conditional sequencing and unit-wait times, Industrial and Engineering research, 52(36), 12950-12792