Short-Term Scheduling of Batch Plants ... - ACS Publications

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Short-Term Scheduling of Batch Plants: Reformulation for Handling Material Transfer at the Same Event Munawar A. Shaik* and Ramsagar Vooradi† Department of Chemical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi−110016, India S Supporting Information *

ABSTRACT: This work presents a new variant of a unit-specific eventbased model for short-term scheduling of batch plants. The motivation behind the proposed model is to further reduce number of events necessary to efficiently solve a given scheduling model. The main novelty in the proposed formulation is that the production and consumption tasks of each intermediate state are modeled to take place at the same event, in contrast to existing batch scheduling models where the consumption tasks normally start at the next event compared to the production tasks. This is the first time such a feature has been attempted in batch process scheduling, although it has been done in scheduling of continuous plants. Also, this is the first attempt where we are able to model the unit-specific events in their true sense, where the first event in each unit actually takes up the first event. The model is primarily applicable and useful for problems involving no recycle streams, although a hybrid approach has been suggested for problems with recycle streams and different reformulations are presented for both cases. In almost all the cases considered, the proposed model gives a reduction in the number events (on the order of two to three) and fewer constraints and variables compared to recent models in the literature. In 77% of the cases, especially for problems with no recycle streams, the proposed feature also results in better computational performance.

1. INTRODUCTION Several modeling approaches have been presented in the batch scheduling literature using different process representations such as state-task-networks (STN),1 resource-task-networks (RTN),2 recipe diagrams (RD),3 state-sequence-networks (SSN),4 and Sgraph approaches.5 The different process representations generally do not have as much impact on the computational performance as that of the different time representations as recently demonstrated by Shaik and Vooradi6 where they presented a model that results in unification of STN and RTN representations. Several models are presented in the literature using different time representations such as discrete-time and continuous-time models. The time horizon is uniformly discretized in discrete time models1 thus allowing easier formulation of different constraints in a simple manner. However, due to the requirement of large number of discrete events the resulting model may be computationally difficult solve or may often give suboptimal or infeasible solutions. To overcome these difficulties several continuous-time models evolved, which can be categorized as slot-based or globalevent-based, precedence-based, and unit-specific event-based formulations. Extensive reviews are available7−12 that present comparisons of different approaches for planning and scheduling. Slot based or global event based models consider nonuniform discretization with unknown durations. The slot based models are classified as process slots13,3 (synchronous) and unit slots14,15 © XXXX American Chemical Society

(asynchronous) depending on whether the tasks across different units are aligned globally or in a unit-specific manner. Global event based models16,17 follow synchronous alignment of tasks and hence can be classified in the same category as slot-based models or single time grid approaches. Sequence- or precedencebased models use different precedence relationships: immediate precedence, general precedence, and mixed immediate and general precedence18−20 for assignment of tasks to units using different possible numbers of batches without the need for explicit use of events or slots. Unit-specific event-based models21,4,22,23 follow asynchronous alignment of tasks and are also classified as unit slot-based models or multiple time grid approaches.24−28 Due to the asynchronous alignment of tasks, the unit-specific event-based models require less events and, hence, offer better computational performance15,29,30 compared to synchronous approaches. Despite the several novel modeling and solution approaches developed recently, it is still difficult to solve large-scale industrial scheduling problems. Hence, there is a large concern to improve the efficiency in solving scheduling problems by developing innovative methods that possibly result in fewer events and/or compact problem size. Castro and Grossmann24 and Castro and Received: Revised: Accepted: Published: A

February 6, 2017 August 31, 2017 September 7, 2017 September 7, 2017 DOI: 10.1021/acs.iecr.7b00519 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research Novais25 proposed multiple time grid (a.k.a. unit-specific eventbased) models for short-term scheduling of batch plants considering unlimited intermediate storage. The model of Castro and Novais25 results in less events due to explicit time grid used for storage, but their model has limited applicability (unlimited storage) unlike the Shaik and Floudas26 model that is valid for finite intermediate storage (FIS) also. Shaik and Floudas23 proposed a unified model using three index variables that efficiently allows tasks to span over multiple events. They proposed a common model that is equally valid for problems with resources and without resources. Vooradi and Shaik30 proposed the concept of active task resulting in compact model formulation compared to the Shaik and Floudas23 model. A unitslot based model was proposed by Susarla et al.15 which can handle different unit wait and storage policies. Seid and Majozi29 proposed conditional alignment of production and consumption tasks leading to fewer events. This concept was further extended by Vooradi and Shaik31 leading to rigorous conditional sequencing and reduction in number of events. However, all the conventional approaches for batch process scheduling in the literature (with the exception of precedence based models where there is no concept of events or slots) are based on the assumption that the consumption task of a given material/state normally occurs at next event/slot compared to the production task, unless there is adequate inventory for the consumption task to start at earlier events. If we allow production and consumption tasks to correspond to the same event, then it is possible to achieve a significant reduction in the number of events, as demonstrated in this work. This feature has been attempted for the first time in this work for batch process scheduling (to the best of authors’ knowledge), although it has been done previously in scheduling of continuous plants. A preliminary version of this work was presented by Vooradi and Shaik.32 In this work, we investigate innovative methods to further reduce the number of events required to efficiently solve a given scheduling problem using unit-specific event-based approaches. Shaik33 presented a comprehensive review of different recent unit-specific event-based models and how these models have evolved with different features leading to further reduction in the number of events and problem size. Motivating Example. Consider an example from the work of Shaik and Floudas23 with the STN shown in Figure 1 with FIS for states S2 and S3.

Figure 2. Gantt chart for motivating example (a) using the SF model or (b) the ideal case.

the first tasks occurring in each stage, unlike previous formulations which require m events. In other words, although the unit-specific event-based approach was adapted in the earlier approaches, the first event in a given unit is still different from N1. The reason is because of the conventional way of defining material balances in the earlier approaches, where the consumption tasks typically start at next event compared to the production tasks, for ease in writing material balances. The sequencing constraints are also written in a conventional way, for instance in the constraint for different tasks in different units used in SF model (given in eq A.1 of Appendix A), if a production task occurs at event n, then the consumption task can start only at the next event n + 1, which need not be true if there is no preceding task occurring in the consumption unit at earlier events. This kind of constraint is used, in some form or other, in all the batch scheduling models in the literature. With the above motivation, in this work we further exploit the fact that the unit-specific events correspond to heterogeneous locations along the time line, thus allowing different tasks to occur at different times and still corresponding to the same event point. The main novelty in this formulation is that the production and consumption tasks of a given state/material are allowed to occur in a sequential manner at the same event point. The proposed model can handle problems with batch splitting and mixing, different storage polices, but with no recycle streams. This paper is organized as follows. The problem definition for short-term scheduling of batch processes is given in section 2. In section 3, the details of proposed mathematical model are presented. Finally, in section 4, several examples are solved to illustrate the proposed approach.

Figure 1. State-task-network representation for example 1.

2. PROBLEM DEFINITION The problem statement is similar to the standard batch scheduling problem extensively addressed in the literature, with the exception of recycle streams. The following data is given related to product price and demands, time horizon, processing times of tasks and plant configuration, equipment and storage capacity limits. The aim is to determine the optimal Gantt chart with detailed production recipe specifying assignment, timing, and batching of tasks for each unit along with inventory levels. The important assumptions are (i) there are no recycle streams, (ii) material cannot wait in the batch unit except at the last event (but material waiting can be handled readily as discussed in the mathematical formulation), and (iii) intermediate states with storage requirements have dedicated storage units (flexible storage can be handled by considering storage as separate task,

23

This example is solved using the Shaik and Floudas model (referred to as SF in this work) with time horizon, H = 8 h, for maximization of profit. The model requires four events to find the optimal solution. In the resulting Gantt chart shown in Figure 2a, it can be observed that there are no tasks occurring in units J3, J4, and J5 at the first event (N1) and, similarly, there are no tasks occurring in units J4 and J5 at the second event (N2). Ideally, in true sense of unit specific events, the first task occurring in a unit should be numbered N1 (or first event) irrespective of when it starts. For example, Figure 2b would be the ideal case for Figure 2a, offering a reduction by two events. But this was not possible with earlier approaches. Remark. For a multistage process with m sequential stages, it is now possible to have a Gantt chart using just one event for all B

DOI: 10.1021/acs.iecr.7b00519 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Capacity Constraint. The minimum and maximum batch size requirements are specified in eq 2.

although not discussed here). Two different objectives can be handled: maximization of profit or minimization of makespan.

Bimin w(i , n , n′) ≤ b(i , n , n′) ≤ Bimax w(i , n , n′)

3. MATHEMATICAL MODEL The base model considered is from the work of Vooradi and Shaik30 (referred to as VS in this work), and new modifications are proposed to handle the issues related to allowing material transfers between production and consumption tasks to occur at the same event. The constraints for allocation, capacity, and duration are same as in the earlier model and are presented here for completeness. Two different frameworks are proposed to handle the material transfers of different intermediate states based on whether these states are involved in recycle streams or not. Because if a state is involved in a recycle loop, then there will be difficulty in monitoring material transfers at same event. Consider a sample STN given in Figure 3, which involves a recycle loop.

∀ i ∈ I , n , n′ ∈ N , n ≤ n′ ≤ n + Δn

(2)

Material Balances. Equation 3 is a material balance for raw materials, and eq 4 is that for products. For raw materials ST(s, n) is the amount of material available at the starting of the consumption task i at event n, and for products, ST(s, n) is the amount of material available at the end of production task i at event n. ST (s , n) = ST0(s)|n = 1 + ST (s , n − 1)|n > 1 +

∑ ρis i ∈ Isc

∑

b(i , n , n′) ∀ s ∈ S R , n ∈ N

n ′∈ N n ≤ n ′≤ n +Δn

(3)

ST (s , n) = STs0|n = 1 + ST (s , n − 1)|n > 1 +

∑ ρis i ϵIsp

The intermediate states S3 and S4 are related to each other in terms of feed and product (i.e., if S3 is a feed state then S4 is a product state and vice versa). So, tasks i2 and i3 cannot be related or sequenced at same event. Since, with respect to state S4 task i2 is the production task and i3 is the consumption task and with respective to state S3 task i3 is the production task and i2 is the consumption task. Therefore, when there are recycle streams, production and consumption tasks cannot occur at the same event, which becomes a limitation of the proposed approach. Although a hybrid approach is suggested for problems having and not having recycle streams, it is not the main focus of this work. In the hybrid approach, for states involved in recycle loops (S ∈ SRC) material transfers are allowed at consecutive events, and for the states not involved in recycle loops (S ∈ SSE) material transfers are allowed at same events. The proposed model can handle all storage policies, i.e. UIS, FIS, NIS (no intermediate storage), and ZW (zero-wait). The material balances, sequencing constraints, and storage constraints are reformulated to effectively handle the proposed concept of allowing material transfers to occur at the same event. Allocation Constraint. At the most, one task can be active in each unit at each event as in eq 1. For the tasks that have dedicated units the allocation constraints need not be written, for Δn = 0. i ∈ Ij

∑

∑

b(i , n′, n) ∀ s ∈ S P , n ∈ N (4)

The material balance for intermediate states that are not involved in recycle loops (s ∈ SSE) is given in eq 5. The important feature in this balance is that the production and consumption tasks of a given state/material are allowed to occur in a sequential manner at the same event point. Equation 5 relates the excess amount at an event (term on LHS) with excess amount available at previous event (first two terms on RHS), amount from producing tasks ending at this event (third term on RHS) and amount from consuming tasks starting at this event (fourth term on RHS). The variable ST(s, n) here represents the excess amount of state s at the start time of consumption task occurring at event n. The excess amount at the first event is calculated based on the initial amount available.

Figure 3. State-task-network representation for a recycle loop.

∑

∑ n ′ϵN n −Δn ≤ n ′≤ n

ST (s , n) = STs0|n = 1 + ST (s , n − 1)|n > 1 +

∑ ρis i ϵIsp

∑

b(i , n′, n) +

∑

∑ ρis i ϵIsc

n ′ϵN n −Δn ≤ n ′≤ n

b(i , n , n′) ∀ s ∈ S SE , n ∈ N

n ′ϵN n ≤ n ′≤ n +Δn

(5)

For the states that are involved in recycle loop (s ∈ SRC), conventional material balances are used where production and consumption tasks occur at consecutive events (n − 1 and n) as given in eq 6. The variable ST(s, n) here represents the excess amount of state s at the start time of consumption task occurring at event n. ST (s , n) = STs0|n = 1 + ST (s , n − 1)|n > 1 +

∑ ρis

i ∈ Isp

w(i , n′, n″) ≤ 1 ∀ j ∈ J , n ∈ N

+

n ′∈ N n ″∈ N n −Δn ≤ n ′≤ n n ≤ n ″≤ n ′+Δn

∑ ρis i ∈ Isc

(1) C

∑

b(i , n′, n − 1)

n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

∑ n ′∈ N n ≤ n ′≤ n +Δn

b(i , n , n′) ∀ s ∈ S RC , n ∈ N (6) DOI: 10.1021/acs.iecr.7b00519 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research Duration Constraints. For the case Δn = 0, where a task occurs over one event, which means it starts and ends at the same event n (hence n′ = n) the end time is calculated from eq 7.

T s(i , n) ≥ T f (i′, n) − M(1 −

∈ Ij ′ , i ≠ i′, j ≠ j′, i ∈ Isc , i′ ∈ Isp

(7)

For the states that are involved in recycle loops (s ∈ S ), the constraint for different tasks in different units is given in eq 14. Here also there could be two possibilities similar to eqs 13a and 13b. T s(i , n + 1) ≥ T f (i′, n) ∀ s ∈ S RC , i , i′, j , j′ ∈ J , n ∈ N , n < N , i ∈ Ij , i′ ∈ Ij , i ≠ i′, j ≠ j′, i ′ ∈ Isc , i′ ∈ Isp (14)

T f (i , n′) ≥ T s(i , n) + αiw(i , n , n′) + βi b(i , n , n′)

The above constraints still assume unconditional sequencing for production and consumption tasks which can be relaxed as discussed in literature.29,31 In eq 15, the total processing time in each unit should be within the specified time horizon.

(8)

T f (i , n′) ≤ T s(i , n) + αiw(i , n , n′) + βi b(i , n , n′) + M(1 − w(i , n , n′)) ∀ i ∈ I , n , n′ ∈ N , n ≤ n′ ≤ n + Δn , Δn > 0

(9)

Sequencing Constraints. The constraint for same task in the same unit remains same and is given by eq 10 for Δn = 0, and by eqs 10 and 11 for Δn > 0. T s(i , n + 1) ≥ T f (i , n) ∀ i ∈ I , n ∈ N , n < N

∑

∑

(10)

w(i , n′, n″))

(11)

n ′∈ N n ≤ n ′≤ n +Δn

(αiw(i , n , n′) + βi b(i , n , n′)) ≤ H

(15)

T s(i , n) ≤ T f (i′, n) + M(2 −

The constraint for different tasks occurring in the same unit remains the same and is given by eq 12. This constraint can be readily modified to handle sequence dependent changeovers, similar to earlier approaches.

∑

w(i′, n′, n)

n ′∈ N n −Δn ≤ n ′≤ n

−

∑

w(i , n , n′))

n ′∈ N n ≤ n ′≤ n +Δn

T s(i , n + 1) ≥ T f (i′, n) ∀ i , i′ ∈ Ij , i ≠ i′, j ∈ Ji , n ∈ N, n < N

∑

i ∈ Ij n ∈ N

Different Storage Policies. The approach would be different based on whether a state is involved in a recycle loop or not. For the states that are not involved in recycle loop, constraint 16, along with constraint 13, enforces the alignment for different tasks taking place in different units with corresponding intermediate states having ZW or NIS or dedicated FIS cases. For the states that are involved in recycle loop constraints 17 and 14 enforce the alignment. As mentioned earlier in this model we assume unconditional alignment of producing and consuming tasks.

n ′∈ N n ″∈ N n −Δn < n ′≤ n n < n ″≤ n ′+Δn

∀ i ∈ I , n ∈ N , n < N , Δn > 0

∑∑ ∀j∈J

T s(i , n + 1) ≤ T f (i , n) + M (1 −

(13b) RC

For Δn > 0, end time of a task spanning over multiple events is calculated from eqs 8 and 9. The end time of a task which is active and ending at an event is equal to its start time plus duration, else its finish time is simply greater than or equal to its start time. The Big-M term in eq 9 is used to relax the constraint if a task does not start at n and ends at n′. Equations 7−9 can be modified to handle unit-wait policies as discussed in the literature.29,31

∀ i ∈ I , n , n′ ∈ N , n ≤ n′ ≤ n + Δn , Δn > 0

w(i′, n′, n))

∀ s ∈ S SE , i , i′, j , j′ ∈ J , n ∈ N , i ∈ Ij , i′

T f (i , n) = T s(i , n) + αiw(i , n , n) + βi b(i , n , n) ∀ i ∈ I , n ∈ N , Δn = 0

∑ n ′∈ N n −Δn ≤ n ′≤ n

∀ s ∈ (S dfis ∪ Snis ∪ S zw ∩ S SE), j , j′ ∈ J , n ∈ N , i (12)

∈ Ij , i′ ∈ Ij , i ≠ i′, j ≠ j′, i ∈ Isc , i′ ∈ Isp ′

The constraint for different tasks in different units would be modeled based on whether a state is involved in a recycle loop or not. For the states which are not involved in recycle loop (s ∈ SSE) the start time of consuming task at event n is should be later than the finish time of producing task at the same event using two possibilities as given in eqs 13a or 13b. As a guideline, eq 13a can be used for states that have dedicated units for the corresponding production and consumption tasks; otherwise, eq 13b can be used for multipurpose units.

T s(i , n + 1) ≤ T f (i′, n) + M(2 −

∑

(16)

w(i′, n′, n)

n ′∈ N n −Δn ≤ n ′≤ n

−

∑

w(i , n + 1, n′))

n ′∈ N n + 1 ≤ n ′≤ n + 1 +Δn

∀ s ∈ (S dfis ∪ Snis ∪ S zw ∩ S RC), j , j′ ∈ J , n < N , i ∈ Ij , i′ ∈ Ij , i ≠ i′, j ≠ j′, i ∈ Isc , i′ ∈ Isp (17) ′ Constraints 18a and 18b impose bounds on excess amount of FIS states depending on whether these states are involved in recycle or not. For the states that are not involved in recycle loops, the

T s(i , n) ≥ T f (i′, n) ∀ s ∈ S SE , i , i′, j , j′ ∈ J , n ∈ N , i ∈ Ij , i′ ∈ Ij ′ , i ≠ i′, j ≠ j′, i ∈ Isc , i′ ∈ Isp (13a) D


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Industrial & Engineering Chemistry Research bound is relaxed in eq 18b at the last event, since the material may continue to wait in the batch unit at last event. ST (s , n) ≤ STsmax ∀ s ∈ (S dfis ∩ S RC), n ∈ N

Max Profit =

s ϵS P

ST (s , n) ≤ STsmax ∀ s ∈ (S dfis ∩ S SE), n ∈ N , n < N (18b)

For ZW states, material cannot wait either in storage or in a processing unit, which is taken care of in eqs 19a and 19b along with the earlier duration and sequencing constraints. For NIS states material can wait only in the processing unit as given in eqs 19c and 19d depending on whether these states are involved in recycle or not. For NIS states not involved in recycle streams, eq 19d is relaxed at the last event, since the material may continue to wait in the same batch unit. Accordingly, the duration constraints and sequencing constraints can be modified for NIS states to allow different unit-wait policies as discussed in the literature.29,31

∑ ∑ ρis n=N

i ∈ Isp

∑

b(i , n′, n) = 0 ∀ s ∈ SZW ∩ S RC

n ′∈ N n −Δn ≤ n ′≤ n

(19b) (19c)

ST (s , n) = 0 ∀ s ∈ Snis ∩ S SE , n < N , n ∈ N

(19d)

Equations 20 and 21 are used to take care of FIS policy by enforcing end time of active production task ending at event n to be later compared to start time of consuming task. For the states involved in recycle loop this condition is applied at the same event, and for the states not involved in recycle loop it is applied for successive events. T f (i′, n) ≥ T s(i , n) − M(1 −

∑

w(i′, n′, n))

n ′∈ N n −Δn ≤ n ′≤ n

∀ s ∈ (S dfis ∩ S RC), j , j′ ∈ J , n ∈ N , i ∈ Ij , i′ ∈ Ij ′ , i ≠ i′, j ≠ j′, i ∈ Isc , i′ ∈ Isp

T f (i′, n) ≥ T s(i , n − 1) − M(1 −

(20)

∑

∀ s ∈ (S dfis ∩ S SE), j , j′ ∈ J , n ∈ N , n > 1, i ∈ Ij , i′ (21)

Bounds on Variables. Since the three index variables are defined only for n′ ≥ n, the other possibilities are eliminated in eq 22a. The upper bounds for start and end times are defined in eq 22b. w(i , n , n′) = 0; b(i , n , n′) = 0 ∀ n′ < n

(22a)

T s(i , n) ≤ H ; T f (i , n) ≤ H

(22b)

ST (s , N ) ≥ Ds ∀ s ∈ S P

(24)

T f (i , N ) ≤ MS ∀ i ∈ I

(25)

4. COMPUTATIONAL RESULTS We consider selected examples from batch scheduling literature to demonstrate the proposed features and enhancements relative to the existing models. In section 4.1, six examples involving no recycle streams are considered with UIS policy for all states with data as given in the work of Vooradi and Shaik.30 Example 1 here is same as example 1 in ref 30, and examples 2−6 are same as examples 4−8 in ref 30. In section 4.2, the same examples are solved using FIS policy, and in section 4.3, the Westenberger− Kallrath scheduling problem is considered to demonstrate the hybrid approach which involves recycle streams. GAMS 23.5 software is used to solve the resulting models with CPLEX 12.2 as the MILP solver, and all computations are done on 2.66 GHz Intel Core 2 Duo processor with 3 GB RAM. Δn and optimality gap are zero in all tables unless otherwise stated. 4.1. UIS Policy and No Recycle Streams. Consider the earlier motivating example discussed in section 1 with a UIS policy and the STN given in Figure 1. Four different instances are considered with varying time horizons (8, 10, 12, and 16 h). The computational results for these cases are shown in Table 1. The

w(i′, n′, n))

n ′∈ N n −Δn ≤ n ′≤ n

∈ Ij ′ , i ≠ i′, j ≠ j′, i ∈ Isc , i′ ∈ Isp

(23)

Important Contributions of This Work. (i) For the first time in this work, material balances and sequencing constraints are reformulated for scheduling of batch plants (to the best of authors’ knowledge) to allow handling of material transfers at the same event, although such concept has been in use for scheduling of continuous processes.34,35 The central idea is that the production and consumption tasks of a given material are modeled to occur in a sequential manner with consumption task starting after production task finishes at the same event. Accordingly material balances, sequencing constraints, and storage constraints are reformulated. Although mathematically it is a small step, conceptually it is significant for scheduling of batch plants, because it results in significant reduction in number of events as shown later. (ii) Also, this is the first attempt where we are able to model the unit-specific events in their true sense, where the first event in each unit actually takes up first event. For instance in Figure 2a, there are three stages (in the STN of Figure 1), and three events are required to carry out first tasks in each stage using earlier approaches, while using the proposed approach only one event is adequate for all first tasks in each stage, as shown in Figure 2b. However, this concept is limited to problems that do not have recycle streams. (iii) A hybrid approach is suggested for problems having with and without recycle streams, where conventional material transfers (i.e., production and consumption tasks occurring at successive events) are used for the states that are involved in recycle loops; and modified material transfers (i.e., both production and consumption tasks happening at same event) can be used for the states not involved in recycle loops. The constraints for handling various storage policies are also reformulated accordingly to accommodate the hybrid approach.

(19a)

ST (s , n) = 0 ∀ s ∈ Snis ∩ S RC , n ∈ N

n=N

If the objective is to minimize makespan (MS), the minimum demand to be met is specified in eq 24. Equation 25 states that all tasks must end before the makespan time. Additionally, in eq 15 H should be replaced with makespan.

(18a)

ST (s , n) = 0 ∀ s ∈ SZW , n ∈ N

∑ Ps ∑ ST(s , n)

Objectives. In eq 23 the objective is maximization of profit which is written differently compared to previous models in literature26,23,30 because the amount produced at the last event is also captured in the variable ST(s, N) itself, in contrast to previous models. E


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Industrial & Engineering Chemistry Research Table 1. Computational Results for Example 1 with UISa model

a

events

RMILP

SF VS this work

4 4 2

2000 1947.5 1947.5

SF VS this work

5 5 3

3000 2914.84 2914.84

SF VS this work

6 6 4

4000 3857.42 3857.42

SF VS this work

9 9 7

6600.9 5956.11 5956.11

SF VS this work

14 14 12

24.236 25.357 25.357

SF VS this work

23 23 21

48.473 50.06 50.06

MILP

CPU time (s)

nodes

binary variables

continuous variables

constraints

68 68 40

89 89 57

87 87 59

117 117 85

106 106 78

145 145 113

163 163 135

229 229 197

258 258 230

374 374 342

429 429 401

626 626 594

binary variables


constraints

12 12 12

74 74 50

101 101 73

15 15 15

104 104 85

140 140 100

12 12 12

105 105 51

139 139 73

16 16 16

102 102 74

137 137 105

32 32 32

174 174 146

241 241 209

Example 1a (H = 8) Maximization of Profit 1840.17 0.01 0 10 1840.17 0.02 0 10 1840.17 0.01 0 10 Example 1b (H = 10) Maximization of Profit 2628.18 0.02 16 15 2628.18 0.04 0 15 2628.18 0.02 0 15 Example 1c (H = 12) Maximization of Profit 3463.62 0.03 27 20 3463.62 0.03 15 20 3463.62 0.03 15 20 Example 1d (H = 16) Maximization of Profit 5038.05 1.28 3790 35 5038.05 0.95 2784 35 5038.05 1.06 2799 35 Example 1e (D4 = 2000 mu) Minimization of Makespan 27.881 7.69 42366 60 27.881 4.04 15338 60 27.881 3.87 14147 60 Example 1f (D4 = 4000 mu) Minimization of Makespan 52.072 7.39 21700 105 52.072 2.27 9419 105 52.072 5.46 25650 105

SF.23 VS.30

Table 2. Computational Results for Examples 2−6 with UISa model

a

events

RMILP ($)

MILP ($)

SF VS this work

5 5 3

13.54 12.63 12.63

10.00 10.00 10.00

SF VS this work

5 5 3

287.13 266.68 266.68

210.00 210.00 210.00

SF VS this work

5 5 2

80.00 68.84 68.84

58.98 58.98 58.98

SF VS this work

6 6 4

400 400 400

400 400 400

SF VS this work

10 10 8

400 400 400

400 400 400

CPU time (s)

nodes

Example 2 (H = 6) 0.01 0 0.01 0 0.04 0 Example 3 (H = 9) 0.02 0 0.02 0 0.02 0 Example 4 (H = 76) 0.01 14 0.01 14 0.01 0 Example 5 (H = 10) 0.02 0 0.01 0 0.01 0 Example 6 (H = 10) 0.03 7 0.03 0 0.03 0

SF.23 VS.30

results using the proposed model are compared with other recent unit specific event based models from literature. When considering profit as the objective, the proposed formulation solves using lesser events, continuous variables, and constraints relative to SF and VS models. The reduction in events is due to the production and consumption tasks being allowed to take place at same events, which is not possible for the SF and VS models as discussed earlier. In the SF and VS models, additionally, tasks that cannot occur at certain events were

identified and the binary variables were eliminated before solving the model, thus, resulting in less number of binary variables. In the proposed approach, it is not possible to determine such tasks that cannot take place at certain events a priori due to the material transfers being allowed to occur at same events. Hence, for same number of events, the number of binary variables would be higher in the proposed model relative to the SF and VS models. For a fair comparison with the proposed model, results F


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Industrial & Engineering Chemistry Research Table 3. Computational Results for Example 1 with FISb model

a

events

RMILP

MILP

SF VS this work

4 4 2

2000 2000 2000

1840.17 1840.17 1840.17

SF VS this work

6 6 4

3973.7 3361.12 3361.12

2628.19 2628.19 2628.19

SF VS this work

8 8 6

4951.24 4419.88 4419.88

3463.62 3463.62 3463.62

SF VS this work

11 11 9

6601.7 6236.04 6236.04

5038.05 5038.05 5038.05

SF VS this work

20 20 18

24.236 24.492 24.492

28.032 27.881 27.881

SF VS this work

29 29 29

48.472 48.789 48.681

54.395 54.395 53.766

CPU time (s)

nodes

binary variables


constraints

68 68 40

117 113 65

106 106 78

189 185 137

144 144 116

261 257 209

201 201 173

369 365 317

372 372 344

698 694 646

543 543 553

1022 1018 1042

Example 1a (H = 8) Maximization of Profit 0.03 0 10 0.03 3 10 0.01 0 10 Example 1b (H = 10) Maximization of Profit 0.16 303 20 0.11 175 20 0.13 217 20 Example 1c (H = 12) Maximization of Profit 1.46 5487 30 0.89 2156 30 0.93 2232 30 Example 1d (H = 16) Maximization of Profit 124.9 560089 45 17.94 90347 45 21.27 104143 45 Example 1e (D4 = 2000 mu) Minimization of Makespan 40000a 29941133 90 40000a 84604531 90 40000a 87031983 90 Example 1f (D4 = 4000 mu) Minimization of Makespan 40000a 20957581 135 40000a 47936042 135 40000a 41526449 145

relative gap

7.25 3.1 3.98 10.4 7.1 6.5

Resource limit reached bSF.23 VS.30

Table 4. Computational Results for Examples 2−6 with FISa model

a

events

RMILP ($)

MILP ($)

SF (Δn = 1) VS (Δn = 1) this work (Δn = 1)

5 5 3

14 14 14

10 10 10

SF (Δn = 2) VS (Δn = 2) this work(Δn = 2)

5 5 3

300 300 300

210 210 210

SF VS this work

5 5 2


6 6 4

400 400 400

400 400 400


10 10 8

400 400 400

400 400 400

80.0 72.65 72.65

58.98 58.98 58.98

CPU time (s) Example 2 (H = 6) 0.05 0.03 0.05 Example 3 (H = 9) 0.09 0.06 0.05 Example 4 (H = 76) 0.02 0.02 0.01 Example 5 (H = 10) 0.05 0.06 0.06 Example 6 (H = 10) 0.16 0.43 0.34

nodes

binary variables


const raints

44 18 17

20 20 20

82 82 58

276 209 133

73 33 26

30 30 30

119 119 85

403 318 194

9 0 0

12 12 12

105 105 51

193 187 85

0 10 0

40 40 40

126 126 98

428 341 229

0 72 0

144 144 144

286 286 258

1004 837 661

SF.23 VS.30

are generated without the prefixing of variables for the VS model and given in Tables S1−S4 (in the Supporting Information). Similarly, by considering makespan as the objective this example is solved for two instances of demands (2000 and 4000 mu) for product S4. The big-M values used in these instances are 50 and 100 h. In the computational results given in Table 1, the proposed model finds the optimal solution using less events in both of the cases. Comparing Tables 1 vs S1 (with and without prefixing of variables), it can be observed that the proposed

model performs either same or better than VS model (wrt CPU time) in 5 out of 6 instances in Table S1 compared to 4 out of 6 instances in Table 1, for the same solution quality. Example 2 considered here is example 4 from the work of Vooradi and Shaik30 which was solved earlier by Maravelias and Grossmann16 and Li et al.28 The results are shown in Tables 2 and S2 for profit maximization, where the proposed formulation finds the optimal solution using three events, with better CPU time in Table S2, whereas SF and VS models require five events. G


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summarized in Table 6, which is quite promising, coupled with the fact that it leads to reduction in number of events and less problem size in almost 100% of the cases considered. 4.3. Westenberger−Kallrath Problem: with Recycle Streams and Different Storage Policies. In this section, to demonstrate the hybrid approach we consider the Westenberger−Kallrath problem36−38,30 which has 17 tasks for processing 19 states (S0−S18) in 9 units with some states having FIS and ZW policies. Two intermediate states (S1, S3) are involved in recycle loop for which the classical material balances are employed at successive events, whereas for all other states the proposed balances are employed at same event (following the hybrid approach). Task 2 produces two states (S2, S3) with flexible fractions for which the material balances are modified, as given in eqs 26 and 27 for state S2 and in eqs 28 and 29 for state S3, so that the production stays within the given bounds (i.e., ρmin is and ρmax is ). For the first event, the first term of the right-hand-side of eq 26 and 27 should be replaced with initial amount of state S2 as shown for state S3 in eq 30. For state S2, eqs 26 and 27 facilitate the material transfers at the same event.

Example 3 considered here is example 5 from the work of Vooradi and Shaik30 which was solved earlier by Li and Floudas27 and Li et al.28 The results are shown in Tables 2 and S2. For this example also the proposed formulation requires two events lesser relative to SF and VS models, with better CPU time in Table S2. Example 4 considered here is example 6 from Vooradi and Shaik30 which was solved earlier by Ierapetritou and Floudas.21 For this problem the proposed model requires three events lesser compared to other models, and gives better RMIP values (along with VS) compared to SF model. Examples 5 and 6 considered here are examples 7 and 8 from the work of Vooradi and Shaik30 which were solved earlier by Susarla et al.15 In these two examples also the proposed formulation requires two events lesser compared to SF and VS models. Comparing Tables 2 vs S2 (with and without prefixing of variables), it can be observed that the proposed model performs either same or better than VS model (wrt CPU time) in all 5 examples (in Table S1), compared to 4 out of 5 examples (in Table 2), for same solution quality. 4.2. FIS Policy and No Recycle Streams. The six examples considered above are solved again but with dedicated FIS policy for some intermediates. The auxiliary storage constraints used in earlier models23,30 are modified to accommodate production and consumption tasks to occur at the same event for states not involved in recycle loop. In example 1, S2 and S3 are FIS states. For profit maximization, the results for FIS case are given in Tables 3 and S3 for with and without prefixing of variables. In all the cases the proposed model solves using two events lesser relative to other models. For example 1a the Gantt chart is as shown in Figure 2b, which is exactly same as the ideal case discussed earlier in the motivation section, where only two events are adequate using the proposed model whereas the SF and VS models require four events. Thus, the Gantt chart for the ideal case is realized using the proposed approach. For makespan minimization the results are shown in Table 3 for two demand instances. For example 1e, the proposed model requires two events lesser, and for example 1f it finds better objective value compared to other models. For examples 1e and 1f, all the models are unable to find the optimal solution within specified CPU time limit (40 000 s). For examples 2−6 the computational results for profit maximization are shown in Tables 4 and S4 with the FIS policy. In example 2, S2 and S3 are FIS states. In example 3, S4 and S5 are FIS states; and in example 4, S2, S4, and S5 are FIS states. In examples 5 and 6, states S2, S3, S4, and S5 are FIS states with different storage capacities. In Table 4 also, in all the problem instances the proposed model results in reduction of two to three events compared to other models. However, for FIS cases higher Δn is required in order to accommodate multiple consumption tasks occurring within the duration corresponding to single production task. Comparing with and without prefixing of variables for FIS case, it can be observed that the proposed model performs either same or better than VS model (wrt CPU time) in 3 out of 6 instances, compared to 4 out of 6 (Tables 3 vs S3) for example 1 (with same or better solution quality). Similarly, comparing Tables 4 vs S4, the proposed model performs either same or better than VS model (wrt CPU time) in 4 out of 5 instances, compared to 3 out of 5 for other examples (with same solution quality). Overall, for problems involving no recycle streams, the proposed model performs better than VS model (wrt CPU time) in 17 out of 22 instances (77.3%) in Tables S1−S4, compared to 15 out of 22 instances (68.2%) in Tables 1−4, as

ST (S 2, n) ≥ ST (S 2, n − 1) + ρi2,min S2

+

∑ ρi ,S2

i ∈ ISc2

∑

∑ ρi ,S2

i ∈ ISc2

b(i 2, n′, n)

b(i , n , n′) ∀ n ∈ N

n ′∈ N n ≤ n ′≤ n +Δn

(26)

ST (S 2, n) ≤ ST (S 2, n − 1) + ρi2,max S2

+

∑ n ′∈ N n −Δn ≤ n ′≤ n

∑

∑

b(i 2, n′, n)

n ′∈ N n −Δn ≤ n ′≤ n

b(i , n , n′) ∀ n ∈ N

n ′∈ N n ≤ n ′≤ n +Δn

(27)

∑ ρi , S3

ST (S 3, n) = ST (S 3, n − 1) +

∑

i ∈ ISc3

b(i , n , n′)

n ′∈ N n ≤ n ′≤ n +Δn

∑

+

b(i 2, n′, n − 1) − (ST (S 2, n − 1)

n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

∑

− ST (S 2, n − 2) −

i ∈ ISc 2

∑

ρi , S2

b(i , n − 1, n′))

n ′∈ N n − 1 ≤ n ′≤ n − 1 +Δn

∀ n ∈ N, n > 2

(28)

ST (S 3, n) = ST (S 3, n − 1) +

∑ ρi ,S3

∑

i ∈ ISc3

b(i , n , n′)

n ′∈ N n ≤ n ′≤ n +Δn

∑

+

b(i 2, n′, n − 1) − (ST (S 2, n − 1) − STs0

n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

−

∑ i ∈ ISc 2

ρi , S2

∑

b(i , n − 1, n′)) ∀ n ∈ N , n = 2

n ′∈ N n − 1 ≤ n ′≤ n − 1 +Δn

(29)

ST (S 3, n) = STS03 +

∀ n ∈ N, n = 1

∑ ρi ,S3

i ∈ ISc3

∑

b(i , n , n′)

n ′∈ N n ≤ n ′≤ n +Δn

(30)

Computational Results. For makespan minimization, 14 cases with different demand distribution of final products (S14− H


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Industrial & Engineering Chemistry Research Table 5. Computational Results for Westenberger−Kallrath Problem (without Prefixing) problem instance

model

events

RMILP (h)

make span (h)

CPU time (s)

binary variables


constraints

P1

VSd this work VS this work VS this work VS this work VS this work VS (Δn = 1) this work (Δn = 1) VS this work VS this work VS (Δn = 1) this work (Δn = 1) VS this work VS this work VS this work VS this work VS this work

5 4 7 5 7 5 6 5 6 5 8 6 7 6 7 5 11 9 9 7 12 11 8 7 10 8 10 9

24 24 24 24 24 24 18.66 18.66 18.66 18.66 16 16 18.66 18.66 18.66 18.66 17.79 17.79 26.66 26.66 30.66 30.66 30 30 28 28 24 24

28 28 28 28 28 28 27 27 26 26 30 30 28 28 28 28 33 33 35 35 40 40 34 34 36 36 36 36

0.048 0.045 0.316 0.238 0.277 0.214 0.312 0.439 0.266 0.528 19.678 37.558 0.597 1.228 0.514 0.392 19748.72 40000a,b 11.206 5.957 6093.39 40000a,c 3.122 8.973 65.843 13.297 357.45 12648.07

70 56 98 70 98 70 96 80 96 80 360 176 112 96 112 80 357 289 162 126 288 264 192 168 240 192 240 216

262 210 359 257 366 262 356 312 362 312 866 436 415 356 422 312 865 705 605 471 1046 959 698 611 872 698 872 785

626 531 881 659 905 670 894 800 914 796 3018 1436 1053 947 1077 796 2850 2359 1667 1334 3042 2833 1986 1777 2514 2041 2514 2305

P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 a

Resource limit reached. bRelative gap: 3.03%. cRelative gap: 2.22%. dVS.30

S18) are considered with same data used in literature.30 The suggested hybrid model is employed with modified material balances and storage constraints for FIS states. The tasks related to products with no demand are removed before hand. For fair comparison with proposed model no prefixing of variables is considered. The results for all problem instances are shown in Table 5. In all the cases the proposed model leads to decrease in number of events by one or two with corresponding decrease in problem size relative to VS model. But due to the presence of recycle streams, the proposed model performs better than the VS model (wrt CPU time) in 6 out 14 instances (42.9%) only. Especially for problem instances P9, P11, and P14, it takes more time to prove optimality. Hence, for problems involving recycle streams, there is still scope for further improvement in the proposed hybrid approach for future work. Overall Performance. The overall performance of the proposed approach across all examples is summarized in Table 6, where it can be observed that the proposed approach performs better than VS model (wrt CPU time) in 23 out of 36 instances (63.9%), coupled with a reduction in number of events of the order of two to three. In the early times, when unit specific event based models were introduced two decades ago, they gave two to three event reduction (in most of the benchmark examples) compared to slot based/global event based models. Now, the proposed approach offers further reduction of another two to three events, which is quite significant. However, since the production and consumption tasks of a given material/state are modeled to occur at the same event, but they have to occur in a sequential manner

Table 6. Comparison of Overall Performance of Proposed Model with VS30 number of instances in which proposed model has same or better CPU time with prefixing (Tables1−4) without prefixing (Tables S1 to S4) without prefixing (Table 5) overall performance (without prefixing)

number of instances in which VS model has better CPU time

For Problems without Recycle 15/22 (68.2%) 7/22 (31.8%) 17/22 (77.3%)

5/22 (22.7%)

For Problems with Recycle 6/14 (42.9%) 8/14 (57.1%) 23/36 (63.9%)

13/36 (36.1%)

with consumption tasks starting after production task finishes, as given in eqs 13 and 14, this could be the reason for slow convergence in some cases. Therefore, it appears that now we have reached a plateau/critical point (in terms of reduction in events) where the problem is perhaps emerging in another dimension (which requires further study and research). Since it gives reduction in number of events in almost 100% of the cases, coupled with better computational performance in majority of cases (77% for problems involving no recycle streams), we hope that it is still a promising direction for further research work. The proposed model should be further customized and extended for sequential multistage processes, to further probe the remark that I


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J = units Ji = units suitable for task i N = events S = states SR = raw materials Sp = final products SSE = intermediates which are not involved in recycle loop SRC = intermediates which are involved in recycle loop Sdfis, SZW, Snis = intermediates with dedicated FIS, ZW, and NIS policies

it is now possible to have a Gantt chart using just one event for all the first tasks occurring in each stage.

5. CONCLUSION A new variant of unit specific event based model is proposed for batch scheduling based on the idea to allow production and consumption tasks of a given state to take place at same event, unlike the earlier approaches. The unit specific events across different units/stages are modeled in their true sense, where the first event in each unit actually takes up first event. The proposed approach is limited to problems that do not have recycle streams, although a hybrid approach has been suggested to handle recycle streams, and different reformulations are presented for both cases. In all the benchmark examples considered, the proposed model results in reduction in number of events required (of the order of two to three) for finding same quality optimal solutions, leading to fewer constraints and variables. For problems having no recycle streams, the proposed model also results in better computational performance in 77% of the cases considered.

Parameters

Bmin = minimum batch size for task i i Bmax = maximum batch size for task i i ST0s = initial amount of state s STmax = maximum capacity of storage for state s s αi = fixed processing time of task i βi = variable term of the processing time of task i ρis = fraction of state s produced (positive) or consumed (negative) by task i H = time horizon Ps = price of state s Δn = maximum number of spanning events M = large positive number Ds = demand for state s

■

APPENDIX A Equation A.1 is a constraint used in the Shaik and Floudas23 model which enforces alignment for different tasks occurring in different units. T s(i , n + 1) ≥ T f (i′, n) − M(1 −

∑

Binary Variables

w(i′, n′, n))

w(i, n, n′) = assignment variable for task i that starts at n and ends at n′

n ′∈ N n −Δn ≤ n ′≤ n

Positive Variables

∀ s , i , i′, j , j′, n ∈ N , n < N , i ∈ Ij , i′

■

∈ Ij ′ , i ≠ i′, j ≠ j′, i ∈ Isc , i′ ∈ Isp

(A.1)

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b00519. Tables S1−S4 (PDF)

■

■

AUTHOR INFORMATION

REFERENCES

(1) Kondili, E.; Pantelides, C. C.; Sargent, R. W. H. A General Algorithm for Short-Term Scheduling of Batch Operations. Part 1. MILP Formulation. Comput. Chem. Eng. 1993, 17, 211. (2) Pantelides, C. C. Unified Frameworks for Optimal Process Planning and Scheduling. Proceedings of the Second International Conference on Foundations of Computer-Aided Process Operations (FOCAPO); 1994; p 253. (3) Sundaramoorthy, A.; Karimi, I. A. A Simpler Better Slot-Based Continuous Time Formulation for Short-Term Scheduling in Multipurpose Batch Plants. Chem. Eng. Sci. 2005, 60, 2679. (4) Majozi, T.; Zhu, X. X. A Novel Continuous-Time MILP Formulation for Multi- Purpose Batch Plants. Ind. Eng. Chem. Res. 2001, 40, 5935. (5) Sanmarti, E.; Holczinger, T.; Puigjaner, L.; Friedler, F. Combinatorial Framework for Effective Scheduling of Multipurpose Batch Plants. AIChE J. 2002, 48, 2557. (6) Shaik, M. A.; Vooradi, R. Unification of STN and RTN Models for Short-Term Scheduling of Batch Plants with Shared Resources. Chem. Eng. Sci. 2013, 98, 104. (7) Floudas, C. A.; Lin, X. Continuous-Time versus Discrete-Time Approaches for Scheduling of Chemical Processes: A Review. Comput. Chem. Eng. 2004, 28, 2109. (8) Mendez, C. A.; Cerda, J.; Grossmann, I. E.; Harjunkoski, I.; Fahl, M. State-of-the-art Review of Optimization Methods for Short-Term Scheduling of Batch Processes. Comput. Chem. Eng. 2006, 30, 913.

Corresponding Author

*E-mail: [email protected]. ORCID

Munawar A. Shaik: 0000-0002-4364-483X Present Address †

Department of Chemical Engineering, National Institute of Technology (NIT), Warangal, India. Notes

The authors declare no competing financial interest.

■

b(i, n, n′) = amount processed by task i that starts at n and ends at n′ ST0(s) = amount of state s required from external resources ST(s, n) = excess amount of state s that needs to be stored at event n Ts(i, n) = time at which task i begins at event n Tf(i, n) = time at which task i ends at event n MS = makespan time

NOMENCLATURE

Indices

i, i′ = tasks j, j′ = units n, n′, n″ = events s = states Sets

I = tasks Ij = tasks suitable in unit j Ips = producing tasks for state s Ics = consuming tasks of state s J


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(31) Vooradi, R.; Shaik, M. A. Rigorous Unit-Specific Event-Based Model for Short-Term Scheduling of Batch Plants with Conditional Sequencing and Unit-Wait Times. Ind. Eng. Chem. Res. 2013, 52, 12950. (32) Vooradi, R.; Shaik, M. A. Advanced Unit-Specific Event Based Modeling Approach for Short-Term Scheduling of Batch Plants. AIChE Annual Meeting; Pittsburgh, USA, 2012. (33) Shaik, M. A. Evolution of Unit-Specific Event Based Models in Batch Process Scheduling. In Synthesis, Design, and Resource Optimization in Batch Chemical Plants; Majozi, T., Seid, E. R., Lee, J.Y., Eds.; CRC press: Taylor & Francis, 2015; Chapter 4, p 65. (34) Shaik, M. A.; Floudas, C. A.; Kallrath, J.; Pitz, H.-J. Production Scheduling of a Large-Scale Industrial Continuous Plant: Short-Term and Medium-Term Scheduling. Comput. Chem. Eng. 2009, 33, 670. (35) Yadav, S.; Shaik, M. A. Short-Term Scheduling of Refinery CrudeOil Operations. Ind. Eng. Chem. Res. 2012, 51, 9287. (36) Kallrath, J. Planning and Scheduling in the Process Industry. OR Spectrum. 2002, 24, 219. (37) Wang, S.; Guignard, M. Hybridizing Discrete- and ContinuousTime Models for Batch Sizing and Scheduling Problems. Comp. Oper. Res. 2006, 33, 971. (38) Janak, S. L.; Floudas, C. A. Improving Unit-Specific Event Based Continuous-Time Approaches for Batch Processes: Integrality Gap and Task Splitting. Comput. Chem. Eng. 2008, 32, 913.

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