Rigorous Unit-Specific Event-Based Model for Short-Term Scheduling

Article pubs.acs.org/IECR

Rigorous Unit-Specific Event-Based Model for Short-Term Scheduling of Batch Plants Using Conditional Sequencing and Unit-Wait Times Ramsagar Vooradi and Munawar A. Shaik* Department of Chemical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India ABSTRACT: Continuous-time models have evolved as a promising tool for formulating problems related to short-term scheduling. This article presents an analysis, advantages, and limitations of recent models proposed in the literature for shortterm scheduling of batch plants based on unit-specific event-based time representation. The purpose of this study is to generalize and enhance previous unit-specific event-based scheduling models for efficient handling of various issues such as nonsimultaneous material transfers, comprehensive sequencing constraints for handling of different storage policies, unit-wait policies, and utility resources. Accordingly, a rigorous unit-specific event-based model has been proposed that allows for conditional sequencing of each production and consumption task, only if the material produced by a given production task is used by a given consumption task. The proposed approach leads to reduction in number of events, and the model can effectively handle scheduling problems with different storage policies: UIS, FIS, NIS, and ZW. It can can also handle different unit-wait policies. The proposed conditional sequencing concept is also extended for efficient handling of utility resources, thus resulting in further reduction in number of events required compared to the published literature. determined. On the other hand, precedence-based models23,24 are formulated based on the concept of batch precedence and are classified into three types: general precedence, immediate precedence, and unit-specific immediate precedence. These formulations do not require postulation of any event points or time slots. These models have shown better computational results for scheduling large scale industrial sequential problems.25 However, handling of storage and resources is not straightforward for these models compared to other classes of models (slot based/event based). Unit-specific event-based models7−10,12,13,15,16 define events on a unit basis by allowing tasks corresponding to the same event point but in different units to take place at different times. 1.1. Review of Unit-Specific Event-Based (a.k.a. UnitSlot or Multiple-Time-Grid) Models. Several continuoustime models have been proposed in the literature for short-term scheduling of multipurpose batch plants based on unit-specific event-based time representation. After Ierapetritou and Floudas7 introduced this concept, the scheduling formulations witnessed a revolutionary change, allowing faster computational times with a lesser number of events for solving benchmark problems. Later, the limitations of this approach were overcome systematically by several researchers. Unit-specific event-based approaches have established advantages and have now replaced the earlier global-event- and process-slot-based approaches. The researchers have now adapted this modeling paradigm, although using other variations such as unit slots or multipletime-grid approaches. Janak et al.8 proposed ways to handle different storage policies and resource constraints by allowing tasks to occur over multiple events and reported better results

1. INTRODUCTION Short-term scheduling of batch plants has been an important research area in the past two decades for achieving industrial objectives such as maximization of profit, minimizing unit idle times, minimization of makespan, and efficient use of limited resources.1−3 Several modeling approaches have been proposed in the literature for short-term scheduling of batch processes. On the basis of the time representation used, these models are classified into two groups: discrete-time models and continuous-time models. Extensive reviews were written by Mendez et al.,1 Floudas and Lin,2 Maravelias,3 Shaik et al.,4 Pitty and Karimi,5 and Sundarmoorthy and Maravelias,6 who presented comparisons of different models and discussed the associated challenges. Due to smaller model size and more flexibility in timing decisions, continuous-time models are well acknowledged in the literature. Based on the model framework, the continuous-time models are further classified into global-eventbased, slot-based, unit-specific event-based, and precedencebased formulations. Among them unit-specific event-based models (a.k.a. unit slots or multiple-time grid) have evolved as a better alternative,4,7−16 as they generally require a lesser number of events to find optimal schedules compared to single-time-grid models (global-event-based17,18 and synchronized/process slot-based19,20 models). Global-event-based models are generally used for networkrepresented processes. Similar to discrete-time models, these models divide the time horizon into unequal parts using events or time slots that are common across all tasks and units.17,18 Slot-based models are derived based on the concept of time slots, which are predefined time intervals with unknown duration. Slot-based models are classified into two types, synchronous slots19−21 and asynchronous slots,14,22 which are similar to global events and unit-specific events, respectively. The time slots are used to align processing tasks of specified products at each stage. The product assignment to each slot and its associated processing times are variables to be © 2013 American Chemical Society

Received: Revised: Accepted: Published: 12950

November 29, 2012 April 26, 2013 August 6, 2013 August 6, 2013 dx.doi.org/10.1021/ie303294k | Ind. Eng. Chem. Res. 2013, 52, 12950−12972

Industrial & Engineering Chemistry Research

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section 1.3) reveals that their model has some limitations and leads to real-time storage violations. 1.2. Definitions. In this section, we introduce some terminology and provide definitions in the context of sequencing constraints for alignment of producing and consuming tasks of a given intermediate state. Explicit sequencing constraints between producing and consuming tasks are generally not required for global-event- and processslot-based models, whereas unit-specific event-based models require explicit sequencing constraints. These constraints can be classified into two types: (i) unconditional sequencing and (ii) conditional sequencing. 1.2.1. Unconditional Sequencing. In unconditional sequencing models7−10,12−15 different tasks in different units are always aligned without monitoring the actual material flows. For instance, the constraint for different tasks in different units used by Vooradi and Shaik15 given in eq B.1 of Appendix B aligns the start time of a consuming task at the next event to be later than the finish time of a producing task at the current event. These models assume that consumption tasks at event n + 1 are always aligned with production tasks at event n irrespective of whether the material produced from a production task is actually used or not. Similarly, for intermediate states having the restriction of either zero-wait (ZW) policy or no intermediate storage (NIS) or dedicated finite intermediate storage (DFIS) cases, constraint B.2 of Appendix B (along with eq B.1) enforces the no-wait condition required for different tasks taking place in different units. However, for the case of DFIS, if material produced by the production tasks at event n is within the storage limit, then the no-wait condition is not required. These two limitations are addressed by Seid and Majozi,16 where they have shown that it is possible to reduce the number of events required by relaxing these assumptions appropriately. 1.2.2. Conditional Sequencing. In conditional sequencing models, production and consumption tasks are aligned conditionally and only if the material produced by a given production task is actually used by a given consumption task, or if the material to be stored is not within the storage capacity. Conditional sequencing can be further classified into two types: partial-conditional sequencing and rigorous-conditional sequencing. Partial-Conditional Sequencing. Recently, Seid and Majozi16 proposed a unit-specific event-based model where the producing and the consuming tasks of an intermediate state are aligned only when a consuming task actually uses the material from a producing task. However, their model aligns all production tasks with all consumption tasks even if a single consumption task uses the material from a production task. This type of alignment is referred to as partial-conditional sequencing here (refer to eqs A.9−A.12 given in Appendix A). The same is true for alignment of producing and consuming tasks of DFIS states, that when the material to be stored is not within its storage capacity, then all producing and all consuming tasks are aligned irrespective of whether such alignment is required for all tasks are not (refer to eqs A.13 and A.14 in Appendix A). Compared to unconditional sequencing, this feature offers a reduction in the number of events required. Rigorous-Conditional Sequencing. As proposed later in this study, here, production and consumption tasks are aligned rigorously by accurately monitoring the material flow from each production task to each consumption task. A comparison of partial vs rigorous conditional sequencing is presented in section 1.3. In this study, it is shown later that it is possible to

compared to the global-event-based model of Maravelias and Grossmann.17 Shaik et al.4 presented an improved version of the model of Ierapetritou and Floudas7 and presented a comparative study of different time representations for unlimited storage policy. Shaik and Floudas10 identified that the model of Ierapetritou and Floudas7 is applicable for only unlimited storage case. They proposed a resource−task network (RTN) based formulation using unit-specific events for handling dedicated finite storage cases without the need for considering storage as a separate task. This model also offered a unification of state−task network (STN) and RTN based models for problems involving no resources. They have shown that the resulting constraints in STN and RTN based models are similar except for the allocation constraints. Castro and Novais11 proposed a multiple-time grid model for short-term scheduling of multistage multiproduct batch plants with an unlimited intermediate storage (UIS) policy. They used explicit time grids for intermediate storage units, which resulted in a lesser number of events compared to the two-index model of Shaik and Floudas.10 However, their model is applicable only for UIS cases, whereas the model of Shaik and Floudas10 is general and is applicable for finite intermediate storage (FIS) cases also. Later, it was identified9,12 that tasks need to be allowed to occur over multiple events (a.k.a. task splitting) for problems involving no resources as well, in addition to the problems involving resource constraints. Shaik and Floudas12 proposed a generic unified modeling approach for both cases of with and without resources, using three-index binary and continuous variables. Their unified model efficiently reduces to the simple case of no resources, when resources are not present. Li and Floudas13 extended the model of Shaik and Floudas12 by considering a postprocessing unit wait policy (i.e., material allowed to wait in unit after processing) and proposed a framework for the determination of the optimal number of event points based on the analysis of critical units and states. Recently, Vooradi and Shaik15 proposed an improved version of the Shaik and Floudas12 model using the concept of active task resulting in a compact model formulation and allowing postprocessing unit waiting similar to the model of Li and Floudas.13 In the models of Shaik and Floudas10,12 nonsimultaneous material transfers are not allowed and they assume no waiting of material in a unit except at the last event, unlike the models of Li and Floudas13 and Vooradi and Shaik,15 which allow postprocessing unit wait policy. Susarla et al.14 proposed a unit-slot-based model that can handle nonsimultaneous material transfers, and various unit-wait and storage policies that required a lesser number of time slots compared to single-time-grid approaches. However, all the unitspecific event-based models in the literature assume unconditional sequencing of producing and consuming tasks (i.e., consumption tasks are always aligned with production tasks irrespective of whether the material produced from a production task is actually used or not). Additionally, all the unit-specific event-based models in the literature assumed unconditional alignment of producing and consuming tasks irrespective of whether there is enough storage capacity or not for storing the amount produced at previous event. Recently, Seid and Majozi16 proposed a unit-specific event-based model with conditional sequencing where production and consumption tasks are aligned conditionally. Although their model reported a lesser number of events to find optimal solutions compared to the published literature for the examples considered in their study, a detailed analysis (presented in 12951

dx.doi.org/10.1021/ie303294k | Ind. Eng. Chem. Res. 2013, 52, 12950−12972


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further reduce the number of events required through rigorous alignment compared to partial alignment as done by Seid and Majozi.16 However, before discussing the proposed model, a detailed analysis of the model of Seid and Majozi16 is presented in section 1.3 to identify its limitations and the potential for improvements. 1.3. Analysis of Seid and Majozi16 (S&M) Formulation. This section presents a detailed analysis of the recent unitspecific event-based model of Seid and Majozi.16 Their model is based on a state−sequence network (SSN). To avoid confusion with the nomenclature used in this work, their model is rewritten using the STN framework as given in Appendix A (referred to as model M1). Advantages. The model M1 has the following advantages over the literature models. (i) For the first time they introduced conditional sequencing where producing and consuming tasks of an intermediate state are aligned only when a consuming task actually uses the material from a producing task. (ii) Similarly, they introduced conditional alignment to handle dedicated finite storage. (iii) Their model requires a lesser number of events to find optimal solutions. Partial-Conditional vs Rigorous-Conditional Sequencing. One of the important issues is that Seid and Majozi16 used partial conditional sequencing and their model aligns a production task with all consumption tasks even if a single consumption task uses material from that production task. Equation A.10 of Appendix A aligns a production task at event n − 1 with all consumption tasks at event n even if a single consumption task uses material from a production task. There is further scope to reduce the number of events if the material flow from each production to each consumption task is monitored accurately. The following case study provides a comparison between the best possible solution (using rigorousconditional sequencing) and the solution obtained by using model M1. Case Study 1. This case study has six tasks occurring in four units processing two raw materials (S1 and S4) and two intermediate states (S2 and S5). The state−task network is given in Figure 1, and relevant problem data are given in Table

Table 1. Data of Parameters for Case Studies and Examples

task task 1 task 2 task 3 task 4 task 1 task 2 task task task task

3 4 5 6

task 1 task 2 task 3

heating reaction 1 reaction 2 reaction 3 separation heating 1 heating 2 reaction 1 reaction 2 reaction 3 separation mixing

Figure 1. State−task network representation for case study 1. task task task task task task

1. The intermediate states S2 and S5 have a maximum storage capacity of 10 mass units (mu), state S2 has an initial storage of 5 mu, and raw materials are available as and when required. The objective is maximization of profit over a given time horizon of 4 h. The price of products S3 and S6 is $1/mu. This case study is solved using model M1 for two event points. The optimal solution gave an objective value of $22. The resulting Gantt chart is shown in Figure 2a. However, the best possible solution using two events is $27, for which the Gantt chart is shown in Figure 2b. In the Gantt chart of Figure 2, event and task numbers are shown inside a box which represents the occurrence as well as the duration of a task. The amount processed is shown above the box. In the Gantt chart shown in Figure 2a, state S2 produced at event n by task i = 1 is consumed by task i = 5 at event n + 1. According to eq A.9

1 2 4 3 5 6 1 2 3 4 5 6 7 8 1 2 3 4 5 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11

1 2 3 4 5 6

1 2 3 4 5 6

task 1 task 2

1 2 3 4 5 6 7

task 3 task 4

12952

i

unit j

αi (h)

Case Study 1 2 1 1 3 2 3 Case Study 2 unit 1 2 unit 2 2 unit 3 2 unit 4 2 unit 3 3 unit 4 3 unit 1 3 unit 2 3 Example 1 unit 1 1.33 unit 2 1.33 unit 3 1 unit 4 0.667 unit 5 0.667 Example 2 heater 0.667 reactor 1 1.334 reactor 2 1.334 reactor 1 1.334 reactor 2 1.334 reactor 1 0.667 reactor 2 0.667 separator 1.3342 Example 3 heater 0.667 heater 1 reactor 1 1.333 reactor 2 1.333 reactor 1 0.667 reactor 2 0.667 reactor 1 1.333 reactor 2 1.333 separator 2 mixer 1 1.333 mixer 2 1.333 Example 4 unit 1 1.666 unit 2 2.333 unit 3 0.669 unit 3 0.667 unit 2 1.332 unit 1 1.5 Example 5 unit 1 2 unit 2 3 unit 3 2 unit 4 3 unit 5 2 unit 1 1 unit 2 3 unit unit unit unit unit unit

1 2 3 2 3 4

Bmin i (mu)

Bmax i (mu)

0 0 0 0 0 0

0 0 0 0 0 0

10 6 6 10 10 15

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

3 9 3 9 1 1 1 1

0.013 33 0.013 33 0.005 0.004 45 0.004 45

0 0 0 0 0

100 150 200 150 150

0.006 67 0.026 64 0.016 65 0.026 64 0.016 65 0.013 32 0.008 325 0.006 66

0 0 0 0 0 0 0 0

100 50 80 50 80 50 80 200

0.006 67 0.01 0.013 33 0.008 89 0.006 67 0.004 45 0.013 3 0.008 89 0.006 67 0.006 67 0.006 67

0 0 0 0 0 0 0 0 0 20 20

100 100 100 150 100 150 100 150 300 200 200

0.077 8 0.066 7 0.077 7 0.033 25 0.055 6 0.025

0 0 0 0 0 0

30 10 30 40 30 20

0 0 0 0 0 0 0

0 0 0 0 0 0 0

260 140 120 120 140 1 1

βi (h/mu)



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Figure 2. Production schedule for case study 1.

variable t(j1,n + 1) is equal to 1. Equation A.10 aligns start times of all the consumption tasks (i = 3 and i = 5) at n + 1 with the finish time of the production task (i = 1) at event n, thus resulting in a suboptimal solution using model M1 for two events due to partial-conditional sequencing, although a global optimal solution can be obtained at higher events. In the Gantt chart of Figure 2b, production and consumption tasks are aligned only when the material produced by the production task is used by the consumption task due to rigorous-conditional sequencing. Hence, task i = 3 is allowed to take place in unit J2. Limitations. Although the S&M model reported a lesser number of events to find optimal solutions compared to the published literature (based on unconditional sequencing) for the examples considered in their study, a detailed analysis reveals that their model has some limitations and leads to realtime storage violations. For instance, their model gives real-time storage violations in some cases due to improper handling of storage. The following subsections present the detailed analysis of some important constraints proposed by Seid and Majozi.16 (i) Preprocessing and Postprocessing Unit Waiting. If a material is allowed to wait in a unit before processing, then it is referred to as preprocessing unit wait (a.k.a. nonsimultaneous material transfer). On the other hand, if a material is allowed to wait in a unit after processing, then it is referred to as postprocessing unit wait in this work. In the model of Seid and Majozi16 eq A.13 activates the binary variable x(s,n) if the amount of state s available at event n exceeds the storage capacity. If an excess amount of material (material higher than the storage capacity) produced by a production task at event n − 1 is consumed by a consumption task at event n, then eq A.14 prevents overlapping of a consumption task at event n − 1 with preprocessing unit wait in a consumption unit at event n. However, there is a chance for overlapping between the preprocessing unit wait at event n and the other active processing tasks occurring at event n − 1 in that unit. Similarly, the constraints proposed for handling of postprocessing unit wait policy are inadequate, as there is a scope for overlapping between postprocessing unit wait at an event and other active processing tasks occurring at that event in a unit. Appropriate sequencing constraints to avoid these overlappings in handling of pre- and postprocessing unit wait times are missing in their model. A case study is presented here to illustrate this issue. Case Study 2. An intermediate state s is produced by tasks i = 1 and 2 and consumed by tasks i = 3 and 4. State s has a maximum storage capacity of 10 mu. Tasks i = 5, 6, 7, and 8 are independent tasks that directly produce final products from raw materials, but they share common units. The state−task network is given in Figure 3, and the task and unit related data are given in Table 1. Raw materials are available as and when required. The price of products is $1/mu.

Figure 3. State−task network representation for case study 2.

For the objective of maximization of profit over a fixed time horizon of 5 h, this case study is solved using two events. Model M1 reported an objective value of $16, but in the Gantt chart shown in Figure 4a there are preprocessing unit wait storage violations (overlapping with other tasks) in one of the processing units (J3 or J4). However, using two events, the optimal solution is shown in Figure 4b with an objective value of $15, which can be obtained by properly aligning production and consumption tasks to avoid storage violations. Task i = 3 is immediately aligned with task i = 1 so that the amount produced by task i = 2 is within the storage limit; hence, task i = 4 is free to start after the finish time of the production task i = 2. (ii) Real-Time Storage Violations. The model of Seid and Majozi16 does not consider explicit storage tasks, but it presents alternate constraints for handling finite storage. However, their model cannot avoid incidences of real-time storage violations due to active production and consumption tasks occurring at the same event, unlike the models of Shaik and Floudas.10,12 Consider Case Study 2 presented above for the objective of maximization of profit. The model of Shaik and Floudas12 requires three events to find a global optimal solution of $15. The Gantt chart is shown in Figure 5b (based on unconditional sequencing). However, the model of Seid and Majozi16 can give a schedule with better objective value ($16) as shown in the sample Gantt chart of Figure 5a, which has a real-time storage violation. Although the amount of intermediate state s produced at each event is within the storage limits, the total production of 12 mu of state s is beyond the storage capacity of 10 mu. From the above analysis, it is clear that the model of Seid and Majozi16 requires suitable modifications and several corrections. In addition, their model is not applicable for problems involving resource constraints. As shown later in this work, beyond partial-conditional sequencing there is further scope to reduce 12953



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Figure 4. Production schedule for case study 2 using two events.

Figure 5. Sample Gantt chart for case study 2 using three events.

Table 2. Comparison of Unit-Specific Event-Based Models issues\models continuous time representation sequencing of production and consumption tasks unit-wait policy

tasks allowed to occur over multiple events utility resources ZW/NIS

Seid and Majozi16

Vooradi and Shaik15

Susarla et al.14

unit-specific events

unit-specific events

unit slots

partial conditional

unconditional (always aligned)

unconditional (always aligned)

• can handle post- and preprocessing unit wait policies • sequencing constraints between unitwait and processing tasks are missing not modeled explicitly

can handle postprocessing unit wait policy

can handle post- and preprocessing unit wait policies

modeled using three-index variables and used Δn to control number of events over which a task can continue can be handled modeled using unconditional sequencing

modeled using 0−1 continuous variables, but without using Δn

not modeled • not modeled explicitly (but can be handled) • if modeled it will be similar to unconditional sequencing

the number of events by monitoring material flow from each production task to each consumption task (i.e., using rigorousconditional sequencing). 1.4. Comparison of Recent Unit-Specific Event-Based Models for Multipurpose Batch Plants. In this section, we present a comparison of recent unit-specific event-based models of Seid and Majozi,16 Vooradi and Shaik,15 and Susarla et al.14 in Table 2 based on different metrics. As discussed earlier, the partial-conditional sequencing used in the Seid and Majozi16 model offers a reduction in the number of events compared to unconditional sequencing based models.14,15 However, for the same number of events the model statistics for unconditional sequencing based models14,15 would be better due to a lesser number of binary and continuous variables. Therefore, this issue would be applicable for rigorousconditional sequencing based models also, as demonstrated in sections 4.1 and 4.2. The Seid and Majozi16 model can handle FIS states, and it can be extended to handle NIS and ZW states as well. However, due to the usage of partial-conditional

not modeled modeled using unconditional sequencing

sequencing, the alignment between production and consumption tasks for ZW states would result in unconditional sequencing. The model proposed by Susarla et al.14 can handle post- and preprocessing unit wait policies, but they also used unconditional sequencing. Among the models compared in Table 2, only the Vooradi and Shaik15 model can handle resources and the model reduces well to the simple case in the absence of resources using a common framework based on the parameter Δn, which controls the number of events over which a task is allowed to continue. In this work, a rigorous unit-specific event-based model is proposed by relaxing the unconditional sequencing assumption of the earlier models and by allowing nonsimultaneous material transfers (postprocessing and preprocessing unit wait policies). Several new constraints are proposed to address the above two issues in a comprehensive manner. Further, the concept of conditional sequencing has been extended for effective handling of NIS and ZW states and also for handling of utility resources. The objective is to explore a framework with rigorous four12954



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index variables and compare the performance of the model with relevance to reduction in number of events and increasing complexity. The rest of the paper is organized as follows. In section 2, the problem statement is presented, followed by a description of the proposed model in section 3. Several benchmark problems are solved in section 4 to demonstrate the effectiveness of the proposed approach.

Bimin w(i , n , n′) ≤ b(i , n , n′) ≤ Bimax w(i , n , n′) ∀ i ∈ I , n , n′ ∈ N , n ≤ n′ ≤ n + Δn

3.3. Material Balances. Equations 3 and 4 are the material balance equations for intermediate states that relate the excess amount of a state at a current event with the excess amount at a previous event, the amount of the state produced by the tasks that are ending at the previous event, and the amount of the state consumed by the tasks that are starting at the current event.

2. PROBLEM STATEMENT The scheduling problem addressed in this work can be stated as follows. Given (i) the scheduling horizon or demands for final products, (ii) the production recipe (i.e., processing times for each task in suitable units, relationship between tasks, states, and units), (iii) processing units and their capacity limits, (iv) storage units along with initial inventories, minimum capacities, and maximum capacities, (v) utilities available and suitable tasks requiring these utilities, and (vi) prices of final products, determine (i) the optimal sequence of tasks taking place in each unit, (ii) the start and end times of different tasks in each unit, (iii) the amount of material processed and stored at each time in each unit, and (iv) utility consumption profiles. The assumptions are as follow: (i) there are no interruptions or failures of units, (ii) there are negligible transfer times between units, (iii) batch processing times are a linear function of batch size, (iv) each storable state has its dedicated storage unit, (v) sequence dependent changeovers and setup times are not considered, (vi) unlimited preprocessing unit wait times are assumed for all states that have either DFIS or NIS policies, and (vii) task specific postprocessing unit wait times are assumed for each task. The different objectives considered are maximization of profit over a given scheduling horizon or minimization of makespan to produce specified demands of final products.

ST(s , n) = ST(s , n − 1) +

i ∈ Ij

∑

∑

+

∑ ρis i ∈ Isc

∑

b(i , n′, n − 1)

n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

∑

b(i , n , n′)

n ′∈ N n ≤ n ′≤ n +Δn

∀ s ∈ SI , n ∈ N , n > 1

ST(s , n) = ST0s +

∑ ρis

i ∈ Isc

(3)

∑

b(i , n , n′)

n ′∈ N n ≤ n ′≤ n +Δn

∀ s ∈ SI , n = 1

(4)

Equations 5 and 6 are the material balance equations for the raw material states. ST(s , n) = ST(s , n − 1) +

∑ ρis

∑

i ∈ Isc

b(i , n , n′)

n ′∈ N n ≤ n ′≤ n +Δn

∀ s ∈ SR , n ∈ N , n > 1 ST(s , n) = ST0(s) +

∑ ρis i ∈ Isc

(5)

∑

b(i , n , n′)

n ′∈ N n ≤ n ′≤ n +Δn

∀ s ∈ SR , n = 1

(6)

Equations 7 and 8 are the material balance equations for the final products. ST(s , n) = ST(s , n − 1) +

∑ ρis

i ∈ Isp

∑

b(i , n′, n − 1) ∀

n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

s ∈ SP , n ∈ N , n > 1

ST(s , n) = ST0s

(7)

∀ s ∈ SP , n = 1, n ∈ N

(8)

3.4. Unit-Wait Times. In this work, we allow unit-wait times for FIS and NIS intermediate states in the processing units and these are classified as (i) postprocessing unit wait times and (ii) preprocessing unit wait times. Postprocessing unit wait is the case where state s can wait in the same unit j after production. Preprocessing unit wait is the case where state s waits in unit j before processing. Here, we assume unlimited wait time for preprocessing unit wait. 3.4.1. Duration Constraints. Similar to Li and Floudas,13 postprocessing unit wait times are handled using duration constraints. If Δn = 0, then the finish time of a task that starts at the same event is calculated from eqs 9 and 10.

w(i , n′, n″) ≤ 1

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

∀ j ∈ J, n ∈ N

∑ ρis

i ∈ Isp

3. MATHEMATICAL FORMULATION The basic framework is adapted from the model of Vooradi and Shaik,15 and several new constraints are proposed to address the issues related to conditional sequencing and different unitwait policies in a comprehensive manner. For instance, fourindex variables are defined so that material flow from each producing task can be tracked, thus yielding further reduction in the number of events required to find optimal solutions. The proposed model can handle different storage policies (UIS, FIS, NIS, and ZW) and resource constraints. The sequencing constraints, storage constraints, and resource constraints are reformulated to enforce rigorous-conditional sequencing and to accommodate various unit-wait policies. The following constraints from eqs 1−15 are similar to those from the model of Vooradi and Shaik15 which are repeated here for completeness. 3.1. Allocation constraints. In every equipment unit, at the most one task can be active at each event as given by constraint 1.

∑

(2)

(1)

3.2. Capacity Constraints. Constraint 2 enforces that the batch size should be within the minimum and maximum limits. If a task is not active, then these constraints enforce zero batch size.

T f (i , n) ≥ T s(i , n) + αiw(i , n , n) + βi b(i , n , n) ∀ i ∈ Isp , s ∉ S zw , n ∈ N , Δn = 0 12955

(9)



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aligned with respect to the producing task. Equation 16 aids in conditional alignment of producing and consuming tasks. The four-index continuous variable b1(i,i′,s,n) is activated only if a consuming task i′ at event n uses material from a producing task i at event n − 1. Equations 17 and 18 are used to calculate the exact amount produced by production task i at event n − 1 that is actually consumed by consumption task i′ at event n using the variable b1(i,i′,s,n).

T f (i , n) ≤ T s(i , n) + αiw(i , n , n) + βi b(i , n , n) + UWiw(i , n , n) ∀ i ∈ Isp , s ∉ S zw , n ∈ N , Δn = 0

(10)

If Δn is nonzero, then the finish time of a task that started at an earlier event is calculated from eqs 11 and 12. T f (i , n′) ≥ T s(i , n) + αiw(i , n , n′) + βi b(i , n , n′) ∀ i ∈ Isp , s ∈ S I , n , n′ ∈ N , n ≤ n′ ≤ n + Δn , Δn > 0

− ∑ ρi ′ s i ′∈ Isc

(11)

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

+

ρis

(12)

3.5. Sequencing Constraints. 3.5.1. Same Task in the Same Unit. The constraint for the same task in the same unit is given by eq 13 for Δn = 0 and by eqs 13 and 14 for Δn > 0. These two constraints enforce that the finish time at the current event is equal to the start time at the next event if the task is active and not ending at current event n.

b(i , n′, n − 1) ≥

∑ b1(i , i′, s , n) i ′∈ Isc

n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

(17)

−ρi ′ s

∑

b(i′, n , n′) ≥

n ′∈ N n ≤ n ′≤ n +Δn

∑ b1(i , i′, s , n) i ∈ Isp

∀ s ∈ S I , s ∉ S zw , s ∉ S nis , i′ ∈ Isc , n ∈ N , n > 1

(13)

(18)

T s(i , n + 1) ≤ T f (i , n) + M ⎛ ⎞ ⎜ ⎟ ∑ ∑ w(i , n′, n″)⎟ ⎜1 − ⎜ ⎟ n ′∈ N n ″∈ N n −Δn < n ′≤ n n < n ″≤ n ′+Δn ⎝ ⎠

3.7. Alignment of Different Tasks in Different Units. Equation 19 activates a four-index binary variable if the material produced by production task at event n − 1 is used by the consumption task at event n. b1(i , i′, s , n) ≤ z(i , i′, s , n)ρis Bimax

(14)

∀ s ∈ S I , s ∉ S nis , i ∈ Isp , i′ ∈ Isc , j , j′

3.5.2. Different Tasks in the Same Unit. Constraint 15 is for different tasks occurring in the same unit, and is same as given by Vooradi and Shaik.15

∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N , n > 1

(19)

The rigorous-conditional sequencing is enforced using eq 20 only if the material produced by a production task at event n − 1 is actually used by a consumption task occurring at event n.

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

∑

(16)

∀ s ∈ S I , s ∉ S zw , s ∉ S nis , i ∈ Isp , n ∈ N , n > 1

∀ i ∈ I, n ∈ N, n < N

∀ i ∈ I , n ∈ N , n < N , Δn > 0

∑ ∑ b1(i , i′, s , n)

∀ s ∈ S I , s ∉ S zw , s ∉ S nis , n ∈ N , n > 1

∀ i ∈ Isp , s ∉ S zw , n , n′ ∈ N , n ≤ n′ ≤ n + Δn , Δn

T s(i , n + 1) ≥ T f (i , n)

b(i′, n , n′) ≤ ST(s , n − 1)

i ′∈ Isc i ∈ Isp

+ UWiw(i , n , n′) + M(1 − w(i , n , n′)) >0

∑ n ′∈ N n ≤ n ′≤ n +Δn

(15)

T s(i′, n) ≥ T f (i , n − 1) − M(1 − z(i , i′, s , n))

The constraint for different tasks in different units of Shaik and Floudas12 and Vooradi and Shaik15 given in eq B.1 of Appendix B always aligns an active production task with the consumption task even though the material produced by the production task is not used by the consumption task (unconditional sequencing). Seid and Majozi16 proposed partial-conditional sequencing, and as discussed in section 1, it is possible to further reduce the number of events through rigorousconditional sequencing by monitoring material flow from each production task to each consumption task using four-index variables so that there is a one-to-one mapping for the actual usage. Accordingly, the following new constraints are proposed for subsequent consumption of a state based on whether there is enough material in storage or not. If there is enough material in the storage at event n, then a consumption task at event n + 1 can start without depending on when the production task finishes at event n. 3.6. Sequencing Constraints Based on Whether There Is Enough Material in Storage or Not. If there is enough material in storage, then the consuming task need not be

∀ s ∈ S I , i ∈ Isp , i′ ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N , n > 1

(20)

Equation 21 prevents real-time storage violations for UIS and FIS states if a consumption task at event n uses material from the storage. ⎛ ⎜ T s(i′, n) ≥ T f (i , n − 2) − M ⎜1 − ⎜ ⎝

∑ n ′∈ N n − 2 −Δn ≤ n ′≤ n − 2

⎞ ⎟ w(i , n′, n − 2)⎟ ⎟ ⎠

∀ s ∈ S I , s ∉ S zw , s ∉ S nis , i ∈ Isp , i′ ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N , n > 1

(21)

Constraints 1−21 can be used for handling of intermediate states with UIS storage policy. To handle dedicated finite storage, auxiliary constraints were proposed by Shaik and Floudas10,12 without the need for considering explicit storage 12956



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tasks as given by eqs B.2 and B.3 in Appendix B. Equation B.2 enforces unconditional alignment between production and consumption tasks if both tasks are active (i.e., production and consumption tasks are aligned even if the total amount of material available for consumption is within the storage limit). In this work, we propose rigorous-conditional sequencing of processing tasks by introducing three four-index variables which appropriately align production and consumption tasks. If there is enough storage capacity for the material produced and stored together, then a consuming task need not be aligned with respect to a producing task. Accordingly, several new constraints are proposed to address this issue in a comprehensive manner. 3.8. Sequencing Constraints Based on Whether There Is Enough Storage Capacity or Not (FIS Storage Policy). In addition to the constraints 1−21, the following constraints are required to handle FIS policy. For FIS states constraint 22 enforces an upper bound on the excess amount stored at each event. Amount produced at the last event can wait in the same unit. ST(s , n) ≤ STmax s

∀ s ∈ S fis , n ∈ N

b2(i , i′, s , n) ≤ ρis Bimax (x(i , i′, s , n) + v(i , i′, s , n)) ∀ s ∈ S fis , i ∈ Isp , i′ ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N , n > 1

If material produced by a production task at event n − 1 is to be consumed immediately by a consumption task at event n, then eq 27 enforces the zero-wait condition and aligns production and consumption tasks at event n − 1 and n. T s(i′, n) ≤ T f (i , n − 1) + M(1 − x(i , i′, s , n)) ∀ s ∈ S fis , i ∈ Isp , i′ ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N , n > 1

∑ ρis ≤

∑

T f (i″ , n − 1) ≤ T f (i , n − 1) + M(1 − v(i , i′ , s , n)) ∀ s ∈ (S fis ∪ S nis), i ∈ Isp , i′ ∈ Isc , j , j′

(22)

∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , i″ ∈ Ij ′ , n ∈ N , n > 1 (28)

Equation 29 enforces the condition that material either waits in a unit or gets consumed, and accordingly only one binary variable can be active.

b(i , n′, n − 1) + ST(s , n − 1)

n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

STmax s

+

x(i , i′, s , n) + v(i , i′, s , n) ≤ 1 ∀ s ∈ S fis , i ∈ Isp , i′ ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′

∑ ∑ b2(i , i′, s , n) i ∈ Isp

i ′∈ Isc

∈ Ij ′ , n ∈ N

∀ s ∈ S fis , n ∈ N , n > 1

(23)

⎛ ⎜ T (i′, n) ≥ T (i , n) − M ⎜1 − ⎜ ⎝ f

∑

b(i , n′, n − 1) ≥

fis

∀s∈S ,i∈ −ρi ′ s

∑ n ′∈ N n ≤ n ′≤ n +Δn

∑ b2(i , i′, s , n) i ′∈ Isc

n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

Isp ,

n ∈ N, n > 1

b(i′, n , n′) ≥

s

∑ n ′∈ N n −Δn ≤ n ′≤ n

⎞ ⎟ w(i′, n′, n)⎟ ∀ ⎟ ⎠

s ∈ S fis , j , j′ ∈ J , n < N , i ∈ Ij , i′ ∈ Ij ′ , i ≠ i′, j ≠ j′, i ∈ Isc , i′

(24)

∈ Isp

(30)

3.9. ZW Storage Policy. For the problems involving ZW states, the model of Seid and Majozi16 does not offer any advantage in terms of further reduction in number of events compared to other models.10,12,14 In this work, the concept of rigorous-conditional sequencing is extended to handle sequencing of ZW and NIS states as well. It can give further reduction in number events required. Accordingly, only those producing and consuming tasks are aligned that are directly related to ZW/NIS states, unlike in the earlier models.10,12,14−16 Consider a sample Gantt chart shown in Figure 6 where one ZW intermediate state is produced by two production tasks ip1 and ip2 and consumed by two consumption tasks ic1 and ic2. To find the optimal schedule given in Figure 6a, the models of Seid and Majozi16 and Vooradi and Shaik15 require three events, whereas the best possible solution can be obtained using two events as shown in Figure 6b.

∑ b2(i , i′, s , n) i ∈ Isp

∀ s ∈ S fis , i′ ∈ Isc , n ∈ N , n > 1

(29)

Equation 30 is used to avoid real-time storage violations for production and consumption tasks occurring at the same event, similar to the model of Shaik and Floudas.12 This constraint is based on the assumption of unconditional sequencing between production and consumption tasks, to avoid the complications arising due to rigorous sequencing.

Equations 24 and 25 are used to calculate the amount produced by a production task i at event n − 1 that needs to be consumed by a consumption task i′ at event n, so as to avoid real time storage violation using the variable b2(i,i′,s,n). This variable is used to align production and consumption tasks using additional sequencing constraints. ρis

(27)

Equation 28 ensures no overlapping between the preprocessing unit wait of material at event n and other tasks occurring in the same unit at previous event n − 1.

In eq 23, a four-index continuous variable, b2(i,i′,s,n) is activated only if the total amount of material available for consumption at event n exceeds the storage capacity. i ∈ Isp

(26)

(25)

If b2(i,i′,s,n) has a nonzero value, then it means that state s produced by task i is beyond the maximum storage capacity and it indicates the amount that should be consumed by task i′. This variable will activate one of the binary variables of x(i,i′,s,n) or v(i,i′,s,n) in eq 26. If x(i,i′,s,n) is 1, then it means that the material produced by task i at event n − 1 is to be immediately consumed by task i′ at event n. Otherwise, if v(i,i′,s,n) is 1, then it means that the material produced by task i at event n − 1 is transferred and can wait in the consuming unit to be consumed by task i′at event n. 12957



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additional storage capacity, but the material can wait in either in the production or consumption unit. ∀ s ∈ (S zw ∪ S nis), n ∈ N

ST(s , n) = 0

For the problems involving ZW states, rigorous-conditional sequencing is proposed for the first time, by using two fourindex variables b1(i,i′,s,n) and z(i,i′,s,n). Here, b1(i,i′,s,n) is the amount of state s consumed immediately by task i′ at event n from the total amount produced by task i at event n − 1. In addition to eqs 1−8, 11, and 13−15, the following constraints are required to handle the ZW storage policy. For ZW states, the duration is exactly equal to the processing time as given in eq 31. T f (i , n) = T s(i , n) + αiw(i , n , n) + βi b(i , n , n)

∈ Ij ′ , n ∈ N , n > 1

T (i , n′) ≤ T (i , n) + αiw(i , n , n′) + βi b(i , n , n′) ∀ i ∈ Isp , s ∈ S zw , n , n′ ∈ N , n ≤ n′ ≤ n + Δn , Δn

∀ s ∈ S nis , i ∈ Isp , i′ ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′

Equation 33 for ZW and NIS states enforces that the amount produced by production tasks at event n − 1 should be consumed by one or more consumption tasks at event n.

∑

b(i , n′, n − 1) =

∈ Ij ′ , n ∈ N , n > 1

i ′∈ Isc

(33)

z(i , i′, s , n) + v(i , i′, s , n) ≤ 1

Equation 34 for ZW and NIS states enforces that the material consumed by consumption task at event n is received from one or more production tasks at event n − 1. −ρi ′ s

∑ n ′∈ N n ≤ n ′≤ n +Δn

b(i′, n , n′) =

∀ s ∈ S nis , i ∈ Isp , i′ ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N

∑ b1(i , i′, s , n) (34)

Equation 35 enforces the rigorous zero-wait condition and aligns the production and consumption tasks at events n − 1 and n, only if the material produced by production task at event n − 1 is consumed by the consumption task at event n. T s(i′, n) ≤ T f (i , n − 1) + M(1 − z(i , i′, s , n)) ∀ s ∈ (S zw ∪ S nis), i ∈ Isp , i′ ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N , n > 1

(40)

3.11. Resource Constraints. The concept of rigorousconditional sequencing is extended to handle tasks consuming the same utility resource (cooling water, steam, limited manpower, etc.). First of all, the balances for utility consumption are modified so that all utility resources are treated in a unified way similar to material states. Generally, in material balance equations a task consumes a state and produces another state, whereas a utility resource is used (shared) by all related tasks at the beginning of each task at event n and at the end of the task at event n the availability of the resource increases (referred as addition) by the same amount (that was consumed at the beginning). The resource addition at the end of event n − 1 along with the excess amount available at event n − 1 becomes the total amount of resource available at event n, and accordingly the resource balances are modified. 3.11.1. Resource Balances. The following is a simple and elegant resource balance equation written for different utility

i ∈ Isp

∀ s ∈ (S zw ∪ S nis), i′ ∈ Isc , n ∈ N , n > 1

(39)

Equation 40 enforces the condition that material either waits in a unit or gets consumed, and accordingly only one of the binary variables can be active, similar to eq 29.

∑ b1(i , i′, s , n)

∀ s ∈ (S zw ∪ S nis), i ∈ Isp , n ∈ N , n > 1

(38)

T s(i′, n) ≥ T f (i , n − 1) − M(1 − v(i , i′, s , n))

(32)

n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

(37)

Equation 39 enforces the rigorous-conditional sequencing by aligning the production and consumption tasks at event n − 1 and n.

+ M(1 − w(i , n , n′))

ρis

∀ s ∈ S zw

∀ s ∈ S nis , i ∈ Isp , i′ ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′

s

>0

n ′∈ N n −Δn ≤ n ′≤ n

b(i , n′, N ) = 0

b1(i , i′, s , n) ≤ (z(i , i′, s , n) + v(i , i′, s , n))ρis Bimax

(31)

For ZW states the finish time at higher Δn is calculated using eqs 15 and 32. f

∑

i ∈ Isp

3.10. NIS Policy. Similar to the ZW states, the concept of rigorous-conditional alignment of processing tasks is extended for NIS states, which can potentially give further reduction in the number of events required. The following constraints are required to enforce rigorous-conditional alignment between processing tasks that produce and consume the NIS states, in addition to eqs 1−15, 20, 28, and 33−36. Equation 38 activates a four-index binary variable if the material produced by a production task at event n − 1 is used by a consumption task at event n. If z(i,i′,s,n) is 1, then it means that the material produced by task i at event n − 1 is immediately consumed by consumption task i′ at event n. If v(i,i′,s,n) is 1, then it means that the material produced by task i at event n − 1 can wait in the consumption unit before being consumed at event n.

Figure 6. Sample Gantt chart for maximization of profit for ZW state.

∀ i ∈ Isp , s ∈ S zw , n ∈ N , Δn = 0

∑

(36)

(35)

Equations 36 and 37 enforce the condition that ZW states are consumed as and when they are produced, and hence, they cannot be stored at any event and that they cannot be produced at the last event point. Similarly, the NIS states have no 12958



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resources: such as cooling water (CW), low pressure steam (LPS), and high pressure steam (HPS). Equation 41 states that the excess amount of a resource at event n is adjusted by the excess amount of resource present at earlier event n − 1, the amount of resource addition at event n − 1, and amount of resource used at event n. In eq 42, the maximum availability for each utility resource is considered as the initial amount available.

γiu

∑ ⎛ ⎜ − ∑ ⎜γiu i ∈ Iu ⎜ ⎝

n ′∈ N n ≤ n ′≤ n +Δn

γi

+ δiu

∑ n ′∈ N n ≤ n ′≤ n +Δn

⎛ ⎜ − ∑ ⎜γiu i ∈ Iu ⎜ ⎝

∑

∑

b(i′, n , n′)

n ′∈ N n ≤ n ′≤ n +Δn

∑ b3(i , i′, s , n)

∀ u ∈ U , i′ ∈ Iu , n ∈ N , n > 1

(45)

Equation 46 activates a four-index binary variable only if the utility released by finishing task at event n − 1 is used by the task starting at event n. b3(i , i′, s , n) ≤ y(i , i′, s , n)AM u0 ∀ u ∈ U , i , i′ ∈ Iu , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′ (46)

Equation 47 enforces the rigorous-conditional sequencing by aligning the finishing time and starting time of two tasks at events n − 1 and n only if the utility released by the finishing task at event n − 1 is used by the task starting at event n.

∀ u ∈ U, n ∈ N, n > 1

T s(i′, n) ≥ T f (i , n − 1) − M(1 − y(i , i′, s , n))

∑

∀ u ∈ U , i , i′ ∈ Iu , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′

w(i , n , n′)

∈ Ij ′ , n ∈ N , n > 1

n ′∈ N n ≤ n ′≤ n +Δn

w(i , n , n′) + δiu

n ′∈ N n ≤ n ′≤ n +Δn

T s(i′, n) ≥ T f (i , n − 2) − M ⎛ ⎞ ⎜ ⎟ ∑ w(i , n′, n − 2)⎟ ⎜1 − ⎜ ⎟ n ′∈ N n − 2 −Δn ≤ n ′≤ n − 2 ⎝ ⎠

(42)

∑

(47)

Equation 48 prevents a real-time violation if the task at event n uses the excess utility available at event n − 1.

∀ u ∈ U, n = 1

n ′∈ N n ≤ n ′≤ n +Δn

∀ u ∈ U , i , i′ ∈ Iu , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N , n > 1

(48)

Because of the unified treatment of resources, these resource balances may overestimate the utility consumption in real time if excess resource is available. However, the actual utility profiles can be accurately plotted from the Gantt charts by using start and finish times of utility consumption as parameters after solving the model.

⎞ ⎟ b(i , n , n′)⎟ ⎟ ⎠

w(i , n , n′) = 0;

b(i , n , n′) = 0

∀ n′ < n (49a)

∑ ∑ b3(i , i′, s , n)

s

T (i , n) ≤ H ;

i ′∈ Iu i ∈ Iu

∀ u ∈ U, n ∈ N, n > 1

′u

∈ Ij ′ , n ∈ N , n > 1

⎞ ⎟ b(i , n , n′)⎟ ⎟ ⎠

≤ AM(u , n − 1) +

w(i′, n , n′) + δi

(44)

i ∈ Iu

3.11.2. Sequencing Constraints Based on Whether There Is Enough Excess Utility Available or Not. Based on the excess amount of a resource available at each event, conditional alignment is enforced among the different tasks related to the same utility. Equation 43 monitors the consumption of utility and a four-index continuous variable, b3(i,i′,s,n), is activated only if the excess amount of a utility available at the previous event is not enough for the consumption tasks at event n. This variable is used to align the utility related tasks using additional sequencing constraints. ⎛ ⎜ ∑ ⎜γiu i ∈ Iu ⎜ ⎝

∑ n ′∈ N n ≤ n ′≤ n +Δn

≥

(41)

AM(u , n) =

′u

w(i , n , n′) + δiu

⎞ ⎟ b(i , n , n′)⎟ ⎟ ⎠

AM u0

∑ b3(i , i′, s , n) i ′∈ Iu

∀ u ∈ U , i ∈ Iu , n ∈ N , n > 1

n ′∈ N n ≤ n ′≤ n +Δn

∑

b(i , n′, n − 1) ≥

n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

⎞ ⎟ b(i , n′, n − 1)⎟ ⎟ ⎠

∑

w(i , n′, n − 1)

∑

+ δiu

AM(u , n) = AM(u , n − 1) ⎛ ⎜ ∑ + ∑ ⎜γiu w(i , n′, n − 1) + δiu n ′∈ N i ∈ Iu ⎜ ⎝ n − 1 −Δn ≤ n ′≤ n − 1

n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

∑ n ′∈ N n − 1 −Δn ≤ n ′≤ n − 1

f

T (i , n) ≤ H

(49b)

3.12. Bounds on Variables. General bounds are added to different variables in eqs 49a and 49b. In eq 49a, the three-index binary and continuous variables that have finishing events earlier than the starting events are fixed at zero before the model is solved. In eq 49b, the upper bound for start and end times of processing tasks are enforced to be the end of the time

(43)

Equations 44 and 45 are used to calculate the amount of utility released by finishing task i at event n − 1 which can be used by task i′ starting at event n, using the variable b3(i,i′,s,n). 12959



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Figure 7. State−task network representation for example 1.

Table 3. Computational Results for Examples 1 and 2 for Maximization of Profit with UIS model

a

events

RMILPa ($)

MILP ($)

V&S this work

4 4

1947.50 2000

1840.17 1840.17

V&S this work

5 5

2914.84 3000.00

2628.18 2628.18

V&S this work

6 6

3857.42 4000.00

3463.62 3463.62

V&S this work

9 9

5956.11 6600.90

5038.05 5038.05

V&S this work

4 4

1730.87 1730.87

1498.57 1498.57

V&S (Δn = 1) this work

6 5

2727.11 2436.69

1962.69 1962.69

V&S this work

7 6

3301.03 3076.62

2658.52 2658.52

V&S this work

8 8

4291.68 4291.68

3738.38 3738.38

CPU time (s)

nodes

binary variables

continuous variables

constraints

10 20

68 79

89 152

15 29

87 101

117 201

20 38

106 125

145 250

35 65

163 194

229 397

18 40

122 162

193 330

60 58

214 209

505 436

42 76

221 256

370 542

50 112

254 350

429 754

Example 1a (H = 8) 0.02 0 0.03 0 Example 1b (H = 10) 0.04 0 0.05 56 Example 1c (H = 12) 0.03 15 0.12 338 Example 1d (H = 16) 0.95 2 784 46.17 261 051 Example 2a (H = 8) 0.03 12 0.03 17 Example 2b (H = 10) 4.52 7 044 0.16 281 Example 2c (H = 12) 1.90 6 155 0.85 1 250 Example 2d (H = 16) 5.94 16 246 14.42 39 653

Reduced mixed-integer linear programming.

horizon. For a given STN, additionally it is possible to identify tasks that cannot occur at certain events and the corresponding binary and continuous variables can be eliminated. For instance, from the STN given for Case Study 1, it can be specified that task i = 6 cannot take place at the first event, because the feed for this task (S5) is not available in the initial inventory. Similarly, we can eliminate occurrences of tasks related to production of intermediates (without specified demands) at the last events. A more detailed procedure for a systematic identification of these tasks can be found in Janak and Floudas.9 3.13. Objective Function. 3.13.1. Maximization of Profit. For the objective of maximization of profit, the total amount of the final products produced by the end of the time horizon is considered along with its price in eq 50.

max profit =

∑ Ps ∑ s ∈ SP

n=N

∑ n ′∈ N n −Δn ≤ n ′≤ n

⎛ ⎜ ⎜ST(s , n) + ⎜ ⎝ ⎞ ⎟ b(i , n′, n)⎟ ⎟ ⎠

ST(s , N ) +

∑ ∑ ρis

n = N i ∈ Isp

∑

b(i , n′, n)) ≥ Ds

n ′∈ N n −Δn ≤ n ′≤ n

∀ s ∈ SP

(51)

T f (i , N ) ≤ MS

∀i∈I

(52)

The parameter H in the relevant constraints earlier must be replaced by either big-M or makespan (MS) appropriately. 3.14. Tightening Constraint. The sum of the durations of all tasks suitable in a unit should be less than the scheduling time horizon as given by eq 53.

∑∑

∑

i ∈ Ij n ∈ N

n ′∈ N n ≤ n ′≤ n +Δn

∀j∈J

∑ ρis

(αiw(i , n , n′) + βi b(i , n , n′)) ≤ H (53)

The resulting model is a mixed-integer linear programming (MILP) formulation which is iteratively solved over different numbers of events until the convergence of a given objective function. Additional iteration is also required over the Δn parameter, similar to the works of Shaik and Floudas12 and Vooradi and Shaik.15 In our experience, for most of the benchmark problems iterations over Δn = 0, 1, and 2 are sufficient to find optimal solutions. 3.15. Important Enhancements of the Proposed Model. The novel contributions of this work are in terms of proposing a rigorous unit-specific event-based model using four-index binary and continuous variables to effectively handle

i ∈ Isp

(50)

3.13.2. Minimization of Makespan. For the objective of minimization of makespan (MS), the specified demand constraints are given by eq 51 and the makespan itself must be greater than the finish time of all tasks by the last event in eq 52. 12960



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Table 4. Computational Results for Example 3 for Maximization of Profit with UIS model

events

RMILP ($)

MILP ($)

V&S this work

5 5

2100.00 2100.00

1583.44 1583.44


8 7

3618.64 3369.68

2358.20 2358.20

V&S this work

7 7

3465.62 3465.62

3041.26 3041.26

V&S this work

10 10

5225.86 5225.86

4262.80 4262.80

CPU time (s)

nodes

Example 3a (H = 8) 0.14 315 4.42 10 679 Example 3b (H = 10) 2346.00 3 823 296 1948.81 5 167 372 Example 3c (H = 12) 0.48 515 1.54 928 Example 3d (H = 16) 6.82 6767 499.64 748 595

binary variables


constraints

30 90

210 284

353 645

115 146

400 414

989 965

52 146

302 414

523 965

85 230

445 609

778 1445

alignment between the processing tasks related to ZW/NIS states. The model can handle postprocessing, preprocessing unit waiting, and nonsimultaneous material transfers in processing units. The proposed rigorous-conditional sequencing concept is extended for efficient handling of utility resources as well.

various issues such as nonsimultaneous material transfers, rigorous sequencing of producing and consuming tasks based on whether there is enough material in the storage or not, and rigorous sequencing of producing and consuming tasks based on whether there is enough storage capacity or not. The material flow from each production task to each consumption task is monitored by using the four-index variables. Accordingly, the sequencing constraints, storage constraints, and resource constraints are reformulated to enforce rigorousconditional sequencing and to accommodate various unit-wait policies. The new constraints are eqs 16−21, 23−29, 33−35, and 38−48. The proposed model can enforce conditional

4. COMPUTATIONAL RESULTS Several benchmark examples are considered from the literature involving short-term scheduling of batch plants with and without resource considerations. The computational performance of the proposed model is compared with the recent 12961



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Table 5. Computational Results for Examples 1−3 for Minimization of Makespan with UIS model

events

RMILP (h)

V&S this work

14 14

25.357 24.236

V&S this work

23 23

50.060 48.472

V&S this work

9 9

18.684 18.684

20 20

45.573 45.573

V&S this work

7 7

11.251 11.251

V&S this work

10 10

14.272 14.272


MILP (h)

CPU time (s)

Table 6. Data of Storage Capacities (mu) for Examples 1−5a

a

nodes

Example 1a (D4 = 2000 mu) 27.881 4.04 15 338 27.881 2452.68 7 579 141 Example 1b (D4 = 4000 mu) 52.072 2.27 9 419 52.072 832.08 2 962 465 Example 2a (D8 = D9 = 200 mu) 19.340 0.49 510 19.340 1.84 505 Example 2b (D8 = 500 mu, D9 = 400 mu) 46.114 5.64 516 46.114 3.66 512 Example 3a (D12 = 100 mu, D13 = 200 mu) 13.366 0.14 65 13.366 0.92 830 Example 3b (D12 = D13 = 250 mu) 17.025 0.40 271 17.025 3.15 1286

state

example 1

example 2

example 3

example 4

example 5

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13

UL 200 250 UL

UL UL UL 100 200 150 200 UL UL

UL UL 100 100 300 150 150 UL 150 150 UL UL UL

UL UL UL 10 5 10 UL UL

UL 10 10 UL UL

binary variables


constraints

60 110

258 309

374 647

105 191

429 516

626 1088

58 130

287 397

497 869

284 328

788 914

1900 2035

52 146

302 414

535 977

85 230

440 609

790 1457

7, and the relevant data are given in Table 1. The initial stock level for all intermediates is assumed to be zero and unlimited storage capacity is assumed for all states. The price of product S4 is $5/mu. For the objective of maximization of profit, this example is solved for four different time horizons (a, H = 8 h; b, H = 10 h; c, H = 12 h; and d, H = 16 h). The model statistics such as the number of binary and continuous variables, the number of constraints, the CPU time required to find a global optimum solution, the number of nodes taken to reach zero integrality gap, and the objective function at the relaxed node are reported in Table 3 for different time horizons. For this example, the proposed formulation requires the same number of events compared to the model of Vooradi and Shaik15 (referred to as V&S in this work). The rigorous-conditional alignment between the production and consumption tasks results in a greater number of binary variables, continuous variables, and constraints. The computational performance of the V&S model is better than that of the proposed model for the same number of events. 4.1.2. Example 2. This is the motivating example from Shaik and Floudas12 in which two different products are produced through five processing stages: heating, reaction 1, reaction 2, reaction 3, and separation, as shown in the STN representation in Figure 8 involving a recycle stream. The relevant data are given in Table 1. Since each of the reaction tasks can take place in two different reactors, each reaction is represented by two separate tasks. The initial stock level for all intermediates is assumed to be zero and UIS is assumed for all states. The price of products 1 and 2 is $10/mu. Computational results are reported in Table 3. For problem instances 2b and 2c, the proposed model requires one event less and Δn = 0 compared to the V&S model, and gives better RMILP and fewer nodes to reach zero integrality gap. 4.1.3. Example 3. This is a relatively complex example from Sundaramoorthy and Karimi20 which has 13 states and 11 tasks that can be performed in six units and involving a recycle stream. The STN for this example is shown in Figure 9, and the relevant data are given in Table 1. The initial stock level for all intermediates is assumed to be zero except for S6 and S7 with 50 mu available each. UIS is assumed for all states. The price of

UL = unlimited storage.

models from the literature. In sections 4.1 and 4.2, benchmark examples having no resource constraints are considered for the objectives of maximization of profit and minimization of makespan, respectively. Here, UIS is considered for all states. In sections 4.3 and 4.4, the same examples are solved using finite intermediate storage (FIS) for both objective functions. The two case studies presented in the Introduction are again solved in section 4.3, along with two additional benchmark examples to highlight nonsimultaneous material transfers. Finally, in section 4.5, two examples with resource constraints are presented. All the examples are solved using GAMS 23.5/ CPLEX 12.2 on a 2.66 GHz Intel Core 2 Duo processor with 3 GB RAM running on the Linux operating system. By default consider Δn as zero and the relative gap as zero wherever they are not mentioned in all the subsequent results. 4.1. Maximization of Profit with UIS and without Resource Considerations. 4.1.1. Example 1. This is a simple example from Shaik and Floudas12 that requires one raw material and produces two intermediates and one final product. The raw material is processed in three sequential tasks, where the first task is suitable in two units (J1 and J2), the second task is suitable in one unit (J3), and the third task is suitable in two units (J4 and J5). The STN for this example is shown in Figure 12962



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Table 7. Computational Results for Examples 1 and 2 for Maximization of Profit with FIS model

events

RMILP ($)

MILP ($)

V&S this work

4 4

2000.00 2000.00

1840.17 1840.17

SLK2 V&S S&M this work

7 6 6 5

4000.00 3361.12 4000.00 3000.00

2628.19 2628.19 2628.19 2628.19


9 8 7 6

4951.20 4419.88 4951.20 4000.00

3463.62 3463.62 3463.62 3463.62


12 11 9 9

6601.7 6236.04 6601.7 6600.90

5038.05 5038.05 5038.05 5038.05

V&S this work

4 4

1730.87 1730.87

1498.57 1498.57

SLK2 V&S (Δn = 1) S&M this work

7 7 6 5

2730.7 2775.41 2730.7 2436.69

1962.69 1962.69 1962.69 1962.69


8 8 7 6

3301.00 3350.09 3301.00 3076.62

2658.52 2658.52 2658.52 2658.52


9 9 8 8

4291.67 4438.96 4291.67 4291.67

3738.38 3738.38 3738.38 3738.38

CPU time (s)

nodes

Example 1a (H = 8) 0.03 3 0.04 9 Example 1b (H = 10) − 2764 0.11 175 − 6215 0.07 63 Example 1c (H = 12) − 284 342 0.89 2 156 − 15 568 0.15 237 Example 1d (H = 16) − 1 628 804 17.94 90 347 − 278 047 61.32 213 315 Example 2a (H = 8) 0.03 14 0.05 23 Example 2b (H = 10) − 19 043 109.2 222 524 − 23 837 0.23 254 Example 2c (H = 12) − 58 065 17.78 54 634 − 39 563 1.53 1 502 Example 2d (H = 16) − 32 351 73.15 206 213 − 138 508 36.12 70 063

binary variables


constraints

10 40

68 91

113 234

60 20 60 57

359 106 205 118

491 185 368 311

80 30 70 74

467 144 239 145

657 257 429 388

110 45 90 125

629 201 307 226

906 365 551 619

18 84

122 204

253 577

72 76 88 122

449 255 383 265

645 724 779 768

84 50 104 160

517 254 448 326

753 569 923 959

96 58 120 236

585 287 513 448

861 648 1067 1341

Figure 10. Gantt chart for example 2b for maximization of profit with FIS.

demand scenarios. In all three examples, M = 50 h is used in big-M constraints in the first instance and M = 100 h is used in the second instance. The computational results are reported in Table 5. In all the problem instances, the proposed model requires the same number of events compared to the V&S model. Hence, the computational performance of the V&S model is better than that of the proposed model. For the problems with UIS policy, the proposed model gave further reduction in the number of events compared to the unconditional sequencing based V&S model, although only in fewer cases. However, both models can be solved to zero optimality gap in reasonable computational time. For the same number of events, the V&S model performs better than the

products S12 and S13 is $5/mu. For the objective of maximization of profit, this example is solved for four different time horizons. The model statistics and computational results for different cases are reported in Table 4. For problem instance 3b our model requires one event less and Δn = 0 compared to the V&S model, and gives a better RMILP value. In all the other instances both models require the same number of events and, hence, V&S model solves faster and requires fewer binary and continuous variables and fewer constraints compared to the proposed model. 4.2. Minimization of Makespan with UIS and without Resource Considerations. For the objective of minimization of makespan all three examples are solved for two different 12963



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Table 8. Computational Results for Examples 3−5 for Maximization of Profit with FIS model

RMILP ($)

MILP ($)


7 6 5 5

2751.00 2750.96 2100.00 2100.00

1583.44 1583.44 1583.44 1583.44


9 9 6 7

3618.68 3618.64 2871.9 3369.68

2337.36c 2345.30c 2345.3c 2358.20


8 7 7 7

3465.62 3465.62 3465.62 3465.62

3041.26 3041.26 3041.26 3041.26

11 11 9 10

5225.86 5644.59 4653.1 5225.86

4241.5c 4262.80 4240.83c 4262.80

V&S S&Mb this work

2 2 2

631.25 650.0 650.0

560.07c 650.0 650.0

S&Mb this work

8 7 8

902.1 902.1 902.1

387.0d 386.0 386.0

S&Mb

2 3 2

24.5 34.01 27.0

22.0c 27.0 27.0

2 3 2 3

16.0 23.2 16.0 24.6

16.0d 16.0d 15.0 15.0


this work S&Mb this work a

events

CPU time (s)

nodes

Example 3a (H = 8) − 388 832 4.51 10 345 − 3 662 6.79 15 310 Example 3b (H = 10) − 872 204 40 000a 39 841 668 − 9 252 4159.45 7 501 570 Example 3c (H = 12) − 3 096 1.91 1 733 − 103 886 20.75 25 586 Example 3d (H = 16) − 705 090 3866.41 2 060 783 − 253 967 804.25 634 591 Example 4 (H = 6) 0.01 0 0.01 0 0.01 0 Example 5 (H = 9) 18.78 56 038 14.11 25 160 87.95 148 288 Case Study 1 (H = 4) 0.01 0 0.03 0 0.01 0 Case Study 2 (H = 5) 0.01 0 0.04 18 0.01 0 0.09 242

binary variables


constraints

102 41 107 210

655 256 418 360

1070 608 889 1165

136 137 131 334

853 457 516 528

1428 1403 1093 1749

119 52 155 334

754 302 604 528

1249 727 1297 1749

170 181 155 520

1051 571 570 780

1786 1755 1224 2625

7 21 16

51 64 55

66 103 109

93 157 182

250 254 292

543 683 792

19 30 24

61 90 57

103 165 115

20 31 28 48

69 101 67 104

107 173 127 222

Resource limit reached (relative gap = 0.018%). bS&M model as reproduced using STN representation. cSuboptimal. dReal-time violation.

objective of maximization of profit with finite dedicated storage capacity for some intermediate states. The relevant data are given in Table 6. Additionally, example 4 (case 1 of Seid and Majozi16) and example 5 are also solved to demonstrate the feasibility of pre- and postprocessing unit wait policies. The computational results of the proposed model are compared with the literature models of Susarla et al.14 (SLK), Seid and Majozi16 (S&M), and Vooradi and Shaik15 (V&S). The computational results for the SLK and S&M models are taken directly from their papers; hence, CPU times for these two models are not directly compared here due to differences in hardware. For the SLK model N events indicates N − 1 slots. In example 1, states S2 and S3 have finite intermediate storage capacity. For the objective of maximization of profit, the computational results for the FIS case are reported in Table 7. For problem instances 1b and 1c, the proposed model requires the minimum number of events and gives better RMILP values compared to the other recent models from the literature. For problem instance 1d, this work and S&M models require nine events whereas the V&S and SLK models require 11 events.


proposed model due to lesser complexity, as also mentioned in section 1.4. 4.3. Maximization of Profit with FIS and without Resource Considerations. The above three examples and the case studies discussed in the Introduction are solved for the 12964



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Figure 12. Production schedule for case 1 of Seid and Majozi16 (example 4 in this work).

Figure 13. STN representation for example 5.

Figure 14. Gantt chart for example 5 using model M1 for maximization of profit with FIS.

Figure 15. Gantt chart for example 5 for maximization of profit with FIS.

Example 2 is solved with finite intermediate storage for states S4, S5, S6, and S7. The computational results are reported in Table 7. For both examples 2b and 2c, the proposed model requires one event less compared to the S&M model and two events less compared to the V&S model. Figure 10 shows the Gantt chart for problem instance 2b. For example 3, the computational results for FIS case are reported in Table 8. In example 3, states S3, S4, S5, S6, S7, S9, and S10 are defined as finite intermediate states. Compared to the results reported for example 3 in Table 4 for UIS, the optimal objective value remains the same in all four scenarios for the FIS case as well, but only using the proposed model. For problem instances 3b and 3d, the S&M and SLK models reported suboptimal solutions (at lower events). The V&S model could not find the optimal solution for example 3b within the specified CPU time limit. Example 4 (case 1 of Seid and Majozi16) has eight states and six tasks that can be performed in three units. The STN for this

example is shown in Figure 11, and the relevant data are given in Table 1. The plant produces two products using different production paths (R1 and R2) by sharing common processing units. Tasks 1 and 6 are suitable in unit 1, tasks 2 and 5 are suitable in unit 2, and tasks 3 and 4 are suitable in unit 3. For the objective of maximization of profit over a given time horizon of 6 h, the computational results are reported in Table 8. The V&S model reported a suboptimal solution of $560.07 as it does not allow nonsimultaneous material transfers. SLK, S&M (as reproduced using the STN version, model M1, given in Appendix A), and the proposed models found an optimal solution of $650, as they allow nonsimultaneous material transfers. However, as discussed in section 1, the S&M model could not handle the pre- and postprocessing unit wait policies accurately. Consider the Gantt chart shown in Figure 12 for example 4 using the proposed model, which is also the same Gantt chart obtained using the S&M model (M1). In this Gantt chart, for the S&M model it appears that the material produced 12965



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Table 9. Computational Results for Examples 1−3 for Minimization of Makespan with FIS model

RMILP (h)

MILP (h)


17 20 13 14

24.2 24.492 24.2 24.236

28.772b 27.881 27.98b 27.881


23 29 22 22 23 24

48.5 48.789 48.5 48.472 48.472 48.472

56.432b 54.395b 53.8b 52.436b 52.236b 52.072


10 9 9 9

18.7 18.684 18.7 18.684

19.340 19.340 19.340 19.340

SLK2 V&S (Δn = 2) S&M this work (Δn = 1)

22 21 21 21

47.4 47.378 46.4 47.378

47.683 47.695b 47.683 47.687b

9 9 7 7

11.3 11.250 11.4 11.250

13.366 13.366 13.366 13.366

11 10 10 10

14.271 14.271 14.271 14.271

17.025 17.025 17.025 17.025

SLK2 V&S S&M this work SLK2 V&S (Δn = 1) S&M this work a

events

CPU time (s)

nodes

binary variables

Example 1a (D4 = 2000 mu) − 328 879 160 40 000a 84 604 531 90 − 4 804 688 130 3537.85 5 722 414 210 Example 1b (D4 = 4000 mu) − 1 522 250 220 40 000a 47 936 042 135 − 8 201 001 220 9317.99 16 304 301 346 40 000a 32 198 988 363 40 000a 21 524 058 380 Example 2a (D8 = D9 = 200 mu) − 175 107 108 1.58 547 58 − 1 452 700 136 2.95 503 274 Example 2b (D8 = 500 mu, D9 = 400 mu) − 302 206 252 40 000a 1 269 217 438 − 1 545 601 328 40 000a 913 878 876 Example 3a (D12 = 100 mu, D13 = 200mu) − 42 218 136 12.41 17 023 74 − 260 155 2.12 1 060 334 Example 3b (D12 = D13 = 250 mu) − 820 170 4.91 652 159 − 15 060 227 6.17 1 036 520


constraints

relative gap (%)

901 372 444 361

1330 694 796 1009

− 3.10 − −

1225 543 750 577 604 631

1828 1018 1345 1625 1702 1779

− 7.1 − − 0.33 0.99

658 287 579 509

983 657 1213 1541

− − − −

1474 967 1359 1387

2279 2841 2941 4459

− 0.66 − 0.65

859 394 605 528

1448 977 1299 1761

− − − −

1057 514 869 780

1806 1591 1911 2637

− − − −

Resource limit reached. bSuboptimal.

Figure 16. Gantt chart for example 1a for minimization of makespan with FIS.

to produce two products. Task i = 1 consumes state S1 and produces state S2; tasks i = 2 and 3 consume state S2 and produce state S3, and tasks i = 4 and 5 consume state S3 and produce product S4. Tasks i = 6 and 7 consume state S1 and produce product S5. The relevant data are given in Table 1. The intermediate states S2 and S3 have a maximum storage capacity of 10 mu. Raw material is available as and when required. The price of products is $1/mu. The objective is maximization of profit over a time horizon of 9 h, and the

by task 2 at event N1 waits in the consumption unit at event N2 for 1 h before processing. However, task 3 (which is not a consuming task for state S5) at event N1 could have overlapped with the preprocessing unit wait at event N2. However, the overlapping did not happen here, only by chance, since the start time of consumption task 5 is exactly equal to the finish time of production task 3. Example 5, given in Figure 13, has four tasks occurring in six units processing one raw material and two intermediate states 12966



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Table 11. State Related Data for Example 6 STmax (kg) s ST0s (kg) ps ($/kg)

F1

F2

I1

I2

I3

P1

P2

1000 400 0

1000 400 0

200 0 0

100 0 0

500 0 0

1000 0 30

1000 0 40

UIS case given earlier in Table 5. For example 2, in the first instance, all the models require the same number of events and report the same objective value compared to the UIS case. In the second instance, the optimal objective values are different for the FIS case compared to the UIS case given in Table 5 due to an unconverged solution. In example 3, for both the UIS and FIS cases the optimal objective values are the same for both scenarios. For example 3a, the proposed model and the S&M model require the same number and lesser number of events compared with unconditional alignment models. 4.5. Examples with FIS and with Resource Considerations. 4.5.1. Example 6. This example from Maravelias and Grossmann17 has four tasks occurring in three units processing seven states as shown in the STN of Figure 17. The corresponding data are given in Tables 10 and 11. Two reactors of type I (R1 and R2) and one reactor of type II (R3) are available, and four reactions are suitable in these reactors. Reactions T1 and T2 require a type I reactor, whereas reactions T3 and T4 require a type II reactor. In addition, the heat required for endothermic reactions T1 and T3 is provided by steam (HS) available in limited amounts. Reactions T2 and T4 are exothermic, and the required cooling water (CW) is also available in limited amounts. Each reactor allows variable batch sizes, where the minimum batch size is half the capacity of the reactor. The processing times and the utility requirements include a fixed term and a variable term that is proportional to the batch size. For the raw materials and final products, unlimited storage is available (say 1000 kg), while for the intermediates, finite storage is available. Based on the resource availability, two different cases are considered: in the first case (example 6a) the availability of both HS and CW is 40 kg/min and in the second case (example 6b) it is 30 kg/min. Also, two different objective functions, maximization of profit and minimization of makespan, are considered. For the objective of maximization of profit both cases are solved over a time horizon of 8 h. The model statistics and computational results are reported in Table 12. The proposed model requires a lesser number of events and Δn = 0 compared to the V&S model. For the objective of minimization of makespan, the two cases given above are solved for a fixed demand of 100 kg of P1 and 80 kg of P2. The computational results are reported in Table 13. For problem instance 6a, the proposed model requires six events and Δn = 0 compared to the V&S model which requires seven events and Δn = 1. In the constraints involving big-M terms, a common value of M = 10 was used.


results are given in Table 8. The model M1 (S&M) reported a better objective value of $387. However, the Gantt chart shown in Figure 14 has unit-wait storage violations in unit J1 at event N2. At event N1 task i = 1 produces 260 mu of intermediate state S2, of which 120 mu of state S2 is consumed by task i = 3 at event N2; the remaining 140 mu is stored in unit J1 at event N2 and consumed by task i = 2 at event N3. However, in unit J1 another instance of task i = 1 has started immediately at event N3 right after the finish time of task i = 1 at event 1. Thus, there is an overlapping of postprocessing unit wait with the processing task in the same unit, as shown using a shaded region in Figure 14. The proposed model reported the optimal solution of $386, and the Gantt chart is shown in Figure 15. The two case studies discussed in section 1 are also solved using the proposed model and the STN version of the Sied and Majozi16 model given in Appendix A (model M1). For the objective of maximization of profit, the computational results are reported in Table 8. For case study 1, the proposed model requires one event less compared to model M1 (S&M) to find the optimal solution. For case study 2, the proposed model reports the optimal solution using two events, whereas model M1 (S&M) reported a better objective value, but has unit-wait storage violations as shown in Figure 4a. For most of the problem instances of FIS, the proposed model offers further reduction in number of events compared to the unconditional sequencing based model of V&S and the partial conditional sequencing based model of S&M. 4.4. Minimization of Makespan with FIS and without Resource Considerations. For the objective of minimization of makespan the computational results for the three examples are given in Table 9 for two demand scenarios. For example 1, compared to the earlier results reported in Table 5 for UIS, the optimal objective value remains the same for FIS as well. For example 1a, the proposed model gave a better objective value of 27.881 h using 14 events requiring 3537.85 s to solve to zero integrality gap. The Gantt chart is shown in Figure 16. The proposed model requires seven events less compared to the V&S model and gives a converged solution. For example 1b, compared to the S&M, SLK, and V&S models, the proposed model gave a better converged optimal value of 52.436 h using 22 events. Using 24 events, the proposed model reported the best objective value of 52.072 h, which is same as that of the Table 10. Data for Example 6a T1 B R1 R2 R3

min

40 25 40

B

max

80 50 80

T2

T3

α

β

α

β

0.5 0.5

0.025 0.4

0.75 0.75

0.0375 0.06

α

0.25

T4 β

α

0.0125

0.5

T1 β

0.025

T2

T3

γiHS

δiHS

γiCW

δiCW

6 4

0.25 0.25

4 3

0.3 0.3

T4

γiHS

δiHS

γiCW

δiCW

8

0.4

4

0.5

B /Bmax in kg, α in h, β in h/kg, γ in kg/min, and δ in kg/min per kg of batch.

a min

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Table 12. Computational Results for Examples 6 and 7 for Maximization of Profit with Resources model

events

RMILP ($)

MILP ($)


6 6

10 713.81 10 713.81

5904.00 5904.00


6 5

8107.30 6213.28

5227.77 5227.77


9 7

21 394.73 18 639.64

13 000 13 000


9 7

24 960.52 19 030.98

16 350 16 350

CPU time (s)

nodes

Example 6a (H = 8) 1.48 1149 1.55 1935 Example 6b (H = 8) 0.56 715 0.09 54 Example 7a (H = 12) 287.29 187 058 1.39 1 078 Example 7b (H = 14) 223.90 170 263 0.36 574

binary variables


constraints

42 182

167 339

639 974

42 140

167 274

639 776

126 242

421 579

1535 1557

148 242

443 579

1677 1557

binary variables


constraints

54 182

200 339

774 981

24 182

149 339

484 981

Table 13. Computational Results for Example 6 for Minimization of Makespan with Resources model

events

RMILP (h)


7 6

5.077 5.077

V&S this work

6 6

5.336 5.077

MILP (h)

CPU time (s)

nodes

Example 6a (Dp1 = 100 kg, Dp2 = 80 kg) 8.50 2.59 1691 8.50 0.36 106 Example 6b (Dp1 = 100 kg, Dp2 = 80 kg) 9.025 0.22 49 9.025 0.28 162

H = 12 h; b, H = 14 h). The model statistics and computational results for these cases are reported in Table 12. In both problem instances, the proposed model requires Δn = 0 and two events fewer compared to the V&S model. The Gantt chart and the resource utilization levels for example 7b are shown in Figure 19.

5. CONCLUSION In this work, we proposed a rigorous unit-specific event-based model for short-term scheduling of batch plants by relaxing the assumptions of unconditional alignment of producing and consuming tasks and the assumptions made related to different unit-wait policies in the earlier models, thus giving further reduction in number of events compared to the case of partialconditional sequencing. The proposed model can effectively handle the cases related to nonsimultaneous material transfers through proper handling of preprocessing and postprocessing unit wait times. The concept of rigorous-conditional alignment has been extended to effectively handle problems involving NIS/ZW states and utility resources, thus offering further reduction in number of events required compared to the published literature for the examples presented in the computational results. Although the number of events can be reduced using the rigorous conditional sequencing, the model complexity however increases and the same issue will be addressed in a future publication.


4.5.2. Example 7. This example from Maravelias and Grossmann17 has 10 tasks occurring in 6 units processing 14 states as shown in the STN representation of Figure 18, and the relevant data are given in Tables 14 and 15. Raw materials F1 and F2, intermediates I1 and I2, final products P1−P3, and WS states have UIS capacity, while states S3 and S4 have FIS, states S2 and S6 have NIS, and states S1 and S5 have ZW policies. Three different renewable utilities are available for this process: with tasks T2, T7, T9, and T10 requiring cold water (CW); tasks T1, T3, T5, and T8 requiring low-pressure steam (LPS); and tasks T4 and T6 requiring high-pressure steam (HPS). The maximum availabilities of CW, LPS, and HPS are 25, 40, and 20 kg/min, respectively. For the objective of maximization of profit, this example is solved for two different time horizons (a, Table 14. Data for Example 7a unit Bmax (kg) α utility γ δ a

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

U1 52 2 LPS 3 2

U2 81 1 CW 4 2

U3 61 1 LPS 4 3

U1 52 2 HPS 3 2

U4 82 2 LPS 8 4

U4 82 2 HPS 4 3

U5 34 4 CW 5 4

U6 42 2 LPS 5 3

U5 32 2 CW 5 3

U6 43 3 CW 3 3

α in h, β in h/kg, γ in kg/min, and δ in kg/min per kg of batch. 12968



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Table 15. State Related Data for Example 7 STmax (kg) s ST0s (kg) Ps ($/kg)

F1

F2

I1

I2

S1

S2

S3

S4

S5

S6

P1

P2

P3

∞ 100 0

∞ 100 0

∞ 0 0

∞ 0 0

0 0 0

0 0 0

15 0 0

40 10 0

0 0 0

0 0 0

∞ 0 1

∞ 0 1

∞ 0 1

Figure 19. Gantt chart and resource utilization levels for example 7b for maximization of profit.

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APPENDIX A: STN VERSION OF SEID AND MAJOZI16 MODEL (MODEL M1) In this section, the state−sequence network (SSN) based model of Seid and Majozi16 is rewritten using the nomenclature of STN based approaches for uniformity in nomenclature with this work. The following nomenclature is used only in this section in addition to the common symbols listed at the end. w(i,n) = binary variable for assignment of task i at event n b(i,n) = batch size of by task i at event n u(i,n) = amount of material stored by task i in unit j (i ∈ Ij) at event n t(j,n) = binary variable associated with usage of state produced by unit j at event n x(s,n) = binary variable associated with availability of storage for state s at event n

A.2. Capacity Constraints

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

(A.2)

A.3. Material Balances

ST(s , n) = ST(s , n − 1) +

i ∈ Ij

∑ ρis b(i , n − 1)

i ∈ Isp

+

∑ ρis b(i , n)

∀ s ∈ (SI ∪ SR ), n ∈ N

i ∈ Isc

(A.3)

ST(s , n) = ST(s , n − 1) +

∑ ρis b(i , n)

i ∈ Isp

∀ s ∈ SP , n ∈ N

(A.4)

A.4. Duration Constraints (Batch Time as a Function of Batch Size)

A.1. Allocation Constraint

∑ w(i , n) ≤ 1

∀ i ∈ I, n ∈ N

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

∀ j ∈ J, n ∈ N

∀ i ∈ I, n ∈ N

(A.1) 12969

(A.5)



Article

A.5. Sequencing Constraints

A.8. Storage Constraints

A.5.1. Same Task in the Same Unit. s

f

T (i , n + 1) ≥ T (i , n)

∀ i ∈ I, n ∈ N

ST(s , n) ≤ STmax + s (A.6)

∀ s ∈ SI , n ∈ N

∑ u(i , n) i ∈ Isp

(A.15)

A.5.2. Different Tasks in the Same Unit. The constraint A.7 for different tasks in the same unit of Seid and Majozi16 has a typo and results in zero objective value. Hence, on the basis of the explanation given in that model, we consider eq A.7a, which is a standard equation for different tasks in the same unit used in the model of Shaik and Floudas.12

u(i , n) ≤ ρis b(i , n − 1) + u(i , n − 1) ∀ s ∈ SI , i ∈ Isp , n ∈ N

u(i , n) ≤ Bimax (1 −

(A.16)

∑ w(i′, n)) i ′∈ Ij

T f (i , n) ≥ T f (i′, n)

∀ i , i′ ∈ Ij , i ≠ i′, j ∈ J , n ∈ N

∀ s ∈ SI , i ∈ Isp , i ∈ Ij , n ∈ N

(A.7)

A.9. Time Horizon Constraints

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

(A.7a)

T f (i , n − 1) ≥ T s(i′, n) − H(1 − w(i , n − 1)) ∀ i , i′ ∈ Ij , i ∈ Isp , i′ ∈ Isc , j ∈ J , n ∈ N

∀s∈S,i∈

i′ ∈

t(j , n)

■

∑ ρis b(i , n − 1)

∀ s ∈ S , j ∈ J , i ∈ Ij , n ∈ N

∀s∈S,i∈

Isp ,

i′ ∈

Isc ,

⎛ ⎜ T s(i , n + 1) ≥ T f (i′, n) − M ⎜1 − ⎜ ⎝

(A.11)

+

⎛ ⎜ T (i , n + 1) ≤ T (i′, n) + M ⎜2 − ⎜ ⎝

∀ s ∈ SI , n ∈ N

i ∈ Isp f

(A.13) f

T (i′, n − 1) ≤ T (i , n − 1) + H(2 − w(i , n − 1)

f

−

∑ n ′∈ N n + 1 ≤ n ′≤ n + 1 +Δn

− w(i′, n)) + H(x(s , n)) ∀ s ∈ SI , i ∈ Isp , i′ ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N

⎞ ⎟ w(i′, n′, n)⎟ ⎟ ⎠

(B.1)

B.2. Storage Constraints

STmax s

s

∑ Bimax (1 − x(s , n))

n ′∈ N n −Δn ≤ n ′≤ n

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

A.7. Sequencing Constraints for FIS Policy

∑ ρis b(i , n − 1) + ST(s , n − 1) ≤

∑

∀ s ∈ SI , j , j′ ∈ J , n ∈ N , n < N , i ∈ Ij , i′ (A.12)

i ∈ Isp

(A.23)

APPENDIX B: ALIGNMENT CONSTRAINTS OF VOORADI AND SHAIK15 AND SHAIK AND FLOUDAS12 MODELS

j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′

∈ Ij ′ , n ∈ N

∀i∈I

B.1. Different Tasks in Different Units

T s(i′, n) ≥ T f (i , n − 2) − M(1 − w(i , n − 2)) I

(A.22)

T f (i , N ) ≤ MS

i ∈ Isp

I

(A.21)

n=N

minimization MS

A.6.2. For the Case When an Intermediate State s Is Produced from More than One Unit. i ′∈ Isc

∀ j ∈ J, n ∈ N

∑ Ps ∑ ST(s , n) s ∈ Sp

(A.10)

− ∑ ρi ′ s b(i′, n) ≤ ST(s , n − 1) +

(A.19)

max profit =

(A.9)

j , j′ ∈ J , i ∈ Ij , i′

∈ Ij ′ , n ∈ N

∀ i ∈ I, n ∈ N

A.11. Objective Function

T (i′, n) ≥ T (i , n − 1) − M(2 − w(i , n − 1) − t(j , n)) Isc ,

T f (i , n) ≤ H

(A.20)

f

Isp ,

(A.18)

i ∈ Ij n ∈ N

ρis b(i , n − 1) ≤ ST(s , n) + Bimax t(j , n)

I

∀ i ∈ I, n ∈ N

∑ ∑ (αiw(i , n) + βi b(i , n)) ≤ H

A.6.1. For the Case When an Intermediate State s Is Produced from One Unit.

∀ s ∈ S I , j ∈ J , i ∈ Ij , i ∈ Isp , n ∈ N

T s(i , n) ≤ H

A.10. Tightening Constraints

(A.8)

A.6. Different Tasks in Different Units

s

(A.17)

∑

w(i′, n′, n)

n ′∈ N n −Δn ≤ n ′≤ n

⎞ ⎟ w(i , n + 1, n′)⎟ ⎟ ⎠

∀ s ∈ S fis , S nis , S zw , j , j′ ∈ J , n ∈ N , n < N , i ∈ Ij , i′ ∈ Ij ′ , i ≠ i′, j ≠ j′, i ∈ Isc , i′ ∈ Isp

(A.14) 12970

(B.2)


Industrial & Engineering Chemistry Research ⎛ ⎜ T (i′, n) ≥ T (i , n) − M ⎜1 − ⎜ ⎝ f

s

∑ n ′∈ N n −Δn ≤ n ′≤ n

Article

⎞ ⎟ w(i′, n′, n)⎟ ⎟ ⎠

Binary Variables

w(i,n,n′) = binary variable for task i that starts at event n and ends at event n′ z(i,i′,s,n) = binary variable for state s which takes a value of 1 if the material produced by task i at event n − 1 is used by task i′ at event n x(i,i′,s,n) = binary variable for state s which takes a value of 1 if the material produced by task i at event n − 1 is consumed immediately by task i′ at event n v(i,i′,s,n) = binary variable for state s which takes a value of 1 if the material produced by task i at event n − 1 can wait in consumption unit before consumed by task i′ at event n y(i,i′,s,n) = binary variable for utility u which takes a value of 1 if the utility released by task i at event n − 1 is used by task i′ at event n

∀ s ∈ s fis , j , j′ ∈ J , n < N , i ∈ Ij , i′ ∈ Ij ′ , i ≠ i′, j ≠ j′, i ∈ Isc , i′ ∈ Isp

■

(B.3)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: (91)-11-26591038. Fax: (91)-11-26581120. Notes

The authors declare no competing financial interest.

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Positive Variables

NOMENCLATURE

Indices

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

I = tasks Ij = tasks which can be performed in unit j Ips = tasks which produce state s Ics = tasks which consume state s Iu = tasks which consume utility u J = units N = total event points postulated in the time horizon S = states SR = states that are raw materials SP = states that are final products SI = states that are intermediates Sfis, Szw, Snis = intermediate states with dedicated finite intermediate storage, zero wait, and no intermediate storage cases, respectively U = utilities

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b(i,n,n′) = batch size of task i that starts at event n and ends at event n′ ST0(s) = initial amount of state s required from external resources ST(s,n) = excess amount of state s that needs to be stored at event n b1(i,i′,s,n) = amount of state s consumed by task i′ at event n from the total amount produced by task i at event n − 1 b2(i,i′,s,n) = excess amount of FIS state s produced by task i at event n − 1, beyond the maximum storage capacity, which is to be either immediately consumed or can wait in consumption unit for consumption by task i′ at event n Ts(i,n) = start time of task i at event n Tf(i,n) = end time of task i at event n AM(u,n) = excess amount of utility u available at event n b3(i,i′,s,n) = amount of utility used by task i′ at event n from the total amount released by task i at event n − 1

REFERENCES

(1) 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. (2) Floudas, C. A.; Lin, X. Continuous-time versus discrete-time approaches for scheduling of chemical processes: A review. Comput. Chem. Eng. 2004, 28, 2109. (3) Maravelias, C. T. A general framework and modeling approach classification for chemical production scheduling. AIChE J. 2012, 58, 1812. (4) Shaik, M. A.; Janak, S. L.; Floudas, C. A. Continuous-time models for short-term scheduling of multipurpose batch plants: A comparative study. Ind. Eng. Chem. Res. 2006, 45, 6190. (5) Pitty, S. S.; Karimi, I. A. Novel MILP models for scheduling permutation flowshops. Chem. Prod. Process Model. 2008, 3, 1. (6) Sundaramoorthy, A.; Maravelias, C. T. Computational study of network-based mixed-integer programming approaches for chemical production scheduling. Ind. Eng. Chem. Res. 2011, 50, 5023. (7) Ierapetritou, M. G.; Floudas, C. A. Effective continuous-time formulation for short-term scheduling. 1. Multipurpose batch processes. Ind. Eng. Chem. Res. 1998, 37, 4341. (8) Janak, S. L.; Lin, X.; Floudas, C. A. Enhanced continuous-time unit-specific event-based formulation for short-term scheduling of multipurpose batch processes: Resource constraints and mixed storage policies. Ind. Eng. Chem. Res. 2004, 43, 2516. (9) 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. (10) Shaik, M. A.; Floudas, C. A. Unit-specific event-based continuous time approach for short-term scheduling of batch plants using RTN framework. Comput. Chem. Eng. 2008, 32, 260.

Parameters

Bmin = minimum batch size of task i i Bmax = maximum batch size of task i i ST0s = initial amount available for state s STmax = maximum storage capacity for state s s αi = fixed processing time of task i βi = linear coefficient of the variable term of the processing time of task i γiu = coefficient of constant term of consumption of utility u by task i δiu = coefficient of variable term of consumption of utility u by task i ρis = fractions of state s produced (ρis ≥ 0) or consumed (ρis ≤ 0) by task i H = short-term scheduling horizon Ps = price of state s Δn = limit on the maximum number of events over which a task is allowed to continue M = large positive number in big-M constraints Ds = demand for state s AM0u = maximum availability of utility u UWi = task-specific maximum unit-wait time 12971



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(11) Castro, P. M.; Novais, A. Q. Short-term scheduling of multistage batch plants with unlimited intermediate storage. Ind. Eng. Chem. Res. 2008, 47, 6126. (12) Shaik, M. A.; Floudas, C. A. Novel unified modeling approach for short-term scheduling. Ind. Eng. Chem. Res. 2009, 48, 2947. (13) Li, J.; Floudas, C. A. Optimal event point determination for short-term scheduling of multipurpose batch plants via unit-specific event-based continuous-time approaches. Ind. Eng. Chem. Res. 2010, 49, 7446. (14) Susarla, N.; Li, J.; Karimi, I. A. A novel approach to scheduling multipurpose batch plants using unit slots. AIChE J. 2010, 56, 1859. (15) Vooradi, R.; Shaik, M. A. Improved three-index unit-specific event-based model for short-term scheduling of batch plants. Comput. Chem. Eng. 2012, 43, 148. (16) Seid, R.; Majozi, T. A robust mathematical formulation for multipurpose batch plants. Chem. Eng. Sci. 2012, 68, 36. (17) Maravelias, C. T.; Grossmann, I. E. New general continuoustime state-task network formulation for short-term scheduling of multipurpose batch plants. Ind. Eng. Chem. Res. 2003, 42, 3056. (18) Castro, P. M.; Barbosa-Povoa, A. P.; Matos, H. A.; Novais, A. Q. Simple continuous-time formulation for short-term scheduling of batch and continuous processes. Ind. Eng. Chem. Res. 2004, 43, 105. (19) Reddy, P. C. P.; Karimi, I. A.; Srinivasan, R. A new continuoustime formulation for scheduling crude oil operations. Chem. Eng. Sci. 2004, 59, 1325. (20) 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. (21) Liu, Y.; Karimi, I. A. Scheduling multistage, multiproduct batch plants with nonidentical parallel units and unlimited intermediated storage. Chem. Eng. Sci. 2007, 62, 1549. (22) Lim, M. F.; Karimi, I. A. Resource-constrained scheduling of parallel production lines using asynchronous slots. Ind. Eng. Chem. Res. 2003, 42, 6832. (23) Mendez, C. A.; Henning, G. P.; Cerda, J. An MILP continuous time approach to short-term scheduling of resource-constrained multistage flowshop batch facilities. Comput. Chem. Eng. 2001, 25, 701. (24) Nadal, S.; Garcia, E. C.; Mendez, C. A.; Puigjaner, L. Material transfer operations in batch scheduling. A critical modeling issue. Ind. Eng. Chem. Res. 2008, 47, 7721. (25) Kopanos, G. M.; Mendez, C. A.; Puigjaner, L. MIP-based decomposition strategies for large-scale scheduling problems in multiproduct multistage batch plants: A benchmark scheduling problem of the pharmaceutical industry. Eur. J. Oper. Res. 2010, 207, 644.

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Rigorous Unit-Specific Event-Based Model for Short-Term Scheduling

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