Enhanced Discrete Time Formulation for a Short-Term Batch Process

Article Cite This: Ind. Eng. Chem. Res. 2019, 58, 14559−14568

pubs.acs.org/IECR

Enhanced Discrete Time Formulation for a Short-Term Batch Process Scheduling Problem with Utility Constraints Yue Wang,†,‡ Xin Jin,*,†,‡ and Shan Lu§ †

School of Information and Control Engineering, Liaoning Shihua University, Fushun 113001, China National Experimental Teaching Demonstration Center of Petrochemical Process Control, Liaoning Shihua University, Fushun 113001, China § Institute of Intelligence Science and Engineering, Shenzhen Polytechnic, Shenzhen 518055, China Downloaded via NOTTINGHAM TRENT UNIV on August 13, 2019 at 08:14:50 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

ABSTRACT: For a short-term production scheduling problem with utility constraints, the discrete-time modeling approach has its inherent advantages compared with the continuous-time approach. Even so, there are still some limitations to the discrete-time approach in calculating utility consumption. This paper aims to enhance the capability of the discrete-time model to calculate utility consumption more accurately. In this paper, a four-dimensional binary variable is introduced on the basis of a basic three-dimensional discrete-time model framework. Constraints crucial to utility consumption calculation, including allocation, batch size, and material balance, are modified significantly. Moreover, production rate fluctuation and safe inventory interval constraints are considered in the proposed model, which can increase the stability and robustness of production and operation. Finally, two cases demonstrate that the proposed four-dimensional discrete-time model is more accurate than the basic three-dimensional model in terms of calculating utility consumption. In addition, when the disturbance of utility supply amount appears, the proposed model can also respond to the disturbance.

1. INTRODUCTION Choice of time representation is an important factor in the structure and character of a model used for production scheduling problems. Currently, there are two classes of the main time representation method for the modeling of a production scheduling problem. One is the continuous-time approach, and the other is the discrete-time approach. In the discrete-time approach, the scheduling horizon is divided into a number of equally spaced intervals, and events can only take place at the beginning or the end of those intervals. With a continuous-time approach, the horizon is divided into fewer intervals, and the spacing is decided as part of solving the problem.1 In the past two decades, the continuous-time approach has been acknowledged as being superior to the discrete-time approach in terms of computational requirements and solution quality.2−4 However, recent computational comparisons and discussions with a number of practitioners lead us to believe that the aforementioned premise is not necessarily valid.1,5 Sundaramoorthy and Maravelias5 performed an extensive © 2019 American Chemical Society

computational comparison between the two time representation approaches in terms of computational requirements, solution robustness, and solution quality. They found that the two time representation approaches both have their typical advantages. However, for different problems, which model is superior changes. Therefore, which time representation approach should be chosen depends on the specific problem, being influenced by different process networks, various objective functions, different scheduling horizons, and a wide range of features (fixed and variable processing times, utilities, holding and backlog costs, intermediate shipments, and setups). For short-term production scheduling of batch processes, the utility problem is of practical significance but has not obtained enough attention. In the production process, utilities Received: Revised: Accepted: Published: 14559

April 12, 2019 July 7, 2019 July 8, 2019 July 8, 2019 DOI: 10.1021/acs.iecr.9b02002 Ind. Eng. Chem. Res. 2019, 58, 14559−14568

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Industrial & Engineering Chemistry Research

utility consideration in general and large-scale industrial sites, Lindholm and Nytzen18 introduced a general framework for production scheduling (PS) and detailed production scheduling (DPS) using a two-level hierarchical discrete-time approach. Lindholm and Giselsson proposed19 a discretetime MIQP model to address utility disturbance in the production process with the aim of minimizing the total economic loss of site. Wang et al.20 proposed a planning and scheduling model framework with a hierarchical structure. In the scheduling layer, a vari-period discrete-time presentation model is formulated to address utility disturbance. The production scheduling problem for a batch process with utility constraints based on the discrete-time approach should draw more attention. Currently, most of the discrete-time models with utility constraints are based on the threedimensional model framework, which has a problem with inaccurately calculating utility consumption. In a batch process, materials are processed batch by batch; namely, a batch of material should not be stored or processed by the next stage until the current processing task has finished. For short-term scheduling of a batch process, most of the durations of the processing tasks are longer than the scheduling periods, which requires that the scheduling model permit the durations of processing tasks to span multiple scheduling periods. However, the basic three-dimensional model framework cannot calculate the amount of material processed or the utility consumption level in the spanned periods. That makes the calculation of the utility consumption level inaccurate and the scheduling solution infeasible. In this paper, we introduce a four-dimensional binary variable based on a basic three-dimensional model framework. The key constraints for utility consumption, including allocation, batch size, and material balance, are modified significantly to enable the proposed model to count the utility consumption level accurately. When a disturbance in the utility supply amount appears, the proposed model can also respond to it. Moreover, production rate fluctuation and safe inventory interval constraints are considered in the proposed model, which increases the stability and robustness of production and operation. This paper is organized as follows: The problem of inaccuracy in the discrete-time model when calculating utility consumption is described in Section 2. Then the proposed model framework with the four-dimensional binary variable is given and elaborated in Section 3. Next, in Section 4, two cases are utilized to demonstrate the effectiveness of the utility constraint under constant and disturbed utility supply. Lastly, we conclude with the major works and finds of this paper.

are supplied to areas or product lines, such as steam, cooling water, and electricity. Usually, these utilities are shared by different areas. When the supply of utility is fluctuant, areas of production are affected seriously and production scheduling cannot be completed, which can even lead to a shutdown of the product line. For a short-term production scheduling problem considering utility constraints, there are some research results based on the continuous-time approach. Méndez and Cerdá6 presented efficient MILP continuous-time formulations for multistage batch process scheduling subject to limited availability of unary and finite renewable resources, including utility. Maravelias and Grossmann7 as well as Castro and Barbosa-Povoa8 considered resource (utility) constraints based on a uniform-time-grid continuous-time representation, and the proposed formulations have been shown to compare favorably to existing continuous-time formulations. Janak et al.2 first introduced utility constraints into a unit-specific, event-based continuoustime model. They developed an enhanced continuous-time formulation for the short-term scheduling of multipurpose batch plants on the basis of formulations proposed by Ierapetritou and Floudas.9 Subsequently, Behdani et al.10 also developed and extended the scheduling formulations of Ierapetritou and Floudas9 to accommodate optimization of utility demands and supplies. On the basis of their work, it is possible to consider and justify the “instantaneous utility consumption” in a scheduling model. Since then, some more research results have been continuously proposed on unitspecific, event-based continuous-time model with utility constraints.11,12 However, some limitations of unit-specific, event-based continuous-time model (USEBCTM) with utility constraints have caused concern among some scholars.2,7,11 Wang and Su13 proposed a unit-specific, event-based and slot-based continuous-time hybrid model framework with utility constraints to eliminate the limitations of USEBCTM, including overstrict constraints on utility resources, multiple counting problems, and so on. The essential cause of the limitations of USEBCTM is the time representation approach. USEBCTM constructs heterogeneous locations of the event points for different units. The utility utilization levels for tasks performed in different units at the same event points are difficult to calculate. To solve this problem, the discrete-time representation approach has an inherent advantage compared with the continuous-time representation approach. The discrete-time approach partitions the scheduling horizon into a known number of periods of equal (and known) lengths, defining a time grid that is common across all units, states, and resources. Processing times are assumed to be constant.5,14−16 Consequently, the discrete-time representation method is based on uniform discretization of time for different units, which makes it easier to handle utility constraints in a uniform model. Although the discrete-time approach has advantages in a production scheduling problem with utility constraints, the research on this problem based on the discrete-time approach has not attracted enough attention. Most of these research results related this problem do not focus on the model’s ability to deal with utility constraints. Velez et al.17 extended the basic discrete-time state-task network (STN) model with variable conversion coefficients (changeovers, utilities, etc.), but they aimed to address mixed integer programs for the scheduling of large-scale chemical production systems. In order to involve

2. INACCURATE PROBLEM OF DISCRETE TIME MODEL IN UTILITY CONSUMPTION CALCULATIONS 2.1. Difference between Material and Utility. Supply of utility is different from supply of material in a batch process scheduling problem. In a batch process, the input material is supplied entirely at the beginning of the processing task, and the output material cannot be obtained until the processing task is finished. However, the utility should be supplied continuously to support the reaction of material and the normal operation of units. Below is a motivating example to illustrate the difference between the supply of utility and the supply of material. 14560

DOI: 10.1021/acs.iecr.9b02002 Ind. Eng. Chem. Res. 2019, 58, 14559−14568

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period t, and the following allocation constraints are formulated:

Figure 1 is the state task network (STN) presentation for the batch process. Tasks 1, 2, and 3 are performed only in units 1,

∑ W (i , j , t ) ≤ 1

∀ j ∈ J, t ∈ T

i ∈ Ij

∑ Figure 1. State task network of the motivating example.

(1)

∑

W (i′, j , t ′) − 1 ≤ M[1 − W (i , j , t )]

i ′∈ Ij t ≤ t ′≤ t + αij − 1

∀ j ∈ J , i ∈ Ij , t ∈ T

2, and 3, respectively. Task 1 consumes utility A, and task 2 consumes utility B to maintain the reactions. Figure 2 presents the Gantt chart of the production schedule and the supply of material and utility according to the above Gantt chart.

(2)

where Ij is the set of tasks that can be performed in unit j, αij is the fixed processing time of task i in unit j and M is a sufficiently large positive number. Constraint 1 states that at most one task can start in any unit in any scheduling period. Constraint 2 further expresses the requirement that if task i starts in unit j in scheduling period t (i.e., W(i, j, t) = 1), then no other tasks can start in the same unit until task i is finished after a duration of αij. Note that the latter constraint becomes trivially satisfied when task i does not start at time t (i.e., W(i, j, t) = 0). To account for batch sizes and mass balances, continuous variable B(i, j, t) is used to represent the amount of material processed by task i in unit j in scheduling period t, and ST(s, t) is the amount of material state s during scheduling period t. The following constraints are introduced to represent the relations among them and the corresponding binary variables:

Figure 2. Gantt chart of the production schedule and the supply of material and utility.

Bijmin W (i , j , t ) ≤ B(i , j , t ) ≤ Bijmax W (i , j , t ) ∀ i ∈ I , j ∈ Ji , t ∈ T

From the motivating example, we find that task 1, performed in unit 1, starts in period 1 and ends in period 4. Task 2, performed in unit 2, starts in period 2 and ends in period 5. The material supply only happens at the beginning of the periods, but the utility should be continuously supplied during the performance of the tasks. However, this key difference between utility and material does not receive enough attention. Most discrete-time scheduling models deal with material and utility supply equally, which leads to inaccurate calculations of utility consumption. 2.2. Constraints Related to Calculating Utility Consumption. Currently, most of the discrete-time scheduling models focuses on computational performance and solution quality. The utility constraints of the scheduling model, in both the discrete-time model and the continuoustime model, are generally neglected, which leads to inaccurate calculations of utility consumption. For the calculation of utility consumption, the most common method supposes that the relationship between the utility consumption level and the amount processed for the task is linear. The amount processed for the task is limited by the batch-size constraint and the material balance constraint. In the batch-size constraint, the binary variable W(i, j, t) from the allocation constraint plays a very important role. Consequently, the allocation constraint, the batch-size constraint, and material balance are the key points for utility consumption. Regarding the discrete-time scheduling model for a batch process, the general basic model framework includes the allocation constraint, the batch-size constraint, and material balance. Allocation constraints involve the allocation of units to tasks. To model these assignments, a general model framework introduces the three-dimensional binary variable W(i, j, t) to determine whether a task (i) starts in unit j at the beginning of

ST(s , t ) = ST(s , t − 1) +

(3)

∑ asi ∑ B(i , j , t − αij) i ∈ IsP

−

j ∈ Ji

∑ bsi ∑ B(i , j , t ) − D(s , t ) i ∈ IsC

j ∈ Ji

s ∈ A , t > 1, t ∈ T

(4)

The most important constraint for calculating the utility consumption level is the utility constraint, given by eq 5. Currently, most of the literature supposes that the utility consumption level is linear with the amount processed for the task. Therefore, the continuous variable B(i, j, t) plays an important role in calculating the utility consumption level.

∑ eiu ∑ B(i , j , t ) ≤ Uutmax i ∈ Iu

j ∈ Ji

∀ u ∈ U, t ∈ T (5)

Continuous variable B(i, j, t) depends on the binary variable W(i, j, t). In the above general discrete-time model framework, the continuous variable B(i, j, t) is not calculated accurately in some scheduling periods caused by the binary variable W(i, j, t), which further leads to inaccurate calculations of the utility consumption level. The root cause of the problem is the binary variable W(i, j, t). As shown in Figure 3, task 1 starts in period 1 and ends in period 3. The variables W(i, j, t), B(i, j, t), and U(i, j, t) in period 1 are 1, Btotal, and Utotal, respectively, on the basis of the general discrete-time model framework. Here, Btotal is the total processed amount for task 1, and Utotal is the total utility consumption for task 1. However, in periods 2 and 3, the variables W(i, j, t), B(i, j, t), and U(i, j, t) are all 0, which is not coincident with the actual situation. The actual situation is that the variables W(i, j, t) and B(i, j, t) should not be 0 because of 14561


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(3)

(4) (5)

(6) Figure 3. Presentation of the inaccurate calculation of utility consumption.

dedicated storage vessels within lower STsmin and upper STsmax bounds. Set of processing tasks i ∈ I. The processing time of task i in unit j is fixed and denoted by αij; different tasks (i) can be performed in the same or different units; parameter αij is an integral multiple of the scheduling period when task i is performed on unit j. Set of equipment units j ∈ J. The set of tasks that can be performed in j is denoted by Ij. The batch size of task i carried out in unit j should be within lower and upper bounds Bijmin and Bijmax, respectively. Set of utilities u ∈ U, with maximum availability Umax ut in period t. The supply amount of the utility can have a disturbance.

In the proposed model, one four-dimensional binary variable, w(i, j, t, t′), is introduced in the allocation constraint. It allows the processed amount of the task, B(i, j, t″), (t ≤ t″ ≤ t′) between period t and t′, to be calculated accurately and further guarantees accuracy in the utility consumption. When a disturbance in the utility supply amount appears, the proposed model can respond to it. The formulations of the proposed scheduling model are provided in the following subsections. 3.1. Allocation Constraints.

the continuous performance of task 1. Moreover, the utility should not be consumed totally in period 1 but should be consumed continuously while performing task 1. As shown in Figure 3, the utility at the shaded area should be moved to the red area. Inaccurate calculation of utility consumption is caused essentially by the binary variable W(i, j, t), which improperly forces the processed amounts of the task in the posterior periods to be 0. The traditional batch process discrete-time scheduling model is generally formulated by the threedimensional binary variable to establish the allocation constraint. When the processing time of the batch is longer than the scheduling period, namely the processing time of the batch spans multiple scheduling periods, the traditional scheduling model neglects the processed amount of the task in the spanned periods. This neglect of the processed amount results in an inaccurate calculation of utility consumption and leads to inaccuracies in the scheduling solution. In the proposed model, one four-dimensional binary variable, w(i, j, t, t′), is introduced to the allocation constraint. It allows the processed amount for the task, B(i, j, t″), (t ≤ t″ ≤ t′) between period t and t′, to be calculated accurately and further guarantees accuracy in the determination of utility consumption.

∑

∑

w(i , j , t ′, t ′ + αij − 1) ≤ 1

i ∈ Ij t − αij + 1 ≤ t ′≤ t t ′≤ T − αij + 1

j ∈ J , t ∈ T , t′ ∈ T

(6)

Allocation constraints state that at most one task can be performed in any unit in any scheduling period. A fourdimensional binary variable, w(i, j, t, t′), is introduced in the allocation constraints to describe accurately the durations of the tasks. If w(i, j, t, t′) = 1, then task i is performed on unit j from period t to period t′. Otherwise, w(i, j, t, t′) = 0. Allocation constraints allow task durations larger than the scheduling periods, but the duration of the task must be an integral multiple of the scheduling period. Parameter αij represents the duration of the scheduling period, during which task i is performed on unit j. Figure 4 presents a matrix of scheduling periods t and t′, as well as the possible situation of variable w(i, j, t′, t′ + αij − 1) when t increases from 1 to T. Here, in order to illustrate clearly, we set T = 8 and αij = 3. On the basis of variable t and the restrictions in constraint 6, all the situations are divided into eight groups, which are separated by dashed lines in Figure 4. In a specific unit (j) and scheduling period (t), constraint 6 requires that the sum of available binary variables w(i, j, t′, t′ + αij − 1) be ≤1. For instance, if t = 4, then t′ could be 2, 3, or 4. Therefore, at this moment, the sum of w(i, j, 2, 4), w(i, j, 3, 5), and w(i, j, 4, 6), where task i can be any task performed in unit j, should not be larger than 1. Because of the addition of one more time dimension in variable w(i, j, t, t′), some redundant binary variables are also introduced. Because the processing time of the batch is fixed (αij), in some binary variables for w(i, j, t, t′), the duration from t to t′ is not equal to fixed processing time αij. That makes these binary variables meaningless. Consequently, constraints 7 and 8 force these meaningless binary variables to be 0. These two constraints express that the situation will not happen if

3. MODEL FRAMEWORK WITH A FOUR-DIMENSIONAL BINARY VARIABLE On the basis of the discrete-time modeling method, an MINLP model is formulated for the short-term batch process scheduling problem. The objective function is to maximize the profit, and the constraints of allocation, material balance, batch size, safe inventory interval, production rate fluctuation, and utility are considered in the MINLP model. The aim of the proposed scheduling model is to determine the sequence of batches (tasks), the processed amount of each task, the beginning and end times of each task, the delivery amount and time of each product, the inventory level of each state, and the amount of utility consumption. The scheduling problem is defined in terms of the following sets, parameters, and hypotheses: (1) A scheduling horizon is uniformly divided into T scheduling periods. (2) Set of states (materials, including feeds, intermediates, and final products) s ∈ S, which can be stored in 14562


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Industrial & Engineering Chemistry Research B(i , j , t ′) ≤ Bijmax w(i , j , t , t + αij − 1) + M[1 − w(i , j , t , t + αij − 1)]

∀ j ∈ J , i ∈ Ij , t ≤ T − αij + 1, t ≤ t ′ ≤ t + αij − 1 (10)

B (i , j , t ) ≤ M

∑

w(i , j , t ′ − αij + 1, t ′)

t ≤ t ′≤ t + αij − 1 t ′≥ αij

∀ j ∈ J , i ∈ Ij , t ∈ T

Batch-size constraints 9 and 10 express that if task i is performed in unit j from period t to period t + αij − 1, namely w(i, j, t, t + αij − 1) = 1, then the processed amount of material, B(i, j, t′), from period t to period t + αij − 1 is limited by the upper and lower bounds. Inversely, if w(i, j, t, t + αij − 1) = 0, then constraints 9 and 10 are relaxed. Constraint 11 states that if there is no task i performed in unit j in period t, then the processed amount of material, B(i, j, t), is forced to be 0. The following equation involves all of the binary variables related with period t:

Figure 4. Period matrix and available four-dimensional binary variable.

task i is active in unit j but the processing time is not equal to αij.

∑ t ′≥ αij

(7)

If the sum is 0, it means that there is no task performed in period t. In this situation, because B(i, j, t) is a floating-point variable and because of constraint 11, variable B(i, j, t) is forced to be 0. Inversely, if

w(i , j , t , t ′) = 0 ∀ j ∈ J , i ∈ Ij , t ∈ T , t ′ > t + αij − 1

w(i , j , t ′ − αij + 1, t ′)

t ≤ t ′≤ t + αij − 1

w(i , j , t , t ′) = 0 ∀ j ∈ J , i ∈ Ij , t ∈ T , t ′ < t + αij − 1

(11)

(8)

∑

As shown in Table 1, constraint 7 forces the variable w(i, j, t, t′) in the “×” area to be 0, and constraint 8 forces the variable w(i, j, t, t′) in the “△”area to be 0. Only the variables w(i, j, t, t′) represented by “□” in the line (t + αij − 1) are meaningful. 3.2. Batch-Size Constraints.

w(i , j , t ′ − αij + 1, t ′) = 1

t ≤ t ′≤ t + αij − 1 t ′≥ αij

then constraint 11 is relaxed. From the above analysis, benefiting from the introduction of the four-dimensional binary variable, the batch-size constraints can calculate the amount of processed material, B(i, j, t), in any scheduling period, including in the spanned periods. That makes the proposed scheduling model have advantages when compared with the traditional three-dimensional scheduling model. 3.3. Material Balance.

B(i , j , t ′) ≥ Bijmin w(i , j , t , t + αij − 1) − M[1 − w(i , j , t , t + αij − 1)] ∀ j ∈ J , i ∈ Ij , t ≤ T − αij + 1, t ≤ t ′ ≤ t + αij − 1 (9)

Table 1. Period Matrix of Meaningful Four-Dimensional Binary Variables

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consumed amount and the delivered amount of state s in the current period. 3.4. Production Rate Fluctuation Constraint.

ST(s , t ) = ST0s +

∑ asi∑ i ∈ IsP

−

∑

B(i , j , t ′) ∑ w(i , j , t ″ , t − 1)

j ∈ Ji t − αij ≤ t ′≤ t − 1

∑ bsi ∑ i ∈ IsC

∑

t ″≤ t ′

j ∈ Ji t ≤ t ′≤ t + αij − 1

− D(s , t )

B(i , j , t ) − x(i , j , t ) ≤ B(i , j , t − 1)

B(i , j , t ′) ∑ w(i , j , t , t ″)

≤ B(i , j , t ) + x(i , j , t )

t ″≥ t ′

s ∈ A, t = 1

(14)

(12)

In order to decrease the influence of production rate fluctuation on the production process, constraint 14 introduces aided continuous variable x(i, j, t), and the corresponding penalized weight, λ, is also set in the objective function. The production rate fluctuation constraint makes the production output in the next period as close to that of the previous period as possible, which avoids large fluctuations in production rate between adjacent periods. 3.5. Inventory Constraints.

ST(s , t ) = ST(s , t − 1) +

∑ asi ∑ i ∈ IsP

−

∑ bsi∑ i ∈ IsC

∑

B(i , j , t ′) ∑ w(i , j , t ″ , t − 1)

j ∈ Ji t − αij ≤ t ′≤ t − 1

∑

t ″≤ t ′

B(i , j , t ′) ∑ w(i , j , t , t ″)

j ∈ Ji t ≤ t ′≤ t + αij − 1

− D(s , t )

t ″≥ t ′

s ∈ A , t > 1, t ∈ T , t ′ ∈ T

(13)

STmin ≤ ST(s , t ) ≤ STmax s s , s ∈ S, t ∈ T

Although the four-dimensional binary variable can calculate the amount of material processed and the utility consumption in each scheduling period, it brings difficulties for formulating the material balance constraints. The traditional three-dimensional model can use the following simple equation to express the supply and output of material: j ∈ Ji

i ∈ IsC

(15)

Constraint 15 limits the inventory level of state s at the end of period t between the maximum inventory bound (STsmax) and minimum inventory bound (STsmin). ub STlb s − z(s , t ) ≤ ST(s , t ) ≤ STs + z(s , t )

∑ asi ∑ B(i , j , t − αij) − ∑ bsi ∑ B(i , j , t ) − D(s , t ) i ∈ IsP

i ∈ I , j ∈ Ji , t > 1

s ∈ S IP , t ∈ T

j ∈ Ji

(16)

In order to increase the robustness of the production process, the safe inventory interval is considered in the inventory constraints. One aided variable, z(s, t), is introduced in constraint 16 to keep the inventory level of state s at the end of period t between the reference maximum bound (STsub) and minimum bound (STslb). This constraint can make the inventory level within the safe inventory interval as high as possible and further prevents the inventory level from being too low or too high. 3.6. Utility Constraint.

In the proposed model, in order to calculate the amount of processed material, B(i, j, t), in each period, we divide the whole batch size into αij sub-batches. As shown in Figure 5,

∑ eiu ∑ B(i , j , t ) ≤ Uutmax i ∈ Iu

∀ u ∈ U, t ∈ T (17)

j ∈ Ji

The relationship between the consumption of utility and the processed amount, B(i, j, t), which requires the consumption the utility, is supposed to be linear. Constraint 17 expresses that the total consumption amount of utility should not be larger than the total amount of utility supplied. Because the variable B(i, j, t) can be calculated in each period on the basis of allocation, batch size, and material balance constraints, here we can easily determine the consumption level of utility in each period. 3.7. Demand Constraints.

Figure 5. Schematic diagram of material balance.

task 1 starts in period 2 and ends in period 3. According to the allocation and batch-size constraints, we know that w(1, j1, 2, 3) should be 1, and B(1, j1, 2) and B(1, j1, 3) should be positive numbers. In the batch process, task 1 should output the processed material after the task is finished. That is, the material processed by task 1 should be stored in period 4. All of this requires that the proposed material balance constraints have the ability to identify which sub-batches should be stored in a specific period. In the same way, the proposed material balance constraints should also have the ability to identify which sub-batches should be supplied for consumption in a specific period. Constraints 12 and 13 are formulated to satisfy the above requirements. They are used for the initial period and subsequent periods, respectively. Constraint 13 expresses that the amount of state s at the end of period t is equal to the amount of state s at the end of period t − 1 plus the output amount of state s in the current period and minus the

D(s , t ) ≤ Ost + Bl(s , t − 1)

s ∈ SP , t ∈ T

Bl(s , t ) = Bl(s , t − 1) + Ost − D(s , t )

(18)

s ∈ SP , t ∈ T (19)

Constraint 18 expresses that the delivery amount of state s in period t should be less than or equal to the sum of the demand in period t and the backlog in period t − 1. The backlog of state s in period t is given by constraint 19, and the corresponding penalty weight, γ, is set in the objective function to prevent late delivery of the order. 3.8. Objective Function. 14564


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Figure 6. State-task network representation of case 1.

Table 2. Related Data for Case 1 heating reaction 1 reaction 2 separation

max

task i

unit j

αij (h)

Bmin (kg) ij

Bmax (kg) ij

eiu

STmin (kg) s

STmax (kg) s

1 2 3 4

heater reactor 1 reactor 2 reactor 3

1 3 1 2

0.5 0.5 0.5 0.5

10 1.3 2 5

2 2 3 

0 0 0 0

unlimited 6 4 unlimited

∑ ∑ [msD(s , t ) − γ Bl(s , t )] − ∑ ∑ θz(s , t ) s ∈ SP t ∈ T

−

s ∈ S IP t ∈ T

∑ ∑ ∑ λx(i , j , t ) i ∈ I j ∈ Ji t > 1

×

∑

w(i , j , t ′ − αij + 1, t ′)

t ≤ t ′≤ t + αij − 2 t ′≥ αij

(20)

The objective function of the proposed model is profit maximization. The first term of the objective function is sale profit and backlog punishment. The last two terms of the objective function are the punishment for surpassing the inventory level and the production rate fluctuation punishment. Parameters γ, θ, and λ are the penalty weights for backlog, surpassing the inventory level, and production rate fluctuation, respectively. In this paper, these parameters are set to be 10, 0.1, and 0.001, respectively.

Figure 7. Scheduling solution of the traditional model under constant utility supply.

4. CASE STUDY In this section, two cases are utilized to demonstrate the effectiveness of the proposed model under constant and disturbed utility supply. The two cases use the same plant flow sheet, which was introduced first by Maravelias and Grossmann7 and has been studied in several studies.21,22 The plant flow sheet of the two cases is described by the state-task network (STN) representation, as shown in Figure 6. The two cases were solved using Lingo 11.0 with a 2.20 GHz computer. 4.1. Case of Constant Utility Supply. As shown in Figure 6, there are four tasks, including heating, reaction 1, reaction 2, and separation, and four material states, including feed A, intermediate hA, intermediate IB, and final product B. Each task has its own specified unit, and each state also has its own specified storage. Data related to this case are given in Table 2. In order to demonstrate the effectiveness of the proposed model, we added two kinds of utilities: high press steam (HPS) and low press steam (LPS) in case 1 and case 2. HPS is consumed by the heating task. LPS is consumed by reaction 1 and reaction 2. The price of product B is $80 per kg. In case 1, the scheduling horizon is 10 h and is divided into 10 scheduling periods (i.e., each scheduling period is 1 h). The

Figure 8. Scheduling solution of the proposed model under constant utility supply.

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Industrial & Engineering Chemistry Research Table 3. Utility Consumption Levels of the Traditional Model under Constant Utility Supply LPS (kg) HPS (kg)

period 1

period 2

period 3

period 4

period 5

period 6

period 7

period 8

period 9

period 10

6 8.8

6 4.86

6 4.86

6 4.86

6 4.86

6 4.86

6 4.86

6 4.86

6 4.86

6 4.86

Table 4. Utility Consumption Levels of the Proposed Model under Constant Utility Supply LPS (kg) HPS (kg)

period 1

period 2

period 3

period 4

period 5

period 6

period 7

period 8

period 9

period 10

6 13.02

6 7.02

6 0

6 12.44

6 2.22

6 0

6 0

0 0

4.29 1

0 0

Table 5. Data of Utility Supply with Disturbance HPS LPS

t=1

t=2

t=3

t=4

t=5

t=6

t=7

t=8

t=9

t = 10

20 7

20 5

20 9

20 8

20 6

20 7

20 7

20 7

20 7

20 7

Table 6. Utility Consumption Levels of the Traditional Model under Disturbed Utility Supply LPS (kg) HPS (kg)

period 1

period 2

period 3

period 4

period 5

period 6

period 7

period 8

period 9

period 10

7 8.12

5 4.86

6 4.86

6 4.86

6 4.86

6 4.86

6 4.86

6 4.86

6 4.86

6 4.86

Table 7. Utility Consumption Levels of the Proposed Model under Disturbed Utility Supply LPS (kg) HPS (kg)

period 1

period 2

period 3

period 4

period 5

period 6

period 7

period 8

period 9

period 10

6 16.6

4.99 2

8.66 5.56

8 5.4

5.96 5.16

6.44 5.06

6.98 0

2.46 1

2.4 1

1.5 1

Figure 9. Scheduling solution of the traditional model under disturbed utility supply.

Figure 10. Scheduling solution of the proposed model under disturbed utility supply.

constant supply amounts of HPS and LPS are 20 and 6 kg/h, respectively. The aim of this case is to demonstrate the effectiveness of the proposed model under constant utility supply through a comparison of the traditional three-dimensional model and the proposed model. The objective functions of the two methods are both profit maximization. The optimal solutions are $1,494.028 and $1,559.066, respectively. Figures 7 and 8 are the optimal solutions of the traditional and proposed models, including the Gantt charts of units and the utility consumption levels. Data on the utility consumption levels of the traditional model and proposed model under constant utility supply are given in Tables 3 and 4. The sequences, durations, and processed amounts for the tasks are illustrated in the Gantt

charts. The lines in Gantt chart represent that tasks are performed in corresponding scheduling periods. The numbers above the lines are the amounts processed in the tasks. As shown in Figure 7, the tasks performed in reactor 1 span three periods. Take the first task performed in reactor 1 as an example. The task starts in period 2 and ends in period 4. The traditional model acknowledges that the amount of material processed is all processed in period 2, and the amount of material processed in periods 3 and 4 are both 0. Then, the utility of LPS is all consumed in period 2, and in periods 3 and 4, LPS is not needed. However, LPS is actually consumed continuously. 14566


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model, which increases the stability and robustness of production and operation based on the scheduling solutions of the proposed four-dimensional model. Through two cases, the proposed model is demonstrated as being more accurate than the basic three-dimensional model, and when a disturbance in utility supply amount is present, the proposed model can respond to the disturbance.

In the LPS subfigure of Figure 7, the black solid line is the scheduling solution of the LPS consumption level. The red dashed line is the actual utility consumption level. We can find that the actual utility consumption level is in conflict with the scheduling solution of the LPS consumption level. The shaded area in Figure 7 is an inaccurate count of the LPS consumption level. Moreover, in periods 3, 4, 6, and 7, the actual utility consumption levels are beyond the maximal LPS supply. This leads to an infeasible scheduling solution in the actual production process. The proposed model can determine the utility consumption and amount of material processed in any period. As shown in Figure 8, the LPS is consumed continuously by reactor 1, which is consistent with the actual situation. Furthermore, because of the accurate determination of LPS needed, the task performed in reactor 2 could be active continuously, because the tasks in reactor 1 and reactor 2 consume LPS simultaneously. The proposed model increases the yield of product B. Therefore, the optimal solution of the proposed model is larger than that of the traditional model ($1,559.066 vs $1,494.028, respectively). 4.2. Case of Disturbed Utility Supply. In this case, the plant flow sheet, scheduling horizon, and scheduling period are the same as those in case 1. The difference between the two cases is that the utility supply in case 2 has a disturbance. The data on the disturbed utility supply are given in Table 5. The data on the utility consumption levels of the traditional model and proposed model with disturbed utility supply are given in Tables 6 and 7. In this case, the optimal solutions of the traditional model and proposed model are $1,467.126 and $1,559.495, respectively. In the LPS subfigure of Figure 9, the black solid line is the scheduling solution for the LPS consumption level. The red dashed line is the actual utility consumption level. The red solid line is the actual LPS amount supplied. From Figure 9, we find that the traditional model cannot respond to utility disturbances. In periods 2 and 6, LPS consumption is beyond the actual amount of LPS supplied. Moreover, the utilization level of LPS based on the traditional model is unsatisfying. In the LPS subfigure of Figure 10, the black solid line is the scheduling solution of the LPS consumption level. The red solid line is the actual amount of LPS supplied. We can see that the proposed model can respond to utility disturbances. As shown in Figure 10, the scheduling solution of the proposed model adjusts the beginning times of the tasks performed in reactor 1 and changes the amount of material processed in the corresponding periods according to the utility amount supplied. The proposed model increases the utilization level of LPS and results in an optimal solution of $1,559.495, which is larger than that of the traditional model ($1,467.126).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +86-135-0043-1319. ORCID

Yue Wang: 0000-0001-7726-5267 Author Contributions

Y.W. conceived, formulated, and verified the scheduling model. Y.W. and X.J. analyzed the results and discussed and corrected the manuscript. All authors have read and approved the final manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Scientific Research Fund of Liaoning Provincial Education Department (Nos. L2017LQN030 and L2017LQN032); by the Talent Scientific Research Fund of Liaoning Shihua University (Nos. 2016XJJ101 and 2016XJJ-102); and by the Youth Innovation Talent Program by the Department of Education of Guangdong Province, China (601821K42050).

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ABBREVIATIONS USEBCTM unit-specific, event-based continuous-time model INDICES i, i′ task j, j′ unit s state u utility t scheduling period

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SETS T set of scheduling periods S all states, including feeds, intermediates, and final products SIP states including intermediates and final products SP states that are final products I tasks Ij tasks that can be performed in unit j IPs tasks that produce state s ICs tasks that consume state s Iu tasks that consume utility u J units Ji units that are suitable for performing task i U utilities

5. CONCLUSION This paper has presented and analyzed the limitations of the existing three-dimensional discrete-time scheduling model in terms of calculation of utility consumption. In order to solve the limitations of the discrete-time model, a four-dimensional binary variable is introduced based on the basic threedimensional model framework. Some crucial constraints for utility consumption, including allocation, batch size, and material balance, are modified significantly to enable the proposed model to determine the utility consumption level accurately. Moreover, production rate fluctuations and safe inventory interval constraints are considered in the proposed

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PARAMETERS minimum batch size of task i performed in unit j maximum batch size of task i performed in unit j initial amount available for state s minimum inventory capacity of state s maximum inventory capacity of state s minimum safe inventory bound of state s

Bmin ij Bmax ij ST0s STmin s STmax s STlbs 14567


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Industrial & Engineering Chemistry Research STub s asi bsi M αij eiu Umax ut Ost ms

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(12) Vooradi, R.; Shaik, M. A. J. C. Improved three-index unitspecific event-based model for short-term scheduling of batch plants. Comput. Chem. Eng. 2012, 43 (20), 148−172. (13) Wang, Y.; Su, H.; Shao, H.; Xie, L. Unit-Specific Event-Based and Slot-Based Hybrid Model Framework with Hierarchical Structure for Short-Term Scheduling. Math. Probl. Eng. 2015, 2015, 1−15. (14) Kondili, E.; Pantelides, C. C.; Sargent, R. W. H. A general algorithm for short-term scheduling of batch operationsI. MILP formulation. Comput. Chem. Eng. 1993, 17 (2), 211−227. (15) Shah, N.; Pantelides, C. C.; Sargent, R. W. H. A general algorithm for short-term scheduling of batch operationsII. Computational issues. Comput. Chem. Eng. 1993, 17 (2), 229−244. (16) Bassett, M. H.; Pekny, J. F.; Reklaitis, G. V. Using Detailed Scheduling To Obtain Realistic Operating Policies for a Batch Processing Facility. Ind. Eng. Chem. Res. 1997, 36 (5), 1717−1726. (17) Velez, S.; Merchan, A. F.; Maravelias, C. T. J. C. E. S. On the solution of large-scale mixed integer programming scheduling models. Chem. Eng. Sci. 2015, 136, 139−157. (18) Lindholm, A.; Johnsson, C.; Quttineh, N.-H.; Lidestam, H.; Henningsson, M.; Wikner, J.; Tang, O.; Nytzén, N.-P.; Forsman, K. Hierarchical scheduling and disturbance management in the process industry. Proceedings of the 7th IFAC Conference on Manufacturing, Modelling, Management, and Control, Saint Petersburg, Russia, June 19−21, 2013; Elsevier, 2013; pp 140−145. (19) Lindholm, A.; Giselsson, P. J. J. o. P. C. Minimization of economical losses due to utility disturbances in the process industry. J. Process Control 2013, 23 (5), 767−777. (20) Wang, Y.; Su, H.; Lu, S.; Xie, L.; Zhu, L. Hierarchical approach of planning and scheduling with demand uncertainty and utility disturbance. Proceedings of the 5th International Symposium on Advanced Control of Industrial Processes (ADCONIP’14), Hiroshima, Japan, May 28−29, 2014. (21) Li, J.; Susarla, N.; Karimi, I.; Shaik, M. A.; Floudas, C. A. An analysis of some unit-specific event-based models for the short-term scheduling of noncontinuous processes. Ind. Eng. Chem. Res. 2010, 49 (2), 633−647. (22) 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 (16), 7446−7469.

maximum safe inventory bound of state s proportion of state s produced by task i proportion of state s consumed by task i large positive number fixed processing time of task i in unit j constant coefficient for utility u consumed by task i maximum supply amount of utility u product demand of state s in period t price of state s

CONTINUOUS VARIABLES ST(s, t) excess amount of state s that needs to be stored in period t B(i, j, t) amount of material processed for task i in unit j in period t D(s, t) delivery amount of material s in period t x(i, j, t) aided continuous variable for decreasing the production rate fluctuation z(s, t) aided continuous variable for limiting the inventory level in a safe interval Bl(s, t) backlog of state s in period t

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BINARY VARIABLES w(i, j, t, t′) binary variable for task i performed in unit j, started in period t, and ended in period t′ REFERENCES

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Enhanced Discrete Time Formulation for a Short-Term Batch Process

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